https://puzzicle.org/math/cc-fourth-grade-math/v/comparing-whole-number-place-valueshttps://puzzicle.org/thumbs/x40b497d9.jpgComparing whole number place valuesSal compares whole numbers written in expanded form, written form, and number form.https://cdn.kastatic.org/ka-youtube-converted/3Xcae0OGavk.mp4/3Xcae0OGavk.mp491yes https://puzzicle.org/math/arithmetic-home/v/comparing-multi-digit-numbers-word-problemshttps://puzzicle.org/thumbs/x48d6cc8f4f0840e4.jpgComparing multi-digit whole numbers word problemsSal solves word problems that involve comparing multi-digit whole numbers.https://cdn.kastatic.org/ka-youtube-converted/PqyS66_mE24.mp4/PqyS66_mE24.mp4352yes https://puzzicle.org/math/cc-fourth-grade-math/v/rounding-whole-numbers-1https://puzzicle.org/thumbs/2600660.jpgRounding whole numbers to nearest hundredSal rounds 24,259 to the nearest hundred.https://cdn.kastatic.org/ka-youtube-converted/_qzs1zozTBo.mp4/_qzs1zozTBo.mp4263yes https://puzzicle.org/math/cc-fourth-grade-math/v/rounding-whole-numbers-2https://puzzicle.org/thumbs/2600661.jpgRounding whole numbers to nearest thousandSal rounds 423,275 to the nearest thousand.https://cdn.kastatic.org/ka-youtube-converted/fh8gkPW_6g4.mp4/fh8gkPW_6g4.mp4148yes https://puzzicle.org/math/cc-fourth-grade-math/v/rounding-to-nearest-ten-hundred-and-thousandhttps://puzzicle.org/thumbs/x2f58281c05cca46c.jpgRounding whole numbers word problemsSal solves word problems by rounding whole numbers to the nearest ten, hundred, or thousand.https://cdn.kastatic.org/ka-youtube-converted/Gg-GD5QfbB4.mp4/Gg-GD5QfbB4.mp4202yes https://puzzicle.org/math/cc-fourth-grade-math/v/rounding-whole-numbers-missing-digithttps://puzzicle.org/thumbs/xaa79450a.jpgRounding whole numbers: missing digitLindsay finds the missing digit in an estimation problem by using a number line.https://cdn.kastatic.org/ka-youtube-converted/Hixy7TX-Nwo.mp4/Hixy7TX-Nwo.mp4353yes https://puzzicle.org/math/cc-fourth-grade-math/v/adding-multi-digit-numbers-with-place-valuehttps://puzzicle.org/thumbs/xfe5a4a1f7a48fc5c.jpgRelate place value to standard algorithm for multi-digit additionTransition from a place value chart to standard algorithm when adding multi-digit numbers.https://cdn.kastatic.org/ka-youtube-converted/DqaHhReVpZI.mp4/DqaHhReVpZI.mp4382yes https://puzzicle.org/math/cc-fourth-grade-math/v/adding-multi-digit-numbers-with-regroupinghttps://puzzicle.org/thumbs/xe7ae47a2d1df7023.jpgAdding multi-digit numbers: 48,029+233,930Add 48,029+233,930 using the standard algorithm.https://cdn.kastatic.org/ka-youtube-converted/TVtdqRNJmiw.mp4/TVtdqRNJmiw.mp4138yes https://puzzicle.org/math/arithmetic-home/v/regrouping-borrowing-twice-examplehttps://puzzicle.org/thumbs/xbedc5cb2.jpgMulti-digit subtraction with regrouping twiceSal using regrouping to subtact 9601-8023.https://cdn.kastatic.org/ka-youtube-converted/OJ-wajo6oa4.mp4/OJ-wajo6oa4.mp4306yes https://puzzicle.org/math/cc-fourth-grade-math/v/understanding-place-value-when-subtractinghttps://puzzicle.org/thumbs/x90ad6246dbd32e70.jpgRelate place value to standard algorithm for multi-digit subtractionTransition from a place value chart to standard algorithm when subtracting multi-digit numbers.https://cdn.kastatic.org/ka-youtube-converted/XAPhD7EDQUk.mp4/XAPhD7EDQUk.mp4398yes https://puzzicle.org/math/cc-fourth-grade-math/v/subtracting-multi-digit-numbers-with-regroupinghttps://puzzicle.org/thumbs/x8f58991ebfadaeed.jpgMulti-digit subtraction: 389,002-76,151Subtract 389,002-76,151 using the standard algorithm.https://cdn.kastatic.org/ka-youtube-converted/buyK1y4rV3E.mp4/buyK1y4rV3E.mp4201yes https://puzzicle.org/math/cc-fourth-grade-math/v/metric-distancehttps://puzzicle.org/thumbs/x58cdb0a7.jpgMetric system: units of distanceSal discusses metric units of length or distance such as millimeter, centimeter, meter, and kilometer.https://cdn.kastatic.org/ka-youtube-converted/I3kQJvR7ZIg.mp4/I3kQJvR7ZIg.mp4416yes https://puzzicle.org/math/cc-fourth-grade-math/v/metric-units-of-volumehttps://puzzicle.org/thumbs/x9bd087b9.jpgMetric system: units of volumeLearn about metric units of volume or capacity such as milliliter and liter.https://cdn.kastatic.org/ka-youtube-converted/LhMEqsL_M5o.mp4/LhMEqsL_M5o.mp4254yes https://puzzicle.org/math/cc-fourth-grade-math/v/converting-weighthttps://puzzicle.org/thumbs/x13181fc4.jpgHow to convert kg to mg and T to ozSal converts kilograms to milligrams and tons to ounces.https://cdn.kastatic.org/ka-youtube-converted/1oFv56vSvvA.mp4/1oFv56vSvvA.mp4264yes https://puzzicle.org/math/cc-fourth-grade-math/v/liters-to-milliliters-exampleshttps://puzzicle.org/thumbs/x3025a20d31aa23f7.jpgConvert liters to millilitersSal converts between liters and milliliters.https://cdn.kastatic.org/ka-youtube-converted/OfTTjQW0CBw.mp4/OfTTjQW0CBw.mp4127yes https://puzzicle.org/math/cc-fourth-grade-math/v/converting-metric-distancehttps://puzzicle.org/thumbs/x53e22e6d.jpgConverting metric units of lengthLearn convert kilometers, meters, and centimeters.https://cdn.kastatic.org/ka-youtube-converted/9iulv2QvKwo.mp4/9iulv2QvKwo.mp4157yes https://puzzicle.org/math/cc-fourth-grade-math/v/metric-system-unit-conversion-exampleshttps://puzzicle.org/thumbs/x4e848a7034350a0b.jpgMulti-step unit conversion examples (metric)Sal solves multi-step unit conversion examples with metric units of measure.https://cdn.kastatic.org/ka-youtube-converted/ODJzxGj_zvU.mp4/ODJzxGj_zvU.mp4323yes https://puzzicle.org/math/cc-fourth-grade-math/v/multiplication-and-division-word-problems-2https://puzzicle.org/thumbs/x2187d181.jpgComparing with multiplication: magicSal solves a multiplication comparison word problem with Harry Potter characters.https://cdn.kastatic.org/ka-youtube-converted/9CZfG3r5JBE.mp4/9CZfG3r5JBE.mp470yes https://puzzicle.org/math/cc-fourth-grade-math/v/comparisons-subtraction-divisionhttps://puzzicle.org/thumbs/x9c85eb98.jpgComparing with multiplication and addition: moneySal solves 2 multiplication comparison word problems.https://cdn.kastatic.org/ka-youtube-converted/ZS157czfx4E.mp4/ZS157czfx4E.mp4395yes https://puzzicle.org/math/cc-fourth-grade-math/v/comparing-areahttps://puzzicle.org/thumbs/x927667dd.jpgComparing areas word problemSal compares the area of two posters using their side-lengths.https://cdn.kastatic.org/ka-youtube-converted/D2Ks5SKU6GM.mp4/D2Ks5SKU6GM.mp4141yes https://puzzicle.org/math/cc-fourth-grade-math/v/multiplying-10shttps://puzzicle.org/thumbs/x6b483ff6.jpgMultiplying 10sLindsay multiplies multiples of 10 by other multiples of 10, such as 30x50.https://cdn.kastatic.org/ka-youtube-converted/uHHnwafYivk.mp4/uHHnwafYivk.mp4308yes https://puzzicle.org/math/cc-fourth-grade-math/v/multiplying-1-digit-numbers-by-10-100-and-1000https://puzzicle.org/thumbs/x618e0032.jpgMultiplying 1-digit numbers by 10, 100, and 1000Lindsay finds a pattern from multiplying 1-digit numbers by 10, 100, and 1000.https://cdn.kastatic.org/ka-youtube-converted/SG4gX-VGzog.mp4/SG4gX-VGzog.mp4377yes https://puzzicle.org/math/cc-fourth-grade-math/v/multiplying-1-digit-numbers-by-multiples-of-10https://puzzicle.org/thumbs/xc4250c8b.jpgMultiplying 1-digit numbers by multiples of 10, 100, and 1000Lindsay finds a pattern from multiplying 1-digit numbers by multiples of 10, 100, and 1000.https://cdn.kastatic.org/ka-youtube-converted/tHQOAvbyRL0.mp4/tHQOAvbyRL0.mp4484yes https://puzzicle.org/math/cc-fourth-grade-math/v/more-ways-to-think-about-multiplyinghttps://puzzicle.org/thumbs/xc0a0f006.jpgUsing area model and properties to multiplySal uses place value, the distributive property, associative property, and an area model to show more ways to multiply.https://cdn.kastatic.org/ka-youtube-converted/NWJinKmWzx8.mp4/NWJinKmWzx8.mp4391yes https://puzzicle.org/math/cc-fourth-grade-math/v/4-by-1-multiplication-with-gridhttps://puzzicle.org/thumbs/xde6ebdd7.jpgMultiplying with area model: 6 x 7981Sal uses an area model to multiply 6x7981.https://cdn.kastatic.org/ka-youtube-converted/UvAOynxGfIU.mp4/UvAOynxGfIU.mp4214yes https://puzzicle.org/math/cc-fourth-grade-math/v/estimating-with-multiplicationhttps://puzzicle.org/thumbs/x09c0495345f64016.jpgEstimating with multiplicationSal estimates to find reasonable products to 1-digit by 2, 3, and 4-digit multiplication expressions.https://cdn.kastatic.org/ka-youtube-converted/eXJb76JUvko.mp4/eXJb76JUvko.mp4189yes https://puzzicle.org/math/cc-fourth-grade-math/v/multiplying-2-digits-by-1-digit-with-partial-productshttps://puzzicle.org/thumbs/x99032ee123e55595.jpgMultiplying 2-digits by 1-digit with partial productsWe can multiply larger numbers by multiplying by the tens and ones separately, then adding the products together. Multiplying with partial products is one way we use the distributive property.https://cdn.kastatic.org/ka-youtube-converted/RSx20wLxctc.mp4/RSx20wLxctc.mp4274yes https://puzzicle.org/math/cc-fifth-grade-math/v/multiplying-using-area-models-and-the-standard-algorithmhttps://puzzicle.org/thumbs/xc8c8cab6547bf583.jpgRelate multiplication with area models to the standard algorithmLearn the concept of multi-digit multiplication. Watch and learn how to break down the process into understandable steps, using two different methods.https://cdn.kastatic.org/ka-youtube-converted/fQ1zZYWjvwc.mp4/fQ1zZYWjvwc.mp4417yes https://puzzicle.org/math/cc-fourth-grade-math/v/3-digit-times-1-digit-example-no-carryinghttps://puzzicle.org/thumbs/x509df20f.jpgMultiplying 3-digit by 1-digitLearn to multiply a 3-digit number by a 1-digit number without regrouping. In this video, we will multiply 4x201.https://cdn.kastatic.org/ka-youtube-converted/4BtXvopHXI8.mp4/4BtXvopHXI8.mp4114yes https://puzzicle.org/math/cc-fourth-grade-math/v/3-digit-times-1-digit-examplehttps://puzzicle.org/thumbs/x4d3274dc.jpgMultiplying 3-digit by 1-digit (regrouping)Learn to multiply a 3-digit number by a 1-digit number using regrouping. In this video, we will multiply 7x253.https://cdn.kastatic.org/ka-youtube-converted/TqRReFvbpXA.mp4/TqRReFvbpXA.mp4135yes https://puzzicle.org/math/cc-fourth-grade-math/v/estimating-2-digit-multiplication-examplehttps://puzzicle.org/thumbs/x541ec875243fd608.jpgEstimating 2-digit multiplicationSal estimates to find reasonable products to 2-digit multiplication expressions.https://cdn.kastatic.org/ka-youtube-converted/FCFYGpY-MM0.mp4/FCFYGpY-MM0.mp4164yes https://puzzicle.org/math/cc-fourth-grade-math/v/area-model-for-multiplicationhttps://puzzicle.org/thumbs/x58e24008.jpgMultiplying with area model: 16 x 27Sal uses an area model to multiply 16x27.https://cdn.kastatic.org/ka-youtube-converted/vbGwcvXgDlg.mp4/vbGwcvXgDlg.mp4527yes https://puzzicle.org/math/cc-fourth-grade-math/v/understanding-multiplication-through-area-modelshttps://puzzicle.org/thumbs/x29584c60.jpgMultiplying with area model: 78 x 65Sal uses an area model to multiply 78x65.https://cdn.kastatic.org/ka-youtube-converted/p0jCw2sqZgs.mp4/p0jCw2sqZgs.mp4298yes https://puzzicle.org/math/cc-fourth-grade-math/v/multiplying-two-2-digit-numbers-using-partial-productshttps://puzzicle.org/thumbs/x36b3930d835242db.jpgMultiplying two 2-digit numbers using partial products26 times 37 is the same as 20 times 37 plus 6 times 37. The distributive property allows us to find each part of the product, like 3 tens times 2 tens, then add them all to get the whole product.https://cdn.kastatic.org/ka-youtube-converted/O2ifHv-3dZA.mp4/O2ifHv-3dZA.mp4266yes https://puzzicle.org/math/arithmetic-home/v/introduction-to-remaindershttps://puzzicle.org/thumbs/x2e6c5e31.jpgIntro to remaindersSal shows how a remainder is what's left over in a division problem.https://cdn.kastatic.org/ka-youtube-converted/BIGX05Mp5nw.mp4/BIGX05Mp5nw.mp4143yes https://puzzicle.org/math/cc-fourth-grade-math/v/quotients-that-are-multiples-of-10https://puzzicle.org/thumbs/xe7f435e0.jpgQuotients that are multiples of 10Lindsay breaks down division problems using multiples of 10.https://cdn.kastatic.org/ka-youtube-converted/UzXoxglkr98.mp4/UzXoxglkr98.mp4373yes https://puzzicle.org/math/cc-fourth-grade-math/v/division-using-understanding-of-place-valuehttps://puzzicle.org/thumbs/x11ff35e2.jpgDivision using place valueSal uses an understanding of place value to divide 5600÷8 and 846÷2.https://cdn.kastatic.org/ka-youtube-converted/xUjRwepN7dk.mp4/xUjRwepN7dk.mp4484yes https://puzzicle.org/math/cc-fourth-grade-math/v/area-models-to-visualize-division-using-place-valuehttps://puzzicle.org/thumbs/xa6e15515.jpgDivision with area modelsSal uses area models to divide 268÷2 and 856÷8.https://cdn.kastatic.org/ka-youtube-converted/FKJjqEdfB9Y.mp4/FKJjqEdfB9Y.mp4498yes https://puzzicle.org/math/cc-fourth-grade-math/v/multi-step-word-problems-with-whole-numbers-exercise-1https://puzzicle.org/thumbs/x06f7f404.jpgDivision word problem: field goalsSal solves a division word problem.https://cdn.kastatic.org/ka-youtube-converted/anlOhNHlqwg.mp4/anlOhNHlqwg.mp4179yes https://puzzicle.org/math/cc-fourth-grade-math/v/multi-step-word-problems-with-whole-numbers-exercise-t2https://puzzicle.org/thumbs/xad8542bb.jpgMultiplication word problem: pizzaSal solves a multiplication word problem.https://cdn.kastatic.org/ka-youtube-converted/cTveNRjWQYo.mp4/cTveNRjWQYo.mp4158yes https://puzzicle.org/math/cc-fourth-grade-math/v/understanding-factor-pairshttps://puzzicle.org/thumbs/x0775563f039594d4.jpgUnderstanding factor pairsUse multiplication and an understanding of area to identify factor pairs for 6 and 16.https://cdn.kastatic.org/ka-youtube-converted/KcKOM7Degu0.mp4/KcKOM7Degu0.mp4249yes https://puzzicle.org/math/cc-fourth-grade-math/v/finding-factors-of-a-numberhttps://puzzicle.org/thumbs/2600703.jpgFinding factors of a numberSal finds the factors of 120.https://cdn.kastatic.org/ka-youtube-converted/vcn2ruTOwFo.mp4/vcn2ruTOwFo.mp4593yes https://puzzicle.org/math/cc-fourth-grade-math/v/reasoning-about-factors-and-multipleshttps://puzzicle.org/thumbs/x6cf07b5dab8b8803.jpgReasoning about factors and multiplesUse the equation 3x5=15 to understand the relationship between factors and multiples.https://cdn.kastatic.org/ka-youtube-converted/g805f4HlV1Y.mp4/g805f4HlV1Y.mp498yes https://puzzicle.org/math/cc-fourth-grade-math/v/finding-factors-and-multipleshttps://puzzicle.org/thumbs/xb863d3f2.jpgFinding factors and multiplesSal uses divisibility rules to determine if numbers are factors of 154 and then finds multiples of 14.https://cdn.kastatic.org/ka-youtube-converted/5xe-6GPR_qQ.mp4/5xe-6GPR_qQ.mp4240yes https://puzzicle.org/math/cc-fourth-grade-math/v/identifying-multipleshttps://puzzicle.org/thumbs/x1a766ffd661cae5e.jpgIdentifying multiplesUse skip counting and division to identify multiples of 9.https://cdn.kastatic.org/ka-youtube-converted/U5L_aaQ6pYQ.mp4/U5L_aaQ6pYQ.mp4240yes https://puzzicle.org/math/cc-fourth-grade-math/v/figuring-out-days-of-the-weekhttps://puzzicle.org/thumbs/x429c2c78.jpgFactors and multiples: days of the weekSal uses factors and multiples to figure out days of the week.https://cdn.kastatic.org/ka-youtube-converted/S7CLLRHe8ik.mp4/S7CLLRHe8ik.mp4322yes https://puzzicle.org/math/cc-fourth-grade-math/v/prime-numbershttps://puzzicle.org/thumbs/85595310.jpgPrime numbersPrime numbers are numbers that have only 2 factors: 1 and themselves. For example, the first 5 prime numbers are 2, 3, 5, 7, and 11. By contrast, numbers with more than 2 factors are call composite numbers.https://cdn.kastatic.org/ka-youtube-converted/mIStB5X4U8M.mp4/mIStB5X4U8M.mp4493yes https://puzzicle.org/math/cc-fourth-grade-math/v/recognizing-prime-numbershttps://puzzicle.org/thumbs/158925083.jpgRecognizing prime and composite numbersCan you recognize the prime numbers in this group of numbers? Which are prime, composite, or neither?https://cdn.kastatic.org/ka-youtube-converted/3h4UK62Qrbo.mp4/3h4UK62Qrbo.mp4280yes https://puzzicle.org/math/cc-fourth-grade-math/v/math-patterns-example-1https://puzzicle.org/thumbs/x8ad2e2cc.jpgMath patterns: tableSal explains a pattern with the number of seats at a table.https://cdn.kastatic.org/ka-youtube-converted/KSrnZMAfwTM.mp4/KSrnZMAfwTM.mp4218yes https://puzzicle.org/math/cc-fourth-grade-math/v/math-patterns-example-2https://puzzicle.org/thumbs/xe158b503.jpgMath patterns: toothpicksSal explores a pattern of creating figures with toothpicks.https://cdn.kastatic.org/ka-youtube-converted/mFftY8Y_pyY.mp4/mFftY8Y_pyY.mp4341yes https://puzzicle.org/math/cc-fourth-grade-math/v/long-division-without-remainderhttps://puzzicle.org/thumbs/xebe85e0b.jpgIntro to long division (no remainders)Watch an introduction to long division with the problem 96÷4.https://cdn.kastatic.org/ka-youtube-converted/KFzcwWTEDDI.mp4/KFzcwWTEDDI.mp4195yes https://puzzicle.org/math/cc-fifth-grade-math/v/division-3-more-long-division-and-remainder-exampleshttps://puzzicle.org/thumbs/23848.jpgLong division with remainders: 2292÷4Learn the long division process for multi-digit numbers, including examples with remainders. The video emphasizes the importance of practice and understanding multiplication tables to tackle any division problem using long division techniques.https://cdn.kastatic.org/ka-youtube-converted/NcADzGz3bSI.mp4/NcADzGz3bSI.mp4607yes https://puzzicle.org/math/cc-fifth-grade-math/v/long-division-with-remainder-examplehttps://puzzicle.org/thumbs/x1a34ebb5.jpgLong division with remainders: 3771÷8To divide 3,771 by 8 using long division, start by dividing the first two digits, 37, by 8. This results in a quotient of 4. Next, bring down the next digit, 7, and divide 57 by 8, which goes 7 times. Finally, bring down the last digit, 1, and divide 11 by 8. This yields a quotient of 1 and a remainder of 3.https://cdn.kastatic.org/ka-youtube-converted/MbpmP1esh-Q.mp4/MbpmP1esh-Q.mp4161yes https://puzzicle.org/math/cc-fourth-grade-math/v/multiplying-2-digit-numbershttps://puzzicle.org/thumbs/x7ea237eb.jpgMultiplying 2-digit numbersLearn to multiply two-digit numbers. In this video, we will multiply 36 times 27.https://cdn.kastatic.org/ka-youtube-converted/DaQlieZH1kk.mp4/DaQlieZH1kk.mp4247yes https://puzzicle.org/math/geometry/v/language-and-notation-of-basic-geometryhttps://puzzicle.org/thumbs/611259516.jpgTerms & labels in geometryExplore geometry fundamentals, including points, line segments, rays, and lines. Understand dimensions and how these elements form shapes and patterns. Learn key geometric terms like colinear points, midpoints, and vertices, and enhance your knowledge of geometry. Improve your skills and discover the world of shapes and space.https://cdn.kastatic.org/ka-youtube-converted/il0EJrY64qE.mp4/il0EJrY64qE.mp4778yes https://puzzicle.org/math/cc-fourth-grade-math/v/lines-line-segments-and-rayshttps://puzzicle.org/thumbs/211177355.jpgLines, line segments, & raysLearn the difference between lines, line segments, and rays.https://cdn.kastatic.org/ka-youtube-converted/JcqCf762y9w.mp4/JcqCf762y9w.mp4218yes https://puzzicle.org/math/cc-seventh-grade-math/v/angle-basicshttps://puzzicle.org/thumbs/263439065.jpgAngles: introductionTwo rays that share the same endpoint form an angle. Learn about angles and the parts of an angle, like the vertex.https://cdn.kastatic.org/ka-youtube-converted/H-de6Tkxej8.mp4/H-de6Tkxej8.mp4409yes https://puzzicle.org/math/cc-fourth-grade-math/v/naming-angleshttps://puzzicle.org/thumbs/x66f2a779.jpgNaming anglesSal names angles based on their vertex and endpoints.https://cdn.kastatic.org/ka-youtube-converted/6vRgqB0PSXE.mp4/6vRgqB0PSXE.mp471yes https://puzzicle.org/math/cc-fourth-grade-math/v/recognizing-angles-exampleshttps://puzzicle.org/thumbs/x1781a3d6.jpgRecognizing anglesSal identifies acute, obtuse, and right angles in diagrams and pictures.https://cdn.kastatic.org/ka-youtube-converted/B0R3MJOrST0.mp4/B0R3MJOrST0.mp4135yes https://puzzicle.org/math/cc-fourth-grade-math/v/acute-right-and-obtuse-angleshttps://puzzicle.org/thumbs/60129393.jpgAcute, right, & obtuse anglesAcute angles measure less than 90 degrees. Right angles measure 90 degrees. Obtuse angles measure more than 90 degrees. Learn about angles types and see examples of each.https://cdn.kastatic.org/ka-youtube-converted/ALhv3Rlydig.mp4/ALhv3Rlydig.mp4332yes https://puzzicle.org/math/cc-fourth-grade-math/v/drawing-acute-right-and-obtuse-angleshttps://puzzicle.org/thumbs/x93c8f042.jpgDrawing acute, right and obtuse anglesLearn how to draw acute, right, and obtuse angles with given points.https://cdn.kastatic.org/ka-youtube-converted/4ZyTVTGVPgE.mp4/4ZyTVTGVPgE.mp4160yes https://puzzicle.org/math/geometry/v/parallel-and-perpendicular-lines-introhttps://puzzicle.org/thumbs/xe3c54f14.jpgParallel & perpendicular lines introParallel lines never intersect, and perpendicular lines intersect at a 90 degree angle. Learn how to identify parallel and perpendicular lines.https://cdn.kastatic.org/ka-youtube-converted/V0xounKGEXs.mp4/V0xounKGEXs.mp4128yes https://puzzicle.org/math/cc-eighth-grade-math/v/identifying-parallel-and-perpendicular-lineshttps://puzzicle.org/thumbs/372432667.jpgParallel & perpendicular linesParallel lines are lines that never intersect, and they form the same angle when they cross another line. Perpendicular lines intersect at a 90-degree angle, forming a square corner. We can identify these lines using angles and symbols in diagrams.https://cdn.kastatic.org/ka-youtube-converted/aq_XL6FrmGs.mp4/aq_XL6FrmGs.mp4215yes https://puzzicle.org/math/cc-fourth-grade-math/v/drawing-lines-exercise-examplehttps://puzzicle.org/thumbs/xfcdfb0ea.jpgDrawing parallel line segmentsSal draws parallel line segments with given points.https://cdn.kastatic.org/ka-youtube-converted/DkZnevdbf0A.mp4/DkZnevdbf0A.mp4115yes https://puzzicle.org/math/cc-fourth-grade-math/v/angle-measurement-and-circle-arcshttps://puzzicle.org/thumbs/x15cb233f.jpgAngle measurement & circle arcsLearn to measure angles as part of a circle.https://cdn.kastatic.org/ka-youtube-converted/D-EIh7NJvtQ.mp4/D-EIh7NJvtQ.mp4457yes https://puzzicle.org/math/cc-fourth-grade-math/v/measuring-angles-circular-protractorhttps://puzzicle.org/thumbs/xc20e9a03.jpgMeasuring angles with a circular protractorLindsay measures angles with a circular protractor.https://cdn.kastatic.org/ka-youtube-converted/Ttji2_wBSEo.mp4/Ttji2_wBSEo.mp4109yes https://puzzicle.org/math/cc-fourth-grade-math/v/angles-in-circles-word-problemhttps://puzzicle.org/thumbs/x7d0b3648.jpgAngles in circles word problemLindsay solves a word problems about angles as part of a circle.https://cdn.kastatic.org/ka-youtube-converted/_jRXtJz3CYc.mp4/_jRXtJz3CYc.mp4199yes https://puzzicle.org/math/cc-fourth-grade-math/v/angle-sense-examplehttps://puzzicle.org/thumbs/xd505def0.jpgIdentifying an angleSal matches pictures of angles to their degrees.https://cdn.kastatic.org/ka-youtube-converted/2mzuFKCuDg4.mp4/2mzuFKCuDg4.mp4123yes https://puzzicle.org/math/cc-fourth-grade-math/v/measuring-angles-in-degreeshttps://puzzicle.org/thumbs/263439060.jpgMeasuring angles in degreesLearn how to measure an angle with a protractor.https://cdn.kastatic.org/ka-youtube-converted/92aLiyeQj0w.mp4/92aLiyeQj0w.mp4502yes https://puzzicle.org/math/cc-fourth-grade-math/v/using-a-protractorhttps://puzzicle.org/thumbs/263439062.jpgMeasuring angles using a protractorLearn to use a virtual protractor to measure angles.https://cdn.kastatic.org/ka-youtube-converted/dw41PMWek6U.mp4/dw41PMWek6U.mp4205yes https://puzzicle.org/math/cc-fourth-grade-math/v/measuring-angleshttps://puzzicle.org/thumbs/211177356.jpgMeasuring angles using a protractor 2Sal measures several angles with a protractor.https://cdn.kastatic.org/ka-youtube-converted/wJ37GJyViU8.mp4/wJ37GJyViU8.mp4245yes https://puzzicle.org/math/cc-fourth-grade-math/v/constructing-angleshttps://puzzicle.org/thumbs/xb6f10aaa.jpgConstructing anglesSal uses a protractor to create 10° and 155° angles.https://cdn.kastatic.org/ka-youtube-converted/QmfoIvgIVlE.mp4/QmfoIvgIVlE.mp4161yes https://puzzicle.org/math/cc-fourth-grade-math/v/decomposing-angleshttps://puzzicle.org/thumbs/x6c2451f9.jpgDecomposing an angleSal finds the measure of angles by adding or subtracting.https://cdn.kastatic.org/ka-youtube-converted/M7NVKImDqGg.mp4/M7NVKImDqGg.mp4103yes https://puzzicle.org/math/cc-fourth-grade-math/v/decomposing-angles-addhttps://puzzicle.org/thumbs/xe1272e01.jpgDecomposing anglesLindsay finds the measures of angles using addition.https://cdn.kastatic.org/ka-youtube-converted/uMj46crbLaw.mp4/uMj46crbLaw.mp4185yes https://puzzicle.org/math/geometry/v/axis-of-symmetryhttps://puzzicle.org/thumbs/x78b40eae.jpgIntro to reflective symmetryIntroduction to the concept of an "axis of symmetry"https://cdn.kastatic.org/ka-youtube-converted/LrTn4cvsewk.mp4/LrTn4cvsewk.mp4167yes https://puzzicle.org/math/cc-fourth-grade-math/v/identifying-symmetrical-figureshttps://puzzicle.org/thumbs/x1f16d320.jpgIdentifying symmetrical figuresLindsay identifies lines of symmetry on three shapes.https://cdn.kastatic.org/ka-youtube-converted/59s_CQjTD6w.mp4/59s_CQjTD6w.mp4333yes https://puzzicle.org/math/cc-fourth-grade-math/v/scalene-isosceles-equilateral-from-anglehttps://puzzicle.org/thumbs/x7eb84d3e.jpgClassifying triangles by anglesYou say you can't categorize a triangle because you don't know the length of the sides. Guess what? If you know the size of the angles you can!https://cdn.kastatic.org/ka-youtube-converted/sZMezOCZr40.mp4/sZMezOCZr40.mp4330yes https://puzzicle.org/math/cc-fourth-grade-math/v/scalene-isosceles-equilateral-acute-right-obtusehttps://puzzicle.org/thumbs/xdb593bbc.jpgClassifying trianglesThe video explores how triangles are classified based on their sides and angles. It introduces the terms scalene, isosceles, and equilateral for side lengths, and acute, right, and obtuse for angles. It emphasizes that triangles can be categorized in multiple ways based on these characteristics.https://cdn.kastatic.org/ka-youtube-converted/D5lZ3thuEeA.mp4/D5lZ3thuEeA.mp4360yes https://puzzicle.org/math/cc-fourth-grade-math/v/recog-triangles-examplehttps://puzzicle.org/thumbs/x3fd4b56d.jpgWorked example: Classifying trianglesWork through sample problems categorizing triangles as scalene, isosceles, equilateral, acute, right, or obtuse.https://cdn.kastatic.org/ka-youtube-converted/NRS2ZJVVzOI.mp4/NRS2ZJVVzOI.mp4143yes https://puzzicle.org/math/cc-fourth-grade-math/v/classifying-shapes-exampleshttps://puzzicle.org/thumbs/xe37249c4.jpgClassifying shapes by line and angles typesSal categorizes shapes based on their sides and angles.https://cdn.kastatic.org/ka-youtube-converted/PiQxA9O7Rd8.mp4/PiQxA9O7Rd8.mp4135yes https://puzzicle.org/math/cc-fourth-grade-math/v/quadrilateral-propertieshttps://puzzicle.org/thumbs/370305272.jpgQuadrilateral propertiesLearn about the properties of quadrilaterals, parallelograms, trapezoids, rhombuses, rectangles, and squares.https://cdn.kastatic.org/ka-youtube-converted/5CeBlu260Rw.mp4/5CeBlu260Rw.mp4307yes https://puzzicle.org/math/cc-fourth-grade-math/v/classifying-shapes-by-lines-and-angleshttps://puzzicle.org/thumbs/xd20b343f.jpgClassifying shapes by lines and anglesLindsay classifies a shape based on hints about its sides and angles.https://cdn.kastatic.org/ka-youtube-converted/xO4n3LfmVag.mp4/xO4n3LfmVag.mp4217yes https://puzzicle.org/math/cc-fourth-grade-math/v/decomposing-a-mixed-numberhttps://puzzicle.org/thumbs/x5939f38d.jpgDecomposing a mixed numberSal uses fraction models to decompose 2 1/4.https://cdn.kastatic.org/ka-youtube-converted/erZe85NrsK0.mp4/erZe85NrsK0.mp4300yes https://puzzicle.org/math/cc-fourth-grade-math/v/equivalent-amount-of-pizzahttps://puzzicle.org/thumbs/x22c750f5.jpgEquivalent fractionsLearn how equivalent fractions represent the same amount, even with different numerators and denominators. We uses a pizza example to show how 1/2, 2/4, and 4/8 are equivalent, as they all represent the same portion.https://cdn.kastatic.org/ka-youtube-converted/N1X0vf5PUz4.mp4/N1X0vf5PUz4.mp4258yes https://puzzicle.org/math/arithmetic-home/v/visualizing-equivalent-fractionshttps://puzzicle.org/thumbs/x885328b8.jpgVisualizing equivalent fractionsSal uses fraction models to show equivalent fractions.https://cdn.kastatic.org/ka-youtube-converted/Bt60JVZRVCI.mp4/Bt60JVZRVCI.mp4225yes https://puzzicle.org/math/cc-fourth-grade-math/v/more-equivalent-fractionshttps://puzzicle.org/thumbs/xa40a399e.jpgMore on equivalent fractionsSal uses fraction models and multiplication to find equivalent fractions.https://cdn.kastatic.org/ka-youtube-converted/yZqVJtDO7gc.mp4/yZqVJtDO7gc.mp4294yes https://puzzicle.org/math/cc-fourth-grade-math/v/finding-common-denominatorshttps://puzzicle.org/thumbs/158925087.jpgFinding common denominatorsLearn all about finding common denominators in fractions. Watch how to use the least common multiple of the denominators to rewrite fractions, making them easier to compare or add.https://cdn.kastatic.org/ka-youtube-converted/lxjmR4pYIVU.mp4/lxjmR4pYIVU.mp4282yes https://puzzicle.org/math/cc-fourth-grade-math/v/common-denominators-35-and-72https://puzzicle.org/thumbs/xf9ec9a64.jpgCommon denominators: 3/5 and 7/2Learn how to rewrite fractions with different denominators as equivalent fractions with a common denominator. Watch how to visually represent this process and apply it to solve problems.https://cdn.kastatic.org/ka-youtube-converted/0EwwoDVJVPQ.mp4/0EwwoDVJVPQ.mp4255yes https://puzzicle.org/math/cc-fourth-grade-math/v/comparing-fractions-visually-with-a-barhttps://puzzicle.org/thumbs/x4e435a2c.jpgComparing fractions: tape diagramSal compares fractions with unlike denominators by drawing barshttps://cdn.kastatic.org/ka-youtube-converted/ory05j2jgBM.mp4/ory05j2jgBM.mp4155yes https://puzzicle.org/math/cc-fourth-grade-math/v/comparing-fractions-on-a-number-linehttps://puzzicle.org/thumbs/xb915b4f5.jpgComparing fractions: number lineSal compares fractions on a number linehttps://cdn.kastatic.org/ka-youtube-converted/7vWzd-ZV6-Q.mp4/7vWzd-ZV6-Q.mp4175yes https://puzzicle.org/math/cc-fourth-grade-math/v/comparing-fractions-visually-with-pieshttps://puzzicle.org/thumbs/xa9de51d5.jpgComparing fractions: fraction modelsSal compares fractions visually with pieshttps://cdn.kastatic.org/ka-youtube-converted/uCtXyhepAmQ.mp4/uCtXyhepAmQ.mp4152yes https://puzzicle.org/math/cc-fourth-grade-math/v/comparing-fractions-with-different-denominatorshttps://puzzicle.org/thumbs/xe9caeb3f.jpgComparing fractions 1 (unlike denominators)Sal compares fractions by finding a common denominator.https://cdn.kastatic.org/ka-youtube-converted/zRjLZROI7wc.mp4/zRjLZROI7wc.mp4256yes https://puzzicle.org/math/arithmetic-home/v/comparing-fractions-2https://puzzicle.org/thumbs/153770147.jpgComparing fractions 2 (unlike denominators)Sal compares fractions by finding a common denominator.https://cdn.kastatic.org/ka-youtube-converted/2dbasvm3iG0.mp4/2dbasvm3iG0.mp4428yes https://puzzicle.org/math/cc-fourth-grade-math/v/adding-fractions-with-like-denominatorshttps://puzzicle.org/thumbs/2600768.jpgAdding fractions with like denominatorsSal adds 3/15+7/15.https://cdn.kastatic.org/ka-youtube-converted/EJjnEau6aeI.mp4/EJjnEau6aeI.mp4198yes https://puzzicle.org/math/cc-fourth-grade-math/v/subtracting-fractionshttps://puzzicle.org/thumbs/2600773.jpgSubtracting fractions with like denominatorsSal subtracts 8/18-5/18.https://cdn.kastatic.org/ka-youtube-converted/UbUdyE1_b9g.mp4/UbUdyE1_b9g.mp4146yes https://puzzicle.org/math/cc-fourth-grade-math/v/fraction-word-problems-1-exercisehttps://puzzicle.org/thumbs/x9b1143b5.jpgFraction word problem: pizzaSal uses fraction models to figure out what fraction of a pizza is left.https://cdn.kastatic.org/ka-youtube-converted/u2hLYcmI5y4.mp4/u2hLYcmI5y4.mp4319yes https://puzzicle.org/math/cc-fourth-grade-math/v/subtracting-fractions-word-problem-1https://puzzicle.org/thumbs/xa613e211.jpgFraction word problem: pianoSal solves a fraction word problem about practicing piano. The fractions have common denominators.https://cdn.kastatic.org/ka-youtube-converted/0njioQqIxKY.mp4/0njioQqIxKY.mp4184yes https://puzzicle.org/math/cc-fourth-grade-math/v/adding-fractions-word-problem-2https://puzzicle.org/thumbs/xcd390cf1.jpgFraction word problem: lizardSal solves a fraction word problem about the length of a lizard. The fractions have common denominators.https://cdn.kastatic.org/ka-youtube-converted/gPbN26bYyN8.mp4/gPbN26bYyN8.mp4149yes https://puzzicle.org/math/cc-fourth-grade-math/v/adding-fractions-word-problem-1https://puzzicle.org/thumbs/x0913bd5c.jpgFraction word problem: spider eyesSal solves a fraction word problem about spider eyes. The fractions have common denominators.https://cdn.kastatic.org/ka-youtube-converted/5r2FYszC_sU.mp4/5r2FYszC_sU.mp4105yes https://puzzicle.org/math/cc-fourth-grade-math/v/multiplying-fractions-and-whole-numbershttps://puzzicle.org/thumbs/x3769a3bc.jpgMultiplying unit fractions and whole numbersLearn how to multiply a whole number by a fraction, using both visual and computational methods. Multiplying 1/2 by 5 can be understood as adding five 1/2's together, resulting in 5/2.https://cdn.kastatic.org/ka-youtube-converted/4PlkCiEXBQI.mp4/4PlkCiEXBQI.mp4188yes https://puzzicle.org/math/cc-fourth-grade-math/v/changing-a-mixed-number-to-an-improper-fractionhttps://puzzicle.org/thumbs/2600700.jpgWriting mixed numbers as improper fractionsSal rewrites 5 1/4 as an improper fraction.https://cdn.kastatic.org/ka-youtube-converted/xkg7370cpjs.mp4/xkg7370cpjs.mp4403yes https://puzzicle.org/math/cc-fourth-grade-math/v/changing-an-improper-fraction-to-a-mixed-numberhttps://puzzicle.org/thumbs/2600701.jpgWriting improper fractions as mixed numbersSal rewrite 7/4 as a mixed number.https://cdn.kastatic.org/ka-youtube-converted/8JyPzTeA_8w.mp4/8JyPzTeA_8w.mp4242yes https://puzzicle.org/math/cc-fourth-grade-math/v/making-line-plots-with-fractional-datahttps://puzzicle.org/thumbs/x4a27cfbc236d6423.jpgMaking line plots with fractional dataThe video is all about creating line plots to represent data sets. It emphasizes the importance of using fractions of a unit (like 1/2, 1/4, 1/8) to accurately depict measurements. It also shows how line plots can help solve problems involving addition and subtraction of fractions.https://cdn.kastatic.org/ka-youtube-converted/NG51xCPse5Y.mp4/NG51xCPse5Y.mp4195yes https://puzzicle.org/math/cc-fourth-grade-math/v/interpreting-line-plotshttps://puzzicle.org/thumbs/xbb328a6d44f13d41.jpgInterpreting line plots with fractionsThe weights on this line plot are to the nearest 1/8 pound, because there are 8 equal spaces between whole numbers. Each dot represents the weight of one baby. Let's find the difference between the two heaviest babies.https://cdn.kastatic.org/ka-youtube-converted/78zFg2F3HsI.mp4/78zFg2F3HsI.mp4165yes https://puzzicle.org/math/cc-fourth-grade-math/v/interpreting-data-in-line-plotshttps://puzzicle.org/thumbs/x1e343920.jpgReading a line plot with fractionsExplore the concept of interpreting data on line plots, particularly focusing on how to read and understand line plots that display measurements in fractions. Understand the importance of comparing data points and calculating differences between them.https://cdn.kastatic.org/ka-youtube-converted/4smVBLi3DxU.mp4/4smVBLi3DxU.mp4213yes https://puzzicle.org/math/arithmetic-home/v/adding-mixed-numbers-2https://puzzicle.org/thumbs/xfa82ffcf.jpgIntro to adding mixed numbersSal adds 2 mixed numbers with common (like) denominators.https://cdn.kastatic.org/ka-youtube-converted/pxX07gUbIQQ.mp4/pxX07gUbIQQ.mp4241yes https://puzzicle.org/math/arithmetic-home/v/subtracting-mixed-numbers-with-like-denominatorshttps://puzzicle.org/thumbs/x551b7da0.jpgIntro to subtracting mixed numbersSal subtracts 2 mixed numbers with common (like) denominators.https://cdn.kastatic.org/ka-youtube-converted/zm3TXDZrifU.mp4/zm3TXDZrifU.mp4289yes https://puzzicle.org/math/cc-fourth-grade-math/v/word-problem-subtracting-fractions-with-like-denominatorshttps://puzzicle.org/thumbs/x99617da1b54c4cc3.jpgSubtracting mixed numbers with like denominators word problemSal solves a word problem involving mixed number subtraction by using a visual model.https://cdn.kastatic.org/ka-youtube-converted/9aD5Z_EMxos.mp4/9aD5Z_EMxos.mp4189yes https://puzzicle.org/math/cc-fourth-grade-math/v/concept-whole-fraction-multhttps://puzzicle.org/thumbs/x848c25e5.jpgMultiplying fractions and whole numbers visuallyLearn the concept of multiplying fractions and whole numbers. Watch how to visually represent this process and practice understanding the relationship between fractions and whole numbers in multiplication.https://cdn.kastatic.org/ka-youtube-converted/XaJQse2u5TQ.mp4/XaJQse2u5TQ.mp4246yes https://puzzicle.org/math/cc-fourth-grade-math/v/fraction-whole-multiplication-concepthttps://puzzicle.org/thumbs/x8320c937.jpgEquivalent fraction and whole number multiplication problemsSal relates mixed numbers to whole number/fraction multiplication problem.https://cdn.kastatic.org/ka-youtube-converted/ZlhrXO1-osA.mp4/ZlhrXO1-osA.mp4164yes https://puzzicle.org/math/cc-fourth-grade-math/v/multiplying-fractions-word-problem-2https://puzzicle.org/thumbs/x487557ed.jpgMultiplying fractions word problem: moviesSal solves a multiplication fraction word problem about movies.https://cdn.kastatic.org/ka-youtube-converted/Yq9W6B7XnmU.mp4/Yq9W6B7XnmU.mp4222yes https://puzzicle.org/math/cc-fourth-grade-math/v/multiplying-fractions-word-problem-3https://puzzicle.org/thumbs/x8737839e.jpgMultiplying fractions word problem: milkSal solves a multiplication fraction word problem about milk.https://cdn.kastatic.org/ka-youtube-converted/GvgJ2_7P9SA.mp4/GvgJ2_7P9SA.mp4139yes https://puzzicle.org/math/cc-fourth-grade-math/v/multiplying-fractions-by-whole-numbers-word-problem-math-4th-grade-khan-academyhttps://puzzicle.org/thumbs/x2c824088.jpgMultiplying fractions by whole numbers word problemLindsay solves a word problem that involves multiplying a fraction by a whole number.https://cdn.kastatic.org/ka-youtube-converted/6PMQdHCtUtA.mp4/6PMQdHCtUtA.mp4141yes https://puzzicle.org/math/cc-fourth-grade-math/v/examples-relating-decimals-and-fractions-in-wordshttps://puzzicle.org/thumbs/x25085dfdfb13c401.jpgRelating decimals and fractions in wordsSal relates equivalent decimals and fractions written in word form.https://cdn.kastatic.org/ka-youtube-converted/5TwOaUrniaI.mp4/5TwOaUrniaI.mp4191yes https://puzzicle.org/math/cc-fourth-grade-math/v/introduction-to-decimalshttps://puzzicle.org/thumbs/x07ef47f9.jpgDecimal place valueSal introduces decimal numbers and understanding place value to the right of the decimal.https://cdn.kastatic.org/ka-youtube-converted/BItpeFXC4vA.mp4/BItpeFXC4vA.mp4386yes https://puzzicle.org/math/cc-fourth-grade-math/v/decimals-as-wordshttps://puzzicle.org/thumbs/xa6903bbe.jpgDecimals as wordsSal writes 0.17 and 40.03 in words. He also writes twenty-four hundredths using numbers.https://cdn.kastatic.org/ka-youtube-converted/qSPwUDmpnJ4.mp4/qSPwUDmpnJ4.mp4338yes https://puzzicle.org/math/cc-fourth-grade-math/v/graphing-tenths-from-0-to-1-math-4th-grade-khan-academyhttps://puzzicle.org/thumbs/x83474085.jpgGraphing tenths from 0 to 1Lindsay graphs 0.6 on a number line.https://cdn.kastatic.org/ka-youtube-converted/7oUBK0cNxV8.mp4/7oUBK0cNxV8.mp491yes https://puzzicle.org/math/cc-fourth-grade-math/v/identifying-tenths-on-a-number-line-math-4th-grade-khan-academyhttps://puzzicle.org/thumbs/x5eacb0ab.jpgIdentifying tenths on a number lineLindsay identifies a point graphed on a number line.https://cdn.kastatic.org/ka-youtube-converted/qb0QSP7Sfz4.mp4/qb0QSP7Sfz4.mp4112yes https://puzzicle.org/math/cc-fourth-grade-math/v/rewriting-fractions-as-decimals-math-4th-grade-khan-academyhttps://puzzicle.org/thumbs/x647578e2.jpgRewriting fractions as decimalsLindsay rewrites two fractions as decimals. The fractions have denominators of 10 and 100.https://cdn.kastatic.org/ka-youtube-converted/4LvGU1a9Z5Q.mp4/4LvGU1a9Z5Q.mp4238yes https://puzzicle.org/math/cc-fourth-grade-math/v/visually-converting-from-tenths-to-hundredthshttps://puzzicle.org/thumbs/x01630e18.jpgVisually converting tenths and hundredthsSal rewrites 8/10 with a denominator of 100.https://cdn.kastatic.org/ka-youtube-converted/zEd67cLFnuQ.mp4/zEd67cLFnuQ.mp4260yes https://puzzicle.org/math/cc-fourth-grade-math/v/decompose-hundredths-on-number-linehttps://puzzicle.org/thumbs/x2effd053.jpgDecomposing hundredths on number lineSal decomposes 26/100 into tenths and hundredths using a number line.https://cdn.kastatic.org/ka-youtube-converted/k4bqucu56Ig.mp4/k4bqucu56Ig.mp4106yes https://puzzicle.org/math/cc-fourth-grade-math/v/graphing-hundredths-from-0-to-01-math-4th-grade-khan-academyhttps://puzzicle.org/thumbs/x67fe7215.jpgGraphing hundredths from 0 to 0.1Lindsay graphs 0.04 on a number line.https://cdn.kastatic.org/ka-youtube-converted/SrL8abwGKyA.mp4/SrL8abwGKyA.mp4159yes https://puzzicle.org/math/cc-fourth-grade-math/v/identifying-hundredths-on-a-number-line-math-4th-grade-khan-academyhttps://puzzicle.org/thumbs/x39416065.jpgIdentifying hundredths on a number lineLindsay identifies decimals graphed on a number line.https://cdn.kastatic.org/ka-youtube-converted/bFjaZ_8pi3c.mp4/bFjaZ_8pi3c.mp4171yes https://puzzicle.org/math/cc-fourth-grade-math/v/decimals-on-a-number-line-2-exampleshttps://puzzicle.org/thumbs/x77c3a934.jpgPlotting decimal numbers on a number lineSal finds decimals with hundredths on a number line.https://cdn.kastatic.org/ka-youtube-converted/gWqYXa7r_V0.mp4/gWqYXa7r_V0.mp4248yes https://puzzicle.org/math/cc-fourth-grade-math/v/fraction-decimal-intuition-exampleshttps://puzzicle.org/thumbs/x24a0750c.jpgWriting a number as a fraction and decimalSal shows the connection between decimals and fractions using a grid diagram and number lines.https://cdn.kastatic.org/ka-youtube-converted/NfU7cmajLAY.mp4/NfU7cmajLAY.mp4277yes https://puzzicle.org/math/cc-fourth-grade-math/v/place-value-for-decimals-greater-than-onehttps://puzzicle.org/thumbs/x155513a9.jpgDecimal place value with regroupingSal discusses decimal numbers that are greater than 1.https://cdn.kastatic.org/ka-youtube-converted/-BYomxn8Rss.mp4/-BYomxn8Rss.mp4415yes https://puzzicle.org/math/cc-fourth-grade-math/v/examples-writing-decimals-and-fractions-greater-than-1-shown-on-gridshttps://puzzicle.org/thumbs/x784640bb38759977.jpgWriting decimals and fractions greater than 1 shown on gridsSal writes decimal numbers and fractions greater than 1 shown on grids. Decimals are limited to tenths and hundredths.https://cdn.kastatic.org/ka-youtube-converted/ek0zu14IP7g.mp4/ek0zu14IP7g.mp4182yes https://puzzicle.org/math/cc-fourth-grade-math/v/writing-decimals-and-fractions-from-number-lineshttps://puzzicle.org/thumbs/x3051d55d5e8d5157.jpgWriting decimals and fractions shown on number linesSal writes decimal numbers and fractions greater than 1 shown on number lines. Decimals are limited to tenths and hundredths.https://cdn.kastatic.org/ka-youtube-converted/68mrr430De0.mp4/68mrr430De0.mp4163yes https://puzzicle.org/math/cc-fourth-grade-math/v/comparing-decimals-visually-examplehttps://puzzicle.org/thumbs/x8f4a7db6.jpgComparing decimals visuallySal compares 0.17 and 0.2 using grid diagrams.https://cdn.kastatic.org/ka-youtube-converted/ftEVCrKfIVM.mp4/ftEVCrKfIVM.mp4220yes https://puzzicle.org/math/cc-fourth-grade-math/v/comparing-numbers-represented-in-different-wayshttps://puzzicle.org/thumbs/x7b0897be.jpgComparing numbers represented different waysExamples of Khan Academy practice problems where you compare decimals, fractions, and number diagrams.https://cdn.kastatic.org/ka-youtube-converted/YM4lE4q0fSk.mp4/YM4lE4q0fSk.mp4460yes https://puzzicle.org/math/cc-fourth-grade-math/v/decimals-on-a-number-linehttps://puzzicle.org/thumbs/158925082.jpgComparing decimal numbers on a number lineSal compares decimals on a number line.https://cdn.kastatic.org/ka-youtube-converted/LpLnmuAyNWg.mp4/LpLnmuAyNWg.mp499yes https://puzzicle.org/math/cc-fourth-grade-math/v/comparing-decimals-with-hundredthshttps://puzzicle.org/thumbs/x818728a2.jpgComparing decimals (tenths and hundredths)Sal compares decimals like 0.7 and 0.09 with greater and less than symbols.https://cdn.kastatic.org/ka-youtube-converted/EX9CdUAMpgE.mp4/EX9CdUAMpgE.mp4209yes https://puzzicle.org/math/4th-engage-ny/v/comparing-decimalshttps://puzzicle.org/thumbs/1807.jpgComparing decimals 3Sal compares decimals like 45.675 and 45.645 with greater and less than symbols.https://cdn.kastatic.org/ka-youtube-converted/HCC96awA-FM.mp4/HCC96awA-FM.mp489yes https://puzzicle.org/math/cc-fourth-grade-math/v/converting-decimals-to-fractions-1-ex-1https://puzzicle.org/thumbs/915208238.jpgRewriting decimals as fractions: 0.15Decimals can be written in fraction form. To convert a decimal to a fraction, place the decimal number over its place value. For example, in 0.6, the six is in the tenths place, so we place 6 over 10 to create the equivalent fraction, 6/10. If needed, simplify the fraction.https://cdn.kastatic.org/ka-youtube-converted/DR2DYe7PI74.mp4/DR2DYe7PI74.mp4116yes https://puzzicle.org/math/cc-fourth-grade-math/v/converting-decimals-to-fractions-1-ex-2https://puzzicle.org/thumbs/914994686.jpgRewriting decimals as fractions: 0.8Sal converts 0.8 to a fraction.https://cdn.kastatic.org/ka-youtube-converted/9P80OLC6wKY.mp4/9P80OLC6wKY.mp453yes https://puzzicle.org/math/cc-fourth-grade-math/v/converting-decimals-to-fractions-1-ex-3https://puzzicle.org/thumbs/915122530.jpgRewriting decimals as fractions: 0.36Sal converts 0.36 to a fraction.https://cdn.kastatic.org/ka-youtube-converted/OS1g4PDdNdM.mp4/OS1g4PDdNdM.mp474yes https://puzzicle.org/math/cc-fourth-grade-math/v/adding-fractions-with-10-and-100-as-denominatorshttps://puzzicle.org/thumbs/x05f0312b.jpgAdding fractions (denominators 10 & 100)Sal adds 3/10+7/100 by finding a common denominator.https://cdn.kastatic.org/ka-youtube-converted/YZD5ifHZILE.mp4/YZD5ifHZILE.mp4265yes https://puzzicle.org/math/cc-fourth-grade-math/v/adding-tenths-to-hundredthshttps://puzzicle.org/thumbs/x48b4dd1b5fbb427f.jpgAdding fractions: 7/10+13/100Sal uses visual models to add 7/10 and 13/100.https://cdn.kastatic.org/ka-youtube-converted/D5yQESfE4Bw.mp4/D5yQESfE4Bw.mp4187yes https://puzzicle.org/math/cc-fourth-grade-math/v/converting-us-fluid-volumehttps://puzzicle.org/thumbs/x8db1ea4b.jpgConverting US customary units of volumeSal converts between gallons, quarts, pints, cups, and fluid ounces.https://cdn.kastatic.org/ka-youtube-converted/9UVgO8zS474.mp4/9UVgO8zS474.mp4157yes https://puzzicle.org/math/cc-fourth-grade-math/v/feet-to-inches-exercise-examplehttps://puzzicle.org/thumbs/xe364a4f7.jpgConverting feet to inchesLearn to convert feet to inches.https://cdn.kastatic.org/ka-youtube-converted/zsI4g5XSlbw.mp4/zsI4g5XSlbw.mp451yes https://puzzicle.org/math/cc-fourth-grade-math/v/time-unit-conversionhttps://puzzicle.org/thumbs/xacafa3ce.jpgConverting units of timeLearn to convert hours, minutes, and seconds.https://cdn.kastatic.org/ka-youtube-converted/-r7JwuSdj5M.mp4/-r7JwuSdj5M.mp4202yes https://puzzicle.org/math/cc-fourth-grade-math/v/change-from-buying-appleshttps://puzzicle.org/thumbs/xaee188ce.jpgWord problem: making changeSal solves how much change Charles will get after his purchase.https://cdn.kastatic.org/ka-youtube-converted/_YXF7WDpWpg.mp4/_YXF7WDpWpg.mp4194yes https://puzzicle.org/math/cc-fourth-grade-math/v/currency-conversion-examplehttps://puzzicle.org/thumbs/x5ad67dda.jpgCurrency conversion word problemSal converts between American dollars and Chinese yuan.https://cdn.kastatic.org/ka-youtube-converted/XcRX-mkjeiM.mp4/XcRX-mkjeiM.mp454yes https://puzzicle.org/math/cc-fourth-grade-math/v/unit-sense-exercise-example-2https://puzzicle.org/thumbs/x5c798014.jpgChoose pounds or ounces to measure weightSal discusses US customary units of mass or weight such as ounce, pound, and ton.https://cdn.kastatic.org/ka-youtube-converted/V1LloSp4PV4.mp4/V1LloSp4PV4.mp475yes https://puzzicle.org/math/cc-fourth-grade-math/v/us-customary-weighthttps://puzzicle.org/thumbs/x81f65552.jpgU.S. customary units: weightSal discusses US customary units of mass or weight such as ounces, pounds, and tons.https://cdn.kastatic.org/ka-youtube-converted/Dj1rbIP8PHM.mp4/Dj1rbIP8PHM.mp4220yes https://puzzicle.org/math/cc-fourth-grade-math/v/us-customary-distancehttps://puzzicle.org/thumbs/xc43fda34.jpgU.S. customary units: distanceSal discusses US customary units of length or distance such as inch, foot, and yard.https://cdn.kastatic.org/ka-youtube-converted/M9bisHkXbKc.mp4/M9bisHkXbKc.mp4311yes https://puzzicle.org/math/cc-fourth-grade-math/v/us-customary-fluid-volumehttps://puzzicle.org/thumbs/xb277b208.jpgU.S. customary units: fluid volumeSal discusses US customary units of volume or capacity such as teaspoon, fluid ounce, cup, pint, quart, and gallon.https://cdn.kastatic.org/ka-youtube-converted/eqKnx5Yk508.mp4/eqKnx5Yk508.mp4441yes https://puzzicle.org/math/cc-fourth-grade-math/v/multi-step-unit-conversion-exampleshttps://puzzicle.org/thumbs/x446e081e3471afcb.jpgMulti-step unit conversion examples (U.S. Customary)Sal solves multi-step unit conversion example using U.S. Customary units of measure.https://cdn.kastatic.org/ka-youtube-converted/5QFwaG5oM_A.mp4/5QFwaG5oM_A.mp4337yes https://puzzicle.org/math/cc-fourth-grade-math/v/time-word-problemhttps://puzzicle.org/thumbs/x3f5d6857.jpgTime word problem: Susan's breakDive into the concept of converting units of time, specifically minutes into hours. Understand the importance the relationship between these units and demonstrates how to apply this knowledge to solve real-world problems involving time intervals.https://cdn.kastatic.org/ka-youtube-converted/UhMM68fq9FA.mp4/UhMM68fq9FA.mp4218yes https://puzzicle.org/math/arithmetic-home/v/rounding-to-the-nearest-10https://puzzicle.org/thumbs/xd886d7e1.jpgRounding to nearest 10This video teaches rounding numbers to the nearest 10 using number lines. It demonstrates rounding with examples like 36, 34, 35, 26, and 12, and provides a rule for rounding when the ones place is 5 or greater. This helps students practice estimation in real-life situations.https://cdn.kastatic.org/ka-youtube-converted/w2M5CzTFYfI.mp4/w2M5CzTFYfI.mp4382yes https://puzzicle.org/math/arithmetic-home/v/rounding-to-the-nearest-100https://puzzicle.org/thumbs/xace78db7.jpgRounding to nearest 100Learners can use a number line to round three digit numbers to the nearest hundred. We learn that the midpoint on the number line helps determine if the number rounds up or round down. In this video, it all comes down to the tens place!https://cdn.kastatic.org/ka-youtube-converted/19yOv4P2ccw.mp4/19yOv4P2ccw.mp4296yes https://puzzicle.org/math/arithmetic-home/v/the-idea-of-divisionhttps://puzzicle.org/thumbs/xdba36435.jpgThe idea of divisionSal uses an array and understanding of multiplication to divide.https://cdn.kastatic.org/ka-youtube-converted/QI6x0KNxiCs.mp4/QI6x0KNxiCs.mp4481yes https://puzzicle.org/math/cc-fifth-grade-math/v/quadrilateral-overviewhttps://puzzicle.org/thumbs/339557900.jpgIntro to quadrilateralThere are many different types of quadrilaterals. A quadrilateral is defined as a two-dimensional shape with four sides, four vertices, and four angles. There are two main types: concave and convex. There are also various subcategories of convex quadrilaterals, such as trapezoids, parallelograms, rectangles, rhombi, and squares.https://cdn.kastatic.org/ka-youtube-converted/Ka82QC4QvGA.mp4/Ka82QC4QvGA.mp4504yes https://puzzicle.org/math/cc-fifth-grade-math/v/quadrilateral-types-exercisehttps://puzzicle.org/thumbs/x819465b4.jpgQuadrilateral typesLearn to identify quadrilaterals such as kites, trapezoids, parallelograms, rhombuses, rectangles, and squares by side length, presence of parallel sides, and angle type.https://cdn.kastatic.org/ka-youtube-converted/wPZIa3SjPF0.mp4/wPZIa3SjPF0.mp4169yes https://puzzicle.org/math/cc-fifth-grade-math/v/classifying-shapeshttps://puzzicle.org/thumbs/x32882801.jpgClassifying quadrilateralsThere are many different relationships between the types of quadrilaterals. Parallelograms have two sets of parallel sides, rhombuses are parallelograms with equal side lengths, and squares are rhombuses with right angles. Though there is overlap, not all quadrilaterals fit into each category.https://cdn.kastatic.org/ka-youtube-converted/1pHhMX0_4Bw.mp4/1pHhMX0_4Bw.mp4148yes https://puzzicle.org/math/4th-grade-illustrative-mathematics/v/solving-word-problems-involving-factors-and-multipleshttps://puzzicle.org/thumbs/x7dc30e3b70022372.jpgSolving word problems involving factors and multiplesLearn how to solve word problems that involve factors and multiples within 100.https://cdn.kastatic.org/ka-youtube-converted/r7rFBk4uBlU.mp4/r7rFBk4uBlU.mp4212yes https://puzzicle.org/math/cc-fourth-grade-math/v/equivalent-fractions-with-modelshttps://puzzicle.org/thumbs/xe04b6d954e97827c.jpgEquivalent fractions with modelsSal uses fraction models and tape diagrams to help identify equivalent fractions.https://cdn.kastatic.org/ka-youtube-converted/vTjgaVmJ8fk.mp4/vTjgaVmJ8fk.mp4336yes https://puzzicle.org/math/cc-fourth-grade-math/v/equivalent-fractions-on-number-lineshttps://puzzicle.org/thumbs/x088ac06e90f8e10f.jpgEquivalent fractions on number linesSal uses a number line to help identify equivalent fractions with different denominators.https://cdn.kastatic.org/ka-youtube-converted/bBkUqkI2TSw.mp4/bBkUqkI2TSw.mp4274yes https://puzzicle.org/math/cc-fourth-grade-math/v/comparing-fractions-word-problemshttps://puzzicle.org/thumbs/x7f990c0c901888f3.jpgComparing fractions word problemsSal practices comparing fractions using real-world contexts.https://cdn.kastatic.org/ka-youtube-converted/NEjIVR9kAFE.mp4/NEjIVR9kAFE.mp4307yes https://puzzicle.org/math/cc-fourth-grade-math/v/common-denominators-14-and-56https://puzzicle.org/thumbs/xbb7dcccd.jpgCommon denominators: 1/4 and 5/6Learn how to find common denominators for fractions with different numerators and denominators by identifying multiples of the original denominators. It demonstrates the process using 1/4 and 5/6, showing that 12 and 24 can be common denominators.https://cdn.kastatic.org/ka-youtube-converted/T8U7hCmUdpo.mp4/T8U7hCmUdpo.mp4499yes https://puzzicle.org/math/arithmetic-home/v/order-fractionshttps://puzzicle.org/thumbs/x65f66810.jpgOrdering fractionsLindsay orders 7/10, 1/3, and 5/6 using common denominators.https://cdn.kastatic.org/ka-youtube-converted/ONsI-yIbsBg.mp4/ONsI-yIbsBg.mp4359yes https://puzzicle.org/math/4th-grade-illustrative-mathematics/v/multiplying-a-unit-fraction-and-a-whole-number-with-fraction-modelshttps://puzzicle.org/thumbs/x5dbe6ca69e524f0b.jpgMultiplying a unit fraction and a whole number with fraction modelsWorked examples on using a tape diagram to determine the product of a whole number and a unit fraction.https://cdn.kastatic.org/ka-youtube-converted/9FmYFzSEqOI.mp4/9FmYFzSEqOI.mp4229yes https://puzzicle.org/math/4th-grade-illustrative-mathematics/v/subtracting-a-fraction-from-a-whole-numberhttps://puzzicle.org/thumbs/xee80e94ce41793a9.jpgSubtracting a fraction from a whole numberLearn two strategies for how to subtract a fraction from a whole number.https://cdn.kastatic.org/ka-youtube-converted/PZtc8BBiMnE.mp4/PZtc8BBiMnE.mp4154yes https://puzzicle.org/math/4th-grade-illustrative-mathematics/v/subtracting-a-mixed-number-from-a-whole-numberhttps://puzzicle.org/thumbs/x736c40f0a2fd3882.jpgSubtracting a mixed number from a whole numberWorked example showing how to subtract a mixed number from a whole number.https://cdn.kastatic.org/ka-youtube-converted/9W9HeeD3Dog.mp4/9W9HeeD3Dog.mp4136yes https://puzzicle.org/math/cc-fourth-grade-math/v/adding-mixed-numbers-with-like-denominatorshttps://puzzicle.org/thumbs/xedb21f38298b8e8d.jpgAdding mixed numbers with like denominatorsSal practices adding mixed numbers with common (like) denominators.https://cdn.kastatic.org/ka-youtube-converted/t32jFmpE-ec.mp4/t32jFmpE-ec.mp4173yes https://puzzicle.org/math/cc-fourth-grade-math/v/mixed-number-subtractionhttps://puzzicle.org/thumbs/x64c1922925855264.jpgSubtracting mixed numbers with like denominatorsSal practices subtracting mixed numbers with common (like) denominators.https://cdn.kastatic.org/ka-youtube-converted/QJZ6mBW8xTU.mp4/QJZ6mBW8xTU.mp4138yes https://puzzicle.org/math/cc-fourth-grade-math/v/mixed-number-addition-with-regroupinghttps://puzzicle.org/thumbs/x833785ba0b451037.jpgMixed number addition with regroupingSal practices adding mixed numbers with common (like) denominators. Items require regrouping.https://cdn.kastatic.org/ka-youtube-converted/tNaKlG_GJGQ.mp4/tNaKlG_GJGQ.mp4220yes https://puzzicle.org/math/cc-fourth-grade-math/v/subtract-mixed-numbers-with-regroupinghttps://puzzicle.org/thumbs/x6e3ce4abf9bb9bcb.jpgSubtracting mixed numbers with regroupingSal practices subtracting mixed numbers with common (like) denominators. Items require regrouping (borrowing).https://cdn.kastatic.org/ka-youtube-converted/TQEg9dMIWFU.mp4/TQEg9dMIWFU.mp4180yes https://puzzicle.org/math/cc-fourth-grade-math/v/decomposing-hudredths-into-tenths-and-hundredthshttps://puzzicle.org/thumbs/xbe6334e8.jpgDecomposing hundredthsSal decomposes 47/100 into tenths and hundredths.https://cdn.kastatic.org/ka-youtube-converted/kAm9x3uWF9I.mp4/kAm9x3uWF9I.mp4250yes https://puzzicle.org/math/cc-fourth-grade-math/v/fraction-multiplcation-on-the-number-linehttps://puzzicle.org/thumbs/x5e8a62f5b4be1859.jpgFraction multiplication on the number lineSal uses number lines to help solve multiplication equations.https://cdn.kastatic.org/ka-youtube-converted/ltKaGC8jfd4.mp4/ltKaGC8jfd4.mp4181yes https://puzzicle.org/math/cc-fourth-grade-math/v/multiply-mixed-numbershttps://puzzicle.org/thumbs/xd7b2de40bb5fa61c.jpgMultiplying mixed numbers by whole numbersSal uses repeated addition to help multiply a whole number by a mixed number.https://cdn.kastatic.org/ka-youtube-converted/hjbSbHr3Fhk.mp4/hjbSbHr3Fhk.mp4144yes https://puzzicle.org/math/4th-grade-illustrative-mathematics/v/ordering-decimals-visually-least-to-greatesthttps://puzzicle.org/thumbs/x90947b37936447c8.jpgOrdering decimals visually: least to greatestLet's order decimals shown with tenths and hundredths grids from least to greatest.https://cdn.kastatic.org/ka-youtube-converted/uvtHTOt17io.mp4/uvtHTOt17io.mp4149yes https://puzzicle.org/math/4th-grade-illustrative-mathematics/v/ordering-decimals-visually-greatest-to-leasthttps://puzzicle.org/thumbs/x4c954ea03b06f818.jpgOrdering decimals visually: greatest to leastLet's order decimals shown with tenths and hundredths grids from greatest to least.https://cdn.kastatic.org/ka-youtube-converted/WlW3l-hM6OA.mp4/WlW3l-hM6OA.mp4256yes https://puzzicle.org/math/4th-grade-illustrative-mathematics/v/locating-large-numbers-on-a-number-linehttps://puzzicle.org/thumbs/xbc90c35356dd281b.jpgLocating large numbers on a number lineLearn how to locate multi-digit numbers on a number line.https://cdn.kastatic.org/ka-youtube-converted/zgNQlR9fJeI.mp4/zgNQlR9fJeI.mp4198yes https://puzzicle.org/math/4th-grade-illustrative-mathematics/v/ordering-multi-digit-numbershttps://puzzicle.org/thumbs/xc55471a58c2b851c.jpgOrdering multi digit numbersPractice using place value to order multi digit number from least to greatest.https://cdn.kastatic.org/ka-youtube-converted/K52kqv87zEg.mp4/K52kqv87zEg.mp491yes https://puzzicle.org/math/4th-grade-illustrative-mathematics/v/finding-the-nearest-multiple-of-1000-10000-and-100000https://puzzicle.org/thumbs/x98f58633cf81b6f5.jpgFinding the nearest multiple of 1,000, 10,000 and 100,000Learn how to find the nearest multiple of 1,000, 10,000, and 100,000 given a multi-digit whole number.https://cdn.kastatic.org/ka-youtube-converted/1a_XdCihuC8.mp4/1a_XdCihuC8.mp4221yes https://puzzicle.org/math/arithmetic-home/v/borrowing-once-example-1https://puzzicle.org/thumbs/xcd3e1c25.jpgMulti-digit subtraction with regrouping: 6798-3359Sal subtracts 6798-3359 using regrouping.https://cdn.kastatic.org/ka-youtube-converted/RRk5qLd__Ro.mp4/RRk5qLd__Ro.mp4225yes https://puzzicle.org/math/arithmetic-home/v/borrowing-once-example-2https://puzzicle.org/thumbs/xf0c43720.jpgMulti-digit subtraction with regrouping: 7329-6278Sal subtracts 7329-6278 using regrouping.https://cdn.kastatic.org/ka-youtube-converted/8_fT1BsTmt4.mp4/8_fT1BsTmt4.mp4268yes https://puzzicle.org/math/4th-grade-illustrative-mathematics/v/estimating-multi-digit-addition-and-subtraction-word-problemshttps://puzzicle.org/thumbs/xf8a8694c80a598c2.jpgEstimating multi-digit addition and subtraction word problemsPractice using estimation as a strategy to add and subtract larger multi-digit numbers.https://cdn.kastatic.org/ka-youtube-converted/Ic4i3L9ivNk.mp4/Ic4i3L9ivNk.mp4138yes https://puzzicle.org/math/cc-seventh-grade-math/v/decimals-and-fractionshttps://puzzicle.org/thumbs/1806.jpgRewriting decimals as fractions: 2.75Here's an example of how we convert a decimal to a simplified fraction. Over time this will become second nature to you.https://cdn.kastatic.org/ka-youtube-converted/NzIoHQfStoM.mp4/NzIoHQfStoM.mp4162yes https://puzzicle.org/math/cc-fourth-grade-math/v/common-fractions-and-decimalshttps://puzzicle.org/thumbs/x95c2808f6820c075.jpgCommon fractions and decimalsExplores the concept of converting common fractions (1/5, 1/4, 1/2) into decimals. Learn the importance of understanding these conversions for real-life applications and for working with multiples of these fractions.https://cdn.kastatic.org/ka-youtube-converted/lh2mp0aqSh8.mp4/lh2mp0aqSh8.mp4333yes https://puzzicle.org/math/5th-grade-foundations-engageny/v/multiplying-whole-numbers-and-applications-6https://puzzicle.org/thumbs/158927009.jpgComparing with multiplication: ageSal solves a multiplication comparison word problem.https://cdn.kastatic.org/ka-youtube-converted/bS6EmYzpou4.mp4/bS6EmYzpou4.mp4159yes https://puzzicle.org/math/4th-grade-illustrative-mathematics/v/solving-word-problems-involving-multiplicative-comparisonhttps://puzzicle.org/thumbs/x959b21ed27e22f5a.jpgSolving word problems involving multiplicative comparisonLearn how to solve word problems involving multiplicative comparison.https://cdn.kastatic.org/ka-youtube-converted/iNPc_UPwEM0.mp4/iNPc_UPwEM0.mp4176yes https://puzzicle.org/math/cc-fourth-grade-math/v/converting-pounds-to-ounceshttps://puzzicle.org/thumbs/23.jpgConverting pounds to ouncesThere are 16 ounces in 1 pound. Learn to convert pounds to ounces.https://cdn.kastatic.org/ka-youtube-converted/IrvEHVn-fks.mp4/IrvEHVn-fks.mp494yes https://puzzicle.org/math/cc-fourth-grade-math/v/converting-yards-into-incheshttps://puzzicle.org/thumbs/16.jpgConverting yards into inchesLearn to convert between yards, feet, and inches.https://cdn.kastatic.org/ka-youtube-converted/8C5kAIKLcZo.mp4/8C5kAIKLcZo.mp4349yes https://puzzicle.org/math/4th-grade-illustrative-mathematics/v/finding-side-lengths-of-rectangles-given-the-perimeterhttps://puzzicle.org/thumbs/x58084f1cad042016.jpgFinding side lengths of rectangles given the perimeterWatch a worked example on finding possible pairs of side lengths of a rectangle given the perimeter.https://cdn.kastatic.org/ka-youtube-converted/NQLhy5DbqQc.mp4/NQLhy5DbqQc.mp4134yes https://puzzicle.org/math/4th-grade-illustrative-mathematics/v/using-multiplication-to-compare-perimeters-of-rectangleshttps://puzzicle.org/thumbs/xe89cc1f43beec74f.jpgUsing multiplication to compare perimeters of rectanglesLearn how to find the perimeter of a rectangle given that the side lengths are a multiple of another rectangle.https://cdn.kastatic.org/ka-youtube-converted/S5rWQJxiBhQ.mp4/S5rWQJxiBhQ.mp4141yes https://puzzicle.org/math/4th-grade-illustrative-mathematics/v/finding-the-perimeter-of-a-rhombushttps://puzzicle.org/thumbs/x5b4dbd5f560993a2.jpgFinding the perimeter of a rhombusLearn how to find the perimeter of a rhombus with fractional side lengths. This example is designed to reinforce skills in fraction arithmetic and multiplication.https://cdn.kastatic.org/ka-youtube-converted/A2Hl7wpOLJ4.mp4/A2Hl7wpOLJ4.mp490yes https://puzzicle.org/math/4th-grade-illustrative-mathematics/v/finding-the-side-length-of-a-parallelogram-given-perimeterhttps://puzzicle.org/thumbs/x9e27011a16939f66.jpgFinding the side length of a parallelogram given perimeterLearn how to find the unknown side length of a parallelogram. As you solve problems, you will apply what you know about adding and subtracting fractions.https://cdn.kastatic.org/ka-youtube-converted/x2nFVwpYefc.mp4/x2nFVwpYefc.mp483yes https://puzzicle.org/math/cc-fourth-grade-math/v/unit-sense-example-for-exercisehttps://puzzicle.org/thumbs/xa2bc2142.jpgFinding reasonable unit of measurement exampleSal chooses reasonable units of time activities like baking cookies.https://cdn.kastatic.org/ka-youtube-converted/OuaBg3Hjqoc.mp4/OuaBg3Hjqoc.mp4227yes https://puzzicle.org/math/cc-fifth-grade-math/v/intro-to-standard-way-of-multiplying-multi-digit-numbershttps://puzzicle.org/thumbs/x0fa45a7026fa5ab7.jpgIntro to standard way of multiplying multi-digit numbersThe standard algorithm for multiplying a multi-digit number by a single digit number involves multiplying each place value by the single digit, and regrouping as necessary. This method is equivalent to breaking the multi-digit number into parts, multiplying each part by the single digit, and adding the results together.https://cdn.kastatic.org/ka-youtube-converted/NuIDwPQLJmk.mp4/NuIDwPQLJmk.mp4265yes https://puzzicle.org/math/cc-fifth-grade-math/v/understanding-the-standard-algorithm-for-multiplicationhttps://puzzicle.org/thumbs/x30f46234cfe82375.jpgUnderstanding the standard algorithm for multiplicationThe standard algorithm for multiplying whole numbers involves breaking the numbers down into their place values and multiplying each place value separately. This process is illustrated through three examples: one complete walkthrough, one where the viewer is asked to identify mistakes in incorrect solutions, and one where the viewer is asked to fill in a missing number.https://cdn.kastatic.org/ka-youtube-converted/GOYnrYee1Y0.mp4/GOYnrYee1Y0.mp4271yes https://puzzicle.org/math/4th-grade-illustrative-mathematics/v/solving-multi-step-problems-multiplication-camping-itemshttps://puzzicle.org/thumbs/x2bf9664349a01fb8.jpgSolving multi-step problems (multiplication): camping itemsPractice solving multi-step step word problems with multiplication.https://cdn.kastatic.org/ka-youtube-converted/h8ClXSzcIXU.mp4/h8ClXSzcIXU.mp4155yes https://puzzicle.org/math/4th-grade-illustrative-mathematics/v/solving-multi-step-problems-multiplication-firewoodhttps://puzzicle.org/thumbs/xb05d142e612f9a09.jpgSolving multi-step problems (multiplication): firewoodPractice solving multi-step step word problems with multiplication.https://cdn.kastatic.org/ka-youtube-converted/SuTdB6pcpeM.mp4/SuTdB6pcpeM.mp4140yes https://puzzicle.org/math/cc-fourth-grade-math/v/introduction-to-division-with-partial-quotientshttps://puzzicle.org/thumbs/xeb7482a2859662d0.jpgIntroduction to division with partial quotients (no remainder)Sal divides multi-digit numbers by a single-digit numbers using partial quotients.https://cdn.kastatic.org/ka-youtube-converted/Ws4Zq-gceSY.mp4/Ws4Zq-gceSY.mp4376yes https://puzzicle.org/math/cc-fourth-grade-math/v/intro-to-remaindershttps://puzzicle.org/thumbs/xcb929e1ca71a635e.jpgIntroduction to remaindersSal shows how a remainder is what's left over in a division problem.https://cdn.kastatic.org/ka-youtube-converted/P1qyjdh_sIw.mp4/P1qyjdh_sIw.mp4289yes https://puzzicle.org/math/cc-fourth-grade-math/v/division-with-partial-quotients-examplehttps://puzzicle.org/thumbs/xa78b245468a7c720.jpgDivision with partial quotients (remainder)Sal divides multi-digit numbers by a single-digit numbers using partial quotients. Answers include remainders.https://cdn.kastatic.org/ka-youtube-converted/gamEPRpt9jo.mp4/gamEPRpt9jo.mp4405yes https://puzzicle.org/math/4th-grade-illustrative-mathematics/v/solving-multi-step-problems-division-bookshttps://puzzicle.org/thumbs/xb091e2faddf3db30.jpgSolving multi-step problems (division): booksPractice solving multi-step step word problems with division.https://cdn.kastatic.org/ka-youtube-converted/ZbiQiubEH3w.mp4/ZbiQiubEH3w.mp4142yes https://puzzicle.org/math/4th-grade-illustrative-mathematics/v/solving-multi-step-problems-division-cookieshttps://puzzicle.org/thumbs/x4b851787c3020908.jpgSolving multi-step problems (division): cookiesPractice solving multi-step step word problems with division.https://cdn.kastatic.org/ka-youtube-converted/3fWW5SsvPq0.mp4/3fWW5SsvPq0.mp4129yes https://puzzicle.org/math/cc-fourth-grade-math/v/2-step-estimation-examplehttps://puzzicle.org/thumbs/xe5e3106c2de872d0.jpg2-step estimation word problemSal uses estimation to solve multi-step word problems involving addition, subtraction, multiplication, and division.https://cdn.kastatic.org/ka-youtube-converted/hzh10qtOxpg.mp4/hzh10qtOxpg.mp4185yes https://puzzicle.org/math/cc-fourth-grade-math/v/estimating-division-that-results-in-non-whole-numbershttps://puzzicle.org/thumbs/x6c251d4a8d28b1c4.jpgEstimating division that results in non-whole numbersSal uses estimation to find quotients that are non-whole numbers.https://cdn.kastatic.org/ka-youtube-converted/VJyk81HmcZQ.mp4/VJyk81HmcZQ.mp4214yes https://puzzicle.org/math/cc-fourth-grade-math/v/estimating-quotientshttps://puzzicle.org/thumbs/x6b03dedfb1635afd.jpgEstimating quotientsSal uses estimation to determine if quotients are reasonable or unreasonable.https://cdn.kastatic.org/ka-youtube-converted/6RT_dE3W-00.mp4/6RT_dE3W-00.mp4245yes https://puzzicle.org/math/4th-grade-illustrative-mathematics/v/composing-and-decomposing-angleshttps://puzzicle.org/thumbs/xf3e6dfa68093da00.jpgComposing and decomposing anglesLearn how to determine the size of an angle based on composing and decomposing angles.https://cdn.kastatic.org/ka-youtube-converted/mg05k5blvf4.mp4/mg05k5blvf4.mp476yes https://puzzicle.org/math/4th-grade-illustrative-mathematics/v/finding-an-unknown-angle-measurehttps://puzzicle.org/thumbs/x36b65adcaebd92e0.jpgFinding an unknown angle measureLearn how to find an unknown angle measure on a diagram, based on composing and decomposing angles.https://cdn.kastatic.org/ka-youtube-converted/lshlkVC57uQ.mp4/lshlkVC57uQ.mp474yes https://puzzicle.org/math/4th-grade-math-eureka-squared-aligned/v/estimating-large-sums-and-differences-word-problemshttps://puzzicle.org/thumbs/xd26e3b9b0ffa07f5.jpgEstimating large sums and differences word problemsGet ready to solve word problems by estimating large sums and differences! Let's round numbers to make addition and subtraction easier. Estimating helps us make smart guesses in real-life situations.https://cdn.kastatic.org/ka-youtube-converted/quGTt2nF4Lg.mp4/quGTt2nF4Lg.mp4177yes https://puzzicle.org/math/arithmetic-home/v/comparing-place-valueshttps://puzzicle.org/thumbs/x19b4a5d9.jpgComparing place valuesSal compares numbers in different place values.https://cdn.kastatic.org/ka-youtube-converted/dZPOCU10TqY.mp4/dZPOCU10TqY.mp4150yes https://puzzicle.org/math/4th-grade-math-eureka-squared-aligned/v/ordering-large-numbers-least-to-greatesthttps://puzzicle.org/thumbs/x55036179cd6e3e6a.jpgOrdering large numbers: least to greatestLet's order whole numbers up to 1 billion from least to greatest using place value understanding.https://cdn.kastatic.org/ka-youtube-converted/ao-Cn3NlqtQ.mp4/ao-Cn3NlqtQ.mp4263yes https://puzzicle.org/math/4th-grade-math-eureka-squared-aligned/v/ordering-large-numbers-greatest-to-leasthttps://puzzicle.org/thumbs/x93e3b3b49ac627d0.jpgOrdering large numbers: greatest to leastLet's order whole numbers up to 1 billion using place value understanding.https://cdn.kastatic.org/ka-youtube-converted/Bz-dxrD4xUw.mp4/Bz-dxrD4xUw.mp4191yes https://puzzicle.org/math/4th-grade-math-eureka-squared-aligned/v/estimating-large-sumshttps://puzzicle.org/thumbs/x29f7e16f5c50f07d.jpgEstimating large sumsLet's estimate sums by rounding numbers to the nearest ten, hundred, or thousand to make adding easier. Estimating helps us check our answers and solve problems quickly in real life—like when shopping or planning a party. Get ready for some fun math tricks!https://cdn.kastatic.org/ka-youtube-converted/ByodvOXFNns.mp4/ByodvOXFNns.mp4186yes https://puzzicle.org/math/4th-grade-math-eureka-squared-aligned/v/estimating-large-differenceshttps://puzzicle.org/thumbs/x71770154f045f6d9.jpgEstimating large differencesLet's round big numbers to the nearest ten, hundred, or thousand to make subtraction easier. Estimating helps us check our answers and solve problems quickly in real life—like when budgeting or figuring out travel time. Let’s make subtraction simple and fun!https://cdn.kastatic.org/ka-youtube-converted/jz6QaBJqDOI.mp4/jz6QaBJqDOI.mp4139yes https://puzzicle.org/math/arithmetic-home/v/2-digit-multiplication-with-gridhttps://puzzicle.org/thumbs/x481bf39b.jpgMultiplying with distributive propertySal uses the distributive property to multiply 87x63.https://cdn.kastatic.org/ka-youtube-converted/3JmnxR8nEi8.mp4/3JmnxR8nEi8.mp4418yes https://puzzicle.org/math/cc-fifth-grade-math/v/estimating-adding-fractions-with-unlike-denominatorshttps://puzzicle.org/thumbs/x20d9230a8c34da99.jpgEstimating adding fractions with unlike denominatorsLearn all about estimating the sum of fractions with unlike denominators. Practice visualizing fractions and using that understanding to make reasonable estimates when adding them together.https://cdn.kastatic.org/ka-youtube-converted/XR_B9AJe1u4.mp4/XR_B9AJe1u4.mp4297yes https://puzzicle.org/math/4th-grade-math-eureka-squared-aligned/v/relating-money-and-place-valuehttps://puzzicle.org/thumbs/x144eb4c92f368a09.jpgRelating money and place valueDiscover how dollars and cents connect to place value in this engaging video using money to explain tenths and hundredths clearly.https://cdn.kastatic.org/ka-youtube-converted/1ZeV2W3rSUs.mp4/1ZeV2W3rSUs.mp4120yes https://puzzicle.org/math/4th-grade-math-eureka-squared-aligned/v/relating-money-to-decimalshttps://puzzicle.org/thumbs/x798a58c97d17face.jpgRelating money to decimalsLearn different ways to represent decimals as fractions, grids, and money in this fun video.https://cdn.kastatic.org/ka-youtube-converted/UeoZoZmUkyg.mp4/UeoZoZmUkyg.mp4208yes https://puzzicle.org/math/4th-grade-math-eureka-squared-aligned/v/adding-decimals-science-experimenthttps://puzzicle.org/thumbs/xc6b3965c3f279e62.jpgAdding decimals: science experimentPractice solving word problems by adding decimals.https://cdn.kastatic.org/ka-youtube-converted/rwIsRzdRlb0.mp4/rwIsRzdRlb0.mp484yes https://puzzicle.org/math/cc-fourth-grade-math/v/comparing-fractions-of-different-wholes-2https://puzzicle.org/thumbs/xaedbf86a.jpgEquivalent fractions and different wholesSal shows that two fractions are equivalent only if they refer to the same whole.https://cdn.kastatic.org/ka-youtube-converted/8VjhtS3KlOY.mp4/8VjhtS3KlOY.mp4366yes https://puzzicle.org/math/cc-fifth-grade-math/v/place-value-with-decimalshttps://puzzicle.org/thumbs/x51d339d2d195dbc9.jpgPlace value with decimalsPlace value and decimals are related. We can begin by reviewing the concept of ones, tens, and hundreds places, and then demonstrates how tenths, hundredths, and thousandths places work. By understanding place value, we can interpret the meaning of each digit in a decimal number.https://cdn.kastatic.org/ka-youtube-converted/wtrrr15mbvQ.mp4/wtrrr15mbvQ.mp4245yes https://puzzicle.org/math/cc-fifth-grade-math/v/multiplying-and-dividing-by-10-100-1000https://puzzicle.org/thumbs/xd4695fbb10631bae.jpgMultiplying and dividing by 10, 100, 1000Learn how multiplying and dividing whole numbers by 10, 100, and 1,000 shifts the digits of the number left or right, respectively. Multiplying increases the place value, while dividing decreases it, making the process easier to understand.https://cdn.kastatic.org/ka-youtube-converted/yVg6LyNgIT8.mp4/yVg6LyNgIT8.mp4524yes https://puzzicle.org/math/cc-fifth-grade-math/v/multiplying-and-dividing-by-powers-of-10https://puzzicle.org/thumbs/x41a68fb542ebc43b.jpgMultiplying and dividing by powers of 10Learn about the patterns when multiplying or dividing numbers by powers of 10. Multiplying shifts digits to the left, while dividing shifts them to the right. The exponent determines the number of places digits shift.https://cdn.kastatic.org/ka-youtube-converted/jBN69fm8_T0.mp4/jBN69fm8_T0.mp4302yes https://puzzicle.org/math/cc-fifth-grade-math/v/introduction-to-powers-of-10https://puzzicle.org/thumbs/xc0c28d82e0324ccc.jpgIntroduction to powers of 10Exponents are a way of simplifying the notation for repeated multiplication. When using a base of 10, the exponent tells you how many times to multiply 10 by itself. Converting between exponential notation and standard notation is straightforward: for example, 10 to the second power is the same as 10 times 10, or 100.https://cdn.kastatic.org/ka-youtube-converted/Xvsqv2oO2ac.mp4/Xvsqv2oO2ac.mp4252yes https://puzzicle.org/math/cc-fifth-grade-math/v/powers-of-10https://puzzicle.org/thumbs/x8f37f955.jpgUsing exponents with powers of 10Explore the concept of multiplying by powers of 10. Learn about the pattern of adding zeros when multiplying by 10 and the use of exponents as a shorthand for repeated multiplication. Discover the connection between the number of times 10 is multiplied and the number of zeros in the product.https://cdn.kastatic.org/ka-youtube-converted/YJdCw2fK-Og.mp4/YJdCw2fK-Og.mp4334yes https://puzzicle.org/math/cc-fifth-grade-math/v/patterns-in-zeros-exercisehttps://puzzicle.org/thumbs/x4bccbc14.jpgExponents and powers of 10 patternsThis lesson explains calculations involving powers of ten. It demonstrates how to multiply and divide by these powers, and explains the results of these calculations, such as adding or removing zeroes from the end of a number. Additionally, the lesson discusses how these calculations can shift the placement of a decimal point to the left or righthttps://cdn.kastatic.org/ka-youtube-converted/pYaIjA4CsKo.mp4/pYaIjA4CsKo.mp4416yes https://puzzicle.org/math/cc-fifth-grade-math/v/thousandths-on-the-number-linehttps://puzzicle.org/thumbs/x53608e02aac6166f.jpgThousandths on the number lineWe can find the value of a point on a number line that falls between two known values, in this case 0.03 and 0.04. It is important to understand place value, specifically the tenths, hundredths, and thousandths places. By dividing the space between the two known values into ten equal segments, it shows that the point is located at 0.038 (or 38 thousandths).https://cdn.kastatic.org/ka-youtube-converted/ABHQJFWZGsk.mp4/ABHQJFWZGsk.mp4140yes https://puzzicle.org/math/cc-fifth-grade-math/v/fractions-as-division-by-powers-of-10https://puzzicle.org/thumbs/x8fdc858a.jpgFractions as division by power of 10Explore the concept of multiplying and dividing numbers by powers of 10. Understand how moving the decimal point to the left or right changes the value of a number, and how this relates to the concept of fractions as division.https://cdn.kastatic.org/ka-youtube-converted/LohXZt7QzCM.mp4/LohXZt7QzCM.mp4307yes https://puzzicle.org/math/cc-fifth-grade-math/v/comparing-place-values-in-decimalshttps://puzzicle.org/thumbs/x8be920b7.jpgComparing decimal place valuesExplore the concept of place value in multi-digit numbers, emphasizing how a digit's value changes based on its position. Understand how a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.https://cdn.kastatic.org/ka-youtube-converted/lR_kUUPL8YY.mp4/lR_kUUPL8YY.mp4340yes https://puzzicle.org/math/cc-fifth-grade-math/v/expanding-out-a-decimal-by-place-valuehttps://puzzicle.org/thumbs/xd4b6caae.jpgWrite decimals in expanded formTo write a decimal in expanded form, we need to break down each digit according to its place value. Start with the whole number portion, identifying the hundreds, tens, and ones places. Then, move on to the tenths, hundredths, and thousandths places. Keep in mind the order of operations when combining the expanded terms.https://cdn.kastatic.org/ka-youtube-converted/he4kcTujy30.mp4/he4kcTujy30.mp4173yes https://puzzicle.org/math/cc-fifth-grade-math/v/decimal-place-value-2https://puzzicle.org/thumbs/1805.jpgDecimals in written form (hundredths)To write a decimal in word form, start by writing out the whole number portion. Next, express the decimal portion as a fraction of hundredths. For clarity, it's often helpful to simplify multiple terms (such as "one tenth and five hundredths") into a single fraction (such as "fifteen hundredths").https://cdn.kastatic.org/ka-youtube-converted/AuD2TX-90Cc.mp4/AuD2TX-90Cc.mp4211yes https://puzzicle.org/math/cc-fifth-grade-math/v/writing-out-a-decimal-in-wordshttps://puzzicle.org/thumbs/x40ad4b83.jpgDecimals in written form (thousandths)Learn how to read and write decimals in different forms, focusing on thousandths. Learn how to decompose a decimal into its parts (20,000, 5/10, and 7/1000) and rewrite it as 20,000 and 507/1000.https://cdn.kastatic.org/ka-youtube-converted/G7QiIkYfeME.mp4/G7QiIkYfeME.mp4202yes https://puzzicle.org/math/cc-fifth-grade-math/v/expressing-decimals-in-multiple-formshttps://puzzicle.org/thumbs/xc6ab61ca967b1e72.jpgExpressing decimals in multiple formsLearn all about understanding how to express decimals in various forms. Learn to break down decimals into place values, show how to write them in expanded form, separate each decimal part, and explain how to say them in words.https://cdn.kastatic.org/ka-youtube-converted/U3w9ppDv_MU.mp4/U3w9ppDv_MU.mp4229yes https://puzzicle.org/math/cc-fifth-grade-math/v/visual-understanding-of-regrouping-decimalshttps://puzzicle.org/thumbs/x3e6af5072fc5f4e5.jpgVisual understanding of regrouping decimalsExplore the concept of place value in decimals, focusing on how to regroup value from one place to another. Understand the relationship between ones, tenths, and hundredths in decimal numbers.https://cdn.kastatic.org/ka-youtube-converted/e_KEaTPcPKg.mp4/e_KEaTPcPKg.mp4319yes https://puzzicle.org/math/cc-fifth-grade-math/v/comparing-decimals-2-examplehttps://puzzicle.org/thumbs/x76e3cf20.jpgComparing decimals: 9.97 and 9.798To compare two decimals, we should start by looking at the largest place value. If the numbers are the same, continue to smaller place values until we find a difference. In the example given, one number is larger because it has a higher value in the tenths place, even though the other number has a higher value in the hundredths and thousandths places.https://cdn.kastatic.org/ka-youtube-converted/f8B2dBQn3_U.mp4/f8B2dBQn3_U.mp4264yes https://puzzicle.org/math/cc-fifth-grade-math/v/comparing-decimals-4https://puzzicle.org/thumbs/158925084.jpgComparing decimals: 156.378 and 156.348This video is all about comparing decimals. Learn how to determine which decimal is greater by comparing each place value, starting from the left and moving right, until a difference is found.https://cdn.kastatic.org/ka-youtube-converted/FwMYoK1QKso.mp4/FwMYoK1QKso.mp4128yes https://puzzicle.org/math/cc-fifth-grade-math/v/comparing-decimals-in-different-representationshttps://puzzicle.org/thumbs/x45985c9a79cb6345.jpgComparing decimals in different representationsThis video is all about comparing decimals. Learn how to determine which of two decimals is larger by converting them into a common format. Practice understanding place value and the importance of each digit's position.https://cdn.kastatic.org/ka-youtube-converted/_tk3nWC5OAA.mp4/_tk3nWC5OAA.mp4268yes https://puzzicle.org/math/cc-fifth-grade-math/v/compare-decimals-word-problemshttps://puzzicle.org/thumbs/x27d33c81597fc209.jpgComparing decimals word problemsExplore the concept of comparing decimals. Learn the importance of aligning place values and starting from the largest place value when comparing. The video also demonstrates that more digits doesn't necessarily mean a larger number.https://cdn.kastatic.org/ka-youtube-converted/ELNooV0ymYk.mp4/ELNooV0ymYk.mp4242yes https://puzzicle.org/math/cc-fifth-grade-math/v/ordering-decimalshttps://puzzicle.org/thumbs/xe6cd87b441d5acc6.jpgOrdering decimalsPractice comparing decimals and learning how to order numbers with decimals from least to greatest. Learn about the importance of examining each decimal place, starting from the left (the largest place value) and moving rightwards.https://cdn.kastatic.org/ka-youtube-converted/RmHmtLQ-pqU.mp4/RmHmtLQ-pqU.mp4211yes https://puzzicle.org/math/cc-fifth-grade-math/v/comparing-decimals-examplehttps://puzzicle.org/thumbs/x231fca9b6108d02c.jpgOrdering decimals through thousandthsLearn all about comparing decimals. It teaches students how to order decimals from least to greatest by comparing each place value, starting from the most significant and moving to the right.https://cdn.kastatic.org/ka-youtube-converted/Dese7hoWZMM.mp4/Dese7hoWZMM.mp4181yes https://puzzicle.org/math/cc-fifth-grade-math/v/rounding-decimalshttps://puzzicle.org/thumbs/1808.jpgWorked example: Rounding decimals to nearest tenthRounding decimals to the nearest tenth can be fun. We can identify the tenths place in the given decimal, 9.564. Then we can look at the digit to the right of the tenths place to determine whether to round up or down. In this case, the answer is to round up to 9.6.https://cdn.kastatic.org/ka-youtube-converted/_MIn3zFkEcc.mp4/_MIn3zFkEcc.mp486yes https://puzzicle.org/math/cc-fifth-grade-math/v/rounding-decimals-on-the-number-linehttps://puzzicle.org/thumbs/xfddb821a92c39916.jpgRounding decimals on the number lineLearn how to round decimals using a number line, visually showing which whole number, tenth, hundredth, or thousandth is closest. Building off intuition of rounding rules and connect rounding to the nearest place value on the number line.https://cdn.kastatic.org/ka-youtube-converted/T3BMIaDUvck.mp4/T3BMIaDUvck.mp4262yes https://puzzicle.org/math/cc-fifth-grade-math/v/rounding-to-the-nearest-tenth-and-hundredthhttps://puzzicle.org/thumbs/xaa5822d8b6021ac6.jpgRounding decimals word problemsThis video dives into the world of rounding decimals. It explores how to adjust decimals to the closest tenth, hundredth, and whole number, all while highlighting the importance of grasping place value.https://cdn.kastatic.org/ka-youtube-converted/IcmAxkEImtI.mp4/IcmAxkEImtI.mp4272yes https://puzzicle.org/math/cc-fifth-grade-math/v/estimating-adding-decimalshttps://puzzicle.org/thumbs/x5bb1dcccd5e31953.jpgEstimating decimal additionLearn all about how to estimate when adding decimals. Practice rounding to the nearest whole number and how this can help us get a close approximation of the actual answer.https://cdn.kastatic.org/ka-youtube-converted/dj3tcEbWBDQ.mp4/dj3tcEbWBDQ.mp4210yes https://puzzicle.org/math/cc-fifth-grade-math/v/introduction-to-adding-decimals-tenthshttps://puzzicle.org/thumbs/xcf0f7c39de0ed576.jpgIntroduction to adding decimals: tenthsThis video introduces the concept of adding decimals, specifically tenths. It explains how to visualize and calculate the addition of tenths using both numerical and visual methods, reinforcing the understanding of decimal addition.https://cdn.kastatic.org/ka-youtube-converted/7I_AILEWTL4.mp4/7I_AILEWTL4.mp4386yes https://puzzicle.org/math/cc-fifth-grade-math/v/adding-decimals-with-ones-and-tenths-partshttps://puzzicle.org/thumbs/xcf7cf31ff1f4c540.jpgAdding decimals with ones and tenths partsLearn how to add decimals, specifically to the tenths. Practice adding whole numbers first, then adding the decimal parts. Learn the concept of carrying over in decimal addition.https://cdn.kastatic.org/ka-youtube-converted/66_Sb1dP81E.mp4/66_Sb1dP81E.mp4365yes https://puzzicle.org/math/cc-fifth-grade-math/v/andding-decimals-with-hundredthshttps://puzzicle.org/thumbs/xb4b36ff6184dd69d.jpgAdding decimals with hundredthsThere are different ways to add decimals involving hundredths. One method is to combine the total number of hundredths, while another involves breaking the numbers down into tenths and hundredths and adding them separately. A more complicated example shows how these techniques can be used to arrive at a whole number plus a fraction. Understanding these concepts is important before moving on to faster methods.https://cdn.kastatic.org/ka-youtube-converted/fWi-i_mdUOI.mp4/fWi-i_mdUOI.mp4417yes https://puzzicle.org/math/cc-fifth-grade-math/v/adding-decimals-with-ones-tenths-and-hundredthshttps://puzzicle.org/thumbs/x5249a8ffcd932f68.jpgAdding decimals with ones, tenths and hundredthsLearn all about adding decimals, specifically in the hundredths place. Learn about aligning decimal points and adding each place value separately, from hundredths to ones. It also introduces a systematic way to add decimals.https://cdn.kastatic.org/ka-youtube-converted/6NBkb98nFOM.mp4/6NBkb98nFOM.mp4413yes https://puzzicle.org/math/cc-sixth-grade-math/v/adding-decimals-example-1https://puzzicle.org/thumbs/xdd0d7ae2.jpgAdding decimals: 9.087+15.31Adding decimals is an essential math skill. To add decimals, such as 9.087 and 15.31, align the decimal points and place values. Fill in any missing digits with zeros to make the numbers the same length. Then, perform the addition as usual, starting with the smallest place value. The result of adding 9.087 and 15.31 is 24.397.https://cdn.kastatic.org/ka-youtube-converted/oLh_sIESQnY.mp4/oLh_sIESQnY.mp4160yes https://puzzicle.org/math/cc-sixth-grade-math/v/adding-decimals-example-2https://puzzicle.org/thumbs/xbcb2a374.jpgAdding decimals: 0.822+5.65Adding decimals is a breeze when you follow these simple steps: First, line up the decimal points and write the numbers one below the other. Next, start adding from the smallest place value (thousandths) and carry over any extra value to the next place if needed. Finally, include the decimal point in your answer. Mastering decimal addition helps build strong math skills for everyday life.https://cdn.kastatic.org/ka-youtube-converted/ErHFQUhZ94I.mp4/ErHFQUhZ94I.mp4106yes https://puzzicle.org/math/cc-sixth-grade-math/v/adding-decimalshttps://puzzicle.org/thumbs/4802227.jpgAdding three decimalsAdding decimals is easy when you follow these steps: line up the decimal points, fill in any missing digits with zeros, and then add the numbers from the smallest place value to the largest. By following this method, you can accurately add decimals and solve real-world problems.https://cdn.kastatic.org/ka-youtube-converted/w616LEmrHVE.mp4/w616LEmrHVE.mp4165yes https://puzzicle.org/math/cc-fifth-grade-math/v/estimating-subtracting-decimalshttps://puzzicle.org/thumbs/x190b7d981962f9b7.jpgEstimating decimal subtractionThis video is all about learning to estimate when subtracting decimals. It emphasizes the importance of rounding to the nearest whole number and doing quick mental calculations to get an approximate answer. It's all about getting comfortable with estimation!https://cdn.kastatic.org/ka-youtube-converted/LQmAGFCJMqo.mp4/LQmAGFCJMqo.mp4208yes https://puzzicle.org/math/cc-fifth-grade-math/v/strategies-for-subtracting-basic-decimalshttps://puzzicle.org/thumbs/xbf77dcbc4bed7035.jpgStrategies for subtracting basic decimalsDive into the world of subtracting decimals, focusing on tenths. It's all about exploring different strategies like using visual aids, hopping along number lines, and splitting numbers into bite-sized chunks.https://cdn.kastatic.org/ka-youtube-converted/N5xWYgeIWmU.mp4/N5xWYgeIWmU.mp4348yes https://puzzicle.org/math/cc-fifth-grade-math/v/strategies-for-subtracting-more-complex-decimals-with-tenthshttps://puzzicle.org/thumbs/x45933de401946e11.jpgStrategies for subtracting more complex decimals with tenthsLearn all about subtraction of decimals, particularly those in the tenths place. The focus is on various strategies like decomposing whole numbers into tenths and using a number line for visualization.https://cdn.kastatic.org/ka-youtube-converted/IE0HQoLrbX4.mp4/IE0HQoLrbX4.mp4526yes https://puzzicle.org/math/cc-fifth-grade-math/v/more-advanced-subtraction-strategies-with-hundredthshttps://puzzicle.org/thumbs/x06d023890a92da69.jpgMore advanced subtraction strategies with hundredthsThis video explores the concept of subtracting decimals, specifically hundredths. It demonstrates different strategies for subtracting decimals, emphasizing the importance of understanding place value and the flexibility of breaking down numbers to make subtraction easier.https://cdn.kastatic.org/ka-youtube-converted/URdTZ38JvOI.mp4/URdTZ38JvOI.mp4302yes https://puzzicle.org/math/cc-fifth-grade-math/v/subtraction-strategies-with-hundredthshttps://puzzicle.org/thumbs/xd56548451a9410b2.jpgSubtraction strategies with hundredthsThis video is all about subtracting decimals, specifically those in the hundredths place. It covers different strategies for subtraction, breaking down the process into manageable steps, and emphasizes the importance of understanding the concept behind the calculations.https://cdn.kastatic.org/ka-youtube-converted/GUK9aup5HOs.mp4/GUK9aup5HOs.mp4339yes https://puzzicle.org/math/cc-fifth-grade-math/v/another-example-subtracting-decimals-to-hundredthshttps://puzzicle.org/thumbs/x10e8640f.jpgSubtracting decimals: 10.1-3.93This video shows how to subtract decimals down to the hundredths place. It emphasizes the importance of aligning decimals and place values, and demonstrates the process of regrouping when subtracting.https://cdn.kastatic.org/ka-youtube-converted/MufbvU4tGh8.mp4/MufbvU4tGh8.mp4113yes https://puzzicle.org/math/cc-fifth-grade-math/v/subtracting-decimals-up-to-hundredthshttps://puzzicle.org/thumbs/x14d899f1.jpgSubtracting decimals: 9.57-8.09Learn to subtract decimals by aligning the decimal points and place values. Regroup numbers for easier subtraction, and then perform the subtraction for each place value. Mastering decimal subtraction helps build strong math skills and real-world problem-solving abilities.https://cdn.kastatic.org/ka-youtube-converted/o31cLUkS23E.mp4/o31cLUkS23E.mp471yes https://puzzicle.org/math/cc-fifth-grade-math/v/estimating-with-multiplying-decimals-and-whole-numbershttps://puzzicle.org/thumbs/xbdafd84387355135.jpgEstimating with multiplying decimals and whole numbersLearn all about the concept of estimating products in multiplication problems involving decimals and whole numbers. Understand the importance of rounding to the nearest whole number for easier computation, and demonstrates how this strategy can help make reasonable estimates.https://cdn.kastatic.org/ka-youtube-converted/d3Qg71bZ4Rw.mp4/d3Qg71bZ4Rw.mp4192yes https://puzzicle.org/math/cc-fifth-grade-math/v/strategies-for-multiplying-decimals-and-whole-numbershttps://puzzicle.org/thumbs/x98562a9c2d7fec19.jpgStrategies for multiplying decimals and whole numbersWe can use different methods for multiplying whole numbers with decimals. The first method involves converting the decimal into a fraction and multiplying the numerators and denominators together. The second method involves breaking the decimal into different place values and multiplying each part by the whole number. Estimating the answer beforehand can help check for accuracyhttps://cdn.kastatic.org/ka-youtube-converted/yueSfjh2ra4.mp4/yueSfjh2ra4.mp4413yes https://puzzicle.org/math/cc-fifth-grade-math/v/multiplying-decimals-and-whole-numbers-with-visualshttps://puzzicle.org/thumbs/xefa4c5fff318bc4a.jpgMultiplying decimals and whole numbers with visualsMultiplying whole numbers and decimals can be fun. It starts with simple examples, such as multiplying 3/10 by 4, and then moves on to more complex examples, such as multiplying 52/100 by 3. In each case, we demonstrate how to use a number line or model to visualize the process and find the product.https://cdn.kastatic.org/ka-youtube-converted/K-bA-k_9Rg0.mp4/K-bA-k_9Rg0.mp4273yes https://puzzicle.org/math/cc-fifth-grade-math/v/strategies-for-multiplying-multi-digit-decimals-by-whole-numbershttps://puzzicle.org/thumbs/xbec7336ee5b8922d.jpgStrategies for multiplying multi-digit decimals by whole numbersLearn how to multiply decimals and whole numbers. The video presents two strategies: converting decimals to tenths and multiplying, then converting back, and using estimation to determine where the decimal should go in the answer. Both methods reinforce understanding of place value.https://cdn.kastatic.org/ka-youtube-converted/RtXpRld8vb4.mp4/RtXpRld8vb4.mp4265yes https://puzzicle.org/math/cc-fifth-grade-math/v/estimating-decimal-multiplicationhttps://puzzicle.org/thumbs/x341feb4f9b12d753.jpgEstimating decimal multiplicationThis video is all about using estimation to simplify the process of multiplying decimals. It emphasizes the idea that rounding numbers to their nearest whole can make multiplication easier and quicker, especially when you're trying to do it in your head!https://cdn.kastatic.org/ka-youtube-converted/1serq4kmeZc.mp4/1serq4kmeZc.mp4239yes https://puzzicle.org/math/cc-fifth-grade-math/v/understanding-decimal-multiplicationhttps://puzzicle.org/thumbs/x59c34c596fe1940b.jpgUnderstanding decimal multiplicationThis video introduces multiplying decimals, emphasizing the role of decimal placement in the product. It highlights the usefulness of estimation in simplifying calculations and understanding the relationship between the numbers.https://cdn.kastatic.org/ka-youtube-converted/005032IwRME.mp4/005032IwRME.mp4160yes https://puzzicle.org/math/cc-fifth-grade-math/v/multiplying-decimals-using-estimationhttps://puzzicle.org/thumbs/x8f65b1eb7d45185c.jpgMultiplying decimals using estimationLearn all about multiplying decimals by using using estimation. Practice ignoring the decimals first, and multiplying the numbers as wholes. Then, use estimation to find where the decimal should go, ensuring the answer is reasonable.https://cdn.kastatic.org/ka-youtube-converted/VQN6MvdJRx4.mp4/VQN6MvdJRx4.mp4218yes https://puzzicle.org/math/cc-fifth-grade-math/v/developing-strategies-for-multiplying-decimalshttps://puzzicle.org/thumbs/x9233b8e2c9b8fe4d.jpgDeveloping strategies for multiplying decimalsThis video is all about understanding how to multiply decimals, specifically focusing on the strategy of converting decimals to fractions to make the multiplication process easier. Multiplying decimals can be made easier by thinking of the decimals as fractions. This helps understand what we are multiplying, and it also lays the groundwork for the standard method of decimal multiplication.https://cdn.kastatic.org/ka-youtube-converted/xBseQOzMB7A.mp4/xBseQOzMB7A.mp4307yes https://puzzicle.org/math/cc-fifth-grade-math/v/division-strategies-for-decimal-quotientshttps://puzzicle.org/thumbs/xbc987a93476de2aa.jpgDivision strategies for decimal quotientsThere are various methods for division when the quotient is not a whole number. Different examples demonstrate how to use fractions, tenths, and reciprocals to multiply instead of divide so we can arrive at the correct answer.https://cdn.kastatic.org/ka-youtube-converted/zlq_jQYD1mA.mp4/zlq_jQYD1mA.mp4544yes https://puzzicle.org/math/cc-fifth-grade-math/v/divide-whole-numbers-with-decimal-quotientshttps://puzzicle.org/thumbs/x841498d4a5021e52.jpgDivide whole numbers with decimal quotients: 5÷2This video teaches how to divide whole numbers to get a decimal quotient. It explains two methods to achieve this, emphasizing that division can result in decimals, not just whole numbers or remainders.https://cdn.kastatic.org/ka-youtube-converted/29qKGbhFnf8.mp4/29qKGbhFnf8.mp4144yes https://puzzicle.org/math/5th-engage-ny/v/strategies-for-dividing-by-tenthshttps://puzzicle.org/thumbs/x883ba0bf760bbec8.jpgStrategies for dividing by tenthsSal uses equivalent fractions to divide both whole numbers and decimals by tenths.https://cdn.kastatic.org/ka-youtube-converted/dToPqCXHNDA.mp4/dToPqCXHNDA.mp4323yes https://puzzicle.org/math/cc-fifth-grade-math/v/multiplying-and-dividing-decimals-by-10-100-1000https://puzzicle.org/thumbs/xbe0687a07ccf2974.jpgMultiplying and dividing decimals by 10, 100, 1000Explore the concept of multiplying and dividing decimals by 10, 100, and 1000. Learn how each digit in a decimal number shifts places when multiplied or divided by these powers of 10, providing a visual and intuitive understanding of the process.https://cdn.kastatic.org/ka-youtube-converted/6fLNcGSa_L4.mp4/6fLNcGSa_L4.mp4285yes https://puzzicle.org/math/cc-fifth-grade-math/v/strategies-for-multiplying-multiples-of-10-100-and-1000https://puzzicle.org/thumbs/x392b1ef21a44a7ad.jpgStrategies for multiplying multiples of 10, 100 and 1000Learn the strategy of estimating in multi-digit multiplication. Watch how to multiply numbers expressed in hundreds, thousands, or tens. Practice this skill by multiplying numbers together and estimating the result, reinforcing their understanding of place value and multiplication.https://cdn.kastatic.org/ka-youtube-converted/iFy0oH-3Pn4.mp4/iFy0oH-3Pn4.mp4400yes https://puzzicle.org/math/cc-fifth-grade-math/v/estimate-multiplying-multi-digit-numbershttps://puzzicle.org/thumbs/xa2131c43fbbd8687.jpgEstimating multi-digit multiplicationLearn the skill of estimating the product of multi-digit numbers. Watch a demonstration of the process. The video emphasizes the importance of estimation in simplifying complex multiplication problemshttps://cdn.kastatic.org/ka-youtube-converted/plzbnyvdaIQ.mp4/plzbnyvdaIQ.mp4158yes https://puzzicle.org/math/cc-fifth-grade-math/v/multiplying-multi-digit-numbers-6742x23https://puzzicle.org/thumbs/x459e2edf1621b728.jpgMultiplying multi-digit numbers: 6,742x23Learn the standard algorithm for multi-digit multiplication. Watch as the instructor breaks down the process step by step, demonstrating how to multiply each digit and carry over numbers.https://cdn.kastatic.org/ka-youtube-converted/lH6ptl4_LCE.mp4/lH6ptl4_LCE.mp4229yes https://puzzicle.org/math/arithmetic-home/v/more-intuition-on-multiplying-decimalshttps://puzzicle.org/thumbs/x5cf28b71.jpgMultiplying decimals: place valueSal uses an understanding of place value to multiply 2.91x3.2.https://cdn.kastatic.org/ka-youtube-converted/4IWfJ7-CYfE.mp4/4IWfJ7-CYfE.mp4389yes https://puzzicle.org/math/cc-fifth-grade-math/v/developing-strategies-for-multiplying-two-digit-decimalshttps://puzzicle.org/thumbs/xc41352bc8de609ea.jpgDeveloping strategies for multiplying 2-digit decimalsLearn to multiply decimals by treating them as fractions. Practice converting decimals to fractions and then multiplying like usual. Then convert the fractions back to decimals to solve!https://cdn.kastatic.org/ka-youtube-converted/ekDTjvF_CJw.mp4/ekDTjvF_CJw.mp4224yes https://puzzicle.org/math/cc-fifth-grade-math/v/minutes-to-hourshttps://puzzicle.org/thumbs/x42f6e58c.jpgConverting units: minutes to hoursLearn how to convert minutes to hours by dividing the minutes by 60. We demonstrate dividing 549 minutes by 60, resulting in 9.15 hours. The video also shows how to express the answer as a mixed number (9 and 3/20).https://cdn.kastatic.org/ka-youtube-converted/DLno3b-Xsmo.mp4/DLno3b-Xsmo.mp4294yes https://puzzicle.org/math/cc-fifth-grade-math/v/ordering-metric-distanceshttps://puzzicle.org/thumbs/x9eceeaf0.jpgConverting units: metric distanceLearn all about converting different metric units to a common unit, in this case, meters. Understand the prefixes like deca, hecto, kilo, milli, and centi, and how they relate to the base unit, meter. This helps us compare and order different measurements.https://cdn.kastatic.org/ka-youtube-converted/mI84WDfhuYA.mp4/mI84WDfhuYA.mp4242yes https://puzzicle.org/math/cc-fifth-grade-math/v/cm-to-metershttps://puzzicle.org/thumbs/xc6caed89.jpgConverting units: centimeters to metersLearn how to convert centimeters to meters by understanding the meaning of the "centi" prefix, which represents 1/100. We convert 37 centimeters to meters by dividing by 100, resulting in 0.37 meters.https://cdn.kastatic.org/ka-youtube-converted/6bQ6HJsAv4A.mp4/6bQ6HJsAv4A.mp4184yes https://puzzicle.org/math/cc-fifth-grade-math/v/converting-gallons-to-quarts-pints-and-cupshttps://puzzicle.org/thumbs/26.jpgConverting units: US volumeLearn how to convert US Customary units of volume (gallons, quarts, pints, and cups). We see the conversion process using an example of converting 3 and 1/2 gallons into cups, emphasizing the relationships between the units.https://cdn.kastatic.org/ka-youtube-converted/4uQtJDf7drU.mp4/4uQtJDf7drU.mp4317yes https://puzzicle.org/math/cc-fifth-grade-math/v/unit-conversion-with-fractionshttps://puzzicle.org/thumbs/153770144.jpgSame length in different unitsThe video dives into the concept of converting US customary units, specifically yards to inches. It emphasizes the importance of understanding the relationships between different units of measurement and demonstrates multiple methods for performing these conversions, including using improper fractions.https://cdn.kastatic.org/ka-youtube-converted/jFSenp9ueaI.mp4/jFSenp9ueaI.mp4348yes https://puzzicle.org/math/cc-fifth-grade-math/v/unit-word-problem-examplehttps://puzzicle.org/thumbs/xacbe5af6.jpgMeasurement word problem: tea partyThe video explores the concept of converting metric units within a real-world context. It emphasizes the importance of understanding and applying conversion factors, particularly between milliliters and liters, to solve multi-step problems.https://cdn.kastatic.org/ka-youtube-converted/PdHSs5_kx24.mp4/PdHSs5_kx24.mp4245yes https://puzzicle.org/math/cc-fifth-grade-math/v/solving-application-problems-involving-units-of-volumehttps://puzzicle.org/thumbs/28.jpgMeasurement word problem: blood driveThe video dives into the concept of converting units within the US Customary system. It emphasizes the process of moving from smaller units (like pints) to larger ones (like gallons), and how this affects the quantity. It's all about understanding the relationships between different units of measurement.https://cdn.kastatic.org/ka-youtube-converted/KgoXx_58QyQ.mp4/KgoXx_58QyQ.mp4209yes https://puzzicle.org/math/cc-fifth-grade-math/v/unit-conversion-word-problemhttps://puzzicle.org/thumbs/x0307b297.jpgMeasurement word problem: distance homeExplore the concept of converting units within the US customary system, specifically focusing on how to apply this skill to solve multi-step, real-world problems. Understand the relationships between different units and using that knowledge to navigate complex scenarios.https://cdn.kastatic.org/ka-youtube-converted/ZS1OZj_oWao.mp4/ZS1OZj_oWao.mp4254yes https://puzzicle.org/math/cc-fifth-grade-math/v/application-problems-involving-units-of-weighthttps://puzzicle.org/thumbs/25.jpgMeasurement word problem: elevatorThis video dives into the concept of converting units within the U.S. Customary system. It emphasizes the importance of understanding how different units relate to each other and how to use these relationships to solve real-world problems. It also highlights the process of converting mixed numbers to improper fractions.https://cdn.kastatic.org/ka-youtube-converted/gnCL_0uA6Ds.mp4/gnCL_0uA6Ds.mp4428yes https://puzzicle.org/math/cc-fifth-grade-math/v/converting-units-of-lengthhttps://puzzicle.org/thumbs/22.jpgMeasurement word problem: running lapsUnderstand how to convert units within the US Customary system to solve word problems. Learn about the importance of having all measurements in the same units to accurately solve problems, and demonstrates how to convert between miles, feet, and yards to determine the solution.https://cdn.kastatic.org/ka-youtube-converted/AGFO-ROxH_I.mp4/AGFO-ROxH_I.mp4484yes https://puzzicle.org/math/cc-fifth-grade-math/v/strategies-for-dividing-multiples-of-10-100-and-1000https://puzzicle.org/thumbs/xb1da7c0dc673d13c.jpgStrategies for dividing multiples of 10, 100, and 1000Learn the concept of multi-digit division estimation. Watch and understand how to divide numbers that are multiples of 10, 100, and 1000.https://cdn.kastatic.org/ka-youtube-converted/dejocl4sYfw.mp4/dejocl4sYfw.mp4428yes https://puzzicle.org/math/cc-fifth-grade-math/v/approximating-multi-digit-divisionhttps://puzzicle.org/thumbs/x8200e92d374e17e1.jpgEstimating multi-digit divisionLearn the concept of estimating multi-digit division problems. Watch examples of how to round numbers to make division easier and practice this skill through guided and independent exercises.https://cdn.kastatic.org/ka-youtube-converted/PssLxfFqHoA.mp4/PssLxfFqHoA.mp4155yes https://puzzicle.org/math/cc-fifth-grade-math/v/introduction-to-dividing-by-2-digitshttps://puzzicle.org/thumbs/xe3ac590d12d186bd.jpgIntroduction to dividing by 2-digitsLearn the art of dividing by two-digit numbers. Watch how to approximate numbers for easier division and practice this skill with examples. The video emphasizes the importance of trial and error in learning multi-digit division.https://cdn.kastatic.org/ka-youtube-converted/KzdbThwGNGI.mp4/KzdbThwGNGI.mp4297yes https://puzzicle.org/math/cc-sixth-grade-math/v/dividing-2-digits-no-remainderhttps://puzzicle.org/thumbs/xa5043558.jpgDividing by 2-digits: 9815÷65In this math lesson, we learn the process of dividing large numbers, specifically 9,815 divided by 65. The technique involves estimation, multiplication, and subtraction to find the quotient. By following these steps, we discover that the result is 151.https://cdn.kastatic.org/ka-youtube-converted/uCBm8iDyg1s.mp4/uCBm8iDyg1s.mp4309yes https://puzzicle.org/math/cc-sixth-grade-math/v/dividing-by-a-two-digit-numberhttps://puzzicle.org/thumbs/x0cf3f261.jpgDividing by 2-digits: 7182÷42Dividing large numbers by two-digit numbers can be done step by step. Start by fitting the divisor into each part of the dividend, then subtract and bring down the next digit. Estimate the number of times the divisor fits into the new number, and repeat the process until there's no remainder. This method simplifies complex division problems.https://cdn.kastatic.org/ka-youtube-converted/xXIG8ouHcsc.mp4/xXIG8ouHcsc.mp4169yes https://puzzicle.org/math/cc-fifth-grade-math/v/dividing-by-a-2-digit-number-with-a-remainderhttps://puzzicle.org/thumbs/x77b881e5e805a7d9.jpgDividing by 2-digits: 4781÷32Learn about dividing large numbers by two-digit numbers can be done step by step. Start by fitting the divisor into each part of the dividend, then subtract and bring down the next digit. Estimate the number of times the divisor fits into the new number, and repeat the process until there's no remainder.https://cdn.kastatic.org/ka-youtube-converted/cSTRd8WOpNE.mp4/cSTRd8WOpNE.mp4253yes https://puzzicle.org/math/cc-fifth-grade-math/v/approximating-dividing-by-decimalshttps://puzzicle.org/thumbs/xbb2207b0bf54fcc8.jpgEstimating decimal divisionEstimating in decimal division involves rounding numbers to make calculations easier. Practice rounding decimals to the nearest whole number or a simple fraction, then perform the division. This won't yield an exact result, but a close approximation, which can be useful in many real-life situations.https://cdn.kastatic.org/ka-youtube-converted/xGbrcOF8Z98.mp4/xGbrcOF8Z98.mp4176yes https://puzzicle.org/math/5th-engage-ny/v/multi-digit-division-strategies-for-decimalshttps://puzzicle.org/thumbs/x94b7a32c8a78d6d5.jpgMulti-digit division strategies for decimalsSal uses understanding of place value, expanded form, and fractions to help divide with decimals.https://cdn.kastatic.org/ka-youtube-converted/bRRtp9XdDrY.mp4/bRRtp9XdDrY.mp4306yes https://puzzicle.org/math/cc-fifth-grade-math/v/divide-whole-numbers-with-decimal-quotients-7812https://puzzicle.org/thumbs/x66052bb48b026436.jpgDivide whole numbers with decimal quotients: 78÷12Practice the concept of dividing whole numbers to get a decimal quotient. Learn how to break down division problems by finding multiples and using remainders. It also shows how to simplify fractions and convert them into decimals.https://cdn.kastatic.org/ka-youtube-converted/shSrtFE2OoM.mp4/shSrtFE2OoM.mp4256yes https://puzzicle.org/math/cc-fifth-grade-math/v/pattern-when-dividing-by-tenths-and-hundredthshttps://puzzicle.org/thumbs/xb46866692e98d206.jpgPattern when dividing by tenths and hundredthsThis video explores the concept of dividing whole numbers by decimals. It emphasizes the idea that dividing by a decimal like 0.1 or 0.01 is the same as multiplying by 10 or 100, respectively. It uses the strategy of converting whole numbers to tenths or hundredths to make the process clearer.https://cdn.kastatic.org/ka-youtube-converted/W6OAc7NKrSI.mp4/W6OAc7NKrSI.mp4286yes https://puzzicle.org/math/cc-fifth-grade-math/v/visually-adding-fractions-with-unlike-denominatorshttps://puzzicle.org/thumbs/x2656f61f.jpgVisually adding fractions: 5/6+1/4To add two fractions with different denominators, you need to find the least common multiple of the denominators. You can then rewrite both fractions with this common denominator, which will allow you to add the numerators together. In the example given, 5/6 and 1/4 are rewritten as 10/12 and 3/12, respectively, resulting in a sum of 13/12.https://cdn.kastatic.org/ka-youtube-converted/fvtv2uYjo_E.mp4/fvtv2uYjo_E.mp4340yes https://puzzicle.org/math/cc-fifth-grade-math/v/visually-subtracting-fractions-with-unlike-denominatorshttps://puzzicle.org/thumbs/xcb71b9b2.jpgVisually subtracting fractions: 3/4-5/8Learn about understanding how to add and subtract fractions with unlike denominators. Practice finding a common denominator, and demonstrates how to visually represent these operations.https://cdn.kastatic.org/ka-youtube-converted/poGjjX_VNbM.mp4/poGjjX_VNbM.mp4226yes https://puzzicle.org/math/cc-fifth-grade-math/v/adding-small-fractions-with-unlike-denominatorshttps://puzzicle.org/thumbs/xbf78b82a.jpgAdding fractions with unlike denominatorsLearn how to add two fractions with different denominators. It can be challenging to combine fractions when the denominators don't match. It is important to find a common denominator. Finally, the resource shows how to find a common multiple of the two denominators in order to convert the fractions so they can be added together.https://cdn.kastatic.org/ka-youtube-converted/bcCLKACsYJ0.mp4/bcCLKACsYJ0.mp4444yes https://puzzicle.org/math/cc-fifth-grade-math/v/subtracting-small-fractions-with-unlike-denominatorshttps://puzzicle.org/thumbs/x8c50f6df.jpgSubtracting fractions with unlike denominatorsTo subtract fractions with different denominators, you need to find a common denominator. This can be done by identifying the least common multiple of the two denominators. After rewriting the fractions so they both have the common denominator, you can subtract the numerators as you would with any two fractions.https://cdn.kastatic.org/ka-youtube-converted/2DPivVFCdqA.mp4/2DPivVFCdqA.mp4301yes https://puzzicle.org/math/cc-fifth-grade-math/v/adding-fractions-with-unlike-denominators-word-problemhttps://puzzicle.org/thumbs/x1a5c8db9.jpgAdding fractions word problem: paintLearn how to add and subtract fractions with unlike denominators through a real-world problem. Watch as the problem is broken down step-by-step, practice finding common denominators, and apply this knowledge to determine if the sum or difference of the fractions meets a specific requirement.https://cdn.kastatic.org/ka-youtube-converted/PKh5B9xyzSc.mp4/PKh5B9xyzSc.mp4347yes https://puzzicle.org/math/cc-fifth-grade-math/v/subtracting-fractions-with-unlike-denominators-word-problemhttps://puzzicle.org/thumbs/x39fd4ded.jpgSubtracting fractions word problem: tomatoesLearn how to solve word problems involving the addition and subtraction of fractions with unlike denominators. Watch examples being worked out step-by-step, and practice applying the same techniques to solve similar problems.https://cdn.kastatic.org/ka-youtube-converted/5fK8HEYNRuQ.mp4/5fK8HEYNRuQ.mp4199yes https://puzzicle.org/math/cc-fifth-grade-math/v/adding-subtracting-mixed-numbers-1-ex-1https://puzzicle.org/thumbs/915061485.jpgAdding mixed numbers: 19 3/18 + 18 2/3To add two mixed numbers, you can separate the whole number and fraction parts. Add the whole numbers together as you normally would, and then find a common denominator to add the fractions. Once you have your sum, you may need to simplify the fraction for your final answer.https://cdn.kastatic.org/ka-youtube-converted/8Eb5MWwcMMY.mp4/8Eb5MWwcMMY.mp4141yes https://puzzicle.org/math/cc-fifth-grade-math/v/adding-subtracting-mixed-numbers-1-ex-2https://puzzicle.org/thumbs/915104659.jpgSubtracting mixed numbers: 7 6/9 - 3 2/5We can subtract two mixed numbers, by converting them to improper fractions. First, multiply the whole number by the denominator and add the numerator to get the improper fraction. Find a common denominator and adjust the numerators accordingly. Subtract the two improper fractions, and simplify the result if possible. Finally, convert the improper fraction back to a mixed number if desiredhttps://cdn.kastatic.org/ka-youtube-converted/zR3gaFobut4.mp4/zR3gaFobut4.mp4151yes https://puzzicle.org/math/cc-fifth-grade-math/v/adding-mixed-numbers-with-regroupinghttps://puzzicle.org/thumbs/xe92731e8.jpgAdding mixed numbers with regroupingLearn how to add and subtract mixed numbers with unlike denominators. Watch the process of finding a common denominator, converting mixed numbers to improper fractions, and adding or subtracting these fractions.https://cdn.kastatic.org/ka-youtube-converted/I01O0r8b-po.mp4/I01O0r8b-po.mp4308yes https://puzzicle.org/math/cc-fifth-grade-math/v/subtracting-mixed-numbers-with-regroupinghttps://puzzicle.org/thumbs/x203af9d9.jpgSubtracting mixed numbers with regrouping (unlike denominators)To subtract mixed numbers, first align the whole numbers and fractions so they can be subtracted separately. If the fractions have different denominators, find a common denominator and convert them accordingly. If the fraction on the bottom is larger, regroup by borrowing from the whole number on top. This will allow you to complete the subtraction.https://cdn.kastatic.org/ka-youtube-converted/r3M68V9Joac.mp4/r3M68V9Joac.mp4231yes https://puzzicle.org/math/cc-fifth-grade-math/v/adding-and-subtracting-multiple-fractions-with-unlike-denominatorshttps://puzzicle.org/thumbs/xfee69324.jpgAdding and subtracting 3 fractionsLearn how to add and subtract fractions with unlike denominators. They watch the process of finding a common denominator, then practice adding and subtracting the fractions to solve the problems.https://cdn.kastatic.org/ka-youtube-converted/HSf9O1Domms.mp4/HSf9O1Domms.mp4323yes https://puzzicle.org/math/cc-fifth-grade-math/v/solving-for-the-missing-fractionhttps://puzzicle.org/thumbs/xa75acfc8.jpgSolving for the missing fractionLearn how to add and subtract fractions with unlike denominators. Watch examples, practice finding common denominators, and apply these skills to solve problems.https://cdn.kastatic.org/ka-youtube-converted/OPpmp-kAuE4.mp4/OPpmp-kAuE4.mp4340yes https://puzzicle.org/math/cc-fifth-grade-math/v/redistributing-trail-mixhttps://puzzicle.org/thumbs/xf5dd90fb.jpgLine plot distribution: trail mixExplore how to interpret data on line plots, specifically focusing on measurements in fractions of a unit. Understand how to redistribute values evenly across a set, and emphasizes the importance of understanding fractions and their operations in this context.https://cdn.kastatic.org/ka-youtube-converted/DtWovvMnPrk.mp4/DtWovvMnPrk.mp4368yes https://puzzicle.org/math/cc-fifth-grade-math/v/fractions-as-divisionhttps://puzzicle.org/thumbs/xdbd32e53.jpgUnderstanding fractions as divisionIn this video, we learn about the relationship between multiplication and division. Let's watch how they can undo each other. We can understand this concept with whole numbers and fractions.https://cdn.kastatic.org/ka-youtube-converted/c-_yrA-GUow.mp4/c-_yrA-GUow.mp4404yes https://puzzicle.org/math/cc-fifth-grade-math/v/fractions-division-mixed-numbershttps://puzzicle.org/thumbs/xf050e223.jpgCreating mixed numbers with fraction divisionLearn how to interpret fractions as division problems. Watch examples of how to break down fractions into whole numbers and remainders. Then see Sal practice turning these into mixed numbers.https://cdn.kastatic.org/ka-youtube-converted/g8yk6zQDAek.mp4/g8yk6zQDAek.mp4310yes https://puzzicle.org/math/mappers/v/my-share-of-soap-as-a-mixed-number-on-a-number-linehttps://puzzicle.org/thumbs/x851dfbb8.jpgMixed numbers on a number lineSal locates 3/2 on a number line.https://cdn.kastatic.org/ka-youtube-converted/V8C15AfL6Gk.mp4/V8C15AfL6Gk.mp4151yes https://puzzicle.org/math/cc-fifth-grade-math/v/creating-a-fraction-through-division-of-whole-numbershttps://puzzicle.org/thumbs/x28264782.jpgCreating a fraction through divisionIn this video, we'll learn about the relationship between division and multiplication using fractions. Watch how dividing by a number is the same as multiplying by its reciprocal. Then watch as Sal practices this concept using a fun, everyday example.https://cdn.kastatic.org/ka-youtube-converted/Mcm0Q3wGhMo.mp4/Mcm0Q3wGhMo.mp4165yes https://puzzicle.org/math/cc-fifth-grade-math/v/multiplying-fractions-by-whole-numbers-on-the-number-linehttps://puzzicle.org/thumbs/xd0821124.jpgMultiplying fractions by whole numbers on a number lineWe can multiply fractions by whole numbers on the number line. We multiply them by adding the fraction multiple times, just like we do with whole numbers.https://cdn.kastatic.org/ka-youtube-converted/QDvKqUc7Mj4.mp4/QDvKqUc7Mj4.mp4111yes https://puzzicle.org/math/cc-fifth-grade-math/v/concept-fraction-whole-number-producthttps://puzzicle.org/thumbs/xab5f5140.jpgMultiplying fractions and whole numbersMultiplying fractions is about combining parts of a whole. When you multiply two fractions together, you're taking a part of a part. When you multiply a fraction by a whole number, you're taking multiple copies of that fraction. In both cases, the result is a new fraction that represents a different part of a whole.https://cdn.kastatic.org/ka-youtube-converted/HiNrFT280_Y.mp4/HiNrFT280_Y.mp4174yes https://puzzicle.org/math/cc-fifth-grade-math/v/constructing-numerical-expressions-examplehttps://puzzicle.org/thumbs/x926c9fdf.jpgConstructing numerical expressionsSometimes, we don't need to solve an equation right away. Instead, we can use the information in a given word problem to construct a numerical expression that represents the information.https://cdn.kastatic.org/ka-youtube-converted/arY-EUZDNfk.mp4/arY-EUZDNfk.mp482yes https://puzzicle.org/math/cc-fifth-grade-math/v/visualizing-fraction-productshttps://puzzicle.org/thumbs/x52cf91fc.jpgMultiplying 2 fractions: fraction modelVisuals can be used to multiply fractions in a variety of ways, such as by using fraction models. By drawing rectangles and dividing them into equal parts, we can use an area model to multiply fractions. help to illustrate how the numerators and denominators of the two fractions interact to produce a product.https://cdn.kastatic.org/ka-youtube-converted/hr_mTd-oJ-M.mp4/hr_mTd-oJ-M.mp4297yes https://puzzicle.org/math/cc-fifth-grade-math/v/concept-fraction-mult-number-linehttps://puzzicle.org/thumbs/xa40b230b.jpgMultiplying 2 fractions: number lineLearn the concept of multiplying fractions by fractions. Watch as the process is visually represented on a number line, helping to understand the 'why' behind the procedure.https://cdn.kastatic.org/ka-youtube-converted/lJdb4_Fuxvw.mp4/lJdb4_Fuxvw.mp4286yes https://puzzicle.org/math/cc-fifth-grade-math/v/multiplying-fractionshttps://puzzicle.org/thumbs/2600759.jpgMultiplying 2 fractions: 5/6 x 2/3When multiplying fractions, you first start with the two fractions you want to multiply. You multiply the numerators (the top numbers) together, and then multiply the denominators (the bottom numbers) together. After putting the two results together as a new fraction, you may need to simplify the fraction in order to express it in its lowest terms.https://cdn.kastatic.org/ka-youtube-converted/CTKMK1ZGLuk.mp4/CTKMK1ZGLuk.mp4146yes https://puzzicle.org/math/cc-fifth-grade-math/v/multiplying-mixed-numbershttps://puzzicle.org/thumbs/2600760.jpgMultiplying mixed numbersMultiplying mixed numbers is similar to multiplying whole numbers, except that you have to account for the fractional parts as well. By converting mixed numbers into improper fractions, you can multiply the two numbers together in a straightforward way. Once you have the product as an improper fraction, you can convert it back into a mixed number.https://cdn.kastatic.org/ka-youtube-converted/RPhaidW0dmY.mp4/RPhaidW0dmY.mp4261yes https://puzzicle.org/math/cc-fifth-grade-math/v/multiplying-fractions-word-problem-4https://puzzicle.org/thumbs/xe872bdba.jpgMultiplying fractions word problem: laundryLearn how to solve word problems involving multiplication of fractions. Watch examples of real-life scenarios where fractions are multiplied, and practice applying the concept to solve similar problems.https://cdn.kastatic.org/ka-youtube-converted/KCehC_3CBBY.mp4/KCehC_3CBBY.mp480yes https://puzzicle.org/math/cc-fifth-grade-math/v/multiplying-fractions-word-problemhttps://puzzicle.org/thumbs/2600762.jpgMultiplying fractions word problem: muffinsLearn how to solve word problems involving the multiplication of fractions. Watch an example of a real-life scenario where fractions need to be multiplied, and then practice applying this concept to similar problems.https://cdn.kastatic.org/ka-youtube-converted/YJgIGwTysk0.mp4/YJgIGwTysk0.mp4219yes https://puzzicle.org/math/cc-fifth-grade-math/v/multiplying-fractions-word-problem-5https://puzzicle.org/thumbs/x6ac69445.jpgMultiplying fractions word problem: bikeThis video is all about understanding how to multiply fractions and mixed numbers. Watch as the steps are explained in a simple and fun way.https://cdn.kastatic.org/ka-youtube-converted/tfjQVtOyoaQ.mp4/tfjQVtOyoaQ.mp4231yes https://puzzicle.org/math/cc-fifth-grade-math/v/multiplication-as-scalinghttps://puzzicle.org/thumbs/xc436768c.jpgMultiplication as scaling with fractionsLearn about the concept of multiplication as scaling. Watch and understand how multiplying fractions can be seen as scaling, or resizing, the value of a number. Visualize this concept with various examples, reinforcing their understanding of multiplication as a scaling process.https://cdn.kastatic.org/ka-youtube-converted/yUYDhmQsiXY.mp4/yUYDhmQsiXY.mp4306yes https://puzzicle.org/math/cc-fifth-grade-math/v/visually-dividing-whole-numbers-by-unit-fractionshttps://puzzicle.org/thumbs/x5fa67069d4745396.jpgVisually dividing whole numbers by unit fractionsIn this video, you'll learn how to divide whole numbers by fractions. You'll watch examples of dividing wholes into fractional pieces and practice counting those pieces. It's all about understanding how many fractional parts make up a whole.https://cdn.kastatic.org/ka-youtube-converted/0lsDqvH5mDs.mp4/0lsDqvH5mDs.mp4148yes https://puzzicle.org/math/cc-fifth-grade-math/v/example-dividing-a-whole-by-a-unit-fractionhttps://puzzicle.org/thumbs/x8eae16da06ae1019.jpgDividing a whole number by a unit fractionIn this video, we'll learn how to divide a whole number by a fraction. We'll watch as three whole units are divided into sections, each being one-fourth of a whole. Then see practice problems counting these sections to understand the result of the division.https://cdn.kastatic.org/ka-youtube-converted/b26JxSoZvic.mp4/b26JxSoZvic.mp4112yes https://puzzicle.org/math/cc-fifth-grade-math/v/visually-dividing-unit-fraction-by-a-whole-numberhttps://puzzicle.org/thumbs/x845b796cc34be653.jpgVisually dividing unit fraction by a whole numberLearn how to divide fractions by whole numbers. You'll watch how dividing a fraction into smaller parts changes the denominator. Then, you'll practice this concept with examples to understand how dividing fractions works.https://cdn.kastatic.org/ka-youtube-converted/RygpFkLCSaM.mp4/RygpFkLCSaM.mp4180yes https://puzzicle.org/math/cc-fifth-grade-math/v/example-diving-a-unit-fraction-by-a-whole-numberhttps://puzzicle.org/thumbs/x1f9a08b9ecfd5604.jpgDividing a unit fraction by a whole numberLearn how to divide fractions by whole numbers. Watch as the instructor visually breaks down a fraction into smaller parts. Then, see practice problems that explain how division changes the value of the original fraction.https://cdn.kastatic.org/ka-youtube-converted/QsDMlHW826U.mp4/QsDMlHW826U.mp4145yes https://puzzicle.org/math/mappers/v/dividing-a-whole-number-by-a-fraction-word-problemhttps://puzzicle.org/thumbs/x32074d93.jpgDividing whole numbers by fractions: word problemLearn to solve a word problem that involves dividing a whole number by a unit fraction.https://cdn.kastatic.org/ka-youtube-converted/xoXYirs2Mzw.mp4/xoXYirs2Mzw.mp4152yes https://puzzicle.org/math/mappers/v/dividing-a-fraction-by-a-whole-number-word-problemhttps://puzzicle.org/thumbs/xedf0d036.jpgDividing fractions by whole numbers: studyingLearn to solve a word problem that involves dividing a unit fraction by a whole number.https://cdn.kastatic.org/ka-youtube-converted/yb7lVnY_VCY.mp4/yb7lVnY_VCY.mp4155yes https://puzzicle.org/math/5th-engage-ny/v/rewriting-a-fraction-as-a-decimalhttps://puzzicle.org/thumbs/xfca0c1cc.jpgRewriting a fraction as a decimal: 3/5To convert a fraction to a decimal, one approach is to rewrite the fraction with a denominator that is a power of ten. This makes it simple to express the fraction as a decimal, with each place to the right of the decimal point corresponding to a different power of ten.https://cdn.kastatic.org/ka-youtube-converted/XHLgY7Z3cb8.mp4/XHLgY7Z3cb8.mp4240yes https://puzzicle.org/math/5th-engage-ny/v/rewriting-fraction-as-a-decimal-to-the-hundredths-placehttps://puzzicle.org/thumbs/xf8dc2c7a.jpgRewriting a fraction as a decimal: 21/60Sal rewrites 21/60 as a decimal.https://cdn.kastatic.org/ka-youtube-converted/E3OfSGjpDoI.mp4/E3OfSGjpDoI.mp4229yes https://puzzicle.org/math/cc-fifth-grade-math/v/translating-expressions-with-parentheseshttps://puzzicle.org/thumbs/x0ce997e7.jpgTranslating expressions with parenthesesDive into the world of writing mathematical expressions from verbal statements. Understand the importance of parentheses in expressions, and how they can change the meaning of calculations.https://cdn.kastatic.org/ka-youtube-converted/ypxHVqE26gI.mp4/ypxHVqE26gI.mp4551yes https://puzzicle.org/math/cc-fifth-grade-math/v/how-we-measure-volumehttps://puzzicle.org/thumbs/x153d2046.jpgVolume introFinding volume, length, and area all involve measuring dimensions in order to quantify a certain aspect of an object or space. Although each property requires different calculations, they all rely on the same fundamental principle: taking measurements and using those measurements to derive a numerical value. When finding volume, you are measuring the amount of space that an object occupies.https://cdn.kastatic.org/ka-youtube-converted/xMz9WFvox9g.mp4/xMz9WFvox9g.mp4471yes https://puzzicle.org/math/cc-fifth-grade-math/v/measuring-volume-with-unit-cubeshttps://puzzicle.org/thumbs/x10cbe797.jpgMeasuring volume with unit cubesExplore the concept of volume, emphasizing that it's a measure of how much space something takes up. Practice the idea of measuring volume in cubic units, like cubic feet or cubic inches, by counting the number of unit cubes that fit into a 3D shape.https://cdn.kastatic.org/ka-youtube-converted/7WloKabLhII.mp4/7WloKabLhII.mp4133yes https://puzzicle.org/math/cc-fifth-grade-math/v/measuring-volume-as-area-times-lengthhttps://puzzicle.org/thumbs/xf932dab5.jpgMeasuring volume as area times lengthTo calculate the volume of a rectangular prism, you can multiply the dimensions (width, depth, and height) together in any order. For example, a prism that is 2cm wide, 3cm deep, and 4cm tall has a volume of 24 cubic centimeters.https://cdn.kastatic.org/ka-youtube-converted/daCT_24RnIY.mp4/daCT_24RnIY.mp4442yes https://puzzicle.org/math/cc-fifth-grade-math/v/volume-of-a-rectangular-prism-or-box-exampleshttps://puzzicle.org/thumbs/x46bdaeef.jpgVolume of a rectangular prismIf you want to know how much stuff you can cram into a box, finding its volume is key. To calculate the volume of a box, you need to know its height, width, and depth. You can find the volume by multiplying these three dimensions together. This formula works regardless of the units you are using (e.g. meters, feet), he method is the same - just make sure your units match up!https://cdn.kastatic.org/ka-youtube-converted/I9efKVtLCf4.mp4/I9efKVtLCf4.mp4119yes https://puzzicle.org/math/cc-fifth-grade-math/v/volume-in-unit-cubes-by-decomposing-shapehttps://puzzicle.org/thumbs/xad903af7.jpgVolume in unit cubes by decomposing shapeExplore the concept of decomposing complex shapes into simpler ones to calculate volume. Understand volume as additive and how to use multiplication of length, width, and height to find the volume of rectangular prisms. Learn how you can approach the same problem differently.https://cdn.kastatic.org/ka-youtube-converted/gm87omItizk.mp4/gm87omItizk.mp4199yes https://puzzicle.org/math/cc-fifth-grade-math/v/volume-through-decompositionhttps://puzzicle.org/thumbs/xe8255aad.jpgVolume through decompositionExplore the concept of finding the volume of complex shapes by decomposing them into simpler, non-overlapping rectangular prisms. Understand the additive nature of volume and demonstrates how to calculate the volume of each prism separately before adding them together for the total volume.https://cdn.kastatic.org/ka-youtube-converted/feNWZEln6Nc.mp4/feNWZEln6Nc.mp4229yes https://puzzicle.org/math/cc-fifth-grade-math/v/volume-word-problem-examplehttps://puzzicle.org/thumbs/x21aac844.jpgVolume word problem: water tankThe video dives into the concept of volume, specifically focusing on how to calculate the volume of a complex shape by subtracting the volume of an object within it from the total volume. Understand the formula for volume and its application in real-world problems.https://cdn.kastatic.org/ka-youtube-converted/OanPzjf2EYY.mp4/OanPzjf2EYY.mp4280yes https://puzzicle.org/math/cc-fifth-grade-math/v/example-finding-area-with-fractional-sideshttps://puzzicle.org/thumbs/x92847262.jpgFinding area with fractional sides 1Learn how to calculate the area of rectangles with fractional side lengths. Watch examples of this concept in action and practice applying it to different problems. The video emphasizes understanding the process, not just getting the answer.https://cdn.kastatic.org/ka-youtube-converted/C3RPDVGZdFw.mp4/C3RPDVGZdFw.mp4227yes https://puzzicle.org/math/cc-fifth-grade-math/v/intuition-for-area-with-fractional-side-lengthshttps://puzzicle.org/thumbs/x9635a341.jpgFinding area with fractional sides 2Learn how to calculate the area of rectangles with fractional side lengths. Watch examples of this concept in action, and then see practice problems applying what was shown to solve similar problems.https://cdn.kastatic.org/ka-youtube-converted/UmSe4rh__iU.mp4/UmSe4rh__iU.mp4422yes https://puzzicle.org/math/cc-fifth-grade-math/v/kites-as-a-mathematical-shapehttps://puzzicle.org/thumbs/x1b577232.jpgKites as a geometric shapeThe video dives into the world of quadrilaterals, specifically focusing on kites. It explores how kites are defined by two pairs of adjacent, congruent sides. It also highlights that the diagonals of a kite intersect at a 90-degree angle, with one line bisecting the other.https://cdn.kastatic.org/ka-youtube-converted/inlMrf2d-k4.mp4/inlMrf2d-k4.mp4338yes https://puzzicle.org/math/cc-fifth-grade-math/v/introduction-to-the-coordinate-planehttps://puzzicle.org/thumbs/xb7cbc96672feae0c.jpgIntroduction to the coordinate planeWhen we use coordinates to locate a point on a coordinate plane, we use two numbers to specify the horizontal (x) and vertical (y) positions. The first number in a coordinate pair tells us how far to move left or right from the origin, while the second number tells us how far to move up or down.https://cdn.kastatic.org/ka-youtube-converted/pAlq9fFwtus.mp4/pAlq9fFwtus.mp4408yes https://puzzicle.org/math/cc-fifth-grade-math/v/graphing-points-exercisehttps://puzzicle.org/thumbs/xba8b84ab.jpgCoordinate plane: graphing pointsThe video is all about understanding the basics of the coordinate plane. It explains how to plot points using ordered pairs, with the first number indicating the horizontal (x-axis) movement and the second number showing the vertical (y-axis) movement.https://cdn.kastatic.org/ka-youtube-converted/4vNloi-zNjU.mp4/4vNloi-zNjU.mp4141yes https://puzzicle.org/math/cc-fifth-grade-math/v/relationships-between-patternshttps://puzzicle.org/thumbs/xee7958b1.jpgGraphing patterns on coordinate planeLearn about understanding numerical patterns. Explore how to generate a second pattern from a given one by applying a rule, in this case, multiplying by 3 and adding 1. Then, graph the pairs of corresponding terms from both patterns on a coordinate plane.https://cdn.kastatic.org/ka-youtube-converted/ayRpoJgph0E.mp4/ayRpoJgph0E.mp4156yes https://puzzicle.org/math/cc-fifth-grade-math/v/interpreting-relationships-between-patternshttps://puzzicle.org/thumbs/x3ddc5e72.jpgInterpreting patterns on coordinate planeExplore the concept of numerical patterns, focusing on how to generate, identify, and graph them on a coordinate plane. Understand the relationship between corresponding terms in two different patterns, and how changes in one pattern affect the other.https://cdn.kastatic.org/ka-youtube-converted/HXg_a9oJ5nA.mp4/HXg_a9oJ5nA.mp4271yes https://puzzicle.org/math/cc-fifth-grade-math/v/interpreting-and-graphing-relationships-between-patternshttps://puzzicle.org/thumbs/x81be10de.jpgInterpreting relationships in ordered pairsExplore the concept of numerical patterns, focusing on how to generate, identify, and graph these patterns on a coordinate plane. Understand the relationships between corresponding terms in two different patterns and how these relationships can be represented as ordered pairs.https://cdn.kastatic.org/ka-youtube-converted/Muba9-W2FOQ.mp4/Muba9-W2FOQ.mp4273yes https://puzzicle.org/math/cc-fifth-grade-math/v/sequence-relationship-examplehttps://puzzicle.org/thumbs/x2d028899.jpgGraphing sequence relationshipsExplore the concept of numerical patterns. Understand how to generate two sequences using given rules, identify relationships between corresponding terms, form ordered pairs from these terms, and graph these pairs on a coordinate plane.https://cdn.kastatic.org/ka-youtube-converted/mqsIJucBn6c.mp4/mqsIJucBn6c.mp4148yes https://puzzicle.org/math/cc-fifth-grade-math/v/interpreting-plotted-pointshttps://puzzicle.org/thumbs/xf713557ca61b7328.jpgInterpreting plotted pointsThe video explores the concept of interpreting data points on a coordinate plane. It emphasizes understanding the relationship between the x and y axes, in this case, representing hours of exercise and screen time.https://cdn.kastatic.org/ka-youtube-converted/9EEJU4Ox-TE.mp4/9EEJU4Ox-TE.mp4156yes https://puzzicle.org/math/cc-fifth-grade-math/v/coordinate-plane-word-problem-examplehttps://puzzicle.org/thumbs/x4eee6cc3.jpgCoordinate plane graphing word problemThe video dives into the concept of using a coordinate plane to solve real-world problems. It emphasizes how to plot points on the plane and interpret their values, and how to calculate the distance between two points by counting blocks in both the horizontal and vertical directions.https://cdn.kastatic.org/ka-youtube-converted/_QKR8I5VE-w.mp4/_QKR8I5VE-w.mp4126yes https://puzzicle.org/math/arithmetic-home/v/rounding-whole-numbers-3https://puzzicle.org/thumbs/2600662.jpgRounding whole numbers to nearest tenSal rounds 1585 to the nearest ten.https://cdn.kastatic.org/ka-youtube-converted/3jBfLaLrk6I.mp4/3jBfLaLrk6I.mp464yes https://puzzicle.org/math/arithmetic-home/v/multiplication-2-the-multiplication-tableshttps://puzzicle.org/thumbs/23826.jpgMultiplication tables for 2-9Introduction to the multiplication 'times tables' from 2-9.https://cdn.kastatic.org/ka-youtube-converted/xO_1bYgoQvA.mp4/xO_1bYgoQvA.mp4927yes https://puzzicle.org/math/arithmetic-home/v/multiplication-6-multiple-digit-numbershttps://puzzicle.org/thumbs/23834.jpgMultiplying multi-digit numbersSal shows lots of examples for how to multiply 2- and 3-digit numbers using "standard algorithm".https://cdn.kastatic.org/ka-youtube-converted/-h3Oqhl8fPg.mp4/-h3Oqhl8fPg.mp4627yes https://puzzicle.org/math/arithmetic-home/v/dividing-3-digit-by-1-digit-without-remainderhttps://puzzicle.org/thumbs/xd8abe311.jpgLong division: 280÷5Learn to divide 280÷5 with long division. The answer wil not have a remainder.https://cdn.kastatic.org/ka-youtube-converted/HY-8ydAbiik.mp4/HY-8ydAbiik.mp4223yes https://puzzicle.org/math/5th-grade-foundations-engageny/v/adding-subtracting-mixed-numbers-0-5-ex-1https://puzzicle.org/thumbs/915206106.jpgAdding mixed numbers with like denominatorsThink of a mixed numbers as a number AND a fraction. You can add numbers....you can add fractions with like denominators. Therefore, you can add mixed numbers! Watch.https://cdn.kastatic.org/ka-youtube-converted/EGy1W24EPEI.mp4/EGy1W24EPEI.mp495yes https://puzzicle.org/math/5th-grade-foundations-engageny/v/adding-subtracting-mixed-numbers-0-5-ex-2https://puzzicle.org/thumbs/915126178.jpgSubtracting mixed numbers with like denominatorsThink of a mixed numbers as a number AND a fraction. You can subtract numbers....you can subtract fractions with like denominators. Therefore, you can subtract mixed numbers! Watch.https://cdn.kastatic.org/ka-youtube-converted/q19UnHEofOA.mp4/q19UnHEofOA.mp4151yes https://puzzicle.org/math/arithmetic-home/v/multiplication-word-problem-example-1https://puzzicle.org/thumbs/xf30d0482.jpgMultiplication word problem: carrotsMultiplication word problem: carrots from Arithmetic (all content)https://cdn.kastatic.org/ka-youtube-converted/fZtUn_THXnk.mp4/fZtUn_THXnk.mp4152yes https://puzzicle.org/math/arithmetic-home/v/division-word-problem-example-1https://puzzicle.org/thumbs/x4aef016e.jpgDivision word problem: row boatDivision word problem: row boat from Arithmetic (all content)https://cdn.kastatic.org/ka-youtube-converted/Z3qRkxzmYnU.mp4/Z3qRkxzmYnU.mp4108yes https://puzzicle.org/math/5th-grade-math-eureka-squared-aligned/v/creating-volume-expressionshttps://puzzicle.org/thumbs/xae7675b829d0d57d.jpgCreating volume expressionsLearn how to write expressions for the volume of a rectangular prism in different ways including the area of the base times the height, and repeated addition of the base.https://cdn.kastatic.org/ka-youtube-converted/NDPTBKbWmy8.mp4/NDPTBKbWmy8.mp4352yes https://puzzicle.org/math/cc-fifth-grade-math/v/multiplication-and-division-relationship-for-fractionshttps://puzzicle.org/thumbs/x060e4e6d013f18ba.jpgMultiplication and division relationship for fractionsWe learn how multiplication and division are related, even when we're dealing with fractions. Watch how dividing by a number is the same as multiplying by its reciprocal. Then watch Sal practice expressing these relationships using both division and multiplication.https://cdn.kastatic.org/ka-youtube-converted/jXA-nMI4Nt4.mp4/jXA-nMI4Nt4.mp4215yes https://puzzicle.org/math/5th-grade-math-ny-next-gen/v/drawing-an-area-model-for-the-product-of-two-unit-fractionshttps://puzzicle.org/thumbs/x0aa96eba85cb8677.jpgDrawing an area model for the product of two unit fractionsA conceptual overview on how to draw an area model that represents a unit fraction of a unit fraction and compare and contrast diagrams.https://cdn.kastatic.org/ka-youtube-converted/o4Lbf7-ffKM.mp4/o4Lbf7-ffKM.mp4271yes https://puzzicle.org/math/5th-grade-math-eureka-squared-aligned/v/multiplying-unit-fractions-area-model-to-expressionhttps://puzzicle.org/thumbs/x47e1b11472d15c7c.jpgMultiplying unit fractions: area model to expressionWorked example on how to represent an area model with a multiplication expression involving a unit fraction and a unit fraction.https://cdn.kastatic.org/ka-youtube-converted/ez4msrCr16Y.mp4/ez4msrCr16Y.mp4113yes https://puzzicle.org/math/5th-grade-math-eureka-squared-aligned/v/multiplying-unit-fractions-expression-to-area-modelhttps://puzzicle.org/thumbs/x1a1dc980fd778c85.jpgMultiplying unit fractions: expression to area modelWorked example on how to represent a multiplication expression involving a unit fraction and a unit fraction with an area model.https://cdn.kastatic.org/ka-youtube-converted/wsxFJsctzXU.mp4/wsxFJsctzXU.mp4126yes https://puzzicle.org/math/5th-grade-math-ny-next-gen/v/fraction-multiplication-area-model-to-expressionhttps://puzzicle.org/thumbs/xe77b60a5c7741bce.jpgFraction multiplication: area model to expressionLearn how to represent an area model with a multiplication expression involving a unit fraction and a non-unit fraction.https://cdn.kastatic.org/ka-youtube-converted/IP-M2bXt0HI.mp4/IP-M2bXt0HI.mp4157yes https://puzzicle.org/math/5th-grade-math-ny-next-gen/v/fraction-multiplication-expression-to-area-modelhttps://puzzicle.org/thumbs/x9ff3a1e5536cce4d.jpgFraction multiplication: expression to area modelLearn how to represent a multiplication expression involving a unit fraction and a non-unit fraction with an area model.https://cdn.kastatic.org/ka-youtube-converted/W4oNi8fmnD4.mp4/W4oNi8fmnD4.mp4162yes https://puzzicle.org/math/cc-fifth-grade-math/v/fraction-division-in-contexthttps://puzzicle.org/thumbs/x8d7dcf069da61a33.jpgFraction division in contextIn this video, learn how to divide fractions by whole numbers. Watch examples of real-life situations where this skill is useful. We'll practice dividing a fraction by a whole number to solve problems.https://cdn.kastatic.org/ka-youtube-converted/dnZ0fykw_w0.mp4/dnZ0fykw_w0.mp4202yes https://puzzicle.org/math/cc-fifth-grade-math/v/dividing-fractions-and-whole-number-word-problemshttps://puzzicle.org/thumbs/x08d1dc45a957f1e5.jpgDividing fractions and whole number word problemsThis video uses tape diagrams to visually represent division with fractions. It shows how to divide a fraction or a whole number into equal parts. Tape diagrams help us see the process in a clear, simple way.https://cdn.kastatic.org/ka-youtube-converted/HUT8eL1sGhk.mp4/HUT8eL1sGhk.mp4266yes https://puzzicle.org/math/5th-grade-math-eureka-squared-aligned/v/multiplying-3-digit-by-2-digit-with-area-modelshttps://puzzicle.org/thumbs/x2f347bb7d138ac30.jpgMultiplying 3-digit by 2-digit with area modelsA worked example showing how to use a diagram to help calculate the product of a 3-digit number and a 2-digit number.https://cdn.kastatic.org/ka-youtube-converted/pkYBNJDxWPQ.mp4/pkYBNJDxWPQ.mp4290yes https://puzzicle.org/math/5th-grade-math-eureka-squared-aligned/v/multiplying-3-digit-by-2-digit-with-partial-productshttps://puzzicle.org/thumbs/x875486c7b01a97c6.jpgMultiplying 3 digit by 2 digit with partial productsA worked example that shows how to multiply a 3-digit number by a 2-digit number with partial products and make a connection between the area model and the written strategy.https://cdn.kastatic.org/ka-youtube-converted/2ru0A_ico_0.mp4/2ru0A_ico_0.mp4372yes https://puzzicle.org/math/5th-grade-illustrative-mathematics/v/multiplying-3-digit-by-2-digit-comparing-standard-algorithm-to-partial-productshttps://puzzicle.org/thumbs/x0d363a5abe0eb912.jpgMultiplying 3-digit by 2-digit: comparing standard algorithm to partial productsA worked example that shows how to use the standard algorithm to multiply a 3-digit number by a 2-digit number, without composing new units.https://cdn.kastatic.org/ka-youtube-converted/bELurwZ4MLg.mp4/bELurwZ4MLg.mp4250yes https://puzzicle.org/math/5th-grade-math-ny-next-gen/v/multiplying-3-digit-by-2-digit-numbers-error-analysishttps://puzzicle.org/thumbs/x49df1a8b6640cd56.jpgMultiplying 3-digit by 2-digit numbers: Error analysisFind the error while multiplying 3-digit by 2-digit numbers by checking each step.https://cdn.kastatic.org/ka-youtube-converted/GartEps0cLc.mp4/GartEps0cLc.mp4108yes https://puzzicle.org/math/5th-grade-math-ny-next-gen/v/3-digit-by-2-digit-multiplication-examplehttps://puzzicle.org/thumbs/xbd04650ac6d626db.jpgMultiplying 3-digit by 2-digit numbersMultiply 3 digit by 2 digit whole numbers to find a product using the standard algorithm.https://cdn.kastatic.org/ka-youtube-converted/3e7WGi_CeAw.mp4/3e7WGi_CeAw.mp4144yes https://puzzicle.org/math/5th-grade-illustrative-mathematics/v/estimating-multi-digit-multiplication-word-problemshttps://puzzicle.org/thumbs/xbc9ee07100febec8.jpgEstimating multi-digit multiplication word problemsUse estimation as a strategy to solve multi-digit multiplication word problems.https://cdn.kastatic.org/ka-youtube-converted/s9pPN7r_cuU.mp4/s9pPN7r_cuU.mp4129yes https://puzzicle.org/math/arithmetic-home/v/partial-quotient-divisionhttps://puzzicle.org/thumbs/258726800.jpgPartial quotient method of division: introductionSal demonstrates an alternate to traditional long division that uses estimationhttps://cdn.kastatic.org/ka-youtube-converted/R486L0M5cWk.mp4/R486L0M5cWk.mp4507yes https://puzzicle.org/math/5th-grade-math-eureka-squared-aligned/v/dividing-by-2-digit-numbers-using-partial-quotientshttps://puzzicle.org/thumbs/x97fe1de016611803.jpgDividing 3-digit by 2-digit numbers using partial quotientsLearn how to use the partial quotient algorithm to divide a 3-digit number by a 2-digit number. Sal shows two ways to solve the same problem.https://cdn.kastatic.org/ka-youtube-converted/JtORV2GLkPI.mp4/JtORV2GLkPI.mp4255yes https://puzzicle.org/math/5th-grade-math-eureka-squared-aligned/v/dividing-4-digit-by-2-digit-numbers-using-partial-quotientshttps://puzzicle.org/thumbs/xf6be7d497f8dcf4a.jpgDividing 4 digit by 2 digit numbers using partial quotientsWorked example showing how to use the partial quotient algorithm to divide a 4-digit number by a 2-digit number.https://cdn.kastatic.org/ka-youtube-converted/tpkGUBZ1lJY.mp4/tpkGUBZ1lJY.mp4404yes https://puzzicle.org/math/5th-grade-math-eureka-squared-aligned/v/estimating-multi-digit-division-word-problemshttps://puzzicle.org/thumbs/xc9ae109226c9f0ca.jpgEstimating multi-digit division word problemsPractice using estimation as a strategy to solve multi-digit division word problems.https://cdn.kastatic.org/ka-youtube-converted/Ag0zq7TImyY.mp4/Ag0zq7TImyY.mp4153yes https://puzzicle.org/math/arithmetic-home/v/decimal-intuition-with-gridshttps://puzzicle.org/thumbs/x193de4d9.jpgWriting decimal numbers shown in gridsSal uses grid diagrams to determine what decimal is represented by a shaded area.https://cdn.kastatic.org/ka-youtube-converted/s9PvbFG2yJA.mp4/s9PvbFG2yJA.mp4186yes https://puzzicle.org/math/cc-fifth-grade-math/v/decimal-multiplication-with-gridshttps://puzzicle.org/thumbs/x25376d934f4123f1.jpgDecimal multiplication with gridsLearn how to use a hundred grid to represent decimal multiplication. Use both vertical and horizontal tenths to represent each multiplier and then use the overlapping areas to find the product.https://cdn.kastatic.org/ka-youtube-converted/3Phgu3ExLmY.mp4/3Phgu3ExLmY.mp4153yes https://puzzicle.org/math/cc-fifth-grade-math/v/visually-dividing-a-whole-number-by-a-decimalhttps://puzzicle.org/thumbs/xa5be050b00312c10.jpgVisually dividing a whole number by a decimalThis video is all about understanding how to divide whole numbers by decimals. It emphasizes the concept of partitioning whole numbers into equal groups of a certain decimal value, and counting how many of those groups we can make.https://cdn.kastatic.org/ka-youtube-converted/aklVb3Z5jx4.mp4/aklVb3Z5jx4.mp4152yes https://puzzicle.org/math/cc-fifth-grade-math/v/dividing-a-whole-number-by-a-decimal-on-a-number-linehttps://puzzicle.org/thumbs/xf5b795a8d75bec5d.jpgDividing a whole number by a decimal on a number lineExplore the concept of dividing whole numbers by decimals on a number line. Understand the process as making jumps on a number line. Each jump represents a decimal value, and the goal is to determine how many jumps it takes to reach a certain whole number.https://cdn.kastatic.org/ka-youtube-converted/PxafebH1u2k.mp4/PxafebH1u2k.mp4114yes https://puzzicle.org/math/cc-fifth-grade-math/v/dividing-whole-numbers-by-decimals-exampleshttps://puzzicle.org/thumbs/xbf41064ead1d20c8.jpgDividing whole numbers by decimals examplesOne approach to dividing whole numbers by decimals is to convert both numbers into the same base, such as tenths or hundredths. Another technique is to multiply the numerator and denominator to eliminate the decimal from the denominator.https://cdn.kastatic.org/ka-youtube-converted/se7rSl-tHkg.mp4/se7rSl-tHkg.mp4278yes https://puzzicle.org/math/cc-fifth-grade-math/v/visually-dividing-decimal-by-whole-numberhttps://puzzicle.org/thumbs/xdfb155f57387494f.jpgDividing a decimal by a whole number with fraction modelsThe video explores the concept of dividing decimals by whole numbers. It emphasizes visualizing the process, using the idea of converting tenths to hundredths to make division more intuitive.https://cdn.kastatic.org/ka-youtube-converted/rYsK3mI0Nto.mp4/rYsK3mI0Nto.mp4129yes https://puzzicle.org/math/cc-fifth-grade-math/v/dividing-a-decimal-by-a-whole-number-on-the-number-linehttps://puzzicle.org/thumbs/x72b5fb2edf65ed05.jpgDividing a decimal by a whole number on the number lineThe video explores the concept of dividing decimals by whole numbers. It emphasizes understanding division as splitting a quantity into equal parts, and uses the number line as a visual tool to demonstrate this. Then connects this concept to real-world situations.https://cdn.kastatic.org/ka-youtube-converted/7Wm0VOTPz5Q.mp4/7Wm0VOTPz5Q.mp489yes https://puzzicle.org/math/cc-fifth-grade-math/v/dividing-a-decimal-by-a-whole-number-examplehttps://puzzicle.org/thumbs/x731aff4073850df2.jpgDividing a decimal by a whole number exampleTo divide a decimal by a whole number, you can convert the decimal into a certain number of hundredths. After dividing the whole number into those hundredths, you can then convert the result back into a decimal form.https://cdn.kastatic.org/ka-youtube-converted/bIRwzhhBzEU.mp4/bIRwzhhBzEU.mp4176yes https://puzzicle.org/math/cc-fifth-grade-math/v/regrouping-decimalshttps://puzzicle.org/thumbs/x68f5d2be774432d5.jpgRegrouping with decimalsExplore the concept of expressing decimals in different forms. Delve into place value, regrouping, and how to represent the same decimal number in different ways using whole numbers. It's all about flexibility in understanding decimals!https://cdn.kastatic.org/ka-youtube-converted/vZmR_21Dwrg.mp4/vZmR_21Dwrg.mp4275yes https://puzzicle.org/math/cc-fifth-grade-math/v/regrouping-with-decimalshttps://puzzicle.org/thumbs/x5bceab99.jpgRegrouping with decimals: 21.3Practice the concept of representing decimals in different forms. Learn about place values, and how regrouping can change the appearance of a number, but not its value.https://cdn.kastatic.org/ka-youtube-converted/3szFVS5p_7A.mp4/3szFVS5p_7A.mp4255yes https://puzzicle.org/math/cc-fifth-grade-math/v/adding-fractions-with-unlike-denominators-introductionhttps://puzzicle.org/thumbs/x1f575daa009e0ec7.jpgAdding fractions with unlike denominators introductionTo add fractions with different denominators, such as 1/2 and 1/3, you need to find a common denominator. The least common multiple of 2 and 3 is 6, so you can rewrite the fractions as 3/6 and 2/6, respectively. This allows you to add the numerators together, resulting in a sum of 5/6.https://cdn.kastatic.org/ka-youtube-converted/zkJ1gOrYhEg.mp4/zkJ1gOrYhEg.mp4242yes https://puzzicle.org/math/cc-fifth-grade-math/v/subtracting-fractions-with-unlike-denominators-introductionhttps://puzzicle.org/thumbs/xdd18d17581f8b2b4.jpgSubtracting fractions with unlike denominators introductionLearn how to subtract fractions with unlike denominators. Watch the process of finding a common denominator, converting the fractions, and then performing the subtraction.https://cdn.kastatic.org/ka-youtube-converted/SMshimc16LM.mp4/SMshimc16LM.mp4216yes https://puzzicle.org/math/cc-fifth-grade-math/v/fraction-multiplication-as-scaling-exampleshttps://puzzicle.org/thumbs/xf9a493ca7b9a3264.jpgFraction multiplication as scaling examplesLearn the concept of multiplication as scaling. Watch examples of how multiplying by a fraction scales a number up or down. Understand that fractions between zero and one make numbers smaller, while fractions greater than one make numbers larger.https://cdn.kastatic.org/ka-youtube-converted/WLvbz6m2Syw.mp4/WLvbz6m2Syw.mp4218yes https://puzzicle.org/math/5th-grade-math-ny-next-gen/v/comparing-products-without-multiplying-fractionshttps://puzzicle.org/thumbs/x5dca92ba20cfea62.jpgComparing products without multiplying (fractions)Learn how to compare the value of products between a fraction and a whole number without calculating.https://cdn.kastatic.org/ka-youtube-converted/lMcKPVhHI7w.mp4/lMcKPVhHI7w.mp487yes https://puzzicle.org/math/cc-fifth-grade-math/v/multiplying-and-dividing-decimals-by-10https://puzzicle.org/thumbs/x3691632a4964cdc0.jpgMultiplying and dividing decimals by 10This video explores the concept of multiplying and dividing decimals by 10, 100, and 1000. It demonstrates how each digit in a number shifts places when multiplied or divided by powers of 10. The video emphasizes understanding place values and how they change in these operations.https://cdn.kastatic.org/ka-youtube-converted/HeiYNlVM-TY.mp4/HeiYNlVM-TY.mp4243yes https://puzzicle.org/math/5th-grade-math-ny-next-gen/v/quadrilateral-relationships-trapezoids-and-parallelogramshttps://puzzicle.org/thumbs/x6a2495e7c1edf6bd.jpgQuadrilateral relationships: Trapezoids and parallelogramsLearn how to determine the relationships between quadrilaterals (rectangles, rhombuses, squares, trapezoids, and parallelograms) by using their properties.https://cdn.kastatic.org/ka-youtube-converted/r6wTVaiwPyQ.mp4/r6wTVaiwPyQ.mp4132yes https://puzzicle.org/math/5th-grade-math-ny-next-gen/v/quadrilateral-relationships-rectangles-and-rhombuseshttps://puzzicle.org/thumbs/x66b60e61844dc31f.jpgQuadrilateral relationships: Rectangles and rhombusesLearn how to determine the relationships between quadrilaterals (rectangles, rhombuses, squares, trapezoids, and parallelograms) by using their properties.https://cdn.kastatic.org/ka-youtube-converted/hdyEkdtB0dA.mp4/hdyEkdtB0dA.mp4252yes https://puzzicle.org/math/arithmetic-home/v/place-value-relationships-examplehttps://puzzicle.org/thumbs/x280befac.jpgPlace value: comparing same digit in different placesLearn the relationship between a "4" in the hundreds place and a "4" in the thousands place.https://cdn.kastatic.org/ka-youtube-converted/PvSx8oJ7PrM.mp4/PvSx8oJ7PrM.mp4160yes https://puzzicle.org/math/5th-grade-math-eureka-squared-aligned/v/representing-multi-step-problems-strip-diagramshttps://puzzicle.org/thumbs/x66594a4c8bcba019.jpgRepresenting multi-step problems: strip diagramsPractice using strip diagrams to model multi-step problems.https://cdn.kastatic.org/ka-youtube-converted/bSoOcDUuG7E.mp4/bSoOcDUuG7E.mp4141yes https://puzzicle.org/math/5th-grade-math-eureka-squared-aligned/v/constructing-numerical-expressions-indian-accenthttps://puzzicle.org/thumbs/x1d37355dd2597676.jpgConstructing numerical expressionsSometimes, we don't need to solve an equation right away. Instead, we can use the information in a given word problem to construct a numerical expression that represents the information.https://cdn.kastatic.org/ka-youtube-converted/YYzP89zrSV8.mp4/YYzP89zrSV8.mp471yes https://puzzicle.org/math/5th-grade-math-eureka-squared-aligned/v/adding-fractions-with-unlike-denominators-indian-accenthttps://puzzicle.org/thumbs/x805a5dc660d6510a.jpgAdding fractions with unlike denominatorsLearn how to add two fractions with different denominators. It can be challenging to combine fractions when the denominators don't match. It is important to find a common denominator. Finally, the resource shows how to find a common multiple of the two denominators in order to convert the fractions so they can be added together.https://cdn.kastatic.org/ka-youtube-converted/2s4vTkBD4tQ.mp4/2s4vTkBD4tQ.mp4421yes https://puzzicle.org/math/5th-grade-math-eureka-squared-aligned/v/adding-and-subtracting-3-fractions-indian-accenthttps://puzzicle.org/thumbs/x84fe62877914aea0.jpgAdding and subtracting 3 fractionsLearn how to add and subtract fractions with unlike denominators. They watch the process of finding a common denominator, then practice adding and subtracting the fractions to solve the problems.https://cdn.kastatic.org/ka-youtube-converted/0O3IG5u_GUk.mp4/0O3IG5u_GUk.mp4310yes https://puzzicle.org/math/5th-grade-math-eureka-squared-aligned/v/solving-for-the-missing-fraction-indian-accenthttps://puzzicle.org/thumbs/x8e8c375c18a0705f.jpgSolving for the missing fractionLearn how to add and subtract fractions with unlike denominators. Watch examples, practice finding common denominators, and apply these skills to solve problems.https://cdn.kastatic.org/ka-youtube-converted/M-43wV3y4YU.mp4/M-43wV3y4YU.mp4329yes https://puzzicle.org/math/cc-sixth-grade-math/v/one-step-equation-intuitionhttps://puzzicle.org/thumbs/948578231.jpgOne-step equations intuitionIn this math lesson, we learn how to solve a balance scale problem with unknown mass. We represent the problem using an algebraic equation, 3x = 9, where x is the unknown mass. By dividing both sides by 3, we find that x = 3, meaning each object has a mass of 3 kilograms.https://cdn.kastatic.org/ka-youtube-converted/kbqO0YTUyAY.mp4/kbqO0YTUyAY.mp4279yes https://puzzicle.org/math/cc-sixth-grade-math/v/adding-decimals-word-problemhttps://puzzicle.org/thumbs/x7b5b2142.jpgAdding decimals word problemAdding decimals is an essential math skill. In this example, we find the total weight of newborn twins. Nadia weighs 7.27 pounds, and Vanessa weighs 8.34 pounds. By carefully lining up the decimals and adding the numbers, we discover their combined weight is 15.61 pounds.https://cdn.kastatic.org/ka-youtube-converted/c4fUaD4g4mc.mp4/c4fUaD4g4mc.mp493yes https://puzzicle.org/math/cc-sixth-grade-math/v/subtracting-decimals-word-problemhttps://puzzicle.org/thumbs/1811.jpgAdding & subtracting decimals word problemIn this math lesson, we learn how to manage a bank account balance by adding and subtracting money with decimals. We start with an initial balance, deposit an amount, and then withdraw cash. By performing these calculations, we determine the final account balance.https://cdn.kastatic.org/ka-youtube-converted/hy_bDS3aHO4.mp4/hy_bDS3aHO4.mp4139yes https://puzzicle.org/math/5th-grade-math-eureka-squared-aligned/v/solving-two-step-word-problems-with-decimals-adding-and-subtractinghttps://puzzicle.org/thumbs/xf811ff94362342e7.jpgSolving two-step word problems with decimals (adding and subtracting)Practice solving two-step word problems that involve decimals. Use different strategies that either add and subtract, subtract then subtract, or subtract and then add.https://cdn.kastatic.org/ka-youtube-converted/1c0bHes5eF4.mp4/1c0bHes5eF4.mp4218yes https://puzzicle.org/math/cc-sixth-grade-math/v/dividing-decimals-completelyhttps://puzzicle.org/thumbs/xa1b83aaa.jpgDividing decimals completelyDividing decimals can be made easier by converting them into whole numbers. To do this, multiply both decimals by the same power of 10. Then, perform long division with the new whole numbers. For example, dividing 6.3 by 0.25 becomes 630 divided by 25, resulting in an answer of 25.2.https://cdn.kastatic.org/ka-youtube-converted/7JPIX3odZrY.mp4/7JPIX3odZrY.mp4404yes https://puzzicle.org/math/5th-grade-math-eureka-squared-aligned/v/understanding-decimal-multiplication-1https://puzzicle.org/thumbs/x1b1a4e6108c6c05c.jpgUnderstanding decimal multiplicationPractice using estimation as a strategy to better understand decimal multiplication. Use the product of whole numbers to help determine the product of decimals with the samedigits in the same order.https://cdn.kastatic.org/ka-youtube-converted/LWE3n23M_m0.mp4/LWE3n23M_m0.mp4238yes https://puzzicle.org/math/5th-grade-math-eureka-squared-aligned/v/reasoning-through-multiplying-decimal-word-problemshttps://puzzicle.org/thumbs/xc223d3fd99fa1c7f.jpgReasoning through multiplying decimal word problemsPractice using estimation and thoughtful reasoning to relate multiplying decimals to multiplying whole numbers.https://cdn.kastatic.org/ka-youtube-converted/JEZB9ARqyw8.mp4/JEZB9ARqyw8.mp4276yes https://puzzicle.org/math/5th-grade-math-eureka-squared-aligned/v/intro-to-quadrilateral-indian-accenthttps://puzzicle.org/thumbs/x938b7bcf345fc617.jpgIntro to quadrilateralThere are many different types of quadrilaterals. A quadrilateral is defined as a two-dimensional shape with four sides, four vertices, and four angles. There are two main types: concave and convex. There are also various subcategories of convex quadrilaterals, such as trapezoids, parallelograms, rectangles, rhombi, and squares.https://cdn.kastatic.org/ka-youtube-converted/Xb8QJqe347k.mp4/Xb8QJqe347k.mp4493yes https://puzzicle.org/math/5th-grade-math-eureka-squared-aligned/v/kites-as-a-geometric-shape-indian-accenthttps://puzzicle.org/thumbs/x9e89e5155eb4970c.jpgKites as a geometric shapeThe video dives into the world of quadrilaterals, specifically focusing on kites. It explores how kites are defined by two pairs of adjacent, congruent sides. It also highlights that the diagonals of a kite intersect at a 90-degree angle, with one line bisecting the other.https://cdn.kastatic.org/ka-youtube-converted/-NWighBs6IU.mp4/-NWighBs6IU.mp4338yes https://puzzicle.org/math/5th-grade-math-eureka-squared-aligned/v/counting-faces-and-edges-of-3d-shapes-indian-accenthttps://puzzicle.org/thumbs/x6a5d80322fd0b646.jpgCounting faces and edges of 3D shapesLearn about shapes! Discover how to count faces and edges on 3D figures. We explore a transparent shape with five faces and another shape, a square pyramid, with eight edges and five faces. It's a colorful journey into geometry!https://cdn.kastatic.org/ka-youtube-converted/S-YfsW94Z7o.mp4/S-YfsW94Z7o.mp4173yes https://puzzicle.org/math/5th-grade-math-eureka-squared-aligned/v/recognizing-common-3d-shapes-indian-accenthttps://puzzicle.org/thumbs/x080f4b9dff202e55.jpgRecognizing common 3D shapesLearn about 3D shapes! A square pyramid has a square base. A rectangular prism is a box shape with all rectangular sides. A triangular prism has triangle faces on opposite sides. A sphere is like a globe. A cylinder looks like a can with circular faces on top and bottom. Shapes are fun!https://cdn.kastatic.org/ka-youtube-converted/pGBfHtUtpmc.mp4/pGBfHtUtpmc.mp4177yes https://puzzicle.org/math/5th-grade-math-eureka-squared-aligned/v/volume-of-a-rectangular-prism-indian-accenthttps://puzzicle.org/thumbs/x102bb681a75d2956.jpgVolume of a rectangular prismIf you want to know how much stuff you can cram into a box, finding its volume is key. To calculate the volume of a box, you need to know its height, width, and depth. You can find the volume by multiplying these three dimensions together. This formula works regardless of the units you are using (e.g. meters, feet), he method is the same - just make sure your units match up!https://cdn.kastatic.org/ka-youtube-converted/tsyxs2DtRqA.mp4/tsyxs2DtRqA.mp4119yes https://puzzicle.org/math/5th-grade-math-eureka-squared-aligned/v/introduction-to-the-coordinate-plane-indian-accenthttps://puzzicle.org/thumbs/x2934594609fe62ec.jpgIntroduction to the coordinate planeWhen we use coordinates to locate a point on a coordinate plane, we use two numbers to specify the horizontal (x) and vertical (y) positions. The first number in a coordinate pair tells us how far to move left or right from the origin, while the second number tells us how far to move up or down.https://cdn.kastatic.org/ka-youtube-converted/K_Em33Bfy_k.mp4/K_Em33Bfy_k.mp4408yes https://puzzicle.org/math/cc-seventh-grade-math/v/identifying-proportional-relationships-from-graphshttps://puzzicle.org/thumbs/x85ad97f71102abf0.jpgIdentifying proportional relationships from graphsWorked example identifying proportional relationships from graphshttps://cdn.kastatic.org/ka-youtube-converted/7IU-TOOSxPM.mp4/7IU-TOOSxPM.mp4185yes https://puzzicle.org/math/cc-sixth-grade-math/v/dividing-fractions-word-problemhttps://puzzicle.org/thumbs/2600767.jpgDividing whole numbers & fractions: t-shirtsIn this math problem, we learn how to calculate the number of baby T-shirts that can be made from 48 yards of fabric. Each T-shirt requires 4/5 yards of fabric. By dividing 48 yards by 4/5 and simplifying, we find out that 60 baby T-shirts can be produced from the given fabric.https://cdn.kastatic.org/ka-youtube-converted/PQsgXNggV7Q.mp4/PQsgXNggV7Q.mp4128yes https://puzzicle.org/math/cc-sixth-grade-math/v/intro-to-multiplying-decimalshttps://puzzicle.org/thumbs/x634fb6a8.jpgIntro to multiplying decimalsMultiplying decimals is easy and fun! To multiply 9 by 0.6, first rewrite 0.6 as 6 divided by 10. Next, multiply 9 by 6, which equals 54. Finally, divide 54 by 10 by moving the decimal point one place left, resulting in the answer 5.4. Practice makes perfect in mastering decimal multiplication!https://cdn.kastatic.org/ka-youtube-converted/STyoP3rCmb0.mp4/STyoP3rCmb0.mp4230yes https://puzzicle.org/math/cc-sixth-grade-math/v/order-operations-introhttps://puzzicle.org/thumbs/x2b14c51a7861494c.jpgOrder of operations introductionThe order of operations is a crucial math concept that ensures consistent results when solving problems with multiple operations. By following PEMDAS (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction), students can accurately evaluate expressions and avoid common mistakes. Mastering this skill is essential for success in mathematics.https://cdn.kastatic.org/ka-youtube-converted/eoYThjIAhOc.mp4/eoYThjIAhOc.mp4273yes https://puzzicle.org/math/cc-sixth-grade-math/v/more-complicated-order-of-operations-examplehttps://puzzicle.org/thumbs/25759.jpgWorked example: Order of operations (PEMDAS)The order of operations (PEMDAS) is essential for solving complex math problems. PEMDAS stands for Parentheses, Exponents, Multiplication and Division (same level), and Addition and Subtraction (same level). By following these steps, you can simplify and accurately solve mathematical expressions, ensuring a correct final answer.https://cdn.kastatic.org/ka-youtube-converted/piIcRV2dx7E.mp4/piIcRV2dx7E.mp4320yes https://puzzicle.org/math/cc-fifth-grade-math/v/u-s-customary-and-metric-unitshttps://puzzicle.org/thumbs/29.jpgU.S. customary and metric unitsThis video dives into the world of measurement units, distinguishing between U.S. customary units and metric units. It explores how these units measure different aspects such as length, weight, mass, and volume.https://cdn.kastatic.org/ka-youtube-converted/y2RHWA7t0WM.mp4/y2RHWA7t0WM.mp4310yes https://puzzicle.org/math/cc-sixth-grade-math/v/ratios-introhttps://puzzicle.org/thumbs/xc75b04f2.jpgIntro to ratiosThe video explains ratios, which show the relationship between two quantities. Using apples and oranges as an example, it demonstrates how to calculate and reduce ratios (6:9 to 2:3) and how to reverse the ratio (9:6 to 3:2).https://cdn.kastatic.org/ka-youtube-converted/bIKmw0aTmYc.mp4/bIKmw0aTmYc.mp4238yes https://puzzicle.org/math/cc-sixth-grade-math/v/ratio-example-problemshttps://puzzicle.org/thumbs/x1ee2e077f63d32e9.jpgBasic ratiosIn this video, the instructor explains ratios using three examples: amusement park ride lines, Katie's book collection, and a picture of apples and bananas. Ratios compare two quantities and can be written as "a to b" or "a:b." The video demonstrates how to find and interpret ratios in various situations.https://cdn.kastatic.org/ka-youtube-converted/IjMn7f6bbLA.mp4/IjMn7f6bbLA.mp4173yes https://puzzicle.org/math/cc-sixth-grade-math/v/ratios-as-fractionshttps://puzzicle.org/thumbs/x771cad22.jpgPart:whole ratiosIn this video, we explore ratios using apples and oranges. We find the simplified ratio of apples to total fruit (2:5) and learn to represent ratios as fractions (2/5). The fraction indicates that 2/5 of the fruit are apples. Key definitions include ratio, simplified ratio, and fraction notation.https://cdn.kastatic.org/ka-youtube-converted/UK-_qEDtvYo.mp4/UK-_qEDtvYo.mp4202yes https://puzzicle.org/math/cc-sixth-grade-math/v/equivalent-ratioshttps://puzzicle.org/thumbs/xb0b46e6a60931f92.jpgEquivalent ratiosOne way to identify equivalent ratios is to determine if you can multiply or divide the corresponding parts of the ratio by the same amount. To do this, look at the two ratios and see if there is a common factor that you can use to scale one ratio to the other. If you can scale one ratio to the other by multiplying or dividing every part of the ratio by the same number, then the two ratios are equivalent.https://cdn.kastatic.org/ka-youtube-converted/eb-GHXCqkhQ.mp4/eb-GHXCqkhQ.mp4283yes https://puzzicle.org/math/cc-sixth-grade-math/v/ratios-for-recipeshttps://puzzicle.org/thumbs/x2427758b379e98c9.jpgEquivalent ratios: recipeAdjusting a cake recipe for different guest numbers involves understanding ratios. Halving ingredients for a smaller cake maintains the same ratio, like eggs to flour (4:3). Ratios ensure consistent taste and quality, regardless of cake size.https://cdn.kastatic.org/ka-youtube-converted/VWO1m0S-a9Y.mp4/VWO1m0S-a9Y.mp4291yes https://puzzicle.org/math/cc-sixth-grade-math/v/ratio-word-problem-exampleshttps://puzzicle.org/thumbs/x03004af138555a6d.jpgEquivalent ratio word problemsThis video teaches solving ratio word problems, using examples like Yoda Soda for guests, fish ratios in a tank, ice cream sundae ingredients, and dog color ratios at a park. Mastering these techniques helps students tackle real-world math challenges.https://cdn.kastatic.org/ka-youtube-converted/WoZ7-wOy-0w.mp4/WoZ7-wOy-0w.mp4362yes https://puzzicle.org/math/cc-sixth-grade-math/v/understanding-equivalent-ratioshttps://puzzicle.org/thumbs/x9b0248c0620f3be9.jpgUnderstanding equivalent ratiosSal discusses what it means for two ratios to be equivalent.Comparing ratios is essential for determining equivalent relationships between quantities. By analyzing and simplifying ratios, we can identify if they have the same proportion, helping us make decisions in real-life situations like mixing paint colors or comparing food recipes.https://cdn.kastatic.org/ka-youtube-converted/4S3Mbl0JrdY.mp4/4S3Mbl0JrdY.mp4345yes https://puzzicle.org/math/cc-sixth-grade-math/v/ratios-and-double-number-lineshttps://puzzicle.org/thumbs/x64933af9b96eb772.jpgRatios and double number linesDouble number lines are a helpful tool for understanding and solving ratio problems. They visually represent the relationship between two quantities, making it easier to find the cost or amount of a single unit. Examples include calculating the price of avocados, building model trains, and making snowballs.https://cdn.kastatic.org/ka-youtube-converted/-Dg9da1BGsM.mp4/-Dg9da1BGsM.mp4353yes https://puzzicle.org/math/cc-sixth-grade-math/v/solving-ratio-problems-with-tables-exercisehttps://puzzicle.org/thumbs/xcfd31e75.jpgSolving ratio problems with tablesEquivalent ratios have the same relationship between their numerators and denominators. To find missing values in tables, maintain the same ratio. Comparing fractions is easier with common numerators or denominators. Constant speed is represented by a constant ratio between distance and time.https://cdn.kastatic.org/ka-youtube-converted/MaMk6-f3T9k.mp4/MaMk6-f3T9k.mp4305yes https://puzzicle.org/math/cc-sixth-grade-math/v/ratios-with-tape-diagramshttps://puzzicle.org/thumbs/x2db404d5087f60bc.jpgRatios with tape diagramsIn the video, we explore ratios, which compare two quantities. We learn to find missing values using given ratios and discover equivalent ratios by multiplying or dividing both parts by the same number. Key examples include quilts, paint mixtures, and spells cast by characters.https://cdn.kastatic.org/ka-youtube-converted/suRIY3ULrQo.mp4/suRIY3ULrQo.mp4270yes https://puzzicle.org/math/cc-sixth-grade-math/v/ratios-and-measurementhttps://puzzicle.org/thumbs/x4ec0f20cc0f50644.jpgRatios and measurementLearn to complete ratio tables and solve unit conversion problems with examples like hours to weeks, yards to miles, millimeters to feet, and dollars per pound to dollars per ounce. Master these techniques to tackle real-world problems with ease.https://cdn.kastatic.org/ka-youtube-converted/0qtIHdda19s.mp4/0qtIHdda19s.mp4417yes https://puzzicle.org/math/cc-sixth-grade-math/v/ratios-on-coordinate-planehttps://puzzicle.org/thumbs/x9e49f2b3548a01a6.jpgRatios on coordinate planeRatios represent the relationship between two quantities and can be visualized on a graph. When the ratio is constant, the points form a straight line, illustrating the connection between the quantities. Examples include bakers using flour, car wash charges, and earnings from shoveling snow.https://cdn.kastatic.org/ka-youtube-converted/yVYJRT5hTSo.mp4/yVYJRT5hTSo.mp4312yes https://puzzicle.org/math/cc-sixth-grade-math/v/ratio-word-problem-exercise-example-1https://puzzicle.org/thumbs/xde4f6af2.jpgPart to whole ratio word problem using tablesDiscover how to solve ratio problems with a real-life example involving indoor and outdoor playtimes. Learn to use ratios to determine the number of indoor and outdoor playtimes in a class with a 2:3 ratio and 30 total playtimes.https://cdn.kastatic.org/ka-youtube-converted/ZJ6y8OVJRw8.mp4/ZJ6y8OVJRw8.mp4155yes https://puzzicle.org/math/cc-sixth-grade-math/v/introduction-to-rateshttps://puzzicle.org/thumbs/x102d9798.jpgIntro to ratesRates are essential in understanding everyday situations, such as speed, wages, and food consumption. They help us measure and compare quantities, like miles per hour or calories per serving. Rates are closely related to ratios and play a significant role in math subjects like algebra and calculus, making them crucial for problem-solving and analysis.https://cdn.kastatic.org/ka-youtube-converted/qGTYSAeLTOE.mp4/qGTYSAeLTOE.mp4330yes https://puzzicle.org/math/cc-sixth-grade-math/v/finding-unit-rateshttps://puzzicle.org/thumbs/1821.jpgSolving unit rate problemJayda delivers newspapers on her paper route, taking 3 hours to distribute 189 newspapers. To calculate her rate per hour, divide the total newspapers (189) by the hours taken (3). This results in Jayda delivering 63 newspapers per hour, showcasing her efficiency and speed.https://cdn.kastatic.org/ka-youtube-converted/Zm0KaIw-35k.mp4/Zm0KaIw-35k.mp4151yes https://puzzicle.org/math/cc-sixth-grade-math/v/finding-unit-priceshttps://puzzicle.org/thumbs/1822.jpgSolving unit price problemNierria earns $75 for tutoring 4 hours. To find her hourly rate, we divide her total earnings by the number of hours worked. By calculating 75 divided by 4, we discover that Nierria makes $18.75 per hour for her tutoring services.https://cdn.kastatic.org/ka-youtube-converted/rpGGMSFO6Ks.mp4/rpGGMSFO6Ks.mp4116yes https://puzzicle.org/math/cc-sixth-grade-math/v/rate-problemshttps://puzzicle.org/thumbs/xa309573541a756a7.jpgRate problemsIn this math lesson, we learn to find unit rates and use them to solve problems. We first calculate the rate for one unit, like cars washed per day or cost per battery. Then, we multiply the unit rate by the desired quantity to find the answer. This method simplifies complex problems and helps us understand real-world situations.https://cdn.kastatic.org/ka-youtube-converted/fpjXtpg_isc.mp4/fpjXtpg_isc.mp4270yes https://puzzicle.org/math/cc-sixth-grade-math/v/practice-computing-and-comparing-rateshttps://puzzicle.org/thumbs/xc67f0ee5.jpgComparing rates exampleA conservationist thinks that squirrels exhibit higher aggression rates when crowded together. To test this, they compare three parks with different squirrel populations and areas. By calculating squirrels per hectare, they find Park C to be the most crowded, followed by Park B and Park A. The conservationist expects the highest aggression rate in Park C due to its crowdedness.https://cdn.kastatic.org/ka-youtube-converted/C7bBZa52h-4.mp4/C7bBZa52h-4.mp4430yes https://puzzicle.org/math/cc-sixth-grade-math/v/describing-the-meaning-of-percenthttps://puzzicle.org/thumbs/4802229.jpgThe meaning of percentIn this math lesson, we learn about percentages by shading 20% of a square. The square is divided into 100 smaller squares in a 10x10 grid. To represent 20%, we shade 20 of these smaller squares, which can be done by coloring two rows of 10 squares each. This visual approach helps us understand the concept of percentages.https://cdn.kastatic.org/ka-youtube-converted/Lvr2YsxG10o.mp4/Lvr2YsxG10o.mp4181yes https://puzzicle.org/math/cc-sixth-grade-math/v/describing-the-meaning-of-percent-2https://puzzicle.org/thumbs/4802230.jpgMeaning of 109%In this lesson, we learn how to represent 109% using shading. We discover that 109% is equivalent to 109 per 100, or 109/100 as a fraction. By shading a complete 10x10 square and an additional 9 squares, we visually demonstrate 109% of a whole square.https://cdn.kastatic.org/ka-youtube-converted/MokI3Fi8jpc.mp4/MokI3Fi8jpc.mp4142yes https://puzzicle.org/math/cc-sixth-grade-math/v/percent-from-fraction-modelshttps://puzzicle.org/thumbs/xbd10d28340636b83.jpgPercents from fraction modelsConverting fractions to percentages is a valuable skill in math. To do this, rewrite the fraction with a denominator of 100, as "percent" means "per 100." For example, 7/10 equals 70/100, or 70%. Additionally, you can add percentages together, such as 100% + 80% = 180%.https://cdn.kastatic.org/ka-youtube-converted/nV6bRozq5Po.mp4/nV6bRozq5Po.mp4262yes https://puzzicle.org/math/cc-sixth-grade-math/v/finding-percentages-with-a-double-number-linehttps://puzzicle.org/thumbs/xd916ed375a5a6d6e.jpgFinding percentages with a double number lineIn this math problem, we determine the percentage of students taking the bus to school in a class of 28. By using a double number line, we find that 21 out of 28 students take the bus, which is 3/4 of the class. Converting this fraction to a percentage, we discover that 75% of the students ride the bus to school.https://cdn.kastatic.org/ka-youtube-converted/bVNnQX5nEFg.mp4/bVNnQX5nEFg.mp4132yes https://puzzicle.org/math/cc-sixth-grade-math/v/finding-the-whole-with-a-tape-diagramhttps://puzzicle.org/thumbs/x775680e37171e129.jpgFinding the whole with a tape diagramKeisha can run 170 meters in one minute today, which is 125% of her distance from three years ago. To find her previous distance, we convert 125% to 5/4 and divide 170 by 5, obtaining 34 meters per fourth. Multiplying 34 by 4 reveals that Keisha could run 136 meters in one minute three years ago.https://cdn.kastatic.org/ka-youtube-converted/M0vf7idRTLA.mp4/M0vf7idRTLA.mp4184yes https://puzzicle.org/math/cc-sixth-grade-math/v/fraction-decimal-and-percent-from-visual-modelhttps://puzzicle.org/thumbs/x722ac3f3a18ed682.jpgFraction, decimal, and percent from visual modelLearn how to convert shaded parts of a square into fractions, decimals, and percentages with ease. Master the skill of representing parts of a whole in different formats, such as simplifying fractions, converting decimals to percentages, and understanding the relationship between them. Boost your math skills and gain confidence in solving real-world problems.https://cdn.kastatic.org/ka-youtube-converted/wwg052FC_Zw.mp4/wwg052FC_Zw.mp4213yes https://puzzicle.org/math/cc-sixth-grade-math/v/representing-a-number-as-a-decimal-percent-and-fractionhttps://puzzicle.org/thumbs/4802231.jpgConverting percents to decimals & fractions exampleConverting percentages to decimals and fractions is a fundamental math skill. To convert 18% into a decimal, simply divide by 100, resulting in 0.18. To express it as a simplified fraction, find the greatest common factor, which in this case is 2, and divide both the numerator and denominator by it, giving us 9/50.https://cdn.kastatic.org/ka-youtube-converted/-gB1y-PMWfs.mp4/-gB1y-PMWfs.mp4212yes https://puzzicle.org/math/cc-sixth-grade-math/v/taking-a-percentage-examplehttps://puzzicle.org/thumbs/xc1057cd3.jpgPercent of a whole numberDiscover how to calculate percentages with a simple example: finding 30% of 6. Learn two methods, using fractions and decimals, to solve this problem. Both approaches lead to the same result, showing that 30% of 6 equals 1.8. Master these techniques for everyday calculations.https://cdn.kastatic.org/ka-youtube-converted/-lUEWEEpmIo.mp4/-lUEWEEpmIo.mp4127yes https://puzzicle.org/math/cc-sixth-grade-math/v/finding-percentages-examplehttps://puzzicle.org/thumbs/x7d62667a.jpgFinding a percentDiscover how to calculate percentages with this simple method. First, write the problem as a fraction, then simplify it. Next, convert the fraction to a percentage by making the denominator 100. Alternatively, divide the numbers to get a decimal and multiply by 100 to find the percentage. Master this skill for everyday use.https://cdn.kastatic.org/ka-youtube-converted/FaDtge_vkbg.mp4/FaDtge_vkbg.mp4213yes https://puzzicle.org/math/cc-sixth-grade-math/v/ways-to-rewrite-a-percentagehttps://puzzicle.org/thumbs/x93ebe427634bd289.jpgWays to rewrite a percentageDiscover how to calculate 2% of 90 using various methods, such as converting percentages to fractions and decimals. Learn to identify correct expressions, like 0.02 times 90 and 2/100 times 90, while eliminating incorrect options. Master percentage calculations with ease and confidence.https://cdn.kastatic.org/ka-youtube-converted/MD5JUwAeg6c.mp4/MD5JUwAeg6c.mp4120yes https://puzzicle.org/math/cc-sixth-grade-math/v/common-percentageshttps://puzzicle.org/thumbs/x3f1d902f592c01a6.jpgFinding common percentagesCalculating percentages in your head is a valuable skill. Common percentages like 1%, 10%, 20%, 25%, 50%, and 400% can be easily converted into fractions (e.g., 1% = 1/100, 10% = 1/10). To find the percentage of a number, simply multiply or divide by the corresponding fraction, making mental math quick and efficient.https://cdn.kastatic.org/ka-youtube-converted/GdwjpJwa8Ys.mp4/GdwjpJwa8Ys.mp4172yes https://puzzicle.org/math/6th-engage-ny/v/solving-percent-problems-3https://puzzicle.org/thumbs/15.jpgPercent word problem: 78 is 15% of what number?In this example, you working with us to find the number that is expressed as a given percentage. We'll create a simple algebraic equation to solve!https://cdn.kastatic.org/ka-youtube-converted/DAikW24_O0A.mp4/DAikW24_O0A.mp4326yes https://puzzicle.org/math/cc-seventh-grade-math/v/another-percent-word-problemhttps://puzzicle.org/thumbs/23458.jpgPercent word problem: guavasWe're putting a little algebra to work to find the full price when you know the discount price in this percent word problem.https://cdn.kastatic.org/ka-youtube-converted/d1oNF88SAgg.mp4/d1oNF88SAgg.mp4378yes https://puzzicle.org/math/6th-engage-ny/v/percent-word-problems-1-example-2https://puzzicle.org/thumbs/xd369a80b.jpgPercent word problem: penguinsYou're given the percent in this example and asked to find the whole number represented by the percent.https://cdn.kastatic.org/ka-youtube-converted/gKywkLHV6Ko.mp4/gKywkLHV6Ko.mp4209yes https://puzzicle.org/math/cc-sixth-grade-math/v/percent-word-problemshttps://puzzicle.org/thumbs/x31086a98.jpgPercent word problem: recycling cansIn this math lesson, we learn how to calculate the percentage of recycled cans. Given that 13 out of 20 cans are recycled, we divide 13 by 20 to get 0.65 as a decimal. Multiplying by 100, we find that 65% of cans are recycled, promoting eco-friendly practices and waste reduction.https://cdn.kastatic.org/ka-youtube-converted/AjYil74WrVo.mp4/AjYil74WrVo.mp4169yes https://puzzicle.org/math/cc-sixth-grade-math/v/conceptual-understanding-of-dividing-fractions-by-fractionshttps://puzzicle.org/thumbs/xa415219d.jpgUnderstanding division of fractionsDividing fractions can be understood using number lines and jumps. To divide a fraction like 8/3 by another fraction like 1/3, count the jumps of 1/3 needed to reach 8/3. Alternatively, multiply 8/3 by the reciprocal of the divisor (3/1) to get the same result. This concept applies to other fractions, such as dividing 8/3 by 2/3.https://cdn.kastatic.org/ka-youtube-converted/f3ySpxX9oeM.mp4/f3ySpxX9oeM.mp4342yes https://puzzicle.org/math/cc-sixth-grade-math/v/visually-dividing-a-fraction-by-a-whole-numberhttps://puzzicle.org/thumbs/xf31f2f8a16d13ba0.jpgDividing a fraction by a whole numberDividing fractions by whole numbers can be done visually or using multiplication. To find the result of 2/3 divided by 5, divide 2/3 into 5 equal parts, each representing 2/15 of the whole. Alternatively, multiply 2/3 by 1/5, resulting in 2/15 as the final answer.https://cdn.kastatic.org/ka-youtube-converted/1YWyTdtofdE.mp4/1YWyTdtofdE.mp4233yes https://puzzicle.org/math/cc-sixth-grade-math/v/meaning-of-the-reciprocalhttps://puzzicle.org/thumbs/x442482abc85a2d5a.jpgMeaning of the reciprocalReciprocals are essential in understanding fractions. To find the reciprocal of a fraction, simply swap the numerator and the denominator. The product of a number and its reciprocal always equals 1. Reciprocals help us determine how many times a specific fraction fits into the whole number 1, making them valuable in various mathematical operations.https://cdn.kastatic.org/ka-youtube-converted/yM9Yrz69Tys.mp4/yM9Yrz69Tys.mp4253yes https://puzzicle.org/math/cc-sixth-grade-math/v/whole-number-divided-by-a-fraction-examplehttps://puzzicle.org/thumbs/x609f7c74bf7d7219.jpgDividing a whole number by a fractionDividing a whole number by a fraction involves multiplying the whole number by the fraction's reciprocal. For example, when dividing 4 by 2/3, multiply 4 by 3/2, resulting in 12/2, which equals 6. This method demonstrates how many sections of the fraction fit into the whole number.https://cdn.kastatic.org/ka-youtube-converted/SPMoNUAbV48.mp4/SPMoNUAbV48.mp4185yes https://puzzicle.org/math/cc-sixth-grade-math/v/dividing-a-whole-number-by-a-fraction-with-reciprocalhttps://puzzicle.org/thumbs/x5a6f5d30a8980c41.jpgDividing a whole number by a fraction with reciprocalDividing by a fraction involves finding the reciprocal and multiplying. For example, when dividing 8 by 7/5, first find the reciprocal of 7/5, which is 5/7. Then, multiply 8 by 5/7, resulting in 40/7 or 5 5/7. This method simplifies fraction division and enhances understanding.https://cdn.kastatic.org/ka-youtube-converted/G7c0CBLWVfs.mp4/G7c0CBLWVfs.mp4216yes https://puzzicle.org/math/cc-sixth-grade-math/v/another-dividing-fractions-examplehttps://puzzicle.org/thumbs/x4829f4ea.jpgDividing fractions: 2/5 ÷ 7/3Dividing fractions is a breeze when you follow these simple steps! First, multiply the first fraction by the reciprocal of the second fraction. Then, multiply the numerators and denominators separately. Voila! You've successfully divided two fractions and found your answer. Keep practicing to master this essential math skill.https://cdn.kastatic.org/ka-youtube-converted/m-pRnFIeoNA.mp4/m-pRnFIeoNA.mp477yes https://puzzicle.org/math/cc-sixth-grade-math/v/dividing-fractions-examplehttps://puzzicle.org/thumbs/2600764.jpgDividing fractions: 3/5 ÷ 1/2Dividing two fractions is the same as multiplying the first fraction by the reciprocal of the second fraction. The first step to dividing fractions is to find the reciprocal (reverse the numerator and denominator) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. Finally, simplify the fractions if needed.https://cdn.kastatic.org/ka-youtube-converted/tnkPY4UqJ44.mp4/tnkPY4UqJ44.mp4216yes https://puzzicle.org/math/cc-sixth-grade-math/v/dividing-mixed-numbers-examplehttps://puzzicle.org/thumbs/xf6a074691b688e71.jpgDividing mixed numbersHow do we divide mixed numbers? We can convert the mixed numbers to improper fractions, then simplify the division problem by finding the reciprocal of one of the fractions. This video also includes a demonstration of different ways of simplifying the final answer.https://cdn.kastatic.org/ka-youtube-converted/tj5AtYoKqvs.mp4/tj5AtYoKqvs.mp4213yes https://puzzicle.org/math/cc-sixth-grade-math/v/writing-fraction-division-story-problemshttps://puzzicle.org/thumbs/x402e317ef1ae977a.jpgWriting fraction division story problemsDarryl spent 24 1/4 hours writing a novel chapter, and we explore situations where dividing 24 1/4 by 3/4 is meaningful. Two examples include calculating the number of days spent writing with 45-minute daily sessions and determining the time spent illustrating if it's 3/4 as long as writing. Creative thinking helps uncover more scenarios.https://cdn.kastatic.org/ka-youtube-converted/UEna8ZEy6k4.mp4/UEna8ZEy6k4.mp4210yes https://puzzicle.org/math/cc-sixth-grade-math/v/area-with-fraction-division-examplehttps://puzzicle.org/thumbs/x9b2a4e6a30477412.jpgArea with fraction division exampleDiscover the length of a yoga mat with a width of 3/5 meters and an area of 1 and 2/25 square meters. Use the formula length = area ÷ width, convert mixed numbers to improper fractions, and simplify. The final dimensions are 3/5 meters wide and 9/5 meters long, perfect for your yoga practice!https://cdn.kastatic.org/ka-youtube-converted/6K6tQulp-x8.mp4/6K6tQulp-x8.mp4199yes https://puzzicle.org/math/cc-sixth-grade-math/v/subtracting-decimals-up-to-thousandths-placehttps://puzzicle.org/thumbs/x2147024f.jpgSubtracting decimals: 39.1 - 0.794Learn to subtract decimals by aligning the decimal points and place values. Regroup numbers for easier subtraction, and then perform the subtraction for each place value. Mastering decimal subtraction helps build strong math skills and real-world problem-solving abilities.https://cdn.kastatic.org/ka-youtube-converted/lDXaju6JoQ0.mp4/lDXaju6JoQ0.mp4126yes https://puzzicle.org/math/cc-sixth-grade-math/v/subtracting-decimalshttps://puzzicle.org/thumbs/4802228.jpgSubtracting decimals: 9.005 - 3.6Subtracting decimals is simple when you line up the decimal points and fill in missing zeros. Regroup when needed by borrowing from the whole number and adding to the tenths place. Mastering this skill helps with everyday calculations and improves overall math understanding.https://cdn.kastatic.org/ka-youtube-converted/Eq4mVCd-yyo.mp4/Eq4mVCd-yyo.mp4128yes https://puzzicle.org/math/cc-sixth-grade-math/v/decimal-multiplication-place-valuehttps://puzzicle.org/thumbs/x4aa02467fdcdfd15.jpgDecimal multiplication place valueWhen we rewrite decimal multiplication using fractions, we can use the commutative and associative properties of multiplication to justify how we place the decimal point in the standard multiplication algorithm.https://cdn.kastatic.org/ka-youtube-converted/hdKDVDjUHhs.mp4/hdKDVDjUHhs.mp4279yes https://puzzicle.org/math/cc-sixth-grade-math/v/more-involved-multiplying-decimals-examplehttps://puzzicle.org/thumbs/xad5ace08.jpgMultiplying challenging decimalsLearn to multiply decimals easily by treating them as whole numbers first. Multiply the numbers without decimals, then count the total decimal places in both original numbers. Add the same number of decimal places to the product, and you'll get the correct answer for the decimal multiplication.https://cdn.kastatic.org/ka-youtube-converted/D5fmcpNygQk.mp4/D5fmcpNygQk.mp4265yes https://puzzicle.org/math/cc-sixth-grade-math/v/dividing-completely-to-get-decimal-answerhttps://puzzicle.org/thumbs/xb4d79b88.jpgDividing whole numbers to get a decimalIn this math lesson, we learn how to divide two numbers and find a decimal answer. We divide 63 by 35, first getting a remainder of 28. To obtain a decimal result, we add a zero to 63, making it 63.0, and then divide 280 by 35. The final answer is 1.8.https://cdn.kastatic.org/ka-youtube-converted/xUDlKV8lJbM.mp4/xUDlKV8lJbM.mp4157yes https://puzzicle.org/math/cc-sixth-grade-math/v/dividing-a-whole-number-by-a-decimalhttps://puzzicle.org/thumbs/x7b24a2d9.jpgDividing a whole number by a decimalDividing a whole number by a decimal can be simplified by multiplying both numbers by 10, shifting the decimal point to the right. For example, dividing 518 by 0.7 becomes 5,180 divided by 7. Using long division, the result is 740, making the process easier and more manageable.https://cdn.kastatic.org/ka-youtube-converted/cfr-yZxTH8Y.mp4/cfr-yZxTH8Y.mp4260yes https://puzzicle.org/math/cc-sixth-grade-math/v/dividing-a-decimal-by-a-whole-numberhttps://puzzicle.org/thumbs/xbe7ce751.jpgDividing a decimal by a whole numberIn this math lesson, we learn how to divide decimals using long division. The key is to place the decimal point correctly and treat it like a regular long division problem. By following these steps, we can easily divide 5.005 by 7, resulting in an answer of 0.715.https://cdn.kastatic.org/ka-youtube-converted/Z_NHrwK6ALE.mp4/Z_NHrwK6ALE.mp4126yes https://puzzicle.org/math/cc-sixth-grade-math/v/dividing-decimals-with-hundredthshttps://puzzicle.org/thumbs/xa7651d1e.jpgDividing decimals with hundredthsDividing decimals can be made simple by multiplying both numbers by the same amount to eliminate decimals. For example, when dividing 30.24 by 0.42, multiply both by 100 to get 3,042 divided by 42. Using long division, the final answer is 72, making the process easy and efficient.https://cdn.kastatic.org/ka-youtube-converted/Hrjr5f5pZ84.mp4/Hrjr5f5pZ84.mp4277yes https://puzzicle.org/math/cc-sixth-grade-math/v/long-division-with-decimalshttps://puzzicle.org/thumbs/x37e1024b.jpgLong division with decimalsHow do we divide a decimal by another decimal? One great approach is first to turn both numbers into whole numbers before proceeding with the standard algorithm. Sal Khan shows us how to do this. When using long division, it is important to keep things organized and in neat columns!https://cdn.kastatic.org/ka-youtube-converted/x48CYVaD0xE.mp4/x48CYVaD0xE.mp4250yes https://puzzicle.org/math/cc-sixth-grade-math/v/dividing-decimalshttps://puzzicle.org/thumbs/1814.jpgDividing by a multi-digit decimalDividing decimals involves a few simple steps. First, convert the divisor into a whole number by shifting the decimal point to the right. Apply the same process to the dividend. Then, perform regular division with the new numbers. Finally, position the decimal point in the quotient to match the dividend.https://cdn.kastatic.org/ka-youtube-converted/Nqts8zW8RxM.mp4/Nqts8zW8RxM.mp4332yes https://puzzicle.org/math/6th-engage-ny/v/divisibility-tests-for-2-3-4-5-6-9-10https://puzzicle.org/thumbs/1245177424.jpgDivisibility tests for 2, 3, 4, 5, 6, 9, 10Worked example of basic divisibility tests.https://cdn.kastatic.org/ka-youtube-converted/Df9h5t64NlQ.mp4/Df9h5t64NlQ.mp4393yes https://puzzicle.org/math/6th-engage-ny/v/recognizing-divisibilityhttps://puzzicle.org/thumbs/2600702.jpgRecognizing divisibilityRecognizing Divisibility.https://cdn.kastatic.org/ka-youtube-converted/2G_Jr_XpnY4.mp4/2G_Jr_XpnY4.mp4386yes https://puzzicle.org/math/6th-engage-ny/v/the-why-of-the-3-divisibility-rulehttps://puzzicle.org/thumbs/884395195.jpgThe why of the 3 divisibility ruleWhy you can add the digits to see if something is divisible by 3https://cdn.kastatic.org/ka-youtube-converted/NehkLV77ITk.mp4/NehkLV77ITk.mp4303yes https://puzzicle.org/math/6th-engage-ny/v/the-why-of-the-9-divisibility-rulehttps://puzzicle.org/thumbs/1245280895.jpgThe why of the 9 divisibility ruleWhy you can test divisibility by 9 by just adding the digitshttps://cdn.kastatic.org/ka-youtube-converted/XAzFGx3Ruig.mp4/XAzFGx3Ruig.mp4308yes https://puzzicle.org/math/cc-sixth-grade-math/v/greatest-common-divisor-factor-exercisehttps://puzzicle.org/thumbs/x0fbf66ba.jpgGreatest common factor examplesThe greatest common factor (GCF) of a set of numbers is the largest factor that all the numbers share. For example, 12, 20, and 24 have two common factors: 2 and 4. The largest is 4, so we say that the GCF of 12, 20, and 24 is 4. GCF is often used to find common denominators.https://cdn.kastatic.org/ka-youtube-converted/bLTfBvkrfsM.mp4/bLTfBvkrfsM.mp4410yes https://puzzicle.org/math/cc-sixth-grade-math/v/greatest-common-divisorhttps://puzzicle.org/thumbs/23860.jpgGreatest common factor explainedThe greatest common divisor (GCD) and greatest common factor (GCF) are the same thing. To find the GCD/GCF of two numbers, list their factors, identify the common factors, and choose the largest one. For example, the GCD/GCF of 12 and 8 is 4. Numbers with a GCD/GCF of 1 are called relatively prime.https://cdn.kastatic.org/ka-youtube-converted/jFd-6EPfnec.mp4/jFd-6EPfnec.mp4380yes https://puzzicle.org/math/cc-sixth-grade-math/v/least-common-multiple-exercisehttps://puzzicle.org/thumbs/x19e7b12a.jpgLeast common multipleThe least common multiple (LCM) is the smallest number that two or more numbers can divide into evenly. To find the LCM, you can use the prime factorization method or list the multiples of each number. Prime factorization involves breaking down numbers into their prime factors and constructing the smallest number with all the factors. Listing multiples involves finding the smallest shared multiple. Both methods help find the LCM of numbers like 18 and 12.https://cdn.kastatic.org/ka-youtube-converted/znmPfDfsir8.mp4/znmPfDfsir8.mp4256yes https://puzzicle.org/math/cc-sixth-grade-math/v/least-common-multiple-exercise-2https://puzzicle.org/thumbs/xafe97682.jpgLeast common multiple: repeating factorsTo find the Least Common Multiple (LCM) of two integers, follow these steps: 1) Find the prime factorization of both numbers. 2) Determine which prime factors are needed for the least common multiple (LCM) to be divisible by both numbers. 3) Multiply the necessary prime factors together to get the least common multiple (LCM).https://cdn.kastatic.org/ka-youtube-converted/QUem_2dkB9I.mp4/QUem_2dkB9I.mp4155yes https://puzzicle.org/math/cc-sixth-grade-math/v/least-common-multiple-lcmhttps://puzzicle.org/thumbs/158925086.jpgLeast common multiple of three numbersThe least common multiple (LCM) is the smallest number that is a multiple of two or more numbers. To find the LCM, you can list the multiples of each number and find the smallest one they share, or use prime factorization to break down the numbers into their prime factors and multiply the highest powers of each factor.https://cdn.kastatic.org/ka-youtube-converted/D6yHKOYJiso.mp4/D6yHKOYJiso.mp4324yes https://puzzicle.org/math/mappers/v/lcm-and-gcf-greatest-common-factor-word-problemshttps://puzzicle.org/thumbs/x59fe3ed2.jpgGCF & LCM word problemsHere we have a couple of word problems--one searching for the least common multiple and the other for the greatest common factor. Just read them with us slowly and follow along. You'll get it.https://cdn.kastatic.org/ka-youtube-converted/1Vb8t7Y-pI0.mp4/1Vb8t7Y-pI0.mp4514yes https://puzzicle.org/math/cc-sixth-grade-math/v/introduction-to-negative-numbershttps://puzzicle.org/thumbs/xb2e450a6862bb2df.jpgIntroduction to negative numbersNegative numbers are essential in math, representing values below zero. On a number line, they appear to the left of zero, opposite positive numbers. They're commonly used in real-world situations like measuring temperatures below freezing. Understanding negative numbers helps us grasp a broader range of mathematical concepts.https://cdn.kastatic.org/ka-youtube-converted/u8UKdNdpkh4.mp4/u8UKdNdpkh4.mp4184yes https://puzzicle.org/math/cc-sixth-grade-math/v/positive-and-negative-decimals-on-a-number-linehttps://puzzicle.org/thumbs/x72f595c1.jpgNegative decimals on the number lineIn this fun math lesson, we learn to locate decimals on a number line. By understanding the scale and recognizing positive and negative numbers, we can accurately place dots at specific decimal points like -0.6, 1.9, and 0.5. This skill is essential for mastering decimals and building a strong foundation in math.https://cdn.kastatic.org/ka-youtube-converted/w23DzA68KPE.mp4/w23DzA68KPE.mp475yes https://puzzicle.org/math/cc-sixth-grade-math/v/points-on-a-number-linehttps://puzzicle.org/thumbs/13179310.jpgDecimals & fractions on the number linePlotting numbers on a number line is a fundamental math skill that helps visualize numerical values. Key points include identifying whole numbers, fractions, and decimals, and placing them accurately on the line. This skill enhances understanding of number relationships and builds a strong foundation for more advanced math concepts.https://cdn.kastatic.org/ka-youtube-converted/uC09taczvOo.mp4/uC09taczvOo.mp4118yes https://puzzicle.org/math/mappers/v/number-line-3-exercise-examplehttps://puzzicle.org/thumbs/x0546ae25.jpgMissing numbers on the number line examplesGiven a few negative numbers on a number line, let's see if we can determine what negative numbers are elsewhere on the number line.https://cdn.kastatic.org/ka-youtube-converted/qW-Ce44ll0Q.mp4/qW-Ce44ll0Q.mp4188yes https://puzzicle.org/math/cc-sixth-grade-math/v/opposite-of-a-numberhttps://puzzicle.org/thumbs/xe940cca2.jpgNumber oppositesOpposites of numbers are essential in understanding number lines and basic math concepts. The opposite of a positive number is its negative counterpart, while the opposite of a negative number is its positive counterpart. Both numbers have the same absolute value but different signs, making them equally distant from zero on the number line.https://cdn.kastatic.org/ka-youtube-converted/2Zk6u7Uk5ow.mp4/2Zk6u7Uk5ow.mp4192yes https://puzzicle.org/math/cc-sixth-grade-math/v/negative-symbol-as-oppositehttps://puzzicle.org/thumbs/x771c48fe.jpgNegative symbol as oppositeOpposites in math involve flipping the sign of a number. A positive number's opposite is the same number but negative, and vice versa. The negative symbol represents the concept of "opposite." When multiple negative symbols are applied, each one flips the number across the number line, resulting in a positive or negative value.https://cdn.kastatic.org/ka-youtube-converted/3-aryZYsoxU.mp4/3-aryZYsoxU.mp4302yes https://puzzicle.org/math/cc-sixth-grade-math/v/compare-rational-numbers-using-a-number-linehttps://puzzicle.org/thumbs/x050d9dcc5093a680.jpgCompare rational numbers using a number lineIn this math tutorial, we learn to compare positive and negative numbers, decimals, and fractions using a number line. By visualizing the numbers on a number line, we can easily determine which number is greater or smaller. This method helps build a strong foundation in understanding number relationships and comparisons.https://cdn.kastatic.org/ka-youtube-converted/6h0Wi8Dcd0E.mp4/6h0Wi8Dcd0E.mp4339yes https://puzzicle.org/math/cc-sixth-grade-math/v/writing-numerical-inequalities-exercisehttps://puzzicle.org/thumbs/xe3dc6ddd.jpgNumerical inequality word problemsIn this math lesson, we learn to compare quantities using inequalities. John makes pasta dough with two types of flour, finding that wheat flour (3/2 cups) is greater than regular flour (3/4 cups). We also compare ocean depths, discovering the Diamantina trench is deeper than the Bering Strait. Lastly, John makes noodles, comparing tapioca flour (11/16 cups) and rice flour (3/8 cups).https://cdn.kastatic.org/ka-youtube-converted/3HD-Ak_a6VE.mp4/3HD-Ak_a6VE.mp4208yes https://puzzicle.org/math/cc-sixth-grade-math/v/ordering-negative-numbershttps://puzzicle.org/thumbs/211177353.jpgOrdering negative numbersOrdering negative numbers can be tricky, but understanding their values is essential. Smaller negative numbers are further from zero, making them less than larger ones. Visualizing them on a number line helps clarify their positions, with the least number on the left and the greatest on the right. Think of owing money or cold temperatures for real-life examples.https://cdn.kastatic.org/ka-youtube-converted/VW6UrPKPInA.mp4/VW6UrPKPInA.mp4167yes https://puzzicle.org/math/cc-sixth-grade-math/v/ordering-rational-numbershttps://puzzicle.org/thumbs/x02e55120.jpgOrdering rational numbersLearn how to order numbers from least to greatest using a number line. Convert fractions and decimals to mixed numbers or whole numbers, then plot them on the number line. Master this skill by practicing with examples like 7/3, -5/2, 0, -2, -12/4, and -3.25.https://cdn.kastatic.org/ka-youtube-converted/i1i2_9wg6N8.mp4/i1i2_9wg6N8.mp4254yes https://puzzicle.org/math/cc-sixth-grade-math/v/absolute-value-of-integershttps://puzzicle.org/thumbs/13179307.jpgAbsolute value examplesThe absolute value of a number represents its distance from zero on a number line, always resulting in a positive value. This concept is essential in mathematics, as it helps to simplify calculations and understand the magnitude of numbers, regardless of their positive or negative sign. Examples include finding the absolute values of 5, -10, and -12.https://cdn.kastatic.org/ka-youtube-converted/r6hS_8nm1jM.mp4/r6hS_8nm1jM.mp4142yes https://puzzicle.org/math/cc-sixth-grade-math/v/meaning-of-absolute-valuehttps://puzzicle.org/thumbs/x0532709128cbedba.jpgMeaning of absolute valueAbsolute value is a fundamental math concept that measures the distance of a number from zero, regardless of its sign. It's always a positive value, as it represents the magnitude of a number without considering its direction. This concept is useful in real-world applications, such as calculating distances and comparing heights.https://cdn.kastatic.org/ka-youtube-converted/zpln5ExhkyI.mp4/zpln5ExhkyI.mp4220yes https://puzzicle.org/math/cc-sixth-grade-math/v/comparing-absolute-valueshttps://puzzicle.org/thumbs/211177352.jpgComparing absolute valuesComparing absolute values helps us understand the distance between numbers and zero on the number line. The absolute value is always positive or zero, and it represents the magnitude of a number. By comparing absolute values, we can determine which number is further from zero, regardless of its positive or negative sign.https://cdn.kastatic.org/ka-youtube-converted/hKkBlcnU9pw.mp4/hKkBlcnU9pw.mp4247yes https://puzzicle.org/math/cc-sixth-grade-math/v/sorting-absolute-values-on-number-linehttps://puzzicle.org/thumbs/xbe02457f.jpgPlacing absolute values on the number lineIn this math lesson, we learn to sort values from least to greatest using a number line. We practice with negative numbers, positive numbers, and absolute values. By plotting these values on the number line, we can easily identify their order and improve our understanding of number relationships.https://cdn.kastatic.org/ka-youtube-converted/MFoXK57sw9k.mp4/MFoXK57sw9k.mp4137yes https://puzzicle.org/math/cc-sixth-grade-math/v/comparing-absolute-values-on-number-linehttps://puzzicle.org/thumbs/x3c41c0ae.jpgComparing absolute values on the number lineIn this math lesson, we explore absolute values and inequalities using a number line with three numbers: a, b, and c. We determine the truth of four inequalities: a < b, |a| > |b|, |a| < |c|, and a < c. By understanding number positions and absolute values, we can easily solve these inequalities.https://cdn.kastatic.org/ka-youtube-converted/29P6bar7nHc.mp4/29P6bar7nHc.mp4236yes https://puzzicle.org/math/cc-sixth-grade-math/v/values-to-make-absolute-value-inequality-truehttps://puzzicle.org/thumbs/xc6d0124b.jpgTesting solutions to absolute value inequalitiesIn this math lesson, we learn how to determine if given values of x satisfy various absolute value inequalities. We examine three different inequalities and test each x value to see if it meets the inequality's conditions. By understanding absolute values and inequalities, we can solve real-world problems and improve our mathematical skills.https://cdn.kastatic.org/ka-youtube-converted/fii9QEVJPas.mp4/fii9QEVJPas.mp4355yes https://puzzicle.org/math/cc-sixth-grade-math/v/interpreting-absolute-valuehttps://puzzicle.org/thumbs/x22385fa0.jpgInterpreting absolute valueIn this weight change scenario, three individuals track their monthly progress. Jenna experiences a 3-pound weight loss, while Sarah gains 2.5 pounds and Bill gains 4 pounds. By plotting these changes on a number line, we can determine that Jenna lost the most weight, Bill had the largest change, and Sarah had the smallest change.https://cdn.kastatic.org/ka-youtube-converted/faUh3bAxomM.mp4/faUh3bAxomM.mp4226yes https://puzzicle.org/math/cc-sixth-grade-math/v/the-coordinate-planehttps://puzzicle.org/thumbs/23630.jpgPoints on the coordinate plane examplesThe coordinate plane is a two-dimension surface formed by two number lines. One number line is horizontal and is called the x-axis. The other number line is vertical number line and is called the y-axis. The two axes meet at a point called the origin. We can use the coordinate plane to graph points, lines, and more.https://cdn.kastatic.org/ka-youtube-converted/VhNkWdLGpmA.mp4/VhNkWdLGpmA.mp4410yes https://puzzicle.org/math/cc-sixth-grade-math/v/points-on-the-coordinate-planehttps://puzzicle.org/thumbs/x08d09f9d.jpgFinding the point not graphedLearn to spot the missing point on a graph! By comparing coordinate pairs to points on a graph, we can identify which pair isn't represented. This skill is key in understanding graphs, a fundamental part of math!https://cdn.kastatic.org/ka-youtube-converted/b9H22F0Qbgw.mp4/b9H22F0Qbgw.mp463yes https://puzzicle.org/math/cc-sixth-grade-math/v/plot-ordered-pairshttps://puzzicle.org/thumbs/396277331.jpgPlotting a point (ordered pair)Plotting ordered pairs on a coordinate plane is like a treasure hunt! The first number tells us how many steps to hop right or left on the x-axis. The second number guides us up or down on the y-axis. So, for (6, -8), we hop 6 steps right and 8 steps down. Voila, we've found our treasure!https://cdn.kastatic.org/ka-youtube-converted/5ctsUsvIp8w.mp4/5ctsUsvIp8w.mp4128yes https://puzzicle.org/math/cc-sixth-grade-math/v/quadrants-of-coordinate-planehttps://puzzicle.org/thumbs/33752747.jpgQuadrants of the coordinate planeThe coordinate plane is divided into four sections, called quadrants. Quadrant I has positive x and y values, Quadrant II has negative x and positive y, Quadrant III has negative x and y, and Quadrant IV has positive x and negative y. The point (-7,7) is in Quadrant II.https://cdn.kastatic.org/ka-youtube-converted/1op92ojA6q0.mp4/1op92ojA6q0.mp4144yes https://puzzicle.org/math/cc-sixth-grade-math/v/graphing-points-and-naming-quadrants-exercisehttps://puzzicle.org/thumbs/x02167ace.jpgPoints and quadrants examplePlotting points on a graph involves moving along the x and y axes. Positive x-values move right, negative x-values move left. Positive y-values move up, negative y-values move down. The graph is divided into four quadrants, helping us locate points easily.https://cdn.kastatic.org/ka-youtube-converted/Jeh5vudjmLI.mp4/Jeh5vudjmLI.mp4116yes https://puzzicle.org/math/cc-sixth-grade-math/v/coordinate-plane-word-problems-exercisehttps://puzzicle.org/thumbs/x52229838.jpgCoordinate plane word problem examplesMilena's journey from home to the mall involves plotting points on a grid, similar to a coordinate plane. The video explains how to calculate the distance between two points, using the x and y coordinates. It also introduces the concept of plotting points in relation to a central point.https://cdn.kastatic.org/ka-youtube-converted/PC_FoyewoIs.mp4/PC_FoyewoIs.mp4287yes https://puzzicle.org/math/geometry-home/v/reflecting-points-exercisehttps://puzzicle.org/thumbs/x1d9ac087.jpgReflecting points in the coordinate planeJust like looking at a mirror image of yourself, but flipped....a reflection point is the mirror point on the opposite side of the axis. Watch this tutorial and reflect :)https://cdn.kastatic.org/ka-youtube-converted/INqaHHZGt8w.mp4/INqaHHZGt8w.mp4215yes https://puzzicle.org/math/cc-sixth-grade-math/v/introduction-to-exponentshttps://puzzicle.org/thumbs/xfe1a8e1f.jpgIntro to exponentsExponents represent repeated multiplication, making numbers grow quickly. For example, 2 to the 3rd power means multiplying three 2's together, resulting in 8. This concept differs from multiplication, which is simply repeated addition. Understanding exponents is essential for mastering higher-level math.https://cdn.kastatic.org/ka-youtube-converted/XZRQhkii0h0.mp4/XZRQhkii0h0.mp4183yes https://puzzicle.org/math/cc-sixth-grade-math/v/the-zeroth-powerhttps://puzzicle.org/thumbs/x21137693.jpgThe zeroth powerSal Khan considers two different ways to think about why a number raised to the zero power equals one: 1) if 2^3 = 1x2x2x2, then 2^0 = 1 times zero twos, which equals 1. 2) By following a pattern of decreasing an exponent by one by dividing by the base, we find that when we get to the 0 power, we end up dividing the base by itself, resulting in 1.https://cdn.kastatic.org/ka-youtube-converted/TWv6f7rwjF4.mp4/TWv6f7rwjF4.mp4272yes https://puzzicle.org/math/cc-sixth-grade-math/v/exponents-of-decimalshttps://puzzicle.org/thumbs/xeeccb03072f11724.jpgExponents of decimalsExponents of decimals can be calculated by multiplying the decimal number by itself as many times as the exponent indicates. When multiplying decimals, count the total number of digits to the right of the decimal points in both numbers and place the same number of digits to the right of the decimal point in the product.https://cdn.kastatic.org/ka-youtube-converted/o_urKmMdGK0.mp4/o_urKmMdGK0.mp4151yes https://puzzicle.org/math/cc-sixth-grade-math/v/powers-of-fractionshttps://puzzicle.org/thumbs/x3603d5d1.jpgPowers of fractionsRaising fractions to a power is a fundamental math skill. To do this, simply multiply the numerators and denominators separately. For example, when raising a fraction like 2/3 to the third power, multiply the numerators (2 × 2 × 2 = 8) and the denominators (3 × 3 × 3 = 27) to get the result 8/27. This method applies to all powers, making it easy to understand and apply.https://cdn.kastatic.org/ka-youtube-converted/oEgeUk_Ix2c.mp4/oEgeUk_Ix2c.mp4107yes https://puzzicle.org/math/cc-sixth-grade-math/v/order-of-operations-with-exponents-exampleshttps://puzzicle.org/thumbs/xf6436b29.jpgOrder of operations examples: exponentsThe order of operations (PEMDAS) is essential for solving math expressions correctly. By following Parentheses, Exponents, Multiplication/Division, and Addition/Subtraction, you ensure accurate results. Understanding the impact of parentheses on calculations helps avoid common mistakes and enhances problem-solving skills.https://cdn.kastatic.org/ka-youtube-converted/_jiI0AV8Vr4.mp4/_jiI0AV8Vr4.mp4214yes https://puzzicle.org/math/cc-sixth-grade-math/v/comparing-exponent-expressionshttps://puzzicle.org/thumbs/xe33cab3e2e0db6d6.jpgComparing exponent expressionsIn this math activity, we arrange three math expressions from smallest to largest. We evaluate the expressions: 2 cubed minus 2, 2 squared plus 3 to the power of 0, and 3 squared. We find their values to be 6, 5, and 9. The correct order is 5, 6, and 9.https://cdn.kastatic.org/ka-youtube-converted/kFElpYl7Te0.mp4/kFElpYl7Te0.mp4129yes https://puzzicle.org/math/algebra/v/what-is-a-variablehttps://puzzicle.org/thumbs/1027163525.jpgWhat is a variable?Variables in math are symbols, often letters, that represent different values in various situations. They help us understand and solve problems with changing values. For example, when calculating total earnings at a job with an hourly wage plus tips, a variable can represent the fluctuating tips, making it easier to determine total income.https://cdn.kastatic.org/ka-youtube-converted/tHYis-DP0oU.mp4/tHYis-DP0oU.mp4198yes https://puzzicle.org/math/cc-sixth-grade-math/v/expression-value-change-exampleshttps://puzzicle.org/thumbs/xfc2a26da.jpgExpression value intuitionDiscover how expressions change as variables within them shift. Learn that increasing x in "100 - x" results in a decreasing value, while decreasing x in "5/x + 5" (x remaining positive) leads to an increase. Observe that "3y/2y" stays constant as y increases, maintaining a value of 3/2.https://cdn.kastatic.org/ka-youtube-converted/UvDcEvDC4vg.mp4/UvDcEvDC4vg.mp4315yes https://puzzicle.org/math/cc-sixth-grade-math/v/expression-terms-factors-and-coefficientshttps://puzzicle.org/thumbs/x1a00fa7a.jpgTerms, factors, & coefficientsIn math expressions, terms are the components added or subtracted, factors are the elements multiplied within each term, and coefficients are the numbers multiplying variables. Understanding these concepts is crucial for effective communication and problem-solving in mathematics.https://cdn.kastatic.org/ka-youtube-converted/9_VCk9tWT0Y.mp4/9_VCk9tWT0Y.mp4441yes https://puzzicle.org/math/algebra/v/combining-like-termshttps://puzzicle.org/thumbs/1027260826.jpgIntro to combining like termsAdding like terms is a fundamental concept in algebra. Coefficients are the numbers in front of variables, and they can be added when the variables are the same. For example, 2x + 3x equals 5x. When dealing with different variables, such as x and y, add them separately, resulting in expressions like 5x + 9y.https://cdn.kastatic.org/ka-youtube-converted/CLWpkv6ccpA.mp4/CLWpkv6ccpA.mp4273yes https://puzzicle.org/math/cc-sixth-grade-math/v/combining-like-terms-1https://puzzicle.org/thumbs/1254705621.jpgCombining like termsIn this lesson, we learned how to add and subtract variables, like apples (a) or x's. We discovered that when we combine or take away the same variable, we can simply count them up or down. For example, 5x minus 2x equals 3x. This helps us understand basic algebra concepts.https://cdn.kastatic.org/ka-youtube-converted/P6_sK8hRWCA.mp4/P6_sK8hRWCA.mp4186yes https://puzzicle.org/math/cc-sixth-grade-math/v/combining-like-terms-2https://puzzicle.org/thumbs/1254724496.jpgCombining like terms exampleIn this lesson, we learned how to simplify algebraic expressions by combining like terms. We focused on understanding the intuition behind adding and subtracting coefficients of the same variables. The result is a simplified expression, making it easier to work with and solve problems.https://cdn.kastatic.org/ka-youtube-converted/1DgDt6gCUKs.mp4/1DgDt6gCUKs.mp4223yes https://puzzicle.org/math/algebra-home/v/combining-like-terms-with-distributionhttps://puzzicle.org/thumbs/x6c739d85.jpgCombining like terms with distributionLearn to expand and simplify an expression like 3(5x+6) + (7x+2)*4.https://cdn.kastatic.org/ka-youtube-converted/5WLjj7Z0AP0.mp4/5WLjj7Z0AP0.mp4139yes https://puzzicle.org/math/cc-sixth-grade-math/v/the-distributive-propertyhttps://puzzicle.org/thumbs/2600690.jpgDistributive property over additionLearn how to apply the distributive law of multiplication over addition and why it works. This is sometimes just called the distributive law or the distributive property.https://cdn.kastatic.org/ka-youtube-converted/gl_-E6iVAg4.mp4/gl_-E6iVAg4.mp4296yes https://puzzicle.org/math/cc-sixth-grade-math/v/the-distributive-property-2https://puzzicle.org/thumbs/2600691.jpgDistributive property over subtractionLearn how to apply the distributive property of multiplication over subtraction and why it works. This is sometimes just called the distributive property or distributive law.https://cdn.kastatic.org/ka-youtube-converted/Badvask-UDU.mp4/Badvask-UDU.mp4131yes https://puzzicle.org/math/cc-sixth-grade-math/v/distributive-property-exercise-2https://puzzicle.org/thumbs/xd74094af.jpgFactor with the distributive propertyThe distributive property helps us factor out the greatest common factor (GCF) in math problems. By finding the GCF of two numbers, we can rewrite the expression as a product, making calculations easier. Understanding this property is essential for simplifying expressions and solving equations.https://cdn.kastatic.org/ka-youtube-converted/t7OkENcBrxs.mp4/t7OkENcBrxs.mp4207yes https://puzzicle.org/math/cc-sixth-grade-math/v/distributive-property-with-variables-exercisehttps://puzzicle.org/thumbs/xd2339082.jpgDistributive property with variablesTo apply the distributive property to an algebraic expression, you multiply each term inside the parentheses by the number or variable outside the parentheses. For example, to simplify 2(x + 3), you would multiply 2 by both x and 3, resulting in 2x + 6.https://cdn.kastatic.org/ka-youtube-converted/Jp25LHI9wII.mp4/Jp25LHI9wII.mp4389yes https://puzzicle.org/math/algebra/v/equivalent-algebraic-expressions-exercisehttps://puzzicle.org/thumbs/x7c5e660c.jpgEquivalent expressionsIn this math lesson, we learn how to find equivalent expressions by combining like terms and factoring. We start with an expression like x + 2 - y + x + 2 and simplify it by adding the x terms and factoring out common factors. This helps us compare expressions and solve problems more easily.https://cdn.kastatic.org/ka-youtube-converted/rHNY01R2VSQ.mp4/rHNY01R2VSQ.mp4118yes https://puzzicle.org/math/cc-sixth-grade-math/v/writing-expressions-with-variables-exampleshttps://puzzicle.org/thumbs/xb9719b96.jpgWriting basic expressions with variablesLearn to write expressions with variables in math! Discover how to represent various operations like addition, subtraction, multiplication, and division using variables. Master the art of creating expressions for real-life situations, such as finding the sum of two numbers or the difference between two values. Boost your algebra skills and become a pro at handling variables!https://cdn.kastatic.org/ka-youtube-converted/640-86yn2wM.mp4/640-86yn2wM.mp4157yes https://puzzicle.org/math/cc-sixth-grade-math/v/algebraic-subtraction-expressionshttps://puzzicle.org/thumbs/xc191c980ab3e56fb.jpgWriting algebraic subtraction expressionsIn this lesson, we learned how to turn word statements into algebraic expressions. We practiced with examples like "two less than x" and "y fewer than three." Understanding how to translate words into math helps us solve real-life problems and makes algebra more fun!https://cdn.kastatic.org/ka-youtube-converted/eFXcsXnKkjw.mp4/eFXcsXnKkjw.mp489yes https://puzzicle.org/math/algebra/v/why-aren-t-we-using-the-multiplication-signhttps://puzzicle.org/thumbs/1027377520.jpgWhy aren't we using the multiplication sign?In algebra, representing multiplication with variables can be tricky due to the similarity between the variable "x" and the multiplication symbol. To avoid confusion, use alternative methods like 2⋅x, 2(x), or 2x. Practice evaluating expressions by substituting given values for variables and following the order of operations.https://cdn.kastatic.org/ka-youtube-converted/vDaIKB19TvY.mp4/vDaIKB19TvY.mp4298yes https://puzzicle.org/math/cc-sixth-grade-math/v/writing-expressions-1https://puzzicle.org/thumbs/xcd612f07.jpgWriting expressions with variablesTo write simple algebraic expressions from mathematical statements, you can follow this procedure: 1. Identify the key terms in the written statement (e.g. "sum," "product," "quantity"). 2. Break the statement down into its component parts. 3. Translate each part into algebraic form, using the appropriate symbols and operations. 4. Simplify the expression if needed.https://cdn.kastatic.org/ka-youtube-converted/Q1vMNyIP4Us.mp4/Q1vMNyIP4Us.mp498yes https://puzzicle.org/math/cc-sixth-grade-math/v/writing-expressions-parentheseshttps://puzzicle.org/thumbs/xc091e2df450dd50d.jpgWriting expressions with parenthesesIn math, using parentheses helps follow the correct order of operations. When multiplying a number by the sum or difference of two terms, like 5 times (x-2) or 10 times (y+3), parentheses ensure the addition or subtraction happens first. This avoids confusion and ensures accurate results.https://cdn.kastatic.org/ka-youtube-converted/cFNbX-2R78M.mp4/cFNbX-2R78M.mp492yes https://puzzicle.org/math/cc-sixth-grade-math/v/writing-expressions-2https://puzzicle.org/thumbs/xbe65fb1e.jpgWriting expressions with variables & parenthesesIn this math lesson, we learned about expressions and how to simplify them step by step. We explored adding and multiplying expressions, like negative numbers and variables. By practicing these skills, we can solve more complex algebra problems with ease and confidence.https://cdn.kastatic.org/ka-youtube-converted/xLYVo_k0_us.mp4/xLYVo_k0_us.mp4165yes https://puzzicle.org/math/cc-sixth-grade-math/v/writing-basic-expressions-from-word-problems-exampleshttps://puzzicle.org/thumbs/x34aac2f6.jpgWriting basic expressions word problemsSusie runs a race at 5 miles per hour for t hours. To find the distance, multiply her speed by time (5t). In a poker game, winnings are split among p players, so each gets $548/p. Phil's net winnings are x dollars minus the $100 entry fee. Hillary makes 48 chocolate chip cookies and y sugar cookies, totaling 48+y cookies.https://cdn.kastatic.org/ka-youtube-converted/lq-2gX3NKCM.mp4/lq-2gX3NKCM.mp4189yes https://puzzicle.org/math/cc-seventh-grade-math/v/more-fancy-expressions-from-word-problemshttps://puzzicle.org/thumbs/x7b5468c2.jpgWriting expressions word problemsLearn how to write expressions with variables to describe situations described in word problems.https://cdn.kastatic.org/ka-youtube-converted/C_KffdI34ZU.mp4/C_KffdI34ZU.mp4282yes https://puzzicle.org/math/algebra/v/variables-and-expressions-1https://puzzicle.org/thumbs/13179305.jpgEvaluating an expression with one variableIn this math lesson, we explore the expression 5 t plus 3 to calculate the cost of participating in a hospital raffle fundraiser. By evaluating the expression for different values of t, the number of tickets purchased, we find the costs for buying 1, 8, and 10 tickets, which are 8, 43, and 53 dollars, respectively.https://cdn.kastatic.org/ka-youtube-converted/AJNDeVt9UOo.mp4/AJNDeVt9UOo.mp4120yes https://puzzicle.org/math/cc-sixth-grade-math/v/evaluate-a-formula-using-substitutionhttps://puzzicle.org/thumbs/13179471.jpgEvaluating expressions with variables: temperatureIn this example we have a formula for converting Celsius temperature to Fahrenheit. Let's substitute the variable with a value (Celsius temp) to get the degrees in Fahrenheit. Great problem to practice with us!https://cdn.kastatic.org/ka-youtube-converted/tC1SfxRvtmM.mp4/tC1SfxRvtmM.mp475yes https://puzzicle.org/math/cc-sixth-grade-math/v/evaluating-expressions-3-exercisehttps://puzzicle.org/thumbs/x27a60b7a.jpgEvaluating expressions with variables: cubesDiscover how to calculate the total surface area of cube-shaped containers with varying side lengths. Learn the formula 6x², where x represents the side length of a cube. Apply this formula to find the surface area of multiple cubes, and add the results to determine the combined surface area that needs painting.https://cdn.kastatic.org/ka-youtube-converted/I9eLKDbc8og.mp4/I9eLKDbc8og.mp4204yes https://puzzicle.org/math/cc-sixth-grade-math/v/evaluating-exponential-expressionshttps://puzzicle.org/thumbs/13179497.jpgEvaluating expressions with variables: exponentsIn this math lesson, we learn to evaluate an algebraic expression with exponents by following the order of operations (PEMDAS). We substitute a given value for the variable, calculate exponents, perform multiplication, and finally, subtraction. By applying these steps, we successfully find the value of the expression.https://cdn.kastatic.org/ka-youtube-converted/0uCslW40VHQ.mp4/0uCslW40VHQ.mp4161yes https://puzzicle.org/math/algebra/v/evaluating-expressions-in-two-variableshttps://puzzicle.org/thumbs/xcce18b8f.jpgEvaluating expressions with two variablesEvaluating expressions with multiple variables involves substituting given values for each variable and simplifying the expression. By replacing variables with their corresponding values, we can easily compute the result of expressions, even for more complex examples with multiple terms and operations.https://cdn.kastatic.org/ka-youtube-converted/S_OX3ByvBSc.mp4/S_OX3ByvBSc.mp4125yes https://puzzicle.org/math/algebra/v/evaluating-expressions-in-two-variables-with-decimals-and-fractionshttps://puzzicle.org/thumbs/x8c62927c.jpgEvaluating expressions with two variables: fractions & decimalsEvaluating expressions with two variables involves substituting the given values for each variable and simplifying the expression. By practicing with examples, we can improve our skills in solving these types of problems, ultimately enhancing our understanding of algebraic expressions and their real-world applications.https://cdn.kastatic.org/ka-youtube-converted/GmD7Czmol0k.mp4/GmD7Czmol0k.mp4207yes https://puzzicle.org/math/cc-sixth-grade-math/v/evaluating-exponent-expressions-with-variableshttps://puzzicle.org/thumbs/x7ca1da5f32237ccb.jpgEvaluating exponent expressions with variablesIn this math lesson, we learn to evaluate expressions with exponents and variables. We practice substituting values for variables and calculating the results. By mastering this skill, we can solve problems involving exponential expressions, enhancing our understanding of algebra and mathematical concepts.https://cdn.kastatic.org/ka-youtube-converted/9K54G5yeR74.mp4/9K54G5yeR74.mp4181yes https://puzzicle.org/math/cc-sixth-grade-math/v/evaluating-expressions-5x2https://puzzicle.org/thumbs/x7fd7ee1eba5733e6.jpgEvaluating expressions like 5x² & ⅓(6)ˣEvaluating expressions with exponents involves substituting a given value for the variable and following the order of operations. Calculate the exponent first, then perform any multiplication or division. Mastering this skill is essential for solving more complex algebraic problems and understanding mathematical relationships.https://cdn.kastatic.org/ka-youtube-converted/ccyZpjKvJV8.mp4/ccyZpjKvJV8.mp486yes https://puzzicle.org/math/cc-sixth-grade-math/v/variables-expressions-and-equationshttps://puzzicle.org/thumbs/583878752.jpgVariables, expressions, & equationsIn algebra, we use variables like x, y, and z to represent unknown values. Expressions are combinations of variables and numbers, while equations equate two expressions. Variables can take on different values depending on the context, and we can evaluate expressions by substituting values for the variables.https://cdn.kastatic.org/ka-youtube-converted/vDqOoI-4Z6M.mp4/vDqOoI-4Z6M.mp4415yes https://puzzicle.org/math/cc-sixth-grade-math/v/testing-solutions-to-equationhttps://puzzicle.org/thumbs/x1923a9f4.jpgTesting solutions to equationsIn this math lesson, we learn how to solve an equation with one variable. We test different values for x to see which one satisfies the equation. By substituting x with the given options, we find the correct solution that makes both sides of the equation equal. Practice makes perfect!https://cdn.kastatic.org/ka-youtube-converted/U9OfQd8poTQ.mp4/U9OfQd8poTQ.mp4225yes https://puzzicle.org/math/cc-sixth-grade-math/v/why-we-do-the-same-thing-to-both-sides-simple-equationshttps://puzzicle.org/thumbs/948454929.jpgSame thing to both sides of equationsIn this fun math problem, we have a balanced scale with a mystery mass on one side and several 1-kilogram masses on the other. To find the mystery mass, we remove the same number of 1-kilogram masses from both sides, keeping the scale balanced. Then, we can easily determine the mystery mass's weight.https://cdn.kastatic.org/ka-youtube-converted/5FBDqY41GjU.mp4/5FBDqY41GjU.mp4152yes https://puzzicle.org/math/cc-sixth-grade-math/v/representing-a-relationship-with-a-simple-equationhttps://puzzicle.org/thumbs/948457910.jpgRepresenting a relationship with an equationIn this lesson, we learn about balancing scales and solving equations. We discover that to find a mystery mass, we can set up an equation using equal signs and balance both sides. By subtracting the same amount from each side, we can solve for the unknown mass, helping us understand the relationship between the two sides of the scale.https://cdn.kastatic.org/ka-youtube-converted/XN48Ba9Ncr8.mp4/XN48Ba9Ncr8.mp4204yes https://puzzicle.org/math/cc-sixth-grade-math/v/intuition-why-we-divide-both-sideshttps://puzzicle.org/thumbs/948526680.jpgDividing both sides of an equationIn this lesson, we learn how to find the mass of a mystery object using a balanced scale. By keeping the scale balanced and taking equal portions from both sides, we can determine the mass of the mystery object. This helps us understand the concept of equality in real-life situations.https://cdn.kastatic.org/ka-youtube-converted/Ye13MIPv6n0.mp4/Ye13MIPv6n0.mp4194yes https://puzzicle.org/math/cc-sixth-grade-math/v/adding-and-subtracting-the-same-thing-from-both-sideshttps://puzzicle.org/thumbs/1032539728.jpgOne-step addition & subtraction equationsIn this lesson, we learn how to solve simple equations with one unknown variable. To find the unknown, we remove extra numbers by doing the opposite operation on both sides of the equation. This keeps the equation balanced and helps us find the value of the unknown variable.https://cdn.kastatic.org/ka-youtube-converted/jWpiMu5LNdg.mp4/jWpiMu5LNdg.mp4129yes https://puzzicle.org/math/cc-sixth-grade-math/v/solving-one-step-equationshttps://puzzicle.org/thumbs/13179465.jpgOne-step addition equationIn this math lesson, we learn to solve simple equations with one variable. We isolate the variable by removing other terms on the same side. For example, if a + 5 = 54, we subtract 5 from both sides to find a = 49. Finally, we check our solution by substituting the value back into the original equation.https://cdn.kastatic.org/ka-youtube-converted/VidnbCEOGdg.mp4/VidnbCEOGdg.mp4114yes https://puzzicle.org/math/cc-sixth-grade-math/v/examples-of-one-step-equations-with-fractions-and-decimalshttps://puzzicle.org/thumbs/x6408645c.jpgOne-step addition & subtraction equations: fractions & decimalsIn this math lesson, we practice solving equations with variables and fractions. We learn to isolate the variable by adding or subtracting terms from both sides of the equation. This helps us find the value of the variable, making the equation true. Keep practicing and have fun!https://cdn.kastatic.org/ka-youtube-converted/XD-FDGdWnR8.mp4/XD-FDGdWnR8.mp4272yes https://puzzicle.org/math/cc-sixth-grade-math/v/solving-one-step-equations-2https://puzzicle.org/thumbs/13179466.jpgOne-step multiplication equationsIn this math lesson, we learn to solve equations with x divided by a number, like x/3 = 14. To find x, we isolate it by multiplying both sides of the equation by the divisor (3). This gives us x = 42. Finally, we check our answer by substituting x back into the original equation to make sure it's correct.https://cdn.kastatic.org/ka-youtube-converted/DqeMQHomwAU.mp4/DqeMQHomwAU.mp4143yes https://puzzicle.org/math/cc-sixth-grade-math/v/simple-equationshttps://puzzicle.org/thumbs/7.jpgOne-step division equationsIn this lesson, we learn how to solve simple algebraic equations systematically. We focus on understanding the meaning of equations, like 7x = 14, and using techniques like dividing both sides of the equation by the coefficient to isolate the variable. This method helps us find the value of the unknown variable, making algebra easier and more approachable.https://cdn.kastatic.org/ka-youtube-converted/9Ek61w1LxSc.mp4/9Ek61w1LxSc.mp4666yes https://puzzicle.org/math/cc-sixth-grade-math/v/one-step-multiplication-and-division-equations-with-fractionshttps://puzzicle.org/thumbs/x6dd22c7e.jpgOne-step multiplication & division equations: fractions & decimalsIn this fun math adventure, we tackle equations with fractions and decimals. We learn to balance equations by applying the same operation on both sides, simplifying fractions, and eliminating decimals. It's a step-by-step journey to mastering algebra!https://cdn.kastatic.org/ka-youtube-converted/a3acutLstF8.mp4/a3acutLstF8.mp4471yes https://puzzicle.org/math/cc-sixth-grade-math/v/one-step-multiplication-fractional-coefficientshttps://puzzicle.org/thumbs/x24d59f9050daac31.jpgOne-step multiplication equations: fractional coefficientsLearn how to solve equations with fractions by using reciprocals. Multiply both sides of the equation by the reciprocal of the fraction to make the coefficient one. This simplifies the equation, making it easier to find the value of the variable.https://cdn.kastatic.org/ka-youtube-converted/vksy7e3hY8Q.mp4/vksy7e3hY8Q.mp4234yes https://puzzicle.org/math/cc-sixth-grade-math/v/finding-mistakes-in-one-step-equationshttps://puzzicle.org/thumbs/xae320dbc.jpgFinding mistakes in one-step equationsIn this lesson, we learned how to spot mistakes in solving algebraic equations. We saw that it's crucial to apply the same operation to both sides of the equation. We also learned the importance of accurate arithmetic in getting the correct answer. Lastly, we saw that not all steps, even if algebraically correct, help simplify the equation.https://cdn.kastatic.org/ka-youtube-converted/CUB5j3Frev0.mp4/CUB5j3Frev0.mp4252yes https://puzzicle.org/math/cc-seventh-grade-math/v/super-yoga-plans-basic-variables-and-equationshttps://puzzicle.org/thumbs/1094227108.jpgEquation word problem: super yoga (1 of 2)Using information from the Super Yoga word problem, explore all the possible combinations and create equations which express the possibilities. Let's figure out which plan is best!https://cdn.kastatic.org/ka-youtube-converted/3tRc6mUumFk.mp4/3tRc6mUumFk.mp4439yes https://puzzicle.org/math/cc-seventh-grade-math/v/super-yoga-plans-solving-one-step-equationshttps://puzzicle.org/thumbs/1094119956.jpgEquation word problem: super yoga (2 of 2)If you're coming to this video before seeing the previous one, back up! Otherwise, we're solving one-step equations to learn which Super Plan is the best value for our budget.https://cdn.kastatic.org/ka-youtube-converted/hq1bUM2tyg0.mp4/hq1bUM2tyg0.mp4460yes https://puzzicle.org/math/cc-sixth-grade-math/v/constructing-basic-equations-exampleshttps://puzzicle.org/thumbs/xdd8bfe76.jpgModeling with one-step equationsAnna wants to throw a pizza party for her birthday. She has $42.50 to spend on pizza boxes that cost $8.50 each. To figure out how many pizzas she can buy, she divides the total money by the cost per box. This concept can also be applied to other situations, like a class selling candy boxes for a fundraiser.https://cdn.kastatic.org/ka-youtube-converted/2REbsY4-S70.mp4/2REbsY4-S70.mp4261yes https://puzzicle.org/math/cc-sixth-grade-math/v/testing-solutions-to-inequalitieshttps://puzzicle.org/thumbs/x9fac0737.jpgTesting solutions to inequalitiesInequalities are like puzzles where you compare two things. You can solve these puzzles by trying different numbers and seeing which ones make the puzzle true. Some numbers will work for one puzzle but not for another. It's all about finding the right fit!https://cdn.kastatic.org/ka-youtube-converted/Yh4TXMVq9eg.mp4/Yh4TXMVq9eg.mp4310yes https://puzzicle.org/math/cc-sixth-grade-math/v/plotting-inequalities-on-a-number-linehttps://puzzicle.org/thumbs/x5c4574ba.jpgPlotting inequalitiesTo plot an inequality, such as x>3, on a number line, first draw a circle over the number (e.g., 3). Then if the sign includes equal to (≥ or ≤), fill in the circle. If the sign does not include equal to (> or <), leave the circle unfilled in. Finally, draw a line going from the circle in the direction of the numbers that make the inequality true.https://cdn.kastatic.org/ka-youtube-converted/ilWDSYnTEFs.mp4/ilWDSYnTEFs.mp4127yes https://puzzicle.org/math/cc-sixth-grade-math/v/inequalities-on-a-number-linehttps://puzzicle.org/thumbs/13179473.jpgPlotting an inequality exampleDrawing a number line helps visualize 'x is less than 4'. We mark 4 with a circle, not a dot, because 4 isn't included. Then, we color the line below 4, showing all values less than 4. Easy peasy, lemon squeezyhttps://cdn.kastatic.org/ka-youtube-converted/dTwZ5N126gw.mp4/dTwZ5N126gw.mp494yes https://puzzicle.org/math/cc-sixth-grade-math/v/real-world-situations-with-inequalitieshttps://puzzicle.org/thumbs/x5a048673.jpgInequalities word problemsInequalities are a handy tool for comparing values. For example, if Eric is shorter than Priti who is 158 cm tall, we write E < 158. If Sydney wears skirts when it's warmer than 25 degrees, we say T > 25. If Hayley finishes a race after Lola's 53 seconds, we write H > 53.https://cdn.kastatic.org/ka-youtube-converted/VEIf5TfceAw.mp4/VEIf5TfceAw.mp4176yes https://puzzicle.org/math/cc-sixth-grade-math/v/dependent-independent-variableshttps://puzzicle.org/thumbs/x0f276e8512307008.jpgDependent & independent variablesWe introduce independent and dependent variables: how to identify the independent/dependent variable in a given setting, and how we write equations with the dependent variable by itself.https://cdn.kastatic.org/ka-youtube-converted/SGC_d7O7_Eg.mp4/SGC_d7O7_Eg.mp4139yes https://puzzicle.org/math/cc-sixth-grade-math/v/dependent-and-independent-variables-exercise-example-2https://puzzicle.org/thumbs/x03fbe709.jpgDependent & independent variables: graphingWhen you answer math quiz questions, each correct answer earns you 5 points. The number of correct answers (x) and total points (y) are related by the equation y = 5x. This relationship can be visualized on a graph.https://cdn.kastatic.org/ka-youtube-converted/0eWm-LY23W0.mp4/0eWm-LY23W0.mp460yes https://puzzicle.org/math/cc-sixth-grade-math/v/equations-relationships-quantitieshttps://puzzicle.org/thumbs/xb4c3652f840b25b2.jpgWriting equations for relationships between quantitiesIn this lesson, we learn how to write equations representing real-world scenarios. First, we explore how to calculate time taken for a walk based on speed and distance. Then, we discover how to determine the amount of water needed in a soup recipe based on the quantity of vegetable stock. Fun, right?https://cdn.kastatic.org/ka-youtube-converted/5T9gSp2W3Oc.mp4/5T9gSp2W3Oc.mp4361yes https://puzzicle.org/math/cc-sixth-grade-math/v/analyzing-relationships-between-variableshttps://puzzicle.org/thumbs/x4fa4a8d08c7b0412.jpgAnalyzing relationships between variablesRava is studying an electric car. She finds out that the more she charges the car, the further it can go. The relationship is simple and direct: more charging time equals more distance. This helps her predict the car's range based on how long it's been charged.https://cdn.kastatic.org/ka-youtube-converted/Pf-s9zNDSCk.mp4/Pf-s9zNDSCk.mp4169yes https://puzzicle.org/math/cc-sixth-grade-math/v/intuition-for-area-of-a-parallelogramhttps://puzzicle.org/thumbs/x5043d50b.jpgArea of a parallelogramDiscover the magic of geometry! The area of a rectangle and a parallelogram can be calculated in the same way. Just multiply the base by the height! This simple trick works because a parallelogram can be rearranged into a rectangle. Geometry is full of surprises!https://cdn.kastatic.org/ka-youtube-converted/hm17lVaor0Q.mp4/hm17lVaor0Q.mp4177yes https://puzzicle.org/math/cc-sixth-grade-math/v/finding-height-of-a-parallelogramhttps://puzzicle.org/thumbs/x96372c65f08c8dce.jpgFinding height of a parallelogramIn this lesson, we learned that the area of a parallelogram equals the base times the height. Given an area of 24 units and a base of 6 units, we used this formula to find the missing height. The answer was 4 units. Isn't math fun?https://cdn.kastatic.org/ka-youtube-converted/SKe3ItfdzTY.mp4/SKe3ItfdzTY.mp4102yes https://puzzicle.org/math/cc-sixth-grade-math/v/intuition-for-area-of-a-trianglehttps://puzzicle.org/thumbs/x11df81ee.jpgArea of a triangleThe area of a rectangle and a parallelogram is found by multiplying the base by the height. For a triangle, the area is half of a parallelogram's, so it's calculated by multiplying the base by the height and then dividing by 2.https://cdn.kastatic.org/ka-youtube-converted/rRTXKQpblEc.mp4/rRTXKQpblEc.mp4330yes https://puzzicle.org/math/cc-sixth-grade-math/v/example-finding-area-of-trianglehttps://puzzicle.org/thumbs/xcc8a19eed3fc9954.jpgFinding area of trianglesThe area of a triangle is found by multiplying one half of the base by the height. In our example, the base is 18 and the height is 6. So, half of 18 is 9, and 9 times 6 equals 54 square units.https://cdn.kastatic.org/ka-youtube-converted/7N5orPxUoGo.mp4/7N5orPxUoGo.mp491yes https://puzzicle.org/math/cc-sixth-grade-math/v/triangle-missing-side-examplehttps://puzzicle.org/thumbs/xf11586ba7868d7f4.jpgTriangle missing side exampleDive into the world of triangles! Learn how to find a missing piece using just the area and height. Apply this knowledge to solve puzzles and explore the magic hidden in these three-sided shapes. Join the triangle adventure!https://cdn.kastatic.org/ka-youtube-converted/wTkH288r84s.mp4/wTkH288r84s.mp4189yes https://puzzicle.org/math/geometry-home/v/area-of-triangle-in-gridhttps://puzzicle.org/thumbs/x33ae35a0.jpgArea of a triangle on a gridLearn how to calculate the area of a triangle on a grid.https://cdn.kastatic.org/ka-youtube-converted/ukPjc3Oyad4.mp4/ukPjc3Oyad4.mp4417yes https://puzzicle.org/math/geometry-home/v/area-of-strange-quadrilateralhttps://puzzicle.org/thumbs/x995b0cd6.jpgArea of a quadrilateral on a gridLearn to break up oddly shaped quadrilaterals into shapes where finding the area is more easily determined.https://cdn.kastatic.org/ka-youtube-converted/gkifo46--JA.mp4/gkifo46--JA.mp4234yes https://puzzicle.org/math/cc-sixth-grade-math/v/area-of-quadrilateral-with-2-parallel-sideshttps://puzzicle.org/thumbs/xfae40098dbd6126d.jpgArea of quadrilateral with 2 parallel sidesLearn how to find the area of a quadrilateral by splitting it into two triangles. Use the formula for triangle area (1/2 base times height) for each triangle. Add the areas together for the total quadrilateral area. It's as easy as pie...or should I say, as easy as area!https://cdn.kastatic.org/ka-youtube-converted/uEIoh8uWs1U.mp4/uEIoh8uWs1U.mp4222yes https://puzzicle.org/math/geometry-home/v/area-of-a-trapezoid-1https://puzzicle.org/thumbs/x9eb86ee6.jpgArea of trapezoidsArea of a trapezoid is found with the formula, A=(a+b)h/2. To find the area of a trapezoid, you need to know the lengths of the two parallel sides (the "bases") and the height. Add the lengths of the two bases together, and then multiply by the height. Finally, divide by 2 to get the area of the trapezoid.https://cdn.kastatic.org/ka-youtube-converted/j3-XYLnxJDY.mp4/j3-XYLnxJDY.mp4288yes https://puzzicle.org/math/geometry-home/v/area-of-a-kitehttps://puzzicle.org/thumbs/xae34441c.jpgArea of kitesTo find the area of a kite, you need to know the lengths of the kite's two diagonals (the lines that cross through the middle of the kite). Multiply the lengths of the two diagonals together, and then divide by 2. This will give you the area of the kite.https://cdn.kastatic.org/ka-youtube-converted/b8q6i_XPyhk.mp4/b8q6i_XPyhk.mp4229yes https://puzzicle.org/math/cc-sixth-grade-math/v/area-comparisonshttps://puzzicle.org/thumbs/x27919a7f.jpgFinding area by rearranging partsDiscover the magic of quadrilaterals and their areas! We explore how a trapezoid's area can match a rectangle's by rearranging its parts. We learn that shapes with the same area can look different. It's a fantastic journey into the world of geometry!https://cdn.kastatic.org/ka-youtube-converted/MamrTJ7V_Vg.mp4/MamrTJ7V_Vg.mp4237yes https://puzzicle.org/math/cc-sixth-grade-math/v/area-breaking-up-shapehttps://puzzicle.org/thumbs/x74fb422f.jpgArea of composite shapesWe can sometimes calculate the area of a complex shape by dividing it into smaller, more manageable parts. In this example, we can determine the area of two triangles, a rectangle, and a trapezoid, and then add up the areas of the four shapes to get the total area.https://cdn.kastatic.org/ka-youtube-converted/loAA3TCNAvU.mp4/loAA3TCNAvU.mp4315yes https://puzzicle.org/math/cc-sixth-grade-math/v/constructing-polygon-on-coordinate-plane-examplehttps://puzzicle.org/thumbs/xd2b8209a.jpgDrawing a quadrilateral on the coordinate plane exampleWe're making a four-sided shape called a quadrilateral using line segments! We use four points on a graph: (0,9), (0,-7), (8,-7), and (8,0). We connect these points to form our quadrilateral. Isn't shape-making fun?https://cdn.kastatic.org/ka-youtube-converted/zxy4KAiSPIc.mp4/zxy4KAiSPIc.mp492yes https://puzzicle.org/math/cc-sixth-grade-math/v/finding-coordinates-of-missing-vertex-examplehttps://puzzicle.org/thumbs/xd94a4f02.jpgCoordinates of a missing vertexThis lesson teaches how to find the missing vertex of a rectangle on a coordinate plane. By understanding that rectangle sides are parallel and vertices form right angles, we can determine that the missing vertex shares its x-coordinate with one vertex and its y-coordinate with another.https://cdn.kastatic.org/ka-youtube-converted/WkiPVButdeg.mp4/WkiPVButdeg.mp4161yes https://puzzicle.org/math/cc-sixth-grade-math/v/area-of-parallelogram-on-coordinate-planehttps://puzzicle.org/thumbs/xbd468ab0.jpgArea of a parallelogram on the coordinate planeIn math, we can find the area of a parallelogram by multiplying the base by the height. The base is one side, and the height is the distance from the base to the opposite side. So, if the base is 12 units and the height is 4 units, the area is 48 square units.https://cdn.kastatic.org/ka-youtube-converted/xTcwnhyPN5E.mp4/xTcwnhyPN5E.mp4128yes https://puzzicle.org/math/cc-sixth-grade-math/v/dimensions-of-rectangle-from-coordinates-examplehttps://puzzicle.org/thumbs/xd521c318.jpgDimensions of a rectangle from coordinatesHow can we determine the length of a side of a rectangle given the coordinates of its vertices? In this example, by plotting the points on a graph, we can see that the length one side is five because the y-coordinate increases by five from one endpoint to the other.https://cdn.kastatic.org/ka-youtube-converted/fEpINE1fjgc.mp4/fEpINE1fjgc.mp4231yes https://puzzicle.org/math/cc-sixth-grade-math/v/coordinates-of-rectangle-examplehttps://puzzicle.org/thumbs/x621a7bcc.jpgCoordinates of rectangle exampleLearn how to sort coordinates from least to greatest using a rectangle diagram. Understand how x, y, and z coordinates relate to each other. Discover that y equals four, z equals three, and x equals two. Master sorting these values in ascending order: x, z, y.https://cdn.kastatic.org/ka-youtube-converted/s2wmcnZi6kE.mp4/s2wmcnZi6kE.mp4142yes https://puzzicle.org/math/cc-sixth-grade-math/v/example-of-shapes-on-a-coordinate-planehttps://puzzicle.org/thumbs/x7442477a5b2dcf62.jpgExample of shapes on a coordinate planePlotting points on a coordinate plane can help us draw shapes like rectangles! The points (1,1), (1,6), (9,6), and (9,1) form a rectangle. The x-coordinate tells us how far right to move, and the y-coordinate tells us how far up to go. The rectangle's height is the difference between the y-coordinates, which is 5.https://cdn.kastatic.org/ka-youtube-converted/8V-PdgS2GKc.mp4/8V-PdgS2GKc.mp4142yes https://puzzicle.org/math/cc-sixth-grade-math/v/analyzing-polygon-on-the-coordinate-planehttps://puzzicle.org/thumbs/x249c50c5.jpgParallelogram on the coordinate planeRemember our discussion of the coordinate plane? Sure you do! Let's graph the given coordinates of three of the polygon vertices, and find where the 4th vertex is.https://cdn.kastatic.org/ka-youtube-converted/8ToCxaB7EOE.mp4/8ToCxaB7EOE.mp4317yes https://puzzicle.org/math/cc-sixth-grade-math/v/volume-of-a-rectangular-prism-with-fractional-cubeshttps://puzzicle.org/thumbs/xc603bd0d.jpgVolume with fractional cubesAnother way of finding the volume of a rectangular prism involves dividing it into fractional cubes, finding the volume of one, and then multiplying that volume by the number of cubes that fit into our rectangular prism.https://cdn.kastatic.org/ka-youtube-converted/tVDslyeLefU.mp4/tVDslyeLefU.mp4258yes https://puzzicle.org/math/cc-sixth-grade-math/v/volume-of-a-rectangular-prism-with-fractional-dimensionshttps://puzzicle.org/thumbs/x5c1fce1f.jpgVolume of a rectangular prism: fractional dimensionsThe video explains how to calculate the volume of a rectangular prism with fractional dimensions. It emphasizes that volume equals the area of the base times the height. To find the volume, multiply the length, width, and height. The video also shows how to simplify fractions during multiplication.https://cdn.kastatic.org/ka-youtube-converted/EJTPGyWqhqc.mp4/EJTPGyWqhqc.mp4255yes https://puzzicle.org/math/cc-sixth-grade-math/v/how-volume-changes-from-changing-dimensionshttps://puzzicle.org/thumbs/x06ec522989f5a1c6.jpgHow volume changes from changing dimensionsIn geometry, altering the dimensions of a rectangular prism impacts its volume. If you double one dimension, the volume doubles. If you double two dimensions, the volume quadruples. When all dimensions are doubled, the volume increases eight times. This is a fundamental concept in understanding volume.https://cdn.kastatic.org/ka-youtube-converted/IShaLZ8CMXY.mp4/IShaLZ8CMXY.mp4257yes https://puzzicle.org/math/cc-sixth-grade-math/v/marbles-in-tank-volumehttps://puzzicle.org/thumbs/xa72132ec.jpgVolume of a rectangular prism: word problemMario has a fish tank shaped like a right rectangular prism. The tank is filled with marbles and water up to 6.4 cm. When the marbles are removed, the water level drops to 5.9 cm. The volume of water displaced by the marbles is found to be 56.16 cubic centimeters.https://cdn.kastatic.org/ka-youtube-converted/CCysGY2cXUo.mp4/CCysGY2cXUo.mp4382yes https://puzzicle.org/math/cc-sixth-grade-math/v/counting-faces-and-edges-of-3d-shapeshttps://puzzicle.org/thumbs/x86473103952799e4.jpgCounting faces and edges of 3D shapesLearn about shapes! Discover how to count faces and edges on 3D figures. We explore a transparent shape with five faces and another shape, a square pyramid, with eight edges and five faces. It's a colorful journey into geometry!https://cdn.kastatic.org/ka-youtube-converted/_y17uHBwa_8.mp4/_y17uHBwa_8.mp4227yes https://puzzicle.org/math/cc-sixth-grade-math/v/recognizing-common-3d-shapeshttps://puzzicle.org/thumbs/x4821014801c816e7.jpgRecognizing common 3D shapesLearn about 3D shapes! A square pyramid has a square base. A rectangular prism is a box shape with all rectangular sides. A triangular prism has triangle faces on opposite sides. A sphere is like a globe. A cylinder looks like a can with circular faces on top and bottom. Shapes are fun!https://cdn.kastatic.org/ka-youtube-converted/tqxQSSzuXX0.mp4/tqxQSSzuXX0.mp4209yes https://puzzicle.org/math/cc-sixth-grade-math/v/nets-of-polyhedrahttps://puzzicle.org/thumbs/x6e267a36.jpgIntro to nets of polyhedraAdmittedly, "nets of polyhedra" sounds like the title of a bad sci-fi movie about man-eating, multi-headed fish. But in reality, nets of polyhedra are just 2D objects that wrap around 3D objects, like wrapping paper around presents.https://cdn.kastatic.org/ka-youtube-converted/ZACf9EecFrY.mp4/ZACf9EecFrY.mp4408yes https://puzzicle.org/math/cc-sixth-grade-math/v/surface-area-from-nethttps://puzzicle.org/thumbs/xbb4d5fe8.jpgSurface area using a net: rectangular prismA polyhedron is a three-dimensional shape that has flat surfaces and straight edges. Learn whether or not a certain net could be folded up into a certain rectangular prism.https://cdn.kastatic.org/ka-youtube-converted/m1ZTnl4CNQg.mp4/m1ZTnl4CNQg.mp4259yes https://puzzicle.org/math/cc-sixth-grade-math/v/finding-surface-area-using-nethttps://puzzicle.org/thumbs/xeb3d20c9.jpgSurface area using a net: triangular prismHere are the steps to compute the surface area of a triangular prism: 1. Find the areas of each of the three rectangular faces, using the formula for the area of a rectangle: length x width. 2. Next, find the area of the two triangular faces, using the formula for the area of a triangle: 1/2 base x height. 3. Add the areas of all five faces together, and you're done!https://cdn.kastatic.org/ka-youtube-converted/mtMNvnm71Z0.mp4/mtMNvnm71Z0.mp4214yes https://puzzicle.org/math/cc-sixth-grade-math/v/surface-area-of-a-box-using-netshttps://puzzicle.org/thumbs/x27da95fa.jpgSurface area of a box using netsDiscover the surface area of a cereal box by visualizing a net. Cut and flatten the box to create a 2D shape. Measure the dimensions, calculate the area of each section, and add them up for the total surface area. Fun and practical!https://cdn.kastatic.org/ka-youtube-converted/vl1uMOYFyf4.mp4/vl1uMOYFyf4.mp4284yes https://puzzicle.org/science/ap-biology/v/surface-area-of-a-boxhttps://puzzicle.org/thumbs/xcd3980f6.jpgSurface area of a box (cuboid)Surface area is total area on the surface of a three-dimensional shape. To find the surface area of a cuboid which has 6 rectangular faces, add the areas of all 6 faces. Or, you can label the length (l), width (w), and height (h) of the cuboid and use the formula: surface area (SA)=2lw+2lh+2hw.https://cdn.kastatic.org/ka-youtube-converted/1iSBNSYhvIU.mp4/1iSBNSYhvIU.mp4145yes https://puzzicle.org/math/cc-sixth-grade-math/v/surface-area-word-problem-examplehttps://puzzicle.org/thumbs/x6a82c2200b76ccc5.jpgSurface area word problem exampleAkira won a science fair and got a golden pyramid trophy. We want to find out how much gold foil was used. The trophy has a square base and four triangular faces. By calculating the area of these shapes, we can find the total surface area covered by the gold foil.https://cdn.kastatic.org/ka-youtube-converted/ODNCH3gSKdA.mp4/ODNCH3gSKdA.mp4275yes https://puzzicle.org/math/cc-sixth-grade-math/v/ways-to-represent-datahttps://puzzicle.org/thumbs/xbcceb7d5.jpgRepresenting dataData can be represented in multiple ways such as tables, bar graphs, histograms, and frequency plots. These methods provide different perspectives of the same information, helping to answer various questions about the data.https://cdn.kastatic.org/ka-youtube-converted/0ZKtsUkrgFQ.mp4/0ZKtsUkrgFQ.mp4498yes https://puzzicle.org/math/cc-sixth-grade-math/v/frequency-tables-and-dot-plotshttps://puzzicle.org/thumbs/x9720769d.jpgFrequency tables & dot plotsFrequency tables and dot plots are handy tools in data representation. Frequency tables show how often each value appears, while dot plots provide a visual depiction of this information. These tools can be used to answer various questions about the data.https://cdn.kastatic.org/ka-youtube-converted/gdE46YSedvE.mp4/gdE46YSedvE.mp4439yes https://puzzicle.org/math/cc-sixth-grade-math/v/understanding-statistical-questionshttps://puzzicle.org/thumbs/xed476f8d.jpgStatistical questionsStatistics is about collecting, presenting, and analyzing data. Variability, a key concept in statistics, refers to how much data points differ from each other. Statistical questions require collecting data with variability to answer. For example, asking about the average number of cars in a parking lot on Monday mornings is a statistical question.https://cdn.kastatic.org/ka-youtube-converted/OjzfQDFf7Uk.mp4/OjzfQDFf7Uk.mp4574yes https://puzzicle.org/math/cc-sixth-grade-math/v/histograms-introhttps://puzzicle.org/thumbs/xbfb17dc0.jpgCreating a histogramA histogram is a graphical display of data using bars of different heights. In a histogram, each bar groups numbers into ranges. Taller bars show that more data falls in that range. A histogram displays the shape and spread of continuous sample data.https://cdn.kastatic.org/ka-youtube-converted/gSEYtAjuZ-Y.mp4/gSEYtAjuZ-Y.mp4442yes https://puzzicle.org/math/cc-sixth-grade-math/v/interpreting-histogramshttps://puzzicle.org/thumbs/xfeb52bb5.jpgInterpreting a histogramIn a cherry pie store, we use a histogram to understand the distribution of cherries on pies. We create buckets for different ranges of cherries, count the pies in each, and can answer questions about our pie inventory and cherry distribution!https://cdn.kastatic.org/ka-youtube-converted/c02vjunQsJM.mp4/c02vjunQsJM.mp4269yes https://puzzicle.org/math/cc-sixth-grade-math/v/statistics-intro-mean-median-and-modehttps://puzzicle.org/thumbs/1152738250.jpgStatistics intro: Mean, median, & modeThe mean (average) of a data set is found by adding all numbers in the data set and then dividing by the number of values in the set. The median is the middle value when a data set is ordered from least to greatest. The mode is the number that occurs most often in a data set.https://cdn.kastatic.org/ka-youtube-converted/h8EYEJ32oQ8.mp4/h8EYEJ32oQ8.mp4534yes https://puzzicle.org/math/cc-sixth-grade-math/v/mean-median-and-modehttps://puzzicle.org/thumbs/380854779.jpgMean, median, & mode exampleHere we give you a set of numbers and then ask you to find the mean, median, and mode. It's your first opportunity to practice with us!https://cdn.kastatic.org/ka-youtube-converted/k3aKKasOmIw.mp4/k3aKKasOmIw.mp4235yes https://puzzicle.org/math/cc-sixth-grade-math/v/using-mean-to-find-missing-valuehttps://puzzicle.org/thumbs/x9cff9877.jpgMissing value given the meanWhen babysitting, you forget one child's age from your records. But, you remember the mean age of the six children is four. Using this mean, you can find the missing age. The total of all ages equals the mean times the number of children. So, you solve for the missing age and find it's three.https://cdn.kastatic.org/ka-youtube-converted/qpbaglogObM.mp4/qpbaglogObM.mp4303yes https://puzzicle.org/math/cc-sixth-grade-math/v/median-and-range-puzzlehttps://puzzicle.org/thumbs/x66204bfe.jpgMedian & range puzzlersIn a car dealership with 11 salespeople, the median number of cars sold is six, and the range of cars sold is four. This means the difference between the highest and lowest number of cars sold is four. Therefore, it's impossible for any salesperson to have sold more than 10 cars.https://cdn.kastatic.org/ka-youtube-converted/0cHCpgQD_8k.mp4/0cHCpgQD_8k.mp4415yes https://puzzicle.org/math/cc-sixth-grade-math/v/impact-on-median-and-mean-when-removing-lowest-value-examplehttps://puzzicle.org/thumbs/x04e04a4c.jpgImpact on median & mean: removing an outlierIn this golf game, Ana's lowest score of 80 was removed due to rule-breaking. This change increased both the mean and median of her remaining scores. However, the mean increased more than the median.https://cdn.kastatic.org/ka-youtube-converted/-2OOBEBq9-4.mp4/-2OOBEBq9-4.mp4337yes https://puzzicle.org/math/cc-sixth-grade-math/v/impact-on-median-and-mean-when-increasing-highest-valuehttps://puzzicle.org/thumbs/xbe8e5c92.jpgImpact on median & mean: increasing an outlierWhen a single high score in a group of four friends increases, the mean score also increases, but the median score stays the same. This is because the median is the middle number, which doesn't change, while the mean is the average of all scores, which does change.https://cdn.kastatic.org/ka-youtube-converted/K1y6Tz_E7oA.mp4/K1y6Tz_E7oA.mp4256yes https://puzzicle.org/math/cc-sixth-grade-math/v/mean-absolute-deviationhttps://puzzicle.org/thumbs/x159a61f7.jpgMean absolute deviation (MAD)Mean absolute deviation (MAD) of a data set is the average distance between each data value and the mean. Mean absolute deviation is a way to describe variation in a data set. Mean absolute deviation helps us get a sense of how "spread out" the values in a data set are.https://cdn.kastatic.org/ka-youtube-converted/GdIkEngwGNU.mp4/GdIkEngwGNU.mp4550yes https://puzzicle.org/math/cc-sixth-grade-math/v/mean-absolute-deviation-examplehttps://puzzicle.org/thumbs/x05e0f25b.jpgMean absolute deviation exampleThis lesson teaches how to calculate the mean and mean absolute deviation (MAD) using a bar graph. The graph shows the number of bubbles blown by four people. The mean is found by adding all the values and dividing by the number of data points. The MAD is calculated by finding the average of the absolute differences from the mean.https://cdn.kastatic.org/ka-youtube-converted/DPuK6ZgBGmE.mp4/DPuK6ZgBGmE.mp4382yes https://puzzicle.org/math/cc-sixth-grade-math/v/calculating-interquartile-range-iqrhttps://puzzicle.org/thumbs/x57fc0c82.jpgInterquartile range (IQR)The IQR describes the middle 50% of values when ordered from lowest to highest. To find the interquartile range (IQR), ​first find the median (middle value) of the lower and upper half of the data. These values are quartile 1 (Q1) and quartile 3 (Q3). The IQR is the difference between Q3 and Q1.https://cdn.kastatic.org/ka-youtube-converted/qLYYHWYr8xI.mp4/qLYYHWYr8xI.mp4372yes https://puzzicle.org/math/cc-sixth-grade-math/v/constructing-a-box-and-whisker-plothttps://puzzicle.org/thumbs/1164226773.jpgConstructing a box plotHere's a word problem that's perfectly suited for a box and whiskers plot to help analyze data. Let's construct one together, shall we?.https://cdn.kastatic.org/ka-youtube-converted/09Cx7xuIXig.mp4/09Cx7xuIXig.mp4498yes https://puzzicle.org/math/cc-sixth-grade-math/v/reading-box-and-whisker-plotshttps://puzzicle.org/thumbs/380860448.jpgReading box plotsIs this some kind of cute cat video? No! Box and whisker plots seek to explain data by showing a spread of all the data points in a sample. The "whiskers" are the two opposite ends of the data. This video is more fun than a handful of catnip.https://cdn.kastatic.org/ka-youtube-converted/b2C9I8HuCe4.mp4/b2C9I8HuCe4.mp4198yes https://puzzicle.org/math/cc-sixth-grade-math/v/box-and-whisker-plot-exercise-examplehttps://puzzicle.org/thumbs/x53757fc0.jpgWorked example: Creating a box plot (odd number of data points)A box-and-whisker plot is a handy tool for visualizing data. By ordering numbers, we can find the range, median, and quartiles. Practice makes perfect when mastering these concepts!https://cdn.kastatic.org/ka-youtube-converted/m6xURMj2ztk.mp4/m6xURMj2ztk.mp4198yes https://puzzicle.org/math/cc-sixth-grade-math/v/another-example-constructing-box-plothttps://puzzicle.org/thumbs/x3e584ad0.jpgWorked example: Creating a box plot (even number of data points)Box-and-whiskers plots help visualize data ranges and medians. First, arrange your numbers from least to greatest. The smallest and largest numbers form the 'whiskers'. The median of the entire data set splits the 'box' in the middle. The medians of the top and bottom halves of the numbers form the 'box' boundaries.https://cdn.kastatic.org/ka-youtube-converted/oajrmwCALmc.mp4/oajrmwCALmc.mp4195yes https://puzzicle.org/math/cc-sixth-grade-math/v/interpreting-box-plotshttps://puzzicle.org/thumbs/xf95ced63.jpgInterpreting box plotsA box and whisker plot is a handy tool to understand the age distribution of students at a party. It helps us identify the minimum, maximum, median, and quartiles of the data. However, it doesn't provide specific details like the exact number of students at certain ages.https://cdn.kastatic.org/ka-youtube-converted/oBREri10ZHk.mp4/oBREri10ZHk.mp4464yes https://puzzicle.org/math/cc-sixth-grade-math/v/comparing-dot-plots-histograms-and-box-plotshttps://puzzicle.org/thumbs/x473b5ffc.jpgComparing dot plots, histograms, and box plotsData can be represented in various ways such as dot plots, histograms, and box plots. Dot plots and box plots are useful for finding the median, while histograms are great for showing the number of values within a specific range.https://cdn.kastatic.org/ka-youtube-converted/s_w3EJ2Jzw0.mp4/s_w3EJ2Jzw0.mp4326yes https://puzzicle.org/math/cc-sixth-grade-math/v/shapes-of-distributionshttps://puzzicle.org/thumbs/xdf97fc58.jpgShapes of distributionsSome distributions are symmetrical, with data evenly distributed about the mean. Other distributions are "skewed," with data tending to the left or right of the mean. We sometimes say that skewed distributions have "tails."https://cdn.kastatic.org/ka-youtube-converted/2oJldeE4JcU.mp4/2oJldeE4JcU.mp4307yes https://puzzicle.org/math/cc-sixth-grade-math/v/examples-analyzing-clusters-gaps-peaks-and-outliers-for-distributionshttps://puzzicle.org/thumbs/x1d0df058.jpgClusters, gaps, peaks & outliersThis video explores the features of distributions in data sets, like clusters, gaps, and peaks. We learn how to identify outliers, which are data points far from the rest. We also discuss how to spot peaks and clusters in data.https://cdn.kastatic.org/ka-youtube-converted/4PXkzcK-b4Y.mp4/4PXkzcK-b4Y.mp4392yes https://puzzicle.org/math/6th-grade-foundations-engageny/v/comparing-with-multiplication-exercisehttps://puzzicle.org/thumbs/xfadbf044.jpgComparing with multiplicationSal solves multiplication comparison problems.https://cdn.kastatic.org/ka-youtube-converted/OCwLwaAQlMA.mp4/OCwLwaAQlMA.mp4137yes https://puzzicle.org/math/arithmetic-home/v/4-digit-times-1-digit-examplehttps://puzzicle.org/thumbs/x3c1ecd5d.jpgMultiplying 4-digit by 1-digit (regrouping)Learn to multiply a 4-digit number by a 1-digit number. In this video, we will multiply 8085 times 9.https://cdn.kastatic.org/ka-youtube-converted/twMdew4Zs8Q.mp4/twMdew4Zs8Q.mp4204yes https://puzzicle.org/math/6th-grade-foundations-engageny/v/associative-law-of-multiplicationhttps://puzzicle.org/thumbs/2600689.jpgAssociative law of multiplicationAssociative Law of Multiplicationhttps://cdn.kastatic.org/ka-youtube-converted/5RzDVNob0-0.mp4/5RzDVNob0-0.mp4135yes https://puzzicle.org/math/6th-grade-foundations-engageny/v/multiplying-a-decimal-by-a-power-of-10https://puzzicle.org/thumbs/153770145.jpgMultiplying a decimal by a power of 10Learn to multiply 0.44 times 1000 using powers of 10.https://cdn.kastatic.org/ka-youtube-converted/mV0RxDZG05A.mp4/mV0RxDZG05A.mp4227yes https://puzzicle.org/math/6th-grade-foundations-engageny/v/dividing-a-decimal-by-a-power-of-10https://puzzicle.org/thumbs/153770148.jpgDividing a decimal by a power of 10Sal talks about why moving the decimal point to the left when dividing by a power of 10 makes sense.https://cdn.kastatic.org/ka-youtube-converted/W_VhVHRal6o.mp4/W_VhVHRal6o.mp4198yes https://puzzicle.org/math/6th-grade-illustrative-math/v/identifying-bases-and-heights-of-parallelogramshttps://puzzicle.org/thumbs/xddba21b4ea9d61fa.jpgIdentifying bases and heights of parallelogramsLearn how to identify the base and height of a parallelogram by looking for the perpendicular relationship between them. We'll practice spotting the height, whether it’s drawn inside the shape or as a dotted line on the outside!https://cdn.kastatic.org/ka-youtube-converted/C3OFtj73N0E.mp4/C3OFtj73N0E.mp4212yes https://puzzicle.org/math/6th-grade-illustrative-math/v/special-polyhedra-cubehttps://puzzicle.org/thumbs/x7db86bb8ad7c5edf.jpgSpecial polyhedra: cubeExplore how the cube functions as a unique polyhedron in which every face is a square and every edge meets at a right angle.https://cdn.kastatic.org/ka-youtube-converted/klMANPlTORs.mp4/klMANPlTORs.mp4413yes https://puzzicle.org/math/6th-grade-illustrative-math/v/volume-of-a-cubehttps://puzzicle.org/thumbs/xcde3bbe65ae2bf72.jpgVolume of a cubeDiscover how to find the volume of a cube by understanding that all its dimensions—length, width, and height—are exactly the same. We’ll see how multiplying the side length by itself three times (V = s^3) allows us to count how many unit cubes fit inside.https://cdn.kastatic.org/ka-youtube-converted/6yYnr8FL1fk.mp4/6yYnr8FL1fk.mp4215yes https://puzzicle.org/math/6th-grade-illustrative-math/v/part-to-part-ratio-modelshttps://puzzicle.org/thumbs/xb37a770e8d63a842.jpgPart-to-part ratio modelsLearn how to use tape diagrams and visual models to compare two different quantities, like the ratio of blue marbles to red marbles. We’ll practice "splitting" a total into parts to see how changing one piece of the ratio affects the entire set.https://cdn.kastatic.org/ka-youtube-converted/3So5vSNtD48.mp4/3So5vSNtD48.mp4138yes https://puzzicle.org/math/cc-sixth-grade-math/v/ratios-with-tape-diagrams-partwholehttps://puzzicle.org/thumbs/x5873e07fce3ea19f.jpgRatios with tape diagrams (part:whole)Find the ratio of open-ended to multiple-choice questions. If the ratio is 5:4, it means for every 5 open-ended questions, there are 4 multiple-choice questions. The video then shows how to use this ratio to complete a table with missing values for multiple-choice and total questions.Read a part-to-whole ratio from a tape diagram showing the ratio of the two parts. Then scale up to generate sets of values that the diagram might represent.https://cdn.kastatic.org/ka-youtube-converted/FX8RIxFN7ZI.mp4/FX8RIxFN7ZI.mp4142yes https://puzzicle.org/math/cc-sixth-grade-math/v/simplify-a-ratio-from-a-tape-diagramhttps://puzzicle.org/thumbs/x4970efbb4ec23069.jpgSimplify a ratio from a tape diagramIn this problem, we have a mixture of yellow and red paint. There are 12 parts yellow and 8 parts red. We want to find the ratio of red paint to the total paint in the mixture. To do this, we add the parts of red and yellow paint together (8 + 12 = 20). The ratio of red paint to total paint is 8:20.https://cdn.kastatic.org/ka-youtube-converted/ElFVoKxX06Y.mp4/ElFVoKxX06Y.mp4161yes https://puzzicle.org/math/6th-grade-illustrative-math/v/missing-information-rate-problemshttps://puzzicle.org/thumbs/x66397d94c78f35b9.jpgMissing information: rate problemsDiscover how to identify the missing piece of information needed to solve a rate problem by organizing known quantities.https://cdn.kastatic.org/ka-youtube-converted/Lwog2qzaS20.mp4/Lwog2qzaS20.mp4236yes https://puzzicle.org/math/6th-grade-illustrative-math/v/modeling-fraction-equations-with-tape-diagrams-bike-routehttps://puzzicle.org/thumbs/x00d454b62fe39734.jpgModeling fraction equations with tape diagrams: bike routeLet's interpret tape diagrams that model fraction division.https://cdn.kastatic.org/ka-youtube-converted/aNywIvg-Q2w.mp4/aNywIvg-Q2w.mp4125yes https://puzzicle.org/math/6th-grade-illustrative-math/v/modeling-fraction-expressions-with-tape-diagrams-pasta-saladhttps://puzzicle.org/thumbs/xd8293535f69d81b0.jpgModeling fraction expressions with tape diagrams: pasta saladLet's create tape diagrams to solve fraction division problems.https://cdn.kastatic.org/ka-youtube-converted/a7RzJ7piDSc.mp4/a7RzJ7piDSc.mp4138yes https://puzzicle.org/math/6th-grade-illustrative-math/v/modeling-expressions-with-mixed-numbers-braceletshttps://puzzicle.org/thumbs/x50f43ba6e5848dc4.jpgModeling expressions with mixed numbers: braceletsLet's interpret problems and use tape diagrams to model solutions.https://cdn.kastatic.org/ka-youtube-converted/JaUu2iS-CuE.mp4/JaUu2iS-CuE.mp4165yes https://puzzicle.org/math/cc-seventh-grade-math/v/dividing-mixed-numbershttps://puzzicle.org/thumbs/2600765.jpgDividing mixed numbersSal shows us how to change mixed numbers into improper fractions, divide, and then change the answer back to a mixed number.https://cdn.kastatic.org/ka-youtube-converted/51-U-Jkq79A.mp4/51-U-Jkq79A.mp4161yes https://puzzicle.org/math/6th-grade-illustrative-math/v/using-volume-for-missing-side-length-mixed-numbershttps://puzzicle.org/thumbs/x1d0c60d9997623d9.jpgUsing volume for missing side length (mixed numbers)Let's learn how to find the missing edge length of a rectangular prism when we already know its volume and two of its dimensions. We'll use the volume formula V = lwh, plug in what we know, and solve for the unknown, getting great practice with fractions and mixed numbers along the way!https://cdn.kastatic.org/ka-youtube-converted/i-xK4mSdGUg.mp4/i-xK4mSdGUg.mp4163yes https://puzzicle.org/math/6th-grade-illustrative-math/v/solving-money-problems-with-decimalshttps://puzzicle.org/thumbs/x2aa0bf2576be3405.jpgSolving money problems with decimalsLet's tackle multi-step money problems with decimals, using a real-world shopping context to find things like total revenue and profit. Along the way, we'll practice estimating our answers to make sure our math is on the right track!https://cdn.kastatic.org/ka-youtube-converted/E64fsviSyV4.mp4/E64fsviSyV4.mp4258yes https://puzzicle.org/math/6th-grade-illustrative-math/v/relating-decimal-multiplication-to-fractions-and-place-valuehttps://puzzicle.org/thumbs/xb222eb9bd2c3cb48.jpgRelating decimal multiplication to fractions and place valueLet's explore how decimals and fractions are just different ways of writing the same numbers, and how switching between the two forms can make multiplication problems easier to solve. We'll practice rewriting expressions in multiple ways so we can choose the approach that works best for us!https://cdn.kastatic.org/ka-youtube-converted/FYN7ElO_6Y0.mp4/FYN7ElO_6Y0.mp4339yes https://puzzicle.org/math/arithmetic-home/v/partial-quotient-method-of-division-2https://puzzicle.org/thumbs/258726799.jpgPartial quotient method of division: example using very large numbersAnother example of doing long division using the partial quotient methodhttps://cdn.kastatic.org/ka-youtube-converted/omFelSZvaJc.mp4/omFelSZvaJc.mp4476yes https://puzzicle.org/math/grade-4-math-tx/v/solving-two-step-word-problems-involving-adding-and-subtracting-decimalshttps://puzzicle.org/thumbs/x889cff953f13e58f.jpgSolving two-step word problems involving adding and subtracting decimalsPractice solving two-step word problems that involve decimals. Use different strategies that either add and subtract, subtract then subtract, or subtract and then add.https://cdn.kastatic.org/ka-youtube-converted/nmYpEiz_89k.mp4/nmYpEiz_89k.mp4171yes https://puzzicle.org/math/6th-grade-illustrative-math/v/multiplying-decimals-word-problemshttps://puzzicle.org/thumbs/x3976c33197f42a0c.jpgMultiplying decimals word problemsPractice multiplying decimals in real-world scenarios.https://cdn.kastatic.org/ka-youtube-converted/D-kaGGEg9yo.mp4/D-kaGGEg9yo.mp4359yes https://puzzicle.org/math/6th-grade-illustrative-math/v/multiplying-decimals-two-step-word-problemshttps://puzzicle.org/thumbs/x537325b4c9ab73f5.jpgMultiplying decimals two-step word problemsPractice solving real world scenarios using decimals and whole numbers. Use your knowledge of multiplying whole numbers and estimation to make multiplying more friendly!https://cdn.kastatic.org/ka-youtube-converted/8Wzjh-CmDHA.mp4/8Wzjh-CmDHA.mp4323yes https://puzzicle.org/math/6th-grade-illustrative-math/v/identifying-equations-from-visual-models-tape-diagramshttps://puzzicle.org/thumbs/x73d83ffa7e56c5be.jpgIdentifying equations from visual models: tape diagramsLearn how to read tape diagrams and translate what we see into one-step equations using variables. We'll work through two examples together, practice writing the equations they represent, and then solve for the unknown to check that our answer makes sense!https://cdn.kastatic.org/ka-youtube-converted/8mrF81QZVF4.mp4/8mrF81QZVF4.mp4228yes https://puzzicle.org/math/6th-grade-illustrative-math/v/one-step-division-equationshttps://puzzicle.org/thumbs/x5f814f842b8c84e4.jpgOne-step division equationsIn this lesson, we learn how to solve simple algebraic equations systematically. We focus on understanding the meaning of equations, like 7x = 14, and using techniques like dividing both sides of the equation by the coefficient to isolate the variable. This method helps us find the value of the unknown variable, making algebra easier and more approachable.https://cdn.kastatic.org/ka-youtube-converted/WhsOJtZazS8.mp4/WhsOJtZazS8.mp4618yes https://puzzicle.org/math/cc-seventh-grade-math/v/solving-percent-problemshttps://puzzicle.org/thumbs/13.jpgSolving percent problemsWe'll use algebra to solve this percent problem.https://cdn.kastatic.org/ka-youtube-converted/TvSKeTFsaj4.mp4/TvSKeTFsaj4.mp4377yes https://puzzicle.org/math/mappers/v/solving-percent-problems-2https://puzzicle.org/thumbs/14.jpgPercent word problem: 100 is what percent of 80?This word problem is expressed simply which can be confusing to some. Take your time and focus on the what is being asked for.https://cdn.kastatic.org/ka-youtube-converted/N4kDzoQOngY.mp4/N4kDzoQOngY.mp4277yes https://puzzicle.org/math/6th-grade-illustrative-math/v/equal-and-equivalent-expressionshttps://puzzicle.org/thumbs/xc6ae6cdb925bd49f.jpgEqual and equivalent expressionsLearn how to simplify expressions and use that skill to figure out when two expressions are always, sometimes, or never equal. We'll work through examples of each case together so we can build a solid understanding of what it really means for two expressions to be equivalent!https://cdn.kastatic.org/ka-youtube-converted/orsTliP9CU8.mp4/orsTliP9CU8.mp4217yes https://puzzicle.org/math/cc-eighth-grade-math/v/powers-of-zerohttps://puzzicle.org/thumbs/xc9cc5314.jpgPowers of zeroAny non-zero number to the zero power equals one. Zero to any positive exponent equals zero. So, what happens when you have zero to the zero power?https://cdn.kastatic.org/ka-youtube-converted/S2PjhiAQMK8.mp4/S2PjhiAQMK8.mp4201yes https://puzzicle.org/math/algebra-home/v/dependent-and-independent-variables-exercise-example-3https://puzzicle.org/thumbs/xe757cceb.jpgDependent & independent variables: equationWe're flipping the last video on its head and doing the opposite. This time we give you the graph and ask you to express it as an equation.https://cdn.kastatic.org/ka-youtube-converted/3ACF7L-7Vsg.mp4/3ACF7L-7Vsg.mp4120yes https://puzzicle.org/math/6th-grade-illustrative-math/v/modeling-distance-over-timehttps://puzzicle.org/thumbs/xf1038ea7a44cb563.jpgModeling distance over timeLearn how to model the relationship between distance and time using three different representations: an equation, a table, and a graph. We'll work through each part of a real problem together and explore how all three representations connect to tell the same story about distance and time!https://cdn.kastatic.org/ka-youtube-converted/e6u9VT4d_vY.mp4/e6u9VT4d_vY.mp4273yes https://puzzicle.org/math/arithmetic-home/v/number-opposites-practicehttps://puzzicle.org/thumbs/x1914a298.jpgNumber opposites challenge problemsSal solves a few challenging problems with number opposites on the number line.https://cdn.kastatic.org/ka-youtube-converted/vRa6XxykfbY.mp4/vRa6XxykfbY.mp4188yes https://puzzicle.org/math/6th-grade-illustrative-math/v/labeling-axes-to-fit-data-setshttps://puzzicle.org/thumbs/xde4ca3351c48b071.jpgLabeling axes to fit data setsLet's learn how to set up a coordinate plane that fits data by carefully choosing the right range and scale for each axis. We'll work through a real example together and see how thinking through these decisions before we start drawing makes graphing so much easier!https://cdn.kastatic.org/ka-youtube-converted/mFN9Wh8K_uw.mp4/mFN9Wh8K_uw.mp4251yes https://puzzicle.org/math/6th-grade-illustrative-math/v/distances-in-the-coordinate-plane-between-reflected-pointshttps://puzzicle.org/thumbs/xb4be9f355b83939b.jpgDistances in the coordinate plane (between reflected points)Let's learn how to find the distance between two points on a coordinate plane when they share the same coordinates but have opposite signs. We'll discover how the axis between the points is always the midpoint, which means we can find the total distance by doubling each point's distance from that axis!https://cdn.kastatic.org/ka-youtube-converted/-GI96_8q2OE.mp4/-GI96_8q2OE.mp4330yes https://puzzicle.org/math/6th-grade-illustrative-math/v/categorical-data-setshttps://puzzicle.org/thumbs/x36d06ea50e839cf7.jpgCategorical data setsLet's explore how data is all around us and learn what makes a data set categorical, including some sneaky examples where numbers are used as labels rather than actual quantities. We'll look at real-world examples together to build our understanding of how to tell the difference between categorical and numerical data!https://cdn.kastatic.org/ka-youtube-converted/Moi2PbI4b48.mp4/Moi2PbI4b48.mp4224yes https://puzzicle.org/math/6th-grade-illustrative-math/v/numerical-data-setshttps://puzzicle.org/thumbs/xd3cde909f336f60c.jpgNumerical data setsexplore how data is all around us and learn what makes a data set numerical, where the values represent actual quantities we can measure or count. We'll work through real-world examples together to deepen our understanding of numerical data and how it differs from categorical data!https://cdn.kastatic.org/ka-youtube-converted/ktaTECHDVy0.mp4/ktaTECHDVy0.mp4204yes https://puzzicle.org/math/6th-grade-illustrative-math/v/meanhttps://puzzicle.org/thumbs/x4298fd56d3b8781c.jpgMeanLet's learn how to calculate the mean by adding up all the values in a data set and dividing by the number of values. We'll also explore how to interpret the mean as a balance point that represents the entire data set with just one number!https://cdn.kastatic.org/ka-youtube-converted/_40su2dEvA4.mp4/_40su2dEvA4.mp4204yes https://puzzicle.org/math/6th-grade-illustrative-math/v/interpreting-meanhttps://puzzicle.org/thumbs/x2e3856042fe75bdc.jpgInterpreting meanLet's build an intuitive understanding of what the mean actually tells us about a data set, no calculations needed! By looking at visual representations and real-world contexts, we'll explore how the mean helps us describe the typical value in a data set and what we can learn by comparing the means of two different groups.https://cdn.kastatic.org/ka-youtube-converted/1DuU8GbTPgk.mp4/1DuU8GbTPgk.mp4165yes https://puzzicle.org/math/6th-grade-illustrative-math/v/interpreting-variabilityhttps://puzzicle.org/thumbs/xd9ff51b28b917565.jpgInterpreting variabilityLet's build an intuitive understanding of what variability tells us about a data set, without doing any MAD calculations! By exploring visual representations, we'll learn how to interpret how spread-out data is around its center and what that reveals about consistency in real-world contexts.https://cdn.kastatic.org/ka-youtube-converted/MlrzY0u_aOg.mp4/MlrzY0u_aOg.mp4188yes https://puzzicle.org/math/6th-grade-illustrative-math/v/medianhttps://puzzicle.org/thumbs/x6d2f767ce8fd49dd.jpgMedianLet's learn what the median is, why we use it as a measure of center, and how to find it by ordering our data and locating the middle value. We'll also explore how the median is especially useful when our data has an outlier that would otherwise throw off our sense of what's typical!https://cdn.kastatic.org/ka-youtube-converted/jQzoJJRgvcI.mp4/jQzoJJRgvcI.mp4253yes https://puzzicle.org/math/cc-sixth-grade-math/v/interpreting-unequal-ratioshttps://puzzicle.org/thumbs/xf891c0d1d760b343.jpgInterpreting unequal ratiosInterpret the ratio of pepper to fish in two recipes to decide which tastes spicier. Akeem's recipe uses 19 grams of ground pepper for every four catfish fillets, while Tisha uses 25 grams for six fillets. To compare their pepper taste, we can find a common number of fillets. For 12 fillets, Akeem uses 57 grams of pepper, and Tisha uses 50 grams. Therefore, Akeem's catfish fillets have a stronger pepper taste.https://cdn.kastatic.org/ka-youtube-converted/u6TNiTf2zhQ.mp4/u6TNiTf2zhQ.mp4191yes https://puzzicle.org/math/6th-grade-math-eureka-squared-aligned/v/euclidean-algorithm-to-find-gcfhttps://puzzicle.org/thumbs/x49dd9ce0615c1aa1.jpgEuclidean algorithm to find GCFLearn the subtraction Euclidean Algorithm for finding the GCF: keep subtracting the smaller number from the bigger one. When the two numbers match, that number is the GCF.https://cdn.kastatic.org/ka-youtube-converted/OwND6sh6FdU.mp4/OwND6sh6FdU.mp4197yes https://puzzicle.org/math/arithmetic-home/v/negative-numbers-introductionhttps://puzzicle.org/thumbs/19647488.jpgIntro to negative numbersMysterious negative numbers! What ARE they? They are numbers less than zero. If you understand the nature of below zero temperatures, you can understand negative numbers. We'll help.https://cdn.kastatic.org/ka-youtube-converted/Hlal9ME2Aig.mp4/Hlal9ME2Aig.mp4576yes https://puzzicle.org/math/6th-grade-math-eureka-squared-aligned/v/constructing-the-coordinate-planehttps://puzzicle.org/thumbs/xe71d85adcf19eb99.jpgConstructing the coordinate planeLearn to build a coordinate plane by choosing the minimum and maximum values and a clear scale for each axis.https://cdn.kastatic.org/ka-youtube-converted/u2Vdc45X5Zw.mp4/u2Vdc45X5Zw.mp4426yes https://puzzicle.org/math/cc-sixth-grade-math/v/order-operations-example-fractions-exponentshttps://puzzicle.org/thumbs/x41b3fbc71efc518a.jpgOrder of operations example: fractions and exponentsThe order of operations is essential for accurately evaluating complex math expressions. By following the steps of parentheses, exponents, multiplication and division, and finally addition and subtraction, you can simplify expressions and find the correct answer. Mastering these steps ensures a strong foundation in mathematics.https://cdn.kastatic.org/ka-youtube-converted/jAVbApE7lJ8.mp4/jAVbApE7lJ8.mp4237yes https://puzzicle.org/math/6th-grade-math-eureka-squared-aligned/v/volume-of-composite-solidshttps://puzzicle.org/thumbs/x967c577e9efb1af7.jpgVolume of composite solidsFind volume by decomposing figures into rectangular prisms.https://cdn.kastatic.org/ka-youtube-converted/epUOumtNaws.mp4/epUOumtNaws.mp4182yes https://puzzicle.org/math/cc-seventh-grade-math/v/introduction-to-proportional-relationshipshttps://puzzicle.org/thumbs/x042dc5b1.jpgIntro to proportional relationshipsTo know if a relationship is proportional, you should look at the ratios between the two variables. If the ratio is always the same, the relationship is proportional. If the ratio changes, the relationship is not proportional.https://cdn.kastatic.org/ka-youtube-converted/qYjiVWwefto.mp4/qYjiVWwefto.mp4230yes https://puzzicle.org/math/cc-seventh-grade-math/v/analyzing-and-identifying-proportional-relationships-ex3https://puzzicle.org/thumbs/xcf658b5a.jpgProportional relationships: spaghettiGiven a table of ratios, watch as we test them for equivalence and determine whether the relationship is proportional.https://cdn.kastatic.org/ka-youtube-converted/jNUz0P5MG9M.mp4/jNUz0P5MG9M.mp463yes https://puzzicle.org/math/cc-seventh-grade-math/v/movie-ticket-proportionalityhttps://puzzicle.org/thumbs/xaef37673.jpgProportional relationships: movie ticketsWhen going to the movies, is the price you pay proportional to the number of tickets you buy?https://cdn.kastatic.org/ka-youtube-converted/-YYTUTrxl28.mp4/-YYTUTrxl28.mp4106yes https://puzzicle.org/math/cc-seventh-grade-math/v/banana-proportionalityhttps://puzzicle.org/thumbs/x48c1f191.jpgProportional relationships: bananasA proportionality problem about eating bananas.https://cdn.kastatic.org/ka-youtube-converted/bJb4n19sFrs.mp4/bJb4n19sFrs.mp4177yes https://puzzicle.org/math/cc-seventh-grade-math/v/identifying-proportional-relationships-visuallyhttps://puzzicle.org/thumbs/x919ece65.jpgProportional relationships: graphsLearn how to tell proportional relationships by drawing graphs. Key idea: the graph of a proportional relationship is a straight line through the origin.https://cdn.kastatic.org/ka-youtube-converted/C7CY4l99S7M.mp4/C7CY4l99S7M.mp4439yes https://puzzicle.org/math/cc-seventh-grade-math/v/proportional-relationships-examplehttps://puzzicle.org/thumbs/x950881e0e1b45d92.jpgProportional relationshipsSal determines which ratios are proportionate to a ratio provided in a context.https://cdn.kastatic.org/ka-youtube-converted/HEmwn5jmy2Y.mp4/HEmwn5jmy2Y.mp4243yes https://puzzicle.org/math/cc-seventh-grade-math/v/proportionality-between-side-length-and-perimeterhttps://puzzicle.org/thumbs/x97c4daf2.jpgIs side length & perimeter proportional?Sal answers the question by drawing a square and thinking about the relationship between side length and perimeter.https://cdn.kastatic.org/ka-youtube-converted/9nH3A4riVxs.mp4/9nH3A4riVxs.mp4159yes https://puzzicle.org/math/cc-seventh-grade-math/v/testing-whether-area-is-proportional-to-side-lengthhttps://puzzicle.org/thumbs/x712a89dd.jpgIs side length & area proportional?Sal answers the question by drawing a square and thinking about the relationship between side length and area.https://cdn.kastatic.org/ka-youtube-converted/Hzz5p0qlObs.mp4/Hzz5p0qlObs.mp4226yes https://puzzicle.org/math/cc-seventh-grade-math/v/constant-of-proportionality-from-equationhttps://puzzicle.org/thumbs/x36bcce972ab1db2d.jpgConstant of proportionality from equationSal identifies the constant of proportionality from equations.https://cdn.kastatic.org/ka-youtube-converted/wXtzBbU_cl8.mp4/wXtzBbU_cl8.mp4143yes https://puzzicle.org/math/cc-seventh-grade-math/v/identifying-constant-of-proportionality-graphicallyhttps://puzzicle.org/thumbs/x10d4fd17bad36cd0.jpgIdentifying constant of proportionality graphicallyThe video explains the concept of the constant of proportionality in graphs. It shows how to calculate it using the formula Y = KX or Y/X = K, where K is the constant. It also demonstrates how to identify lines with specific constants on a graph.https://cdn.kastatic.org/ka-youtube-converted/jihKUWH4Hfc.mp4/jihKUWH4Hfc.mp4277yes https://puzzicle.org/math/cc-seventh-grade-math/v/proportionality-constant-from-tablehttps://puzzicle.org/thumbs/x25cab8c41c3a0b83.jpgConstant of proportionality from table (with equations)Sal identifies the constant of proportionality from table.https://cdn.kastatic.org/ka-youtube-converted/5Ilo0pw3iL8.mp4/5Ilo0pw3iL8.mp4105yes https://puzzicle.org/math/cc-seventh-grade-math/v/comparing-proportionality-constantshttps://puzzicle.org/thumbs/x632f1595cd506367.jpgComparing proportionality constantsSal compares constants of proportionality in various forms, such as graphs, equations, contexts, diagrams, and tables.https://cdn.kastatic.org/ka-youtube-converted/4g5ZEGoYtOA.mp4/4g5ZEGoYtOA.mp4333yes https://puzzicle.org/math/cc-seventh-grade-math/v/interpret-proportionality-constantshttps://puzzicle.org/thumbs/xe6bdf10fa34e4b6c.jpgInterpret proportionality constantsSal interpret what the constant of proportionality means in a context.https://cdn.kastatic.org/ka-youtube-converted/FXpJQXzoXwI.mp4/FXpJQXzoXwI.mp4200yes https://puzzicle.org/math/cc-seventh-grade-math/v/find-an-unknown-in-a-proportionhttps://puzzicle.org/thumbs/153770146.jpgWorked example: Solving proportionsLearn the reasoning behind solving proportions. We'll put some algebra to work to get our answers, too.https://cdn.kastatic.org/ka-youtube-converted/GO5ajwbFqVQ.mp4/GO5ajwbFqVQ.mp4440yes https://puzzicle.org/math/cc-seventh-grade-math/v/writing-proportionshttps://puzzicle.org/thumbs/211185145.jpgWriting proportions exampleSome examples of writing two ratios and setting them equal to each other to solve proportion word problems.https://cdn.kastatic.org/ka-youtube-converted/96ZEmUbnuU8.mp4/96ZEmUbnuU8.mp4351yes https://puzzicle.org/math/cc-seventh-grade-math/v/find-an-unknown-in-a-proportion-2https://puzzicle.org/thumbs/153770142.jpgProportion word problem: cookiesA recipe for oatmeal cookies calls for 2 cups of flour for every 3 cups of oatmeal. How much flour is needed for a big batch of cookies that uses 9 cups of oatmeal?https://cdn.kastatic.org/ka-youtube-converted/4ywTWCaLmXE.mp4/4ywTWCaLmXE.mp4348yes https://puzzicle.org/math/cc-seventh-grade-math/v/using-proportion-to-solve-for-variablehttps://puzzicle.org/thumbs/x648638d3.jpgProportion word problem: hot dogsMika can eat 21 hot dogs in 66 minutes. She wants to know how many minutes it would take her to eat 35 hot dogs if she can keep up the same pace.https://cdn.kastatic.org/ka-youtube-converted/s8_14yxp1lQ.mp4/s8_14yxp1lQ.mp4298yes https://puzzicle.org/math/cc-seventh-grade-math/v/equations-of-proportional-relationshipshttps://puzzicle.org/thumbs/x7d23b788.jpgEquations for proportional relationshipsLearn how to write a proportional equation y=kx where k is the so-called "constant of proportionality".https://cdn.kastatic.org/ka-youtube-converted/qcz1Cm_-l50.mp4/qcz1Cm_-l50.mp4301yes https://puzzicle.org/math/cc-seventh-grade-math/v/constructing-an-equation-for-a-proportional-relationshiphttps://puzzicle.org/thumbs/x20c99318.jpgWriting proportional equations from tablesWriting an equation to describe the relationship between the number of scoops in an ice cream cone and the pricehttps://cdn.kastatic.org/ka-youtube-converted/dBGeri4eFv4.mp4/dBGeri4eFv4.mp4167yes https://puzzicle.org/math/7th-engage-ny/v/examples-interpreting-graphs-of-proportional-relationshipshttps://puzzicle.org/thumbs/x6dbbed4b.jpgInterpreting graphs of proportional relationshipsLearn how to read and understand graphs of proportional relationshipshttps://cdn.kastatic.org/ka-youtube-converted/muZmOiiukQE.mp4/muZmOiiukQE.mp4452yes https://puzzicle.org/math/cc-seventh-grade-math/v/determining-rates-with-fractionshttps://puzzicle.org/thumbs/x4861d7fa.jpgRates with fractionsSal solves an example problem with rates with fractions.https://cdn.kastatic.org/ka-youtube-converted/2DBBKArGfus.mp4/2DBBKArGfus.mp4307yes https://puzzicle.org/math/cc-seventh-grade-math/v/exploring-scale-copieshttps://puzzicle.org/thumbs/x76b3e7a99b5e6187.jpgExploring scale copiesSal uses virtual manipulatives to identify figures that scale proportionally.https://cdn.kastatic.org/ka-youtube-converted/z2E_GpucfeA.mp4/z2E_GpucfeA.mp4112yes https://puzzicle.org/math/cc-seventh-grade-math/v/corresponding-points-and-sides-of-scaled-shapeshttps://puzzicle.org/thumbs/xc50cc9fa6fa67a46.jpgCorresponding points and sides of scaled shapesSal identifies corresponding sides and corresponding points in scale copies of figures.https://cdn.kastatic.org/ka-youtube-converted/FeogTR_omiM.mp4/FeogTR_omiM.mp4175yes https://puzzicle.org/math/cc-seventh-grade-math/v/identifying-scaled-copieshttps://puzzicle.org/thumbs/x575ccd7ff0de3724.jpgIdentifying scale copiesSal looks at side measures on figures to determine if they are scale copies.https://cdn.kastatic.org/ka-youtube-converted/QBiTspq2Wck.mp4/QBiTspq2Wck.mp4244yes https://puzzicle.org/math/7th-engage-ny/v/identifying-scale-factorshttps://puzzicle.org/thumbs/x75f35bd8098b3fe6.jpgIdentifying scale factorsSal looks at a figure and a scale copy of the figure to determine what scale factor was used to create the scale copy.https://cdn.kastatic.org/ka-youtube-converted/Z3SXp8jDbdU.mp4/Z3SXp8jDbdU.mp4175yes https://puzzicle.org/math/cc-seventh-grade-math/v/identifying-values-in-scale-drawingshttps://puzzicle.org/thumbs/xa0f1defc3a6e9244.jpgIdentifying values in scale copiesSal identifies missing side lengths in scale drawings.https://cdn.kastatic.org/ka-youtube-converted/YCjc3uRTq7I.mp4/YCjc3uRTq7I.mp4229yes https://puzzicle.org/math/cc-seventh-grade-math/v/ratio-word-problem-exercise-example-2https://puzzicle.org/thumbs/x8db3094d.jpgScale drawing: centimeters to kilometersSal uses ratios to figure out how far apart two cities should be on a map.https://cdn.kastatic.org/ka-youtube-converted/tOd2T72eJME.mp4/tOd2T72eJME.mp472yes https://puzzicle.org/math/cc-seventh-grade-math/v/constructing-scale-drawingshttps://puzzicle.org/thumbs/xb773c610.jpgMaking a scale drawingAn urban planner needs your help in creating a scale drawing. Let's use our knowledge about scale factor, length, and area to assist.https://cdn.kastatic.org/ka-youtube-converted/f-gCYOCFMYc.mp4/f-gCYOCFMYc.mp4198yes https://puzzicle.org/math/cc-seventh-grade-math/v/scale-drawings-examplehttps://puzzicle.org/thumbs/x20cf89b2.jpgInterpreting a scale drawingUnderstand how a scale drawing is converted into real numbers using the scale factor.https://cdn.kastatic.org/ka-youtube-converted/byjmR7JBXKc.mp4/byjmR7JBXKc.mp4266yes https://puzzicle.org/math/cc-seventh-grade-math/v/scale-drawing-example-2https://puzzicle.org/thumbs/x6b901e6c.jpgSolving a scale drawing word problemSee how we solve a word problem by using a scale drawing and finding the scale factor.https://cdn.kastatic.org/ka-youtube-converted/EgOyJJxuuP8.mp4/EgOyJJxuuP8.mp4351yes https://puzzicle.org/math/cc-seventh-grade-math/v/zero-pairs-worked-examplehttps://puzzicle.org/thumbs/x324537f1206956c8.jpgZero pairs worked exampleWhen we have a positive unit and a negative unit, they add up to zero. We call that a zero pair. Let's use integer chips to represent the number -2, but not just with 2 negative chips. This will open the door for us to add and subtract with negative numbers.https://cdn.kastatic.org/ka-youtube-converted/hwK33GvGIK4.mp4/hwK33GvGIK4.mp4127yes https://puzzicle.org/math/cc-seventh-grade-math/v/adding-with-integer-chipshttps://puzzicle.org/thumbs/x83da0698c825f538.jpgAdding with integer chipsLet's model and evaluate sums using integer chip diagrams. When all the integer chips are negative, we can just count up our total negatives. When we have a positive integer chip and negative integer chip, those pair up to equal zero. So our sum is whatever chips are left over.https://cdn.kastatic.org/ka-youtube-converted/BqfDKV0gPRE.mp4/BqfDKV0gPRE.mp4226yes https://puzzicle.org/math/cc-seventh-grade-math/v/adding-negative-numbers-on-number-line-exampleshttps://puzzicle.org/thumbs/x13e7e4f6.jpgAdding negative numbers on the number lineSal shows how to model 6 + (-2) using vertical number lines.https://cdn.kastatic.org/ka-youtube-converted/3CKpidALDEg.mp4/3CKpidALDEg.mp4325yes https://puzzicle.org/math/arithmetic-home/v/adding-negative-numbershttps://puzzicle.org/thumbs/153770154.jpgAdding negative numbers exampleUse a number line to add -15 + (-46) + (-29)https://cdn.kastatic.org/ka-youtube-converted/AcpIO62x2oo.mp4/AcpIO62x2oo.mp4341yes https://puzzicle.org/math/cc-seventh-grade-math/v/signs-of-sums-on-a-number-linehttps://puzzicle.org/thumbs/xaa50e176d0198ec1.jpgSigns of sums on a number lineLet's build intuition for the sign of sum involving negative and positive numbers. We'll decide whether each sum ends to the left or right of 0. Then we'll evaluate exactly to see whether the intuition worked out.https://cdn.kastatic.org/ka-youtube-converted/VBEQgXLhBzU.mp4/VBEQgXLhBzU.mp4444yes https://puzzicle.org/math/cc-seventh-grade-math/v/subtracting-with-integer-chipshttps://puzzicle.org/thumbs/xe9d6449e2ebe97f4.jpgSubtracting with integer chipsLet's model subtraction expressions using positive and negative integer chips. To subtract, we'll take away the appropriate number of chips. What if we don't have enough chips to take away? We can add as many zero pairs as we need, because they don't change the value of the expression.https://cdn.kastatic.org/ka-youtube-converted/yTqhzgpt_9Y.mp4/yTqhzgpt_9Y.mp4330yes https://puzzicle.org/math/cc-seventh-grade-math/v/adding-the-opposite-with-integer-chipshttps://puzzicle.org/thumbs/xeb730da455448485.jpgAdding the opposite with integer chipsUse integer chips to explore the conjecture that subtracting a number gives the same result as adding the opposite of the number.https://cdn.kastatic.org/ka-youtube-converted/0O4hGprr71g.mp4/0O4hGprr71g.mp4311yes https://puzzicle.org/math/cc-seventh-grade-math/v/adding-the-opposite-with-number-lineshttps://puzzicle.org/thumbs/x57519ddea4bca03b.jpgAdding the opposite with number linesExplore the claim that subtracting a number is the same as adding its opposite using number lines. We move to the right on the number line when we add a positive number. We move to the left on the number line both when we subtract a positive number and when we add a negative number. So which direction do we draw the arrow when we subtract a negative number on the number line?https://cdn.kastatic.org/ka-youtube-converted/S5oBJ9aL9QQ.mp4/S5oBJ9aL9QQ.mp4297yes https://puzzicle.org/math/algebra-basics/v/adding-and-subtracting-negative-number-exampleshttps://puzzicle.org/thumbs/xded1ea52.jpgAdding & subtracting negative numbersLearn how to add and subtract negative numbers. The problems solved in this video are 2 - 3 = -1 and -2 - 3 = -5 and -2 + 3 = 1 and 2 - (-3) = 5 and -2 - (-3).https://cdn.kastatic.org/ka-youtube-converted/NQSN00zL5gg.mp4/NQSN00zL5gg.mp4247yes https://puzzicle.org/math/cc-seventh-grade-math/v/why-subtracting-a-negative-equivalent-to-adding-a-positivehttps://puzzicle.org/thumbs/1244788705.jpgSubtracting a negative = adding a positiveFind out why subtracting a negative number is the same as adding the absolute value of that number.https://cdn.kastatic.org/ka-youtube-converted/lBscLuttQq0.mp4/lBscLuttQq0.mp493yes https://puzzicle.org/math/cc-seventh-grade-math/v/equivalent-expressions-with-negative-rationalshttps://puzzicle.org/thumbs/xa2ed7747d165fc27.jpgEquivalent expressions with negative numbersUnderstand that we can commute and associate terms as long as we keep the signs with the terms. We can also apply the additive identity property and the additive inverse property.https://cdn.kastatic.org/ka-youtube-converted/Shvr_SgLPJo.mp4/Shvr_SgLPJo.mp4613yes https://puzzicle.org/math/cc-seventh-grade-math/v/integer-equations-to-describe-diagramhttps://puzzicle.org/thumbs/xb8d99f61.jpgNumber equations & number linesInteger equations to describe diagramhttps://cdn.kastatic.org/ka-youtube-converted/Ddvw2wEBfpc.mp4/Ddvw2wEBfpc.mp4163yes https://puzzicle.org/math/cc-seventh-grade-math/v/graphing-negative-number-addition-and-subtraction-expressionshttps://puzzicle.org/thumbs/x3cfb7e6fd1638784.jpgGraphing negative number addition and subtraction expressionsGraph addition and subtraction expressions involving negative numbers on the number line. When we add a positive number, we move to the right on the number line. When we add a negative number, we move to the left. When we subtract a number, we move in the opposite direction as we would to add the same number.https://cdn.kastatic.org/ka-youtube-converted/LwUMnbV303k.mp4/LwUMnbV303k.mp4234yes https://puzzicle.org/math/cc-seventh-grade-math/v/interpreting-integer-expressionshttps://puzzicle.org/thumbs/x79fbd639.jpgInterpreting numeric expressions exampleSal chooses the correct interpretation of the integer expression 3 + (-7).https://cdn.kastatic.org/ka-youtube-converted/ahib7LGDysQ.mp4/ahib7LGDysQ.mp4146yes https://puzzicle.org/math/cc-seventh-grade-math/v/absolute-value-as-distance-between-numbershttps://puzzicle.org/thumbs/x2149e09f.jpgAbsolute value as distance between numbersIn this video, we think about what |a-b| really means, and we verify that |a-b| = |b-a| by looking at an example.https://cdn.kastatic.org/ka-youtube-converted/t4xOkpP8FgE.mp4/t4xOkpP8FgE.mp4330yes https://puzzicle.org/math/cc-seventh-grade-math/v/interpreting-absolute-value-as-distancehttps://puzzicle.org/thumbs/x2357089f.jpgInterpreting absolute value as distanceIn this video, we work through a bunch of examples that stretch our thinking on absolute value.https://cdn.kastatic.org/ka-youtube-converted/K759mIqpvOU.mp4/K759mIqpvOU.mp4654yes https://puzzicle.org/math/cc-seventh-grade-math/v/associative-commutative-with-negativeshttps://puzzicle.org/thumbs/x7fab7267219829a6.jpgAssociative and commutative properties of addition with negativesThe commutative and associative properties apply to addition, but not to subtraction. Rewriting subtraction as addition of the opposite is a superpower that suddenly lets us use the commutative and associative properties (without changing the value of the expression) in expressions that previously had subtraction involved.https://cdn.kastatic.org/ka-youtube-converted/uHWDJHerbGQ.mp4/uHWDJHerbGQ.mp4298yes https://puzzicle.org/math/cc-seventh-grade-math/v/adding-fractions-with-different-signshttps://puzzicle.org/thumbs/13179314.jpgAdding fractions with different signsUse a number line to add fractions with different signs.https://cdn.kastatic.org/ka-youtube-converted/QS1LMomm0Gk.mp4/QS1LMomm0Gk.mp4382yes https://puzzicle.org/math/cc-seventh-grade-math/v/adding-and-subtracting-fractions-with-negativeshttps://puzzicle.org/thumbs/x4eac6603b14fea42.jpgAdding and subtracting fractions with negativesUse the properties of addition to strategically add and subtract fractions with negatives and unlike denominators. We can rewrite the subtraction as adding the opposite so that the addition properties will apply.https://cdn.kastatic.org/ka-youtube-converted/nhJ60MCLeCU.mp4/nhJ60MCLeCU.mp4225yes https://puzzicle.org/math/cc-seventh-grade-math/v/comparing-rational-numbershttps://puzzicle.org/thumbs/158922111.jpgComparing rational numbersComparing Rational Numbershttps://cdn.kastatic.org/ka-youtube-converted/VZOHWaw5dqM.mp4/VZOHWaw5dqM.mp4419yes https://puzzicle.org/math/cc-seventh-grade-math/v/adding-and-subtracting-a-decimal-percentage-and-fractionhttps://puzzicle.org/thumbs/x6f64c4e8.jpgAdding & subtracting rational numbers: 79% - 79.1 - 58 1/10Sal evaluates 79% - 79.1 - 58 1/10.https://cdn.kastatic.org/ka-youtube-converted/FSpSrZQzzVs.mp4/FSpSrZQzzVs.mp4269yes https://puzzicle.org/math/cc-seventh-grade-math/v/adding-fractions-decimals-and-percentageshttps://puzzicle.org/thumbs/x45f7c965.jpgAdding & subtracting rational numbers: 0.79 - 4/3 - 1/2 + 150%Sal evaluates 0.79 - 4/3 - 1/2 + 150%.https://cdn.kastatic.org/ka-youtube-converted/xEDnwEOOf7Y.mp4/xEDnwEOOf7Y.mp4274yes https://puzzicle.org/math/cc-seventh-grade-math/v/why-a-negative-times-a-negative-is-a-positivehttps://puzzicle.org/thumbs/867458280.jpgWhy a negative times a negative is a positiveUse the distributive property to understand the products of negative numbers.https://cdn.kastatic.org/ka-youtube-converted/rK4sXm_MPWo.mp4/rK4sXm_MPWo.mp4347yes https://puzzicle.org/math/cc-seventh-grade-math/v/why-a-negative-times-a-negative-makes-intuitive-sensehttps://puzzicle.org/thumbs/1244872792.jpgWhy a negative times a negative makes senseUse the repeated addition model of multiplication to give an understanding of multiplying negative numbers.https://cdn.kastatic.org/ka-youtube-converted/pzQY-9Nmtws.mp4/pzQY-9Nmtws.mp4374yes https://puzzicle.org/math/cc-seventh-grade-math/v/multiplying-positive-and-negative-numbershttps://puzzicle.org/thumbs/862036603.jpgMultiplying positive & negative numbersLearn some rules of thumb for multiplying positive and negative numbers.https://cdn.kastatic.org/ka-youtube-converted/47wjId9k2Hs.mp4/47wjId9k2Hs.mp4334yes https://puzzicle.org/math/cc-seventh-grade-math/v/dividing-positive-and-negative-numbershttps://puzzicle.org/thumbs/862107118.jpgDividing positive and negative numbersDiscover the basics of dividing with negative numbers.https://cdn.kastatic.org/ka-youtube-converted/bQ-KR3clFgs.mp4/bQ-KR3clFgs.mp4292yes https://puzzicle.org/math/cc-seventh-grade-math/v/converting-fractions-to-decimals-ex2https://puzzicle.org/thumbs/915095642.jpgFraction to decimal: 11/25Sal writes 11/25 as a decimal.https://cdn.kastatic.org/ka-youtube-converted/sCVyvfOLI6U.mp4/sCVyvfOLI6U.mp493yes https://puzzicle.org/math/cc-seventh-grade-math/v/converting-fractions-to-decimals-examplehttps://puzzicle.org/thumbs/158925085.jpgWorked example: Converting a fraction (7/8) to a decimalLearn how to write the fraction, 7/8 as a decimal.https://cdn.kastatic.org/ka-youtube-converted/Y1V5mZaMfTk.mp4/Y1V5mZaMfTk.mp4137yes https://puzzicle.org/math/cc-seventh-grade-math/v/converting-fractions-to-decimals-ex1https://puzzicle.org/thumbs/915084910.jpgFraction to decimal with roundingSal writes 16/21 as a rounded decimal.https://cdn.kastatic.org/ka-youtube-converted/NM8qTo361ic.mp4/NM8qTo361ic.mp4195yes https://puzzicle.org/math/cc-seventh-grade-math/v/multiplying-negative-and-positive-fractionshttps://puzzicle.org/thumbs/x747283e7.jpgMultiplying positive and negative fractionsSee examples of multiplying and dividing fractions with negative numbers.https://cdn.kastatic.org/ka-youtube-converted/pi3WWQ0q6Lc.mp4/pi3WWQ0q6Lc.mp4273yes https://puzzicle.org/math/cc-seventh-grade-math/v/examples-of-dividing-negative-fractionshttps://puzzicle.org/thumbs/x91b6f5a7.jpgDividing negative fractionsDividing by a fraction is the same as multiplying by its reciprocal. To multiply fractions, we multiply the numerators together and the denominators together.https://cdn.kastatic.org/ka-youtube-converted/H0q9Fqb8YT4.mp4/H0q9Fqb8YT4.mp4251yes https://puzzicle.org/math/cc-seventh-grade-math/v/negative-signs-in-numerators-and-denominatorshttps://puzzicle.org/thumbs/x6e35dcba.jpgNegative signs in fractionsSal finds equivalent expressions to -g/h.https://cdn.kastatic.org/ka-youtube-converted/9eSPhvhuInw.mp4/9eSPhvhuInw.mp4308yes https://puzzicle.org/math/arithmetic-home/v/making-sense-of-hairy-fractionshttps://puzzicle.org/thumbs/x372e1287.jpgSimplifying complex fractionsLearn how to take complex looking fractions and make them much, much simpler.https://cdn.kastatic.org/ka-youtube-converted/K2b8iMPY11I.mp4/K2b8iMPY11I.mp4319yes https://puzzicle.org/math/cc-seventh-grade-math/v/rational-number-expressionshttps://puzzicle.org/thumbs/xfbb8940b.jpgExpressions with rational numbersLearn to compare expressions with positive and negative fractions.https://cdn.kastatic.org/ka-youtube-converted/a_Wi-6SRBTc.mp4/a_Wi-6SRBTc.mp4359yes https://puzzicle.org/math/cc-seventh-grade-math/v/arithmetic-properties-with-negative-numbershttps://puzzicle.org/thumbs/xc98aaf32.jpgEquivalent expressions with negative numbers (multiplication and division)In this video, we figure out whether or not some expressions are equivalent. To do this, we think about properties of multiplication and division.https://cdn.kastatic.org/ka-youtube-converted/hmtJV49AWio.mp4/hmtJV49AWio.mp4502yes https://puzzicle.org/math/algebra/v/why-dividing-by-zero-is-undefinedhttps://puzzicle.org/thumbs/871510490.jpgWhy dividing by zero is undefinedAs much as we would like to have an answer for "what's 1 divided by 0?" it's sadly impossible to have an answer. The reason, in short, is that whatever we may answer, we will then have to agree that that answer times 0 equals to 1, and that cannot be ​true, because anything times 0 is 0.https://cdn.kastatic.org/ka-youtube-converted/SQzjzStU1RQ.mp4/SQzjzStU1RQ.mp4248yes https://puzzicle.org/math/cc-seventh-grade-math/v/order-of-operations-1https://puzzicle.org/thumbs/13179327.jpgOrder of operations exampleSimplify this tricky expression using the order of operations. Expression include negative numbers and exponents.https://cdn.kastatic.org/ka-youtube-converted/gjrGd9TjjnY.mp4/gjrGd9TjjnY.mp4276yes https://puzzicle.org/math/cc-seventh-grade-math/v/negative-number-word-problems-1https://puzzicle.org/thumbs/xefd871cb.jpgNegative number word problem: temperaturesUse a number line to solve a word problem that includes a negative number.https://cdn.kastatic.org/ka-youtube-converted/Oo2vGhVkvDo.mp4/Oo2vGhVkvDo.mp4262yes https://puzzicle.org/math/cc-seventh-grade-math/v/finding-initial-temperature-from-temperature-changeshttps://puzzicle.org/thumbs/x2ae483b7.jpgNegative number word problem: AlaskaIn this video, we figure out the temperature in Fairbanks, Alaska by adding and subtracting negative numbers.https://cdn.kastatic.org/ka-youtube-converted/t4jDcc8IIBM.mp4/t4jDcc8IIBM.mp4218yes https://puzzicle.org/math/cc-seventh-grade-math/v/interpreting-negative-number-statementshttps://puzzicle.org/thumbs/xbfc0f373.jpgInterpreting negative number statementsThe examples in this video use negative numbers to represent real world situations. What are these negative numbers really telling us?https://cdn.kastatic.org/ka-youtube-converted/ioH0PUdUW3M.mp4/ioH0PUdUW3M.mp4257yes https://puzzicle.org/math/cc-seventh-grade-math/v/interpreting-multiplication-and-division-of-negative-numbershttps://puzzicle.org/thumbs/x630b911f.jpgInterpreting multiplication & division of negative numbersLet's think about the meaning of negative numbers in different situations.https://cdn.kastatic.org/ka-youtube-converted/mPLCXCscqI8.mp4/mPLCXCscqI8.mp4275yes https://puzzicle.org/math/cc-seventh-grade-math/v/adding-integers-find-the-missing-valuehttps://puzzicle.org/thumbs/x0b4107d070724b99.jpgAdding integers: find the missing valueUse number lines to find missing addends in addition equations with integers. When we add a positive value, we move to the right on the number line. When we add a negative value, we move to the left.https://cdn.kastatic.org/ka-youtube-converted/jA1vyOrzlOE.mp4/jA1vyOrzlOE.mp4182yes https://puzzicle.org/math/cc-seventh-grade-math/v/subtracting-integers-find-the-missing-valuehttps://puzzicle.org/thumbs/xc219bb304356f328.jpgSubtracting integers: find the missing valueUse number lines to find missing numbers in subtraction equations with integers. Arrows to the left equivalently mean adding a negative number and subtracting a positive number of the same magnitude. Arrows to the right equivalently mean adding a positive number and subtracting a negative number of the same magnitude.https://cdn.kastatic.org/ka-youtube-converted/HsPEStiHQS0.mp4/HsPEStiHQS0.mp4414yes https://puzzicle.org/math/cc-seventh-grade-math/v/substitution-with-negative-numbershttps://puzzicle.org/thumbs/x3c5fb2bf.jpgSubstitution with negative numbersPractice substituting positive and negative values for variables.https://cdn.kastatic.org/ka-youtube-converted/XkRD9lv_y44.mp4/XkRD9lv_y44.mp4478yes https://puzzicle.org/math/cc-seventh-grade-math/v/understanding-variables-representing-integers-on-number-linehttps://puzzicle.org/thumbs/x9db5c19c.jpgOrdering expressionsLet's get some practice thinking about adding and subtracting variables representing positive and negative numbers on the number line.https://cdn.kastatic.org/ka-youtube-converted/wRBPzExOH2A.mp4/wRBPzExOH2A.mp4410yes https://puzzicle.org/math/algebra/v/combining-like-terms-and-the-distributive-propertyhttps://puzzicle.org/thumbs/1027316723.jpgCombining like terms with negative coefficients & distributionWe've learned about order of operations and combining like terms. Let's layer the distributive property on top of this.https://cdn.kastatic.org/ka-youtube-converted/3NHSwiv_pSE.mp4/3NHSwiv_pSE.mp4247yes https://puzzicle.org/math/algebra/v/combining-like-terms-3https://puzzicle.org/thumbs/1254690345.jpgCombining like terms with negative coefficientsThis example of combining like terms in an expression gets a little hairy. Pay attention.https://cdn.kastatic.org/ka-youtube-converted/FNnmseBlvaY.mp4/FNnmseBlvaY.mp4279yes https://puzzicle.org/math/algebra/v/simplifying-expressions-involving-rational-numbershttps://puzzicle.org/thumbs/x1750bfec.jpgCombining like terms with rational coefficientsLearn how to rewrite algebraic expressions by combining like terms. The expressions in this video have decimal and fraction coefficients.https://cdn.kastatic.org/ka-youtube-converted/rtNuo7R3scY.mp4/rtNuo7R3scY.mp4256yes https://puzzicle.org/math/cc-seventh-grade-math/v/factoring-algebraic-expressionshttps://puzzicle.org/thumbs/xa57a0279.jpgThe distributive property with variablesLearn how to apply the distributive property to factor out the greatest common factor from an algebraic expression like 2+4x.https://cdn.kastatic.org/ka-youtube-converted/I6TBBzIvgB8.mp4/I6TBBzIvgB8.mp4374yes https://puzzicle.org/math/algebra/v/factoring-linear-binomialshttps://puzzicle.org/thumbs/xa1e260c1.jpgFactoring with the distributive propertySal shows how to factor the expression 4x+18 into the expression 2(2x+9).https://cdn.kastatic.org/ka-youtube-converted/mbb3msmX2xs.mp4/mbb3msmX2xs.mp4208yes https://puzzicle.org/math/cc-seventh-grade-math/v/equivalent-expressions-with-distribution-and-negative-numbershttps://puzzicle.org/thumbs/x71d68e5f.jpgEquivalent expressions: negative numbers & distributionLearn how to identify equivalent expressions using your knowledge of the distributive property and negative numbers.https://cdn.kastatic.org/ka-youtube-converted/Xtcylk5r0g8.mp4/Xtcylk5r0g8.mp4240yes https://puzzicle.org/math/cc-seventh-grade-math/v/interpreting-linear-expressions-example-2https://puzzicle.org/thumbs/xd74549ff.jpgInterpreting linear expressions: flowersLet's practice matching expressions to their meaning in this example of interpreting linear expressions.https://cdn.kastatic.org/ka-youtube-converted/edVnqL2Z59A.mp4/edVnqL2Z59A.mp4122yes https://puzzicle.org/math/cc-seventh-grade-math/v/interpreting-linear-expressions-examplehttps://puzzicle.org/thumbs/xdc8d5607.jpgInterpreting linear expressions: diamondsWe match the expressions to their meaning in this example. We're reinforcing our knowledge of linear expressions.https://cdn.kastatic.org/ka-youtube-converted/ao9cx8JlJIU.mp4/ao9cx8JlJIU.mp4112yes https://puzzicle.org/math/cc-seventh-grade-math/v/why-we-do-the-same-thing-to-both-sides-two-step-equationshttps://puzzicle.org/thumbs/948504421.jpgIntro to two-step equationsHere's how we solve a two step equation. It begins with the concept of equality: what we do to one side of the equation must be done to the other.https://cdn.kastatic.org/ka-youtube-converted/_y_Q3_B2Vh8.mp4/_y_Q3_B2Vh8.mp4312yes https://puzzicle.org/math/cc-seventh-grade-math/v/equations-2https://puzzicle.org/thumbs/8.jpgTwo-step equations intuitionThis example demonstrates how we solve an equation expressed such: ax + b = c. It's a little more complicated than previous examples, but you can do it!https://cdn.kastatic.org/ka-youtube-converted/XoEn1LfVoTo.mp4/XoEn1LfVoTo.mp4521yes https://puzzicle.org/math/cc-seventh-grade-math/v/solving-equations-1https://puzzicle.org/thumbs/26777.jpgWorked example: two-step equationsSal solves the equation -16 = x/4 + 2. It takes two steps because he first has to subtract 2 from both sides and then multiply both sides by 4.https://cdn.kastatic.org/ka-youtube-converted/p5e5mf_G3FI.mp4/p5e5mf_G3FI.mp4184yes https://puzzicle.org/math/cc-seventh-grade-math/v/two-step-equations-with-decimals-and-fractionshttps://puzzicle.org/thumbs/xe1652ad7.jpgTwo-step equations with decimals and fractionsLearn how to solve equations that involve decimals and fractions. The equations shown in this video are called "two-step" equations because they each take two steps to solve.https://cdn.kastatic.org/ka-youtube-converted/BOIA9wsM4ok.mp4/BOIA9wsM4ok.mp4471yes https://puzzicle.org/math/cc-seventh-grade-math/v/two-step-equationshttps://puzzicle.org/thumbs/23616.jpgTwo-step equations with decimals and fractionsLet's practice some two step equations, some of which require merging terms and using the distributive property.https://cdn.kastatic.org/ka-youtube-converted/tuVd355R-OQ.mp4/tuVd355R-OQ.mp4830yes https://puzzicle.org/math/cc-seventh-grade-math/v/mistakes-in-solving-equationshttps://puzzicle.org/thumbs/xf5d268b2.jpgFind the mistake: two-step equationsWhen doing algebra, we sometimes make mistakes! It's important to be able to check our own work (and other people's work) for correctness.https://cdn.kastatic.org/ka-youtube-converted/6q8mk7z72AU.mp4/6q8mk7z72AU.mp4228yes https://puzzicle.org/math/cc-seventh-grade-math/v/basic-linear-equation-word-problemhttps://puzzicle.org/thumbs/1243194170.jpgTwo-step equation word problem: computersLearn how to construct and solve a basic linear equation to solve a word problem.https://cdn.kastatic.org/ka-youtube-converted/roHvNNFXr4k.mp4/roHvNNFXr4k.mp4286yes https://puzzicle.org/math/cc-seventh-grade-math/v/application-problems-with-equation-in-one-variablehttps://puzzicle.org/thumbs/13179470.jpgTwo-step equation word problem: gardenHere's a nifty word problem in which we find the dimensions of a garden given only the perimeter. Let's create an equation to solve!https://cdn.kastatic.org/ka-youtube-converted/iimpwYBiKNg.mp4/iimpwYBiKNg.mp4122yes https://puzzicle.org/math/cc-seventh-grade-math/v/linear-equation-word-problem-examplehttps://puzzicle.org/thumbs/xdfc35e50.jpgTwo-step equation word problem: orangesLearn how to solve a word problem by writing an equation to model the situation. In this video, we use the linear equation 210(t-5) = 41,790.https://cdn.kastatic.org/ka-youtube-converted/xKH1Evwu150.mp4/xKH1Evwu150.mp4359yes https://puzzicle.org/math/cc-seventh-grade-math/v/inequalities-using-multiplication-and-divisionhttps://puzzicle.org/thumbs/23662.jpgOne-step inequalities examplesOur discussion of linear inequalities begins with multiplying and dividing by negative numbers. Pay attention for the word "swap." Super important!https://cdn.kastatic.org/ka-youtube-converted/PNXozoJWsWc.mp4/PNXozoJWsWc.mp4627yes https://puzzicle.org/math/cc-seventh-grade-math/v/one-step-inequalities-2https://puzzicle.org/thumbs/13179475.jpgOne-step inequalities: -5c ≤ 15In addition to solving the inequality, we'll graph the solution. Remember to swap if you mutiply both sides of the inequality by a negative number.https://cdn.kastatic.org/ka-youtube-converted/D1cKk48kz-E.mp4/D1cKk48kz-E.mp4174yes https://puzzicle.org/math/cc-seventh-grade-math/v/inequalitieshttps://puzzicle.org/thumbs/26815.jpgOne-step inequality word problemInequalities are more than abstract concepts and exercises. They help solve real life problems. Here's an example.https://cdn.kastatic.org/ka-youtube-converted/FZ2APP6-grU.mp4/FZ2APP6-grU.mp4129yes https://puzzicle.org/math/cc-seventh-grade-math/v/solving-inequalitieshttps://puzzicle.org/thumbs/26814.jpgTwo-step inequalitiesTwo-step inequalities are slightly more complicated than one-step inequalities (duh!). This is a worked example of solving ⅔>-4y-8⅓.https://cdn.kastatic.org/ka-youtube-converted/y7QLay8wrW8.mp4/y7QLay8wrW8.mp4272yes https://puzzicle.org/math/cc-seventh-grade-math/v/interpreting-inequalitieshttps://puzzicle.org/thumbs/26813.jpgTwo-step inequality word problem: applesWe'll talk you through this fun and challenging inequality problem.https://cdn.kastatic.org/ka-youtube-converted/wo7DSaPP8hQ.mp4/wo7DSaPP8hQ.mp4399yes https://puzzicle.org/math/cc-seventh-grade-math/v/writing-and-using-inequalities-3https://puzzicle.org/thumbs/26821.jpgTwo-step inequality word problem: R&BLet's tackle this word problem together. We'll interpret the information and then construct a linear inequality to solve it.https://cdn.kastatic.org/ka-youtube-converted/pbLiN8D9gAk.mp4/pbLiN8D9gAk.mp4121yes https://puzzicle.org/math/cc-seventh-grade-math/v/circles-radius-diameter-and-circumferencehttps://puzzicle.org/thumbs/24957.jpgRadius, diameter, circumference & πLearn how the number Pi allows us to relate the radius, diameter, and circumference of a circle.https://cdn.kastatic.org/ka-youtube-converted/jyLRpr2P0MQ.mp4/jyLRpr2P0MQ.mp4665yes https://puzzicle.org/math/cc-seventh-grade-math/v/parts-of-a-circlehttps://puzzicle.org/thumbs/370305280.jpgLabeling parts of a circleRadius, diameter, center, and circumference--all are parts of a circle. Let's go through each and understand how they are defined.https://cdn.kastatic.org/ka-youtube-converted/04N79tItPEA.mp4/04N79tItPEA.mp4121yes https://puzzicle.org/math/cc-seventh-grade-math/v/area-of-a-circlehttps://puzzicle.org/thumbs/370305273.jpgArea of a circleThe area of a circle is pi times the radius squared (A = π r²). Learn how to use this formula to find the area of a circle when given the diameter.https://cdn.kastatic.org/ka-youtube-converted/ZyOhRgnFmIY.mp4/ZyOhRgnFmIY.mp4242yes https://puzzicle.org/math/cc-seventh-grade-math/v/relating-circumference-and-areahttps://puzzicle.org/thumbs/x450a23885a563c76.jpgRelating circumference and areaSal uses formulas and a specific example to see how area and circumference relate.https://cdn.kastatic.org/ka-youtube-converted/-jqWW22FMdI.mp4/-jqWW22FMdI.mp4291yes https://puzzicle.org/math/cc-seventh-grade-math/v/partial-circle-area-and-arc-lengthhttps://puzzicle.org/thumbs/x6f26a88613dd9b5a.jpgPartial circle area and arc lengthSal finds the area of a semicircle and the arc length of a partial circle.https://cdn.kastatic.org/ka-youtube-converted/YjWCDdNlXxc.mp4/YjWCDdNlXxc.mp4180yes https://puzzicle.org/math/geometry-home/v/perimeter-and-area-of-a-non-standard-polygonhttps://puzzicle.org/thumbs/372432668.jpgPerimeter & area of composite shapesSal finds perimeter and area of a non-standard polygon.https://cdn.kastatic.org/ka-youtube-converted/7S1MLJOG-5A.mp4/7S1MLJOG-5A.mp4183yes https://puzzicle.org/math/geometry/v/solid-geometry-volumehttps://puzzicle.org/thumbs/60129394.jpgVolume of triangular prism & cubeUsing the formulas for the volume of triangular prism and cube to solve some solid geometry problems.https://cdn.kastatic.org/ka-youtube-converted/ZqzAOZ9pP9Q.mp4/ZqzAOZ9pP9Q.mp4236yes https://puzzicle.org/math/algebra-home/v/growing-by-a-percentagehttps://puzzicle.org/thumbs/23456.jpgGrowing by a percentageIn this example we grow a whole number by a percentage of itself. Growing by percentage is a common skill often used when figuring how much is owed or earned with interest.https://cdn.kastatic.org/ka-youtube-converted/X2jVap1YgwI.mp4/X2jVap1YgwI.mp4582yes https://puzzicle.org/math/cc-seventh-grade-math/v/percent-word-problem-exampleshttps://puzzicle.org/thumbs/x367a706c14a18040.jpgPercent word problem: magic clubSal solves percent word problems including percent comparisons and percent of change.https://cdn.kastatic.org/ka-youtube-converted/e1Cd0XL5OEs.mp4/e1Cd0XL5OEs.mp4479yes https://puzzicle.org/math/cc-seventh-grade-math/v/tax-discount-and-tip-exampleshttps://puzzicle.org/thumbs/xc0cdcf37b1a134e9.jpgPercent word problems: tax and discountSal solves percent word problems involving tax, tip, and discounts.https://cdn.kastatic.org/ka-youtube-converted/jb_RwR_Eso4.mp4/jb_RwR_Eso4.mp4348yes https://puzzicle.org/math/cc-seventh-grade-math/v/scale-factors-and-areahttps://puzzicle.org/thumbs/x2c11f01900e1665c.jpgScale factors and areaSal explore how scale factor affects the area of a scaled figure.https://cdn.kastatic.org/ka-youtube-converted/h-qwno7n5F4.mp4/h-qwno7n5F4.mp4364yes https://puzzicle.org/math/cc-seventh-grade-math/v/basic-probabilityhttps://puzzicle.org/thumbs/185786265.jpgIntro to theoretical probabilityWe give you an introduction to probability through the example of flipping a quarter and rolling a die.https://cdn.kastatic.org/ka-youtube-converted/uzkc-qNVoOk.mp4/uzkc-qNVoOk.mp4498yes https://puzzicle.org/math/cc-seventh-grade-math/v/simple-probabilityhttps://puzzicle.org/thumbs/380860447.jpgSimple probability: yellow marbleIn order to find the probability of picking a yellow marble from a bag, we have to first determine the number of possible outcomes, and then how many of them meet our constraints.https://cdn.kastatic.org/ka-youtube-converted/yUaI0JriZtY.mp4/yUaI0JriZtY.mp4176yes https://puzzicle.org/math/cc-seventh-grade-math/v/probability-1-module-exampleshttps://puzzicle.org/thumbs/60107867.jpgSimple probability: non-blue marbleIn this example we are figuring out the probability of randomly picking a non-blue marble from a bag. Again, we'll have to think about the possible outcomes first.https://cdn.kastatic.org/ka-youtube-converted/mLE-SlOZToc.mp4/mLE-SlOZToc.mp4596yes https://puzzicle.org/math/cc-seventh-grade-math/v/experimental-probabilityhttps://puzzicle.org/thumbs/xfc5cab7c.jpgExperimental probabilityBased on past experience, we can make reasonable estimates of the likelihood of future events.https://cdn.kastatic.org/ka-youtube-converted/RdehfQJ8i_0.mp4/RdehfQJ8i_0.mp4415yes https://puzzicle.org/math/cc-seventh-grade-math/v/compound-sample-spaceshttps://puzzicle.org/thumbs/x905af91a.jpgSample spaces for compound eventsExplore the notion of a "sample space". See a sample space represented as a tree diagram, table, and list.https://cdn.kastatic.org/ka-youtube-converted/PR-A3UAO7_0.mp4/PR-A3UAO7_0.mp4550yes https://puzzicle.org/math/cc-seventh-grade-math/v/events-and-outcomes-2https://puzzicle.org/thumbs/27092.jpgDie rolling probabilityWe're thinking about the probability of rolling doubles on a pair of dice. Let's create a grid of all possible outcomes.https://cdn.kastatic.org/ka-youtube-converted/O4Qnsubo2tg.mp4/O4Qnsubo2tg.mp4315yes https://puzzicle.org/math/cc-seventh-grade-math/v/probability-from-compound-sample-spacehttps://puzzicle.org/thumbs/x5634e573.jpgProbability of a compound eventLearn how to use sample space diagrams to find probabilities.https://cdn.kastatic.org/ka-youtube-converted/6zWPgvEMVlE.mp4/6zWPgvEMVlE.mp4303yes https://puzzicle.org/math/cc-seventh-grade-math/v/tree-diagram-to-count-outcomeshttps://puzzicle.org/thumbs/x31590e02.jpgCount outcomes using tree diagramTree diagrams display all the possible outcomes of an event. Each branch in a tree diagram represents a possible outcome. Tree diagrams can be used to find the number of possible outcomes and calculate the probability of possible outcomes.https://cdn.kastatic.org/ka-youtube-converted/Zxvc6iPKdec.mp4/Zxvc6iPKdec.mp4271yes https://puzzicle.org/math/cc-seventh-grade-math/v/counting-pot-and-flower-scenarioshttps://puzzicle.org/thumbs/xcc826c9d.jpgCounting outcomes: flower potsFind the number of ways you can put four types of flowers into three types of pots.https://cdn.kastatic.org/ka-youtube-converted/B_vAlneziHo.mp4/B_vAlneziHo.mp4155yes https://puzzicle.org/math/cc-seventh-grade-math/v/comparing-theoretical-to-experimental-probabiliteshttps://puzzicle.org/thumbs/xfb9f61b0.jpgTheoretical and experimental probabilitiesCompare expected probabilities to what really happens when we run experiments.https://cdn.kastatic.org/ka-youtube-converted/tXlcE_K_C-Y.mp4/tXlcE_K_C-Y.mp4422yes https://puzzicle.org/math/cc-seventh-grade-math/v/intuitive-sense-of-probabilitieshttps://puzzicle.org/thumbs/x5f14b7ab.jpgIntuitive sense of probabilitiesThink about what probabilities really mean. What does a probability of 0 mean? How about 1?https://cdn.kastatic.org/ka-youtube-converted/KFgvOQtH0Z0.mp4/KFgvOQtH0Z0.mp4531yes https://puzzicle.org/math/cc-seventh-grade-math/v/making-predictions-with-probabilityhttps://puzzicle.org/thumbs/xd724ea03.jpgMaking predictions with probabilityPredict the number of times a spinner will land on an elephant. The theoretical probability helps us get close with our prediction, but we can't know for sure until we actually spin the spinner.https://cdn.kastatic.org/ka-youtube-converted/8bK-xfh8-rY.mp4/8bK-xfh8-rY.mp4305yes https://puzzicle.org/math/cc-seventh-grade-math/v/constructing-probability-model-from-observationshttps://puzzicle.org/thumbs/x94fb291d.jpgProbability models example: frozen yogurtModel the probability of a frozen yogurt line having 0, 1, or 2 people in it.https://cdn.kastatic.org/ka-youtube-converted/BIpsQIJUCC8.mp4/BIpsQIJUCC8.mp4445yes https://puzzicle.org/math/cc-seventh-grade-math/v/reasonable-sampleshttps://puzzicle.org/thumbs/x6db9c771.jpgReasonable samplesTo make a valid conclusion, you'll need a representaive, not skewed, sample.https://cdn.kastatic.org/ka-youtube-converted/aKAw-DyZa6g.mp4/aKAw-DyZa6g.mp4257yes https://puzzicle.org/math/cc-seventh-grade-math/v/comparing-swim-times-at-the-olympicshttps://puzzicle.org/thumbs/xba691b00.jpgComparing distributions with dot plots (example problem)Sal examines two distributions in dot plots to draw conclusions about the times of Olympic swimmers.https://cdn.kastatic.org/ka-youtube-converted/KXDOOmquZag.mp4/KXDOOmquZag.mp4240yes https://puzzicle.org/math/cc-seventh-grade-math/v/complementary-and-supplementary-angleshttps://puzzicle.org/thumbs/263439064.jpgComplementary & supplementary anglesLearn about complementary and supplementary angles, as well as the definitions of adjacent and straight angles.https://cdn.kastatic.org/ka-youtube-converted/BTnAlNSgNsY.mp4/BTnAlNSgNsY.mp4511yes https://puzzicle.org/math/cc-seventh-grade-math/v/angles-at-the-intersection-of-two-lineshttps://puzzicle.org/thumbs/263439063.jpgVertical anglesBy using our knowledge of supplementary, adjacent, and vertical angles, we can solve problems involving the intersection of two lines. Including this one!https://cdn.kastatic.org/ka-youtube-converted/_7aUxFzTG5w.mp4/_7aUxFzTG5w.mp4439yes https://puzzicle.org/math/cc-seventh-grade-math/v/solving-for-unknown-angleshttps://puzzicle.org/thumbs/xb15e7f87.jpgFind measure of vertical anglesGiven two intersecting lines and the measure of vertical angles, watch as we solve to find the measure of the remaining angles.https://cdn.kastatic.org/ka-youtube-converted/TOobWKx1xe0.mp4/TOobWKx1xe0.mp4180yes https://puzzicle.org/math/cc-seventh-grade-math/v/pie-splittinghttps://puzzicle.org/thumbs/xa2a27151.jpgFind measure of angles word problemSolve this word problem to find the measure of angles. In this example you'll split up a pie (don't forget to share!)https://cdn.kastatic.org/ka-youtube-converted/3u9Ux29MSXE.mp4/3u9Ux29MSXE.mp4274yes https://puzzicle.org/math/cc-seventh-grade-math/v/example-using-algebra-to-find-measure-of-supplementary-angleshttps://puzzicle.org/thumbs/x2e4ad4ea.jpgEquation practice with supplementary anglesGiven the algebraic expressions that represent a pair of supplementary angles, Sal forms and solves an equation.https://cdn.kastatic.org/ka-youtube-converted/zNtbi4OJs9I.mp4/zNtbi4OJs9I.mp4157yes https://puzzicle.org/math/cc-seventh-grade-math/v/example-using-algebra-to-find-measure-of-complementary-angleshttps://puzzicle.org/thumbs/x0fe8e133.jpgEquation practice with complementary anglesGiven the algebraic expressions that represent a pair of complementary angles, learn how to form and solve an equation to find an unknown angle.https://cdn.kastatic.org/ka-youtube-converted/XDOH9Ul2OOM.mp4/XDOH9Ul2OOM.mp4153yes https://puzzicle.org/math/cc-seventh-grade-math/v/constructing-right-isosceleshttps://puzzicle.org/thumbs/x3961d04a.jpgConstruct a right isosceles triangleCan you build a triangle that is both a right triangle and an isosceles triangle?https://cdn.kastatic.org/ka-youtube-converted/lohMwoq3WFA.mp4/lohMwoq3WFA.mp486yes https://puzzicle.org/math/cc-seventh-grade-math/v/constructing-triangles-with-constraintshttps://puzzicle.org/thumbs/xa363a49e.jpgConstruct a triangle with constraintsHere's a challenge: in this problem we are given constraints and asked to construct a triangle. It can be done! You'll learn about degenerate triangles, too.https://cdn.kastatic.org/ka-youtube-converted/E78ji1uBtSA.mp4/E78ji1uBtSA.mp4239yes https://puzzicle.org/math/cc-seventh-grade-math/v/triangle-inqequality-theoremhttps://puzzicle.org/thumbs/1242820000.jpgTriangle inequality theoremIntuition behind the triangle inequality theoremhttps://cdn.kastatic.org/ka-youtube-converted/KlKYvbigBqs.mp4/KlKYvbigBqs.mp4352yes https://puzzicle.org/math/7th-engage-ny/v/ordering-triangle-sides-and-angleshttps://puzzicle.org/thumbs/xac0d898b.jpgOrdering triangle sides and angles exampleSal first solves a problem where he orders the sides of a triangle given the angles, then solves a problem where he orders the angles of a triangle given the sides.https://cdn.kastatic.org/ka-youtube-converted/I5eTOBsvUK8.mp4/I5eTOBsvUK8.mp4169yes https://puzzicle.org/math/geometry/v/vertical-slice-of-rectangular-pyramidhttps://puzzicle.org/thumbs/x938fa80f.jpgSlicing a rectangular pyramidWhat happens when you slice vertically into a rectangular pyramid? What kind of geometric shape results?https://cdn.kastatic.org/ka-youtube-converted/hoa1RBk4dTo.mp4/hoa1RBk4dTo.mp4210yes https://puzzicle.org/math/geometry/v/ways-to-cut-a-cubehttps://puzzicle.org/thumbs/x73ceab49.jpgWays to cross-section a cubeLearn about all the different ways we can obtain cross-sections of a cube.https://cdn.kastatic.org/ka-youtube-converted/aSokFEpoJFM.mp4/aSokFEpoJFM.mp4443yes https://puzzicle.org/math/cc-seventh-grade-math/v/circumference-from-areahttps://puzzicle.org/thumbs/x875f3feb.jpgFinding circumference of a circle when given the areaLearn how to find the circumference, the distance around a circle, when given the area.https://cdn.kastatic.org/ka-youtube-converted/Rcb7ZUTOQ1I.mp4/Rcb7ZUTOQ1I.mp4113yes https://puzzicle.org/math/cc-seventh-grade-math/v/area-of-shaded-regionhttps://puzzicle.org/thumbs/x1f3c3d07.jpgArea of a shaded regionHere's a fun one: find the area of a shaded region where you first determine the area of a square and then the area of a circle.https://cdn.kastatic.org/ka-youtube-converted/gbGghzo7yEM.mp4/gbGghzo7yEM.mp491yes https://puzzicle.org/math/cc-seventh-grade-math/v/radius-change-impacthttps://puzzicle.org/thumbs/x057e031d.jpgImpact of increasing the radiusIf we change the radius of a circle, how does the circumference and area change?https://cdn.kastatic.org/ka-youtube-converted/GB9Rqsah2gs.mp4/GB9Rqsah2gs.mp4211yes https://puzzicle.org/math/cc-seventh-grade-math/v/volume-of-gold-ringhttps://puzzicle.org/thumbs/xd26b50ec.jpgVolume word problem: gold ringSee if you can find the incremental volume of a ring in cubic inches. You'll need to put to use your knowledge of how to measure volume.https://cdn.kastatic.org/ka-youtube-converted/ViFLPsLTO1k.mp4/ViFLPsLTO1k.mp4160yes https://puzzicle.org/math/cc-seventh-grade-math/v/crates-in-boxcarhttps://puzzicle.org/thumbs/x9b271d72.jpgVolume of rectangles inside rectanglesHow many crates can we fit into a boxcar? We'll explain that there's more than one way to solve for the volume in this problem.https://cdn.kastatic.org/ka-youtube-converted/AxHAQ5l45FE.mp4/AxHAQ5l45FE.mp4287yes https://puzzicle.org/math/7th-grade-foundations-engageny/v/usain-bolt-s-average-speedhttps://puzzicle.org/thumbs/1254749169.jpgFinding average speed or rateUsing the formula for finding distance we can determine Usian Bolt's average speed, or rate, when he broke the world record in 2009 in the 100m. Watch.https://cdn.kastatic.org/ka-youtube-converted/mt6Nq0dzFjo.mp4/mt6Nq0dzFjo.mp4795yes https://puzzicle.org/math/7th-grade-foundations-engageny/v/inequalities-in-one-variable-1-exercisehttps://puzzicle.org/thumbs/xc0d659df.jpgInequality word problem: one variableThis time we're creating a variable to represent a number, and then writing an inequality. We're building on our knowledge.https://cdn.kastatic.org/ka-youtube-converted/aINeMQH77jg.mp4/aINeMQH77jg.mp4147yes https://puzzicle.org/math/7th-grade-foundations-engageny/v/identifying-percent-amount-and-basehttps://puzzicle.org/thumbs/153770143.jpgIdentifying percent amount and baseSal identifies the percent, amount, and base in a percent problem.https://cdn.kastatic.org/ka-youtube-converted/JaScdH47PYg.mp4/JaScdH47PYg.mp4152yes https://puzzicle.org/math/cc-seventh-grade-math/v/identifying-corresponding-parts-of-scaled-copieshttps://puzzicle.org/thumbs/x459c0342676e7ea6.jpgIdentifying corresponding parts of scaled copiesIdentifying corresponding parts of scaled copies from 7th grade mathhttps://cdn.kastatic.org/ka-youtube-converted/o07enwTfSpU.mp4/o07enwTfSpU.mp4151yes https://puzzicle.org/math/cc-seventh-grade-math/v/identifying-scale-factors-examplehttps://puzzicle.org/thumbs/x99c36e33e74c162f.jpgIdentifying scale factor in drawingsIdentify the scale factor used to create a scale copy.https://cdn.kastatic.org/ka-youtube-converted/ZeY1mgog2YQ.mp4/ZeY1mgog2YQ.mp486yes https://puzzicle.org/math/7th-grade-illustrative-math/v/corresponding-angles-in-scaled-figureshttps://puzzicle.org/thumbs/xdea105d2193935de.jpgCorresponding angles in scaled figuresStudents learn to compare two scale drawings of the same object made with different scales. Identify how the lengths, widths, or areas change as the scale changes.https://cdn.kastatic.org/ka-youtube-converted/6x5Ekb3TV70.mp4/6x5Ekb3TV70.mp4216yes https://puzzicle.org/math/7th-grade-illustrative-math/v/corresponding-angles-and-distances-in-scaled-figureshttps://puzzicle.org/thumbs/x76db62100159f634.jpgCorresponding angles and distances in scaled figuresStudents learn that two figures may have either corresponding side lengths with equivalent ratios OR corresponding angle measures equal, and still NOT be scaled copies. We must see that both characteristics are present to be certain that two shapes are scaled copies.https://cdn.kastatic.org/ka-youtube-converted/igH_ogbq1UY.mp4/igH_ogbq1UY.mp4210yes https://puzzicle.org/math/7th-grade-illustrative-math/v/corresponding-distances-in-scaled-figureshttps://puzzicle.org/thumbs/xc72d0c0e6a1244b0.jpgCorresponding distances in scaled figuresLearn how to find the measurements of distances on scaled figures that are not side lengths.https://cdn.kastatic.org/ka-youtube-converted/-3gU16Etzaw.mp4/-3gU16Etzaw.mp4149yes https://puzzicle.org/math/cc-seventh-grade-math/v/interpreting-scale-factors-in-drawingshttps://puzzicle.org/thumbs/x59907ba1ca7a8d9c.jpgInterpreting scale factors in drawingsSal interprets a scale factor to determine if a scale copy will be larger or smaller than the original figure.https://cdn.kastatic.org/ka-youtube-converted/Rgi1z9r5hOk.mp4/Rgi1z9r5hOk.mp486yes https://puzzicle.org/math/7th-grade-illustrative-math/v/reciprocal-scale-factorshttps://puzzicle.org/thumbs/xdb270652215b795a.jpgReciprocal scale factorsExplore how scale factors relate in scaled copies. Learn why the scale factor from the copy back to the original is the reciprocal of the scale factor from the original to the copy.https://cdn.kastatic.org/ka-youtube-converted/ocEor9YWaXQ.mp4/ocEor9YWaXQ.mp4149yes https://puzzicle.org/math/cc-seventh-grade-math/v/scaled-drawings-worked-exampleshttps://puzzicle.org/thumbs/xcfa89ee54e1b4d14.jpgScale drawingsSal finds a missing side length in a scale drawing when given either a set of corresponding side lengths or one side length and the scale factor.https://cdn.kastatic.org/ka-youtube-converted/XUOPGqKGbjk.mp4/XUOPGqKGbjk.mp4213yes https://puzzicle.org/math/cc-seventh-grade-math/v/creating-scale-drawingshttps://puzzicle.org/thumbs/x01989a848c05e686.jpgCreating scale drawingsSal uses real world contexts to create scale drawings on a grid.https://cdn.kastatic.org/ka-youtube-converted/TASA6V5fGT8.mp4/TASA6V5fGT8.mp4138yes https://puzzicle.org/math/7th-grade-illustrative-math/v/comparing-different-scaleshttps://puzzicle.org/thumbs/x0f434f9940990e67.jpgComparing different scalesStudents learn to compare two scale drawings of the same object made with different scales. Identify how the lengths, widths, or areas change as the scale changes.https://cdn.kastatic.org/ka-youtube-converted/ofRwYtHPGSQ.mp4/ofRwYtHPGSQ.mp4233yes https://puzzicle.org/math/7th-grade-illustrative-math/v/equivalent-scaleshttps://puzzicle.org/thumbs/x4c8dabc5f1400d37.jpgEquivalent scalesLearn how to identify equivalent scales with different units.https://cdn.kastatic.org/ka-youtube-converted/5HhDK7TH4Ag.mp4/5HhDK7TH4Ag.mp4125yes https://puzzicle.org/math/7th-grade-illustrative-math/v/writing-scales-with-and-without-unitshttps://puzzicle.org/thumbs/x16dbb65ea55e53f8.jpgWriting scales with and without unitsLearn how to write scales with and without units.https://cdn.kastatic.org/ka-youtube-converted/LPgqrInAxqk.mp4/LPgqrInAxqk.mp4155yes https://puzzicle.org/math/7th-grade-illustrative-math/v/comparing-scales-with-different-unitshttps://puzzicle.org/thumbs/x86fed3cf9432379e.jpgComparing scales with different unitsLearn how to compare scales that have different units.https://cdn.kastatic.org/ka-youtube-converted/s7_jHLMXvys.mp4/s7_jHLMXvys.mp4122yes https://puzzicle.org/math/cc-seventh-grade-math/v/example-identifying-a-proportional-relatioship-given-a-constant-of-proportionalihttps://puzzicle.org/thumbs/x08a5abf4706ddad6.jpgConstant of proportionality from tablesIdentifying proportionality constants by looking at tables of values.https://cdn.kastatic.org/ka-youtube-converted/rzDQ_ZIpi84.mp4/rzDQ_ZIpi84.mp4154yes https://puzzicle.org/math/cc-seventh-grade-math/v/identifying-the-constant-of-proportionalityhttps://puzzicle.org/thumbs/x15418d80a6af0b42.jpgIdentifying the constant of proportionality from equationThe constant of proportionality is a key concept in math. It's the number that we multiply one variable by to get another in a proportional relationship. This can be seen in everyday situations like cooking or in equations.https://cdn.kastatic.org/ka-youtube-converted/rj7DweP8e58.mp4/rj7DweP8e58.mp4231yes https://puzzicle.org/math/cc-seventh-grade-math/v/example-writing-proportional-equationshttps://puzzicle.org/thumbs/xbc9814cbfad8288a.jpgWriting proportional equationsSal figures out the constant of proportionality when dealing with a constant rate.https://cdn.kastatic.org/ka-youtube-converted/MRFrQXDZim0.mp4/MRFrQXDZim0.mp4130yes https://puzzicle.org/math/7th-grade-illustrative-math/v/using-equations-and-tables-to-represent-proportional-relationshipshttps://puzzicle.org/thumbs/x4b25a03394be3666.jpgUsing equations and tables to represent proportional relationshipsLearn to make connections between equations and tables in proportional relationships.https://cdn.kastatic.org/ka-youtube-converted/OccfgY3juXE.mp4/OccfgY3juXE.mp489yes https://puzzicle.org/math/7th-grade-illustrative-math/v/analyzing-proportionality-using-equations-and-tables-bike-rentalhttps://puzzicle.org/thumbs/x31f8b1035a008531.jpgAnalyzing proportionality using equations and tables: bike rentalLearn about analyzing proportionality using equations and tables in a bike rental scenario.https://cdn.kastatic.org/ka-youtube-converted/pXA_8bRSjGM.mp4/pXA_8bRSjGM.mp4233yes https://puzzicle.org/math/7th-grade-illustrative-math/v/analyzing-proportionality-using-equations-and-tables-distancehttps://puzzicle.org/thumbs/x2d174db7ceaa9822.jpgAnalyzing proportionality using equations and tables: distanceLearn about analyzing proportionality using equations and tables in a distance scenario.https://cdn.kastatic.org/ka-youtube-converted/qMcdQDfolA8.mp4/qMcdQDfolA8.mp4142yes https://puzzicle.org/math/cc-seventh-grade-math/v/constant-of-proportionality-from-graphhttps://puzzicle.org/thumbs/x3cde70670ae56928.jpgConstant of proportionality from graphIn this lesson, we learn about the constant of proportionality in a graph. It's the number we multiply by x to get y. In our example, when x increases, y increases three times as much. So, our constant of proportionality is 3.https://cdn.kastatic.org/ka-youtube-converted/n_UHN9t3cNg.mp4/n_UHN9t3cNg.mp4110yes https://puzzicle.org/math/cc-seventh-grade-math/v/interpreting-graphs-of-proportional-relationshipshttps://puzzicle.org/thumbs/xf0607b55dc5b25fd.jpgInterpreting graphs of proportional relationshipsWorked example interpreting graphs of proportional relationshipshttps://cdn.kastatic.org/ka-youtube-converted/ApstXGJtRrE.mp4/ApstXGJtRrE.mp4162yes https://puzzicle.org/math/cc-seventh-grade-math/v/comparing-constants-of-proportionalityhttps://puzzicle.org/thumbs/x8ccefb0267cc2b7f.jpgComparing constants of proportionalityWorking example comparing constants of proportionality.https://cdn.kastatic.org/ka-youtube-converted/oZxbLuJ1U5w.mp4/oZxbLuJ1U5w.mp4272yes https://puzzicle.org/math/7th-grade-illustrative-math/v/interpreting-constant-of-proportionality-in-graphshttps://puzzicle.org/thumbs/xc0936937fcdedda7.jpgInterpreting constant of proportionality in graphsLearn how to identify and interpret the constant of proportionality from a graph of a proportional relationship, including what the special point (1, k) means in a real-world context.https://cdn.kastatic.org/ka-youtube-converted/e_1M6jOLv68.mp4/e_1M6jOLv68.mp4146yes https://puzzicle.org/math/7th-grade-illustrative-math/v/constant-of-proportionality-in-both-directionshttps://puzzicle.org/thumbs/xc4a88d0794954878.jpgConstant of proportionality in both directionsLearn how to use graphs of proportional relationships to identify and interpret constants of proportionality.https://cdn.kastatic.org/ka-youtube-converted/D42p5dXb4-s.mp4/D42p5dXb4-s.mp4117yes https://puzzicle.org/math/7th-grade-illustrative-math/v/comparing-constants-of-proportionality-graphicallyhttps://puzzicle.org/thumbs/xf603b35d88aea0cd.jpgComparing constants of proportionality graphicallyLearn how to use graphs of proportional relationships to identify and interpret constants of proportionality.https://cdn.kastatic.org/ka-youtube-converted/b3VcacQ90hE.mp4/b3VcacQ90hE.mp4199yes https://puzzicle.org/math/7th-grade-illustrative-math/v/explaining-the-relationship-between-diameter-and-circumference-ordered-pairshttps://puzzicle.org/thumbs/x03ee9d053d7b3ea0.jpgExplaining the relationship between diameter and circumference: ordered pairsLearn to generalize that the relationship between diameter and circumference is proportional and that the constant of proportionality is a little more than 3.https://cdn.kastatic.org/ka-youtube-converted/HfsOzAx_bbo.mp4/HfsOzAx_bbo.mp4142yes https://puzzicle.org/math/7th-grade-illustrative-math/v/explaining-the-relationship-between-diameter-and-circumference-from-a-graphhttps://puzzicle.org/thumbs/xc10b39030bc04f6b.jpgExplaining the relationship between diameter and circumference from a graphLearn to generalize that the relationship between diameter and circumference is proportional and that the constant of proportionality is a little more than 3.https://cdn.kastatic.org/ka-youtube-converted/nuUsOGJCFCI.mp4/nuUsOGJCFCI.mp4233yes https://puzzicle.org/math/7th-grade-illustrative-math/v/using-the-relationship-between-diameter-and-circumference-to-find-an-incorrect-mhttps://puzzicle.org/thumbs/x915db6868b920264.jpgIdentifying incorrect circle measurementsLearn to generalize that the relationship between diameter and circumference is proportional and that the constant of proportionality is a little more than 3.https://cdn.kastatic.org/ka-youtube-converted/iXHGujDaEqI.mp4/iXHGujDaEqI.mp4121yes https://puzzicle.org/math/7th-grade-illustrative-math/v/using-the-relationship-between-diameter-and-circumference-to-find-reasonable-apphttps://puzzicle.org/thumbs/x0136accb61421384.jpgApproximating diameter and circumferenceLearn to generalize that the relationship between diameter and circumference is proportional and that the constant of proportionality is a little more than 3.https://cdn.kastatic.org/ka-youtube-converted/Q4mpvsjJGyI.mp4/Q4mpvsjJGyI.mp4124yes https://puzzicle.org/math/cc-seventh-grade-math/v/finding-radius-and-diameter-from-circumferencehttps://puzzicle.org/thumbs/x395b2629.jpgRadius & diameter from circumferenceSal finds the radius and diameter of a circle given the circumference.https://cdn.kastatic.org/ka-youtube-converted/yWGeFDXgQvg.mp4/yWGeFDXgQvg.mp4216yes https://puzzicle.org/math/7th-grade-illustrative-math/v/estimating-area-of-a-circle-with-upper-and-lower-boundshttps://puzzicle.org/thumbs/xfac73d70e07be1cb.jpgEstimating area of a circle with upper and lower boundsLearn how to estimate the area inside different circles on a grid.https://cdn.kastatic.org/ka-youtube-converted/Qm1vXiVDyoY.mp4/Qm1vXiVDyoY.mp4165yes https://puzzicle.org/math/7th-grade-illustrative-math/v/estimating-area-of-a-circle-using-polygons-high-estimatehttps://puzzicle.org/thumbs/xb68b7fd79d033814.jpgEstimating area of a circle using polygons: high estimateLearn how to estimate the area inside different circles on a grid.https://cdn.kastatic.org/ka-youtube-converted/7cW2Jf_dOLM.mp4/7cW2Jf_dOLM.mp4190yes https://puzzicle.org/math/7th-grade-illustrative-math/v/estimating-area-of-a-circle-using-polygons-low-estimatehttps://puzzicle.org/thumbs/xca9e4e583efc6039.jpgEstimating area of a circle using polygons: low estimateLearn how to estimate the area inside different circles on a grid.https://cdn.kastatic.org/ka-youtube-converted/nscKNWbt2sE.mp4/nscKNWbt2sE.mp4123yes https://puzzicle.org/math/7th-grade-illustrative-math/v/expressing-a-fractional-increase-as-a-fraction-of-the-originalhttps://puzzicle.org/thumbs/xb98450221fcb8fdb.jpgExpressing a fractional increase as a fraction of the originalLearn about expressing a fractional increase as a fraction of the original by working through a piano practice example.https://cdn.kastatic.org/ka-youtube-converted/3AhufdthqEc.mp4/3AhufdthqEc.mp4124yes https://puzzicle.org/math/7th-grade-illustrative-math/v/writing-expressions-for-a-fraction-of-a-quantityhttps://puzzicle.org/thumbs/x7bde6d373f5818f0.jpgWriting expressions for a fraction of a quantityLearn about writing equivalent expressions for a remaining quantity.https://cdn.kastatic.org/ka-youtube-converted/Qn5AEUWHbb8.mp4/Qn5AEUWHbb8.mp488yes https://puzzicle.org/math/7th-grade-illustrative-math/v/interpreting-a-tape-diagram-showing-a-fractional-increase-2https://puzzicle.org/thumbs/xabc7a92a9e37f910.jpgInterpreting a tape diagram showing a fractional increaseLearn about interpreting tape diagrams that represent fractional increases by matching them to real-world contexts.https://cdn.kastatic.org/ka-youtube-converted/jmdaLhmXkoI.mp4/jmdaLhmXkoI.mp4148yes https://puzzicle.org/math/cc-eighth-grade-math/v/converting-a-fraction-to-a-repeating-decimalhttps://puzzicle.org/thumbs/xa06e19a6.jpgConverting a fraction to a repeating decimalTo convert a fraction like 19/27 into a decimal, you should divide 27 into 19, and use the remainder to figure out the decimal. The decimal begins 0.703703..., and the notation for a repeating decimal like this is to write the numbers that repeat and then put a line above them. The first six digits of the decimal should be included in the answer.https://cdn.kastatic.org/ka-youtube-converted/Y2-tz27nKoQ.mp4/Y2-tz27nKoQ.mp4265yes https://puzzicle.org/math/7th-grade-illustrative-math/v/expressing-a-percent-decrease-as-a-percentage-of-the-originalhttps://puzzicle.org/thumbs/x725600cc9a3846c7.jpgExpressing a percent decrease as a percentage of the originalLearn about expressing a percent decrease as a percentage of the original amount using a library book checkout example.https://cdn.kastatic.org/ka-youtube-converted/kjSQSBne0wg.mp4/kjSQSBne0wg.mp4123yes https://puzzicle.org/math/7th-grade-illustrative-math/v/writing-expressions-for-a-percent-increasehttps://puzzicle.org/thumbs/x7eeb241903b4eb66.jpgWriting expressions for a percent increaseLearn about writing equivalent expressions for a total amount after a percent increase using a blanket collection example.https://cdn.kastatic.org/ka-youtube-converted/lwctkmKBDxA.mp4/lwctkmKBDxA.mp4120yes https://puzzicle.org/math/7th-grade-illustrative-math/v/interpreting-a-tape-diagram-showing-a-fractional-increasehttps://puzzicle.org/thumbs/xf9f50973bae9bd5d.jpgInterpreting a tape diagram to find percent increaseLearn about using a tape diagram to describe one quantity as a percent greater than another using a plant height example.https://cdn.kastatic.org/ka-youtube-converted/n21E7OzXOwI.mp4/n21E7OzXOwI.mp4100yes https://puzzicle.org/math/cc-seventh-grade-math/v/rational-number-word-problem-example-1https://puzzicle.org/thumbs/x39d05dd5.jpgRational number word problem: iceWord problems force us to put concepts to work using real-world applications. In this example, determine the volume of frozen water and express the answer as a fraction.https://cdn.kastatic.org/ka-youtube-converted/MZpULgKhaEU.mp4/MZpULgKhaEU.mp4201yes https://puzzicle.org/math/mappers/v/converting-decimals-to-percents-ex-2https://puzzicle.org/thumbs/914998704.jpgConverting decimals to percents: 1.501Sal converts 1.501 to a percent.https://cdn.kastatic.org/ka-youtube-converted/3_caioiRu5I.mp4/3_caioiRu5I.mp4108yes https://puzzicle.org/math/mappers/v/converting-percents-to-decimals-ex-1https://puzzicle.org/thumbs/xf7f8fc80.jpgConverting percents to decimals: 59.2%Decimals can be written in percent form. ​Per-cent means per-100. So, we multiply the decimal by 100 to get an equivalent percent. Then, we add a percent sign (%). For example, 0.8 can be converted to percent form by solving 0.8✕100. So, 0.8=80%. Another way to think of multiplying by 100 is moving the decimal two places to the right.https://cdn.kastatic.org/ka-youtube-converted/MkpbtCRwcCE.mp4/MkpbtCRwcCE.mp483yes https://puzzicle.org/math/7th-grade-illustrative-math/v/finding-the-sales-tax-percentage-from-a-pricehttps://puzzicle.org/thumbs/x545319b671fb969f.jpgFinding the sales tax percentage from a priceLearn about calculating the sales tax percentage by comparing the original and final price of a bubble tea.https://cdn.kastatic.org/ka-youtube-converted/qNg4GZF3sQM.mp4/qNg4GZF3sQM.mp492yes https://puzzicle.org/math/7th-grade-illustrative-math/v/finding-the-discount-percentage-from-a-sale-pricehttps://puzzicle.org/thumbs/xd539a4fb57f00da7.jpgFinding the discount percentage from a sale priceLearn about calculating the discount percentage by comparing the original and sale price of snow boots.https://cdn.kastatic.org/ka-youtube-converted/OaZp01AhE5Q.mp4/OaZp01AhE5Q.mp4108yes https://puzzicle.org/math/7th-grade-illustrative-math/v/finding-the-percent-increase-in-pricinghttps://puzzicle.org/thumbs/x04b89027003c842b.jpgFinding the percent increase in pricingLearn about calculating the percent increase by comparing the original and new price of musical tickets.https://cdn.kastatic.org/ka-youtube-converted/7c8eR_CCx-o.mp4/7c8eR_CCx-o.mp488yes https://puzzicle.org/math/7th-grade-illustrative-math/v/finding-percent-errorhttps://puzzicle.org/thumbs/x8f81cde968646a2b.jpgFinding percent errorLearn how to find the percent error of an estimate.https://cdn.kastatic.org/ka-youtube-converted/WQLcC5SpbYY.mp4/WQLcC5SpbYY.mp473yes https://puzzicle.org/math/7th-grade-illustrative-math/v/applying-percent-error-pedometerhttps://puzzicle.org/thumbs/x9fa49aa34ad94f90.jpgApplying percent error: pedometerLearn about applying percent error to find an erroneous estimate.https://cdn.kastatic.org/ka-youtube-converted/oeJf2EfGox8.mp4/oeJf2EfGox8.mp4123yes https://puzzicle.org/math/7th-grade-illustrative-math/v/applying-percent-error-skateboardhttps://puzzicle.org/thumbs/xb1a2d63fbda266bd.jpgApplying percent error: skateboardLearn about applying percent error to find an actual amount.https://cdn.kastatic.org/ka-youtube-converted/cWF8zgvqqMk.mp4/cWF8zgvqqMk.mp4202yes https://puzzicle.org/math/7th-grade-illustrative-math/v/applying-percent-error-with-an-underestimatehttps://puzzicle.org/thumbs/x3222143ad74e12a7.jpgApplying percent error with an underestimateLearn about applying percent error with an underestimate to find an actual amount.https://cdn.kastatic.org/ka-youtube-converted/1iqIpSlEH6E.mp4/1iqIpSlEH6E.mp4186yes https://puzzicle.org/math/cc-seventh-grade-math/v/u09-l1-t2-we4-adding-real-numbers-applicationhttps://puzzicle.org/thumbs/466047489.jpgRational number word problem: checking accountYour checking account is overdrawn. Not good. Use your knowledge of decimals and adding negative and positive numbers to bring your account back into the black.https://cdn.kastatic.org/ka-youtube-converted/fFdOr8U4mnI.mp4/fFdOr8U4mnI.mp4195yes https://puzzicle.org/math/7th-grade-illustrative-math/v/applying-signed-number-addition-to-money-situations-writing-equationshttps://puzzicle.org/thumbs/x12706ede88342f82.jpgApplying signed number addition to money situations: writing equationsLearn about writing equations with negative numbers to represent a real-world situation involving spending from a fund.https://cdn.kastatic.org/ka-youtube-converted/OLuI0KtQHkc.mp4/OLuI0KtQHkc.mp4128yes https://puzzicle.org/math/7th-grade-illustrative-math/v/applying-signed-number-addition-to-find-a-final-balancehttps://puzzicle.org/thumbs/xf6dda9b54d1dea16.jpgApplying signed number addition to find a final balanceLearn about adding positive and negative integers to find a final balance using a theater ticket sales example.https://cdn.kastatic.org/ka-youtube-converted/pfitOXYt6Fs.mp4/pfitOXYt6Fs.mp457yes https://puzzicle.org/math/7th-grade-illustrative-math/v/applying-signed-number-addition-to-elevation-situations-writing-equationshttps://puzzicle.org/thumbs/x7013804ad8586887.jpgApplying signed number addition to elevation situations: writing equationsLearn about writing equations with negative numbers to represent a real-world situation involving a change in elevation.https://cdn.kastatic.org/ka-youtube-converted/TmHMtGa1fYU.mp4/TmHMtGa1fYU.mp4116yes https://puzzicle.org/math/7th-grade-illustrative-math/v/applying-multiplication-of-signed-numbers-to-motion-writing-equationshttps://puzzicle.org/thumbs/x8f749d795a593e38.jpgApplying multiplication of signed numbers to motion: writing equationsLearn about writing multiplication equations with negative numbers to represent a drone's change in position using velocity and time.https://cdn.kastatic.org/ka-youtube-converted/UYaAQny-lQo.mp4/UYaAQny-lQo.mp4112yes https://puzzicle.org/math/7th-grade-illustrative-math/v/applying-multiplication-of-signed-numbers-to-motion-calculating-final-positionhttps://puzzicle.org/thumbs/x01e9f64cabfd91f3.jpgApplying multiplication of signed numbers to motion: calculating final positionLearn about multiplying a negative rate by a positive time to find a final position using a delivery robot example.https://cdn.kastatic.org/ka-youtube-converted/o2GympSkzhI.mp4/o2GympSkzhI.mp471yes https://puzzicle.org/math/7th-grade-illustrative-math/v/relating-multiplication-and-division-with-negative-numbershttps://puzzicle.org/thumbs/xf9cdad23fc39da20.jpgRelating multiplication and division with negative numbersLearn about dividing a negative number by a decimal by using the relationship between multiplication and division.https://cdn.kastatic.org/ka-youtube-converted/KUKdzmAM2b8.mp4/KUKdzmAM2b8.mp489yes https://puzzicle.org/math/cc-seventh-grade-math/v/introduction-to-order-of-operationshttps://puzzicle.org/thumbs/25757.jpgIntro to order of operationsThis example shows the steps and clarifies the purpose of order of operations: to have ONE way to interpret a mathematical statement.https://cdn.kastatic.org/ka-youtube-converted/ClYdw4d4OmA.mp4/ClYdw4d4OmA.mp4580yes https://puzzicle.org/math/cc-seventh-grade-math/v/multistep-word-problems-example-3https://puzzicle.org/thumbs/xd55ad2ce.jpgRational number word problem: cabBudgeting money is one of the most valuable math skills you'll learn. Practice with us as we figure out how much money is left after a cab fare.https://cdn.kastatic.org/ka-youtube-converted/-sSDb_wZqKQ.mp4/-sSDb_wZqKQ.mp4128yes https://puzzicle.org/math/7th-grade-illustrative-math/v/matching-addition-equations-with-rational-numbers-to-real-world-situationshttps://puzzicle.org/thumbs/x5f84633696437aca.jpgMatching addition equations with rational numbers to real-world situationsLearn about identifying real-world situations that can be represented by an equation with negative numbers.https://cdn.kastatic.org/ka-youtube-converted/bZ-PfX9BvEs.mp4/bZ-PfX9BvEs.mp496yes https://puzzicle.org/math/7th-grade-illustrative-math/v/writing-multiplication-equations-with-rational-numbers-to-represent-situationshttps://puzzicle.org/thumbs/x4f94e8e49e03ff4f.jpgWriting multiplication equations with rational numbers to represent situationsLearn about writing an equation with a negative fraction to represent a real-world situation involving a descending platform.https://cdn.kastatic.org/ka-youtube-converted/SFBr8Agkl4w.mp4/SFBr8Agkl4w.mp477yes https://puzzicle.org/math/7th-grade-illustrative-math/v/writing-addition-equations-with-rational-numbers-to-represent-situationshttps://puzzicle.org/thumbs/x1d12b08b177907ff.jpgWriting addition equations with rational numbers to represent situationsLearn about writing an equation with negative numbers to represent a real-world situation involving a rising water level.https://cdn.kastatic.org/ka-youtube-converted/-hVfZpHSVeo.mp4/-hVfZpHSVeo.mp4116yes https://puzzicle.org/math/7th-grade-illustrative-math/v/reading-tape-diagrams-and-identifying-equivalent-equationshttps://puzzicle.org/thumbs/xe366a8764a154591.jpgReading tape diagrams and identifying equivalent equationsLearn how to match a tape diagram to its equivalent equations by identifying the parts of the diagram and translating them into algebraic expressions.https://cdn.kastatic.org/ka-youtube-converted/i4TrTPucvrs.mp4/i4TrTPucvrs.mp4234yes https://puzzicle.org/math/7th-grade-illustrative-math/v/reasoning-with-tape-diagrams-to-solve-equationshttps://puzzicle.org/thumbs/x94047009de26fb5a.jpgReasoning with tape diagrams to solve equationsLearn how to use a tape diagram to solve an equation of the form a(x + b) = c by finding the value of each equal part and solving for x.https://cdn.kastatic.org/ka-youtube-converted/LWHBG5daCUs.mp4/LWHBG5daCUs.mp4114yes https://puzzicle.org/math/7th-grade-illustrative-math/v/reasoning-with-tape-diagrams-to-connect-situations-and-equationshttps://puzzicle.org/thumbs/xb66e2b0cfb95c490.jpgReasoning with tape diagrams to connect situations and equationsLearn how to read a tape diagram and match its structure to the real-world situation it represents.https://cdn.kastatic.org/ka-youtube-converted/6a2HDER0Sgk.mp4/6a2HDER0Sgk.mp4242yes https://puzzicle.org/math/cc-seventh-grade-math/v/plotting-inequalities-on-number-linehttps://puzzicle.org/thumbs/xd542624c8cfcf35f.jpgPlotting inequalities on a number lineGraph inequalities on a number line, using filled in points at the boundary for ≤ or ≥ and arrows in the appropriate direction. Make sense of the inequalities in context.https://cdn.kastatic.org/ka-youtube-converted/xe6TO45kLL4.mp4/xe6TO45kLL4.mp4318yes https://puzzicle.org/math/7th-grade-illustrative-math/v/identifying-identical-triangle-copieshttps://puzzicle.org/thumbs/x48d20f354609b8cf.jpgIdentifying identical triangle copiesLearn how to identify identical triangles by checking that the same side length connects the same two angles, even when the triangle is rotated or flipped.https://cdn.kastatic.org/ka-youtube-converted/zs7mLUcatCI.mp4/zs7mLUcatCI.mp4107yes https://puzzicle.org/math/7th-grade-illustrative-math/v/comparing-triangles-by-analyzing-angle-and-side-arrangementshttps://puzzicle.org/thumbs/x7f834abb754315e4.jpgComparing triangles by analyzing angle and side arrangementsLearn how to compare two triangles by checking whether a given angle and side lengths are arranged in the same way, not just whether the same measurements appear.https://cdn.kastatic.org/ka-youtube-converted/Vp5OLgy_7ps.mp4/Vp5OLgy_7ps.mp4187yes https://puzzicle.org/math/7th-grade-illustrative-math/v/determining-triangle-uniquenesshttps://puzzicle.org/thumbs/x7ed679550ab6861e.jpgDetermining triangle uniquenessLearn how to determine whether a given angle and two side lengths describe exactly one triangle, more than one triangle, or no triangle at all.https://cdn.kastatic.org/ka-youtube-converted/D8z6FzeNntk.mp4/D8z6FzeNntk.mp4124yes https://puzzicle.org/math/7th-grade-illustrative-math/v/determining-if-a-triangle-is-possible-given-two-angle-measureshttps://puzzicle.org/thumbs/xe09603fa388af6a5.jpgDetermining if a triangle is possible given two angle measuresLearn how to determine whether two given angles can form a valid triangle.https://cdn.kastatic.org/ka-youtube-converted/M8NSMsWkqzs.mp4/M8NSMsWkqzs.mp453yes https://puzzicle.org/math/7th-grade-illustrative-math/v/finding-the-volume-of-a-prism-with-a-polygon-basehttps://puzzicle.org/thumbs/x9e6cf6eb7afb4651.jpgFinding the volume of a prism with a polygon baseLearn how to find the volume of a prism with a trapezoidal base by first calculating the area of the base and then multiplying by the height.https://cdn.kastatic.org/ka-youtube-converted/pJJe4PktTn8.mp4/pJJe4PktTn8.mp4136yes https://puzzicle.org/math/7th-grade-illustrative-math/v/finding-a-missing-dimension-of-a-prism-using-the-volume-formulahttps://puzzicle.org/thumbs/x3b2dad60a97872cf.jpgFinding a missing dimension of a prism using the volume formulaLearn how to find the height of a prism by using the grid to find the area of the base and then working backwards from the volume formula.https://cdn.kastatic.org/ka-youtube-converted/RvCJat52CGU.mp4/RvCJat52CGU.mp489yes https://puzzicle.org/math/7th-grade-illustrative-math/v/finding-the-volume-of-a-prism-with-a-polygon-base-on-a-gridhttps://puzzicle.org/thumbs/x804ced09ff2ed6d5.jpgFinding the volume of a prism with a polygon base on a gridLearn how to find the volume of a hexagonal prism by using a grid to calculate the area of the base and then multiplying by the height.https://cdn.kastatic.org/ka-youtube-converted/0X8Oo013zIc.mp4/0X8Oo013zIc.mp4152yes https://puzzicle.org/math/7th-grade-illustrative-math/v/making-a-prediction-based-on-results-from-a-single-experimenthttps://puzzicle.org/thumbs/x919b6a4c442a1840.jpgMaking a prediction based on results from a single experimentLearn how to use experimental data to estimate the probability of an outcome and predict what is most likely to happen next.https://cdn.kastatic.org/ka-youtube-converted/lv9qmWir61I.mp4/lv9qmWir61I.mp4133yes https://puzzicle.org/math/7th-grade-illustrative-math/v/comparing-two-experiments-to-predict-the-better-chancehttps://puzzicle.org/thumbs/xccc5edf98115253b.jpgComparing two experiments to predict the better chanceLearn how to compare experimental probabilities given as fractions with different denominators to determine which outcome is more likely.https://cdn.kastatic.org/ka-youtube-converted/rp4sEjQbvno.mp4/rp4sEjQbvno.mp4161yes https://puzzicle.org/math/7th-grade-illustrative-math/v/classifying-outcomes-using-likelihood-termshttps://puzzicle.org/thumbs/xc0608f26a32b4500.jpgClassifying outcomes using likelihood termsLearn how to use the number of favorable outcomes out of total outcomes to identify whether an event is unlikely, likely, certain, or impossible.https://cdn.kastatic.org/ka-youtube-converted/weyF8fSxoXQ.mp4/weyF8fSxoXQ.mp4115yes https://puzzicle.org/math/7th-grade-illustrative-math/v/matching-a-probability-value-to-a-likelihood-termhttps://puzzicle.org/thumbs/xefa0c2256fdb6017.jpgMatching a probability value to a likelihood termLearn how to match a probability value to the correct likelihood term, from impossible to certain.https://cdn.kastatic.org/ka-youtube-converted/pW7iL7MdXmA.mp4/pW7iL7MdXmA.mp494yes https://puzzicle.org/math/7th-grade-illustrative-math/v/comparing-two-probability-values-expressed-in-different-formshttps://puzzicle.org/thumbs/x027645dc59692bee.jpgComparing two probability values expressed in different formsLearn how to compare probabilities expressed in different forms, such as percents and decimals, to determine which outcome is more likely.https://cdn.kastatic.org/ka-youtube-converted/4KUg8dcCnQI.mp4/4KUg8dcCnQI.mp474yes https://puzzicle.org/math/7th-grade-illustrative-math/v/identifying-valid-simulationshttps://puzzicle.org/thumbs/xe6f331f230f9b79e.jpgIdentifying valid simulationsLearn how to evaluate whether a simulation accurately models a real-world situation by checking that the probability in the simulation matches the probability in the situation.https://cdn.kastatic.org/ka-youtube-converted/sgpbnajY250.mp4/sgpbnajY250.mp4176yes https://puzzicle.org/math/7th-grade-illustrative-math/v/selecting-a-valid-simulation-for-a-multi-step-eventhttps://puzzicle.org/thumbs/xaea073e7905851aa.jpgSelecting a valid simulation for a multi-step eventLearn how to choose a simulation that correctly models a real-world probability by checking that both the probability and the number of trials match the situation.https://cdn.kastatic.org/ka-youtube-converted/rMvfxpoAaOA.mp4/rMvfxpoAaOA.mp4124yes https://puzzicle.org/math/7th-grade-illustrative-math/v/estimating-probability-from-a-simulationhttps://puzzicle.org/thumbs/x16c9e0c00cbdf1b6.jpgEstimating probability from a simulationLearn how to use simulation results to estimate the probability of a compound event by identifying how many trials produced the desired outcome.https://cdn.kastatic.org/ka-youtube-converted/4xK62sct1fY.mp4/4xK62sct1fY.mp4160yes https://puzzicle.org/math/7th-grade-illustrative-math/v/determining-whether-a-sample-is-representative-using-centerhttps://puzzicle.org/thumbs/xd0e0eab65fcea2f9.jpgDetermining whether a sample is representative using centerLearn how to calculate a sample mean and compare it to a population mean to determine whether a sample is a reasonable representation of the population.https://cdn.kastatic.org/ka-youtube-converted/zrwLq2Decck.mp4/zrwLq2Decck.mp4178yes https://puzzicle.org/math/7th-grade-illustrative-math/v/identifying-the-most-representative-sample-using-shape-center-and-spreadhttps://puzzicle.org/thumbs/x70805a8e961173aa.jpgIdentifying the most representative sample using shape, center, and spreadLearn how to identify the most representative sample by comparing the shape and spread of sample dot plots to the population dot plot.https://cdn.kastatic.org/ka-youtube-converted/QPqigb5jgR0.mp4/QPqigb5jgR0.mp4185yes https://puzzicle.org/math/7th-grade-illustrative-math/v/estimating-a-population-count-from-a-random-samplehttps://puzzicle.org/thumbs/x3c46154d4d096f69.jpgEstimating a population count from a random sampleLearn how to use survey data from a sample to make a proportional estimate about a larger population.https://cdn.kastatic.org/ka-youtube-converted/SpCYrTAzhjc.mp4/SpCYrTAzhjc.mp4157yes https://puzzicle.org/math/7th-grade-illustrative-math/v/comparing-populations-using-madhttps://puzzicle.org/thumbs/xa20d8ee974cbcc1c.jpgComparing populations using MADLearn how to determine whether two sample means are meaningfully different by comparing the difference in means to twice the larger MAD.https://cdn.kastatic.org/ka-youtube-converted/QhmATB1SkQs.mp4/QhmATB1SkQs.mp4190yes https://puzzicle.org/math/7th-grade-illustrative-math/v/comparing-populations-using-iqrhttps://puzzicle.org/thumbs/x5ce09b4ce3cb9f24.jpgComparing populations using IQRLearn how to use box plots to compare the medians and IQRs of two samples and decide whether the difference in typical values is meaningful.https://cdn.kastatic.org/ka-youtube-converted/s013HOLgMWM.mp4/s013HOLgMWM.mp4236yes https://puzzicle.org/math/7th-grade-illustrative-math/v/drawing-conclusions-from-sample-datahttps://puzzicle.org/thumbs/xbc9dfb6a851524a9.jpgDrawing conclusions from sample dataLearn how to identify what information is needed to determine whether a difference in sample medians is meaningful, and why the IQR is required to make that judgment.https://cdn.kastatic.org/ka-youtube-converted/E3DrQ7bfh7s.mp4/E3DrQ7bfh7s.mp4110yes https://puzzicle.org/math/cc-seventh-grade-math/v/introduction-proportional-relationshipshttps://puzzicle.org/thumbs/xbec0045658d07365.jpgIntroduction to proportional relationshipsProportional relationships are relationships between two variables where their ratios are equivalent. Another way to think about them is that, in a proportional relationship, one variable is always a constant value times the other. That constant is known as the "constant of proportionality".https://cdn.kastatic.org/ka-youtube-converted/l-HtxhClZ-0.mp4/l-HtxhClZ-0.mp4312yes https://puzzicle.org/math/7th-grade-math-eureka-squared-aligned/v/relating-side-lengths-in-similar-figureshttps://puzzicle.org/thumbs/xe51649b42c20dad7.jpgRelating side lengths in similar figuresGiven a pair of similar figures, write proportion statements that relate a side with an unknown length to known side lengths, using both ratios within and between the similar shapes. Then solve for the unknown side length.https://cdn.kastatic.org/ka-youtube-converted/zuV24mxOEfU.mp4/zuV24mxOEfU.mp4267yes https://puzzicle.org/math/geometry/v/similar-triangle-basicshttps://puzzicle.org/thumbs/284859451.jpgIntro to triangle similaritySal explains what it means for triangles to be similar, and how this follows from the definition of similarity.https://cdn.kastatic.org/ka-youtube-converted/9ThXDY9Y3oU.mp4/9ThXDY9Y3oU.mp4557yes https://puzzicle.org/math/7th-grade-math-eureka-squared-aligned/v/checking-triangle-similarity-with-side-length-ratioshttps://puzzicle.org/thumbs/xe93ac197a6317522.jpgChecking triangle similarity with side length ratiosGiven a triangle with the side lengths labeled, determine which other triangles are similar to it, based on ratios of side lengths between the figures.https://cdn.kastatic.org/ka-youtube-converted/5hhHMEp1Zng.mp4/5hhHMEp1Zng.mp4267yes https://puzzicle.org/math/geometry/v/similarity-example-problemshttps://puzzicle.org/thumbs/284859450.jpgSolving similar trianglesSal solves two problems where a missing side length is found by proving that triangles are similar and using this to find the measure.https://cdn.kastatic.org/ka-youtube-converted/R-6CAr_zEEk.mp4/R-6CAr_zEEk.mp4457yes https://puzzicle.org/math/cc-seventh-grade-math/v/exponents-with-negative-baseshttps://puzzicle.org/thumbs/x77666c47.jpgExponents with negative basesLearn to what we know about negative numbers to determine how negative bases with exponents are affected and what patterns develop. Also learn how order of operations affect the pattern.https://cdn.kastatic.org/ka-youtube-converted/vEZea0EThus.mp4/vEZea0EThus.mp4270yes https://puzzicle.org/math/cc-seventh-grade-math/v/powers-of-1-and-1https://puzzicle.org/thumbs/xb66d1c1a.jpg1 and -1 to different powersDifferent exponents affect the value of a number: when raised to the power of zero, any number equals one; when raised to an even power, negative numbers yield positive results; and when raised to an odd power, negative numbers yield negative results.https://cdn.kastatic.org/ka-youtube-converted/jYOfMszfzAQ.mp4/jYOfMszfzAQ.mp4368yes https://puzzicle.org/math/cc-seventh-grade-math/v/multistep-word-problems-example-1https://puzzicle.org/thumbs/xf59a9468.jpgRational number word problem: school reportFirst visualize this word problem then use subtraction and multiplication of decimals and fractions to get at the answer.https://cdn.kastatic.org/ka-youtube-converted/GwycEivqYYI.mp4/GwycEivqYYI.mp4220yes https://puzzicle.org/math/cc-seventh-grade-math/v/multistep-word-problems-example-2https://puzzicle.org/thumbs/x602a9681.jpgRational number word problem: cosmeticsRemember units of measurement? Convert minutes into hours and put your knowledge of fractions to work in this word problem.https://cdn.kastatic.org/ka-youtube-converted/QfsK8ZyC_bE.mp4/QfsK8ZyC_bE.mp4145yes https://puzzicle.org/math/cc-seventh-grade-math/v/rational-number-word-problem-example-2https://puzzicle.org/thumbs/x19dfbdfd.jpgRational number word problem: computersIn this word problem, we'll compare the weight of two objects using a ratio of integers. We'll practice simplifying a fraction, too.https://cdn.kastatic.org/ka-youtube-converted/oqAgaGeBOAU.mp4/oqAgaGeBOAU.mp4153yes https://puzzicle.org/math/cc-seventh-grade-math/v/rational-number-word-problem-example-3https://puzzicle.org/thumbs/x608113aa.jpgRational number word problem: stockUse your knowledge of proportions and fractions to determine shareholders profit in this word problem.https://cdn.kastatic.org/ka-youtube-converted/KiVGac1aBt8.mp4/KiVGac1aBt8.mp4248yes https://puzzicle.org/math/cc-seventh-grade-math/v/proof-vertical-angles-are-equalhttps://puzzicle.org/thumbs/263439066.jpgVertical angles are congruent proofProving that vertical angles are equalhttps://cdn.kastatic.org/ka-youtube-converted/wRBMmiNHQaE.mp4/wRBMmiNHQaE.mp4291yes https://puzzicle.org/math/cc-seventh-grade-math/v/using-algebra-to-find-the-measures-of-vertical-angleshttps://puzzicle.org/thumbs/x67c17e9a.jpgEquation practice with vertical anglesGiven algebraic expressions that represent a pair of vertical angles, Sal forms and solves an equation.https://cdn.kastatic.org/ka-youtube-converted/2CZrkdtgeNU.mp4/2CZrkdtgeNU.mp4154yes https://puzzicle.org/math/7th-grade-math-eureka-squared-aligned/v/modeling-two-step-equationshttps://puzzicle.org/thumbs/x89a6dc7b61763998.jpgModeling two step equationsLet's use algebra tiles to model and solve two-step equations with integer coefficients in one variable. We'll see how each step of the model connects with how we can show the same step algebraically.https://cdn.kastatic.org/ka-youtube-converted/w3XjNDBc5sI.mp4/w3XjNDBc5sI.mp4231yes https://puzzicle.org/math/7th-grade-math-eureka-squared-aligned/v/writing-two-step-word-problemshttps://puzzicle.org/thumbs/x21173d321f0837d3.jpgWriting two step word problemsLet's write some contexts to represent a two-step equation and inequality in one variable. We'll consider what makes sense with a negative rate of change and what the unknown we'd be solving for in each case would be.https://cdn.kastatic.org/ka-youtube-converted/nMXNCvmhomY.mp4/nMXNCvmhomY.mp4293yes https://puzzicle.org/math/7th-grade-math-eureka-squared-aligned/v/modeling-two-step-inequalitieshttps://puzzicle.org/thumbs/xfbc387dea89fd93d.jpgModeling two step inequalitiesWe can use algebra tiles to model and solve two-step inequalities with integer coefficients in one variable in the same way that we model equations. Let's solve an inequality with a positive coefficient, but negative constants.https://cdn.kastatic.org/ka-youtube-converted/t1DiyNTwMLU.mp4/t1DiyNTwMLU.mp4178yes https://puzzicle.org/math/7th-grade-math-eureka-squared-aligned/v/modeling-inequalities-with-negative-coefficientshttps://puzzicle.org/thumbs/xc616985760ca27b4.jpgModeling inequalities with negative coefficientsHow do we make negative 3 equal groups? We'll use algebra tiles to model and solve an inequality of the form Ax + B is greater than C, where A is negative. Let's connect the model to the algebraic representation to really understand what happens with the inequality symbol when we divide or multiply both sides of the inequality by a negative number.https://cdn.kastatic.org/ka-youtube-converted/npLlPecY8Ic.mp4/npLlPecY8Ic.mp4391yes https://puzzicle.org/math/7th-grade-math-eureka-squared-aligned/v/relating-diameter-and-circumferencehttps://puzzicle.org/thumbs/x3650e6f98d15f99e.jpgRelating diameter and circumferenceWrite algebraic expressions relating pi and the different parts of a circle.https://cdn.kastatic.org/ka-youtube-converted/s2vnzKp3EPQ.mp4/s2vnzKp3EPQ.mp4432yes https://puzzicle.org/math/geometry/v/informal-argument-for-the-area-of-circle-formulahttps://puzzicle.org/thumbs/x778456ed.jpgArea of a circle intuitionUsing triangles to create an informal argument for the area of a circle formula.https://cdn.kastatic.org/ka-youtube-converted/a2L27dTOyHk.mp4/a2L27dTOyHk.mp4566yes https://puzzicle.org/math/7th-grade-math-eureka-squared-aligned/v/area-of-composite-figure-with-circleshttps://puzzicle.org/thumbs/xe26ece7269c17cb6.jpgArea of composite figure with circlesFind the area of a composite figure by splitting it into a triangle and fractions of circles.https://cdn.kastatic.org/ka-youtube-converted/nuOD3-4ZOlg.mp4/nuOD3-4ZOlg.mp4225yes https://puzzicle.org/math/7th-grade-math-eureka-squared-aligned/v/area-composite-figure-parallelogramshttps://puzzicle.org/thumbs/x3843fcd6513844f3.jpgArea of composite figure with parallelogramsFind the area of a composite figure by splitting it into congruent parallelograms and rectangles.https://cdn.kastatic.org/ka-youtube-converted/lHxLRfxqdN4.mp4/lHxLRfxqdN4.mp4119yes https://puzzicle.org/math/7th-grade-math-eureka-squared-aligned/v/total-and-lateral-surface-area-using-a-net-rectangular-pyramidhttps://puzzicle.org/thumbs/x9b27361786920930.jpgTotal and lateral surface area using a net rectangular pyramidA rectangular pyramid has 5 faces, a rectangular base and 4 triangular lateral faces. Given the net of a rectangular pyramid, let's determine the lateral and total surface area of the figure. The total surface area is the area of all the faces, and the lateral surface area leaves out the base.https://cdn.kastatic.org/ka-youtube-converted/rKSZA9sHNlY.mp4/rKSZA9sHNlY.mp4229yes https://puzzicle.org/math/7th-grade-math-eureka-squared-aligned/v/lateral-surface-area-using-a-net-triangular-prismhttps://puzzicle.org/thumbs/xca67fe7f8d820429.jpgLateral surface area using a net triangular prismGiven the net of a triangular prism, we'll determine the lateral surface area of the figure. The lateral surface area is the sum of the area of all of the faces except for the 2 parallel triangular bases.https://cdn.kastatic.org/ka-youtube-converted/fFVFOdPRxxk.mp4/fFVFOdPRxxk.mp4142yes https://puzzicle.org/math/7th-grade-math-eureka-squared-aligned/v/volume-of-rectangular-pyramid-part-1-cubehttps://puzzicle.org/thumbs/xcf1390864d5d8df1.jpgVolume of rectangular pyramids using cubesExplore why the formula for the volume of rectangular pyramids has a 1/3 in it! We'll start by investigating with a cube.https://cdn.kastatic.org/ka-youtube-converted/tDNCy8dalPY.mp4/tDNCy8dalPY.mp491yes https://puzzicle.org/math/7th-grade-math-eureka-squared-aligned/v/volume-of-rectangular-pyramid-part-2-rectangular-prismhttps://puzzicle.org/thumbs/xfe9b10e15c2f2354.jpgVolume of rectangular pyramids using rectangular prismsExplore why the formula for the volume of a rectangular pyramid has a 1/3 in it! We'll investigate a general rectangular prism.https://cdn.kastatic.org/ka-youtube-converted/PPey8fjuH84.mp4/PPey8fjuH84.mp4268yes https://puzzicle.org/math/cc-eighth-grade-math/v/comparing-proportional-relationships-exercise-2https://puzzicle.org/thumbs/xedb42f94.jpgRates & proportional relationships exampleLet's compare unit rates in equations and graphs. Learn how a change in 'x' affects 'y' in an equation like y = 6.5x, and see how this compares to the rate of change in a graph. Uncover why one might increase at a slower pace than the other.https://cdn.kastatic.org/ka-youtube-converted/kHQCDrWTDv8.mp4/kHQCDrWTDv8.mp485yes https://puzzicle.org/math/cc-eighth-grade-math/v/comparing-proportional-relationships-exercisehttps://puzzicle.org/thumbs/x1ce04c40.jpgRates & proportional relationships: gas mileageMaria and Nadia calculate their speed on a road trip to Toronto using distance and time. We compare different speeds using equations and tables. Then, we reinforce the concept by comparing the revolution speeds of records.https://cdn.kastatic.org/ka-youtube-converted/F6rtQczAYco.mp4/F6rtQczAYco.mp4199yes https://puzzicle.org/math/7th-grade-math-eureka-squared-aligned/v/combining-discountshttps://puzzicle.org/thumbs/xdf3304cf3c490071.jpgCombining discountsLet's calculate prices with multiple discounts applied to each. We'll explore how the order of the discounts affects the final price.https://cdn.kastatic.org/ka-youtube-converted/wR-qdDkOzC8.mp4/wR-qdDkOzC8.mp4270yes https://puzzicle.org/math/7th-grade-math-eureka-squared-aligned/v/comparing-discountshttps://puzzicle.org/thumbs/xe170683fa18c8862.jpgComparing discountsLet's compare the prices and discounts available at different stores and determine the lowest price available for a purchase for different quantities of the object. We'll also consider other factors besides price that might influence where we choose to make a purchase.https://cdn.kastatic.org/ka-youtube-converted/ZxCrTlHnUu0.mp4/ZxCrTlHnUu0.mp4242yes https://puzzicle.org/math/algebra-1-eureka-squared-aligned/v/calculating-simple-compound-interesthttps://puzzicle.org/thumbs/xed3bba74863d1e86.jpgCalculating simple & compound interestCalculate and compare simple interest and compound interest earnings.https://cdn.kastatic.org/ka-youtube-converted/79HS3N1IBt8.mp4/79HS3N1IBt8.mp4394yes https://puzzicle.org/math/7th-grade-math-eureka-squared-aligned/v/compound-events-tree-diagramhttps://puzzicle.org/thumbs/x3b1c6fb6.jpgCompound events example with tree diagramSal figures out the probability of flipping three coins and getting at least two tails.https://cdn.kastatic.org/ka-youtube-converted/4fBXIcO0It4.mp4/4fBXIcO0It4.mp4144yes https://puzzicle.org/math/precalculus/v/independent-events-2https://puzzicle.org/thumbs/27095.jpgIndependent events example: test takingHave you ever taken a test and discovered you have no choice but to guess on a couple of problems? In this example problem, we are considering the probability of two independent events occurring.https://cdn.kastatic.org/ka-youtube-converted/VWAfEbgf1Po.mp4/VWAfEbgf1Po.mp4284yes https://puzzicle.org/math/precalculus/v/compound-probability-of-independent-eventshttps://puzzicle.org/thumbs/191769945.jpgCompound probability of independent eventsLearn how to calculate probabilities of coin flips using two methods: listing all possible outcomes or multiplying independent events. Understand what independent events are and how to avoid the gambler's fallacy. Review fractions and how the sum of all probabilities equals one.https://cdn.kastatic.org/ka-youtube-converted/xSc4oLA9e8o.mp4/xSc4oLA9e8o.mp4360yes https://puzzicle.org/math/statistics-probability/v/lebron-asks-about-the-chances-of-making-10-free-throwshttps://puzzicle.org/thumbs/753767269.jpgFree-throw probabilityOur friend and Cleveland Cavalier, LeBron James, asks Sal how to determine the probability of making 10 free throws in a row. Hint: the answer is surprising!https://cdn.kastatic.org/ka-youtube-converted/iMqCSgqzmiQ.mp4/iMqCSgqzmiQ.mp4425yes https://puzzicle.org/math/statistics-probability/v/lebron-asks-about-free-throws-versus-three-pointershttps://puzzicle.org/thumbs/1022209423.jpgThree-pointer vs free-throw probabilityOur friend and Cleveland Cavalier, LeBron James, asks Sal if there's a high probability of making three free throws in a row or one three-pointer. Before solving the problem, jot down what you think the answer will be!https://cdn.kastatic.org/ka-youtube-converted/PddbEVNMgTY.mp4/PddbEVNMgTY.mp4347yes https://puzzicle.org/math/precalculus/v/probability-with-playing-cards-and-venn-diagramshttps://puzzicle.org/thumbs/185786264.jpgProbability with Venn diagramsWant to learn some probability? This video explains the probability of drawing a Jack or a Heart from a deck of 52 cards. It uses a Venn diagram to illustrate the concept of overlapping events and how to calculate the combined probability. Key definitions include "equally likely events" and "overlapping events".https://cdn.kastatic.org/ka-youtube-converted/obZzOq_wSCg.mp4/obZzOq_wSCg.mp4602yes https://puzzicle.org/math/precalculus/v/addition-rule-for-probabilityhttps://puzzicle.org/thumbs/188066789.jpgAddition rule for probabilityVenn diagrams and the addition rule for probabilityhttps://cdn.kastatic.org/ka-youtube-converted/QE2uR6Z-NcU.mp4/QE2uR6Z-NcU.mp4643yes https://puzzicle.org/math/probability/v/random-numbers-for-experimental-probabilityhttps://puzzicle.org/thumbs/x8e152fdf6ff0b58e.jpgRandom numbers for experimental probabilityUsing a list of random number to calculate an experimental probability.https://cdn.kastatic.org/ka-youtube-converted/vjGINFbV8Cs.mp4/vjGINFbV8Cs.mp4449yes https://puzzicle.org/math/ap-statistics/v/random-number-list-to-run-experimenthttps://puzzicle.org/thumbs/x67e16aadb45da1f7.jpgRandom number list to run experimentUsing a list of random numbers to simulate multiple trials of an experiment.https://cdn.kastatic.org/ka-youtube-converted/2AVLfSRpmfg.mp4/2AVLfSRpmfg.mp4451yes https://puzzicle.org/math/algebra-basics/v/adding-integers-with-different-signshttps://puzzicle.org/thumbs/13179312.jpgAdding numbers with different signsUse a number line to add 15 + (-46) + 29https://cdn.kastatic.org/ka-youtube-converted/NrVvu7cM8_o.mp4/NrVvu7cM8_o.mp4332yes https://puzzicle.org/math/arithmetic-home/v/negative-number-practicehttps://puzzicle.org/thumbs/x2ef60d58.jpgAddition & subtraction: find the missing value (part 1 of 2)Sal solves some missing value problems: -1 = 7 - __ and -10 + __ = -5 and 4 = -2 + __ and 6 + __ = -8https://cdn.kastatic.org/ka-youtube-converted/-4bTgmmWI9k.mp4/-4bTgmmWI9k.mp4333yes https://puzzicle.org/math/arithmetic-home/v/even-more-negative-number-practicehttps://puzzicle.org/thumbs/xf26670ad.jpgAddition & subtraction: find the missing value (part 2 of 2)Sal solves more missing value problems: 4 = -8 - __ and __ - (-2) = -7 and -4 = __ - 9https://cdn.kastatic.org/ka-youtube-converted/KNGa11O2uLE.mp4/KNGa11O2uLE.mp4400yes https://puzzicle.org/math/mappers/v/adding-and-subtracting-three-fractionshttps://puzzicle.org/thumbs/xeff72154.jpgAdding & subtracting fractionsAdd and subtract negative fractions with unlike denominators.https://cdn.kastatic.org/ka-youtube-converted/9tmtDBpqq9s.mp4/9tmtDBpqq9s.mp4156yes https://puzzicle.org/math/cc-eighth-grade-math/v/raising-a-number-to-the-0th-and-1st-powerhttps://puzzicle.org/thumbs/x2bba124f.jpgThe 0 & 1st powerDifferent ways of thinking about exponents. Raising a number to an exponent means multiplying that number by itself a certain number of times. Any non-zero number raised to the zero power will be equal to one, and that any number raised to the first power will be equal to itself.https://cdn.kastatic.org/ka-youtube-converted/dAvosUEUH6I.mp4/dAvosUEUH6I.mp4307yes https://puzzicle.org/math/cc-eighth-grade-math/v/multiplying-and-dividing-even-and-odd-numbers-of-negativeshttps://puzzicle.org/thumbs/x4940c781.jpgEven & odd numbers of negativesWe can figure out whether multiplication and division problems give us a positive or negative result by thinking about how many negative numbers are used in the computation.https://cdn.kastatic.org/ka-youtube-converted/XHHYA2Ug9lk.mp4/XHHYA2Ug9lk.mp4426yes https://puzzicle.org/math/cc-eighth-grade-math/v/thinking-about-the-sign-of-expressionshttps://puzzicle.org/thumbs/x0fa0d50a.jpgSign of expressions challenge problemsSome examples that test our understanding of what happens when we multply or divide a bunch of positive or negative numbers.https://cdn.kastatic.org/ka-youtube-converted/Pms4cBWwPSU.mp4/Pms4cBWwPSU.mp4519yes https://puzzicle.org/math/cc-eighth-grade-math/v/exponent-properties-involving-productshttps://puzzicle.org/thumbs/23686.jpgExponent properties with productsTo simplify expressions with exponents, there are a few properties that may help. One is that when two numbers with the same base are multiplied, the exponents can be added. Another is that when a number with an exponent is raised to another exponent, the exponents can be multiplied.https://cdn.kastatic.org/ka-youtube-converted/zM_p7tfWvLU.mp4/zM_p7tfWvLU.mp4840yes https://puzzicle.org/math/cc-eighth-grade-math/v/products-and-exponents-raised-to-an-exponent-propertieshttps://puzzicle.org/thumbs/x7c5d0111.jpgExponent properties with parenthesesLearn two exponent properties: (ab)^c = (a^c)*(b^c) and (a^b)^c = a ^ (b*c). See WHY they work and HOW to use them. In other words, multiplying two numbers, then raising the product to an exponent is the same as raising each number to that exponent and then multiplying. Raising a number to an exponent and then to another exponent equals raising the base to the product of the two exponents.https://cdn.kastatic.org/ka-youtube-converted/SwqOrUWzDY8.mp4/SwqOrUWzDY8.mp4358yes https://puzzicle.org/math/cc-eighth-grade-math/v/exponent-properties-involving-quotientshttps://puzzicle.org/thumbs/23688.jpgExponent properties with quotientsLearn how to simplify expressions like (5^6)/(5^2). Also learn how 1/(a^b) is the same as a^-b. Towards the end of the video, we practice simplifying more complex expressions like (25 * x * y^6)/(20 * y^5 * x^2).https://cdn.kastatic.org/ka-youtube-converted/AR1uqNbjM5s.mp4/AR1uqNbjM5s.mp4562yes https://puzzicle.org/math/cc-eighth-grade-math/v/negative-exponentshttps://puzzicle.org/thumbs/x9ecf1c5c.jpgNegative exponentsNegative exponents can be rewritten in two ways. Firstly, start with 1 and divide it by 2 the same number of times as the exponent. Secondly, take the reciprocal of the base and raise it to the positive exponent.https://cdn.kastatic.org/ka-youtube-converted/JnpqlXN9Whw.mp4/JnpqlXN9Whw.mp4434yes https://puzzicle.org/math/cc-eighth-grade-math/v/negative-exponent-intuitionhttps://puzzicle.org/thumbs/25767.jpgNegative exponent intuitionHow do negative exponents work? Let's build our intuition about why a^(-b) = 1/(a^b) and how this definition keeps exponent rules consistent. Continue the pattern of decreasing exponents by dividing by 'a', and see how it extends to zero and negative powers. While we're at it, we'll see why a^0 =1.https://cdn.kastatic.org/ka-youtube-converted/Tqpcku0hrPU.mp4/Tqpcku0hrPU.mp4278yes https://puzzicle.org/math/cc-eighth-grade-math/v/multiplying-multiples-of-powers-of-10https://puzzicle.org/thumbs/x36950ef2.jpgMultiplying multiples of powers of 10Let's multiply (9 * 10^9) (-2 * 10^-3) using the power of exponents! Change the order of multiplication to make it easier, multiply the non-powers first. and then simplify the powers of 10. Remember, multiplying by a negative changes the sign of the product. It's all about using exponent properties to simplify the product.https://cdn.kastatic.org/ka-youtube-converted/tV8cMNYi4q4.mp4/tV8cMNYi4q4.mp4161yes https://puzzicle.org/math/cc-eighth-grade-math/v/approximating-with-powers-of-10https://puzzicle.org/thumbs/xc45ba77a.jpgApproximating with powers of 10How much larger was the world population than the US population in 2014? We can use division and powers of 10 to get a sense of just how many times as large the world population was. Is there really such a big difference between 10 to the 8th and 10 to the 9th? You bet there is!https://cdn.kastatic.org/ka-youtube-converted/Xl724v0CU7E.mp4/Xl724v0CU7E.mp4250yes https://puzzicle.org/math/cc-eighth-grade-math/v/scientific-notationhttps://puzzicle.org/thumbs/4802223.jpgScientific notation examplesScientific notation is a way of writing very large or very small numbers. A number is written in scientific notation when a number between 1 and 10 is multiplied by a power of 10. For example, 650,000,000 can be written in scientific notation as 6.5 ✕ 10^8. Created by Sal Khan and CK-12 Foundation.https://cdn.kastatic.org/ka-youtube-converted/i6lfVUp5RW8.mp4/i6lfVUp5RW8.mp4686yes https://puzzicle.org/math/cc-eighth-grade-math/v/scientific-notation-ihttps://puzzicle.org/thumbs/13179507.jpgScientific notation example: 0.0000000003457Can you imagine if you had to do calculations with very, very small numbers? How would you handle all those zeros to the right of the decimal? Thank goodness for scientific notation!https://cdn.kastatic.org/ka-youtube-converted/6phoVfGKKec.mp4/6phoVfGKKec.mp4252yes https://puzzicle.org/math/algebra-basics/v/scientific-notation-example-2https://puzzicle.org/thumbs/13179508.jpgMultiplying in scientific notation exampleScientific notation shows a number (greater than or equal to 1 and less than 10) times a power of 10. To multiply two numbers in scientific notation, we can rearrange the equation with the associative and commutative properties. If the final product is not in scientific notation, we can regroup a factor of 10.https://cdn.kastatic.org/ka-youtube-converted/xxAFh-qHPPA.mp4/xxAFh-qHPPA.mp4323yes https://puzzicle.org/math/cc-eighth-grade-math/v/multiplying-and-dividing-in-scientific-notationhttps://puzzicle.org/thumbs/xdf6e3aee.jpgMultiplying & dividing in scientific notationIn order to simplify multiplication and division using scientific notation, you should multiply and divide numbers with the same base, and add or subtract the exponents. Through this process, complex expressions can be simplified into a single value multiplied with 10 to a certain power. As an example, 7 times 10 to the fifth over 2 times 10 to the negative 2 times 2.5 times 10 to the ninth can be simplified to 1.4 times 10 to the negative 2. Scientific notation helps to simplify complex equations that involve multiplying and dividing numbers with the same base.https://cdn.kastatic.org/ka-youtube-converted/497oIjqRPco.mp4/497oIjqRPco.mp4173yes https://puzzicle.org/math/cc-eighth-grade-math/v/multiplying-in-scientific-notationhttps://puzzicle.org/thumbs/466009718.jpgMultiplying three numbers in scientific notationMultiplying really big or really small numbers is much easier when using scientific notation. When we multiply numbers in scientific notation, we can commute and associate the factors. That lets us multiply the decimal factors first. Then we can use exponent properties to multiply the powers of 10.https://cdn.kastatic.org/ka-youtube-converted/67jn5Zv-myg.mp4/67jn5Zv-myg.mp4455yes https://puzzicle.org/math/cc-eighth-grade-math/v/subtracting-in-scientific-notationhttps://puzzicle.org/thumbs/x0c719c2c.jpgSubtracting in scientific notationLearn how to subtract numbers written in scientific notation. The problem solved in this video is (4.1 * 10^-2) - (2.6 * 10^-3).https://cdn.kastatic.org/ka-youtube-converted/ios3QL9t9LQ.mp4/ios3QL9t9LQ.mp4253yes https://puzzicle.org/math/cc-eighth-grade-math/v/simplifying-a-complicated-expression-into-scientific-notationhttps://puzzicle.org/thumbs/x80584388.jpgSimplifying in scientific notation challengeLet's explore how to simplify expressions with numbers in scientific notation. We'll transform numbers into scientific notation, multiply and divide them, and then reassemble them. It's a cool and useful method for handling large or small numbers!https://cdn.kastatic.org/ka-youtube-converted/EbmgLiSVACU.mp4/EbmgLiSVACU.mp4352yes https://puzzicle.org/math/cc-eighth-grade-math/v/calculating-red-blood-cells-in-the-body-using-scientific-notationhttps://puzzicle.org/thumbs/x767b7883.jpgScientific notation word problem: red blood cellsVampires and math students want to know: How many red blood cells are in the a human body? We can find the answer using scientific notation.https://cdn.kastatic.org/ka-youtube-converted/0lOpqmTdtzk.mp4/0lOpqmTdtzk.mp4286yes https://puzzicle.org/math/cc-eighth-grade-math/v/scientific-notation-3https://puzzicle.org/thumbs/26898.jpgScientific notation word problem: U.S. national debtEver wonder what your part of the national debt is? It might surprise you. What isn't surprising is that you can use scientific notation and division to figure out the answer.https://cdn.kastatic.org/ka-youtube-converted/XJBwJjP2_hM.mp4/XJBwJjP2_hM.mp4381yes https://puzzicle.org/math/cc-eighth-grade-math/v/scientific-notation-3-newhttps://puzzicle.org/thumbs/13179509.jpgScientific notation word problem: speed of lightIt is possible to simplify multiplication and division using scientific notation. This can be used to calculate the distance between the sun and the earth, which is 1.5 times 10 to the 11th power meters. This is an incredibly large distance and difficult to visualize. Scientific notation can be used to simplify calculations and understand large numbers. This involves using the commutative property to rearrange the numbers and multiplying the units, and then adding the exponents to simplify the equation.https://cdn.kastatic.org/ka-youtube-converted/dD71fFNwVJw.mp4/dD71fFNwVJw.mp4267yes https://puzzicle.org/math/8th-engage-ny/v/sets-overviewhttps://puzzicle.org/thumbs/x2563c48d.jpgIntroduction to geometric transformationsHow are virtual sets built in the computer? Pixar uses geometric transformations in animation to manipulate the position, orientation, and size of objects. This process involves story reels, set dressing, and camera staging. Each shot is carefully crafted, using math as the foundation!https://cdn.kastatic.org/ka-youtube-converted/lR0kyOdT86E.mp4/lR0kyOdT86E.mp4186yes https://puzzicle.org/math/geometry/v/identifying-transformationshttps://puzzicle.org/thumbs/x3d83ba250d08dc20.jpgIdentifying transformationsLet's look at four types of transformations: rotations (spinning a shape around a point), translations (shifting a shape), reflections (flipping a shape over a line), and dilations (shrinking or expanding a shape). We practice identifying these transformations in different pairs of figures.https://cdn.kastatic.org/ka-youtube-converted/K7QTjFdUIyk.mp4/K7QTjFdUIyk.mp4227yes https://puzzicle.org/math/geometry/v/translating-pointshttps://puzzicle.org/thumbs/x154b1155a0199523.jpgTranslating pointsLet's practice translations on a coordinate plane. A translation is a way to move a point or shape by a certain number of units in a certain direction. For example, we can translate a point by moving it 5 units left and 3 units up. We describe this translation algebraically using the coordinates (x-5, y+3).https://cdn.kastatic.org/ka-youtube-converted/SxxzfojT1Pw.mp4/SxxzfojT1Pw.mp4231yes https://puzzicle.org/math/geometry/v/drawing-image-of-translationhttps://puzzicle.org/thumbs/x03952f21.jpgTranslating shapesIn translations, we slide a shape around on a grid. We use the letter "T" to represent translations. We move every point of the shape a certain distance left or right, and up or down, to create a new shape that's the same size and shape as the original. We call the new shape the image.https://cdn.kastatic.org/ka-youtube-converted/g_ijRF_NMLk.mp4/g_ijRF_NMLk.mp4115yes https://puzzicle.org/math/geometry/v/formal-translation-tool-examplehttps://puzzicle.org/thumbs/x38f28d5c.jpgDetermining translationsWe can find the translation by figuring out how much the x- and y-coordinates need to change to map one triangle onto the other. We map each point to its corresponding point in the other triangle. Moving left means the translation of x is negative, and moving down means the translation of y is negative.https://cdn.kastatic.org/ka-youtube-converted/BrYEuO7fwMw.mp4/BrYEuO7fwMw.mp492yes https://puzzicle.org/math/geometry/v/rotating-pointshttps://puzzicle.org/thumbs/x0dcf739db7b81a07.jpgRotating pointsPositive rotation angles mean we turn counterclockwise. Negative angles are clockwise. We can think of a 60 degree turn as 1/3 of a 180 degree turn. A 90 degree turn is 1/4 of the way around a full circle. The angle goes from the center to first point, then from the center to the image of the point.https://cdn.kastatic.org/ka-youtube-converted/0NYbQPcdxg8.mp4/0NYbQPcdxg8.mp4110yes https://puzzicle.org/math/geometry/v/defining-rotation-examplehttps://puzzicle.org/thumbs/x5a934725.jpgDetermining rotationsTo see the angle of rotation, we draw lines from the center to the same point in the shape before and after the rotation. Counterclockwise rotations have positive angles, while clockwise rotations have negative angles. Then we estimate the angle. For example, 30 degrees is 1/3 of a right angle.https://cdn.kastatic.org/ka-youtube-converted/6pH-7pciYU8.mp4/6pH-7pciYU8.mp4177yes https://puzzicle.org/math/geometry/v/reflecting-pointshttps://puzzicle.org/thumbs/x2b4f6a857249e629.jpgReflecting pointsWe can plot points after reflecting them across a line, like the x-axis or y-axis. Reflections create mirror images of points, keeping the same distance from the line. When we reflect across the y-axis, the image point is the same height, but has the opposite position from left to right.https://cdn.kastatic.org/ka-youtube-converted/F77xiSlCehI.mp4/F77xiSlCehI.mp4225yes https://puzzicle.org/math/geometry/v/reflecting-shapeshttps://puzzicle.org/thumbs/xc5d517f510449447.jpgReflecting shapesLet's reflect a quadrilateral across the x-axis. To do this, we find new points (A', B', C', D') by keeping the same x-coordinates and changing the y-coordinates to their opposite signs. This creates a flipped image of the original quadrilateral.https://cdn.kastatic.org/ka-youtube-converted/lfBDOQddTD0.mp4/lfBDOQddTD0.mp4158yes https://puzzicle.org/math/geometry/v/determining-reflectionshttps://puzzicle.org/thumbs/xae5156462a07bf1d.jpgDetermining reflectionsA line of reflection is an imaginary line that flips one shape onto another. We find this line by finding the halfway points between matching points on the source and image triangles. All of the halfway points are on the line. Once we find that line, it shows how one triangle reflects onto the other.https://cdn.kastatic.org/ka-youtube-converted/435Ipnmp7bA.mp4/435Ipnmp7bA.mp4100yes https://puzzicle.org/math/geometry/v/finding-measures-using-rigid-transformationshttps://puzzicle.org/thumbs/x9324fd9746376a81.jpgFinding measures using rigid transformationsFinding measures after a rigid transformation, like a reflection, is pretty simple! Since the shape and size stay the same, the lengths of corresponding sides and angle measures remain unchanged. Area and perimeter depend on the side lengths, so they stay the same too. So, if we know the measures of the original figure, we can use those same measures for the transformed figure.https://cdn.kastatic.org/ka-youtube-converted/7y4FGCvNgB8.mp4/7y4FGCvNgB8.mp4296yes https://puzzicle.org/math/geometry/v/rigid-transformations-preserved-propertieshttps://puzzicle.org/thumbs/x89713b044e601426.jpgRigid transformations: preserved propertiesRigid transformations, like rotations and reflections, change a shape's position but keep its size and shape. These transformations preserve side lengths, angle measures, perimeter, and area. But they might not keep the same coordinates or relationships to lines outside the figure.https://cdn.kastatic.org/ka-youtube-converted/brccWoewXiI.mp4/brccWoewXiI.mp4420yes https://puzzicle.org/math/geometry/v/mapping-shapeshttps://puzzicle.org/thumbs/x43371bfd817e219c.jpgMapping shapesLet's find the right sequence of rigid transformations (like rotations, translations, and reflections) to map one triangle onto another. Different sequences can work, but order matters. So, it's important to test each one to see if it maps the triangles correctly.https://cdn.kastatic.org/ka-youtube-converted/QOXBJ2X1_9k.mp4/QOXBJ2X1_9k.mp4531yes https://puzzicle.org/math/geometry/v/testing-congruence-by-transformations-examplehttps://puzzicle.org/thumbs/xdfa0c35c.jpgCongruent shapes & transformationsIf we can map one figure onto another using rigid transformations, they are congruent. They are still congruent if we need to use more than one transformation to map it. They aren't if we use a transformation that changes the size of the shape.https://cdn.kastatic.org/ka-youtube-converted/KZMRhWdzEPo.mp4/KZMRhWdzEPo.mp4103yes https://puzzicle.org/math/geometry/v/another-congruence-by-transformation-examplehttps://puzzicle.org/thumbs/x5172fee3.jpgNon-congruent shapes & transformationsCongruent shapes are the same size and shape. Rigid transformations, like translations, keep shapes congruent, but dilations are not rigid transformations because they change the size. So, if we use a dilation to map one shape onto another, they are not congruent.https://cdn.kastatic.org/ka-youtube-converted/7Rb3V-HyVD8.mp4/7Rb3V-HyVD8.mp4123yes https://puzzicle.org/math/8th-engage-ny/v/sat-math-s6-easierhttps://puzzicle.org/thumbs/x8e8c18af.jpgCongruence and similarity — Basic exampleWatch Sal work through a basic congruence and similarity problem.https://cdn.kastatic.org/ka-youtube-converted/XcXmLHra3aA.mp4/XcXmLHra3aA.mp4259yes https://puzzicle.org/math/8th-engage-ny/v/sat-math-s6-harderhttps://puzzicle.org/thumbs/x1adedf5e.jpgCongruence and similarity — Harder exampleWatch Sal work through a harder Congruence and similarity problem.https://cdn.kastatic.org/ka-youtube-converted/nKpHg25_AdM.mp4/nKpHg25_AdM.mp4198yes https://puzzicle.org/math/cc-eighth-grade-math/v/angles-formed-by-parallel-lines-and-transversalshttps://puzzicle.org/thumbs/263439061.jpgAngles, parallel lines, & transversalsParallel lines are lines in the same plane that go in the same direction and never intersect. When a third line, called a transversal, crosses these parallel lines, it creates angles. Some angles are equal, like vertical angles (opposite angles) and corresponding angles (same position at each intersection).https://cdn.kastatic.org/ka-youtube-converted/H-E5rlpCVu4.mp4/H-E5rlpCVu4.mp4427yes https://puzzicle.org/math/cc-eighth-grade-math/v/figuring-out-angles-between-transversal-and-parallel-lineshttps://puzzicle.org/thumbs/xc5b28d56.jpgMissing angles with a transversalWhen a third line called a transversal crosses two parallel lines, we can find the measures of angles using properties like corresponding angles, vertical angles, and supplementary angles. If we know just one of the angle measurements, these properties help us find all the missing angle measurements.https://cdn.kastatic.org/ka-youtube-converted/2WjGD3LZEWo.mp4/2WjGD3LZEWo.mp4136yes https://puzzicle.org/math/cc-eighth-grade-math/v/using-algebra-to-find-measures-of-angles-formed-from-transversalhttps://puzzicle.org/thumbs/xa92ee636.jpgMeasures of angles formed by a transversalWhen parallel lines are crossed by another line called a transversal, special angles are formed. We can find the measures of these angles using algebra and the relationships between the angles, like congruent angles and supplementary angles. Once we solve for the variable, we use that value to figure out the angle measures.https://cdn.kastatic.org/ka-youtube-converted/Ld7Vxb5XV6A.mp4/Ld7Vxb5XV6A.mp4281yes https://puzzicle.org/math/geometry/v/proof-sum-of-measures-of-angles-in-a-triangle-are-180https://puzzicle.org/thumbs/265540637.jpgAngles in a triangle sum to 180° proofThe sum of the interior angle measures of a triangle always adds up to 180°. We can draw a line parallel to the base of any triangle through its third vertex. Then we use transversals, vertical angles, and corresponding angles to rearrange those angle measures into a straight line, proving that they must add up to 180°.https://cdn.kastatic.org/ka-youtube-converted/6s1CI3uuhko.mp4/6s1CI3uuhko.mp4268yes https://puzzicle.org/math/geometry/v/triangle-angle-example-1https://puzzicle.org/thumbs/265540642.jpgTriangle exterior angle exampleAn exterior angle is formed outside a triangle by extending a side. Its measure is equal to the sum of the two non-adjacent interior angle measures, thanks to the fact that all angle measures in a triangle add up to 180 degrees.https://cdn.kastatic.org/ka-youtube-converted/hmj3_zbz2eg.mp4/hmj3_zbz2eg.mp4363yes https://puzzicle.org/math/cc-eighth-grade-math/v/triangle-angle-example-2https://puzzicle.org/thumbs/265540638.jpgWorked example: Triangle angles (intersecting lines)Learn how to find unknown angles in triangles and intersecting lines. We use concepts like supplementary angles (angles that add up to 180°), vertical angles (opposite angles made by two crossing lines), and exterior angles (angles outside a triangle). There are lots of ways to think about this problem.https://cdn.kastatic.org/ka-youtube-converted/eTwnt4G5xE4.mp4/eTwnt4G5xE4.mp4295yes https://puzzicle.org/math/cc-eighth-grade-math/v/triangle-angle-example-3https://puzzicle.org/thumbs/265540640.jpgWorked example: Triangle angles (diagram)We have a right triangle that's split into several smaller right triangles. The angle measures in any triangle always add up to 180 degrees. By using this fact and looking at complementary (adding up to 90 degrees) and supplementary (adding up to 180 degrees) angles, we can find the missing angle measures in the triangles.https://cdn.kastatic.org/ka-youtube-converted/iqeGTtyzQ1I.mp4/iqeGTtyzQ1I.mp4342yes https://puzzicle.org/math/cc-eighth-grade-math/v/challenging-triangle-angle-problemhttps://puzzicle.org/thumbs/265540635.jpgTriangle angle challenge problemLearn about the sum of exterior angles in a polygon, specifically a pentagon. We find out that the sum of exterior angles is always the same, no matter the shape of the pentagon. To understand this, we use interior angles and divide the pentagon into triangles.https://cdn.kastatic.org/ka-youtube-converted/95logvV8nXY.mp4/95logvV8nXY.mp4517yes https://puzzicle.org/math/cc-eighth-grade-math/v/finding-more-angleshttps://puzzicle.org/thumbs/265540639.jpgTriangle angle challenge problem 2Find angle measures in triangles when the given measures are algebraic expressions. The key idea is that the sum of all angles in a triangle is always 180 degrees. We can also use parallel lines and transversals to find angle measures.https://cdn.kastatic.org/ka-youtube-converted/0gzSreH8nUI.mp4/0gzSreH8nUI.mp4365yes https://puzzicle.org/math/geometry/v/bhaskara-s-proof-of-pythagorean-theorem-avihttps://puzzicle.org/thumbs/1199369003.jpgBhaskara's proof of the Pythagorean theoremThe 12th century Indian mathematician Bhaskara developed an elegant visual proof of the Pythagorean Theorem. Bhaskara uses a square and four congruent right triangles, rearranged in two ways, to prove this theorem. He shows that in a right triangle, the square of the longest side (hypotenuse) is equal to the sum of the squares of the other two sides.https://cdn.kastatic.org/ka-youtube-converted/1ul8g55dYA4.mp4/1ul8g55dYA4.mp4543yes https://puzzicle.org/math/geometry/v/thinking-about-dilationshttps://puzzicle.org/thumbs/x1854dd51.jpgDilating shapes: expandingLet's dilate a rectangle around a point P using a scale factor 1 2/3. This changes the size of the rectangle while keeping its shape. Because 1 2/3 is bigger than 1, most of the points get farther from P. Point P stays in place, because a distance of 0 times any scale factor is still 0.https://cdn.kastatic.org/ka-youtube-converted/6dyWKD_JPhI.mp4/6dyWKD_JPhI.mp4159yes https://puzzicle.org/math/cc-eighth-grade-math/v/scaling-down-a-triangle-by-halfhttps://puzzicle.org/thumbs/x86399361.jpgDilating shapes: shrinking by 1/2Dilations are transformations that change the size of a shape and its distance from the center of dilation. When the center is the origin, we can change the distance by multiplying the x- and y-coordinates by the scale factor. That's how we find the new positions of the points after dilation.https://cdn.kastatic.org/ka-youtube-converted/qlWZJ21O63s.mp4/qlWZJ21O63s.mp4146yes https://puzzicle.org/math/geometry/v/dilating-points-examplehttps://puzzicle.org/thumbs/xd9ea6a29c4a9812f.jpgDilating pointsDilations are a way to stretch or shrink shapes around a point called the center of dilation. The amount we stretch or shrink is called the scale factor. If the scale factor is greater than 1, the shape stretches. If it's between 0 and 1, the shape shrinks.https://cdn.kastatic.org/ka-youtube-converted/N0_HfmxdbCg.mp4/N0_HfmxdbCg.mp4187yes https://puzzicle.org/math/geometry/v/dilation-scale-factorhttps://puzzicle.org/thumbs/x76dc81a4313d7166.jpgDilations: scale factorDilations make a shape bigger or smaller. A scale factor tells us how much to multiply by the side lengths to change the size. The scale factor is the ratio of the side length in the new shape (image) to the side length in the corresponding side in the original shape (pre-image). Each pair of corresponding sides has the same factor.https://cdn.kastatic.org/ka-youtube-converted/b4ORXN7a0-w.mp4/b4ORXN7a0-w.mp4299yes https://puzzicle.org/math/geometry/v/dilations-and-propertieshttps://puzzicle.org/thumbs/xb836462d55b30b93.jpgDilations and propertiesDilations can change the lengths of a shape and its distance from the center, but they keep the angle measures the same. What does this mean about coordinates, side lengths, perimeter, and area?https://cdn.kastatic.org/ka-youtube-converted/chdnDLbSbU8.mp4/chdnDLbSbU8.mp4323yes https://puzzicle.org/math/geometry/v/similarity-postulateshttps://puzzicle.org/thumbs/284859449.jpgTriangle similarity postulates/criteriaSal reviews all the different ways we can determine that two triangles are similar. This is similar to the congruence criteria, only for similarity!https://cdn.kastatic.org/ka-youtube-converted/7bO0TmJ6Ba4.mp4/7bO0TmJ6Ba4.mp4734yes https://puzzicle.org/math/geometry/v/similar-triangle-example-problemshttps://puzzicle.org/thumbs/289578994.jpgDetermining similar trianglesMultiple examples where Sal analyzes the similarity of triangles.https://cdn.kastatic.org/ka-youtube-converted/Ly86lwq_2gc.mp4/Ly86lwq_2gc.mp4563yes https://puzzicle.org/math/geometry/v/testing-similarity-through-transformationshttps://puzzicle.org/thumbs/xfc5aad45.jpgSimilar shapes & transformationsTwo shapes are similar if we can change one shape into the other using rigid transformations (like moving or rotating) and dilations (making it bigger or smaller). Other kinds of transformations can change the angles or the ratios of lengths in a figure. If we need those types of transformations, the shapes are not similar.https://cdn.kastatic.org/ka-youtube-converted/H-YuW_uckz4.mp4/H-YuW_uckz4.mp4158yes https://puzzicle.org/math/cc-eighth-grade-math/v/the-pythagorean-theoremhttps://puzzicle.org/thumbs/2600799.jpgIntro to the Pythagorean theoremThe Pythagorean theorem is a cornerstone of math that helps us find the missing side length of a right triangle. In a right triangle with sides A, B, and hypotenuse C, the theorem states that A² + B² = C². The hypotenuse is the longest side, opposite the right angle.https://cdn.kastatic.org/ka-youtube-converted/AA6RfgP-AHU.mp4/AA6RfgP-AHU.mp4646yes https://puzzicle.org/math/cc-eighth-grade-math/v/pythagorean-theorem-2https://puzzicle.org/thumbs/26903.jpgPythagorean theorem exampleThe Pythagorean theorem states that in a right triangle, the sum of the squares of the two shorter sides equals the square of the longest side (the hypotenuse). We can apply the theorem to find the missing side length of a right triangle, even when the missing length is one of the shorter sides.https://cdn.kastatic.org/ka-youtube-converted/O64YFlX1_aI.mp4/O64YFlX1_aI.mp4257yes https://puzzicle.org/math/cc-eighth-grade-math/v/pythagorean-theorem-3https://puzzicle.org/thumbs/26766.jpgPythagorean theorem word problem: carpetIf we draw the diagonal on a rectangular carpet, then its length, width, and diagonal form a right triangle. Since we know the length and diagonal measurements, we can use the Pythagorean theorem to solve for the missing width of the carpet.https://cdn.kastatic.org/ka-youtube-converted/T0IOrRETWhI.mp4/T0IOrRETWhI.mp4180yes https://puzzicle.org/math/geometry/v/pythagorean-theorem-with-right-trianglehttps://puzzicle.org/thumbs/xb26b2a89796216fc.jpgPythagorean theorem with isosceles triangleTo find the value of a base (x) in an isosceles triangle, first split the triangle into two congruent right triangles by drawing an altitude. Then, use the Pythagorean theorem to create an equation involving x. Finally, solve the equation to find the unknown base, x.https://cdn.kastatic.org/ka-youtube-converted/TjOlZr_2uW4.mp4/TjOlZr_2uW4.mp4228yes https://puzzicle.org/math/cc-eighth-grade-math/v/pythagorean-theorem-1https://puzzicle.org/thumbs/26902.jpgPythagorean theorem word problem: fishing boatIf we assume that the mast of a boat is at a right angle to the deck, then we can model the length of a rope stretched between the mast and deck as the hypotenuse of a right triangle. Then we can use the Pythagorean theorem to relate the lengths.https://cdn.kastatic.org/ka-youtube-converted/JVrkLIcA2qw.mp4/JVrkLIcA2qw.mp4272yes https://puzzicle.org/math/algebra-basics/v/soccer-thiagohttps://puzzicle.org/thumbs/x19541b96.jpgThiago asks: How much time does a goalkeeper have to react to a penalty kick?Sal uses the Pythagorean theorem to answer a question posed by a soccer superstar!https://cdn.kastatic.org/ka-youtube-converted/XtJwkIacpi8.mp4/XtJwkIacpi8.mp4725yes https://puzzicle.org/math/geometry/v/garfield-s-proof-of-the-pythagorean-theoremhttps://puzzicle.org/thumbs/1199652424.jpgGarfield's proof of the Pythagorean theoremFormer U.S. President James Garfield wrote a proof of the Pythagorean theorem. He used a trapezoid made of two identical right triangles and half of a square to show that the sum of the squares of the two shorter sides equals the square of the longest side of a right triangle.https://cdn.kastatic.org/ka-youtube-converted/EINpkcphsPQ.mp4/EINpkcphsPQ.mp4536yes https://puzzicle.org/math/geometry/v/pythagorean-theorem-proof-using-similarityhttps://puzzicle.org/thumbs/290279020.jpgPythagorean theorem proof using similarityProof of the Pythagorean Theorem using similarityhttps://cdn.kastatic.org/ka-youtube-converted/LrS5_l-gk94.mp4/LrS5_l-gk94.mp4592yes https://puzzicle.org/math/geometry/v/another-pythagorean-theorem-proofhttps://puzzicle.org/thumbs/1199568849.jpgAnother Pythagorean theorem proofVisually proving the Pythagorean Theoremhttps://cdn.kastatic.org/ka-youtube-converted/rcBaqkGp7CA.mp4/rcBaqkGp7CA.mp4623yes https://puzzicle.org/math/algebra/v/equations-3https://puzzicle.org/thumbs/9.jpgIntro to equations with variables on both sidesLearn how to solve the equation 2x + 3 = 5x - 2 with the variable on both sides. We start by visualizing the equation, then isolate the variable by performing the same operations on both sides. Finally, we solve the equation to find the value of the variable.https://cdn.kastatic.org/ka-youtube-converted/f15zA0PhSek.mp4/f15zA0PhSek.mp4533yes https://puzzicle.org/math/algebra/v/solving-equations-2https://puzzicle.org/thumbs/26778.jpgEquations with variables on both sides: 20-7x=6x-6Solving equations like 20 - 7x = 6x - 6 with the variable on both sides involves a few steps! First, we add or subtract terms from both sides to separate constants and variables to different sides of the equation. Then, we simplify to isolate the variable. Finally, we check our answer by plugging it back into the original equation.https://cdn.kastatic.org/ka-youtube-converted/1c5HY3z4k8M.mp4/1c5HY3z4k8M.mp4246yes https://puzzicle.org/math/algebra/v/solving-equations-with-the-distributive-property-2https://puzzicle.org/thumbs/26780.jpgEquation with variables on both sides: fractionsTo solve the equation (3/4)x + 2 = (3/8)x - 4, we first eliminate fractions by multiplying both sides by the least common multiple of the denominators. Then, we add or subtract terms from both sides of the equation to group the x-terms on one side and the constants on the other. Finally, we solve and check as normal.https://cdn.kastatic.org/ka-youtube-converted/PL9UYj2awDc.mp4/PL9UYj2awDc.mp4290yes https://puzzicle.org/math/algebra/v/ex-2-multi-step-equationhttps://puzzicle.org/thumbs/1242815904.jpgEquation with the variable in the denominatorThe variable x is in the denominator in 7 - 10/x = 2 + 15/x. We can multiply both sides of the equation by x to turn it into a more familiar, linear form without fractions. From there, we isolate the variable to solve. P.S. x cannot equal zero, because then the equation would be undefined.https://cdn.kastatic.org/ka-youtube-converted/Z7C69xP08d8.mp4/Z7C69xP08d8.mp4204yes https://puzzicle.org/math/algebra/v/solving-equations-with-the-distributive-propertyhttps://puzzicle.org/thumbs/26779.jpgEquations with parenthesesWhen an equation such as -9 - (9x - 6) = 3(4x + 6) has parentheses, we can distribute without changing the value of each side. Then combine like terms. Next, we move all the x-terms to one side and the constants to the other. Finally, we solve for x.https://cdn.kastatic.org/ka-youtube-converted/YZBStgZGyDY.mp4/YZBStgZGyDY.mp4364yes https://puzzicle.org/math/algebra/v/number-of-solutions-to-linear-equationshttps://puzzicle.org/thumbs/x60d48e64.jpgNumber of solutions to equationsA linear equation could have exactly 1, 0, or infinite solutions. If we can solve the equation and get something like x=b where b is a specific number, then we have one solution. If we end up with a statement that's always false, like 3=5, then there's no solution. If we end up with a statement that's always true, like 5=5, then there are infinite solutions.https://cdn.kastatic.org/ka-youtube-converted/qsL_5Y8uWPU.mp4/qsL_5Y8uWPU.mp4328yes https://puzzicle.org/math/algebra/v/equation-special-caseshttps://puzzicle.org/thumbs/13179469.jpgWorked example: number of solutions to equationsSal attempts to solve 8(3x + 10) = 28x - 14 - 4x only to find that the equation has no solution.https://cdn.kastatic.org/ka-youtube-converted/zKotuhQWIRg.mp4/zKotuhQWIRg.mp4115yes https://puzzicle.org/math/algebra/v/number-of-solutions-to-linear-equations-ex-2https://puzzicle.org/thumbs/x62c9e8fa.jpgCreating an equation with no solutionsSal shows how to complete the equation -11x + 4 = __x + __ so that it has no solutions.https://cdn.kastatic.org/ka-youtube-converted/uQs100shv-A.mp4/uQs100shv-A.mp483yes https://puzzicle.org/math/algebra/v/number-of-solutions-to-linear-equations-ex-3https://puzzicle.org/thumbs/xdfacc6f3.jpgCreating an equation with infinitely many solutionsSal shows how to complete the equation 4(x - 2) + x = 5x + __ so that it has infinitely many solutions.https://cdn.kastatic.org/ka-youtube-converted/Dq0xFgQB9qo.mp4/Dq0xFgQB9qo.mp4100yes https://puzzicle.org/math/cc-eighth-grade-math/v/graphing-proportional-relationships-examplehttps://puzzicle.org/thumbs/x13c3e512.jpgGraphing proportional relationships: unit rateIn proportional relationships, the unit rate is the slope of the line. Changes in x lead to steady changes in y when there's a proportional relationship. We can use the unit rate to write and graph an equation of the line that represents the relationship.https://cdn.kastatic.org/ka-youtube-converted/1F7LAJEVp-U.mp4/1F7LAJEVp-U.mp4271yes https://puzzicle.org/math/cc-eighth-grade-math/v/graphing-proportional-relationships-example-2https://puzzicle.org/thumbs/x5794411e.jpgGraphing proportional relationships from a tableLet's graph a proportional relationship from a table of values. The graph of a proportional relationship is a line, so we can graph from any 2 points in the table. The slope of the line represents the unit rate, so changes in x and y values determine the slope.https://cdn.kastatic.org/ka-youtube-converted/R32xBmxXj2E.mp4/R32xBmxXj2E.mp472yes https://puzzicle.org/math/cc-eighth-grade-math/v/graphing-proportional-relationships-example-3https://puzzicle.org/thumbs/x683c6b9a.jpgGraphing proportional relationships from an equationLet's graph the equation y = 2.5x. For every increase of 1 in x, y increases by 2.5. We call this the "unit rate" or "slope". The graph shows a proportional relationship because y changes at a constant rate as x changes and because y is 0 when x is 0.https://cdn.kastatic.org/ka-youtube-converted/LoKEPEPaNm4.mp4/LoKEPEPaNm4.mp4189yes https://puzzicle.org/math/8th-engage-ny/v/determining-a-linear-equation-by-trying-out-values-from-a-tablehttps://puzzicle.org/thumbs/x5fa37e27.jpgTesting solutions to linear equationsSal figures out which equation was used to generate a table of x and y values.https://cdn.kastatic.org/ka-youtube-converted/WU9g4CvZyR0.mp4/WU9g4CvZyR0.mp4216yes https://puzzicle.org/math/algebra/v/plotting-x-y-relationshipshttps://puzzicle.org/thumbs/33752746.jpgGraphing a linear equation: 5x+2y=20We can graph a the linear equation like 5x + 2y = 20 by rewriting it so y is isolated, then plugging in x values to find their corresponding y-values in a table. We can then graph those x-y pairs as points on a graph.https://cdn.kastatic.org/ka-youtube-converted/86NwKBcOlow.mp4/86NwKBcOlow.mp4269yes https://puzzicle.org/math/algebra/v/finding-x-intercept-of-a-linehttps://puzzicle.org/thumbs/x6bca96e9.jpgx-intercept of a lineSal determines the x-intercept of a linear equation from a graph. Afterwards, he checks his work by plugging values back into the equation.https://cdn.kastatic.org/ka-youtube-converted/xiIQQNufFuU.mp4/xiIQQNufFuU.mp4103yes https://puzzicle.org/math/algebra/v/finding-intercepts-for-a-linear-function-from-a-tablehttps://puzzicle.org/thumbs/xe3701afe.jpgIntercepts from a tableThe y-intercept is the y-coordinate when x=0, and the x-intercept is the x-coordinate when y=0. The y-intercept is not in the table. Since the table represents a line, there's a constant rate of change of y with respect to x. So we can find that pattern and fill in skipped values from the table to find the y-intercept.https://cdn.kastatic.org/ka-youtube-converted/CGZZINHT0I8.mp4/CGZZINHT0I8.mp4240yes https://puzzicle.org/math/algebra/v/x-and-y-interceptshttps://puzzicle.org/thumbs/26802.jpgIntercepts from an equationLet's graph -5x + 4y = 20 from its intercepts. Intercepts are the places where a line crosses the x- and y-axes. When a line crosses the x-axis, the y value is 0. That's the x-intercept. When a line crosses the y-axis, the x value is 0. That's the y-intercept. The line passes through those points.https://cdn.kastatic.org/ka-youtube-converted/xGmef7lFc5w.mp4/xGmef7lFc5w.mp4248yes https://puzzicle.org/math/cc-eighth-grade-math/v/x-and-y-intercepts-2https://puzzicle.org/thumbs/26803.jpgWorked example: intercepts from an equationLet's find the x- and y-intercepts of the equation 2y + 1/3x = 12, where a line crosses the x-axis (the x-intercept) and the y-axis (the y-intercept) on a graph. To find the x-intercept, we make y equal to 0 and solve the equation for x. To find the y-intercept, we make x equal to 0 and solve for y.https://cdn.kastatic.org/ka-youtube-converted/405boztgZig.mp4/405boztgZig.mp4205yes https://puzzicle.org/math/cc-eighth-grade-math/v/slope-intuition-examplehttps://puzzicle.org/thumbs/x4d473b5b.jpgSlope & direction of a lineA line that's flat has a slope of 0. If a line goes up as you move to the right, it has a positive slope. The steeper it is, the bigger the slope. If a line goes down as you move to the right, it has a negative slope. The steeper it is, the more negative the slope.https://cdn.kastatic.org/ka-youtube-converted/6_9xNMtwnfs.mp4/6_9xNMtwnfs.mp4155yes https://puzzicle.org/math/algebra/v/slope-of-a-linehttps://puzzicle.org/thumbs/26800.jpgWorked example: slope from graphThe slope of a line is rise over run. Learn how to calculate the slope of the line in a graph by finding the change in y and the change in x.https://cdn.kastatic.org/ka-youtube-converted/R948Tsyq4vA.mp4/R948Tsyq4vA.mp4280yes https://puzzicle.org/math/cc-eighth-grade-math/v/graphical-slope-of-a-linehttps://puzzicle.org/thumbs/33752751.jpgSlope of a line: negative slopeSlope is like a hill's steepness. We find it by dividing the vertical change (rise) by the horizontal change (run). If we move right on a graph and go up, the slope is positive. If we go down, it's negative. We can find the slope between any two points on a line, and it's always the same.https://cdn.kastatic.org/ka-youtube-converted/81SseQCpGws.mp4/81SseQCpGws.mp4240yes https://puzzicle.org/math/algebra/v/slope-of-a-line-2https://puzzicle.org/thumbs/33752752.jpgWorked example: slope from two pointsLet's find the slope of the line that goes through the ordered pairs (4,2) and (-3, 16). The slope, or steepness, of a line is found by dividing the vertical change (rise) by the horizontal change (run). The formula is slope =(y₂ - y₁)/(x₂ - x₁), where (x₁, y₁) and (x₂, y₂) are the coordinates of two points on the line.https://cdn.kastatic.org/ka-youtube-converted/WkspBxrzuZo.mp4/WkspBxrzuZo.mp4432yes https://puzzicle.org/math/algebra/v/slope-of-a-line-3https://puzzicle.org/thumbs/33752753.jpgSlope of a horizontal lineWhen two points have the same y-value, it means they lie on a horizontal line. The slope of such a line is 0, and you will also find this by using the slope formula.https://cdn.kastatic.org/ka-youtube-converted/CFSHq099Mx0.mp4/CFSHq099Mx0.mp4305yes https://puzzicle.org/math/algebra/v/graphing-a-line-in-slope-intercept-formhttps://puzzicle.org/thumbs/26804.jpgGraph from slope-intercept equationTo graph a linear equation in slope-intercept form, we can use the information given by that form. For example, y=2x+3 tells us that the slope of the line is 2 and the y-intercept is at (0,3). This gives us one point the line goes through, and the direction we should continue from that point to draw the entire line.https://cdn.kastatic.org/ka-youtube-converted/uk7gS3cZVp4.mp4/uk7gS3cZVp4.mp4181yes https://puzzicle.org/math/algebra/v/graphs-using-slope-intercept-formhttps://puzzicle.org/thumbs/23638.jpgSlope-intercept equation from graphLearn to write equations in slope-intercept form for three different lines.https://cdn.kastatic.org/ka-youtube-converted/9wOalujeZf4.mp4/9wOalujeZf4.mp4671yes https://puzzicle.org/math/algebra/v/equation-of-a-line-1https://puzzicle.org/thumbs/33752754.jpgSlope-intercept equation from slope & pointLearn how to write an equation in slope-intercept form (y=mx+b) for the line with a slope of -3/4 that goes through the point (0,8). We identify the slope (m) and y-intercept (b) to create our equation y = (-3/4)*x + 8.https://cdn.kastatic.org/ka-youtube-converted/AqFwKecNaTk.mp4/AqFwKecNaTk.mp4226yes https://puzzicle.org/math/algebra/v/equation-of-a-line-3https://puzzicle.org/thumbs/33752756.jpgSlope-intercept equation from two pointsGiven two points on a line, we can write an equation for that line by finding the slope between those points, then solving for the y-intercept in the slope-intercept equation y=mx+b. In this example, we write an equation of the line that passes through the points (-1,6) and (5,-4).https://cdn.kastatic.org/ka-youtube-converted/XMJ72mtMn4Y.mp4/XMJ72mtMn4Y.mp4402yes https://puzzicle.org/math/cc-eighth-grade-math/v/slope-intercept-form-from-tablehttps://puzzicle.org/thumbs/xdf2c138e.jpgSlope-intercept form from a tableLearn how to write an equation of the line that matches up to a table of values.https://cdn.kastatic.org/ka-youtube-converted/3Ayt7mOd_To.mp4/3Ayt7mOd_To.mp4266yes https://puzzicle.org/math/cc-eighth-grade-math/v/converting-to-slope-intercept-formhttps://puzzicle.org/thumbs/26806.jpgConverting to slope-intercept formLearn to convert equations like 4x + 2y = -8 into slope-intercept form.https://cdn.kastatic.org/ka-youtube-converted/V6Xynlqc_tc.mp4/V6Xynlqc_tc.mp4307yes https://puzzicle.org/math/cc-eighth-grade-math/v/linear-equations-in-slope-intercept-formhttps://puzzicle.org/thumbs/23646.jpgSlope-intercept form problemsLearn how to solve problems involving writing an equation in slope-intercept form.https://cdn.kastatic.org/ka-youtube-converted/5fkh01mClLU.mp4/5fkh01mClLU.mp4898yes https://puzzicle.org/math/geometry/v/similar-triangles-to-prove-that-the-slope-is-constant-for-a-linehttps://puzzicle.org/thumbs/x208e430a.jpgProving slope is constant using similaritySal uses a clever proof involving similar triangles to show that slope is constant for a line.https://cdn.kastatic.org/ka-youtube-converted/24WMbh1BBKc.mp4/24WMbh1BBKc.mp4476yes https://puzzicle.org/math/algebra/v/idea-behind-point-slope-formhttps://puzzicle.org/thumbs/x917b105d.jpgIntro to point-slope formPoint-slope is the general form y-y₁=m(x-x₁) for linear equations. It emphasizes the slope of the line and a point on the line (that is not the y-intercept). Watch this video to learn more about it and see some examples.https://cdn.kastatic.org/ka-youtube-converted/K_OI9LA54AA.mp4/K_OI9LA54AA.mp4367yes https://puzzicle.org/math/algebra/v/trolls-tolls-and-systems-of-equationshttps://puzzicle.org/thumbs/1095856196.jpgSystems of equations: trolls, tolls (1 of 2)A troll forces us to use algebra to figure out the make-up of his currency. We end up setting up a system of equations.https://cdn.kastatic.org/ka-youtube-converted/OcNt-36QKu8.mp4/OcNt-36QKu8.mp4411yes https://puzzicle.org/math/algebra/v/solving-the-troll-riddle-visuallyhttps://puzzicle.org/thumbs/1095783741.jpgSystems of equations: trolls, tolls (2 of 2)Solving the system of equations visually. Now we can save the prince/princess.https://cdn.kastatic.org/ka-youtube-converted/H-HfmyUzPw8.mp4/H-HfmyUzPw8.mp4341yes https://puzzicle.org/math/algebra/v/testing-a-solution-for-a-system-of-equationshttps://puzzicle.org/thumbs/33752762.jpgTesting a solution to a system of equationsSal checks whether (-1,7) is a solution of the system: x+2y=13 and 3x-y=-11.https://cdn.kastatic.org/ka-youtube-converted/SkMNREAMNvc.mp4/SkMNREAMNvc.mp4158yes https://puzzicle.org/math/cc-eighth-grade-math/v/solving-linear-systems-by-graphinghttps://puzzicle.org/thumbs/23674.jpgSystems of equations with graphingWhen solving systems of linear equations, one method is to graph both equations on the same coordinate plane. The intersection of the two lines represents a solution that satisfies both equations. Other, more mathematical, methods may also be used.https://cdn.kastatic.org/ka-youtube-converted/5a6zpfl50go.mp4/5a6zpfl50go.mp4510yes https://puzzicle.org/math/cc-eighth-grade-math/v/graphings-systems-of-equationshttps://puzzicle.org/thumbs/33752761.jpgSystems of equations with graphing: 5x+3y=7 & 3x-2y=8To solve a system of linear equations by graphing, you need to graph each equation separately. Find two points on each line and connect them. The point where the two lines intersect is the solution to the system of equations.https://cdn.kastatic.org/ka-youtube-converted/MRAIgJmRmag.mp4/MRAIgJmRmag.mp4395yes https://puzzicle.org/math/algebra/v/solving-systems-graphicallyhttps://puzzicle.org/thumbs/1112700719.jpgSystems of equations with graphing: y=7/5x-5 & y=3/5x-1Sal graphs the following system of equations and solves it by looking for the intersection point: y=7/5x-5 and y=3/5x-1.https://cdn.kastatic.org/ka-youtube-converted/VhH2nEDCd68.mp4/VhH2nEDCd68.mp4142yes https://puzzicle.org/math/cc-eighth-grade-math/v/graphical-systems-application-problemhttps://puzzicle.org/thumbs/33752765.jpgSystems of equations with graphing: choresGraphical Systems Application Problemhttps://cdn.kastatic.org/ka-youtube-converted/3mimxluSVBo.mp4/3mimxluSVBo.mp4319yes https://puzzicle.org/math/8th-engage-ny/v/solving-systems-by-elimination-2https://puzzicle.org/thumbs/26882.jpgSystems of equations with elimination: 3t+4g=6 & -6t+g=6Sal solves the system of equations 3t + 4g = 6 and -6t + g = 6 using elimination.https://cdn.kastatic.org/ka-youtube-converted/Y6JsEja15Vk.mp4/Y6JsEja15Vk.mp4225yes https://puzzicle.org/math/8th-engage-ny/v/addition-elimination-method-1https://puzzicle.org/thumbs/33752772.jpgSystems of equations with elimination: x+2y=6 & 4x-2y=14Learn to use elimination to solve the system of equations x + 2y = 6 and 4x - 2y = 14.https://cdn.kastatic.org/ka-youtube-converted/0P0SCQf-hWQ.mp4/0P0SCQf-hWQ.mp4218yes https://puzzicle.org/math/8th-engage-ny/v/addition-elimination-method-2https://puzzicle.org/thumbs/33752773.jpgSystems of equations with elimination: -3y+4x=11 & y+2x=13Sal solves the system of equations -3y + 4x = 11 and y + 2x = 13 using elimination.https://cdn.kastatic.org/ka-youtube-converted/hJOqGkWElNs.mp4/hJOqGkWElNs.mp4266yes https://puzzicle.org/math/8th-engage-ny/v/addition-elimination-method-3https://puzzicle.org/thumbs/33752774.jpgSystems of equations with elimination: 2x-y=14 & -6x+3y=-42Sal solves the system of equations 2x - y = 14 and -6x + 3y = -42 using elimination.https://cdn.kastatic.org/ka-youtube-converted/mKu0L46Qed4.mp4/mKu0L46Qed4.mp4305yes https://puzzicle.org/math/8th-engage-ny/v/solving-systems-by-elimination-3https://puzzicle.org/thumbs/26883.jpgSystems of equations with elimination: 4x-2y=5 & 2x-y=2.5Sal solves the system of equations 4x - 2y = 5 and 2x - y = 2.5 using elimination.https://cdn.kastatic.org/ka-youtube-converted/Dqp6xOeR3Ls.mp4/Dqp6xOeR3Ls.mp4297yes https://puzzicle.org/math/algebra/v/simple-elimination-practicehttps://puzzicle.org/thumbs/1112791956.jpgSystems of equations with elimination: x-4y=-18 & -x+3y=11Sal solves the following system of equations by eliminating x: x-4y=-18 and -x+3y=11.https://cdn.kastatic.org/ka-youtube-converted/NPXTkj75-AM.mp4/NPXTkj75-AM.mp4359yes https://puzzicle.org/math/8th-engage-ny/v/systems-with-elimination-practicehttps://puzzicle.org/thumbs/1115141430.jpgSystems of equations with elimination: 6x-6y=-24 & -5x-5y=-60Sal solves the system of equations 6x - 6y = -24 and -5x - 5y = -60 using elimination.https://cdn.kastatic.org/ka-youtube-converted/u5dPUHjagSI.mp4/u5dPUHjagSI.mp4331yes https://puzzicle.org/math/cc-eighth-grade-math/v/the-substitution-methodhttps://puzzicle.org/thumbs/33752769.jpgSystems of equations with substitution: 2y=x+7 & x=y-4When solving a system of equations using substitution, you can isolate one variable and substitute it with an expression from another equation. This will allow you to solve for one variable, which you can then use to solve for the other.https://cdn.kastatic.org/ka-youtube-converted/uzyd_mIJaoc.mp4/uzyd_mIJaoc.mp4279yes https://puzzicle.org/math/cc-eighth-grade-math/v/solving-systems-by-substitution-2https://puzzicle.org/thumbs/26879.jpgSystems of equations with substitution: y=4x-17.5 & y+2x=6.5Learn to solve the system of equations y = 4x - 17.5 and y + 2x = 6.5 using substitution.https://cdn.kastatic.org/ka-youtube-converted/wB3QCk0MGuw.mp4/wB3QCk0MGuw.mp4246yes https://puzzicle.org/math/algebra/v/practice-using-substitution-for-systemshttps://puzzicle.org/thumbs/1116433408.jpgSystems of equations with substitution: -3x-4y=-2 & y=2x-5Learn to solve the system of equations -3x - 4y = -2 and y = 2x - 5 using substitution.https://cdn.kastatic.org/ka-youtube-converted/GWZKz4F9hWM.mp4/GWZKz4F9hWM.mp4261yes https://puzzicle.org/math/cc-eighth-grade-math/v/solving-systems-by-substitution-3https://puzzicle.org/thumbs/26880.jpgSystems of equations with substitution: 9x+3y=15 & y-x=5Learn to solve the system of equations 9x + 3y = 15 and y - x = 5 using substitution.https://cdn.kastatic.org/ka-youtube-converted/0BgUKHTW37E.mp4/0BgUKHTW37E.mp4364yes https://puzzicle.org/math/cc-eighth-grade-math/v/substitution-method-2https://puzzicle.org/thumbs/33752770.jpgSystems of equations with substitution: y=-5x+8 & 10x+2y=-2Learn to solve the system of equations y = -5x + 8 and 10x + 2y = -2 using substitution.https://cdn.kastatic.org/ka-youtube-converted/2VeqrZ_PMiY.mp4/2VeqrZ_PMiY.mp4224yes https://puzzicle.org/math/cc-eighth-grade-math/v/solving-systems-by-substitution-1https://puzzicle.org/thumbs/26878.jpgSystems of equations with substitution: y=-1/4x+100 & y=-1/4x+120Learn to solve the system of equations y = -1/4x + 100 and y = -1/4x + 120 using substitution.https://cdn.kastatic.org/ka-youtube-converted/HJV_HY0Sh0s.mp4/HJV_HY0Sh0s.mp4224yes https://puzzicle.org/math/algebra/v/using-a-system-of-equations-to-find-the-price-of-apples-and-orangeshttps://puzzicle.org/thumbs/x9cf8703a.jpgSystems of equations with elimination: apples and orangesSal solves a word problem about the price of apples and oranges by creating a system of equations and solving it.https://cdn.kastatic.org/ka-youtube-converted/gRntusF_tVI.mp4/gRntusF_tVI.mp4306yes https://puzzicle.org/math/algebra/v/solving-systems-by-eliminationhttps://puzzicle.org/thumbs/26881.jpgSystems of equations with elimination: TV & DVDSal solves a word problem about the weights of TVs and DVDs by creating a system of equations and solving it.https://cdn.kastatic.org/ka-youtube-converted/z1hz8-Kri1E.mp4/z1hz8-Kri1E.mp4351yes https://puzzicle.org/math/algebra/v/king-s-cupcakes-solving-systems-by-eliminationhttps://puzzicle.org/thumbs/1112847811.jpgSystems of equations with elimination: King's cupcakesSal uses simple elimination to figure out how many cupcakes are eaten by children and adults.https://cdn.kastatic.org/ka-youtube-converted/xCIHAjsZCE0.mp4/xCIHAjsZCE0.mp4540yes https://puzzicle.org/math/8th-engage-ny/v/addition-elimination-method-4https://puzzicle.org/thumbs/33752775.jpgSystems of equations with elimination: Sum/difference of numbersSal finds two numbers that sum to 70 and have a difference of 24.https://cdn.kastatic.org/ka-youtube-converted/DxSmZ0MDSxM.mp4/DxSmZ0MDSxM.mp4158yes https://puzzicle.org/math/algebra/v/how-many-bags-of-potato-chips-do-people-eathttps://puzzicle.org/thumbs/1114711148.jpgSystems of equations with elimination: potato chipsSal solves another system of equations using elimination.https://cdn.kastatic.org/ka-youtube-converted/VuJEidLhY1E.mp4/VuJEidLhY1E.mp4561yes https://puzzicle.org/math/algebra/v/understanding-systems-of-equations-examplehttps://puzzicle.org/thumbs/x6f9a8813.jpgSystems of equations with elimination: coffee and croissantsSal solves a word problem about the price of coffee and croissant by creating a system of equations and solving it.https://cdn.kastatic.org/ka-youtube-converted/grmmM_KhQ4U.mp4/grmmM_KhQ4U.mp4255yes https://puzzicle.org/math/algebra/v/substitution-method-3https://puzzicle.org/thumbs/33752771.jpgSystems of equations with substitution: coinsSal solves a word problem about the number of nickels and quarters in a piggy bank by creating a system of equations and solving it.https://cdn.kastatic.org/ka-youtube-converted/v6L8--MlnKo.mp4/v6L8--MlnKo.mp4358yes https://puzzicle.org/math/algebra/v/solving-systems-with-substitutionhttps://puzzicle.org/thumbs/1115394533.jpgSystems of equations with substitution: potato chipsTo solve a system of equations using substitution... 1. Isolate one of the variables in one of the equations, e.g. rewrite 2x+y=3 as y=3-2x. 2. You can now express the isolated variable using the other one. *Substitute* that expression into the second equation, e.g. rewrite x+2y=5 as x+2(3-2x)=5. 3. Now you have an equation with one variable! Solve it, and use what you got to find the other variable.https://cdn.kastatic.org/ka-youtube-converted/2EwPpga_XPw.mp4/2EwPpga_XPw.mp4321yes https://puzzicle.org/math/8th-engage-ny/v/algebraic-word-problemhttps://puzzicle.org/thumbs/26781.jpgSystems of equations with substitution: shelvesSal solves a system of equations to figure out how long a couple of different shelves will be.https://cdn.kastatic.org/ka-youtube-converted/cNlwi6lUCEM.mp4/cNlwi6lUCEM.mp4482yes https://puzzicle.org/math/algebra/v/ex-2-age-word-problemhttps://puzzicle.org/thumbs/1242705398.jpgAge word problem: ImranSal solves the following age word problem: In 40 years, Imran will be 11 times as old as he is right now. How old is he right now?https://cdn.kastatic.org/ka-youtube-converted/KyHvVJWjW6Y.mp4/KyHvVJWjW6Y.mp4223yes https://puzzicle.org/math/algebra/v/ex-3-age-word-problemhttps://puzzicle.org/thumbs/1242500916.jpgAge word problem: Ben & WilliamSal solves the following age word problem: William is 4 times as old as Ben. 12 years ago, William was 7 times as old as Ben. How old is Ben now?https://cdn.kastatic.org/ka-youtube-converted/RGbA2IyJILY.mp4/RGbA2IyJILY.mp4388yes https://puzzicle.org/math/algebra/v/ex-1-age-word-problemhttps://puzzicle.org/thumbs/1242638298.jpgAge word problem: Arman & DiyaSal solves the following age word problem: Arman is 18. Diya is 2. How many years will it take for Arman to be 3 times as old as Diya?https://cdn.kastatic.org/ka-youtube-converted/W-5liMGKgHA.mp4/W-5liMGKgHA.mp4321yes https://puzzicle.org/math/algebra/v/consistent-and-inconsistent-systemshttps://puzzicle.org/thumbs/33752763.jpgSolutions to systems of equations: consistent vs. inconsistentA consistent system of equations has at least one solution, and an inconsistent system has no solution. Watch an example of analyzing a system to see if it's consistent or inconsistent.https://cdn.kastatic.org/ka-youtube-converted/Ix8Nne-a-KQ.mp4/Ix8Nne-a-KQ.mp4329yes https://puzzicle.org/math/algebra/v/inconsistent-systems-of-equationshttps://puzzicle.org/thumbs/1116349757.jpgSystems of equations number of solutions: fruit prices (1 of 2)Sal gives an example of a system of equations that has no solution!https://cdn.kastatic.org/ka-youtube-converted/NJRyxmommqQ.mp4/NJRyxmommqQ.mp4565yes https://puzzicle.org/math/algebra/v/infinite-solutions-to-systemshttps://puzzicle.org/thumbs/1116483367.jpgSystems of equations number of solutions: fruit prices (2 of 2)Sal gives an example of a system of equations that has infinite solutions!https://cdn.kastatic.org/ka-youtube-converted/MpRUEGD2-_c.mp4/MpRUEGD2-_c.mp4428yes https://puzzicle.org/math/8th-engage-ny/v/solving-systems-by-graphing-3https://puzzicle.org/thumbs/26877.jpgSystems of equations number of solutions: y=3x+1 & 2y+4=6xSal graphs a system of equations to find the number of solutions it has.https://cdn.kastatic.org/ka-youtube-converted/F5Nb6cIRZLU.mp4/F5Nb6cIRZLU.mp4308yes https://puzzicle.org/math/algebra/v/independent-and-dependent-systemshttps://puzzicle.org/thumbs/561902042.jpgSolutions to systems of equations: dependent vs. independentA dependent system of equations has infinite solutions, and an independent system has a single solution. Watch an example of analyzing a system to see if it's dependent or independent.https://cdn.kastatic.org/ka-youtube-converted/WSpF5uvApLA.mp4/WSpF5uvApLA.mp4310yes https://puzzicle.org/math/algebra/v/practice-thinking-about-number-of-solutions-to-systemshttps://puzzicle.org/thumbs/1116353804.jpgNumber of solutions to a system of equations graphicallySal determines how many solutions the following system of equations has by considering its graph: 10x-2y=4 and 10x-2y=16.https://cdn.kastatic.org/ka-youtube-converted/SuB1gkto9LU.mp4/SuB1gkto9LU.mp4396yes https://puzzicle.org/math/8th-engage-ny/v/constructing-solutions-to-systems-of-equationshttps://puzzicle.org/thumbs/x736f77fe.jpgForming systems of equations with different numbers of solutionsSal write an equation that along with the equation 4x + 5y = 2 forms a system of equations with infinitely many solutions.https://cdn.kastatic.org/ka-youtube-converted/Z96vkuybvjE.mp4/Z96vkuybvjE.mp4381yes https://puzzicle.org/math/8th-engage-ny/v/comparing-celsius-and-farenheit-temperature-scaleshttps://puzzicle.org/thumbs/33.jpgComparing Celsius and Fahrenheit temperature scalesComparing Celsius and Fahrenheit temperature scaleshttps://cdn.kastatic.org/ka-youtube-converted/aASUZqJCHHA.mp4/aASUZqJCHHA.mp4170yes https://puzzicle.org/math/8th-engage-ny/v/converting-farenheit-to-celsiushttps://puzzicle.org/thumbs/153770151.jpgConverting Fahrenheit to CelsiusTo convert from Celsius to Fahrenheit, we can use the formula F = 9/5C + 32, where C is the temperature in degrees Celsius and F is the temperature in degrees Fahrenheit. To convert from Fahrenheit to Celsius we can use the formula C = 5/9(F - 32).https://cdn.kastatic.org/ka-youtube-converted/NqbkvEK46xM.mp4/NqbkvEK46xM.mp4178yes https://puzzicle.org/math/algebra/v/what-is-a-functionhttps://puzzicle.org/thumbs/x2ec1ec8a.jpgWhat is a function?Functions assign a single output for each of their inputs. In this video, we see examples of various kinds of functions.https://cdn.kastatic.org/ka-youtube-converted/kvGsIo1TmsM.mp4/kvGsIo1TmsM.mp4477yes https://puzzicle.org/math/algebra/v/understanding-function-notation-example-1https://puzzicle.org/thumbs/x1292edd2.jpgWorked example: Evaluating functions from equationLearn how to evaluate f(x)=49-x² at x=5.https://cdn.kastatic.org/ka-youtube-converted/Id6UovYjd-M.mp4/Id6UovYjd-M.mp459yes https://puzzicle.org/math/algebra/v/understanding-function-notation-example-2https://puzzicle.org/thumbs/xbdf4c75d.jpgWorked example: Evaluating functions from graphEvaluating a function at x=-1 using the graph of that function.https://cdn.kastatic.org/ka-youtube-converted/kzYtx_AqzjM.mp4/kzYtx_AqzjM.mp444yes https://puzzicle.org/math/algebra/v/difference-between-equations-and-functionshttps://puzzicle.org/thumbs/1254705646.jpgEquations vs. functionsEquations and functions are not the same thing, but they can be related in several ways. Watch Jesse Roe and Sal talk about the difference between equations and functionshttps://cdn.kastatic.org/ka-youtube-converted/l3iXON1xEC4.mp4/l3iXON1xEC4.mp4258yes https://puzzicle.org/math/algebra2/v/solving-for-a-variable-2https://puzzicle.org/thumbs/26783.jpgManipulating formulas: temperatureSal takes the formula for converting from Fahrenheit temperature to Celsius, and solves it for F so it serves to convert from Celsius to Fahrenheit.https://cdn.kastatic.org/ka-youtube-converted/BR5yFOt0zao.mp4/BR5yFOt0zao.mp4161yes https://puzzicle.org/math/cc-eighth-grade-math/v/testing-if-a-relationship-is-a-functionhttps://puzzicle.org/thumbs/26791.jpgTesting if a relationship is a functionLearn to determine if points on a graph represent a function.https://cdn.kastatic.org/ka-youtube-converted/3SO1BQQ9_1E.mp4/3SO1BQQ9_1E.mp4143yes https://puzzicle.org/math/cc-eighth-grade-math/v/relations-and-functionshttps://puzzicle.org/thumbs/357917981.jpgRelations and functionsLearn to determine if a relation given by a set of ordered pairs is a function.https://cdn.kastatic.org/ka-youtube-converted/Uz0MtFlLD-k.mp4/Uz0MtFlLD-k.mp4417yes https://puzzicle.org/math/algebra/v/graphical-relations-and-functionshttps://puzzicle.org/thumbs/357917979.jpgRecognizing functions from graphChecking whether a given set of points can represent a function. For the set to represent a function, each domain element must have one corresponding range element at most.https://cdn.kastatic.org/ka-youtube-converted/qGmJ4F3b5W8.mp4/qGmJ4F3b5W8.mp4247yes https://puzzicle.org/math/cc-eighth-grade-math/v/recognizing-functions-example-3https://puzzicle.org/thumbs/x9cd7b997.jpgChecking if a table represents a functionSal determines if y is a function of x from looking at a table.https://cdn.kastatic.org/ka-youtube-converted/lEhf75ma7Ww.mp4/lEhf75ma7Ww.mp476yes https://puzzicle.org/math/algebra/v/recognizing-functions-example-1https://puzzicle.org/thumbs/xe1f24412.jpgRecognizing functions from verbal descriptionChecking whether y can be described as a function of x if y is always three more than twice x.https://cdn.kastatic.org/ka-youtube-converted/tBRW-CyJE6k.mp4/tBRW-CyJE6k.mp491yes https://puzzicle.org/math/algebra/v/functional-relationships-1https://puzzicle.org/thumbs/26790.jpgRecognizing functions from tableChecking whether a table of people and their heights can represent a function that assigns a height to a name.https://cdn.kastatic.org/ka-youtube-converted/5cK86VKoBPw.mp4/5cK86VKoBPw.mp4280yes https://puzzicle.org/math/algebra/v/recognizing-functions-example-5https://puzzicle.org/thumbs/xe4c1673b.jpgRecognizing functions from verbal description word problemsChecking whether a description of the price of an order can be represented as a function of the shipping cost.https://cdn.kastatic.org/ka-youtube-converted/z45UEiPaE8c.mp4/z45UEiPaE8c.mp4203yes https://puzzicle.org/math/cc-eighth-grade-math/v/recognizing-functions-example-4https://puzzicle.org/thumbs/xce131edd.jpgChecking if an equation represents a functionSal determines if y is a function of x from looking at an equation.https://cdn.kastatic.org/ka-youtube-converted/QWLcNxQ3KvQ.mp4/QWLcNxQ3KvQ.mp4213yes https://puzzicle.org/math/algebra/v/recognizing-functions-example-2https://puzzicle.org/thumbs/x5b15777f.jpgDoes a vertical line represent a function?Explaining why a vertical line doesn't represent a function.https://cdn.kastatic.org/ka-youtube-converted/_npwsLh0vws.mp4/_npwsLh0vws.mp4100yes https://puzzicle.org/math/geometry/v/cylinder-volume-and-surface-areahttps://puzzicle.org/thumbs/60129395.jpgCylinder volume & surface areaA cylinder's volume is π r² h, and its surface area is 2π r h + 2π r². Learn how to use these formulas to solve an example problem.https://cdn.kastatic.org/ka-youtube-converted/gL3HxBQyeg0.mp4/gL3HxBQyeg0.mp4487yes https://puzzicle.org/science/ap-biology/v/volume-of-a-spherehttps://puzzicle.org/thumbs/372432663.jpgVolume of a sphereThe formula for the volume of a sphere is V = 4/3 π r³, where V = volume and r = radius. The radius of a sphere is half its diameter. So, to calculate the volume of a sphere given the diameter of the sphere, you can first calculate the radius, then the volume.https://cdn.kastatic.org/ka-youtube-converted/IelS2vg7JO8.mp4/IelS2vg7JO8.mp4155yes https://puzzicle.org/math/geometry/v/volume-cone-examplehttps://puzzicle.org/thumbs/x8cc61186.jpgVolume of a coneThe formula for the volume of a cone is V=1/3hπr². Learn how to use this formula to solve an example problem.https://cdn.kastatic.org/ka-youtube-converted/hC6zx9WAiC4.mp4/hC6zx9WAiC4.mp4344yes https://puzzicle.org/math/cc-eighth-grade-math/v/recognizing-linear-functionshttps://puzzicle.org/thumbs/26798.jpgRecognizing linear functionsLearn to recognize if a function is linear.https://cdn.kastatic.org/ka-youtube-converted/AZroE4fJqtQ.mp4/AZroE4fJqtQ.mp4241yes https://puzzicle.org/math/cc-eighth-grade-math/v/linear-and-nonlinear-functions-example-3https://puzzicle.org/thumbs/x7f5ef1b0.jpgLinear & nonlinear functions: tableLearn to determine if a table of values represents a linear function.https://cdn.kastatic.org/ka-youtube-converted/Pt-Tn6L60-o.mp4/Pt-Tn6L60-o.mp4146yes https://puzzicle.org/math/cc-eighth-grade-math/v/linear-and-nonlinear-functions-example-1https://puzzicle.org/thumbs/xf9937022.jpgLinear & nonlinear functions: word problemLearn to determine if the relationship described in a word problem is a function.https://cdn.kastatic.org/ka-youtube-converted/OWPVZoxNe-U.mp4/OWPVZoxNe-U.mp4215yes https://puzzicle.org/math/cc-eighth-grade-math/v/linear-and-nonlinear-functions-example-2https://puzzicle.org/thumbs/xa6a89fbd.jpgLinear & nonlinear functions: missing valueLearn to find the missing value in a table to make sure it represents a linear equation.https://cdn.kastatic.org/ka-youtube-converted/HUnF0mZmMKc.mp4/HUnF0mZmMKc.mp4133yes https://puzzicle.org/math/cc-eighth-grade-math/v/interpreting-a-graph-exercise-examplehttps://puzzicle.org/thumbs/x98c9e2aa.jpgInterpreting a graph exampleLearn to interpret the graphs of a linear functions.https://cdn.kastatic.org/ka-youtube-converted/YyjpeoxdUXc.mp4/YyjpeoxdUXc.mp4143yes https://puzzicle.org/math/cc-eighth-grade-math/v/comparing-features-of-functions-1https://puzzicle.org/thumbs/x198644d9.jpgComparing linear functions: equation vs. graphSal is given the formula of a linear function and the graph of another, and is asked to determine which function increases faster.https://cdn.kastatic.org/ka-youtube-converted/fZO-JylMFqY.mp4/fZO-JylMFqY.mp4143yes https://puzzicle.org/math/cc-eighth-grade-math/v/comparing-features-of-functions-3https://puzzicle.org/thumbs/xfd98bb7a.jpgComparing linear functions: same rate of changeSal is given a table of values of a linear function and four linear graphs, and is asked to determine which graph has the same rate of change as the function represented in the table.https://cdn.kastatic.org/ka-youtube-converted/wl2iQAuQl7Y.mp4/wl2iQAuQl7Y.mp4156yes https://puzzicle.org/math/cc-eighth-grade-math/v/comparing-features-of-functions-5https://puzzicle.org/thumbs/x63490941.jpgComparing linear functions: faster rate of changeSal is given a table of values of a linear function and four linear graphs, and is asked to determine which graph increases faster than the function represented in the table.https://cdn.kastatic.org/ka-youtube-converted/QYoNHOjr7Rg.mp4/QYoNHOjr7Rg.mp4220yes https://puzzicle.org/math/cc-eighth-grade-math/v/constructing-scatter-plothttps://puzzicle.org/thumbs/x486a060d.jpgConstructing a scatter plotDoes the time an exam happens affects the scores? Aubrey gathered info. Then she made a scatter plot with the time of day on the bottom (on the horizontal axis) and the scores on the side (on the vertical axis). She didn't find a clear pattern.https://cdn.kastatic.org/ka-youtube-converted/sHbX58y5D4U.mp4/sHbX58y5D4U.mp4152yes https://puzzicle.org/math/cc-eighth-grade-math/v/scatter-plot-interpretinghttps://puzzicle.org/thumbs/x905613d7.jpgExample of direction in scatterplotsThere are two graphs. One shows the relationship between study time and test grades. Do you get better grades if you study more? The other graph shows the relationship between shoe size and grades. Do you get better grades if you wear larger shoes?https://cdn.kastatic.org/ka-youtube-converted/Jpbm5YgciqI.mp4/Jpbm5YgciqI.mp4146yes https://puzzicle.org/math/cc-eighth-grade-math/v/trends-in-smoking-scatter-plothttps://puzzicle.org/thumbs/x95ffaffd.jpgScatter plot: smokersIn a negative linear association, as one variable goes up, the other variable goes down at a constant rate. In this case, the percent of adults who smoke drops 0.5% each time the year goes up. In the right graph, the points go down by about 0.5% each year. There should be no points far away from the main trend.https://cdn.kastatic.org/ka-youtube-converted/F5uiFXECnVY.mp4/F5uiFXECnVY.mp4155yes https://puzzicle.org/math/cc-eighth-grade-math/v/estimating-the-line-of-best-fit-exercisehttps://puzzicle.org/thumbs/x61240c40.jpgEstimating the line of best fit exerciseThe "line of best fit" is a line that shows the pattern of data points. If we can find a good line, it means there is a linear trend. If not, it means there is no linear trend. We can't ignore points that don't fit the trend.https://cdn.kastatic.org/ka-youtube-converted/ioieTr41L24.mp4/ioieTr41L24.mp477yes https://puzzicle.org/math/cc-eighth-grade-math/v/smoking-1945-extrapolationhttps://puzzicle.org/thumbs/x47ddceeb.jpgLine of best fit: smoking in 1945The scatter plot shows how many adults in America smoked from year to year. We can guess how many smoked in 1945 by drawing a line that slopes down through the points. Then we see how high the line would be 20 years earlier.https://cdn.kastatic.org/ka-youtube-converted/d4WE4SnhBDU.mp4/d4WE4SnhBDU.mp4287yes https://puzzicle.org/math/cc-eighth-grade-math/v/interpreting-trend-linehttps://puzzicle.org/thumbs/x5409e0ad.jpgInterpreting a trend lineThe graph shows how studying affects test scores. The line slope of 15 means that for each extra hour of studying, there is usually a 15-point increase in test score. But it's not a guarantee. Some students may do better or worse than the trend.https://cdn.kastatic.org/ka-youtube-converted/mfX_yUvwJho.mp4/mfX_yUvwJho.mp4208yes https://puzzicle.org/math/cc-eighth-grade-math/v/interpreting-linear-graphshttps://puzzicle.org/thumbs/370285953.jpgLinear graphs word problemsSal discusses how we would go about making sense of the direction of a linear graph that represents a relationship between two real-world quantities.https://cdn.kastatic.org/ka-youtube-converted/xR9r38mZjK4.mp4/xR9r38mZjK4.mp4305yes https://puzzicle.org/math/cc-eighth-grade-math/v/interpreting-features-of-linear-functions-examplehttps://puzzicle.org/thumbs/x97976348.jpgLinear graphs word problem: catsSal finds the y-intercept and the slope of a linear relationship representing someone accumulating cats! He then interprets what the y-intercept and the slope mean in that context.https://cdn.kastatic.org/ka-youtube-converted/qo5jU_V6JVo.mp4/qo5jU_V6JVo.mp4163yes https://puzzicle.org/math/cc-eighth-grade-math/v/interpreting-features-of-linear-functions-example-2https://puzzicle.org/thumbs/x77591b48.jpgLinear equations word problems: volcanoSal finds the y-intercept and the slope of a linear relationship representing someone climbing a volcano. He then interprets what the y-intercept and the slope mean in that context.https://cdn.kastatic.org/ka-youtube-converted/eLUmpX_h9sw.mp4/eLUmpX_h9sw.mp4358yes https://puzzicle.org/math/cc-eighth-grade-math/v/slope-examplehttps://puzzicle.org/thumbs/26801.jpgLinear equations word problems: earningsSal finds the slope of a linear relationship between the number of work hours and the money earned. He then interprets what this slope means in that context.https://cdn.kastatic.org/ka-youtube-converted/jTCZfMMcHBo.mp4/jTCZfMMcHBo.mp4243yes https://puzzicle.org/math/cc-eighth-grade-math/v/exploring-linear-relationshipshttps://puzzicle.org/thumbs/26797.jpgModeling with linear equations: snowSal uses a linear equation to model the amount of snow on the ground.https://cdn.kastatic.org/ka-youtube-converted/qPx7i1jwXX4.mp4/qPx7i1jwXX4.mp4298yes https://puzzicle.org/math/cc-eighth-grade-math/v/comparing-features-of-functions-2https://puzzicle.org/thumbs/x6b95f0e9.jpgComparing linear functions word problem: climbSal is given the formula and a table of values that represent two people climbing a wall, and is asked to determine which one started out higher.https://cdn.kastatic.org/ka-youtube-converted/MKErxh58RME.mp4/MKErxh58RME.mp4284yes https://puzzicle.org/math/cc-eighth-grade-math/v/comparing-features-of-functions-4https://puzzicle.org/thumbs/xda657261.jpgComparing linear functions word problem: walkSal is given a table of values that represents four people walking to school, and is asked to determine which one started out farther from the school.https://cdn.kastatic.org/ka-youtube-converted/NOsnG2DkGKM.mp4/NOsnG2DkGKM.mp4224yes https://puzzicle.org/math/cc-eighth-grade-math/v/comparing-features-of-functions-6https://puzzicle.org/thumbs/x09dee3a1.jpgComparing linear functions word problem: workSal is given a table of values that represents a person walking to work, and is asked to determine which verbal description represents someone starting at the same distance from work.https://cdn.kastatic.org/ka-youtube-converted/kJIqoTw_Og8.mp4/kJIqoTw_Og8.mp4226yes https://puzzicle.org/math/cc-eighth-grade-math/v/application-problem-with-graphhttps://puzzicle.org/thumbs/33752750.jpgLinear function example: spending moneySal solves an interesting application problem using a linear model.https://cdn.kastatic.org/ka-youtube-converted/ljROszF_rqs.mp4/ljROszF_rqs.mp4322yes https://puzzicle.org/math/cc-eighth-grade-math/v/word-problem-solving-4https://puzzicle.org/thumbs/23644.jpgModeling with linear equations: gym membership & lemonadeConstructing linear equations to solve word problemshttps://cdn.kastatic.org/ka-youtube-converted/5EdbPz1ZVn0.mp4/5EdbPz1ZVn0.mp4605yes https://puzzicle.org/math/cc-eighth-grade-math/v/fitting-a-line-to-datahttps://puzzicle.org/thumbs/23654.jpgFitting a line to dataSal creates a scatter plot and then fits a line to data on the median California family income.https://cdn.kastatic.org/ka-youtube-converted/OhUkMQtBGmE.mp4/OhUkMQtBGmE.mp4468yes https://puzzicle.org/math/cc-eighth-grade-math/v/two-way-frequency-tables-and-venn-diagramshttps://puzzicle.org/thumbs/x1d9a21e3.jpgTwo-way frequency tables and Venn diagramsSometimes data belongs to more than one category. For example, candies might have chocolate, coconut, both, or neither. We can use Venn diagrams and two-way tables. Venn diagrams show sets and their overlaps. Two-way tables organize data in rows and columns. Both methods help show relationships between categories.https://cdn.kastatic.org/ka-youtube-converted/l5MrtV7ZN88.mp4/l5MrtV7ZN88.mp4383yes https://puzzicle.org/math/cc-eighth-grade-math/v/two-way-relative-frequency-tableshttps://puzzicle.org/thumbs/xb908feeb.jpgTwo-way relative frequency tablesRelative frequencies show how often something happens compared to the total number of times it could happen. In our example, we calculate the relative frequency of accidents for SUVs by dividing the number of SUVs that had accidents by the total number of SUVs. This gives us a percentage or fraction that tells us how common accidents are for that type of vehicle.https://cdn.kastatic.org/ka-youtube-converted/_ETPMszULXc.mp4/_ETPMszULXc.mp4268yes https://puzzicle.org/math/cc-eighth-grade-math/v/interpreting-two-way-tableshttps://puzzicle.org/thumbs/xaae200df.jpgInterpreting two-way tablesTwo-way tables let us sort a group in two ways. For example, we see how men and women voted in the 2012 US presidential election. We can compare the percentages of men and women who voted for each candidate. Two-way tables help us understand how categories relate.https://cdn.kastatic.org/ka-youtube-converted/MarqSlyz-lU.mp4/MarqSlyz-lU.mp4104yes https://puzzicle.org/math/algebra/v/introduction-to-square-rootshttps://puzzicle.org/thumbs/xa58e57f6.jpgIntro to square rootsLearn about the square root symbol (the principal root) and what it means to find a square root. Also learn how to solve simple square root equations.https://cdn.kastatic.org/ka-youtube-converted/mbc3_e5lWw0.mp4/mbc3_e5lWw0.mp4324yes https://puzzicle.org/math/algebra/v/introduction-to-cube-rootshttps://puzzicle.org/thumbs/x63d365e2.jpgIntro to cube rootsLearn the meaning of cube roots and how to find them. Also learn how to find the cube root of a negative number.https://cdn.kastatic.org/ka-youtube-converted/87_qIofPwhg.mp4/87_qIofPwhg.mp4481yes https://puzzicle.org/math/cc-eighth-grade-math/v/finding-cube-rootshttps://puzzicle.org/thumbs/126711578.jpgWorked example: Cube root of a negative numberLearn how to find the cube root of negative 512 by breaking it down into prime factors. When we find groups of three of the same factor, we know that's a factor of the cube root. It helps to remember that -1*-1*-1 is -1, so the cube root of -1 is itself.https://cdn.kastatic.org/ka-youtube-converted/DKh16Th8x6o.mp4/DKh16Th8x6o.mp4258yes https://puzzicle.org/math/algebra/v/finding-square-root-of-decimalhttps://puzzicle.org/thumbs/x4451110d.jpgSquare root of decimalLearn how to find the square root of a decimal number. The problem solved in this video is p^2 = 0.81.https://cdn.kastatic.org/ka-youtube-converted/tRHLEWSUjrQ.mp4/tRHLEWSUjrQ.mp4184yes https://puzzicle.org/math/cc-eighth-grade-math/v/dimensions-of-cube-from-volumehttps://puzzicle.org/thumbs/xc5cba819.jpgDimensions of a cube from its volumeWhen we know the volume of a cube, we can use the cube root of the volume to find the length of each side. We'll need to factor the volume and find 3 equal groups of factors. The value of one group is the length of one side of the cube.https://cdn.kastatic.org/ka-youtube-converted/GpOxsRF1Cjc.mp4/GpOxsRF1Cjc.mp4215yes https://puzzicle.org/math/algebra/v/introduction-to-rational-and-irrational-numbershttps://puzzicle.org/thumbs/x2d1f5fa7.jpgIntro to rational & irrational numbersLearn the difference between rational and irrational numbers, learn how to identify them, and discover why some of the most famous numbers in mathematics, like Pi and e, are actually irrational. Did you know that there's always an irrational number between any two rational numbers?https://cdn.kastatic.org/ka-youtube-converted/cLP7INqs3JM.mp4/cLP7INqs3JM.mp4355yes https://puzzicle.org/math/algebra/v/recognizing-irrational-numbershttps://puzzicle.org/thumbs/x00ce2e4a.jpgClassifying numbers: rational & irrationalWe can write any rational number as the ratio of two integers. We cannot write irrational numbers, such as the square root of 8 and pi, in this way. Learn other forms, such as decimals, in which these types of numbers can appear.https://cdn.kastatic.org/ka-youtube-converted/d9pO2z2qvXU.mp4/d9pO2z2qvXU.mp4138yes https://puzzicle.org/math/cc-eighth-grade-math/v/categorizing-numbershttps://puzzicle.org/thumbs/xd693fe85.jpgClassifying numbersThere are many categories we can use to classify numbers. Some of those categories are rational numbers, irrational numbers, integers, and whole numbers. Rational numbers are represented as a fraction of two integers, while irrational numbers cannot be represented as a fraction of two integers. Integers are positive and negative numbers that don’t involve fractions or decimals. Whole numbers are a subset of integers — they are non-negative integers.https://cdn.kastatic.org/ka-youtube-converted/-QHff5pRdM8.mp4/-QHff5pRdM8.mp4529yes https://puzzicle.org/math/cc-eighth-grade-math/v/approximating-square-roots-2https://puzzicle.org/thumbs/x83a7e947.jpgApproximating square rootsLearn how to find the approximate values of square roots. The examples used in this video are √32, √55, and √123. The technique used is to compare the squares of whole numbers to the number we're taking the square root of.https://cdn.kastatic.org/ka-youtube-converted/Lh7NMBPFVZw.mp4/Lh7NMBPFVZw.mp4288yes https://puzzicle.org/math/cc-eighth-grade-math/v/approximating-square-rootshttps://puzzicle.org/thumbs/125518448.jpgApproximating square roots to hundredthsLet's approximate the square root of 45 without a calculator. We'll explore how to find the perfect squares around 45 and use them to make an educated guess. Then, we'll refine our guess by squaring it to see how close we get to 45.https://cdn.kastatic.org/ka-youtube-converted/EFVrAk61xjE.mp4/EFVrAk61xjE.mp4432yes https://puzzicle.org/math/cc-eighth-grade-math/v/coverting-repeating-decimals-to-fractions-1https://puzzicle.org/thumbs/351466917.jpgConverting repeating decimals to fractions (part 1 of 2)The process of converting a repeating decimal to a fraction can be broken down into a few easy steps. To start, set the decimal equal to a variable. Multiply the decimal by 10 and subtract the original decimal from it. Finally, divide both sides by 9 to obtain the fractional form of the decimal. For example, 0.7 repeating would be 7/9, and 1.2 repeating would be 11/9.https://cdn.kastatic.org/ka-youtube-converted/FPNhCVZlbJs.mp4/FPNhCVZlbJs.mp4243yes https://puzzicle.org/math/cc-eighth-grade-math/v/coverting-repeating-decimals-to-fractions-2https://puzzicle.org/thumbs/351405953.jpgConverting repeating decimals to fractions (part 2 of 2)Repeated decimals can be converted into fractions by shifting the decimal to the right and subtracting the decimals. To do this, multiply the number by 10 to the second power, then subtract. For example, 0.363636 repeating is 4/11 and 0.7141414 repeating is 707/990. Another example is 3.257257257 repeating, which is 3254/999. This calculation can be done in the head or by borrowing. After the subtraction, the numerator and denominator can be reduced and the fraction can be simplified.https://cdn.kastatic.org/ka-youtube-converted/Ihws0d-WLzU.mp4/Ihws0d-WLzU.mp4546yes https://puzzicle.org/math/cc-eighth-grade-math/v/sorting-irrational-numbers-involving-radicalshttps://puzzicle.org/thumbs/xc4d6526d.jpgComparing irrational numbers with radicalsLearn how to sort a bunch of numbers (4√2 2√3 3√2 √17 3√3 5) from least to greatest without using a calculator.https://cdn.kastatic.org/ka-youtube-converted/KibTbfkoPTs.mp4/KibTbfkoPTs.mp4329yes https://puzzicle.org/math/cc-eighth-grade-math/v/comparing-values-with-calculatorhttps://puzzicle.org/thumbs/x44e869e5.jpgComparing values with calculatorLet's use the technique of squaring to compare values. We first compare 22.9% to an arbitrary value, 0.3, and then square 0.3 to compare it to the square root of 0.45. This helps us understand which value is larger without needing a calculator.https://cdn.kastatic.org/ka-youtube-converted/PuRsXST9EZU.mp4/PuRsXST9EZU.mp4241yes https://puzzicle.org/math/8th-engage-ny/v/example-finding-distance-with-pythagorean-theoremhttps://puzzicle.org/thumbs/x6c0e0c41583be82e.jpgFinding distance with Pythagorean theoremSal finds the distance between two points with the Pythagorean theorem.https://cdn.kastatic.org/ka-youtube-converted/apkT6rVE8qo.mp4/apkT6rVE8qo.mp4221yes https://puzzicle.org/math/geometry/v/distance-formulahttps://puzzicle.org/thumbs/23756.jpgDistance formulaLearn how to find the distance between two points by using the distance formula, which is an application of the Pythagorean theorem. We can rewrite the Pythagorean theorem as d=√((x_2-x_1)²+(y_2-y_1)²) to find the distance between any two points.https://cdn.kastatic.org/ka-youtube-converted/nyZuite17Pc.mp4/nyZuite17Pc.mp4579yes https://puzzicle.org/math/cc-eighth-grade-math/v/area-of-an-isosceles-trianglehttps://puzzicle.org/thumbs/x3249b1b3b5e14fc2.jpgUse Pythagorean theorem to find area of an isosceles triangleWe can find the area of an isosceles triangle using the Pythagorean theorem. Once we recognize the triangle as isosceles, we divide it into congruent right triangles. Then we use the theorem to find the height. Last, we calculate the area with the formula: 1/2 × base × height.https://cdn.kastatic.org/ka-youtube-converted/yZ4CbopnVBg.mp4/yZ4CbopnVBg.mp4240yes https://puzzicle.org/math/geometry/v/pythagoriean-theorem-in-3dhttps://puzzicle.org/thumbs/x5a21eae7.jpgPythagorean theorem in 3DDiscover how to find the length of an edge in a 3D shape using the Pythagorean theorem! This fun math lesson explores right pyramids and rectangular prisms, guiding you through solving for unknown lengths by applying the theorem multiple times.https://cdn.kastatic.org/ka-youtube-converted/1UQ5IbihJNI.mp4/1UQ5IbihJNI.mp4356yes https://puzzicle.org/math/geometry/v/applying-volume-of-solidshttps://puzzicle.org/thumbs/xd5c92d1bf1929ee0.jpgApplying volume of solidsCalculate the volume of a grain hopper and then use the given rate to solve an applied problem.https://cdn.kastatic.org/ka-youtube-converted/dp3my9MnmOM.mp4/dp3my9MnmOM.mp4455yes https://puzzicle.org/math/geometry-home/v/perimeter-and-area-basicshttps://puzzicle.org/thumbs/283213901.jpgPerimeter & areaPerimeter is the distance around the outside of a shape. For example, if you walk around the edge of a playground, you're measuring its perimeter. Area is the amount of space inside a shape. For example, if you want to put new tiles on your kitchen floor, you need to know the area of the room to know how many tiles to buy.https://cdn.kastatic.org/ka-youtube-converted/LoaBd-sPzkU.mp4/LoaBd-sPzkU.mp4505yes https://puzzicle.org/math/8th-grade-illustrative-math/v/identifying-rigid-transformationshttps://puzzicle.org/thumbs/x80eeb454428561c6.jpgIdentifying rigid transformationsDescribe the movement of shapes informally and formally using the terms “clockwise,” “counterclockwise,” “translations,” “rotations,” and “reflections” of figures.https://cdn.kastatic.org/ka-youtube-converted/8mkTsVgSJe4.mp4/8mkTsVgSJe4.mp4396yes https://puzzicle.org/math/8th-grade-illustrative-math/v/rotating-line-segmentshttps://puzzicle.org/thumbs/x973676fe440d470b.jpgRotating line segmentsDraw and label a diagram of a line segment rotated 90 degrees clockwise or counterclockwise about a given center.https://cdn.kastatic.org/ka-youtube-converted/Wmb3ZVfVUuc.mp4/Wmb3ZVfVUuc.mp4287yes https://puzzicle.org/math/geometry/v/introduction-to-transformationshttps://puzzicle.org/thumbs/x7856e9c3.jpgRigid transformations introTransformations in math involve changing a shape's position or which way the shape points. There are three main types: translations (moving the shape), rotations (turning the shape), and reflections (flipping the shape like a mirror image). Rigid transformations keep the shape's size and angles the same. The image is the shape in its new position and direction.https://cdn.kastatic.org/ka-youtube-converted/XiAoUDfrar0.mp4/XiAoUDfrar0.mp4443yes https://puzzicle.org/math/8th-grade-illustrative-math/v/rigid-transformations-of-parallel-lineshttps://puzzicle.org/thumbs/xc49d2ace8ebfabf7.jpgRigid transformations of parallel linesDiscover why parallel lines stay parallel even after you rotate, reflect, or translate them.https://cdn.kastatic.org/ka-youtube-converted/PbLM99ay7LI.mp4/PbLM99ay7LI.mp4220yes https://puzzicle.org/math/8th-grade-illustrative-math/v/alternate-interior-angleshttps://puzzicle.org/thumbs/x03a2c099a54d7fb1.jpgAlternate interior anglesIdentify alternate interior angles made by a transversal connecting two parallel lines and use that information to find missing angle measures.https://cdn.kastatic.org/ka-youtube-converted/mbxSy9TTTbM.mp4/mbxSy9TTTbM.mp4253yes https://puzzicle.org/math/8th-grade-illustrative-math/v/dilating-points-on-a-circular-gridhttps://puzzicle.org/thumbs/x9d851b8004f8b9a1.jpgDilating points on a circular gridIdentify the image of a dilation on a circular grid given a scale factor and center of dilation, and determine the scale factor of a dilation by comparing the distances of original and image points from the center of dilation.https://cdn.kastatic.org/ka-youtube-converted/Uc-UqvEhcQw.mp4/Uc-UqvEhcQw.mp4186yes https://puzzicle.org/math/8th-grade-illustrative-math/v/dilating-figures-on-a-circular-gridhttps://puzzicle.org/thumbs/x04ac4eadc36cc6e1.jpgDilating figures on a circular gridIdentify the image of a dilation on a circular grid given a scale factor and center of dilation, and determine the scale factor of a dilation by comparing the distances of original and image points from the center of dilation.https://cdn.kastatic.org/ka-youtube-converted/B0fCqf_FDPU.mp4/B0fCqf_FDPU.mp4134yes https://puzzicle.org/math/8th-grade-illustrative-math/v/identifying-center-of-dilation-with-no-gridhttps://puzzicle.org/thumbs/xa55a1ad414b16cd2.jpgIdentifying center of dilation with no gridIdentify the center of dilation without a grid by drawing rays through corresponding vertices and determining where they intersect.https://cdn.kastatic.org/ka-youtube-converted/0-PM-gN4rog.mp4/0-PM-gN4rog.mp4107yes https://puzzicle.org/math/8th-grade-illustrative-math/v/dilating-figures-with-no-gridhttps://puzzicle.org/thumbs/x4990e32ae03ca0ec.jpgDilating figures with no gridUnderstand how to create and describe dilations without a grid, using a center and scale factor, and explain how the scale factor affects the size and position of a figure.https://cdn.kastatic.org/ka-youtube-converted/7ng5GV40W9Q.mp4/7ng5GV40W9Q.mp4100yes https://puzzicle.org/math/geometry/v/dilations-introhttps://puzzicle.org/thumbs/x4095f44e644d564a.jpgDilations introDilations are a type of non-rigid transformation. They change the size of a shape by scaling it up or down, making it bigger or smaller. Unlike rigid transformations, dilations do not keep the shape's size the same.https://cdn.kastatic.org/ka-youtube-converted/Kv-lyVmQ5ps.mp4/Kv-lyVmQ5ps.mp491yes https://puzzicle.org/math/cc-eighth-grade-math/v/comparing-side-lengths-after-dilationhttps://puzzicle.org/thumbs/x1eac271a.jpgSide lengths after dilationIn a dilation, if the scale factor is greater than 1, the shape will stretch away from the center of dilation, getting larger. If the scale factor is between 0 and 1, the shape will shrink towards the center, getting smaller. All the lengths in the figure change by the same scale factor.https://cdn.kastatic.org/ka-youtube-converted/2yjSAarzWF8.mp4/2yjSAarzWF8.mp4133yes https://puzzicle.org/math/8th-grade-illustrative-math/v/determining-similarity-in-polygonshttps://puzzicle.org/thumbs/x38cb3631e04f21e6.jpgDetermining similarity in polygonsUnderstand that similar polygons have congruent corresponding angles and proportional corresponding side lengths, and use both conditions together to determine whether two polygons are similar.https://cdn.kastatic.org/ka-youtube-converted/PZEUpMiOn6M.mp4/PZEUpMiOn6M.mp4220yes https://puzzicle.org/math/8th-grade-illustrative-math/v/identifying-similar-triangleshttps://puzzicle.org/thumbs/x7e0662cecf4192f0.jpgIdentifying similar trianglesIdentify similar triangles by finding two pairs of congruent angles.https://cdn.kastatic.org/ka-youtube-converted/E7KfcxpKF7c.mp4/E7KfcxpKF7c.mp4163yes https://puzzicle.org/math/8th-grade-illustrative-math/v/introducing-triangle-similarityhttps://puzzicle.org/thumbs/x1eeea9da473fd69a.jpgIntroducing triangle similarityIntroduce triangle similarity by explaining that similar triangles have the same shape but different sizes. Show that similarity can be determined by verifying either that corresponding angles are congruent or that corresponding sides are proportional.https://cdn.kastatic.org/ka-youtube-converted/AZSUyrOqFBg.mp4/AZSUyrOqFBg.mp4465yes https://puzzicle.org/math/8th-grade-illustrative-math/v/finding-missing-lengths-in-slope-triangleshttps://puzzicle.org/thumbs/x2852efd7c7410824.jpgFinding missing lengths in slope trianglesCalculate missing side lengths in similar triangles by using proportional reasoning within triangles and scale factors between triangles.https://cdn.kastatic.org/ka-youtube-converted/MKl9otX8gwA.mp4/MKl9otX8gwA.mp4145yes https://puzzicle.org/math/8th-grade-illustrative-math/v/side-lengths-in-similar-triangleshttps://puzzicle.org/thumbs/xa15952ef65933850.jpgSide lengths in similar trianglesCalculate missing side lengths in similar triangles by using proportional reasoning within triangles and scale factors between triangles.https://cdn.kastatic.org/ka-youtube-converted/5GrAoRoV7co.mp4/5GrAoRoV7co.mp4278yes https://puzzicle.org/math/algebra/v/introduction-to-slopehttps://puzzicle.org/thumbs/xf91ad68e.jpgIntro to slopeSlope tells us how steep a line is. It's like measuring how quickly a hill goes up or down. We find the slope by seeing how much we go up or down (vertical change) for each step to the right (horizontal change). If a line goes up 2 steps for every 1 step to the right, its slope is 2.https://cdn.kastatic.org/ka-youtube-converted/MeU-KzdCBps.mp4/MeU-KzdCBps.mp4416yes https://puzzicle.org/math/8th-grade-illustrative-math/v/drawing-lines-for-a-given-slopehttps://puzzicle.org/thumbs/xb3b4fd4b2c6e43d6.jpgDrawing lines for a given slopeDefine slope as the ratio of vertical to horizontal change, draw lines with specific slopes, and justify the similarity of slope triangles.https://cdn.kastatic.org/ka-youtube-converted/BI1aS8-PdyU.mp4/BI1aS8-PdyU.mp4127yes https://puzzicle.org/math/8th-grade-illustrative-math/v/finding-points-on-a-line-when-given-an-equationhttps://puzzicle.org/thumbs/xb18be98d27e98a71.jpgFinding points on a line when given an equationVerify whether a point lies on a line by substituting its coordinates into the line's equation and confirming the resulting ratio is equivalent.https://cdn.kastatic.org/ka-youtube-converted/7eeaXv6brQ8.mp4/7eeaXv6brQ8.mp4137yes https://puzzicle.org/math/8th-grade-illustrative-math/v/writing-the-equation-of-a-line-with-a-positive-slopehttps://puzzicle.org/thumbs/x92dff0810cd001dc.jpgWriting the equation of a line with a positive slopeCreate an equation of a line with a positive slope on the coordinate plane using knowledge of similar triangles.https://cdn.kastatic.org/ka-youtube-converted/Q53TQP6eimk.mp4/Q53TQP6eimk.mp4430yes https://puzzicle.org/math/8th-grade-illustrative-math/v/interpreting-and-comparing-proportional-relationshipshttps://puzzicle.org/thumbs/x1e48ece36b726bac.jpgInterpreting and comparing proportional relationshipsPractice comparing the speeds of four wind-up robots by reading their distance-vs-time graphs and figuring out which one is the slowest.https://cdn.kastatic.org/ka-youtube-converted/txRmzpNHtYI.mp4/txRmzpNHtYI.mp486yes https://puzzicle.org/math/8th-grade-illustrative-math/v/comparing-scale-in-proportional-relationship-graphshttps://puzzicle.org/thumbs/xc37deb5297ec4883.jpgComparing scale in proportional relationship graphsPractice graphing proportional relationships from equations like y = kx on coordinate planes with varying axis scales. Learn to recognize when differently-scaled graphs actually represent the same relationship.https://cdn.kastatic.org/ka-youtube-converted/CdMUBpFP2ro.mp4/CdMUBpFP2ro.mp4170yes https://puzzicle.org/math/8th-grade-illustrative-math/v/comparing-proportional-and-nonproportional-linear-relationshipshttps://puzzicle.org/thumbs/x0b6e964bf4bcbfc2.jpgComparing proportional and nonproportional linear relationshipsCompare a proportional relationship to a nonproportional linear one using a cup-stacking scenario, testing whether stack height is proportional to the number of cups. Find the rate of change from a graph and interpret the vertical-axis intercept as a starting value rather than zero.https://cdn.kastatic.org/ka-youtube-converted/0ESzVmo0hJY.mp4/0ESzVmo0hJY.mp4228yes https://puzzicle.org/math/8th-grade-illustrative-math/v/using-equations-to-represent-linear-relationshipshttps://puzzicle.org/thumbs/x30190501c24f3357.jpgUsing equations to represent linear relationshipsWrite an equation for a linear relationship from a real-world context, and connect it to the same relationship shown as a verbal description, table, and graph. Find the rate of change and initial value from the context, then see how the equation, table, and graph all tell the same story.https://cdn.kastatic.org/ka-youtube-converted/qGRtaBart38.mp4/qGRtaBart38.mp452yes https://puzzicle.org/math/8th-grade-illustrative-math/v/interpreting-slope-and-y-intercepthttps://puzzicle.org/thumbs/xc240148546b7a224.jpgInterpreting slope and y-interceptGiven the graph of a linear relationship in a real-world context, interpret what the slope and vertical intercept mean within that story. The focus is on explaining what each feature represents, not on computing values or writing an equation.https://cdn.kastatic.org/ka-youtube-converted/PrjV4zv8Xck.mp4/PrjV4zv8Xck.mp4146yes https://puzzicle.org/math/8th-grade-illustrative-math/v/comparing-graphs-of-linear-relationships-with-different-y-interceptshttps://puzzicle.org/thumbs/x69c9b603e665e525.jpgComparing graphs of linear relationships with different y-interceptsCompare two linear relationships with the same rate of change but different initial values, graphed together in a real-world context. See how equal slopes produce parallel lines, with the gap between them reflecting the difference in starting values.https://cdn.kastatic.org/ka-youtube-converted/6Nig8sO_a3A.mp4/6Nig8sO_a3A.mp4129yes https://puzzicle.org/math/algebra/v/introduction-to-interceptshttps://puzzicle.org/thumbs/x9cb60d6a.jpgIntro to interceptsLearn what x- and y-intercepts are. The equations used in this video are y = 0.5x - 3 and 5x + 6y = 30.https://cdn.kastatic.org/ka-youtube-converted/LNSB0N6esPU.mp4/LNSB0N6esPU.mp4393yes https://puzzicle.org/math/algebra/v/slope-intercept-intro-exampleshttps://puzzicle.org/thumbs/xbe126b648573616d.jpgWorked examples: slope-intercept introWorked examples practicing the relationship between slope, y-intercept, and linear equation in slope-intercept form.https://cdn.kastatic.org/ka-youtube-converted/qgsNNqmlLoA.mp4/qgsNNqmlLoA.mp4226yes https://puzzicle.org/math/algebra/v/checking-ordered-pair-solutions-to-equations-1https://puzzicle.org/thumbs/x9a47991d.jpgSolutions to 2-variable equationsThe solution of an equation with one variable is a number. How does the solution of a 2-variable equation look like? Well, it is an ordered pair. Learn more about it and how to test solutions to 2-variable equations.https://cdn.kastatic.org/ka-youtube-converted/qk69pR91R00.mp4/qk69pR91R00.mp4155yes https://puzzicle.org/math/algebra/v/checking-ordered-pair-solutions-to-equations-2https://puzzicle.org/thumbs/x6e2674bb.jpgWorked example: solutions to 2-variable equationsHow do you check if an ordered pair is a solution to a given equation? You need to plug in the numbers and see what equality results. Watch this video to see a worked example.https://cdn.kastatic.org/ka-youtube-converted/d3VMo1VWFvc.mp4/d3VMo1VWFvc.mp4146yes https://puzzicle.org/math/8th-grade-illustrative-math/v/writing-and-graphing-linear-equationshttps://puzzicle.org/thumbs/x634a21346bd2d17a.jpgWriting and graphing linear equationsWrite an equation in the form Ax + By = C from a real-world context where two quantities trade off to keep a fixed total, then graph the relationship. This form fits naturally when neither variable depends on the other, since both simply vary together to maintain that constant total.https://cdn.kastatic.org/ka-youtube-converted/LH5xq2SPAeU.mp4/LH5xq2SPAeU.mp4272yes https://puzzicle.org/math/8th-grade-illustrative-math/v/determining-solutions-to-2-variable-equations-and-graphshttps://puzzicle.org/thumbs/x49f8282296fc12d3.jpgDetermining solutions to 2-variable equations and graphsCheck whether ordered pairs are solutions to a two-variable linear equation by substituting and verifying. Connect this back to the graph: every point on the line is a solution, and any point off the line is not.https://cdn.kastatic.org/ka-youtube-converted/jOS45lOWXdI.mp4/jOS45lOWXdI.mp483yes https://puzzicle.org/math/8th-grade-illustrative-math/v/calculating-solutions-to-a-linear-equationhttps://puzzicle.org/thumbs/xe0520e7003312201.jpgCalculating solutions to a linear equationGiven a two-variable linear equation and a value for one variable, calculate the value of the other variable that makes the pair a solution. This builds the substitution skill needed to find missing coordinates and complete solution tables.https://cdn.kastatic.org/ka-youtube-converted/JAhG2bRH55I.mp4/JAhG2bRH55I.mp4110yes https://puzzicle.org/math/algebra/v/graphing-solutions-to-2-variable-linear-equations-1https://puzzicle.org/thumbs/x7686ff2c.jpgCompleting solutions to 2-variable equationsGiven a 2-variable equation and the x or y values of a solution, Sal finds the value of the other variable in the solution.https://cdn.kastatic.org/ka-youtube-converted/EHR-YDwrrhM.mp4/EHR-YDwrrhM.mp4431yes https://puzzicle.org/math/algebra/v/descartes-and-cartesian-coordinateshttps://puzzicle.org/thumbs/578962249.jpgIntro to the coordinate planeDescartes created a system of two perpendicular axes (the x and y axes) which could be used to plot points in a plane. This enabled the visualization of algebraic equations in geometric form. For example, a linear equation like y = 2x + 1 could be plotted as a line on the coordinate plane, while a quadratic equation like y = x^2 could be plotted as a parabola. This linked algebra and geometry through the use of graphs.https://cdn.kastatic.org/ka-youtube-converted/N4nrdf0yYfM.mp4/N4nrdf0yYfM.mp4682yes https://puzzicle.org/math/algebra/v/slope-from-equationhttps://puzzicle.org/thumbs/x7c5573fa2851b201.jpgSlope from equationWorked examples of finding the slope of a line given its equation, using many forms of equations.https://cdn.kastatic.org/ka-youtube-converted/G4r9PpYQiL8.mp4/G4r9PpYQiL8.mp4379yes https://puzzicle.org/math/algebra-1-eureka-squared-aligned/v/equation-with-variables-on-both-sides-fractions-indian-accenthttps://puzzicle.org/thumbs/x2dd6070ceda1ccc6.jpgEquation with variables on both sides: fractionsTo solve the equation (3/4)x + 2 = (3/8)x - 4, we first eliminate fractions by multiplying both sides by the least common multiple of the denominators. Then, we add or subtract terms from both sides of the equation to group the x-terms on one side and the constants on the other. Finally, we solve and check as normal.https://cdn.kastatic.org/ka-youtube-converted/MWHBSLGI-Tk.mp4/MWHBSLGI-Tk.mp4289yes https://puzzicle.org/math/cc-eighth-grade-math/v/sum-consecutive-integershttps://puzzicle.org/thumbs/x32215d81.jpgSums of consecutive integersWhen solving a math problems involving consecutive integers, we can use algebra to set up an equation to solve for the smallest of the four integers.https://cdn.kastatic.org/ka-youtube-converted/8CJ6Qdcoxsc.mp4/8CJ6Qdcoxsc.mp4319yes https://puzzicle.org/math/cc-eighth-grade-math/v/sum-of-consecutive-odd-integershttps://puzzicle.org/thumbs/26782.jpgSum of integers challengeThe sum of three consecutive odd integers is 231. First let's identify the integers. If the smallest is x, what is the value of the next odd integer? Then we set up an equation, adding all three integers to get their total. We can solve for x, the smallest integer. Once we know the smallest, what is the largest integer?https://cdn.kastatic.org/ka-youtube-converted/d8De3xcVmnw.mp4/d8De3xcVmnw.mp4273yes https://puzzicle.org/math/algebra/v/systems-word-problem-with-no-solutionhttps://puzzicle.org/thumbs/xa7908d53.jpgSystem of equations word problem: no solutionSystems of equations can be used to solve many real-world problems. In this video, we solve a problem about a toy factory. In this case, the problem has no viable solution, which means the information describes an impossible situation.https://cdn.kastatic.org/ka-youtube-converted/SRn3WhhS6vs.mp4/SRn3WhhS6vs.mp4303yes https://puzzicle.org/math/8th-grade-illustrative-math/v/rules-of-input-output-tableshttps://puzzicle.org/thumbs/x360eafae76e6b5ca.jpgRules of input-output tablesLearn how input-output diagrams work, where a rule transforms each input into a corresponding output. Practice identifying the rule from given pairs and applying it to find new outputs.https://cdn.kastatic.org/ka-youtube-converted/h8vFG2Ov43c.mp4/h8vFG2Ov43c.mp484yes https://puzzicle.org/math/cc-eighth-grade-math/v/understanding-function-notation-example-3https://puzzicle.org/thumbs/x5860a542.jpgFunction notation exampleSal uses function notation to help Frank figure out how much water he can put in his balloon.https://cdn.kastatic.org/ka-youtube-converted/IXRMVcoqRRQ.mp4/IXRMVcoqRRQ.mp4133yes https://puzzicle.org/math/algebra/v/create-function-from-equationhttps://puzzicle.org/thumbs/x0c7d59f7.jpgObtaining a function from an equationEquations and functions are not the same thing, but they can be related in several ways. In this video, we obtain a function from an equation. The function represents the same relationship between the quantities in the equation.https://cdn.kastatic.org/ka-youtube-converted/OOim0QPsJ9o.mp4/OOim0QPsJ9o.mp4184yes https://puzzicle.org/math/8th-grade-illustrative-math/v/graphs-of-functions-in-contexthttps://puzzicle.org/thumbs/xbaa8c569ce9e817b.jpgGraphs of functions in contextInterpret points on graphs, including some that don't represent functions, and draw graphs to model real-world contexts. Explain which quantity depends on which, identifying which variable is a function of the other.https://cdn.kastatic.org/ka-youtube-converted/sVxImbpK8AI.mp4/sVxImbpK8AI.mp4122yes https://puzzicle.org/math/8th-grade-illustrative-math/v/interpreting-piecewise-linear-functionshttps://puzzicle.org/thumbs/x7381d074929b3ea3.jpgInterpreting piecewise linear functionsCalculate the rate of change across different segments of a piecewise linear graph and interpret what each rate means in context. Practice modeling a nonlinear function using a piecewise linear approximation.https://cdn.kastatic.org/ka-youtube-converted/8OeKOVeIZMQ.mp4/8OeKOVeIZMQ.mp4131yes https://puzzicle.org/math/8th-grade-illustrative-math/v/relating-height-and-volume-of-cylindershttps://puzzicle.org/thumbs/x36f913dff566a550.jpgRelating height and volume of cylindersExplore how height and volume relate as a linear function in the context of filling cylinders.https://cdn.kastatic.org/ka-youtube-converted/xSNzGtTEC1U.mp4/xSNzGtTEC1U.mp4224yes https://puzzicle.org/math/8th-grade-illustrative-math/v/calculating-dimensions-of-a-cylinderhttps://puzzicle.org/thumbs/x6a8d4eb0b326c0dd.jpgCalculating dimensions of a cylinderUse the cylinder volume formula to solve for a missing dimension, height or radius, when the volume and other dimension are given. Work through a table of cylinder dimensions and look for patterns.https://cdn.kastatic.org/ka-youtube-converted/0tc_fga-Tq0.mp4/0tc_fga-Tq0.mp4100yes https://puzzicle.org/math/8th-grade-illustrative-math/v/calculating-dimensions-of-a-conehttps://puzzicle.org/thumbs/x936d531dd3666b82.jpgCalculating dimensions of a coneGiven the volume of a cone and one dimension, use the cone volume formula to find the missing dimension. Explain the reasoning behind each step of the calculation.https://cdn.kastatic.org/ka-youtube-converted/7p3YR2wYT7U.mp4/7p3YR2wYT7U.mp496yes https://puzzicle.org/math/8th-grade-illustrative-math/v/estimating-volume-of-hemisphereshttps://puzzicle.org/thumbs/xe903e13906de6142.jpgEstimating volume of hemispheresCalculate the volumes of a cylinder and cone sharing the same radius and height, using them as upper and lower bounds for a hemisphere with that same radius. Use this reasoning to estimate the hemisphere's volume.https://cdn.kastatic.org/ka-youtube-converted/zFYso4nHDUc.mp4/zFYso4nHDUc.mp4266yes https://puzzicle.org/math/cc-eighth-grade-math/v/example-estimating-from-regression-linehttps://puzzicle.org/thumbs/xdc4d346d7f8b65c6.jpgEstimating with linear regression (linear models)A line of best fit is a straight line that shows the relationship between two sets of data. We can use the line to make predictions. To find the best equation for the line, we look at the slope and the y-intercept. Remember, this is just a model, so it's not always perfect!https://cdn.kastatic.org/ka-youtube-converted/TWkYalFn8lo.mp4/TWkYalFn8lo.mp4203yes https://puzzicle.org/math/8th-grade-illustrative-math/v/exploring-negative-exponents-with-a-base-of-10https://puzzicle.org/thumbs/xe32dca70e931d530.jpgExploring negative exponents with a base of 10Build intuition for negative exponents using base 10, seeing how the same exponent rules extend into negative territory. Use patterns from repeated multiplication and division by 10 to make sense of what negative exponents mean.https://cdn.kastatic.org/ka-youtube-converted/777jxRvzmNc.mp4/777jxRvzmNc.mp4115yes https://puzzicle.org/math/8th-grade-illustrative-math/v/exponent-properties-with-parentheses-indian-accenthttps://puzzicle.org/thumbs/xce8b849c8e4fa8fa.jpgExponent properties with parenthesesLearn two exponent properties: (ab)^c = (a^c)*(b^c) and (a^b)^c = a ^ (b*c). See WHY they work and HOW to use them. In other words, multiplying two numbers, then raising the product to an exponent is the same as raising each number to that exponent and then multiplying. Raising a number to an exponent and then to another exponent equals raising the base to the product of the two exponents.https://cdn.kastatic.org/ka-youtube-converted/RUwsqBQmozA.mp4/RUwsqBQmozA.mp4253yes https://puzzicle.org/math/8th-grade-illustrative-math/v/negative-exponents-indian-accenthttps://puzzicle.org/thumbs/x1b05d8532e54ed79.jpgNegative exponentsNegative exponents can be rewritten in two ways. Firstly, start with 1 and divide it by 2 the same number of times as the exponent. Secondly, take the reciprocal of the base and raise it to the positive exponent.https://cdn.kastatic.org/ka-youtube-converted/G-v0wwQuGXA.mp4/G-v0wwQuGXA.mp4434yes https://puzzicle.org/math/algebra/v/multiplying-and-dividing-powers-with-integer-exponentshttps://puzzicle.org/thumbs/x51aac3781797c497.jpgMultiplying & dividing powers (integer exponents)For any base a and any integer exponents n and m, aⁿ⋅aᵐ=aⁿ⁺ᵐ. For any nonzero base, aⁿ/aᵐ=aⁿ⁻ᵐ. These are worked examples for using these properties with integer exponents.https://cdn.kastatic.org/ka-youtube-converted/CZ5ne_mX5_I.mp4/CZ5ne_mX5_I.mp4264yes https://puzzicle.org/math/algebra/v/powers-of-products-and-quotients-integer-exponentshttps://puzzicle.org/thumbs/x77dba0123ee2609c.jpgPowers of products & quotients (integer exponents)For any integers a and b and for any exponents n, (a⋅b)ⁿ=aⁿ⋅bⁿ and (a/b)ⁿ=aⁿ/bⁿ. These are worked examples for using these properties with integer exponents.https://cdn.kastatic.org/ka-youtube-converted/dC1ojsMi1yU.mp4/dC1ojsMi1yU.mp4404yes https://puzzicle.org/math/8th-grade-illustrative-math/v/exponent-properties-with-products-indian-accenthttps://puzzicle.org/thumbs/x8204aa93c32cc151.jpgExponent properties with productsTo simplify expressions with exponents, there are a few properties that may help. One is that when two numbers with the same base are multiplied, the exponents can be added. Another is that when a number with an exponent is raised to another exponent, the exponents can be multiplied.https://cdn.kastatic.org/ka-youtube-converted/pQ5hkjM-E1E.mp4/pQ5hkjM-E1E.mp4762yes https://puzzicle.org/math/8th-grade-illustrative-math/v/representing-numbers-using-powers-of-10https://puzzicle.org/thumbs/xd7edb4f6fd20ef42.jpgRepresenting numbers using powers of 10Learn how to write very large and very small numbers as multiples of powers of 10. Practice converting numbers into this form to make their size easier to understand and compare.https://cdn.kastatic.org/ka-youtube-converted/9QuCWeJOgwI.mp4/9QuCWeJOgwI.mp4257yes https://puzzicle.org/math/8th-grade-illustrative-math/v/small-numbers-on-the-number-linehttps://puzzicle.org/thumbs/xb150d95f0607ff7e.jpgSmall numbers on the number lineUse a number line to place very small numbers, coordinating their decimal forms with multiples of powers of 10, including negative exponents. Practice matching the two representations to build a sense of scale.https://cdn.kastatic.org/ka-youtube-converted/25eC1INldxI.mp4/25eC1INldxI.mp493yes https://puzzicle.org/math/8th-grade-illustrative-math/v/large-numbers-on-the-number-linehttps://puzzicle.org/thumbs/x67c4ca83d55553a0.jpgLarge numbers on the number lineUse a number line to place very large numbers, coordinating their decimal forms with multiples of powers of 10. Practice matching the two representations to build a sense of scale.https://cdn.kastatic.org/ka-youtube-converted/XHofFxtaMKI.mp4/XHofFxtaMKI.mp4102yes https://puzzicle.org/math/8th-grade-illustrative-math/v/solving-problems-using-powers-of-10-in-contexthttps://puzzicle.org/thumbs/x91a53676a6ed0286.jpgSolving problems using powers of 10 in contextWork through real-world problems involving very large or very small quantities, identifying what information is needed to solve them. Use exponent rules and powers of 10 to organize the numbers and work out a solution.https://cdn.kastatic.org/ka-youtube-converted/xAOJ5sJslWU.mp4/xAOJ5sJslWU.mp4167yes https://puzzicle.org/math/8th-grade-illustrative-math/v/finding-the-area-of-a-tilted-squarehttps://puzzicle.org/thumbs/x272b80a734f8eb6e.jpgFinding the area of a tilted squareFind the area of a tilted square drawn on a grid, where its sides don't run along the gridlines. Practice using the grid to reason about area when the usual side-length approach doesn't directly apply.https://cdn.kastatic.org/ka-youtube-converted/P_Y5W7BXCH4.mp4/P_Y5W7BXCH4.mp4133yes https://puzzicle.org/math/8th-grade-illustrative-math/v/relating-side-lengths-and-areas-of-squareshttps://puzzicle.org/thumbs/xa84d9da78f15f3b2.jpgRelating side lengths and areas of squaresIntroduce square roots as the side length of a square with a given area. Estimate a square's side length by comparing it to squares with known, nearby areas.https://cdn.kastatic.org/ka-youtube-converted/G7uDwEvu_Wk.mp4/G7uDwEvu_Wk.mp491yes https://puzzicle.org/math/8th-grade-illustrative-math/v/finding-side-lengths-of-triangleshttps://puzzicle.org/thumbs/xb1cbe5860d862c04.jpgFinding side lengths of trianglesDiscover the Pythagorean theorem by exploring patterns in the side lengths of right triangles. Introduce the terms legs and hypotenuse along the way.https://cdn.kastatic.org/ka-youtube-converted/q2kKuZMqpv4.mp4/q2kKuZMqpv4.mp4167yes https://puzzicle.org/math/8th-grade-illustrative-math/v/converting-repeating-decimals-to-fractions-part-1-of-2https://puzzicle.org/thumbs/x88fb36b6bdb70d06.jpgConverting repeating decimals to fractions (part 1 of 2)Converting repeating decimals to fractions (part 1 of 2) (Indian Accent)https://cdn.kastatic.org/ka-youtube-converted/GMXAckRulSU.mp4/GMXAckRulSU.mp4242yes https://puzzicle.org/math/8th-grade-illustrative-math/v/converting-repeating-decimals-to-fractions-part-2-of-2-indiahttps://puzzicle.org/thumbs/xf2cfbfea21ff221e.jpgConverting repeating decimals to fractions (part 2 of 2)Converting repeating decimals to fractions (part 2 of 2) (Indian Accent)https://cdn.kastatic.org/ka-youtube-converted/4hrdbrDtT_M.mp4/4hrdbrDtT_M.mp4545yes https://puzzicle.org/math/8th-grade-illustrative-math/v/solving-aspect-ratio-problems-with-the-pythagorean-theoremhttps://puzzicle.org/thumbs/x55a2e859eb7c2298.jpgSolving aspect ratio problems with the Pythagorean theoremSolve an aspect-ratio problem end to end with the Pythagorean theorem, setting up an equation from a given ratio and diagonal length. Solve for the unknown, find the actual side lengths, and finish by calculating the area.https://cdn.kastatic.org/ka-youtube-converted/R32NVZowWMY.mp4/R32NVZowWMY.mp4232yes https://puzzicle.org/math/cc-eighth-grade-math/v/number-sets-2https://puzzicle.org/thumbs/27078.jpgWorked example: classifying numbersHow do repeating decimals fit into the number sets? We'll explore if a number like 3.4028 repeating is a real number, a rational number, or both. We'll use multiplication and subtraction, plus a bit of algebra, to convince ourselves of whether or not there is a fraction representation of the number.https://cdn.kastatic.org/ka-youtube-converted/qfQv8GzyjB4.mp4/qfQv8GzyjB4.mp4412yes https://puzzicle.org/math/geometry/v/equilateral-and-isosceles-example-problemshttps://puzzicle.org/thumbs/281570418.jpgIsosceles & equilateral triangles problemsIsosceles triangles have two congruent sides and two congruent base angles. Equilateral triangles have all side lengths equal and all angle measures equal. We use these properties to find missing angles in composite figures. The problems are partly from Art of Problem Solving, by Richard Rusczyk.https://cdn.kastatic.org/ka-youtube-converted/7FTNWE7RTfQ.mp4/7FTNWE7RTfQ.mp4387yes https://puzzicle.org/math/geometry/v/example-identifying-the-center-of-dilationhttps://puzzicle.org/thumbs/xb795568ffba397e7.jpgDilations: centerDetermining the center of dilation, given a figure and its image under a dilation.https://cdn.kastatic.org/ka-youtube-converted/rDj0WNzp-f8.mp4/rDj0WNzp-f8.mp4163yes https://puzzicle.org/math/geometry/v/similarity-example-where-same-side-plays-different-roleshttps://puzzicle.org/thumbs/289578995.jpgSolving similar triangles: same side plays different rolesSal finds a missing side length in a problem where the same side plays different roles in two similar triangles.https://cdn.kastatic.org/ka-youtube-converted/Tal_fgREll0.mp4/Tal_fgREll0.mp4365yes https://puzzicle.org/math/algebra/v/standard-form-for-linear-equationshttps://puzzicle.org/thumbs/x03fa0ba8.jpgIntro to linear equation standard formThe standard form for linear equations in two variables is Ax+By=C. For example, 2x+3y=5 is a linear equation in standard form. When an equation is given in this form, it's pretty easy to find both intercepts (x and y). This form is also very useful when solving systems of two linear equations.https://cdn.kastatic.org/ka-youtube-converted/6CFE60iP2Ug.mp4/6CFE60iP2Ug.mp4487yes https://puzzicle.org/math/algebra/v/examples-of-slopes-and-equations-of-horizontal-and-vertical-lineshttps://puzzicle.org/thumbs/xc9aa084efcd925fa.jpgHorizontal & vertical linesWorked examples identifying the equations and slope of horizontal and vertical lines.https://cdn.kastatic.org/ka-youtube-converted/J43CIbKpdWc.mp4/J43CIbKpdWc.mp4315yes https://puzzicle.org/math/algebra/v/positive-and-negative-slopehttps://puzzicle.org/thumbs/x77a4dfa1.jpgPositive & negative slopeSal analyzes what it means for a slope to be positive or negative (spoiler: it affects the direction of the line!)https://cdn.kastatic.org/ka-youtube-converted/EQoNfxToez0.mp4/EQoNfxToez0.mp4302yes https://puzzicle.org/math/algebra/v/slope-intercept-formhttps://puzzicle.org/thumbs/x989ab2a9.jpgIntro to slope-intercept formSlope-intercept form (y=mx+b) of linear equations highlights the slope (m) and the y-intercept (b) of a line. Watch this video to learn more about it and see some examples.https://cdn.kastatic.org/ka-youtube-converted/IL3UCuXrUzE.mp4/IL3UCuXrUzE.mp4540yes https://puzzicle.org/math/geometry/v/parallel-lineshttps://puzzicle.org/thumbs/26808.jpgParallel lines from equationSal determines which pairs out of a few given linear equations are parallel.https://cdn.kastatic.org/ka-youtube-converted/T-aCweuimis.mp4/T-aCweuimis.mp4228yes https://puzzicle.org/math/geometry/v/parallel-lines-2https://puzzicle.org/thumbs/26809.jpgParallel lines from equation (example 2)Sal determines which pairs out of a few given linear equations are parallel.https://cdn.kastatic.org/ka-youtube-converted/lOWXZFP8Vuc.mp4/lOWXZFP8Vuc.mp4183yes https://puzzicle.org/math/geometry/v/parallel-lines-3https://puzzicle.org/thumbs/26810.jpgParallel lines from equation (example 3)Sal determines which pairs out of a few given linear equations are parallel.https://cdn.kastatic.org/ka-youtube-converted/y5yNi08cr6I.mp4/y5yNi08cr6I.mp4162yes https://puzzicle.org/math/algebra/v/point-slope-and-slope-intercept-form-from-two-pointshttps://puzzicle.org/thumbs/x51bcef9f.jpgPoint-slope & slope-intercept equationsPoint-slope is the general form y-y₁=m(x-x₁) for linear equations. It emphasizes the slope of the line and a point on the line (that is not the y-intercept). We can rewrite an equation in point-slope form to be in slope-intercept form y=mx+b, to highlight the same line's slope and y-intercept.https://cdn.kastatic.org/ka-youtube-converted/LtpXvUCrgrM.mp4/LtpXvUCrgrM.mp4427yes https://puzzicle.org/math/algebra/v/solving-systems-graphically-exampleshttps://puzzicle.org/thumbs/xf2261778.jpgSystems of equations with graphing: exact & approximate solutionsSal solves a system of two linear equations in standard form, and then approximates the solution of a system whose solution isn't clearly visible.https://cdn.kastatic.org/ka-youtube-converted/o4pbAQgJYjI.mp4/o4pbAQgJYjI.mp4356yes https://puzzicle.org/math/algebra/v/solving-systems-by-graphing-2https://puzzicle.org/thumbs/26876.jpgNumber of solutions to a system of equationsSal is given three lines on the coordinate plane, and identifies one system of two lines that has a single solution, and one system that has no solution.https://cdn.kastatic.org/ka-youtube-converted/BNHLzEv6Mjg.mp4/BNHLzEv6Mjg.mp4206yes https://puzzicle.org/math/algebra/v/examples-algebraically-analyzing-solutions-to-systemshttps://puzzicle.org/thumbs/x7a8c8c42.jpgNumber of solutions to a system of equations algebraicallySal solves several examples where he reasons about the number of solutions of systems of equations using algebraic reasoning.https://cdn.kastatic.org/ka-youtube-converted/ogEddosP1G8.mp4/ogEddosP1G8.mp4361yes https://puzzicle.org/math/algebra/v/understanding-systems-of-equations-example-2https://puzzicle.org/thumbs/x2a69cbaa.jpgHow many solutions does a system of linear equations have if there are at least two?Sal answers this question for you!https://cdn.kastatic.org/ka-youtube-converted/bzseeIVEslA.mp4/bzseeIVEslA.mp4132yes https://puzzicle.org/math/algebra/v/system-of-equations-distances-word-problemhttps://puzzicle.org/thumbs/xc63ae76c.jpgSystem of equations word problem: walk & rideSystems of equations can be used to solve many real-world problems. In this video, we solve a problem about distances walking and riding bus to school.https://cdn.kastatic.org/ka-youtube-converted/Q0tTfe2lKIc.mp4/Q0tTfe2lKIc.mp4420yes https://puzzicle.org/math/algebra/v/system-with-not-enough-informationhttps://puzzicle.org/thumbs/x8883c043.jpgSystem of equations word problem: infinite solutionsSystems of equations can be used to solve many real-world problems. In this video, we solve a problem about a vegetable farmer. In this case, the problem has infinite solutions, which means there's not enough information to find a single solution.https://cdn.kastatic.org/ka-youtube-converted/njb6xYF0GZ0.mp4/njb6xYF0GZ0.mp4301yes https://puzzicle.org/math/algebra/v/graphing-linear-functions-1https://puzzicle.org/thumbs/xe11cda5c.jpgLinear functions word problem: fuelSal is given a verbal description of a real-world relationship involving a truck's fuel consumption, and is asked to draw the graph that represents this relationship.https://cdn.kastatic.org/ka-youtube-converted/AAmVITyDXbc.mp4/AAmVITyDXbc.mp4330yes https://puzzicle.org/math/cc-eighth-grade-math/v/graphing-linear-functions-2https://puzzicle.org/thumbs/x5a10148b.jpgLinear functions word problem: poolSal is given a verbal description of a real-world relationship involving a pool being filled with water, and is asked to draw the graph that represents this relationship.https://cdn.kastatic.org/ka-youtube-converted/zSVrr8K_CiI.mp4/zSVrr8K_CiI.mp4192yes https://puzzicle.org/math/cc-eighth-grade-math/v/constructing-linear-functions-1https://puzzicle.org/thumbs/x32996275.jpgLinear functions word problem: icebergSal is given a verbal description of a real-world relationship involving a melting iceberg, and is asked to find the formula of the function that represents this relationship.https://cdn.kastatic.org/ka-youtube-converted/W7H-VcaSSu8.mp4/W7H-VcaSSu8.mp4395yes https://puzzicle.org/math/cc-eighth-grade-math/v/constructing-linear-functions-2https://puzzicle.org/thumbs/x40f2b455.jpgLinear functions word problem: paintSal is given a verbal description of a real-world relationship involving a person painting his room, and is asked to find the formula of the function that represents this relationship.https://cdn.kastatic.org/ka-youtube-converted/5c9N_1PEfHw.mp4/5c9N_1PEfHw.mp4503yes https://puzzicle.org/math/algebra/v/the-beauty-of-algebrahttps://puzzicle.org/thumbs/563207734.jpgThe beauty of algebraAlgebra is a language that helps us understand the world around us. It's not just about numbers, it's about abstract ideas that can be applied to many areas like economics, physics, and even the fundamental structure of the universe.https://cdn.kastatic.org/ka-youtube-converted/kpCJyQ2usJ4.mp4/kpCJyQ2usJ4.mp4607yes https://puzzicle.org/math/algebra/v/why-all-the-letters-in-algebrahttps://puzzicle.org/thumbs/1254552768.jpgWhy all the letters in algebra?In algebra, we often want to work with general concepts instead of specific numbers. Using letters allows us to represent quantities that could change or that we don't know the value of yet. Letters can also be used to generalize formulas or equations.https://cdn.kastatic.org/ka-youtube-converted/Tm98lnrlbMA.mp4/Tm98lnrlbMA.mp4184yes https://puzzicle.org/math/algebra2/v/adding-and-subtracting-polynomials-1https://puzzicle.org/thumbs/26920.jpgAdding polynomialsLearn how to simplify polynomials by combining like terms! Discover the power of adding and subtracting terms with the same degree of x. Uncover the magic of removing parentheses and grouping similar terms together to simplify complex expressions. Master the art of polynomial simplification!https://cdn.kastatic.org/ka-youtube-converted/ahdKdxsTj8E.mp4/ahdKdxsTj8E.mp4121yes https://puzzicle.org/math/algebra2/v/subtracting-polynomialshttps://puzzicle.org/thumbs/13179515.jpgSubtracting polynomialsMaster the art of adding and subtracting polynomials! Learn how to distribute negative signs across terms, combine like terms, and simplify expressions. This skill is key to understanding algebra and making math easier.https://cdn.kastatic.org/ka-youtube-converted/5ZdxnFspyP8.mp4/5ZdxnFspyP8.mp4127yes https://puzzicle.org/math/algebra2/v/multiply-monomials-introhttps://puzzicle.org/thumbs/xcef25551.jpgMultiplying monomialsLearn how to multiply monomials like a pro! Discover how to multiply numbers and variables separately, then combine them for the final answer. Explore how exponent properties come into play when multiplying variables. Dive into examples with different variables for variety.https://cdn.kastatic.org/ka-youtube-converted/7EplN4Ch98Q.mp4/7EplN4Ch98Q.mp4196yes https://puzzicle.org/math/algebra-home/v/multiplying-and-dividing-monomials-1https://puzzicle.org/thumbs/26915.jpgMultiplying monomials to find area: two variablesSal expresses the area of a rectangle with length 4xy and width 2y as a monomial.https://cdn.kastatic.org/ka-youtube-converted/p_61XhXdlxI.mp4/p_61XhXdlxI.mp4112yes https://puzzicle.org/math/algebra-home/v/multiplying-monomialshttps://puzzicle.org/thumbs/13179516.jpgMultiplying monomials to find areaLearn to express the area of a rectangle with length 4y and width 2y as a monomial.https://cdn.kastatic.org/ka-youtube-converted/ssY1dFl7d30.mp4/ssY1dFl7d30.mp465yes https://puzzicle.org/math/algebra-home/v/missing-values-in-monomial-multiplicationhttps://puzzicle.org/thumbs/x0ae662f7.jpgMultiplying monomials challengeSal finds the values of coefficients a and b that make (3x^a)(bx^4)=-24x^6 true for all x-values.https://cdn.kastatic.org/ka-youtube-converted/mzULbezO6iA.mp4/mzULbezO6iA.mp4212yes https://puzzicle.org/math/algebra-home/v/dividing-monomialshttps://puzzicle.org/thumbs/13179522.jpgSimplifying rational expressions (old video)Sal simplifies (28b^6)/(7b) as 4b^5.https://cdn.kastatic.org/ka-youtube-converted/4Eax48gDg9g.mp4/4Eax48gDg9g.mp456yes https://puzzicle.org/math/algebra2/v/area-model-of-monomial-polynomial-multiplicationhttps://puzzicle.org/thumbs/xb60519c0.jpgMultiplying monomials by polynomials: area modelDiscover how to calculate the area of complex shapes using algebra! By breaking down a rectangle into smaller parts, we can find the total area by multiplying the height and width of each part. This method introduces us to the concept of multiplying monomials by polynomials.https://cdn.kastatic.org/ka-youtube-converted/bkZYILrmFRU.mp4/bkZYILrmFRU.mp494yes https://puzzicle.org/math/algebra2/v/multiplying-monomials-by-polynomialshttps://puzzicle.org/thumbs/13179517.jpgMultiplying monomials by polynomialsDiscover how to multiply monomials by polynomials using the distributive property. Learn to simplify expressions by multiplying coefficients and adding exponents. Get a handle on negative terms and see how they affect the final result. It's all about breaking down complex problems into simpler steps!https://cdn.kastatic.org/ka-youtube-converted/pD2-H15ucNE.mp4/pD2-H15ucNE.mp4163yes https://puzzicle.org/math/algebra-home/v/missing-values-in-monomial-polynomial-multiplicationhttps://puzzicle.org/thumbs/xffe234f4.jpgMultiplying monomials by polynomials challengeSal finds the values of coefficients c, d, and f that make -2y(y²+cy-3)=dy³+12y²+fy true for all y-values.https://cdn.kastatic.org/ka-youtube-converted/NADPP4wlo1Q.mp4/NADPP4wlo1Q.mp4258yes https://puzzicle.org/math/algebra/v/difference-of-squares-pattern-for-simple-binomialshttps://puzzicle.org/thumbs/x7f0427db.jpgSpecial products of the form (x+a)(x-a)Sal introduces difference of squares expressions. For example, (x+3)(x-3) is expanded as x²-9.https://cdn.kastatic.org/ka-youtube-converted/JfuisfEdnjk.mp4/JfuisfEdnjk.mp4294yes https://puzzicle.org/math/algebra/v/special-polynomials-products-1https://puzzicle.org/thumbs/13179520.jpgSpecial products of the form (ax+b)(ax-b)Sal expands the difference of squares (2x+8)(2x-8) as 4x²-64.https://cdn.kastatic.org/ka-youtube-converted/h6HmHjkA034.mp4/h6HmHjkA034.mp4149yes https://puzzicle.org/math/algebra/v/pattern-for-squaring-simple-binomialshttps://puzzicle.org/thumbs/x2269fa10.jpgSquaring binomials of the form (x+a)²Sal introduces perfect square expressions. For example, (x+7)² is expanded as x²+14x+49.https://cdn.kastatic.org/ka-youtube-converted/xH_GllPuymc.mp4/xH_GllPuymc.mp4343yes https://puzzicle.org/math/algebra/v/square-a-binomialhttps://puzzicle.org/thumbs/13179521.jpgSquaring binomials of the form (ax+b)²Sal expands the perfect square (7x+10)² as 49x^2+140x+100.https://cdn.kastatic.org/ka-youtube-converted/xjkbR7Gjgjs.mp4/xjkbR7Gjgjs.mp4218yes https://puzzicle.org/math/algebra/v/factor-completely-common-factorhttps://puzzicle.org/thumbs/xc451daa13b239dcf.jpgFactoring completely with a common factorWe can factor quadratics by first pulling out a common factor so the result looks like a(x+b)(x+c).https://cdn.kastatic.org/ka-youtube-converted/FoTD41xzF5Q.mp4/FoTD41xzF5Q.mp4240yes https://puzzicle.org/math/algebra/v/factoring-quadratics-common-factorhttps://puzzicle.org/thumbs/xa72408d04d9c47df.jpgFactoring quadratics with a common factorWe can factor quadratics by first pulling out a common factor so the result looks like a(x+b)(x+c).https://cdn.kastatic.org/ka-youtube-converted/m6uOqU91ypI.mp4/m6uOqU91ypI.mp4185yes https://puzzicle.org/math/algebra/v/difference-of-squares-introhttps://puzzicle.org/thumbs/x7514a13c1dd4ff7b.jpgDifference of squares introWhen an expression can be viewed as the difference of two perfect squares, i.e. a²-b², then we can factor it as (a+b)(a-b). For example, x²-25 can be factored as (x+5)(x-5). This method is based on the pattern (a+b)(a-b)=a²-b², which can be verified by expanding the parentheses in (a+b)(a-b).https://cdn.kastatic.org/ka-youtube-converted/HLNSouzygw0.mp4/HLNSouzygw0.mp4294yes https://puzzicle.org/math/algebra/v/identifying-perfect-square-trinomialshttps://puzzicle.org/thumbs/x2957b9f5.jpgIdentifying perfect square formSal shows how we can identify that a trinomial has the "perfect square" form.https://cdn.kastatic.org/ka-youtube-converted/4STGvuoBi5U.mp4/4STGvuoBi5U.mp4277yes https://puzzicle.org/math/algebra/v/perfect-square-factorization-introhttps://puzzicle.org/thumbs/x0c1a3582bef51e96.jpgPerfect square factorization introWhen an expression has the general form a²+2ab+b², then we can factor it as (a+b)². For example, x²+10x+25 can be factored as (x+5)². This method is based on the pattern (a+b)²=a²+2ab+b², which can be verified by expanding the parentheses in (a+b)(a+b).https://cdn.kastatic.org/ka-youtube-converted/JX5Zvh6swmo.mp4/JX5Zvh6swmo.mp4319yes https://puzzicle.org/math/algebra/v/factoring-perfect-square-trinomialshttps://puzzicle.org/thumbs/13179582.jpgFactoring perfect squaresSal factors 25x^2-30x+9 as (5x-3)^2 or as (-5x+3)^2.https://cdn.kastatic.org/ka-youtube-converted/liRNTieIU_k.mp4/liRNTieIU_k.mp4294yes https://puzzicle.org/math/algebra/v/u09-l2-t1-we1-factoring-special-products-1https://puzzicle.org/thumbs/26989.jpgFactoring perfect squares: negative common factorSal factors -4t^2-12t-9 as -1(2t+3)^2.https://cdn.kastatic.org/ka-youtube-converted/XuwldEyWjH0.mp4/XuwldEyWjH0.mp4232yes https://puzzicle.org/math/algebra/v/factoring-simple-quadratic-expressionhttps://puzzicle.org/thumbs/x0da5d332.jpgFactoring quadratics as (x+a)(x+b)Sal factors x²-3x-10 as (x+2)(x-5) using the sum-product form: (x+a)(x+b)=x²+(a+b)x+a*b.https://cdn.kastatic.org/ka-youtube-converted/D3a8NnpQ2vU.mp4/D3a8NnpQ2vU.mp4394yes https://puzzicle.org/math/algebra/v/factoring-polynomials-1https://puzzicle.org/thumbs/1228073357.jpgFactoring quadratics as (x+a)(x+b) (example 2)Sal factors x^2-14x+40 as (x-4)(x-10) and x^2-x-12 as (x+3)(x-4).https://cdn.kastatic.org/ka-youtube-converted/1kfq0aR3ASs.mp4/1kfq0aR3ASs.mp4260yes https://puzzicle.org/math/algebra/v/factoring-quadratic-expressionshttps://puzzicle.org/thumbs/23710.jpgMore examples of factoring quadratics as (x+a)(x+b)Can't get enough of Sal factoring simple quadratics? Here's a handful of examples just for you!https://cdn.kastatic.org/ka-youtube-converted/eF6zYNzlZKQ.mp4/eF6zYNzlZKQ.mp4990yes https://puzzicle.org/math/algebra/v/solving-for-constants-in-perfect-square-polynomialhttps://puzzicle.org/thumbs/x052132a2.jpgFactoring perfect squares: missing valuesSal analyzes the factorization of x^2+5x+c as (x+d)^2 to find the values of the missing coefficients c and d.https://cdn.kastatic.org/ka-youtube-converted/oIslHyWJHEY.mp4/oIslHyWJHEY.mp4159yes https://puzzicle.org/math/precalculus/v/simplifying-rational-expressions-introductionhttps://puzzicle.org/thumbs/23782.jpgReducing rational expressions to lowest termsLearn how to simplify rational expressions, which are fractions with variable expressions in the numerator and denominator. This video also shows how to factor the expressions and cancel out common factors, just like with rational numbers. Don't forget to exclude values of x that would make the denominator zero, since those would make the expression undefined.https://cdn.kastatic.org/ka-youtube-converted/7Uos1ED3KHI.mp4/7Uos1ED3KHI.mp4922yes https://puzzicle.org/math/math3/v/simplifying-rational-expressions-w-common-monomial-factorshttps://puzzicle.org/thumbs/x71dfabf5.jpgSimplifying rational expressions: common monomial factorsSal simplifies the rational expressions (14x²+7x)/(14x) and (17z³+17z²)/(34z³-51z²) by taking common factors and canceling them.https://cdn.kastatic.org/ka-youtube-converted/XFwQV-KCudw.mp4/XFwQV-KCudw.mp4419yes https://puzzicle.org/math/precalculus/v/adding-and-subtracting-rational-expressions-with-like-denominatorshttps://puzzicle.org/thumbs/xbecbe152.jpgAdding & subtracting rational expressions: like denominatorsSal adds 6/(2x²-7) + (-3x-8)/(2x²-7) and subtracts (9x²+3)/(14x²-9) - (-3x²+5)/(14x²-9).https://cdn.kastatic.org/ka-youtube-converted/rmYlCuiC5uY.mp4/rmYlCuiC5uY.mp4225yes https://puzzicle.org/math/precalculus/v/algebraic-expression-adding-fractionshttps://puzzicle.org/thumbs/xe973c6c7.jpgIntro to adding rational expressions with unlike denominatorsSal rewrites a/b+c/d as a single rational expression.https://cdn.kastatic.org/ka-youtube-converted/w7NhLkQynS8.mp4/w7NhLkQynS8.mp4164yes https://puzzicle.org/math/precalculus/v/adding-rational-expression-w-unlike-denominatorshttps://puzzicle.org/thumbs/x220777a7.jpgAdding rational expression: unlike denominatorsLearn how to add rational expressions with different denominators by finding a common denominator, multiplying the numerators and denominators by the same factors, and simplifying the result. A rational expression is a fraction with polynomials in the numerator and denominator.https://cdn.kastatic.org/ka-youtube-converted/DrE_iiw1Mvk.mp4/DrE_iiw1Mvk.mp4311yes https://puzzicle.org/math/precalculus/v/subtracting-rational-expressions-w-unlike-denominatorshttps://puzzicle.org/thumbs/xe3210011.jpgSubtracting rational expressions: unlike denominatorsSal rewrites (-5x)/(8x+7)-(6x³)/(3x+1) as (-48x⁴-42x³-15x²-5x)/(8x+7)(3x+1).https://cdn.kastatic.org/ka-youtube-converted/evmDZkDvlNw.mp4/evmDZkDvlNw.mp4287yes https://puzzicle.org/math/precalculus/v/subtracting-rational-expressions-w-factored-denominatorshttps://puzzicle.org/thumbs/x282a4135.jpgSubtracting rational expressions: factored denominatorsSal subtracts two rational expressions whose denominators are factored. The denominators aren't the same but they share a factor.https://cdn.kastatic.org/ka-youtube-converted/j8cKFzj5V6g.mp4/j8cKFzj5V6g.mp4288yes https://puzzicle.org/math/precalculus/v/multiplying-and-dividing-rational-expressions-monomial-numerator-denominatorhttps://puzzicle.org/thumbs/xda27881a.jpgMultiplying & dividing rational expressions: monomialsSal multiplies (6x³/5) X (2/3x) and divides (2x⁴/7) ➗ (5x⁴/4).https://cdn.kastatic.org/ka-youtube-converted/6nALFmvvgds.mp4/6nALFmvvgds.mp4356yes https://puzzicle.org/math/math3/v/multiplying-and-dividing-rational-expressions-2https://puzzicle.org/thumbs/27015.jpgMultiplying rational expressionsSal multiplies and simplifies (a²-4)/(a²-1) X (a+1)/(a+2).https://cdn.kastatic.org/ka-youtube-converted/f-wz_ZzSDdg.mp4/f-wz_ZzSDdg.mp4292yes https://puzzicle.org/math/math3/v/multiplying-and-dividing-rational-expressions-3https://puzzicle.org/thumbs/27016.jpgDividing rational expressionsSal divides and simplifies (2p+6)/(p+5) ➗ (10)/(4p+20).https://cdn.kastatic.org/ka-youtube-converted/gcnk8TnzsLc.mp4/gcnk8TnzsLc.mp4250yes https://puzzicle.org/math/math3/v/multiplying-and-dividing-rational-expressions-1https://puzzicle.org/thumbs/27014.jpgMultiplying rational expressions: multiple variablesSal multiplies and simplifies (3x²y)/(2ab) X (14a²b)/(18xy²).https://cdn.kastatic.org/ka-youtube-converted/3GL69IA2q4s.mp4/3GL69IA2q4s.mp4218yes https://puzzicle.org/math/math3/v/dividing-rational-expressions-w-unknown-expressionhttps://puzzicle.org/thumbs/xdc5c1290.jpgDividing rational expressions: unknown expressionSal finds the polynomial D for which (20y²-80)/D ÷ (4y²-8y)/(y³+9y²)=1 is true for all values of y for which it's defined.https://cdn.kastatic.org/ka-youtube-converted/54ZFp4CwWEg.mp4/54ZFp4CwWEg.mp4362yes https://puzzicle.org/math/geometry/v/midpoint-formulahttps://puzzicle.org/thumbs/23758.jpgMidpoint formulaLearn how to use the midpoint formula to find the midpoint of a line segment on the coordinate plane, or find the endpoint of a line segment given one point and the midpoint.https://cdn.kastatic.org/ka-youtube-converted/Ez_-RwV9WVo.mp4/Ez_-RwV9WVo.mp4401yes https://puzzicle.org/math/geometry/v/volume-of-pyramids-intuitionhttps://puzzicle.org/thumbs/x3e58b9e685faa0ed.jpgVolume of pyramids intuitionThe volume of a pyramid is a fraction of the volume of the rectangular prism that encloses it. We can find out what that fraction is by cutting a prism into several pyramids.https://cdn.kastatic.org/ka-youtube-converted/fD52yzLF8DA.mp4/fD52yzLF8DA.mp4509yes https://puzzicle.org/math/geometry/v/volumes-of-cones-intuitionhttps://puzzicle.org/thumbs/xe65e41d409141ca6.jpgVolumes of cones intuitionUse Cavalieri's Principle to establish that the formula for the volume of a cone is the same as the formula for the volume of a pyramid (1/3 * base area * height).https://cdn.kastatic.org/ka-youtube-converted/LmBViMOeYXg.mp4/LmBViMOeYXg.mp4342yes https://puzzicle.org/math/algebra-home/v/steps-when-solving-equationshttps://puzzicle.org/thumbs/xb929087f.jpgDescribing steps when solving equationsSal lists the steps necessary in order to solve a linear equation. This is useful for thinking more strategically about equations.https://cdn.kastatic.org/ka-youtube-converted/LzYJVsvqS50.mp4/LzYJVsvqS50.mp4136yes https://puzzicle.org/math/algebra-home/v/missing-stepshttps://puzzicle.org/thumbs/x9c0b1421.jpgFiguring out missing algebraic stepThere is a hidden step in Steven's solution to solving an equation. There are different ways to solve an equation. To find Steven's missing step, we can see what we would do in the equation. Then we can see if another step will get us to the work Sten is showing. If not, we can try solving a different way.https://cdn.kastatic.org/ka-youtube-converted/RXgj09NXNWo.mp4/RXgj09NXNWo.mp4114yes https://puzzicle.org/math/algebra-home/v/one-step-inequalitieshttps://puzzicle.org/thumbs/13179474.jpgOne-step inequality involving additionHow to solve and graph one-step inequalities.https://cdn.kastatic.org/ka-youtube-converted/17a443nL7Qw.mp4/17a443nL7Qw.mp4114yes https://puzzicle.org/math/algebra-home/v/inequalities-using-addition-and-subtractionhttps://puzzicle.org/thumbs/23660.jpgInequalities using addition and subtractionInequalities Using Addition and Subtractionhttps://cdn.kastatic.org/ka-youtube-converted/UTs4uZhu5t8.mp4/UTs4uZhu5t8.mp4468yes https://puzzicle.org/math/algebra/v/multi-step-inequalities-3https://puzzicle.org/thumbs/153770149.jpgInequalities with variables on both sidesSal solves the inequality -3p-7<p+9, draws the solution on a number line and checks a few values to verify the solution.https://cdn.kastatic.org/ka-youtube-converted/SgKBBUFaGb4.mp4/SgKBBUFaGb4.mp4338yes https://puzzicle.org/math/algebra/v/multi-step-inequalities-2https://puzzicle.org/thumbs/13179489.jpgInequalities with variables on both sides (with parentheses)Sal solves the inequality 5x+7>3(x+1), draws the solution on a number line and checks a few values to verify the solution.https://cdn.kastatic.org/ka-youtube-converted/XOAn5z8mkvI.mp4/XOAn5z8mkvI.mp4227yes https://puzzicle.org/math/algebra/v/multi-step-inequalitieshttps://puzzicle.org/thumbs/547403341.jpgMulti-step inequalitiesSal solves several multi-step linear inequalities.https://cdn.kastatic.org/ka-youtube-converted/xOxvyeSl0uA.mp4/xOxvyeSl0uA.mp4482yes https://puzzicle.org/math/algebra/v/2-variable-linear-equations-graphshttps://puzzicle.org/thumbs/x82b3a036.jpgTwo-variable linear equations introLearn about a class of equations in two variables that's called "linear equations." They are called that way because their graph is a line. These are the most basic and probably most useful equations you will ever know!https://cdn.kastatic.org/ka-youtube-converted/AOxMJRtoR2A.mp4/AOxMJRtoR2A.mp4501yes https://puzzicle.org/math/algebra/v/less-obvious-slope-intercept-formhttps://puzzicle.org/thumbs/xbb26a1d9b606a0a4.jpgSlope and y-intercept from equationIn an equation in slope-intercept form (y=mx+b) the slope is m and the y-intercept is b. We can also rewrite certain equations to look more like slope-intercept form. For example, y=x can be rewritten as y=1x+0, so its slope is 1 and its y-intercept is 0.https://cdn.kastatic.org/ka-youtube-converted/Z65mz__8DQ0.mp4/Z65mz__8DQ0.mp4250yes https://puzzicle.org/math/algebra/v/solving-systems-of-equations-by-multiplicationhttps://puzzicle.org/thumbs/23680.jpgSystems of equations with elimination (and manipulation)In some cases, we need to slightly manipulate a system of equations before we can solve it using the elimination method. See how it's done in this video.https://cdn.kastatic.org/ka-youtube-converted/wYrxKGt_bLg.mp4/wYrxKGt_bLg.mp4721yes https://puzzicle.org/math/algebra/v/why-we-do-the-same-thing-to-both-sides-basic-systemshttps://puzzicle.org/thumbs/948327843.jpgWhy can we subtract one equation from the other in a system of equations?The example of a scale where we try to achieve balance helps to explain why we can subtract one equation from the other in a system of equations.https://cdn.kastatic.org/ka-youtube-converted/h9ZgZimXn2Q.mp4/h9ZgZimXn2Q.mp4493yes https://puzzicle.org/math/algebra/v/graphing-inequalitieshttps://puzzicle.org/thumbs/23672.jpgIntro to graphing two-variable inequalitiesLearn how to graph two-variable linear inequalities like y≤4x+3.https://cdn.kastatic.org/ka-youtube-converted/unSBFwK881s.mp4/unSBFwK881s.mp4483yes https://puzzicle.org/math/algebra/v/graphing-linear-inequalities-in-two-variables-example-2https://puzzicle.org/thumbs/2600589.jpgGraphing two-variable inequalitiesSal graphs the inequality y<3x+5.https://cdn.kastatic.org/ka-youtube-converted/FnrqBgot3jM.mp4/FnrqBgot3jM.mp4183yes https://puzzicle.org/math/algebra/v/graphing-linear-inequalities-in-two-variables-3https://puzzicle.org/thumbs/26874.jpgTwo-variable inequalities from their graphsSal is given a graph and he analyzes it to find the two-variable inequality it represents.https://cdn.kastatic.org/ka-youtube-converted/YBYu5aZPLeg.mp4/YBYu5aZPLeg.mp4200yes https://puzzicle.org/math/ap-statistics/v/sample-variancehttps://puzzicle.org/thumbs/1163970906.jpgSample varianceWe delve into measuring variability in quantitative data, focusing on calculating sample variance and population variance. The importance of using a sample size minus one (n-1) for a more accurate estimate is highlighted. The distinction between sample mean and population mean is also clarified.https://cdn.kastatic.org/ka-youtube-converted/iHXdzfF7UEs.mp4/iHXdzfF7UEs.mp4638yes https://puzzicle.org/math/ap-statistics/v/sample-standard-deviation-and-biashttps://puzzicle.org/thumbs/x6ad248d1.jpgSample standard deviation and biasSal shows an example of calculating standard deviation and bias.https://cdn.kastatic.org/ka-youtube-converted/DNAnQBhGpRw.mp4/DNAnQBhGpRw.mp4573yes https://puzzicle.org/math/ap-statistics/v/visual-standard-deviationhttps://puzzicle.org/thumbs/xfc2e7c96cfe32c07.jpgVisually assessing standard deviationWorked examples visually assessing the standard distribution.https://cdn.kastatic.org/ka-youtube-converted/-iCMQoCJFBY.mp4/-iCMQoCJFBY.mp4229yes https://puzzicle.org/math/algebra/v/origins-of-algebrahttps://puzzicle.org/thumbs/565175118.jpgOrigins of algebraWhere did the word "Algebra" and its underlying ideas come from? Algebra, a key branch of mathematics, has a rich history. The term 'algebra' comes from an Arabic word meaning 'restoration' or 'completion'. Significant contributions to algebra were made by Diophantus in Greece, Brahmagupta in India, and al-Khwarizmi in Baghdad, who is often credited with giving algebra its name.https://cdn.kastatic.org/ka-youtube-converted/_LDR1_Prveo.mp4/_LDR1_Prveo.mp4438yes https://puzzicle.org/math/algebra/v/abstract-nesshttps://puzzicle.org/thumbs/565149394.jpgAbstract-nessAbstraction is like taking a real thing and turning it into an idea. We can use a cube as an example. Even though cubes can look different in real life, the abstract idea of what a cube is stays the same.https://cdn.kastatic.org/ka-youtube-converted/ZOYRb2sYrL0.mp4/ZOYRb2sYrL0.mp4420yes https://puzzicle.org/math/algebra/v/creativity-important-in-algebrahttps://puzzicle.org/thumbs/x12ce07a329691eb8.jpgCreativity break: Why is creativity important in algebra?Algebra is a creative problem-solving tool, similar to art. Just as an artist uses different techniques and brushes to create a masterpiece, algebra combines various mathematical tools to solve complex problems. It's not about memorizing solutions, but about using your knowledge in innovative ways.https://cdn.kastatic.org/ka-youtube-converted/SjZfW07DbMo.mp4/SjZfW07DbMo.mp482yes https://puzzicle.org/math/algebra/v/creativity-stem-jobshttps://puzzicle.org/thumbs/x76a1dad8aa3e70d2.jpgCreativity break: Why is creativity important in STEM jobs?Experts share why creativity plays an important role in STEM jobs.https://cdn.kastatic.org/ka-youtube-converted/ynvn3jqxcX4.mp4/ynvn3jqxcX4.mp497yes https://puzzicle.org/math/algebra/v/why-zero-divided-by-zero-is-undefined-indeterminatehttps://puzzicle.org/thumbs/871532281.jpgThe problem with dividing zero by zeroOne can argue that 0/0 is ​0, because 0 divided by anything is 0. Another one can argue that 0/0 is ​1, because anything divided by itself is 1. And that's exactly the problem! Whatever we say 0/0 equals to, we contradict one crucial property of numbers or another. To avoid "breaking math," we simply say that 0/0 is undetermined.https://cdn.kastatic.org/ka-youtube-converted/PDReqvXfkBA.mp4/PDReqvXfkBA.mp4146yes https://puzzicle.org/math/algebra/v/undefined-and-indeterminatehttps://puzzicle.org/thumbs/871539458.jpgUndefined & indeterminate expressionsRevisiting the problems of dividing any number by zero and dividing zero by zero. Using general mathematical considerations, we see why those are undefined and indeterminate problems.https://cdn.kastatic.org/ka-youtube-converted/lHdlHTsXbZg.mp4/lHdlHTsXbZg.mp4452yes https://puzzicle.org/math/algebra/v/why-we-do-the-same-thing-to-both-sides-multi-step-equationshttps://puzzicle.org/thumbs/948601082.jpgWhy we do the same thing to both sides: Variable on both sidesWe will use a scale to try to achieve balance and help explain why we do the same thing to both sides of an equation, even when the variable appears on both sides.https://cdn.kastatic.org/ka-youtube-converted/vkhYFml0w6c.mp4/vkhYFml0w6c.mp4456yes https://puzzicle.org/math/algebra/v/reasoning-with-linear-equationshttps://puzzicle.org/thumbs/xfe28aa3bc9f3418d.jpgReasoning with linear equationsWhen we perform operations to manipulate equations, some operations produce equivalent equations, while others don't necessarily produce equivalent equations. When we're solving an equation, we need to use operations that guarantee equivalence.https://cdn.kastatic.org/ka-youtube-converted/BV-TGIMUXaw.mp4/BV-TGIMUXaw.mp4460yes https://puzzicle.org/math/algebra/v/linear-equations-with-unknown-coefficientshttps://puzzicle.org/thumbs/xdb214b95.jpgLinear equations with unknown coefficientsSal solves the equations ax+3x=bx+5 and a(5-x)=bx-8 for x. Note that these equations include other unknowns (a and b) but we solve them for x.https://cdn.kastatic.org/ka-youtube-converted/adPgapI-h3g.mp4/adPgapI-h3g.mp4335yes https://puzzicle.org/math/algebra/v/why-algebra-importanthttps://puzzicle.org/thumbs/x8f0df416e19126f4.jpgWhy is algebra important to learn?Professionals share why algebra is important to learn.https://cdn.kastatic.org/ka-youtube-converted/H4g0GZsgq0I.mp4/H4g0GZsgq0I.mp446yes https://puzzicle.org/math/algebra/v/using-inequalities-to-solve-problemshttps://puzzicle.org/thumbs/xe2ceae424fad445f.jpgUsing inequalities to solve problemsWe can use inequalities to solve problems in a given context.https://cdn.kastatic.org/ka-youtube-converted/E6rn-YD_2_Q.mp4/E6rn-YD_2_Q.mp4220yes https://puzzicle.org/math/algebra/v/compund-inequalitieshttps://puzzicle.org/thumbs/13179490.jpgCompound inequalities: ORSal solves the compound inequality 5z+7<27 OR -3z≤18.https://cdn.kastatic.org/ka-youtube-converted/0YErxSShF0A.mp4/0YErxSShF0A.mp4271yes https://puzzicle.org/math/algebra/v/compund-inequalities-2https://puzzicle.org/thumbs/13179491.jpgCompound inequalities: ANDSal solves the compound inequality 3y+7<2y AND 4y+8>-48.https://cdn.kastatic.org/ka-youtube-converted/d2cnQ5ahHgE.mp4/d2cnQ5ahHgE.mp4313yes https://puzzicle.org/math/algebra/v/compound-inequalities-4https://puzzicle.org/thumbs/13179493.jpgA compound inequality with no solutionSal solves the compound inequality 5x-3<12 AND 4x+1>25, only to realize there's no x-value that makes both inequalities true.https://cdn.kastatic.org/ka-youtube-converted/ZF_cZ-GX9PI.mp4/ZF_cZ-GX9PI.mp4161yes https://puzzicle.org/math/algebra/v/compound-inequalities-3https://puzzicle.org/thumbs/153754478.jpgDouble inequalitiesSal solves the double inequality -16≤3x+5≤20, which is the same as the compound inequality -16≤3x+5 AND 3x+5≤20.https://cdn.kastatic.org/ka-youtube-converted/cvB8b4AACyE.mp4/cvB8b4AACyE.mp4268yes https://puzzicle.org/math/algebra/v/compound-inequalitieshttps://puzzicle.org/thumbs/23666.jpgCompound inequalities examplesSal solves several compound linear inequalities.https://cdn.kastatic.org/ka-youtube-converted/A3xPhzs-KBI.mp4/A3xPhzs-KBI.mp4705yes https://puzzicle.org/math/algebra/v/dimensional-analysis-units-algebraicallyhttps://puzzicle.org/thumbs/x49104b67.jpgIntro to dimensional analysisSal shows how we can treat units of measurement algebraically, and use these tools in order to convert between different units of the same quantity.https://cdn.kastatic.org/ka-youtube-converted/hIAdCTNi1S8.mp4/hIAdCTNi1S8.mp4389yes https://puzzicle.org/math/algebra/v/thinking-about-reasonable-units-to-describe-a-ratehttps://puzzicle.org/thumbs/xa36759d8.jpgSame rate with different unitsSal shows how we can describe the rate 50 km per hour in a variety of different units, using dimensional analysis.https://cdn.kastatic.org/ka-youtube-converted/d5lcGCbV5cM.mp4/d5lcGCbV5cM.mp4595yes https://puzzicle.org/math/algebra/v/realize-liked-algebrahttps://puzzicle.org/thumbs/x9c052287ee65491c.jpgCreativity break: When did you first realize that you liked algebraExperts share when they first realized that they liked algebra and mathematics.https://cdn.kastatic.org/ka-youtube-converted/4l1t-_Eb2MI.mp4/4l1t-_Eb2MI.mp4102yes https://puzzicle.org/math/algebra/v/appropriate-unitshttps://puzzicle.org/thumbs/xfc37075b5d039482.jpgDefining appropriate quantities for modelingExamples for different ways to define how quantities of interest are calculated in a modeling problem.https://cdn.kastatic.org/ka-youtube-converted/717zfRdog9s.mp4/717zfRdog9s.mp4371yes https://puzzicle.org/math/algebra/v/interpreting-units-in-formulashttps://puzzicle.org/thumbs/xdc3ada78.jpgFormulas and units: Volume of a poolWhen using formulas to calculate real-world quantities, we need to make sure our units are consistent. In this video, the base area of a pool is given in square meters while its height is given in centimeters. In order to use the formula for volume, we need to convert one of the measurements to units that match the other measurement.https://cdn.kastatic.org/ka-youtube-converted/nziP-8PeZqY.mp4/nziP-8PeZqY.mp4200yes https://puzzicle.org/math/algebra/v/interpreting-units-in-formulas-novelhttps://puzzicle.org/thumbs/xc8a101e6.jpgFormulas and units: Comparing ratesWhen using formulas to calculate real-world quantities, we need to make sure our units are consistent. In this video, one growth rate is given in centimeters per week and the other is given in millimeters per day. In order to compare which rate is faster, we need to convert one of the rates to units that match the other rate.https://cdn.kastatic.org/ka-youtube-converted/IuvsQjW1qa4.mp4/IuvsQjW1qa4.mp4222yes https://puzzicle.org/math/algebra/v/reporting-measurementshttps://puzzicle.org/thumbs/x39704acdb26c489a.jpgReporting measurementsWe should think about the appropriate level of precision of different measurements in modeling problems.https://cdn.kastatic.org/ka-youtube-converted/UEe81kJtY8A.mp4/UEe81kJtY8A.mp4363yes https://puzzicle.org/math/algebra/v/using-units-to-solve-problems-toy-factoryhttps://puzzicle.org/thumbs/xb97ff210e26f3ff6.jpgUsing units to solve problems: Toy factoryIn word problems that involve multiple quantities, we can use the units of the quantities to guide our solution. In this video, we find the value of toys produced at a factory using information that involves many different quantities, not all of which are useful for our problem.https://cdn.kastatic.org/ka-youtube-converted/ocOXYiborsw.mp4/ocOXYiborsw.mp4183yes https://puzzicle.org/math/algebra/v/using-units-to-solve-problems-road-triphttps://puzzicle.org/thumbs/xa84111c831c35125.jpgUsing units to solve problems: Road tripIn word problems that involve multiple quantities, we can use the units of the quantities to guide our solution. In this video, we find the cost of fuel for a road trip, using information that involves many different quantities, not all of which are useful for our problem.https://cdn.kastatic.org/ka-youtube-converted/_UAQJw_q4p4.mp4/_UAQJw_q4p4.mp4234yes https://puzzicle.org/math/algebra/v/unit-conversion-example-drug-dosagehttps://puzzicle.org/thumbs/1622153.jpgUsing units to solve problems: Drug dosageFuture doctors and nurses out there, take note. This unit conversion word problem deals with converting drug dosage units, something that is commonly done in hospitals. Give it a try with us.https://cdn.kastatic.org/ka-youtube-converted/ScvuRb6vsz4.mp4/ScvuRb6vsz4.mp4644yes https://puzzicle.org/math/algebra/v/math-creativity-changing-worldhttps://puzzicle.org/thumbs/x10e3af1862147e0b.jpgCreativity break: How are math and creativity changing the world?Experts share examples of how math and creativity are changing the world.https://cdn.kastatic.org/ka-youtube-converted/nFnRRdvaS9g.mp4/nFnRRdvaS9g.mp4116yes https://puzzicle.org/math/algebra/v/graph-from-point-slopehttps://puzzicle.org/thumbs/x895c45614b6db41b.jpgGraphing a line given point and slopePractice graphing a line given its slope and a point the line passes through.https://cdn.kastatic.org/ka-youtube-converted/5mgH-_5UJ54.mp4/5mgH-_5UJ54.mp4131yes https://puzzicle.org/math/algebra/v/slope-from-tablehttps://puzzicle.org/thumbs/x99364552bda42485.jpgCalculating slope from tablesPractice calculating the slope of a line given some points on the line shown in a table.https://cdn.kastatic.org/ka-youtube-converted/pGsicLglzY8.mp4/pGsicLglzY8.mp4159yes https://puzzicle.org/math/algebra/v/slope-intercepts-contexthttps://puzzicle.org/thumbs/x5028a8b98bc6bc16.jpgSlope, x-intercept, y-intercept meaning in contextPractice determining what each of the slope, x-intercept, and y-intercept represent in a given linear relationship.https://cdn.kastatic.org/ka-youtube-converted/XQlaZuh7E8w.mp4/XQlaZuh7E8w.mp4240yes https://puzzicle.org/math/algebra/v/slope-intercept-meaninghttps://puzzicle.org/thumbs/xa831a69862ad1434.jpgSlope and intercept meaning in contextPractice determining which feature of a linear model (the slope, the x-intercept, or the y-intercept) is useful for answering a given question in context.https://cdn.kastatic.org/ka-youtube-converted/TYbN--4KHIk.mp4/TYbN--4KHIk.mp4140yes https://puzzicle.org/math/algebra/v/slope-intercept-meaning-tablehttps://puzzicle.org/thumbs/xbf6179b0a74c3a3e.jpgSlope and intercept meaning from a tableGiven a linear relationship in a table, Sal graphs the relationship to explore what the slope and intercepts mean in the given context.https://cdn.kastatic.org/ka-youtube-converted/nc9XHXlxSyM.mp4/nc9XHXlxSyM.mp4416yes https://puzzicle.org/math/algebra/v/find-slope-intercepts-tablehttps://puzzicle.org/thumbs/xd76ee344ace70dc8.jpgFinding slope and intercepts from tablesGiven a linear relationship in a table, practice finding the slope and intercepts.https://cdn.kastatic.org/ka-youtube-converted/g98OfN5cWy4.mp4/g98OfN5cWy4.mp4324yes https://puzzicle.org/math/algebra/v/comparing-linear-rates-examplehttps://puzzicle.org/thumbs/x74532e0e2faa26b7.jpgComparing linear rates exampleCompare the positions of two creatures moving at constant speed and determine when one catches up with the other.https://cdn.kastatic.org/ka-youtube-converted/AoMjn2qKvXs.mp4/AoMjn2qKvXs.mp4483yes https://puzzicle.org/math/algebra/v/apply-creativity-algebrahttps://puzzicle.org/thumbs/x9b5c46a2ee2957db.jpgCreativity break: How do you apply creativity in algebraExperts share how we can apply creativity in algebra, mathematics, and problem solving.https://cdn.kastatic.org/ka-youtube-converted/2vAMs-KM7UA.mp4/2vAMs-KM7UA.mp4100yes https://puzzicle.org/math/algebra/v/linear-equation-word-problemshttps://puzzicle.org/thumbs/xd699d2b73e7bff12.jpgLinear equation word problemsLinear equation word problemshttps://cdn.kastatic.org/ka-youtube-converted/w6R8rywmgek.mp4/w6R8rywmgek.mp4221yes https://puzzicle.org/math/algebra/v/construct-linear-equation-contexthttps://puzzicle.org/thumbs/x794eb725ba1416d2.jpgConstructing linear equations from contextGiven a written description of a linear relationship in a some context, write an equation that represents the linear relationship described.https://cdn.kastatic.org/ka-youtube-converted/hBpI9IfmMKg.mp4/hBpI9IfmMKg.mp4187yes https://puzzicle.org/math/algebra/v/standard-form-ruleshttps://puzzicle.org/thumbs/xc5589921cbd30812.jpgClarifying standard form rulesClarifying standard form ruleshttps://cdn.kastatic.org/ka-youtube-converted/PfSsLjnnS60.mp4/PfSsLjnnS60.mp4289yes https://puzzicle.org/math/algebra/v/converting-from-slope-intercept-to-standard-formhttps://puzzicle.org/thumbs/xf4b68996.jpgConverting from slope-intercept to standard formWe can rewrite an equation in slope-intercept form (y=mx+b) to be in standard form (Ax+By=C) instead. In this example, we rewrite the slope-intercept equation y=2/3x+4/7 in standard form.https://cdn.kastatic.org/ka-youtube-converted/XOIhNVeLfWs.mp4/XOIhNVeLfWs.mp4309yes https://puzzicle.org/math/algebra/v/point-slope-and-standard-formhttps://puzzicle.org/thumbs/26807.jpgWriting linear equations in all formsSal finds the equation of a line that passes through (-3,6) and (6,0) in point-slope, slope-intercept, and standard form.https://cdn.kastatic.org/ka-youtube-converted/-6Fu2T_RSGM.mp4/-6Fu2T_RSGM.mp4476yes https://puzzicle.org/math/algebra/v/get-into-creative-zonehttps://puzzicle.org/thumbs/x8ec4a97e981ddb6e.jpgCreativity break: What do you do to get into your creative zone?Experts share what they do to get into their creative zone.https://cdn.kastatic.org/ka-youtube-converted/7WjlO5u3Vqo.mp4/7WjlO5u3Vqo.mp497yes https://puzzicle.org/math/algebra/v/setting-up-systemhttps://puzzicle.org/thumbs/x0227afd4a0dd6307.jpgSetting up a system of equations from context example (pet weights)Practice writing a system of linear equations that fits the constraints in a word problem.https://cdn.kastatic.org/ka-youtube-converted/HftghcUXaEg.mp4/HftghcUXaEg.mp4119yes https://puzzicle.org/math/algebra/v/setting-up-systems-weight-pricehttps://puzzicle.org/thumbs/x711262ce541d80e9.jpgSetting up a system of linear equations example (weight and price)Practice writing a system of linear equations that fits the constraints in a word problem.https://cdn.kastatic.org/ka-youtube-converted/pL4PXSXH-R8.mp4/pL4PXSXH-R8.mp4252yes https://puzzicle.org/math/algebra/v/interpret-points-systemshttps://puzzicle.org/thumbs/x3922ec65b7d6722c.jpgInterpreting points in context of graphs of systemsGiven a graph of a system of equations and a context, practice interpreting what various points represent in that system.https://cdn.kastatic.org/ka-youtube-converted/_hW1OzRlx50.mp4/_hW1OzRlx50.mp4339yes https://puzzicle.org/math/algebra/v/elimination-strategieshttps://puzzicle.org/thumbs/x82f52d65de01cb07.jpgElimination strategiesPractice identifying strategies for eliminating variables in a system of equations.https://cdn.kastatic.org/ka-youtube-converted/5jgSW5Jb-Y8.mp4/5jgSW5Jb-Y8.mp4436yes https://puzzicle.org/math/algebra/v/identifying-equivalent-systems-of-equationshttps://puzzicle.org/thumbs/x146f77c8.jpgWorked example: equivalent systems of equationsSal analyzes a couple of systems of equations and determines whether they have the same solution as a third given system.https://cdn.kastatic.org/ka-youtube-converted/XstL_x4ucm4.mp4/XstL_x4ucm4.mp4331yes https://puzzicle.org/math/algebra/v/identifying-non-equivalent-systems-of-equationshttps://puzzicle.org/thumbs/x1b00a657.jpgWorked example: non-equivalent systems of equationsSal analyzes a couple of systems of equations and determines whether they have the same solution as a third given system.https://cdn.kastatic.org/ka-youtube-converted/PSDQdwL1T0M.mp4/PSDQdwL1T0M.mp4270yes https://puzzicle.org/math/algebra/v/reasoning-system-equationshttps://puzzicle.org/thumbs/x0fb7b0cb44857460.jpgReasoning with systems of equationsWhen we perform operations on a system of equations, some operations produce an equivalent system, while others don't necessarily produce an equivalent system. When we're solving a system of equations, we need to use operations that guarantee equivalence.https://cdn.kastatic.org/ka-youtube-converted/s1RB-IgAHr8.mp4/s1RB-IgAHr8.mp4516yes https://puzzicle.org/math/algebra/v/graphing-inequalities-1https://puzzicle.org/thumbs/33752759.jpgTesting solutions to inequalitiesSal checks if the ordered pairs (3,5) and (1,-7) are solutions of the inequality 5x-3y≥25.https://cdn.kastatic.org/ka-youtube-converted/2oGsLdAWxlk.mp4/2oGsLdAWxlk.mp4366yes https://puzzicle.org/math/algebra/v/testing-solutions-for-a-system-of-inequalitieshttps://puzzicle.org/thumbs/33752767.jpgTesting solutions to systems of inequalitiesSal checks whether the ordered pair (2,5) is a solution of the following system: y≥2x+1 and x>1.https://cdn.kastatic.org/ka-youtube-converted/XzYNh2wpO0A.mp4/XzYNh2wpO0A.mp475yes https://puzzicle.org/math/algebra/v/graphical-system-of-inequalitieshttps://puzzicle.org/thumbs/33752766.jpgIntro to graphing systems of inequalitiesLearn how to graph systems of two-variable linear inequalities, like "y>x-8 and y<5-x."https://cdn.kastatic.org/ka-youtube-converted/CA4S7S-3Lg4.mp4/CA4S7S-3Lg4.mp4336yes https://puzzicle.org/math/algebra/v/graphing-systems-of-inequalities-2https://puzzicle.org/thumbs/26891.jpgGraphing systems of inequalitiesSal graphs the solution set of the system "y≥2x+1 and y<2x-5 and x>1."https://cdn.kastatic.org/ka-youtube-converted/YjT3QYfoy4Q.mp4/YjT3QYfoy4Q.mp4204yes https://puzzicle.org/math/algebra/v/constructing-two-variable-linear-inequality-word-problemhttps://puzzicle.org/thumbs/xd477103a.jpgWriting two-variable inequalities word problemSal solves a word problem about scores in a chess tournament by creating a two-variable linear inequality.https://cdn.kastatic.org/ka-youtube-converted/MHgi8ZQCG0I.mp4/MHgi8ZQCG0I.mp4142yes https://puzzicle.org/math/algebra/v/solving-two-variable-linear-inequality-word-problemhttps://puzzicle.org/thumbs/xc18c0e7b.jpgSolving two-variable inequalities word problemGiven a two-variable linear inequality that models a context about watering plants, Sal finds how many flowers can be watered.https://cdn.kastatic.org/ka-youtube-converted/ysdY1iX_XCs.mp4/ysdY1iX_XCs.mp4222yes https://puzzicle.org/math/algebra/v/analyzing-two-variable-linear-inequalityhttps://puzzicle.org/thumbs/xecc8be0c.jpgInterpreting two-variable inequalities word problemSal analyzes a two-variable linear inequality that models a context about YouTube videos.https://cdn.kastatic.org/ka-youtube-converted/TTYDbGXgcCk.mp4/TTYDbGXgcCk.mp4260yes https://puzzicle.org/math/algebra/v/u06-l3-t1-we3-graphing-systems-of-inequalitieshttps://puzzicle.org/thumbs/2600587.jpgModeling with systems of inequalitiesSal models a real-world context into an algebraic system of linear inequalities and graphs it.https://cdn.kastatic.org/ka-youtube-converted/BUmLw5m6F9s.mp4/BUmLw5m6F9s.mp4585yes https://puzzicle.org/math/algebra/v/creating-system-of-inequalities-word-problemhttps://puzzicle.org/thumbs/x78626c28.jpgWriting systems of inequalities word problemSal models a word problem about baking cupcakes and muffins by creating a system of linear inequalities.https://cdn.kastatic.org/ka-youtube-converted/jRnkxSfwIpU.mp4/jRnkxSfwIpU.mp4240yes https://puzzicle.org/math/algebra/v/interpreting-system-of-inequalities-in-contexthttps://puzzicle.org/thumbs/xd4f8409b.jpgSolving systems of inequalities word problemGiven a system of linear inequalities that models a context about making chairs and tables, Sal finds how many can be made.https://cdn.kastatic.org/ka-youtube-converted/ZTKAn9YNbNY.mp4/ZTKAn9YNbNY.mp4265yes https://puzzicle.org/math/algebra/v/graph-of-system-of-inequalities-word-problemhttps://puzzicle.org/thumbs/x5cab75bc.jpgGraphs of systems of inequalities word problemGiven the graph of a two-variable linear inequality that models a context about chopping vegetables, Sal finds if there's enough time to chop!https://cdn.kastatic.org/ka-youtube-converted/C_7Tqk9fw4k.mp4/C_7Tqk9fw4k.mp4215yes https://puzzicle.org/math/algebra/v/interpreting-visual-linear-inequality-word-problemhttps://puzzicle.org/thumbs/x7f6281ad.jpgGraphs of two-variable inequalities word problemGiven the graph of a two-variable linear inequality that models a context about dog biscuits, Sal finds if the dog can get enough biscuits.https://cdn.kastatic.org/ka-youtube-converted/DhiiGFuUE9I.mp4/DhiiGFuUE9I.mp4299yes https://puzzicle.org/math/algebra/v/expand-creative-skillshttps://puzzicle.org/thumbs/x13a6d24bd477485a.jpgCreativity break: What can we do to expand our creative skills?Experts share their advice on how to expand our creative skills.https://cdn.kastatic.org/ka-youtube-converted/SFUG5KpIf_Q.mp4/SFUG5KpIf_Q.mp488yes https://puzzicle.org/math/algebra/v/evaluate-discrete-functionhttps://puzzicle.org/thumbs/x5b89a73430127844.jpgEvaluating discrete functionsGiven the graph of a discrete function, Sal shows how to evaluate the function for a few different values.https://cdn.kastatic.org/ka-youtube-converted/Ddw8u7nNmKs.mp4/Ddw8u7nNmKs.mp4200yes https://puzzicle.org/math/algebra/v/evaluating-a-function-expressionhttps://puzzicle.org/thumbs/x4edf0df7.jpgWorked example: evaluating expressions with function notationEvaluating -2⋅f(-6)+g(1) given the graphs of f and g.https://cdn.kastatic.org/ka-youtube-converted/uaPm3Tpuxbc.mp4/uaPm3Tpuxbc.mp483yes https://puzzicle.org/math/algebra/v/finding-input-given-function-output-formulahttps://puzzicle.org/thumbs/x19e813a3.jpgWorked example: matching an input to a function's output (equation)Sal finds the input value for which f(t)=13, given that f(t)=-2t+5.https://cdn.kastatic.org/ka-youtube-converted/tJBQZ3PUkPo.mp4/tJBQZ3PUkPo.mp4100yes https://puzzicle.org/math/algebra/v/matching-function-input-to-output-with-graphhttps://puzzicle.org/thumbs/x71621631.jpgWorked example: matching an input to a function's output (graph)Sal finds the input value for which g(x)=-2 given the graph of g.https://cdn.kastatic.org/ka-youtube-converted/0Rmmynff5Yc.mp4/0Rmmynff5Yc.mp467yes https://puzzicle.org/math/algebra/v/different-inputs-giving-same-value-for-functionhttps://puzzicle.org/thumbs/x4773a085.jpgWorked example: two inputs with the same output (graph)Sal finds the input value other than -5 for which f(x)=f(-5), given the graph of f.https://cdn.kastatic.org/ka-youtube-converted/qSktOkaW6EA.mp4/qSktOkaW6EA.mp4108yes https://puzzicle.org/math/algebra/v/interpreting-function-notation-example-1https://puzzicle.org/thumbs/xf396d2ea.jpgFunction notation word problem: bankLearn how to interpret expressions that contain a function within a real-world context. In this video, the function we interpret models an account balance over time.https://cdn.kastatic.org/ka-youtube-converted/HERb3x0aw6c.mp4/HERb3x0aw6c.mp4204yes https://puzzicle.org/math/algebra/v/interpreting-function-notation-example-2https://puzzicle.org/thumbs/xeb7c9083.jpgFunction notation word problem: beachLearn how to interpret expressions that contain a function within a real-world context. In this video, the function we interpret models the number of people at the beach over time.https://cdn.kastatic.org/ka-youtube-converted/94Gnto5G1PU.mp4/94Gnto5G1PU.mp4156yes https://puzzicle.org/math/algebra/v/introduction-to-interval-notationhttps://puzzicle.org/thumbs/xbf5f3043.jpgIntervals and interval notationIntroducing intervals, which are bounded sets of numbers and are very useful when describing domain and range. We can use interval notation to show that a value falls between two endpoints. For example, -3≤x≤2, [-3,2], and {x∈ℝ|-3≤x≤2} all mean that x is between -3 and 2 and could be either endpoint.https://cdn.kastatic.org/ka-youtube-converted/UJQkqV2zGv0.mp4/UJQkqV2zGv0.mp4576yes https://puzzicle.org/math/algebra/v/domain-of-a-function-introhttps://puzzicle.org/thumbs/xb1c520a6.jpgWhat is the domain of a function?Functions assign outputs to inputs. The domain of a function is the set of all possible inputs for the function. For example, the domain of f(x)=x² is all real numbers, and the domain of g(x)=1/x is all real numbers except for x=0. We can also define special functions whose domains are more limited.https://cdn.kastatic.org/ka-youtube-converted/-DTMakGDZAw.mp4/-DTMakGDZAw.mp4405yes https://puzzicle.org/math/algebra/v/range-of-a-functionhttps://puzzicle.org/thumbs/x1d008f0a.jpgWhat is the range of a function?The range of a function is the set of all possible outputs the function can produce. Some functions (like linear functions) can have a range of all real numbers, but lots of functions have a more limited set of possible outputs.https://cdn.kastatic.org/ka-youtube-converted/96uHMcHWD2E.mp4/96uHMcHWD2E.mp4399yes https://puzzicle.org/math/algebra/v/domain-and-range-from-graphshttps://puzzicle.org/thumbs/x1410f50a.jpgWorked example: domain and range from graphFinding the domain and the range of a function that is given graphically.https://cdn.kastatic.org/ka-youtube-converted/sXP7VhU1gYE.mp4/sXP7VhU1gYE.mp4214yes https://puzzicle.org/math/algebra/v/determine-values-domainhttps://puzzicle.org/thumbs/x8acbe884682908ed.jpgDetermining whether values are in domain of functionSal shows how to test whether or not a value is or isn't in the domain of a function.https://cdn.kastatic.org/ka-youtube-converted/LdbrNNheFg8.mp4/LdbrNNheFg8.mp4297yes https://puzzicle.org/math/algebra/v/find-domain-functionhttps://puzzicle.org/thumbs/x034ca9c03c9a9c64.jpgExamples finding the domain of functionsSal shows how to algebraically find the domain of a few different functions.https://cdn.kastatic.org/ka-youtube-converted/4ZWbeESjv4M.mp4/4ZWbeESjv4M.mp4265yes https://puzzicle.org/math/algebra/v/domain-of-a-modeling-function-example-1https://puzzicle.org/thumbs/x3607e05a.jpgWorked example: determining domain word problem (real numbers)Determining the domain of a function that models the height of a plant over time.https://cdn.kastatic.org/ka-youtube-converted/2DnQEaNTd08.mp4/2DnQEaNTd08.mp4139yes https://puzzicle.org/math/algebra/v/domain-of-a-modeling-function-example-2https://puzzicle.org/thumbs/x9b3adf42.jpgWorked example: determining domain word problem (positive integers)Determining the domain of a function that models the price of candy bars.https://cdn.kastatic.org/ka-youtube-converted/AiW7syKXfJM.mp4/AiW7syKXfJM.mp4154yes https://puzzicle.org/math/algebra/v/domain-of-a-modeling-function-example-3https://puzzicle.org/thumbs/xcba2e3a7.jpgWorked example: determining domain word problem (all integers)Determining the domain of a function that models going up and down a ladder.https://cdn.kastatic.org/ka-youtube-converted/d1O54EQ1DSE.mp4/d1O54EQ1DSE.mp4334yes https://puzzicle.org/math/algebra/v/relative-minima-maximahttps://puzzicle.org/thumbs/xfc106389.jpgIntroducing minimum and maximum pointsSal explains all about minimum and maximum points, both absolute and relative.https://cdn.kastatic.org/ka-youtube-converted/Hoyv3-BMAGc.mp4/Hoyv3-BMAGc.mp4330yes https://puzzicle.org/math/algebra/v/identifying-relative-and-absolute-maxima-and-minimahttps://puzzicle.org/thumbs/x8aa33a5c.jpgWorked example: absolute and relative extremaExtrema is the general name for maximum and minimum points. This video shows how to identify relative and absolute extrema in the graph of a function.https://cdn.kastatic.org/ka-youtube-converted/xmrhZ5ySaD0.mp4/xmrhZ5ySaD0.mp4298yes https://puzzicle.org/math/algebra/v/increasing-decreasing-positive-and-negative-intervalshttps://puzzicle.org/thumbs/xbbb7485d.jpgIncreasing, decreasing, positive or negative intervalsFunction values can be positive or negative, and they can increase or decrease as the input increases. Here we introduce these basic properties of functions.https://cdn.kastatic.org/ka-youtube-converted/KxOp3s9ottg.mp4/KxOp3s9ottg.mp4356yes https://puzzicle.org/math/algebra/v/when-a-function-is-positive-or-negativehttps://puzzicle.org/thumbs/xdf96166c.jpgWorked example: positive & negative intervalsFinding the positive or negative intervals of a function from its graph.https://cdn.kastatic.org/ka-youtube-converted/eDv7dk9uNmM.mp4/eDv7dk9uNmM.mp470yes https://puzzicle.org/math/algebra/v/interpreting-function-graphs-word-problemshttps://puzzicle.org/thumbs/xadbc5422.jpgGraph interpretation word problem: temperatureWhen a function models a real-world context, we can learn a lot about the content from the function's graph. In this video, we consider a graph that models temperature over time.https://cdn.kastatic.org/ka-youtube-converted/VtBRkjRua0I.mp4/VtBRkjRua0I.mp4235yes https://puzzicle.org/math/algebra/v/interpreting-features-of-functions-2-example-1https://puzzicle.org/thumbs/xef8bffa5.jpgGraph interpretation word problem: basketballWhen a function models a real-world context, we can learn a lot about the context from the function's graph. In this video, we interpret the y-intercept of a graph that models a basketball free throw.https://cdn.kastatic.org/ka-youtube-converted/M3PTPN7NH8M.mp4/M3PTPN7NH8M.mp4230yes https://puzzicle.org/math/algebra/v/how-creative-in-algebrahttps://puzzicle.org/thumbs/x2e1ac5b89b77ae04.jpgCreativity break: How can people get creative in algebraExperts share how we can get creative in algebra.https://cdn.kastatic.org/ka-youtube-converted/DXPz0-HCK80.mp4/DXPz0-HCK80.mp467yes https://puzzicle.org/math/algebra/v/introduction-to-average-rate-of-changehttps://puzzicle.org/thumbs/x3beaa235.jpgIntroduction to average rate of changeWhat's the average rate of change of a function over an interval?https://cdn.kastatic.org/ka-youtube-converted/oT6LclcJ-I8.mp4/oT6LclcJ-I8.mp4332yes https://puzzicle.org/math/algebra/v/average-rate-of-change-example-1https://puzzicle.org/thumbs/x50b7170d.jpgWorked example: average rate of change from graphFinding the interval in a function's graph where the function has an average rate of change of -4.https://cdn.kastatic.org/ka-youtube-converted/TEXSW-o8674.mp4/TEXSW-o8674.mp4135yes https://puzzicle.org/math/algebra/v/average-rate-of-change-example-3https://puzzicle.org/thumbs/x765678cf.jpgWorked example: average rate of change from tableFinding the average rate of change of a function over the interval -5<x<-2, given a table of values of the function.https://cdn.kastatic.org/ka-youtube-converted/U1lQbh21y1k.mp4/U1lQbh21y1k.mp4140yes https://puzzicle.org/math/algebra/v/average-rate-of-change-from-table-word-problemhttps://puzzicle.org/thumbs/xfdc54aa7.jpgAverage rate of change word problem: tableAverage rate of change tells us how much the function changed per a single time unit, over a specific interval. It has many real-world applications. In this video, we compare the average rate of change of temperature over different time periods.https://cdn.kastatic.org/ka-youtube-converted/FBWGRbHf7rU.mp4/FBWGRbHf7rU.mp4220yes https://puzzicle.org/math/algebra/v/average-rate-of-change-from-graph-word-problemhttps://puzzicle.org/thumbs/x8ae36eb0.jpgAverage rate of change word problem: graphAverage rate of change tells us how much the function changed per a single time unit, over a specific interval. It has many real-world applications. In this video, we find the average rate of descent of a skydiver over a specific time interval.https://cdn.kastatic.org/ka-youtube-converted/-6EqUILZ1yw.mp4/-6EqUILZ1yw.mp4418yes https://puzzicle.org/math/algebra/v/introduction-to-function-inverseshttps://puzzicle.org/thumbs/26633.jpgIntro to inverse functionsSal explains what inverse functions are. Then he explains how to algebraically find the inverse of a function and looks at the graphical relationship between inverse functions.https://cdn.kastatic.org/ka-youtube-converted/W84lObmOp8M.mp4/W84lObmOp8M.mp4545yes https://puzzicle.org/math/algebra/v/understanding-inverse-functionshttps://puzzicle.org/thumbs/xd5d133d9.jpgInputs & outputs of inverse functionsSal explains that if f(a)=b, then f ⁻¹(b)=a, or in other words, the inverse function of f outputs a when its input is b.https://cdn.kastatic.org/ka-youtube-converted/KzaPBzFFLRM.mp4/KzaPBzFFLRM.mp4379yes https://puzzicle.org/math/algebra/v/understanding-function-inverses-examplehttps://puzzicle.org/thumbs/x2835aa81.jpgGraphing the inverse of a linear functionSal is given a line segment on the coordinate plane, and he graphs the inverse of the function represented by that segment.https://cdn.kastatic.org/ka-youtube-converted/KPeS2gdizhQ.mp4/KPeS2gdizhQ.mp4137yes https://puzzicle.org/math/algebra/v/function-inverse-example-1https://puzzicle.org/thumbs/26634.jpgFinding inverse functions: linearSal finds the inverses of f(x)=-x+4 and g(x)=-2x-1.https://cdn.kastatic.org/ka-youtube-converted/wSiamij_i_k.mp4/wSiamij_i_k.mp4404yes https://puzzicle.org/math/algebra/v/explicit-and-recursive-definitions-of-sequenceshttps://puzzicle.org/thumbs/1383721311.jpgSequences introSequences are ordered lists of numbers (called "terms"), like 2,5,8. Some sequences follow a specific pattern that can be used to extend them indefinitely. For example, 2,5,8 follows the pattern "add 3," and now we can continue the sequence. Sequences can have formulas that tell us how to find any term in the sequence. For example, 2,5,8,... can be represented by the formula 2+3(n-1).https://cdn.kastatic.org/ka-youtube-converted/KRFiAlo7t1E.mp4/KRFiAlo7t1E.mp4498yes https://puzzicle.org/math/algebra/v/arithmetic-sequenceshttps://puzzicle.org/thumbs/1383600767.jpgIntro to arithmetic sequencesSal introduces arithmetic sequences and their main features, the initial term and the common difference. He gives various examples of such sequences, defined explicitly and recursively.https://cdn.kastatic.org/ka-youtube-converted/_cooC3yG_p0.mp4/_cooC3yG_p0.mp4427yes https://puzzicle.org/math/algebra/v/extending-arithmetic-sequenceshttps://puzzicle.org/thumbs/xf18f69c4.jpgExtending arithmetic sequencesSal extends the arithmetic sequences -8, -14, -20, -26,... and 2, -1, -4, -7, -10,... to find their next terms.https://cdn.kastatic.org/ka-youtube-converted/EU0c6qrrevA.mp4/EU0c6qrrevA.mp467yes https://puzzicle.org/math/algebra/v/using-arithmetic-sequences-formulashttps://puzzicle.org/thumbs/x3fb279c6.jpgUsing arithmetic sequences formulasSal finds terms of arithmetic sequences using their explicit and recursive formulas.https://cdn.kastatic.org/ka-youtube-converted/8eSUbi_aYL4.mp4/8eSUbi_aYL4.mp4218yes https://puzzicle.org/math/algebra/v/using-recursive-formula-for-arithmetic-sequencehttps://puzzicle.org/thumbs/x0ffef60ab9a90d3f.jpgWorked example: using recursive formula for arithmetic sequenceExample finding the 4th term in a recursively defined arithmetic sequence.https://cdn.kastatic.org/ka-youtube-converted/v1ucHH06AxM.mp4/v1ucHH06AxM.mp4172yes https://puzzicle.org/math/algebra/v/recursive-formula-for-arithmetic-sequencehttps://puzzicle.org/thumbs/xfcb4de1d.jpgRecursive formulas for arithmetic sequencesSal finds the recursive formula of the arithmetic sequence 4, 3⅘, 3⅗, 3⅖,...https://cdn.kastatic.org/ka-youtube-converted/lBtb30SjU2Q.mp4/lBtb30SjU2Q.mp4186yes https://puzzicle.org/math/algebra/v/explicit-formulas-for-arithmetic-sequenceshttps://puzzicle.org/thumbs/xf3b43d64.jpgExplicit formulas for arithmetic sequencesSal finds explicit formulas of arithmetic sequences given the first few terms of those sequences. He also explores equivalent forms of such formulas.https://cdn.kastatic.org/ka-youtube-converted/ViLt2WI0XSg.mp4/ViLt2WI0XSg.mp4377yes https://puzzicle.org/math/algebra/v/finding-the-100th-term-in-a-sequencehttps://puzzicle.org/thumbs/26789.jpgArithmetic sequence problemSal finds the 100th term in the sequence 15, 9, 3, -3...https://cdn.kastatic.org/ka-youtube-converted/C5titprQAc4.mp4/C5titprQAc4.mp4357yes https://puzzicle.org/math/algebra/v/recursive-formulas-for-arithmetic-sequenceshttps://puzzicle.org/thumbs/x2b3ba86f.jpgConverting recursive & explicit forms of arithmetic sequencesSal is given an arithmetic sequence in explicit form and he converts it to recursive form. Then he does so the other way around!https://cdn.kastatic.org/ka-youtube-converted/GA_yxxeFYBU.mp4/GA_yxxeFYBU.mp4368yes https://puzzicle.org/math/algebra/v/geometric-sequences-introductionhttps://puzzicle.org/thumbs/23698.jpgIntro to geometric sequencesSal introduces geometric sequences and their main features, the initial term and the common ratio.https://cdn.kastatic.org/ka-youtube-converted/pXo0bG4iAyg.mp4/pXo0bG4iAyg.mp4645yes https://puzzicle.org/math/algebra/v/extending-geometric-sequenceshttps://puzzicle.org/thumbs/x3a8fd24d.jpgExtending geometric sequencesSal finds the next term in the geometric sequence -1/32, 1/8, -1/2, 2,...https://cdn.kastatic.org/ka-youtube-converted/yZ-GufE_uyA.mp4/yZ-GufE_uyA.mp4134yes https://puzzicle.org/math/algebra/v/using-explicit-formulas-of-geometric-sequenceshttps://puzzicle.org/thumbs/x4ebafa27.jpgUsing explicit formulas of geometric sequencesSal finds the 5th term in the geometric sequence whose explicit formula is 3(-¼)^(i-1).https://cdn.kastatic.org/ka-youtube-converted/HODIXLzIMIk.mp4/HODIXLzIMIk.mp497yes https://puzzicle.org/math/algebra/v/using-recursive-formulas-of-geometric-sequenceshttps://puzzicle.org/thumbs/x0e585ad5.jpgUsing recursive formulas of geometric sequencesSal finds the 4th term in the sequence whose recursive formula is a(1)=-⅛, a(i)=2a(i-1).https://cdn.kastatic.org/ka-youtube-converted/RkJBZALhXRA.mp4/RkJBZALhXRA.mp4226yes https://puzzicle.org/math/algebra/v/explicit-and-recursive-formulas-for-geometric-sequenceshttps://puzzicle.org/thumbs/x0c2d51aa.jpgExplicit & recursive formulas for geometric sequencesSal finds an explicit formula of a geometric sequence given the first few terms of the sequences. Then he explores equivalent forms the explicit formula and finds the corresponding recursive formula.https://cdn.kastatic.org/ka-youtube-converted/8a1a5A3CfdQ.mp4/8a1a5A3CfdQ.mp4338yes https://puzzicle.org/math/algebra/v/converting-an-explicit-function-to-a-recursive-functionhttps://puzzicle.org/thumbs/x9e46afe9.jpgConverting recursive & explicit forms of geometric sequencesSal solves the following problem: The explicit formula of a geometric sequence is g(x)=9*8^(x-1). Find the recursive formula of the sequence.https://cdn.kastatic.org/ka-youtube-converted/Iq7a2vEsT-o.mp4/Iq7a2vEsT-o.mp4390yes https://puzzicle.org/math/algebra/v/modeling-situations-with-arithmetic-and-geometric-sequenceshttps://puzzicle.org/thumbs/x7149cb33.jpgSequences word problemsSal solves two word problems about modeling real-world situations with arithmetic and geometric sequences.https://cdn.kastatic.org/ka-youtube-converted/yYGf7xn7TyM.mp4/yYGf7xn7TyM.mp4343yes https://puzzicle.org/math/algebra/v/recursive-formulas-for-sequenceshttps://puzzicle.org/thumbs/x901988c5.jpgEvaluating sequences in recursive formSal shows how to evaluate a sequence that is defined with a recursive formula. This definition gives the base case and then defines how to find the subsequent terms using the base case.https://cdn.kastatic.org/ka-youtube-converted/Kjli0Gunkds.mp4/Kjli0Gunkds.mp4547yes https://puzzicle.org/math/algebra/v/sequences-and-domainhttps://puzzicle.org/thumbs/x0c1b541142bd4db2.jpgSequences and domainWe can generate the same sequence with different functions and different domains.https://cdn.kastatic.org/ka-youtube-converted/TymkzM7J1PY.mp4/TymkzM7J1PY.mp4376yes https://puzzicle.org/math/algebra/v/shifting-absolute-value-graphshttps://puzzicle.org/thumbs/x0d6890ee04b473f5.jpgShifting absolute value graphsThe graph of y=|x-h|+k is the graph of y=|x| shifted h units to the right and k units up. See worked examples practicing this relationship.https://cdn.kastatic.org/ka-youtube-converted/Wri26sPEBoI.mp4/Wri26sPEBoI.mp4372yes https://puzzicle.org/math/algebra/v/reflecting-and-scaling-absolute-value-functionhttps://puzzicle.org/thumbs/x9c13b4008f199f84.jpgScaling & reflecting absolute value functions: equationThe graph of y=k|x| is the graph of y=|x| scaled by a factor of |k|. If k<0, it's also reflected (or "flipped") across the x-axis. In this worked example, we find the equation of an absolute value function from a description of the transformation performed on y=|x|.https://cdn.kastatic.org/ka-youtube-converted/lPcahMeUYcU.mp4/lPcahMeUYcU.mp4253yes https://puzzicle.org/math/algebra/v/scaling-and-reflecting-absolute-value-graphshttps://puzzicle.org/thumbs/x42691af83250ee2e.jpgScaling & reflecting absolute value functions: graphThe graph of y=k|x| is the graph of y=|x| scaled by a factor of |k|. If k<0, it's also reflected (or "flipped") across the x-axis. In this worked example, we find the equation of an absolute value function from its graph.https://cdn.kastatic.org/ka-youtube-converted/Yyfk0rwuZSE.mp4/Yyfk0rwuZSE.mp4208yes https://puzzicle.org/math/algebra/v/graphing-absolute-value-functionshttps://puzzicle.org/thumbs/x196b1cad71b2fcb4.jpgGraphing absolute value functionsWe can graph any absolute value equation of the form y=k|x-a|+h by thinking about function transformations (horizontal shifts, vertical shifts, reflections, and scalings).https://cdn.kastatic.org/ka-youtube-converted/v6Ct2OEmGNU.mp4/v6Ct2OEmGNU.mp4371yes https://puzzicle.org/math/algebra/v/piecewise-function-examplehttps://puzzicle.org/thumbs/x931bced2.jpgIntroduction to piecewise functionsA piecewise function is a function built from pieces of different functions over different intervals. For example, we can make a piecewise function f(x) where f(x) = -9 when -9 < x ≤ -5, f(x) = 6 when -5 < x ≤ -1, and f(x) = -7 when -1https://cdn.kastatic.org/ka-youtube-converted/tedzsRH0Jas.mp4/tedzsRH0Jas.mp4229yes https://puzzicle.org/math/algebra/v/evaluating-piecewise-functions-examplehttps://puzzicle.org/thumbs/x84b71dc7.jpgWorked example: evaluating piecewise functionsA piecewise function is a function that is defined in separate "pieces" or intervals. For each region or interval, the function may have a different equation or rule that describes it. We can evaluate piecewise functions (find the value of the function) by using their formulas or their graphs.https://cdn.kastatic.org/ka-youtube-converted/hg2HR9zJFq4.mp4/hg2HR9zJFq4.mp4264yes https://puzzicle.org/math/algebra/v/graphing-piecewise-functionhttps://puzzicle.org/thumbs/x5db71766.jpgWorked example: graphing piecewise functionsA piecewise function is a function that is defined in separate "pieces" or intervals. For each region or interval, the function may have a different equation or rule that describes it. We can graph a piecewise function by graphing each individual piece.https://cdn.kastatic.org/ka-youtube-converted/PQiXRrT_14o.mp4/PQiXRrT_14o.mp4270yes https://puzzicle.org/math/algebra/v/domain-and-range-for-piecewise-step-functionhttps://puzzicle.org/thumbs/xd42a1ea3.jpgWorked example: domain & range of step functionFinding the domain and range of a piecewise function that is constant in each segment. Such functions are called "step functions."https://cdn.kastatic.org/ka-youtube-converted/4n7TIvRHuDs.mp4/4n7TIvRHuDs.mp4164yes https://puzzicle.org/math/algebra/v/domain-and-range-for-piecewise-linear-functionhttps://puzzicle.org/thumbs/xae914f2c.jpgWorked example: domain & range of piecewise linear functionsFinding the domain and range of a piecewise function where each segment is linear.https://cdn.kastatic.org/ka-youtube-converted/rpI-X9Gn5a4.mp4/rpI-X9Gn5a4.mp4445yes https://puzzicle.org/math/algebra/v/understanding-square-rootshttps://puzzicle.org/thumbs/2600694.jpgUnderstanding square rootsLearn how square root means what number multiplied by itself will result in the given number.https://cdn.kastatic.org/ka-youtube-converted/ROIfbUQrSY4.mp4/ROIfbUQrSY4.mp481yes https://puzzicle.org/math/algebra/v/5th-rootshttps://puzzicle.org/thumbs/x7fb0bb70951d6897.jpg5th roots5th rootshttps://cdn.kastatic.org/ka-youtube-converted/8wrtyW1RwNk.mp4/8wrtyW1RwNk.mp4261yes https://puzzicle.org/math/algebra/v/simplifying-square-roots-1https://puzzicle.org/thumbs/x18b09b45.jpgSimplifying square rootsRoots are nice, but we prefer dealing with regular numbers as much as possible. So, for example, instead of √4 we prefer dealing with 2. What about roots that aren't equal to an integer, like √20? Still, we can write 20 as 4⋅5 and then use known properties to write √(4⋅5) as √4⋅√5, which is 2√5. We *simplified* √20.https://cdn.kastatic.org/ka-youtube-converted/cw3mp8oNASk.mp4/cw3mp8oNASk.mp4189yes https://puzzicle.org/math/algebra/v/simplifying-square-roots-comment-responsehttps://puzzicle.org/thumbs/126711576.jpgSimplifying square roots (variables)A worked example of simplifying radical with a variable in it. In this example, we simplify 3√(500x³).https://cdn.kastatic.org/ka-youtube-converted/qFFhdLlX220.mp4/qFFhdLlX220.mp4256yes https://puzzicle.org/math/algebra/v/simplifying-square-root-expressionshttps://puzzicle.org/thumbs/x0f8a3173e33fabc3.jpgSimplifying square-root expressionsWorked examples of taking expressions with square roots and taking all of the perfect squares out of the square roots. For example, 2√(7x)⋅3√(14x²) can be written as 42x√(2x).https://cdn.kastatic.org/ka-youtube-converted/3MijXB91DVo.mp4/3MijXB91DVo.mp4510yes https://puzzicle.org/math/algebra/v/exponential-growth-functionshttps://puzzicle.org/thumbs/23694.jpgIntro to exponential functionsAn exponential function represents the relationship between an input and output, where we use repeated multiplication on an initial value to get the output for any given input. Exponential functions can grow or decay very quickly. Exponential functions are often used to model things in the real world, such as populations, radioactive materials, and compound interest.https://cdn.kastatic.org/ka-youtube-converted/6WMZ7J0wwMI.mp4/6WMZ7J0wwMI.mp4461yes https://puzzicle.org/math/algebra/v/exponential-vs-linear-growthhttps://puzzicle.org/thumbs/xe7727adf4e991d4c.jpgExponential vs. linear growthFor constant increments in x, a linear growth would increase by a constant difference, and an exponential growth would increase by a constant ratio.https://cdn.kastatic.org/ka-youtube-converted/_vlXdx-CqM0.mp4/_vlXdx-CqM0.mp4185yes https://puzzicle.org/math/algebra/v/linear-exponential-modelshttps://puzzicle.org/thumbs/x9c757281.jpgExponential vs. linear models: verbalLinear growth occurs at a constant rate, with equal increments added or subtracted over time, while exponential growth involves a constant multiplier that drives an increase or decrease over time. We can look at the type of change over time to see if given example represents linear or exponential growth.https://cdn.kastatic.org/ka-youtube-converted/8OHEgD6YMBw.mp4/8OHEgD6YMBw.mp4208yes https://puzzicle.org/math/algebra/v/exponential-vs-linear-modelshttps://puzzicle.org/thumbs/x72dd6e08bd99f9cc.jpgExponential vs. linear models: tableDetermining whether real world model is linear or exponential, where the model is given as a table.https://cdn.kastatic.org/ka-youtube-converted/tV0NNJ6ndgk.mp4/tV0NNJ6ndgk.mp4206yes https://puzzicle.org/math/algebra/v/exponential-expressions-word-problems-numericalhttps://puzzicle.org/thumbs/x7f3f1edc036a1eb0.jpgExponential expressions word problems (numerical)Given the description of a real-world context, we write a calculation of a certain measure. The expression is exponential because it involves repeated multiplication.https://cdn.kastatic.org/ka-youtube-converted/L70UJVjc-bc.mp4/L70UJVjc-bc.mp4296yes https://puzzicle.org/math/algebra/v/initial-value-and-common-ratio-of-exponential-functionshttps://puzzicle.org/thumbs/x93e0cc1c.jpgInitial value & common ratio of exponential functionsIn an exponential function of the form f(x) = a*b^x, the initial value is usually taken to be the value of f(0), or "a". The common ratio refers to the rate of change in an exponential function. In the form given above, the common ratio is "b". For example, in the function f(x) = 2*3^x, the initial value is 2 and the common ratio is 3.https://cdn.kastatic.org/ka-youtube-converted/G2WybA4Hf7Y.mp4/G2WybA4Hf7Y.mp4327yes https://puzzicle.org/math/algebra/v/exponential-expressions-word-problems-algebraichttps://puzzicle.org/thumbs/xecc63cbc09fbd99f.jpgExponential expressions word problems (algebraic)Given a real-world context that involves repeated multiplication, we model it with an exponential function.https://cdn.kastatic.org/ka-youtube-converted/Z3YVZzCEi_A.mp4/Z3YVZzCEi_A.mp4216yes https://puzzicle.org/math/algebra/v/interpreting-exponential-expression-word-problemhttps://puzzicle.org/thumbs/x178f92593558693f.jpgInterpreting exponential expression word problemGiven an exponential function that models a real-world context, we interpret it to see what each part of the function represents in the real world.https://cdn.kastatic.org/ka-youtube-converted/Pud5ygp6H38.mp4/Pud5ygp6H38.mp4165yes https://puzzicle.org/math/algebra/v/graphing-exponential-functionshttps://puzzicle.org/thumbs/364954029.jpgExponential function graphWe can graph an exponential function, like y=5ˣ, by picking a few inputs (x-values) and finding their corresponding outputs (y-values). We'll see that an exponential function has a horizontal asymptote in one direction and rapidly changes in the other direction.https://cdn.kastatic.org/ka-youtube-converted/9SOSfRNCQZQ.mp4/9SOSfRNCQZQ.mp4332yes https://puzzicle.org/math/algebra/v/graphs-of-exponential-growthhttps://puzzicle.org/thumbs/x522467b5f5f47522.jpgGraphs of exponential growthIdentifying which graph represents a given exponential function.https://cdn.kastatic.org/ka-youtube-converted/6rX2VNybXEE.mp4/6rX2VNybXEE.mp4261yes https://puzzicle.org/math/algebra/v/exponential-vs-linear-growth-over-timehttps://puzzicle.org/thumbs/x068cd70493de8f71.jpgExponential vs. linear growth over timeIf we compare linear and exponential growth, we will see that over time, *any* exponential growth will surpass *any* linear growth, no matter how steep it is.https://cdn.kastatic.org/ka-youtube-converted/cwnke_pjX90.mp4/cwnke_pjX90.mp4305yes https://puzzicle.org/math/algebra/v/exponential-decay-introhttps://puzzicle.org/thumbs/x915aa47aea7bedc6.jpgExponential decay introBoth exponential growth and decay functions involve repeated multiplication by a constant factor. However, the difference lies in the size of that factor: - In an exponential growth function, the factor is greater than 1, so the output will increase (or "grow") over time. - In an exponential decay function, the factor is between 0 and 1, so the output will decrease (or "decay") over time.https://cdn.kastatic.org/ka-youtube-converted/v4IdaXvyE7U.mp4/v4IdaXvyE7U.mp4492yes https://puzzicle.org/math/algebra/v/graphing-basic-exponential-functionshttps://puzzicle.org/thumbs/x0c59dbab.jpgGraphing exponential growth & decayGraphing the exponential functions y=27⋅(⅓)ˣ and y=-30⋅2ˣ.https://cdn.kastatic.org/ka-youtube-converted/RVv0Jgi3Pbw.mp4/RVv0Jgi3Pbw.mp4246yes https://puzzicle.org/math/algebra/v/writing-functions-exponential-decayhttps://puzzicle.org/thumbs/x9c458523bb3a9600.jpgWriting functions with exponential decayWe can write a function to model exponential decay in a context.https://cdn.kastatic.org/ka-youtube-converted/cgZJU38j8-U.mp4/cgZJU38j8-U.mp4240yes https://puzzicle.org/math/algebra/v/writing-exponential-functionshttps://puzzicle.org/thumbs/xcd02b076.jpgWriting exponential functionsWriting the exponential function whose initial value is -2 and common ratio is 1/7.https://cdn.kastatic.org/ka-youtube-converted/UgLfAb_aFt4.mp4/UgLfAb_aFt4.mp4100yes https://puzzicle.org/math/algebra/v/constructing-linear-and-exponential-functions-from-datahttps://puzzicle.org/thumbs/x6c7009f6.jpgWriting exponential functions from tablesBoth linear equations and exponential equations represent relationships between two variables. However, the way that the variables are related to each other in each type of equation is different. A linear equation can be thought of as representing repeated addition on an initial value, while an exponential equation can be thought of as representing repeated multiplication on an initial value.https://cdn.kastatic.org/ka-youtube-converted/Qst1UVtq8pE.mp4/Qst1UVtq8pE.mp4317yes https://puzzicle.org/math/algebra/v/constructing-linear-and-exponential-functions-from-graphhttps://puzzicle.org/thumbs/x94bf277a.jpgWriting exponential functions from graphsGiven a graph of a line, we can write a linear function in the form y=mx+b by identifying the slope (m) and y-intercept (b) in the graph. GIven a graph of an exponential curve, we can write an exponential function in the form y=ab^x by identifying the common ratio (b) and y-intercept (a) in the graph.https://cdn.kastatic.org/ka-youtube-converted/gFdh_rE2XgU.mp4/gFdh_rE2XgU.mp4445yes https://puzzicle.org/math/algebra/v/analyzing-tables-of-exponential-functionshttps://puzzicle.org/thumbs/x8643e384.jpgAnalyzing tables of exponential functionsHere is how we can write an equation for an exponential function from a table of values: 1. Determine the common ratio. For example, if we see that every time x increases by 1, y is multiplied by 2, then the common ratio is 2. 2. Find the initial value of the function, or the y-intercept. This is the y-value when x=0. 3. Write the equation in the form y = ab^x, where a is the initial value and b is the common ratio.https://cdn.kastatic.org/ka-youtube-converted/yOqhKOicqhU.mp4/yOqhKOicqhU.mp4440yes https://puzzicle.org/math/algebra/v/analyzing-graph-of-exponential-functionhttps://puzzicle.org/thumbs/xd6c50c3c.jpgAnalyzing graphs of exponential functionsGiven the graph of an exponential function, Sal finds the formula of the function and a value that is outside the graph.https://cdn.kastatic.org/ka-youtube-converted/fe1Hsqyetzk.mp4/fe1Hsqyetzk.mp4246yes https://puzzicle.org/math/algebra/v/analyzing-exponential-graph-with-negative-initial-valuehttps://puzzicle.org/thumbs/xe38de1db.jpgAnalyzing graphs of exponential functions: negative initial valueGiven the graph of an exponential function with a negative initial value, Sal finds the formula of the function and solves an equation.https://cdn.kastatic.org/ka-youtube-converted/No1LKevjF6U.mp4/No1LKevjF6U.mp4367yes https://puzzicle.org/math/algebra/v/modeling-ticket-fines-with-exponential-functionhttps://puzzicle.org/thumbs/xd71282d5.jpgModeling with basic exponential functions word problemSal solves a word problem where he models the growth of a speeding ticket fine over time as an exponential function, and then interprets this model.https://cdn.kastatic.org/ka-youtube-converted/2iZXFd_ZImA.mp4/2iZXFd_ZImA.mp4337yes https://puzzicle.org/math/algebra/v/linear-and-exponential-growth-from-datahttps://puzzicle.org/thumbs/x774a385f.jpgLinear vs. exponential growth: from dataSal constructs functions that model the growth of trees over time. To do that, he identifies which growth is linear and which is exponential.https://cdn.kastatic.org/ka-youtube-converted/721RrH6auoU.mp4/721RrH6auoU.mp4357yes https://puzzicle.org/math/algebra/v/exponential-model-from-datahttps://puzzicle.org/thumbs/xe9df159d.jpgLinear vs. exponential growth: from data (example 2)Sal constructs a function that models cooling water. To do that, he decides whether the function is linear or exponential.https://cdn.kastatic.org/ka-youtube-converted/edwZDu94wNs.mp4/edwZDu94wNs.mp4416yes https://puzzicle.org/math/algebra/v/polynomials-introhttps://puzzicle.org/thumbs/x34127be666be8d0e.jpgPolynomials introThis introduction to polynomials covers common terminology like terms, degree, standard form, monomial, binomial and trinomial.https://cdn.kastatic.org/ka-youtube-converted/Vm7H0VTlIco.mp4/Vm7H0VTlIco.mp4648yes https://puzzicle.org/math/algebra/v/multiply-monomials-polynomials-area-modelhttps://puzzicle.org/thumbs/x0877cbe9e5c706b7.jpgMultiply monomials by polynomials: Area modelWe can use an area model to multiply a polynomial by a single term.https://cdn.kastatic.org/ka-youtube-converted/wznE1wlmhR0.mp4/wznE1wlmhR0.mp4246yes https://puzzicle.org/math/algebra/v/area-model-for-multiplying-binomialshttps://puzzicle.org/thumbs/x58a3c2e0.jpgMultiplying binomials: area modelSal expresses the area of a rectangle whose height is x+2 and width is x+3.https://cdn.kastatic.org/ka-youtube-converted/oOTFGdjhqqM.mp4/oOTFGdjhqqM.mp4235yes https://puzzicle.org/math/algebra/v/multiplying-simple-binomialshttps://puzzicle.org/thumbs/xc1613c4e.jpgMultiplying binomials introSal expresses (x-4)(x+7) as the standard trinomial x²+3x-28 and discusses how the general product (x+a)(x+b) can be written as x²+(a+b)x+a*b.https://cdn.kastatic.org/ka-youtube-converted/Xy8NKUoyy98.mp4/Xy8NKUoyy98.mp4287yes https://puzzicle.org/math/algebra/v/multiplying-binomialshttps://puzzicle.org/thumbs/13179518.jpgMultiplying binomialsSal expresses the product (3x+2)(5x-7) as 15x²-11x-14.https://cdn.kastatic.org/ka-youtube-converted/ZMLFfTX615w.mp4/ZMLFfTX615w.mp4347yes https://puzzicle.org/math/algebra/v/factors-and-divisibility-in-algebrahttps://puzzicle.org/thumbs/xef13fe32.jpgIntro to factors & divisibilitySal explains what it means for a polynomial to be a factor of another polynomial, and what it means for a polynomial to be divisible by another polynomial.https://cdn.kastatic.org/ka-youtube-converted/SjN3_xCJamA.mp4/SjN3_xCJamA.mp4326yes https://puzzicle.org/math/algebra/v/factor-by-grouping-and-factoring-completelyhttps://puzzicle.org/thumbs/23714.jpgIntro to groupingSal introduces the method of grouping, which is very useful in factoring quadratics whose leading coefficient is not 1.https://cdn.kastatic.org/ka-youtube-converted/X7B_tH4O-_s.mp4/X7B_tH4O-_s.mp4837yes https://puzzicle.org/math/algebra/v/factoring-trinomials-by-grouping-4https://puzzicle.org/thumbs/26986.jpgFactoring quadratics by groupingSal factors 4y^2+4y-15 as (2y-3)(2y+5) by grouping.https://cdn.kastatic.org/ka-youtube-converted/u1SAo2GiX8A.mp4/u1SAo2GiX8A.mp4235yes https://puzzicle.org/math/algebra/v/factoring-trinomials-by-grouping-5https://puzzicle.org/thumbs/26987.jpgFactoring quadratics: common factor + groupingSal factors 35k^2+100k-15 as 5(k+3)(7k-1).https://cdn.kastatic.org/ka-youtube-converted/R-rhSQzFJL0.mp4/R-rhSQzFJL0.mp4286yes https://puzzicle.org/math/algebra/v/factoring-trinomials-by-grouping-6https://puzzicle.org/thumbs/26988.jpgFactoring quadratics: negative common factor + groupingSal factors -12f^2-38f+22 as -2(2f-1)(3f+11).https://cdn.kastatic.org/ka-youtube-converted/d-2Lcp0QKfI.mp4/d-2Lcp0QKfI.mp4317yes https://puzzicle.org/math/algebra/v/combine-ways-of-thinking-in-problem-solvinghttps://puzzicle.org/thumbs/xe95b118c5ccddcda.jpgCreativity break: How can we combine ways of thinking in problem solving?Experts share how they combine ways of thinking to solve complex problems.https://cdn.kastatic.org/ka-youtube-converted/jAyqOcNurGU.mp4/jAyqOcNurGU.mp4151yes https://puzzicle.org/math/algebra/v/factoring-to-produce-difference-of-squareshttps://puzzicle.org/thumbs/x828e9a9a.jpgFactoring difference of squares: leading coefficient ≠ 1Sal factors 45x^2-125 as 5(3x+5)(3x-5).https://cdn.kastatic.org/ka-youtube-converted/jmbg-DKWuc4.mp4/jmbg-DKWuc4.mp4143yes https://puzzicle.org/math/algebra/v/equivalent-expressions-to-differences-of-squareshttps://puzzicle.org/thumbs/x80330a14.jpgFactoring difference of squares: analyzing factorizationSal analyzes two different factorizations of 16x^2-64 and determines whether they are correct.https://cdn.kastatic.org/ka-youtube-converted/TK-U1p9O6Nc.mp4/TK-U1p9O6Nc.mp4189yes https://puzzicle.org/math/algebra/v/finding-a-shared-binomial-factorhttps://puzzicle.org/thumbs/x5d12a4f8.jpgFactoring difference of squares: shared factorsSal finds the binomial factor shared by m^2-4m-45 and 6m^2-150.https://cdn.kastatic.org/ka-youtube-converted/xYQR2LNZLeU.mp4/xYQR2LNZLeU.mp4263yes https://puzzicle.org/math/algebra/v/shared-binomial-factor-between-perfect-square-and-difference-of-squareshttps://puzzicle.org/thumbs/x67daeb72.jpgFactoring perfect squares: shared factorsSal finds the binomial factor shared by 4x^2+12x+9 and 4x^2-9.https://cdn.kastatic.org/ka-youtube-converted/6jZZ6EeE-j0.mp4/6jZZ6EeE-j0.mp4266yes https://puzzicle.org/math/algebra/v/strategy-in-factoring-quadratics-1https://puzzicle.org/thumbs/x797baa764c018152.jpgStrategy in factoring quadratics (part 1 of 2)There are a lot of methods to factor quadratics, which apply on different occasions and conditions. After learning all of them in separate, let's think strategically about which method is useful for a given quadratic expression we want to factor.https://cdn.kastatic.org/ka-youtube-converted/2hHyY1eyHQs.mp4/2hHyY1eyHQs.mp4432yes https://puzzicle.org/math/algebra/v/strategy-in-factoring-quadratics-2https://puzzicle.org/thumbs/x82bceac4add1dc85.jpgStrategy in factoring quadratics (part 2 of 2)There are a lot of methods to factor quadratics, which apply on different occasions and conditions. After learning all of them in separate, let's think strategically about which method is useful for a given quadratic expression we want to factor.https://cdn.kastatic.org/ka-youtube-converted/ahPdiNQGXRA.mp4/ahPdiNQGXRA.mp4521yes https://puzzicle.org/math/algebra/v/parabolas-introhttps://puzzicle.org/thumbs/x3ffe2e08365a9116.jpgParabolas introGraphs of quadratic functions all have the same shape which we call "parabola." All parabolas have shared characteristics. For example, they are all symmetric about a line that passes through their vertex. This video covers this and other basic facts about parabolas.https://cdn.kastatic.org/ka-youtube-converted/BGz3pkoGPag.mp4/BGz3pkoGPag.mp4494yes https://puzzicle.org/math/algebra/v/interpret-parabola-contexthttps://puzzicle.org/thumbs/x10ab3ee853e280f3.jpgInterpreting a parabola in contextGiven a parabola that models a context, we can relate key features of the parabola — like the y-intercept, vertex, and x-intercepts — to what they represent in the given context.https://cdn.kastatic.org/ka-youtube-converted/_Bk6XkV9O_0.mp4/_Bk6XkV9O_0.mp4154yes https://puzzicle.org/math/algebra/v/interpret-a-quadratic-graphhttps://puzzicle.org/thumbs/xfeb705aa7b75cee2.jpgInterpret a quadratic graphWe can interpret what the features of a graph of a quadratic model mean in terms of a given context.https://cdn.kastatic.org/ka-youtube-converted/Tqxu53deWCo.mp4/Tqxu53deWCo.mp4157yes https://puzzicle.org/math/algebra/v/zero-product-propertyhttps://puzzicle.org/thumbs/x556b284b8e003b36.jpgZero product propertyThe zero product property states that if a⋅b=0 then either a or b equal zero. This basic property helps us solve equations like (x+2)(x-5)=0.https://cdn.kastatic.org/ka-youtube-converted/-lWVpoPaPBc.mp4/-lWVpoPaPBc.mp4437yes https://puzzicle.org/math/algebra/v/graphing-quadratics-in-factored-formhttps://puzzicle.org/thumbs/x8456dafc88a8915f.jpgGraphing quadratics in factored formAn example for a quadratic function in factored form is y=½(x-6)(x+2). We can analyze this form to find the x-intercepts of the graph, as well as find the vertex.https://cdn.kastatic.org/ka-youtube-converted/EV57jv7JKCs.mp4/EV57jv7JKCs.mp4363yes https://puzzicle.org/math/algebra/v/quadratic-word-problems-factored-formhttps://puzzicle.org/thumbs/x9b8c4618ba35e3de.jpgQuadratic word problems (factored form)Solving a problem where a quadratic function (given in factored form) models the height of a launched rocket.https://cdn.kastatic.org/ka-youtube-converted/xVZA6NOQA7A.mp4/xVZA6NOQA7A.mp4335yes https://puzzicle.org/math/algebra/v/simple-quadratic-equationhttps://puzzicle.org/thumbs/150088074.jpgSolving quadratics by taking square rootsSal solves the equation 2x^2+3=75 by isolating x^2 and taking the square root of both sides.https://cdn.kastatic.org/ka-youtube-converted/RweAgQwLdMs.mp4/RweAgQwLdMs.mp4138yes https://puzzicle.org/math/algebra/v/solving-quadratics-by-taking-square-rootshttps://puzzicle.org/thumbs/x7a55a0a8.jpgSolving quadratics by taking square roots examplesSal solves the equation (x+3)²-4=0 and finds the x-intercepts of f(x)=(x-2)²-9.https://cdn.kastatic.org/ka-youtube-converted/VTlvg4wJ1X0.mp4/VTlvg4wJ1X0.mp4312yes https://puzzicle.org/math/algebra/v/determining-mistakes-in-steps-examplehttps://puzzicle.org/thumbs/xf7eacd77.jpgQuadratics by taking square roots: strategySal analyzes a given solution of a quadratic equation, and finds where and what was the error in that process.https://cdn.kastatic.org/ka-youtube-converted/jtPvDycHVQw.mp4/jtPvDycHVQw.mp4105yes https://puzzicle.org/math/algebra/v/order-of-steps-exercise-examplehttps://puzzicle.org/thumbs/x947b1dd9.jpgSolving quadratics by taking square roots: with stepsSal discusses the exact order of steps in the process of solving the equation 3(x+6)^2=75.https://cdn.kastatic.org/ka-youtube-converted/naVFTNWVlQU.mp4/naVFTNWVlQU.mp4265yes https://puzzicle.org/math/algebra/v/vertex-form-introhttps://puzzicle.org/thumbs/x920ee4c2b9181777.jpgVertex form introductionOne of the common forms for quadratic functions is called vertex form, because it highlights the coordinates of the vertex of the function's graph.https://cdn.kastatic.org/ka-youtube-converted/_QqhuLixNEk.mp4/_QqhuLixNEk.mp4373yes https://puzzicle.org/math/algebra/v/graphing-a-parabola-in-vertex-formhttps://puzzicle.org/thumbs/xac1e66a2.jpgGraphing quadratics: vertex formLearn how to graph any quadratic function that is given in vertex form. Here, Sal graphs y=-2(x-2)²+5.https://cdn.kastatic.org/ka-youtube-converted/7QMoNY6FzvM.mp4/7QMoNY6FzvM.mp4196yes https://puzzicle.org/math/algebra/v/quadratic-word-problems-in-vertex-formhttps://puzzicle.org/thumbs/x6e498f765d2f765d.jpgQuadratic word problems (vertex form)Given a quadratic function that models the height of an object being launched from a platform, we analyze the function to answer questions like "what is the height of the platform?" or "when does the object reach its maximum height?"https://cdn.kastatic.org/ka-youtube-converted/lUA1LYCoIAg.mp4/lUA1LYCoIAg.mp4423yes https://puzzicle.org/math/algebra/v/example-1-solving-a-quadratic-equation-by-factoringhttps://puzzicle.org/thumbs/26992.jpgSolving quadratics by factoringSal solves the equation s^2-2s-35=0 by factoring the expression on the left as (s+5)(s-7) and finding the s-values that make each factor equal to zero.https://cdn.kastatic.org/ka-youtube-converted/2ZzuZvz33X0.mp4/2ZzuZvz33X0.mp4382yes https://puzzicle.org/math/algebra/v/solving-quadratics-by-dividing-and-factoringhttps://puzzicle.org/thumbs/x9d3269d7.jpgSolving quadratics by factoring: leading coefficient ≠ 1Sal solves 6x²-120x+600=0 by first dividing by 6 and then factoring.https://cdn.kastatic.org/ka-youtube-converted/KbFwLvCOBUI.mp4/KbFwLvCOBUI.mp4275yes https://puzzicle.org/math/algebra/v/using-structure-to-solve-quadraticshttps://puzzicle.org/thumbs/x3579f012.jpgSolving quadratics using structureSal solves (2x-3)^2=4x-6 by substituting p for 2x-3 and obtaining the simpler equation p^2=2p.https://cdn.kastatic.org/ka-youtube-converted/ZIqW_sXymrM.mp4/ZIqW_sXymrM.mp4235yes https://puzzicle.org/math/algebra/v/example-4-solving-a-quadratic-equation-by-factoringhttps://puzzicle.org/thumbs/13179588.jpgQuadratic equations word problem: triangle dimensionsSal solves a geometry problem using a quadratic equation.https://cdn.kastatic.org/ka-youtube-converted/STcsaKuW-24.mp4/STcsaKuW-24.mp4501yes https://puzzicle.org/math/algebra/v/example-3-solving-a-quadratic-equation-by-factoringhttps://puzzicle.org/thumbs/26994.jpgQuadratic equations word problem: box dimensionsSal solves a volume problem using a quadratic equation.https://cdn.kastatic.org/ka-youtube-converted/vl9o9XEfXtw.mp4/vl9o9XEfXtw.mp4354yes https://puzzicle.org/math/algebra/v/using-the-quadratic-formulahttps://puzzicle.org/thumbs/23724.jpgThe quadratic formulaThe quadratic formula helps us solve any quadratic equation. First, we bring the equation to the form ax²+bx+c=0, where a, b, and c are coefficients. Then, we plug these coefficients in the formula: (-b±√(b²-4ac))/(2a) . See examples of using the formula to solve a variety of equations.https://cdn.kastatic.org/ka-youtube-converted/i7idZfS8t8w.mp4/i7idZfS8t8w.mp4992yes https://puzzicle.org/math/algebra/v/quadratic-formula-2https://puzzicle.org/thumbs/27003.jpgWorked example: quadratic formula (example 2)Sal solves the equation -x^2+8x=1 by first bringing it to standard form and then using the quadratic formula.https://cdn.kastatic.org/ka-youtube-converted/CLrImGKeuEI.mp4/CLrImGKeuEI.mp4400yes https://puzzicle.org/math/algebra/v/applying-the-quadratic-formulahttps://puzzicle.org/thumbs/150088073.jpgWorked example: quadratic formula (negative coefficients)Sal solves -3x^2+10x-3=0 by plugging a=-3, b=10, c=-3 in the quadratic formula. Then he multiplies everything by -1 and solves again. The results are the same!https://cdn.kastatic.org/ka-youtube-converted/_E3gx_yQhHQ.mp4/_E3gx_yQhHQ.mp4291yes https://puzzicle.org/math/algebra/v/quadratic-formula-3https://puzzicle.org/thumbs/27004.jpgUsing the quadratic formula: number of solutionsSal determines how many solutions the equation x²+14x+49=0 has by considering its quadratic formula, and more specifically, its discriminant.https://cdn.kastatic.org/ka-youtube-converted/XUvKjC21fYU.mp4/XUvKjC21fYU.mp4299yes https://puzzicle.org/math/algebra/v/solving-quadratic-equations-by-completing-the-squarehttps://puzzicle.org/thumbs/23722.jpgCompleting the squareSome quadratic expressions can be factored as perfect squares. For example, x²+6x+9=(x+3)². However, even if an expression isn't a perfect square, we can turn it into one by adding a constant number. For example, x²+6x+5 isn't a perfect square, but if we add 4 we get (x+3)². This, in essence, is the method of *completing the square*https://cdn.kastatic.org/ka-youtube-converted/bNQY0z76M5A.mp4/bNQY0z76M5A.mp4846yes https://puzzicle.org/math/algebra/v/ex1-completing-the-squarehttps://puzzicle.org/thumbs/26998.jpgWorked example: Completing the square (intro)Sal completes x²-44x into a perfect square.https://cdn.kastatic.org/ka-youtube-converted/VvuuRpJbbHE.mp4/VvuuRpJbbHE.mp4202yes https://puzzicle.org/math/algebra/v/rewriting-quadratics-as-perfect-squareshttps://puzzicle.org/thumbs/xfa51b4a3.jpgWorked example: Rewriting expressions by completing the squareSal rewrites x²+16x+9 as (x+8)²-55 by completing the square.https://cdn.kastatic.org/ka-youtube-converted/sh-MP-dVhD4.mp4/sh-MP-dVhD4.mp4182yes https://puzzicle.org/math/algebra/v/solving-quadratics-by-completing-the-squarehttps://puzzicle.org/thumbs/xf7adc86f.jpgWorked example: Rewriting & solving equations by completing the squareSal solves x²-2x-8=0 by rewriting the equation as (x-1)²-9=0 (which is done by completing the square!).https://cdn.kastatic.org/ka-youtube-converted/KouDAzYl_bc.mp4/KouDAzYl_bc.mp4379yes https://puzzicle.org/math/algebra/v/solve-completing-square-integer-solutionshttps://puzzicle.org/thumbs/x540f247ec7770468.jpgSolve by completing the square: Integer solutionsWe can use the strategy of completing the square to solve quadratic equations. If the solutions are integers, we can solve by factoring as well.https://cdn.kastatic.org/ka-youtube-converted/eU41vG9z35M.mp4/eU41vG9z35M.mp4389yes https://puzzicle.org/math/algebra/v/solve-completing-square-non-integer-solutionshttps://puzzicle.org/thumbs/x22ce3fc5ae0cc6c9.jpgSolve by completing the square: Non-integer solutionsWe can use the strategy of completing the square to solve quadratic equations even when the solutions aren't integers.https://cdn.kastatic.org/ka-youtube-converted/Ldp9_K3L4Q8.mp4/Ldp9_K3L4Q8.mp4242yes https://puzzicle.org/math/algebra/v/completing-the-square-to-solve-quadratic-equationshttps://puzzicle.org/thumbs/150088071.jpgWorked example: completing the square (leading coefficient ≠ 1)Sal solves the equation 4x^2+40x-300=0 by completing the square.https://cdn.kastatic.org/ka-youtube-converted/TV5kDqiJ1Os.mp4/TV5kDqiJ1Os.mp4344yes https://puzzicle.org/math/algebra/v/ex2-completing-the-squarehttps://puzzicle.org/thumbs/26999.jpgSolving quadratics by completing the square: no solutionSal solves the equation 4x^2+40x+280=0 by completing the square, only to find there's no solution for this equation.https://cdn.kastatic.org/ka-youtube-converted/6agzj3A9IgA.mp4/6agzj3A9IgA.mp4324yes https://puzzicle.org/math/algebra/v/proof-of-quadratic-formulahttps://puzzicle.org/thumbs/23726.jpgProof of the quadratic formulaSal proves the quadratic formula using the method of completing the square.https://cdn.kastatic.org/ka-youtube-converted/r3SEkdtpobo.mp4/r3SEkdtpobo.mp4464yes https://puzzicle.org/math/algebra/v/strategy-in-solving-quadratic-equationshttps://puzzicle.org/thumbs/x75ddcb01a69632a1.jpgStrategy in solving quadratic equationsBased on the initial form of a quadratic equation, we can determine which solution methods are and aren't appropriate.https://cdn.kastatic.org/ka-youtube-converted/_MllyJivas4.mp4/_MllyJivas4.mp4350yes https://puzzicle.org/math/algebra/v/ex3-completing-the-squarehttps://puzzicle.org/thumbs/27000.jpgFinding the vertex of a parabola in standard formSal rewrites the equation y=-5x^2-20x+15 in vertex form (by completing the square) in order to identify the vertex of the corresponding parabola.https://cdn.kastatic.org/ka-youtube-converted/IbI-l7mbKO4.mp4/IbI-l7mbKO4.mp4340yes https://puzzicle.org/math/algebra/v/graphing-a-parabola-using-roots-and-vertexhttps://puzzicle.org/thumbs/xe26c2892.jpgGraphing quadratics: standard formLearn how to graph any quadratic function that is given in standard form. Here, Sal graphs y=5x²-20x+15.https://cdn.kastatic.org/ka-youtube-converted/MQtsRYPx3v0.mp4/MQtsRYPx3v0.mp4281yes https://puzzicle.org/math/algebra/v/application-problem-with-quadratic-formulahttps://puzzicle.org/thumbs/150009952.jpgQuadratic word problem: ballSal solves a word problem about a ball being shot in the air. The equation for the height of the ball as a function of time is modeled by a quadratic function.https://cdn.kastatic.org/ka-youtube-converted/OZtqz_xw0SQ.mp4/OZtqz_xw0SQ.mp4353yes https://puzzicle.org/math/algebra/v/rewriting-a-quadratic-function-to-find-roots-and-vertexhttps://puzzicle.org/thumbs/x96c7edd8.jpgForms & features of quadratic functionsDifferent forms of quadratic functions reveal different features of those functions. Here, Sal rewrites f(x)=x²-5x+6 in factored form to reveal its zeros and in vertex form to reveal its vertex.https://cdn.kastatic.org/ka-youtube-converted/4Bc5-HRop5Y.mp4/4Bc5-HRop5Y.mp4509yes https://puzzicle.org/math/algebra/v/strategy-with-forms-of-quadraticshttps://puzzicle.org/thumbs/xabfbf7e7fd3b373c.jpgWorked examples: Forms & features of quadratic functionsChoosing whether factored form, vertex form or standard form of a quadratic is best in different situations.https://cdn.kastatic.org/ka-youtube-converted/eRbgHCaWQQE.mp4/eRbgHCaWQQE.mp4402yes https://puzzicle.org/math/algebra/v/quadratic-functions-2https://puzzicle.org/thumbs/26996.jpgVertex & axis of symmetry of a parabolaSal rewrites a quadratic equation in vertex form and shows how it reveals the vertex of the corresponding parabola.https://cdn.kastatic.org/ka-youtube-converted/dfoXtodyiIA.mp4/dfoXtodyiIA.mp4442yes https://puzzicle.org/math/algebra/v/finding-features-of-quadratic-functionshttps://puzzicle.org/thumbs/xb8f6b949.jpgFinding features of quadratic functionsSal finds the zeros, the vertex, & the line of symmetry of quadratic functions given in vertex form, factored form, & standard form.https://cdn.kastatic.org/ka-youtube-converted/V-BAdpvejTw.mp4/V-BAdpvejTw.mp4503yes https://puzzicle.org/math/algebra/v/interpret-quadratic-models-factored-formhttps://puzzicle.org/thumbs/x1ba131f975c73839.jpgInterpret quadratic models: Factored formGiven a quadratic function that models a relationship, we can rewrite the function to reveal certain properties of the relationship. Factored form helps us identify the x-intercepts or zeros of the function.https://cdn.kastatic.org/ka-youtube-converted/ITBRCLusU1s.mp4/ITBRCLusU1s.mp4240yes https://puzzicle.org/math/algebra/v/interpret-quadratic-modelshttps://puzzicle.org/thumbs/xb055979a96a7ee4e.jpgInterpret quadratic models: Vertex formGiven a quadratic function that models a relationship, we can rewrite the function to reveal certain properties of the relationship.https://cdn.kastatic.org/ka-youtube-converted/JdWEdb90V1s.mp4/JdWEdb90V1s.mp4353yes https://puzzicle.org/math/algebra/v/creativity-role-lifehttps://puzzicle.org/thumbs/xa372f275cca4b470.jpgCreativity break: How does creativity play a role in your everyday life?Experts share what role creativity plays in their everyday lives.https://cdn.kastatic.org/ka-youtube-converted/eU8IfaoD-v8.mp4/eU8IfaoD-v8.mp4144yes https://puzzicle.org/math/algebra/v/comparing-features-of-quadratic-functionshttps://puzzicle.org/thumbs/xae2fb360.jpgComparing features of quadratic functionsSal compares the y-intercepts, the zeros, and the concavity of quadratic functions given graphically and algebraically.https://cdn.kastatic.org/ka-youtube-converted/UdLYAqN0gNY.mp4/UdLYAqN0gNY.mp4251yes https://puzzicle.org/math/algebra/v/comparing-features-of-functions-2-example-2https://puzzicle.org/thumbs/x8d256629.jpgComparing maximum points of quadratic functionsGiven several quadratic functions represented in different forms, Sal finds the one with the lowest maximum value.https://cdn.kastatic.org/ka-youtube-converted/4Bx06GFyhUA.mp4/4Bx06GFyhUA.mp4239yes https://puzzicle.org/math/algebra/v/shifting-and-scaling-parabolashttps://puzzicle.org/thumbs/x78bacc55.jpgIntro to parabola transformationsSal discusses how we can shift and scale the graph of a parabola to obtain any other parabola, and how this affects the equation of the parabola.https://cdn.kastatic.org/ka-youtube-converted/99v51U3HSCU.mp4/99v51U3HSCU.mp4481yes https://puzzicle.org/math/algebra/v/example-translating-parabolahttps://puzzicle.org/thumbs/x8a38e1f66105044e.jpgShifting parabolasThe graph of y=(x-k)²+h is the resulting of shifting (or translating) the graph of y=x², k units to the right and h units up. For example, y=(x-3)²-4 is the result of shifting y=x² 3 units to the right and -4 units up, which is the same as 4 units down.https://cdn.kastatic.org/ka-youtube-converted/ZmVOR6n_fzY.mp4/ZmVOR6n_fzY.mp4282yes https://puzzicle.org/math/algebra/v/scaling-and-reflecting-parabolashttps://puzzicle.org/thumbs/x510a7b8e94445c84.jpgScaling & reflecting parabolasThe graph of y=k⋅x² is the graph of y=x² scaled by a factor of |k|. If k<0, it's also reflected (or "flipped") across the x-axis. In this worked example, we find the equation of a parabola from its graph.https://cdn.kastatic.org/ka-youtube-converted/AMsbZLxd3lE.mp4/AMsbZLxd3lE.mp4285yes https://puzzicle.org/math/algebra/v/sum-and-product-of-rational-numbershttps://puzzicle.org/thumbs/x1ff9a2a1.jpgProof: sum & product of two rationals is rationalSal proves that the sum, or the product, of any two rational numbers will always be a rational number.https://cdn.kastatic.org/ka-youtube-converted/HKUJkMQsGkM.mp4/HKUJkMQsGkM.mp4166yes https://puzzicle.org/math/algebra/v/proof-that-rational-times-irrational-is-irrationalhttps://puzzicle.org/thumbs/xf3863cc0.jpgProof: product of rational & irrational is irrationalThe product of any rational number and any irrational number will always be an irrational number. This allows us to quickly conclude that 3π is irrational.https://cdn.kastatic.org/ka-youtube-converted/KT32CsdEZEY.mp4/KT32CsdEZEY.mp4195yes https://puzzicle.org/math/algebra/v/proof-that-sum-of-rational-and-irrational-is-irrationalhttps://puzzicle.org/thumbs/xeb700d9f.jpgProof: sum of rational & irrational is irrationalThe sum of any rational number and any irrational number will always be an irrational number. This allows us to quickly conclude that ½+√2 is irrational.https://cdn.kastatic.org/ka-youtube-converted/pPM72fPwIjw.mp4/pPM72fPwIjw.mp4204yes https://puzzicle.org/math/algebra/v/sums-and-products-of-irrational-numbershttps://puzzicle.org/thumbs/xda6c575074a2c44a.jpgSums and products of irrational numbersThe sum of two irrational numbers can be rational and it can be irrational. It depends on which irrational numbers we're talking about exactly. The same goes for products for two irrational numbers. This video covers this fact with various examples.https://cdn.kastatic.org/ka-youtube-converted/16-GZWi66CI.mp4/16-GZWi66CI.mp4357yes https://puzzicle.org/math/algebra/v/recognizing-rational-and-irrational-expressions-examplehttps://puzzicle.org/thumbs/x85d5add2.jpgWorked example: rational vs. irrational expressionsSal shows how to determine whether the following expressions are rational or irrational: 9 + √(45), √(45)/ (3*√(5)), and 3*√(9).https://cdn.kastatic.org/ka-youtube-converted/hTJgK-wZMcE.mp4/hTJgK-wZMcE.mp4156yes https://puzzicle.org/math/algebra/v/recognizing-rational-and-irrational-expressions-w-unknownshttps://puzzicle.org/thumbs/x78710541.jpgWorked example: rational vs. irrational expressions (unknowns)Sal determines whether expressions with unknown rational/irrational numbers are rational or irrational.https://cdn.kastatic.org/ka-youtube-converted/NC6HXlrH53Y.mp4/NC6HXlrH53Y.mp4386yes https://puzzicle.org/math/algebra/v/proof-that-square-root-of-2-is-irrationalhttps://puzzicle.org/thumbs/x8924c336.jpgProof: √2 is irrationalSal proves that the square root of 2 is an irrational number, i.e. it cannot be given as the ratio of two integers.https://cdn.kastatic.org/ka-youtube-converted/mX91_3GQqLY.mp4/mX91_3GQqLY.mp4467yes https://puzzicle.org/math/algebra/v/proof-that-square-root-of-prime-number-is-irrationalhttps://puzzicle.org/thumbs/x5725b03f.jpgProof: square roots of prime numbers are irrationalSal proves that the square root of any prime number must be an irrational number. For example, because of this proof we can quickly determine that √3, √5, √7, or √11 are irrational numbers.https://cdn.kastatic.org/ka-youtube-converted/W-Nio466Ek4.mp4/W-Nio466Ek4.mp4448yes https://puzzicle.org/math/algebra/v/proof-that-there-is-an-irrational-number-between-any-two-rational-numbershttps://puzzicle.org/thumbs/x04436acf.jpgProof: there's an irrational number between any two rational numbersSal proves that when given any two rational numbers, no matter how close, we can find an irrational number that lies between them.https://cdn.kastatic.org/ka-youtube-converted/hxqo_DiL3pw.mp4/hxqo_DiL3pw.mp4333yes https://puzzicle.org/math/algebra/v/creativity-in-algebrahttps://puzzicle.org/thumbs/xa0e13c97c9f29692.jpgCreativity in algebraHear from professionals as they talk about creativity in algebra and mathematics.https://cdn.kastatic.org/ka-youtube-converted/nOBZM5QYcBc.mp4/nOBZM5QYcBc.mp4352yes https://puzzicle.org/math/algebra/v/creative-algebra-at-workhttps://puzzicle.org/thumbs/x55ae315ce1fa0cf6.jpgCreative algebra at workHear from experts as they talk about creative algebra and mathematics at work.https://cdn.kastatic.org/ka-youtube-converted/Ep7diUpHuhs.mp4/Ep7diUpHuhs.mp4493yes https://puzzicle.org/math/algebra/v/the-future-of-creativity-in-algebrahttps://puzzicle.org/thumbs/x90da36ecf98bb584.jpgThe future of creativity in algebraHear from professionals and students as they talk about the future of creativity in algebra.https://cdn.kastatic.org/ka-youtube-converted/Q-apRGBr3Xg.mp4/Q-apRGBr3Xg.mp4573yes https://puzzicle.org/math/algebra-1-eureka-squared-aligned/v/combining-like-terms-with-negative-coefficients-indian-accenthttps://puzzicle.org/thumbs/x728a776289c36232.jpgCombining like terms with negative coefficientsThis example of combining like terms in an expression gets a little hairy. Pay attention.https://cdn.kastatic.org/ka-youtube-converted/WnrV31asOS0.mp4/WnrV31asOS0.mp4282yes https://puzzicle.org/math/algebra-1-eureka-squared-aligned/v/adding-and-subtracting-polynomials-of-degree-twohttps://puzzicle.org/thumbs/xac21546ae99eca29.jpgAdding and subtracting polynomials of degree twoWhen given two polynomials of degree two, add or subtract to simplify.https://cdn.kastatic.org/ka-youtube-converted/kieyGh0SDQk.mp4/kieyGh0SDQk.mp4190yes https://puzzicle.org/math/algebra-1-eureka-squared-aligned/v/equations-with-variables-on-both-sides-20-7x6x-6-indian-accenthttps://puzzicle.org/thumbs/x114c2d766b5ae816.jpgEquations with variables on both sides: 20-7x=6x-6Solving equations like 20 - 7x = 6x - 6 with the variable on both sides involves a few steps! First, we add or subtract terms from both sides to separate constants and variables to different sides of the equation. Then, we simplify to isolate the variable. Finally, we check our answer by plugging it back into the original equation. Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now! (add course link here) Khan Academy India is a nonprofit organization with the mission of providing a free, world-class education for anyone, anywhere. We have videos and exercises that have been translated into multiple Indian languages, and 15 million people around the globe learn on Khan Academy every month. Support Us: https://india.khanacademy.org/donate Created byhttps://cdn.kastatic.org/ka-youtube-converted/0ftMWNI7UGk.mp4/0ftMWNI7UGk.mp4246yes https://puzzicle.org/math/algebra2/v/example-of-solving-for-a-variablehttps://puzzicle.org/thumbs/2402503.jpgManipulating formulas: perimeterSal rewrites the formula for the perimeter of a rectangle so it is solved for width.https://cdn.kastatic.org/ka-youtube-converted/Aig1hkq3OsU.mp4/Aig1hkq3OsU.mp4418yes https://puzzicle.org/math/algebra2/v/rearrange-formulas-to-isolate-specific-variableshttps://puzzicle.org/thumbs/13179472.jpgManipulating formulas: areaSal rewrites the formula for the area of a triangle so it is solved for height.https://cdn.kastatic.org/ka-youtube-converted/eTSVTTg_QZ4.mp4/eTSVTTg_QZ4.mp4149yes https://puzzicle.org/math/algebra-home/v/absolute-value-equationshttps://puzzicle.org/thumbs/23668.jpgIntro to absolute value equations and graphsTo solve absolute value equations, find x values that make the expression inside the absolute value positive or negative the constant. To graph absolute value functions, plot two lines for the positive and negative cases that meet at the expression's zero. The graph is v-shaped.https://cdn.kastatic.org/ka-youtube-converted/u6zDpUL5RkU.mp4/u6zDpUL5RkU.mp4641yes https://puzzicle.org/math/algebra-1-eureka-squared-aligned/v/solving-absolute-value-equationshttps://puzzicle.org/thumbs/x97d9e2fcd70bce73.jpgSolving absolute value equationsUnderstand why an absolute value equation can have from 0 to 2 solutions. Solve absolute value equations with different numbers of solutions.https://cdn.kastatic.org/ka-youtube-converted/6KrKv4Z-aH4.mp4/6KrKv4Z-aH4.mp4406yes https://puzzicle.org/math/algebra-home/v/absolute-value-equations-1https://puzzicle.org/thumbs/26785.jpgWorked example: absolute value equations with one solutionTo solve absolute value problems with one solution, identify expression in absolute value bars. Recall absolute value is zero only if expression is zero. Make expression equal right-hand side. Use algebra to find x values that satisfy equation. Check solutions by plugging in. Graph solutions on number line. Mark points.https://cdn.kastatic.org/ka-youtube-converted/GwjiR2_7A7Y.mp4/GwjiR2_7A7Y.mp4156yes https://puzzicle.org/math/algebra-home/v/absolute-value-equation-examplehttps://puzzicle.org/thumbs/xe7acd9ae.jpgWorked example: absolute value equation with two solutionsSolving the equation 8|x+7|+4 = -6|x+7|+6 which has two possible solutions.https://cdn.kastatic.org/ka-youtube-converted/UvtWf4TVawE.mp4/UvtWf4TVawE.mp4290yes https://puzzicle.org/math/algebra-home/v/absolute-value-equation-with-no-solutionhttps://puzzicle.org/thumbs/x492a306c.jpgWorked example: absolute value equations with no solutionSolving the equation 4|x+10|+4 = 6|x+10|+10 to find that it has no possible solution.https://cdn.kastatic.org/ka-youtube-converted/15s6B7K9paA.mp4/15s6B7K9paA.mp4230yes https://puzzicle.org/math/algebra-home/v/absolute-value-inequalitieshttps://puzzicle.org/thumbs/4802220.jpgIntro to absolute value inequalitiesTo solve inequalities with absolute values, use a number line to see how far the absolute value is from zero. Split into two cases: when it is positive or negative. Solve each case with algebra. The answer is both cases together, in intervals or words.https://cdn.kastatic.org/ka-youtube-converted/iI_2Piwn_og.mp4/iI_2Piwn_og.mp4793yes https://puzzicle.org/math/algebra-home/v/absolute-value-inequalities-example-1https://puzzicle.org/thumbs/26816.jpgSolving absolute value inequalities 1Sal solves the inequality |h|-19.5 < -12.https://cdn.kastatic.org/ka-youtube-converted/TvUCe6Bomy4.mp4/TvUCe6Bomy4.mp4207yes https://puzzicle.org/math/algebra-home/v/absolute-value-inequalities-example-2https://puzzicle.org/thumbs/13179495.jpgSolving absolute value inequalities 2Sal solves the inequality |p-12|+4 < 14.https://cdn.kastatic.org/ka-youtube-converted/YKQLZOq4HTA.mp4/YKQLZOq4HTA.mp4279yes https://puzzicle.org/math/algebra-home/v/absolute-value-inequalities-example-3https://puzzicle.org/thumbs/26818.jpgSolving absolute value inequalities: no solutionSal solves the inequality |y|+22 ≤ 13.5 to find that it has no solution.https://cdn.kastatic.org/ka-youtube-converted/y9MGpOGQVqQ.mp4/y9MGpOGQVqQ.mp4155yes https://puzzicle.org/math/statistics-probability/v/population-standard-deviationhttps://puzzicle.org/thumbs/x4a3e3b6e.jpgPopulation standard deviationThe population standard deviation is a measure of how much variation there is among individual data points in a population. It's a way of quantifying how spread out the data is from its mean. A small standard deviation means that the data points are generally close to the mean, while a large standard deviation means that the data is more dispersed.https://cdn.kastatic.org/ka-youtube-converted/PWiWkqHmum0.mp4/PWiWkqHmum0.mp4485yes https://puzzicle.org/math/ap-statistics/v/comparing-distributionshttps://puzzicle.org/thumbs/xbb2baca6c0bfe77f.jpgExample: Comparing distributionsCompare distributions using the features of shape, center, spread, and outliers.https://cdn.kastatic.org/ka-youtube-converted/lT3LRjtSbJc.mp4/lT3LRjtSbJc.mp4438yes https://puzzicle.org/math/algebra-1-eureka-squared-aligned/v/two-variable-linear-equations-intro-indian-accenthttps://puzzicle.org/thumbs/xd02f9efa5c5542f2.jpgTwo-variable linear equations introLearn about a class of equations in two variables that's called "linear equations." They are called that way because their graph is a line. These are the most basic and probably most useful equations you will ever know! Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now! Khan Academy India is a nonprofit organization with the mission of providing a free, world-class education for anyone, anywhere. We have videos and exercises that have been translated into multiple Indian languages, and 15 million people around the globe learn on Khan Academy every month. Support Us: https://india.khanacademy.org/donate Created byhttps://cdn.kastatic.org/ka-youtube-converted/HKnvMMPiyZw.mp4/HKnvMMPiyZw.mp4501yes https://puzzicle.org/math/algebra-1-eureka-squared-aligned/v/intro-to-linear-equation-standard-form-indian-accenthttps://puzzicle.org/thumbs/x8867b3d8c81527c5.jpgIntro to linear equation standard formThe standard form for linear equations in two variables is Ax+By=C. For example, 2x+3y=5 is a linear equation in standard form. When an equation is given in this form, it's pretty easy to find both intercepts (x and y). This form is also very useful when solving systems of two linear equations. Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now! Khan Academy India is a nonprofit organization with the mission of providing a free, world-class education for anyone, anywhere. We have videos and exercises that have been translated into multiple Indian languages, and 15 million people around the globe learn on Khan Academy every month. Support Us: https://india.khanacademy.org/donate Created byhttps://cdn.kastatic.org/ka-youtube-converted/7L3tm5a2Pus.mp4/7L3tm5a2Pus.mp4487yes https://puzzicle.org/math/algebra-1-eureka-squared-aligned/v/writing-linear-equations-in-two-variables-given-a-tablehttps://puzzicle.org/thumbs/x150cac52890ce0c7.jpgWriting linear equations in two variables given a tableWhen given a table of values, write the linear equation that represents this data in slope-intercept form.https://cdn.kastatic.org/ka-youtube-converted/v_aEGYeIG9w.mp4/v_aEGYeIG9w.mp4183yes https://puzzicle.org/math/ap-statistics/v/calculating-residual-examplehttps://puzzicle.org/thumbs/x6cf80c68c23af38d.jpgCalculating residual exampleWe look at an example scenario that includes understanding least squares regression, interpreting the regression equation, calculating residuals, and interpreting the significance of positive and negative residuals in relation to the regression line.https://cdn.kastatic.org/ka-youtube-converted/50ezMTE_BuA.mp4/50ezMTE_BuA.mp4292yes https://puzzicle.org/math/ap-statistics/v/residual-plotshttps://puzzicle.org/thumbs/xec01b05aa4e7daa5.jpgResidual plotsCreating and analyzing residual plots based on regression lines.https://cdn.kastatic.org/ka-youtube-converted/VamMrPZ-8fc.mp4/VamMrPZ-8fc.mp4372yes https://puzzicle.org/math/ap-statistics/v/correlation-coefficient-intuition-exampleshttps://puzzicle.org/thumbs/xf4b1ea36.jpgExample: Correlation coefficient intuitionSal explains the intuition behind correlation coefficients and does a problem where he matches correlation coefficients to scatter plots.https://cdn.kastatic.org/ka-youtube-converted/-Y-M9aD_ccQ.mp4/-Y-M9aD_ccQ.mp4441yes https://puzzicle.org/math/ap-statistics/v/calculating-correlation-coefficient-rhttps://puzzicle.org/thumbs/x934981fcf1616c2c.jpgCalculating correlation coefficient rThe most common way to calculate the correlation coefficient (r) is by using technology, but using the formula can help us understand how r measures the direction and strength of the linear association between two quantitative variables.https://cdn.kastatic.org/ka-youtube-converted/u4ugaNo6v1Q.mp4/u4ugaNo6v1Q.mp4742yes https://puzzicle.org/math/statistics-probability/v/video-games-and-violence-bivariate-datahttps://puzzicle.org/thumbs/x7f0764fa.jpgCategorical data exampleWe can explore the relationship between two categorical variables with two-way tables to see if there is an association between the variables. In this example, we see if data from a sample suggests an associate between video games and violence.https://cdn.kastatic.org/ka-youtube-converted/ahyWUV7AwKw.mp4/ahyWUV7AwKw.mp4201yes https://puzzicle.org/math/statistics-probability/v/analyzing-trends-categorical-datahttps://puzzicle.org/thumbs/x49a1fd7f.jpgAnalyzing trends in categorical dataSal solves an example where he is asked to calculate relative frequencies and analyze trends in categorical data.https://cdn.kastatic.org/ka-youtube-converted/TXGhMDbbnzQ.mp4/TXGhMDbbnzQ.mp4534yes https://puzzicle.org/math/algebra2/v/graphical-equations-introhttps://puzzicle.org/thumbs/x90275c239bedf23d.jpgSolving equations by graphing: introSome equations are hard to solve exactly with algebraic tools. We can always solve an equation by graphing it, although the solution might not be exact. This is an example of how to solve a relatively simple equation graphically.https://cdn.kastatic.org/ka-youtube-converted/573yqfOoMwE.mp4/573yqfOoMwE.mp4267yes https://puzzicle.org/math/algebra2/v/graphically-solving-equations-calculatorhttps://puzzicle.org/thumbs/xe6aff4650331a1a7.jpgSolving equations by graphing: graphing calculatorWe can approximate the solutions to any equation by graphing both sides of the equation and looking for intersection points. If we have a graphing calculator, we can get a very good approximation of the solution.https://cdn.kastatic.org/ka-youtube-converted/hlFhHN67ICE.mp4/hlFhHN67ICE.mp4231yes https://puzzicle.org/math/algebra-1-eureka-squared-aligned/v/determining-the-effects-on-fx-x-when-replaced-by-fx-d-or-fx-chttps://puzzicle.org/thumbs/xe706906e88b5258c.jpgDetermining the effects on f(x) = x when replaced by f(x) + d or f(x - c)Determine the effects on f(x) = x when replaced by f(x) + d or f(x - c).https://cdn.kastatic.org/ka-youtube-converted/AxkLjDmD_oU.mp4/AxkLjDmD_oU.mp4207yes https://puzzicle.org/math/algebra-home/v/reflecting-and-compressing-functionshttps://puzzicle.org/thumbs/x92138f98.jpgReflecting & compressing functionsGiven the graphs of functions f and g, where g is the result of reflecting & compressing f by a factor of 3, Sal finds g(x) in terms of f(x).https://cdn.kastatic.org/ka-youtube-converted/Upw1b0EohIM.mp4/Upw1b0EohIM.mp4261yes https://puzzicle.org/math/algebra-1-eureka-squared-aligned/v/determining-the-effects-on-fx-x-when-replaced-by-afx-or-fbxhttps://puzzicle.org/thumbs/xdbd5174755a281cc.jpgDetermining the effects on f(x) = x when replaced by af(x) or f(bx)Determine the effects on f(x) when replaced by af(x) or f(bx).https://cdn.kastatic.org/ka-youtube-converted/nUYbTIAwH8w.mp4/nUYbTIAwH8w.mp4200yes https://puzzicle.org/math/algebra-1-eureka-squared-aligned/v/determining-the-effects-on-fx-x-multiple-transformationshttps://puzzicle.org/thumbs/x81f7ac9cf2f18a4d.jpgDetermining the effects on f(x) = x (multiple transformations)Determine the effects on f(x) = x with multiple transformations.https://cdn.kastatic.org/ka-youtube-converted/c26tyBzYnsw.mp4/c26tyBzYnsw.mp4317yes https://puzzicle.org/math/algebra2/v/line-and-parabola-systemhttps://puzzicle.org/thumbs/x4af415d1c1bb39db.jpgQuadratic systems: a line and a parabolaA system of equations that contains one linear equation and one quadratic equations can be solved both graphically and algebraically. Each method has its pros and cons. See an example using both methods.https://cdn.kastatic.org/ka-youtube-converted/Cy1Pxz_wLfA.mp4/Cy1Pxz_wLfA.mp4372yes https://puzzicle.org/math/algebra2/v/systems-of-nonlinear-equations-3https://puzzicle.org/thumbs/x9e7fad26.jpgQuadratic systems: a line and a circleSal solves the system y=x+1 and x^2+y^2=25.https://cdn.kastatic.org/ka-youtube-converted/swFohliPgmQ.mp4/swFohliPgmQ.mp4254yes https://puzzicle.org/math/algebra-1-eureka-squared-aligned/v/sat-math-p8-easierhttps://puzzicle.org/thumbs/xb049a70f.jpgLinear and quadratic systems — Basic exampleWatch Sal work through a basic Linear and quadratic systems problem.https://cdn.kastatic.org/ka-youtube-converted/c9H60WSptdE.mp4/c9H60WSptdE.mp4290yes https://puzzicle.org/math/algebra-1-eureka-squared-aligned/v/sat-math-p8-harderhttps://puzzicle.org/thumbs/x0c79f02f.jpgLinear and quadratic systems — Harder exampleWatch Sal work through a harder Linear and quadratic systems problem.https://cdn.kastatic.org/ka-youtube-converted/OL024xNFuro.mp4/OL024xNFuro.mp4353yes https://puzzicle.org/math/algebra-1-eureka-squared-aligned/v/writing-a-quadratic-function-to-fit-data-estimate-solutions-and-make-predictionshttps://puzzicle.org/thumbs/x855cb8445c50bcfa.jpgWriting a quadratic function to fit data, estimate solutions, and make predictionsPractice writing quadratic functions to fit data, estimate solutions and make predictions from a table of values.https://cdn.kastatic.org/ka-youtube-converted/99XW8CxKzao.mp4/99XW8CxKzao.mp4449yes https://puzzicle.org/math/statistics-probability/v/comparing-models-to-fit-datahttps://puzzicle.org/thumbs/x69fec2ac.jpgComparing models to fit data exampleSal determines if a quadratic or exponential model fits the data better, then uses the model to make a prediction.https://cdn.kastatic.org/ka-youtube-converted/2EwTHdg-xgw.mp4/2EwTHdg-xgw.mp4245yes https://puzzicle.org/math/algebra-1-eureka-squared-aligned/v/recursive-arithmetic-sequencehttps://puzzicle.org/thumbs/xe8cb4f7c79e03195.jpgRecursive formula for arithmetic sequenceFind the recursive formula for an arithmetic sequence by using the common difference and solving for the designated initial term.https://cdn.kastatic.org/ka-youtube-converted/ddRPUzAXAG4.mp4/ddRPUzAXAG4.mp4189yes https://puzzicle.org/math/algebra2/v/simplifying-exponent-expression-with-divisionhttps://puzzicle.org/thumbs/x578b37c4.jpgRewriting quotient of powers (rational exponents)Sal rewrites the expression m^(7/9) / m^(1/3) as a single exponential term m^(4/9).https://cdn.kastatic.org/ka-youtube-converted/0z-yIFzpunM.mp4/0z-yIFzpunM.mp4151yes https://puzzicle.org/math/algebra2/v/transforming-exponential-graphshttps://puzzicle.org/thumbs/x0540fde8.jpgTransforming exponential graphsGiven the graph of y=2ˣ, Sal graphs y=2⁻ˣ-5, which is a horizontal reflection and shift of y=2ˣ.https://cdn.kastatic.org/ka-youtube-converted/83gM3ufe1Nw.mp4/83gM3ufe1Nw.mp4185yes https://puzzicle.org/math/algebra2/v/transforming-exponential-graphs-2https://puzzicle.org/thumbs/x84ac3a80.jpgTransforming exponential graphs (example 2)Given the graph of y=2ˣ, Sal graphs y=(-1)2ˣ⁺³+4, which is a vertical reflection and a shift of y=2ˣ.https://cdn.kastatic.org/ka-youtube-converted/ja4WRZ4DrAw.mp4/ja4WRZ4DrAw.mp4318yes https://puzzicle.org/math/algebra2/v/graphing-exponential-functions-interactivehttps://puzzicle.org/thumbs/x9ee7b980.jpgGraphing exponential functionsSal graphs y=-2*3ˣ+5 using our interactive graphing widget.https://cdn.kastatic.org/ka-youtube-converted/vlSEpdWf2hQ.mp4/vlSEpdWf2hQ.mp4195yes https://puzzicle.org/math/algebra-1-eureka-squared-aligned/v/identifying-key-features-of-exponential-functionshttps://puzzicle.org/thumbs/x6a44d01fdd585c21.jpgIdentifying key features of exponential functionsWhen given an exponential function, identify key features.https://cdn.kastatic.org/ka-youtube-converted/mPxxlOKqPqQ.mp4/mPxxlOKqPqQ.mp4274yes https://puzzicle.org/math/algebra-1-eureka-squared-aligned/v/determining-and-representing-the-domain-and-range-of-exponential-functionshttps://puzzicle.org/thumbs/xfcbcef1d7de4c58c.jpgDetermining and representing the domain and range of exponential functionsGiven an exponential function of the form f(x) = a*b^x in algebraic or graphed form, determine its domain and range.https://cdn.kastatic.org/ka-youtube-converted/sOLWlpwvfFI.mp4/sOLWlpwvfFI.mp4390yes https://puzzicle.org/math/algebra-1-eureka-squared-aligned/v/finding-simple-interest-for-one-year-comparing-quantities-class-7-india-khan-acahttps://puzzicle.org/thumbs/xc53bdf431c4bc02a.jpgFinding simple interest for one yearLet's learn what the 'interest rate per annum' means and how we can calculate simple interest for a year using the rate of interest.https://cdn.kastatic.org/ka-youtube-converted/qWnEEKqeWL0.mp4/qWnEEKqeWL0.mp4324yes https://puzzicle.org/math/algebra-1-eureka-squared-aligned/v/finding-simple-interest-for-many-years-comparing-quantities-class-7-india-khan-ahttps://puzzicle.org/thumbs/x341e110afa8ac9c5.jpgFinding simple interest for many yearsHave you wondered why we call simple interest 'simple'? Also, let's understand the basics behind the famous formula for calculating simple interest.https://cdn.kastatic.org/ka-youtube-converted/T6wBHgOxUhc.mp4/T6wBHgOxUhc.mp4484yes https://puzzicle.org/math/algebra2/v/interpreting-time-in-exponential-modelshttps://puzzicle.org/thumbs/x720cd5ae.jpgInterpreting time in exponential modelsSal finds the time interval over which a quantity changes by a given factor in various exponential models.https://cdn.kastatic.org/ka-youtube-converted/zFksanIexHI.mp4/zFksanIexHI.mp4403yes https://puzzicle.org/math/algebra2/v/interpreting-change-in-exponential-modelshttps://puzzicle.org/thumbs/x2052fba3.jpgInterpreting change in exponential modelsSal finds the factor by which a quantity changes over a single time unit in various exponential models.https://cdn.kastatic.org/ka-youtube-converted/uoN_Ax3dfQo.mp4/uoN_Ax3dfQo.mp4414yes https://puzzicle.org/math/algebra-1-eureka-squared-aligned/v/writing-exponential-functions-to-fit-data-and-make-predictionshttps://puzzicle.org/thumbs/x3528ab7725c634e5.jpgWriting exponential functions to fit data and make predictionsWhen given data, use exponential functions to predict a value.https://cdn.kastatic.org/ka-youtube-converted/7E9tV6SL5AU.mp4/7E9tV6SL5AU.mp4469yes https://puzzicle.org/math/ap-statistics/v/introduction-to-residuals-and-least-squares-regressionhttps://puzzicle.org/thumbs/x1dd8ed47d6d9272f.jpgIntroduction to residuals and least squares regressionIntroduction to residuals and least squares regressionhttps://cdn.kastatic.org/ka-youtube-converted/yMgFHbjbAW8.mp4/yMgFHbjbAW8.mp4460yes https://puzzicle.org/math/geometry/v/perpendicular-lineshttps://puzzicle.org/thumbs/26811.jpgPerpendicular lines from equationSal determines which pairs out of a few given linear equations are perpendicular.https://cdn.kastatic.org/ka-youtube-converted/0671cRNjeKI.mp4/0671cRNjeKI.mp4209yes https://puzzicle.org/math/geometry/v/perpendicular-lines-2https://puzzicle.org/thumbs/26812.jpgWriting equations of perpendicular linesGiven line A and point P, Sal finds the equation of the line perpendicular to A that passes through P.https://cdn.kastatic.org/ka-youtube-converted/TsEhZRT16LU.mp4/TsEhZRT16LU.mp4181yes https://puzzicle.org/math/geometry/v/find-the-equation-of-a-linehttps://puzzicle.org/thumbs/xbded91b9.jpgWriting equations of perpendicular lines (example 2)Sal finds the equation of a line perpendicular to a line given in slope-intercept form that passes through a specific point.https://cdn.kastatic.org/ka-youtube-converted/8m8hFcFJ-JM.mp4/8m8hFcFJ-JM.mp4105yes https://puzzicle.org/math/geometry/v/proving-that-parallel-lines-have-the-same-slopehttps://puzzicle.org/thumbs/x9e89ebd8.jpgProof: parallel lines have the same slopeSal proves that parallel lines have the same slope using triangle similarity.https://cdn.kastatic.org/ka-youtube-converted/9ZYoIliBXYE.mp4/9ZYoIliBXYE.mp4349yes https://puzzicle.org/math/geometry/v/proof-that-perpendicular-lines-have-negative-reciprocal-slopehttps://puzzicle.org/thumbs/x058b41f0.jpgProof: perpendicular lines have opposite reciprocal slopesTo prove that perpendicular lines have opposite reciprocal slopes, draw the two lines and label key points. By identifying the two right triangles that are formed, one can use the definition of slope to calculate the slopes of the two lines. Finally, using the properties of similar right triangles, show that the two slopes are indeed opposite reciprocals of each other.https://cdn.kastatic.org/ka-youtube-converted/HyThzLRuqXo.mp4/HyThzLRuqXo.mp4535yes https://puzzicle.org/math/ap-statistics/v/bivariate-relationship-linearity-strength-and-directionhttps://puzzicle.org/thumbs/xd2233b61519abb0d.jpgBivariate relationship linearity, strength and directionDescribe a bivariate relationship's linearity, strength, and directionhttps://cdn.kastatic.org/ka-youtube-converted/30LcZqRfPRY.mp4/30LcZqRfPRY.mp4493yes https://puzzicle.org/math/ap-statistics/v/interpreting-slope-of-regression-linehttps://puzzicle.org/thumbs/x092457deb73afcbe.jpgInterpreting slope of regression lineWe look at an example scatter plot showing the relationship between hours studying and test scores. We interpret scatter plots, investigate the relationship between variables, interpret the slope of a trend line, and make predictions based on the trend line's slope and intercept.https://cdn.kastatic.org/ka-youtube-converted/PplggM0KtJ8.mp4/PplggM0KtJ8.mp4178yes https://puzzicle.org/math/ap-statistics/v/interpreting-y-intercept-in-regression-modelhttps://puzzicle.org/thumbs/xe58ed121b4d756b7.jpgInterpreting y-intercept in regression modelInterpreting y-intercept in regression modelhttps://cdn.kastatic.org/ka-youtube-converted/00YVR5TSZqw.mp4/00YVR5TSZqw.mp4156yes https://puzzicle.org/math/probability/v/correlation-and-causalityhttps://puzzicle.org/thumbs/203846247.jpgCorrelation and causalityUnderstanding why correlation does not imply causality (even though many in the press and some researchers often imply otherwise)https://cdn.kastatic.org/ka-youtube-converted/ROpbdO-gRUo.mp4/ROpbdO-gRUo.mp4645yes https://puzzicle.org/math/probability/v/regression-residual-introhttps://puzzicle.org/thumbs/x798e2a79d4b18400.jpgIntroduction to residuals and least-squares regressionIn linear regression, a residual is the difference between the actual value and the value predicted by the model (y-ŷ) for any given point. A least-squares regression model minimizes the sum of the squared residuals.https://cdn.kastatic.org/ka-youtube-converted/VqD-nf1YUks.mp4/VqD-nf1YUks.mp4289yes https://puzzicle.org/math/algebra2/v/avg-rate-of-change-of-polynomialshttps://puzzicle.org/thumbs/xa25f278a0eb3ede9.jpgFinding average rate of change of polynomialsLearn how to calculate the average rate of change of a function over a specific interval. Discover how changes in the function's value relate to changes in x. Use tables and visuals to understand the concept better. This is key to mastering polynomial functions in algebra.https://cdn.kastatic.org/ka-youtube-converted/neZeuTgEoMA.mp4/neZeuTgEoMA.mp4241yes https://puzzicle.org/math/algebra2/v/sign-of-avg-rate-of-change-polynomialhttps://puzzicle.org/thumbs/x8198387faa4f2462.jpgSign of average rate of change of polynomialsDiscover how to find the average rate of change in polynomials. Learn to identify intervals with positive average rates of change by comparing function values at different points. See how this concept applies to real-life situations, making math fun and practical!https://cdn.kastatic.org/ka-youtube-converted/9COoRE8pc5k.mp4/9COoRE8pc5k.mp4336yes https://puzzicle.org/math/algebra2/v/basic-fractional-exponentshttps://puzzicle.org/thumbs/x77aefe04.jpgIntro to rational exponentsWhat does it mean to take a number by a power which is a unit fraction? For example, what is the result of 3 raised to ½?https://cdn.kastatic.org/ka-youtube-converted/lZfXc4nHooo.mp4/lZfXc4nHooo.mp4300yes https://puzzicle.org/math/algebra2/v/rewriting-roots-as-rational-exponentshttps://puzzicle.org/thumbs/x7c8829c5.jpgRewriting roots as rational exponentsSal solves several problems about the equivalence of expressions with roots and rational exponents. For example, rewrite ⁶√(g⁵) as g^⅚.https://cdn.kastatic.org/ka-youtube-converted/gH4IsIEYof0.mp4/gH4IsIEYof0.mp4259yes https://puzzicle.org/math/algebra2/v/solving-for-a-fractional-exponenthttps://puzzicle.org/thumbs/xe5b92d45.jpgExponential equation with rational answerA worked example of rewriting a radical as an exponent. In this example, we solve for the unknown in 3ᵃ = ⁵√(3²).https://cdn.kastatic.org/ka-youtube-converted/6OFwfxmhtE8.mp4/6OFwfxmhtE8.mp4242yes https://puzzicle.org/math/algebra2/v/fractional-exponent-expressions-2https://puzzicle.org/thumbs/26914.jpgRewriting mixed radical and exponential expressionsSal rewrites (r^(2/3)s^3)^2*√(20r^4s^5), once as an exponential expression and once as a radical expression.https://cdn.kastatic.org/ka-youtube-converted/4F6cFLnAAFc.mp4/4F6cFLnAAFc.mp4451yes https://puzzicle.org/math/algebra2/v/fractional-exponents-with-numerators-other-than-1https://puzzicle.org/thumbs/x6f810954.jpgEvaluating fractional exponentsSal shows how to evaluate 64^(2/3) and (8/27)^(-2/3).https://cdn.kastatic.org/ka-youtube-converted/S34NM0Po0eA.mp4/S34NM0Po0eA.mp4354yes https://puzzicle.org/math/algebra2/v/negative-fractional-exponent-exampleshttps://puzzicle.org/thumbs/x22d2b44c.jpgEvaluating fractional exponents: negative unit-fractionHow to evaluate powers that are negative unit fractions, like 9 raised to -½ and 27 raised to -⅓.https://cdn.kastatic.org/ka-youtube-converted/tn53EdOr6Rw.mp4/tn53EdOr6Rw.mp4182yes https://puzzicle.org/math/algebra2/v/negative-fractional-exponent-examples-2https://puzzicle.org/thumbs/x86a0dd64.jpgEvaluating fractional exponents: fractional baseSal shows how to evaluate (25/9)^(1/2) and (81/256)^(-1/4).https://cdn.kastatic.org/ka-youtube-converted/bIFdW0NZ9W4.mp4/bIFdW0NZ9W4.mp4204yes https://puzzicle.org/math/algebra2/v/simplifying-numerical-expression-with-rational-exponentshttps://puzzicle.org/thumbs/xf177d976.jpgEvaluating quotient of fractional exponentsSal simplifies the complicated expression 256^(4/7) / 2^(4/7) until he finds that the expression is equal to 16.https://cdn.kastatic.org/ka-youtube-converted/KGSvigZQKZY.mp4/KGSvigZQKZY.mp4230yes https://puzzicle.org/math/algebra2/v/simplifying-with-exponent-propertieshttps://puzzicle.org/thumbs/xf256dbf5.jpgEvaluating mixed radicals and exponentsA worked example of calculating an expression that has both a radical and an exponent. In this example, we evaluate 6^(1/2)⋅(⁵√6)³.https://cdn.kastatic.org/ka-youtube-converted/Ht-YXje4R2g.mp4/Ht-YXje4R2g.mp4171yes https://puzzicle.org/math/algebra2/v/simplifying-an-exponential-expressionhttps://puzzicle.org/thumbs/xb357b82e.jpgRewriting exponential expressions as A⋅BᵗSal simplifies the expression 10*9^(0.5t+2)*5^(3t) as 810*375^t.https://cdn.kastatic.org/ka-youtube-converted/Y6wNiYcuCoE.mp4/Y6wNiYcuCoE.mp4310yes https://puzzicle.org/math/algebra2/v/rewriting-an-exponential-expression-in-a-hairier-wayhttps://puzzicle.org/thumbs/xc9227a8a.jpgEquivalent forms of exponential expressionsSal rewrites (1/32)*2^t as 32*1024^(t/10-1).https://cdn.kastatic.org/ka-youtube-converted/vSijVSL3ChU.mp4/vSijVSL3ChU.mp4512yes https://puzzicle.org/math/algebra-basics/v/simplifying-square-root-expressions-no-variableshttps://puzzicle.org/thumbs/xc5f2930c.jpgSimplifying square-root expressions: no variablesWhen square roots have the same value inside the radical, we can combine like terms. First we simplify the radical expressions by removing all factors that are perfect squares from inside the radicals. Then we can see whether we can combine terms or not. If there is only one term, there is nothing to combine.https://cdn.kastatic.org/ka-youtube-converted/74iuGIaBgRc.mp4/74iuGIaBgRc.mp4380yes https://puzzicle.org/math/algebra-home/v/adding-and-simplifying-radicalshttps://puzzicle.org/thumbs/127771046.jpgSimplifying radical expressions (addition)A worked example of simplifying an expression that is a sum of several radicals. In this example, we simplify √(2x²)+4√8+3√(2x²)+√8.https://cdn.kastatic.org/ka-youtube-converted/egNq4tSfi1I.mp4/egNq4tSfi1I.mp4281yes https://puzzicle.org/math/algebra2/v/interpreting-change-in-exponential-models-with-manipulationhttps://puzzicle.org/thumbs/x8c4967da.jpgInterpreting change in exponential models: with manipulationSal analyzes the rate of change of various exponential models, where the function that models the situation needs some manipulation.https://cdn.kastatic.org/ka-youtube-converted/q20v5ZBsTk8.mp4/q20v5ZBsTk8.mp4261yes https://puzzicle.org/math/algebra2/v/interpreting-change-in-exponential-models-changing-unitshttps://puzzicle.org/thumbs/x2501cf19.jpgInterpreting change in exponential models: changing unitsSal analyzes the rate of change of various exponential models for different time units by manipulating the functions that model the situations.https://cdn.kastatic.org/ka-youtube-converted/SM6XZj5dPQQ.mp4/SM6XZj5dPQQ.mp4262yes https://puzzicle.org/math/algebra-1-fl-best/v/intro-to-simple-interest-comparing-quantities-class-7-india-math-khan-academyhttps://puzzicle.org/thumbs/x4f1218d729f52396.jpgIntro to simple interestWhat is the meaning of paying interest when we borrow money? What is the meaning of the word "principal" here?https://cdn.kastatic.org/ka-youtube-converted/Rmol9IzSDXk.mp4/Rmol9IzSDXk.mp4166yes https://puzzicle.org/math/algebra-2-fl-best/v/intro-to-compound-interest-class-8-india-math-khan-academyhttps://puzzicle.org/thumbs/xb5756734b955851f.jpgIntro to compound interestLet's understand how compound interest is different from simple interest. Let's also see how compound interest is simply a special case of percentage increase.https://cdn.kastatic.org/ka-youtube-converted/inayVc1w72M.mp4/inayVc1w72M.mp4482yes https://puzzicle.org/math/algebra-2-fl-best/v/solving-compound-interest-problems-class-8-india-math-khan-academyhttps://puzzicle.org/thumbs/xf9ad53d97701deec.jpgSolved example: compound interestIn a previous video, we learned that compound interest is just a special case of percentage increase. Here, let's learn how to solve problems involving compound interest by solving an example question.https://cdn.kastatic.org/ka-youtube-converted/cf-Pab4YR40.mp4/cf-Pab4YR40.mp4889yes https://puzzicle.org/math/algebra2/v/subtracting-polynomials-and-closurehttps://puzzicle.org/thumbs/xaf4f324c.jpgPolynomial subtractionLearn how to subtract polynomials. Discover how distributing a negative sign changes the signs of all terms in a polynomial. Understand that when you subtract polynomials, you still get a polynomial, showing that the set of polynomials is 'closed' under subtraction.https://cdn.kastatic.org/ka-youtube-converted/xgoqcc2-RS4.mp4/xgoqcc2-RS4.mp4170yes https://puzzicle.org/math/algebra2/v/area-model-for-negative-termshttps://puzzicle.org/thumbs/xa5dc6194d8c22038.jpgArea model for multiplying polynomials with negative termsDiscover how to multiply monomials by polynomials using area models. This method works even when dealing with negative terms! By visualizing the process, we can understand why we multiply different terms and how negative areas affect the total area.https://cdn.kastatic.org/ka-youtube-converted/_PC1JMBZKUQ.mp4/_PC1JMBZKUQ.mp4317yes https://puzzicle.org/math/algebra2/v/multiplying-polynomials-using-area-modelhttps://puzzicle.org/thumbs/xe407348a.jpgMultiplying binomials by polynomials: area modelDiscover the magic of multiplying binomials by polynomials using an area model! This method transforms complex algebra into simple rectangles, making it easier to understand. By breaking down the big rectangle into smaller ones, we can find the area and thus the product of the polynomials.https://cdn.kastatic.org/ka-youtube-converted/C9m5VEEyno8.mp4/C9m5VEEyno8.mp4390yes https://puzzicle.org/math/algebra2/v/more-multiplying-polynomialshttps://puzzicle.org/thumbs/13179530.jpgMultiplying binomials by polynomialsLearn how to multiply binomials by polynomials with ease! This lesson breaks down the process into simple steps, using the distributive property to multiply each term. You'll master combining like terms to simplify expressions, turning complex polynomials into manageable math.https://cdn.kastatic.org/ka-youtube-converted/D6mivA_8L8U.mp4/D6mivA_8L8U.mp4154yes https://puzzicle.org/math/algebra2/v/polynomial-division-introhttps://puzzicle.org/thumbs/x45f51d0760a90a0e.jpgPolynomial division introductionWhen we divide the polynomial p(x) by q(x) we are basically asking "what should we multiply by q(x) to get p(x)?" If this sounds familiar, it's because it's very similar to dividing numbers! In this introduction we see how some quotients end up as a polynomial, while other times we have a remainder and cannot express the quotient as a polynomial. This is very similar to quotients of integers!https://cdn.kastatic.org/ka-youtube-converted/3eowXt_LNbg.mp4/3eowXt_LNbg.mp4327yes https://puzzicle.org/math/algebra2/v/dividing-polynomials-by-xhttps://puzzicle.org/thumbs/xfdec982c2246bbeb.jpgDividing polynomials by x (no remainders)Discover how to divide polynomials by 'x' in two different ways. Finding the quotient (x⁴-2x³+5x)/x is the same as asking "what should we multiply by x to get x⁴-2x³+5x?" First, we distribute '1/x' to each term in the polynomial. Second, we factor out an 'x' from each term. Both methods simplify complex expressions,https://cdn.kastatic.org/ka-youtube-converted/HUdA6LdCRk8.mp4/HUdA6LdCRk8.mp4269yes https://puzzicle.org/math/algebra2/v/polynomial-divided-by-monomialhttps://puzzicle.org/thumbs/13179523.jpgDivide polynomials by x (with remainders)Learn how to simplify complex expressions by dividing polynomials by 'x'. Discover how to break down the numerator, distribute the division, and use exponent properties. Master the art of simplifying fractions and handling negative exponents. Sal demonstrates by dividing (18x^4-3x^2+6x-4) by 6x.https://cdn.kastatic.org/ka-youtube-converted/UquFdMg6Z_U.mp4/UquFdMg6Z_U.mp4285yes https://puzzicle.org/math/ap-statistics/v/identifying-individuals-variables-and-categorical-variables-in-a-data-sethttps://puzzicle.org/thumbs/x6b023a43115966cd.jpgIdentifying individuals, variables and categorical variables in a data setThe concept of variables in data sets comes to life through an exploration of categorical and quantitative variables. Using nutritional data from a coffee shop as an example, the lesson highlights how variables can represent diverse aspects of a data set, such as drink type, calorie count, sugar content, and caffeine amount.https://cdn.kastatic.org/ka-youtube-converted/EqeVXI4WNHM.mp4/EqeVXI4WNHM.mp4160yes https://puzzicle.org/math/ap-statistics/v/median-histogramhttps://puzzicle.org/thumbs/xa3c51d8e2fcf899f.jpgMedian in a histogramWorked example from Khan Academy finding the interval in a histogram containing the media of a data set.https://cdn.kastatic.org/ka-youtube-converted/xTwDmnEEb9E.mp4/xTwDmnEEb9E.mp4148yes https://puzzicle.org/math/ap-statistics/v/estimating-mean-median-displayshttps://puzzicle.org/thumbs/xec8235ec63b789f8.jpgEstimating mean and median in data displaysEstimating mean and median in data displayshttps://cdn.kastatic.org/ka-youtube-converted/rb-T5ppxJk0.mp4/rb-T5ppxJk0.mp4275yes https://puzzicle.org/math/ap-statistics/v/example-describing-a-distributionhttps://puzzicle.org/thumbs/xf2e81b25614f71f5.jpgExample: Describing a distributionLearn how to describe a distribution of quantitative data by discussing its shape, center, spread, and potential outliers.https://cdn.kastatic.org/ka-youtube-converted/XLNAkLuokzY.mp4/XLNAkLuokzY.mp4341yes https://puzzicle.org/math/ap-statistics/v/types-studies-aphttps://puzzicle.org/thumbs/xeaaf31ee356c9c79.jpgTypes of studiesWe can distinguish between experiments and observational studies, and further describe types of observational studies: retrospective, prospective, and sample surveys.https://cdn.kastatic.org/ka-youtube-converted/zmXBbHDUgY4.mp4/zmXBbHDUgY4.mp4319yes https://puzzicle.org/math/ap-statistics/v/worked-example-identifying-observational-studyhttps://puzzicle.org/thumbs/xa69c3c6076363b1a.jpgWorked example identifying observational studyWorked example identifying observational studyhttps://cdn.kastatic.org/ka-youtube-converted/tHFyaoIrV14.mp4/tHFyaoIrV14.mp4404yes https://puzzicle.org/math/probability/v/invalid-conclusions-studies-examplehttps://puzzicle.org/thumbs/x88df25b9f097e577.jpgInvalid conclusions from studies exampleExample understanding when a conclusion about causality can be made from a study. Discussion of experimental and observational studies.https://cdn.kastatic.org/ka-youtube-converted/xSvvjdZjgl8.mp4/xSvvjdZjgl8.mp4219yes https://puzzicle.org/math/probability/v/causality-from-studyhttps://puzzicle.org/thumbs/x93ca171b7d55cf2c.jpgCan causality be established from this study?Thinking about whether a causal relationship can be established or if there are confounding variables.https://cdn.kastatic.org/ka-youtube-converted/yWva9DtV0Rk.mp4/yWva9DtV0Rk.mp4214yes https://puzzicle.org/math/ap-statistics/v/marginal-distribution-and-conditional-distributionhttps://puzzicle.org/thumbs/x4e33fc0d5833c2a8.jpgMarginal and conditional distributionsWe investigate distributions using a two-way table and then explain the concept of marginal distribution, both in counts and percentages, to understand the distribution of each variable individually. Finally, we cover conditional distribution, where we look at the relationship between variables and understand how one variable impacts the distribution of another.https://cdn.kastatic.org/ka-youtube-converted/Iw9fEYIpPMA.mp4/Iw9fEYIpPMA.mp4317yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/numerical-and-categorical-datahttps://puzzicle.org/thumbs/xe16ed34fb60b2f1f.jpgNumerical and categorical dataLearn how to tell whether data is numerical or categorical by looking at the types of answers a statistical question produces.https://cdn.kastatic.org/ka-youtube-converted/EYMfpriHkcg.mp4/EYMfpriHkcg.mp4136yes https://puzzicle.org/math/ap-statistics/v/classifying-distributionshttps://puzzicle.org/thumbs/xccf3bee87725bcf8.jpgClassifying shapes of distributionsWhen we describe shapes of distributions, we commonly use words like symmetric, left-skewed, right-skewed, bimodal, and uniform. Not every distribution fits one of these descriptions, but they are still a useful way to summarize the overall shape of many distributions.https://cdn.kastatic.org/ka-youtube-converted/Y53_8WRrPzg.mp4/Y53_8WRrPzg.mp4268yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/using-iqr-and-mad-to-compare-variabilityhttps://puzzicle.org/thumbs/x25ab550a46da6fad.jpgUsing IQR and MAD to compare variabilityCompare two data sets with nearly the same measure of center and use IQR and MAD to determine which has greater variability.https://cdn.kastatic.org/ka-youtube-converted/6_Pmwrx_kg0.mp4/6_Pmwrx_kg0.mp4248yes https://puzzicle.org/math/statistics-probability/v/range-variance-and-standard-deviation-as-measures-of-dispersionhttps://puzzicle.org/thumbs/23764.jpgMeasures of spread: range, variance & standard deviationRange, variance, and standard deviation all measure the spread or variability of a data set in different ways. The range is easy to calculate—it's the difference between the largest and smallest data points in a set. Standard deviation is the square root of the variance. Standard deviation is a measure of how spread out the data is from its mean.https://cdn.kastatic.org/ka-youtube-converted/E4HAYd0QnRc.mp4/E4HAYd0QnRc.mp4754yes https://puzzicle.org/math/statistics-probability/v/variance-of-a-populationhttps://puzzicle.org/thumbs/1162004491.jpgVariance of a populationPopulation variance is a measure of how spread out a group of data points is. Specifically, it quantifies the average squared deviation from the mean. So, if all data points are very close to the mean, the variance will be small; if data points are spread out over a wide range, the variance will be larger.https://cdn.kastatic.org/ka-youtube-converted/dvoHB9djouc.mp4/dvoHB9djouc.mp4485yes https://puzzicle.org/math/statistics-probability/v/how-parameters-change-as-data-is-shifted-and-scaledhttps://puzzicle.org/thumbs/x025cb6ad95d9c9b9.jpgHow parameters change as data is shifted and scaledSee how transforming a data set by adding, subtracting, multiplying, or dividing a constant affects measures of center and spread.https://cdn.kastatic.org/ka-youtube-converted/JFlI4Vxtzxo.mp4/JFlI4Vxtzxo.mp4438yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/standard-deviation-in-contexthttps://puzzicle.org/thumbs/xc0f44e2b009a2e71.jpgStandard deviation in contextInterpret the standard deviation in the context of a problem.https://cdn.kastatic.org/ka-youtube-converted/nNhzI4GdcKg.mp4/nNhzI4GdcKg.mp4154yes https://puzzicle.org/math/ap-statistics/v/judging-outliers-in-a-datasethttps://puzzicle.org/thumbs/xc73e4b612b2ff150.jpgJudging outliers in a datasetUsing the inter-quartile range (IQR) to judge outliers in a dataset.https://cdn.kastatic.org/ka-youtube-converted/FRlTh5HQORA.mp4/FRlTh5HQORA.mp4501yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/investigating-outliershttps://puzzicle.org/thumbs/xa4491f7c15cda491.jpgInvestigating outliersDetermine when values are considered outliers, investigate their source, and determine if they should be excluded from the data.https://cdn.kastatic.org/ka-youtube-converted/BT5x_BEib-4.mp4/BT5x_BEib-4.mp4172yes https://puzzicle.org/math/ap-statistics/v/mean-and-standard-deviation-versus-median-and-iqrhttps://puzzicle.org/thumbs/xe556bac7306cc461.jpgMean and standard deviation versus median and IQRLearn to choose the "preferred" measures of center and spread when outliers are present in a set of data.https://cdn.kastatic.org/ka-youtube-converted/qNKOi08NxHs.mp4/qNKOi08NxHs.mp4479yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/graphs-of-linear-equationshttps://puzzicle.org/thumbs/23634.jpgGraphing a linear equation: y=2x+7Learn how to create a graph of the linear equation y = 2x + 7.https://cdn.kastatic.org/ka-youtube-converted/rgvysb9emcQ.mp4/rgvysb9emcQ.mp4790yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/interpreting-linear-equations-in-different-formshttps://puzzicle.org/thumbs/x3b2e1939604a1ab1.jpgInterpreting linear equations in different formsExplore what standard form and slope-intercept form each tell us about a linear relationship and its graph.https://cdn.kastatic.org/ka-youtube-converted/-cRbQpUmj6Q.mp4/-cRbQpUmj6Q.mp4301yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/using-technology-to-graph-and-evaluate-functionshttps://puzzicle.org/thumbs/x0723159c0026881f.jpgUsing technology to graph and evaluate functionsUse technology to graph and evaluate functions given in function notation.https://cdn.kastatic.org/ka-youtube-converted/lMiWwRukSKY.mp4/lMiWwRukSKY.mp4176yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/comparing-graphshttps://puzzicle.org/thumbs/xd3e962be04eb2402.jpgComparing graphsAnalyze and interpret functions, using graphs, equations (such as f(x)=g(x)), and function notation, to compare their key features and understand what their relationships mean in context.https://cdn.kastatic.org/ka-youtube-converted/t-LuEry9P3o.mp4/t-LuEry9P3o.mp4176yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/exploring-key-features-of-functionshttps://puzzicle.org/thumbs/x00f13c460a3f890d.jpgExploring key features of functionsInterpret key features of a graph.https://cdn.kastatic.org/ka-youtube-converted/PYJdiguNWKQ.mp4/PYJdiguNWKQ.mp4222yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/interpreting-a-piecewise-function-graphhttps://puzzicle.org/thumbs/xd37f8ac55ee5c7d8.jpgInterpreting a piecewise function graphLearn how to read a piecewise function graph by interpreting movie ticket prices for different age ranges, including what the open and closed circles mean at each boundary.https://cdn.kastatic.org/ka-youtube-converted/G1SH5MJvfoM.mp4/G1SH5MJvfoM.mp4128yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/comparing-guesses-and-absolute-errorshttps://puzzicle.org/thumbs/x6669803dc09ebb97.jpgComparing guesses and absolute errorsAnalyze and interpret a scatter plot of numerical guesses versus their absolute errors, calculating the errors from a target value and generalizing the relationship in words and/or equations.https://cdn.kastatic.org/ka-youtube-converted/Y3oJbKTnVXQ.mp4/Y3oJbKTnVXQ.mp4280yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/finding-and-interpreting-inverse-functionshttps://puzzicle.org/thumbs/x3e530684eb6dc77a.jpgFinding and interpreting inverse functionsFind and interpret an inverse function in terms of situations.https://cdn.kastatic.org/ka-youtube-converted/fvJZWSrdSOM.mp4/fvJZWSrdSOM.mp4178yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/comparing-linear-and-exponential-relationshipshttps://puzzicle.org/thumbs/x2a9416c840661655.jpgComparing linear and exponential relationshipsAnalyze and compare linear and exponential relationships using numerical calculations and by interpreting graphs that illustrate their respective growth patterns.https://cdn.kastatic.org/ka-youtube-converted/exCgXqE6ZF0.mp4/exCgXqE6ZF0.mp4212yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/modeling-car-value-over-time-with-exponential-decayhttps://puzzicle.org/thumbs/x1ef7795a0dac7d4b.jpgModeling car value over time with exponential decayInterpret an exponential decay graph that models the value of a car over time, including what the graph suggests about the car’s value in the long run and what the y-intercept represents.https://cdn.kastatic.org/ka-youtube-converted/no0tMoIHLwg.mp4/no0tMoIHLwg.mp4152yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/exploring-exponential-situations-as-functionshttps://puzzicle.org/thumbs/x2d16e879f950a45e.jpgExploring exponential situations as functionsDecide if a relationship is a function by looking at descriptions, tables, equations, or graphs, and use function notation to write equations for exponential relationships.https://cdn.kastatic.org/ka-youtube-converted/YhEBIHPG9cM.mp4/YhEBIHPG9cM.mp496yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/interpreting-exponential-functionshttps://puzzicle.org/thumbs/x5ce936490527bc26.jpgInterpreting exponential functionsAnalyze exponential functions by interpreting and describing exponential relationships using function notation and using graphing technology to explore domains and behaviors of exponential functions.https://cdn.kastatic.org/ka-youtube-converted/8XicDs22wlw.mp4/8XicDs22wlw.mp4206yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/calculating-and-interpreting-rates-of-change-for-exponential-functionshttps://puzzicle.org/thumbs/x0e03a55d2202170c.jpgCalculating and interpreting rates of change for exponential functionsCalculate the average rate of change over intervals to explain how it reflects the behavior of exponential functions and to understand how exponential growth differs from the constant rate of change in linear functions.https://cdn.kastatic.org/ka-youtube-converted/FQtcvJiVmWQ.mp4/FQtcvJiVmWQ.mp4268yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/comparing-exponential-functions-with-the-same-growth-factorhttps://puzzicle.org/thumbs/xfd210b64fd74cdc4.jpgComparing exponential functions with the same growth factorUse equations and graphs to compare exponential functions and explore how changing the parameters a and b in an exponential function affects its graph.https://cdn.kastatic.org/ka-youtube-converted/3-BIGCWr8Ow.mp4/3-BIGCWr8Ow.mp4119yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/comparing-exponential-functions-with-the-same-initial-valuehttps://puzzicle.org/thumbs/x8e1f7b837fa993ce.jpgComparing exponential functions with the same initial valueUse equations and graphs to compare exponential functions and explore how changing the parameters a and b in an exponential function affects its graph.https://cdn.kastatic.org/ka-youtube-converted/6UwuYpY9IxY.mp4/6UwuYpY9IxY.mp4121yes https://puzzicle.org/math/algebra2/v/graphically-solving-equations-word-problemshttps://puzzicle.org/thumbs/x0e2500ebf3a509df.jpgSolving equations by graphing: word problemsWe can approximate the solutions of any equation by graphing both sides of the equation and looking for intersection point. See how we apply this idea to solve some word problems.https://cdn.kastatic.org/ka-youtube-converted/wlhuoE56t3c.mp4/wlhuoE56t3c.mp4510yes https://puzzicle.org/math/algebra-home/v/word-problem-solving-exponential-growth-and-decayhttps://puzzicle.org/thumbs/23700.jpgExponential growth & decay word problemsHow do you solve word problems involving exponential growth and decay? In this video, you will learn how to use a table and a formula to find the percentage of a radioactive substance that remains after a certain time. You will also see how a common ratio, which is the factor by which the quantity changes every time period, determines the rate of change. You will use a calculator to apply the formula and get the answers.https://cdn.kastatic.org/ka-youtube-converted/m5Tf6vgoJtQ.mp4/m5Tf6vgoJtQ.mp4441yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/exploring-different-compounding-intervalshttps://puzzicle.org/thumbs/xda3b77e9fa084729.jpgExploring different compounding intervalsCalculate and compare the effects of different interest rates and compounding intervals to determine the better investment option.https://cdn.kastatic.org/ka-youtube-converted/VordmbxdG3A.mp4/VordmbxdG3A.mp4479yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/exploring-changes-over-equal-intervalshttps://puzzicle.org/thumbs/xf78b0afe30e02c95.jpgExploring changes over equal intervalsCompare how functions change over equal input intervals to determine whether a situation is linear or exponential. Linear functions change by equal differences over equal intervals, while exponential functions change by equal factors.https://cdn.kastatic.org/ka-youtube-converted/JcN8Dl1Y1tg.mp4/JcN8Dl1Y1tg.mp4255yes https://puzzicle.org/math/algebra-1-illustrative-math/v/analyzing-data-to-model-functionshttps://puzzicle.org/thumbs/x95d791bff4888b71.jpgAnalyzing data to model functionsSal demonstrates the process of matching a table of values to a specific function type: linear, quadratic, or exponential.https://cdn.kastatic.org/ka-youtube-converted/v7gmPAARnSw.mp4/v7gmPAARnSw.mp4362yes https://puzzicle.org/math/algebra-1-illustrative-math/v/building-quadratic-functions-from-geometric-patternshttps://puzzicle.org/thumbs/x7cc9cdb35759333d.jpgBuilding quadratic functions from geometric patternsRepresent growing patterns of squares using expressions and functions. Compare different expressions that represent the same pattern and match quadratic rules to the patterns they create.https://cdn.kastatic.org/ka-youtube-converted/DrrYX7VhGLE.mp4/DrrYX7VhGLE.mp486yes https://puzzicle.org/math/algebra-home/v/comparing-exponentials-quadraticshttps://puzzicle.org/thumbs/x51b990d2.jpgComparing growth of exponential & quadratic modelsSal discusses two functions that model the shipment rate of cars. One function is quadratic and the other is exponential. Which one will eventually exceed the other?https://cdn.kastatic.org/ka-youtube-converted/EuwKjpUpkhs.mp4/EuwKjpUpkhs.mp4420yes https://puzzicle.org/math/algebra-1-illustrative-math/v/comparing-quadratic-and-exponential-functionshttps://puzzicle.org/thumbs/x197842e5ebe3db0b.jpgComparing quadratic and exponential functionsCompare quadratic and exponential functions using tables, graphs, calculations, and growth factors to understand how each type of function grows. Identify when an exponential becomes greater and explain why exponential growth eventually surpasses quadratic growth.https://cdn.kastatic.org/ka-youtube-converted/RDkH6qG-E6U.mp4/RDkH6qG-E6U.mp4310yes https://puzzicle.org/math/algebra-1-illustrative-math/v/modeling-free-fall-with-quadratic-functionshttps://puzzicle.org/thumbs/xa4d18f6766eb393f.jpgModeling free fall with quadratic functionsUse equations, tables, and graphs to model free-fall situations. Interpret and write quadratic functions that describe height over time.https://cdn.kastatic.org/ka-youtube-converted/kPzBt5gUFuY.mp4/kPzBt5gUFuY.mp4155yes https://puzzicle.org/math/algebra-home/v/domain-and-range-of-a-function-given-a-formulahttps://puzzicle.org/thumbs/364981333.jpgDomain and range of quadratic functionsSal finds the domain and the range of f(x)=3x^2+6x-2.https://cdn.kastatic.org/ka-youtube-converted/za0QJRZ-yQ4.mp4/za0QJRZ-yQ4.mp4487yes https://puzzicle.org/math/algebra-1-illustrative-math/v/choosing-an-appropriate-domain-for-a-quadratic-functionhttps://puzzicle.org/thumbs/xe4e43c7908c4f531.jpgChoosing an appropriate domain for a quadratic functionChoose an appropriate domain for a quadratic function that represents a context, and explain why it was selected.https://cdn.kastatic.org/ka-youtube-converted/aEBI6aO9-1E.mp4/aEBI6aO9-1E.mp4240yes https://puzzicle.org/math/algebra-1-illustrative-math/v/interpreting-graphs-of-quadratic-functionshttps://puzzicle.org/thumbs/x52c2f1eaf0167721.jpgInterpreting graphs of quadratic functionsConnect how a quadratic expression in factored form corresponds to the x- and y-intercepts of its graph.https://cdn.kastatic.org/ka-youtube-converted/t_pzO1ydVbo.mp4/t_pzO1ydVbo.mp4136yes https://puzzicle.org/math/algebra-1-illustrative-math/v/graphing-quadratics-in-factored-form-when-a-1https://puzzicle.org/thumbs/x1f6bbfe9480d8c91.jpgGraphing quadratics in factored form when a=1Graph quadratic functions written in factored form, f(x) = a(x-p)(x-q), when a = 1, and identify the vertex and y-intercept.https://cdn.kastatic.org/ka-youtube-converted/Mk_hLTGabTQ.mp4/Mk_hLTGabTQ.mp4241yes https://puzzicle.org/math/algebra-1-illustrative-math/v/sat-math-p10-easierhttps://puzzicle.org/thumbs/x2708bc6f.jpgInterpreting nonlinear expressions — Basic exampleWatch Sal work through a basic Interpreting nonlinear expressions problem.https://cdn.kastatic.org/ka-youtube-converted/YfYiS2Bfx20.mp4/YfYiS2Bfx20.mp4127yes https://puzzicle.org/math/algebra-1-illustrative-math/v/quadratic-equation-from-geometric-contexthttps://puzzicle.org/thumbs/xafa3561c1c8ad6b8.jpgQuadratic equation from geometric contextWrite a quadratic equation from a situation and interpret what a solution means in context.https://cdn.kastatic.org/ka-youtube-converted/au5T5cznr5k.mp4/au5T5cznr5k.mp4188yes https://puzzicle.org/math/algebra-1-illustrative-math/v/determining-the-number-of-solutions-of-a-quadratic-graphicallyhttps://puzzicle.org/thumbs/x617210543009ae91.jpgDetermining the number of solutions of a quadratic graphicallyUse graphs to determine whether quadratic equations have two solutions, one solution, or no real solutions.https://cdn.kastatic.org/ka-youtube-converted/EQwrOE86YPw.mp4/EQwrOE86YPw.mp4195yes https://puzzicle.org/math/algebra-1-illustrative-math/v/why-you-cant-divide-by-a-variable-when-solving-quadraticshttps://puzzicle.org/thumbs/x4c421a9807140240.jpgWhy you can't divide by a variable when solving quadraticsLearn why dividing both sides of a quadratic equation by a variable can eliminate solutions, and see how factoring reveals all solutions instead.https://cdn.kastatic.org/ka-youtube-converted/PL9X7Ww8aFU.mp4/PL9X7Ww8aFU.mp4223yes https://puzzicle.org/math/algebra-1-illustrative-math/v/solving-by-completing-the-square-rational-solutionshttps://puzzicle.org/thumbs/xc9a4c925512f4e2a.jpgSolving by completing the square: rational solutionsExpress a quadratic equation in the form (x+p)^2=q, and use completing the square to solve quadratic equations in which the squared term has a coefficient of 1.https://cdn.kastatic.org/ka-youtube-converted/wupekgmKRbA.mp4/wupekgmKRbA.mp4283yes https://puzzicle.org/math/algebra-1-illustrative-math/v/exact-vs-approximate-solutions-to-quadratic-equationshttps://puzzicle.org/thumbs/x056ab8790eaecae5.jpgExact vs. approximate solutions to quadratic equationsSolve a quadratic equation by completing the square and by graphing, then compare the exact and approximate solutions.https://cdn.kastatic.org/ka-youtube-converted/qsk_5WCPEy8.mp4/qsk_5WCPEy8.mp4336yes https://puzzicle.org/math/algebra-1-illustrative-math/v/rewriting-a-quadratic-function-from-vertex-form-to-standard-formhttps://puzzicle.org/thumbs/xdb2a02453900a4cc.jpgRewriting a quadratic function from vertex form to standard formGiven a quadratic function in vertex form, rewrite it in standard form.https://cdn.kastatic.org/ka-youtube-converted/NbNfSIG65Qc.mp4/NbNfSIG65Qc.mp4170yes https://puzzicle.org/math/algebra2/v/terms-coefficients-and-exponents-in-a-polynomialhttps://puzzicle.org/thumbs/13179510.jpgThe parts of polynomial expressionsDiscover the magic of polynomials! Learn to identify terms, coefficients, and exponents in a polynomial. Understand that terms are the parts being added, coefficients are the numbers multiplying the powers of x, and exponents are the powers to which x is raised. Dive into the world of polynomials and make math fun!https://cdn.kastatic.org/ka-youtube-converted/REiDXCN0lGU.mp4/REiDXCN0lGU.mp4156yes https://puzzicle.org/math/algebra2/v/interpreting-expressions-with-multiple-variables-resistorshttps://puzzicle.org/thumbs/xa067e8a28a42697f.jpgInterpreting expressions with multiple variables: ResistorsGiven the expression for the total resistance in an electric circuit with two resistors, analyze the expression to tell what happens to the total resistance when the resistance of one resistor increases and the resistance of the other resistor stays the same.https://cdn.kastatic.org/ka-youtube-converted/8ffMjFXiW9A.mp4/8ffMjFXiW9A.mp4207yes https://puzzicle.org/math/algebra2/v/interpreting-expressions-with-multiple-variables-cylinderhttps://puzzicle.org/thumbs/x1bc6c490b5f36156.jpgInterpreting expressions with multiple variables: CylinderGiven the vale and the expression for the radius of a cylinder, find the radius of a cylinder with the same volume and 100 times the height. This involves analyzing the expression for the radius to see how changing the height affects the radius.https://cdn.kastatic.org/ka-youtube-converted/k00KEcYijo8.mp4/k00KEcYijo8.mp4215yes https://puzzicle.org/math/algebra-1-tx/v/adding-and-subtracting-polynomials-of-degree-onehttps://puzzicle.org/thumbs/x3a2560f8a6da50d7.jpgAdding and subtracting polynomials of degree oneExpand previous knowledge of combining like terms to this new context with more than one variable.https://cdn.kastatic.org/ka-youtube-converted/4MRaGUNKw8U.mp4/4MRaGUNKw8U.mp4293yes https://puzzicle.org/math/algebra-1-tx/v/dividing-polynomials-of-degree-onehttps://puzzicle.org/thumbs/x953db4fde352a6b6.jpgDividing polynomials of degree oneExpand previous knowledge of division to this new context involving more than one term. With and without remainders are in the exercise, so the video should cover remainders.https://cdn.kastatic.org/ka-youtube-converted/LtCd09uHFnc.mp4/LtCd09uHFnc.mp4407yes https://puzzicle.org/math/algebra-1-tx/v/domain-and-range-of-lines-segments-and-rayshttps://puzzicle.org/thumbs/xefb4ca8a30451db2.jpgDomain and range of lines, segments, and raysCompare and contrast domain and range given graphs of line segment and line.https://cdn.kastatic.org/ka-youtube-converted/wd19aPWRRKA.mp4/wd19aPWRRKA.mp4267yes https://puzzicle.org/math/algebra-1-tx/v/domain-and-range-from-graphs-of-quadratic-and-exponential-functionshttps://puzzicle.org/thumbs/xe722bfcbf46256ce.jpgDomain and range from graphs of quadratic and exponential functionsCompare and contrast domain and range given graphs of partial quadratic and exponential functions, where the ends of the shown graph vary among points, and arrows.https://cdn.kastatic.org/ka-youtube-converted/jArkTYaZuek.mp4/jArkTYaZuek.mp4207yes https://puzzicle.org/math/algebra-home/v/direct-and-inverse-variationhttps://puzzicle.org/thumbs/62754976.jpgIntro to direct & inverse variationSal explains what it means for quantities to vary directly or inversely, and gives many examples of both types of variation.https://cdn.kastatic.org/ka-youtube-converted/92U67CUy9Gc.mp4/92U67CUy9Gc.mp4594yes https://puzzicle.org/math/algebra-home/v/proportionality-constant-for-direct-variationhttps://puzzicle.org/thumbs/124291560.jpgProportionality constant for direct variationWorked example: y is directly proportional to x, and y=30 when x=6. Find the value of x when y=45.https://cdn.kastatic.org/ka-youtube-converted/c6-FJRda_Vc.mp4/c6-FJRda_Vc.mp4104yes https://puzzicle.org/math/algebra-home/v/direct-variation-1https://puzzicle.org/thumbs/26794.jpgDirect variation word problem: filling gasWorked example: Model a context about filling gas with a direct variation equation.https://cdn.kastatic.org/ka-youtube-converted/rSadG6EtJmY.mp4/rSadG6EtJmY.mp4219yes https://puzzicle.org/math/algebra-1-tx/v/writing-equations-of-parallel-lineshttps://puzzicle.org/thumbs/x431a7475d4b78eb4.jpgWriting equations of parallel linesWrite the equation of a line that passes through a given point and is parallel to a given line.https://cdn.kastatic.org/ka-youtube-converted/i3Jsfn8dd80.mp4/i3Jsfn8dd80.mp4278yes https://puzzicle.org/math/probability/v/statistical-significance-experimenthttps://puzzicle.org/thumbs/xfcac80a7.jpgStatistical significance of experimentSal determines if the results of an experiment about advertising are statistically significant.https://cdn.kastatic.org/ka-youtube-converted/jLFeqQxGtOc.mp4/jLFeqQxGtOc.mp4484yes https://puzzicle.org/math/algebra-1-tx/v/writing-a-system-of-linear-equations-from-tableshttps://puzzicle.org/thumbs/x9fac2e33fa41b85f.jpgWriting a system of linear equations from tablesWrite a system of two linear equations from tables.https://cdn.kastatic.org/ka-youtube-converted/B86zMf0u864.mp4/B86zMf0u864.mp4329yes https://puzzicle.org/math/algebra-1-tx/v/writing-a-system-of-linear-equations-from-a-graphhttps://puzzicle.org/thumbs/x6af67034e0e70150.jpgWriting a system of linear equations from a graphWrite a system of two linear equations from a graph.https://cdn.kastatic.org/ka-youtube-converted/ODYSynkDNnQ.mp4/ODYSynkDNnQ.mp4261yes https://puzzicle.org/math/algebra-1-tx/v/estimating-graphically-with-systems-in-real-world-problemshttps://puzzicle.org/thumbs/x0ee2551fa79974fd.jpgEstimating graphically with systems in real world problemsEstimate the solution to a real-world system of linear equations by graphing both equations, identifying the intersection point, and comparing costs using the graph.https://cdn.kastatic.org/ka-youtube-converted/9p7sVDEof5Q.mp4/9p7sVDEof5Q.mp4190yes https://puzzicle.org/math/algebra2/v/polynomial-divisionhttps://puzzicle.org/thumbs/23780.jpgIntro to long division of polynomialsLearn how to divide polynomials, also known as algebraic long division. This video starts with simple examples and gradually moves to more complex ones, demonstrating how to divide quadratics by linear factors. The process involves looking at the highest degree terms, dividing, and subtracting to simplify expressions.https://cdn.kastatic.org/ka-youtube-converted/FXgV9ySNusc.mp4/FXgV9ySNusc.mp4729yes https://puzzicle.org/math/algebra2/v/quad-div-by-linear-no-remainderhttps://puzzicle.org/thumbs/xdfd8519b61001565.jpgDividing quadratics by linear expressions (no remainders)We can divide polynomials similarly to how we divide integers. For example, when we divide (x²+7x+10) by (x+2), we are asking "what can we multiply by (x+2) to get (x²+7x+10)?" We can answer this question in many ways. One is with using factorization, and another one is long division.https://cdn.kastatic.org/ka-youtube-converted/dRvLVdQ8cw4.mp4/dRvLVdQ8cw4.mp4244yes https://puzzicle.org/math/algebra2/v/quad-div-by-linear-with-remainderhttps://puzzicle.org/thumbs/xe8d44657d914ffe9.jpgDividing quadratics by linear expressions with remaindersLearn how to simplify complex math problems by dividing quadratics by linear factors with remainders. Discover two methods for simplifying these kinds of math problems: factoring the numerator or using algebraic long division. This process helps break down tricky equations, making them easier to solve.https://cdn.kastatic.org/ka-youtube-converted/-veSAbfer7Q.mp4/-veSAbfer7Q.mp4296yes https://puzzicle.org/math/algebra2/v/quad-div-by-linear-missing-x-termhttps://puzzicle.org/thumbs/x0c9165d8371b07d6.jpgDividing quadratics by linear expressions with remainders: missing x-termAn interesting case in polynomial division is when one of the terms is missing. The video explains how to divide a quadratic expression, like (x²+1), by a linear one, such as (x+2). It shows two methods: re-expressing the numerator and using algebraic long division. Both methods lead to the same answer.https://cdn.kastatic.org/ka-youtube-converted/CDJGU9P_pGU.mp4/CDJGU9P_pGU.mp4283yes https://puzzicle.org/math/algebra-1-tx/v/rewriting-expressions-with-exponents-challenge-1https://puzzicle.org/thumbs/xf0e2b4b0fb9b9a03.jpgRewriting expressions with exponents challenge 1Expand previous knowledge of isolated exponent rules to rewriting expressions using multiple rules in the same problem.https://cdn.kastatic.org/ka-youtube-converted/NsNhpzljMFE.mp4/NsNhpzljMFE.mp4308yes https://puzzicle.org/math/algebra-1-tx/v/exponent-rule-challenge-2https://puzzicle.org/thumbs/xf6979f999f54ef2f.jpgRewriting expressions with exponents challenge 2Expand previous knowledge of isolated exponent rules to rewriting expressions using multiple rules in the same problem.https://cdn.kastatic.org/ka-youtube-converted/Ne8_dXWjeeU.mp4/Ne8_dXWjeeU.mp4290yes https://puzzicle.org/math/algebra-2-tx/v/writing-a-quadratic-when-given-the-vertex-and-another-pointhttps://puzzicle.org/thumbs/xa1dbf5a2e76e1c1a.jpgWriting a quadratic when given the vertex and another pointGiven the vertex of a quadratic function and another point on the graph, write the equation for that function in vertex form.https://cdn.kastatic.org/ka-youtube-converted/04vzdWS3Q7k.mp4/04vzdWS3Q7k.mp4318yes https://puzzicle.org/math/algebra-1-tx/v/writing-a-quadratic-function-from-a-graphhttps://puzzicle.org/thumbs/xac2455cfda81760a.jpgWriting a quadratic function from a graphWhen given a graph of a quadratic function, write its equation.https://cdn.kastatic.org/ka-youtube-converted/xUtzpjM4xyE.mp4/xUtzpjM4xyE.mp4317yes https://puzzicle.org/math/algebra-1-tx/v/writing-a-quadratic-function-from-solutionshttps://puzzicle.org/thumbs/xaf44af9711632f18.jpgWriting a quadratic function from solutionsWhen given the real solutions of a quadratic function, write its equation.https://cdn.kastatic.org/ka-youtube-converted/Wv-SiowM4M8.mp4/Wv-SiowM4M8.mp4283yes https://puzzicle.org/math/algebra-1-tx/v/graphing-arithmetic-sequenceshttps://puzzicle.org/thumbs/x3ec74f2d88ba92f9.jpgGraphing arithmetic sequencesConnecting existing knowledge of arithmetic sequences expressed in recursive function form to their representation graphically.https://cdn.kastatic.org/ka-youtube-converted/2OU-pb7CCT0.mp4/2OU-pb7CCT0.mp4263yes https://puzzicle.org/math/algebra-1-tx/v/graphing-geometric-sequenceshttps://puzzicle.org/thumbs/x30bbcb43daaab271.jpgGraphing geometric sequencesConnecting existing knowledge of geometric sequences expressed in recursive function form to their representation graphically.https://cdn.kastatic.org/ka-youtube-converted/a37pYSmcU7c.mp4/a37pYSmcU7c.mp4235yes https://puzzicle.org/math/algebra-1-tx-teks-imra-2026-publisher-initiated-changes/v/horizontal-and-vertical-lines-equations-and-slopehttps://puzzicle.org/thumbs/xd5310b991eb3872d.jpgHorizontal and vertical lines: equations and slope (IMRA 2026 publisher-initiated change)Write the equation of a horizontal and vertical line, and determine whether its slope is zero or undefined. IMRA 2026 publisher-initiated changehttps://cdn.kastatic.org/ka-youtube-converted/gKG2fKmeEY4.mp4/gKG2fKmeEY4.mp4270yes https://puzzicle.org/math/algebra-1-tx-teks-imra-2026-publisher-initiated-changes/v/finding-the-zero-of-a-linear-functionhttps://puzzicle.org/thumbs/x56432438250b9bdf.jpgFinding the zero of a linear function (IMRA 2026 publisher-initiated change)Find the zero of a linear function, also called the x-intercept, by reading it from a graph and by solving the equation algebraically.https://cdn.kastatic.org/ka-youtube-converted/DVObvLWb720.mp4/DVObvLWb720.mp4271yes https://puzzicle.org/math/algebra-1-tx-teks-imra-2026-publisher-initiated-changes/v/interpreting-the-zero-of-a-linear-function-in-a-real-world-problemhttps://puzzicle.org/thumbs/xb53a0b68d5c3bda8.jpgInterpreting the zero of a linear function in a real world problem (IMRA 2026 publisher-initiated change)Find the zero of a linear function that models a real-world situation,, and explain what it means in context.https://cdn.kastatic.org/ka-youtube-converted/o58Du_cErSI.mp4/o58Du_cErSI.mp4110yes https://puzzicle.org/math/algebra-1-tx-teks-imra-2026-publisher-initiated-changes/v/fitting-a-line-to-data-using-a-graphing-calculatorhttps://puzzicle.org/thumbs/xa32348d946651ecf.jpgFitting a line to data using a graphing calculator (IMRA 2026 publisher-initiated change)Sal uses a graphing calculator (Desmos) to fit a line to a person's weekly walking distance using linear regression, then uses the model to make a prediction. IMRA 2026 publisher-initiated change.https://cdn.kastatic.org/ka-youtube-converted/Bnzg9wtyCq8.mp4/Bnzg9wtyCq8.mp4373yes https://puzzicle.org/math/algebra-1-tx-teks-imra-2026-publisher-initiated-changes/v/worked-example-rewriting-and-solving-equations-by-completing-the-squarehttps://puzzicle.org/thumbs/x7ec99b7dc5aca17c.jpgWorked example: Rewriting and solving equations by completing the square (IMRA 2026 publisher-initiated change)Rewrite and solve quadratic equations by completing the square. IMRA 2026 publisher-initiated changehttps://cdn.kastatic.org/ka-youtube-converted/Q0rT65JxXjo.mp4/Q0rT65JxXjo.mp4378yes https://puzzicle.org/math/algebra2/v/introduction-to-i-and-imaginary-numbershttps://puzzicle.org/thumbs/145918385.jpgIntro to the imaginary numbersLearn about the imaginary unit, "i", a unique number defined as the square root of -1. It's a key part of complex numbers, which are in the form a + bi. The powers of "i" cycle through a set of values.https://cdn.kastatic.org/ka-youtube-converted/ysVcAYo7UPI.mp4/ysVcAYo7UPI.mp4320yes https://puzzicle.org/math/algebra2/v/imaginary-roots-of-negative-numbershttps://puzzicle.org/thumbs/150088079.jpgSimplifying roots of negative numbersDiscover the magic of the imaginary unit 'i'! This lesson dives into simplifying the square root of negative numbers using 'i', the principal square root of -1. We'll explore how to rewrite negative numbers as products, and use prime factorization to simplify roots.https://cdn.kastatic.org/ka-youtube-converted/s03qez-6JMA.mp4/s03qez-6JMA.mp4244yes https://puzzicle.org/math/algebra2/v/calculating-i-raised-to-arbitrary-exponentshttps://puzzicle.org/thumbs/145918384.jpgPowers of the imaginary unitThe imaginary unit i is defined such that i²=-1. So what's i³? i³=i²⋅i=-i. What's i⁴? i⁴=i²⋅i²=(-1)²=1. What's i⁵? i⁵=i⁴⋅i=1⋅i=i. Discover how the powers of 'i' cycle through values, making it possible to calculate high exponents of 'i' easily.https://cdn.kastatic.org/ka-youtube-converted/QiwfF83NWNA.mp4/QiwfF83NWNA.mp4380yes https://puzzicle.org/math/algebra2/v/i-as-the-principal-root-of-1-a-little-technicalhttps://puzzicle.org/thumbs/148079183.jpgi as the principal root of -1The video explores the intriguing concept of imaginary numbers, specifically the imaginary unit 'i'. 'i' is equal to the square root of -1, and every complex number can be expressed as a + bi, where a and b are real numbers. This video clarifies misconceptions about square roots of negative numbers.https://cdn.kastatic.org/ka-youtube-converted/rYG1D5lUE4I.mp4/rYG1D5lUE4I.mp4404yes https://puzzicle.org/math/algebra2/v/complex-number-introhttps://puzzicle.org/thumbs/xe9c076b5.jpgIntro to complex numbersSal explains how we obtain complex numbers by adding real numbers and imaginary numbers.https://cdn.kastatic.org/ka-youtube-converted/SP-YJe7Vldo.mp4/SP-YJe7Vldo.mp4284yes https://puzzicle.org/math/algebra2/v/adding-complex-numbershttps://puzzicle.org/thumbs/150088080.jpgAdding complex numbersLearn how to break down the process of adding complex numbers into simple steps. First, add the real parts. Then, do the same with the imaginary parts. The result? A new complex number! Learn and try some examples with Sal.https://cdn.kastatic.org/ka-youtube-converted/SfbjqVyQljk.mp4/SfbjqVyQljk.mp471yes https://puzzicle.org/math/algebra2/v/subtracting-complex-numbershttps://puzzicle.org/thumbs/150088077.jpgSubtracting complex numbersLearn how to break down the process of subtracting complex numbers into simple steps. Distribute the negative sign to get rid of parentheses, then add up the real and imaginary parts separately. Voila! You've got your answer in the form of a + bi.https://cdn.kastatic.org/ka-youtube-converted/tvXRaZbIjO8.mp4/tvXRaZbIjO8.mp4113yes https://puzzicle.org/math/algebra2/v/multiplying-complex-numbershttps://puzzicle.org/thumbs/150088076.jpgMultiplying complex numbersDiscover how to multiply complex numbers! Just like multiplying regular numbers, you can use the distributive property or FOIL method. Remember, the imaginary unit 'i' squared equals -1. So, when you multiply complex numbers like 1-3i and 2+5i, you get a new complex number: 17-i.https://cdn.kastatic.org/ka-youtube-converted/cWn6g8Qqvs4.mp4/cWn6g8Qqvs4.mp4332yes https://puzzicle.org/math/algebra2/v/complex-roots-from-the-quadratic-formulahttps://puzzicle.org/thumbs/150088081.jpgSolving quadratic equations: complex rootsSolving quadratic equations can lead to complex solutions. Listen to Sal break down the process using the quadratic formula and standard form. Then, learn how to verify these complex solutions, ensuring you've got the right answer. It's all about turning tricky problems into solvable equations!https://cdn.kastatic.org/ka-youtube-converted/dnjK4DPqh0k.mp4/dnjK4DPqh0k.mp4615yes https://puzzicle.org/math/algebra2/v/factor-high-deg-poly-introhttps://puzzicle.org/thumbs/xcb872f1c2aba4146.jpgIntroduction to factoring higher degree polynomialsUnpack the process of factoring monomials in algebra. Learn to simplify third-degree polynomials and tackle fourth-degree monomials. Understand the structure of introductory algebra and apply it to higher degree polynomials. Explore the concept of factoring multiple times and delve into the difference of squares. It's all about breaking down complex expressions into simpler parts!https://cdn.kastatic.org/ka-youtube-converted/7y_2jQlnKag.mp4/7y_2jQlnKag.mp4353yes https://puzzicle.org/math/algebra2/v/algebraic-factoring-by-greatest-common-monomial-factorhttps://puzzicle.org/thumbs/xaae59186.jpgTaking common factor from binomialExplore the process of factoring polynomials using the greatest common monomial factor. This involves breaking down coefficients and powers of variables to find the largest common factor, and then rewriting the expression with this common factor factored out. It's an essential skill for simplifying and solving algebraic expressionshttps://cdn.kastatic.org/ka-youtube-converted/iTn3TTF0IXk.mp4/iTn3TTF0IXk.mp4298yes https://puzzicle.org/math/algebra2/v/factoring-and-the-distributive-property-3https://puzzicle.org/thumbs/26982.jpgTaking common factor from trinomialBreak down the process of taking common factors from trinomials. Learn how to identify the greatest common factor of a trinomial expression and use it to simplify the expression. Follow along as Sal factors 4x⁴y-8x³y-2x² as 2x²(2x²y-4xy-1) by taking the greatest common factor.https://cdn.kastatic.org/ka-youtube-converted/MZl6Mna0leQ.mp4/MZl6Mna0leQ.mp4354yes https://puzzicle.org/math/algebra2/v/factoring-polynomial-with-area-modelhttps://puzzicle.org/thumbs/x5424a175.jpgTaking common factor: area modelThe video explores taking common factors in algebra using an area model. It demonstrates how to find the greatest common monomial factor of terms and uses the area of a rectangle as a visual aid. By factoring out the length and width, you can better grasp the process of factoring polynomials.https://cdn.kastatic.org/ka-youtube-converted/tYknkDjp-bQ.mp4/tYknkDjp-bQ.mp4327yes https://puzzicle.org/math/algebra2/v/factor-high-deg-polyhttps://puzzicle.org/thumbs/xc1951d0a2ffae6d9.jpgFactoring higher degree polynomialsFactoring higher degree polynomials involves breaking down complex expressions into simpler parts. This process includes identifying common factors, using the distributive property, and recognizing perfect squares. Sal demonstrates how to factor a partially factored polynomial and how to factor a third degree polynomial by grouping.https://cdn.kastatic.org/ka-youtube-converted/AUBX_aHStQY.mp4/AUBX_aHStQY.mp4238yes https://puzzicle.org/math/algebra2/v/factoring-perfect-square-polynomialhttps://puzzicle.org/thumbs/x1520bb8a.jpgFactoring higher degree polynomials: Common factorDiscover the art of factoring higher degree polynomials. We start by pulling out common factors, then spot perfect squares. The key is seeing patterns and using them to simplify complex expressions. Sal demonstrates by factoring 16x^3+24x^2+9x as (x)(4x+3)^2.https://cdn.kastatic.org/ka-youtube-converted/BFW2lHobO4E.mp4/BFW2lHobO4E.mp4219yes https://puzzicle.org/math/algebra2/v/id-quad-patternshttps://puzzicle.org/thumbs/x91cf610e0c621e01.jpgIdentifying quadratic patternsFactoring using structure involves recognizing patterns in expressions, like the difference of squares or perfect square trinomials, and using these patterns to break down complex expressions into simpler ones. As an example, you’ll determine whether the polynomial 9x⁸+6x⁴y+y² can be factored using the perfect square pattern or the difference of squares pattern (or neither).https://cdn.kastatic.org/ka-youtube-converted/OcavJJA8Uyg.mp4/OcavJJA8Uyg.mp4350yes https://puzzicle.org/math/algebra2/v/factor-w-substitutionhttps://puzzicle.org/thumbs/xd2d2ed4ed28d8978.jpgFactorization with substitutionUnravel the mystery of algebraic expressions with factorization using substitution! This lesson explores how to simplify complex expressions by identifying patterns and substituting variables. By using U+V² and U+V x U-V structures, you can easily transform and factor expressions!https://cdn.kastatic.org/ka-youtube-converted/js5cP-rq0FI.mp4/js5cP-rq0FI.mp4313yes https://puzzicle.org/math/algebra2/v/factoring-special-products-2https://puzzicle.org/thumbs/26990.jpgFactoring using the perfect square patternSal factors 25x^4-30x^2+9 as (5x^2-3)^2.https://cdn.kastatic.org/ka-youtube-converted/o-ZbdYVGehI.mp4/o-ZbdYVGehI.mp4145yes https://puzzicle.org/math/algebra2/v/factoring-difference-of-squareshttps://puzzicle.org/thumbs/13179583.jpgFactoring using the difference of squares patternIf we expand (a+b)(a-b) we will get a²-b². Factorization goes the other way: suppose we have an expression that is the difference of two squares, like x²-25 or 49x²-y², then we can factor is using the roots of those squares. For example, x²-25 can be factored as (x+5)(x-5). This is an extremely useful method that is used throughout math.https://cdn.kastatic.org/ka-youtube-converted/tvnOWIoeeaU.mp4/tvnOWIoeeaU.mp4109yes https://puzzicle.org/math/algebra2/v/geo-series-introhttps://puzzicle.org/thumbs/x5b09b14bc0585679.jpgGeometric series introductionLearn how money grows in a bank account with geometric series! Discover how each deposit grows by a fixed percentage every year, creating a pattern. This pattern forms a geometric series, a useful concept in finance and business. Keep depositing and watch your money multiply!https://cdn.kastatic.org/ka-youtube-converted/BdsURIUQ04k.mp4/BdsURIUQ04k.mp4377yes https://puzzicle.org/math/algebra2/v/deriving-formula-for-sum-of-finite-geometric-serieshttps://puzzicle.org/thumbs/x59f9c2a4.jpgFinite geometric series formulaA finite geometric series can be solved using the formula a(1-rⁿ)/(1-r). Sal demonstrates how to derive a formula for the sum of the first 'n' terms of such a series, emphasizing the importance of understanding the number of terms being summed.https://cdn.kastatic.org/ka-youtube-converted/Cf0-6ngH2gs.mp4/Cf0-6ngH2gs.mp4435yes https://puzzicle.org/math/algebra2/v/three-examples-of-evaluating-finite-geometric-serieshttps://puzzicle.org/thumbs/xe9cd3f25.jpgWorked examples: finite geometric seriesGet a front row seat to Sal's step-by-step solutions to finite geometric series. Learn how to find the sum of the first 50 terms of a series by multiplying each term by a common ratio. Discover how to apply the formula for the sum of a finite geometric series. Plus, explore how to handle series with a change in sign.https://cdn.kastatic.org/ka-youtube-converted/DY9Q3qNmZnw.mp4/DY9Q3qNmZnw.mp4415yes https://puzzicle.org/math/algebra2/v/poly-div-by-linearhttps://puzzicle.org/thumbs/x97920bb5395034a1.jpgDividing polynomials by linear expressionsLearn how to divide polynomials by linear factors! This tutorial guides you through the process of algebraic long division, focusing on highest degree terms first. You'll master the art of subtracting polynomials, handling remainders, and rewriting expressions in different forms.https://cdn.kastatic.org/ka-youtube-converted/G2VBRwXq1q4.mp4/G2VBRwXq1q4.mp4204yes https://puzzicle.org/math/algebra2/v/poly-div-by-linear-missing-termhttps://puzzicle.org/thumbs/xe64842d7cab9c7d2.jpgDividing polynomials by linear expressions: missing termThis lesson guides you through the process of dividing polynomials by linear factors, showing you how to neatly organize your work and avoid common errors. You'll learn how to handle missing degree terms and ensure correct subtraction. The result? You'll confidently simplify and rewrite complex rational expressions.https://cdn.kastatic.org/ka-youtube-converted/QTZ5Dn0t6M4.mp4/QTZ5Dn0t6M4.mp4272yes https://puzzicle.org/math/algebra2/v/factor-w-poly-divhttps://puzzicle.org/thumbs/x0b2e705bd9a9e53b.jpgFactoring using polynomial divisionThis video breaks down the process of dividing polynomials by linear factors. It starts with a given polynomial and a known factor, then uses polynomial division to rewrite the expression as a product of linear factors.https://cdn.kastatic.org/ka-youtube-converted/I9PiXAbjnMQ.mp4/I9PiXAbjnMQ.mp4358yes https://puzzicle.org/math/algebra2/v/factor-w-poly-div-missing-termhttps://puzzicle.org/thumbs/x2e3087588cbe3086.jpgFactoring using polynomial division: missing termIf we know one linear factor of a higher degree polynomial, we can use polynomial division to find other factors of the polynomial. For example, we can use the fact that (x+6) is a factor of (x³+9x²-108) in order to completely factor the polynomial. We just need to be careful because the polynomial has no x-term.https://cdn.kastatic.org/ka-youtube-converted/AkY3DGzrvk8.mp4/AkY3DGzrvk8.mp4238yes https://puzzicle.org/math/algebra2/v/polynomial-remainder-theoremhttps://puzzicle.org/thumbs/xdb4ca463.jpgIntro to the Polynomial Remainder TheoremThe Polynomial Remainder Theorem allows us to determine whether a linear expression is a factor of a polynomial expression easily. It tells us the remainder when a polynomial is divided by \[x - a\] is \[f(a)\]. This means if \[x - a\] is a factor of the polynomial, the remainder is zero. It's a neat trick to quickly find remainders without doing long division!https://cdn.kastatic.org/ka-youtube-converted/MwG6QD352yc.mp4/MwG6QD352yc.mp4403yes https://puzzicle.org/math/algebra2/v/polynomial-remainder-theorem-examplehttps://puzzicle.org/thumbs/x908ffb83.jpgRemainder theorem: finding remainder from equationThe Polynomial Remainder Theorem simplifies the process of finding the remainder when dividing a polynomial by \[x - a\]. Instead of long division, you just evaluate the polynomial at \[a\]. This method saves time and space, making polynomial division more manageable.https://cdn.kastatic.org/ka-youtube-converted/MHtTP6vc4RU.mp4/MHtTP6vc4RU.mp4212yes https://puzzicle.org/math/algebra2/v/remainder-theorem-exampleshttps://puzzicle.org/thumbs/x4fb30cebf726e2c3.jpgRemainder theorem examplesThe polynomial remainder theorem says that for a polynomial p(x) and a number a, the remainder on division by (x-a) is p(a). This might not be very clear right now, but you will understand this much better after watching these examples.https://cdn.kastatic.org/ka-youtube-converted/dBOpC_-PDTA.mp4/dBOpC_-PDTA.mp4332yes https://puzzicle.org/math/algebra2/v/polynomial-remainder-theorem-to-test-factorhttps://puzzicle.org/thumbs/xd36a1a36.jpgRemainder theorem: checking factorsLearn how to determine if an expression is a factor of a polynomial by dividing the polynomial by the expression. If the remainder is zero, the expression is a factor. The video also demonstrates how to quickly calculate the remainder using the theorem.https://cdn.kastatic.org/ka-youtube-converted/JAdNNJynWM4.mp4/JAdNNJynWM4.mp4206yes https://puzzicle.org/math/algebra2/v/constructing-a-polynomial-that-has-a-certain-factorhttps://puzzicle.org/thumbs/x7ea74388.jpgRemainder theorem: finding coefficientsLearn how to find the value of 'c' that makes 'x - 5' a factor of a polynomial 'p(x)'. Understand how setting 'p(5)' equal to zero and solving for 'c' unveils the mystery. Grasp the concept of the Polynomial Remainder Theorem with a real example.https://cdn.kastatic.org/ka-youtube-converted/e0lBUViss8E.mp4/e0lBUViss8E.mp4219yes https://puzzicle.org/math/algebra2/v/polynomial-remainder-theorem-proofhttps://puzzicle.org/thumbs/x1f4eea1e.jpgProof of the Polynomial Remainder TheoremThe PRT (Polynomial Remainder Theorem) may seem crazy to prove, but Sal shows how you can do it in less than six minutes!https://cdn.kastatic.org/ka-youtube-converted/NIazpCER9oM.mp4/NIazpCER9oM.mp4342yes https://puzzicle.org/math/algebra2/v/polynomial-zeros-introductionhttps://puzzicle.org/thumbs/xeb690eddfd61151c.jpgZeros of polynomials introductionThe zeros of a polynomial p(x) are all the x-values that make the polynomial equal to zero. They are interesting to us for many reasons, one of which is that they tell us about the x-intercepts of the polynomial's graph. We will also see that they are directly related to the factors of the polynomial.https://cdn.kastatic.org/ka-youtube-converted/EtzwLlkoAnM.mp4/EtzwLlkoAnM.mp4309yes https://puzzicle.org/math/algebra2/v/plot-polynomial-zeroshttps://puzzicle.org/thumbs/xa02384672e2b08f2.jpgZeros of polynomials: plotting zerosWhen we are given a polynomial in factored form, we can quickly find the polynomial's zeros. Then, we can represent them as the x-intercepts of the polynomial's graph.https://cdn.kastatic.org/ka-youtube-converted/5qQUN1fugXQ.mp4/5qQUN1fugXQ.mp4199yes https://puzzicle.org/math/algebra2/v/polynomial-equation-from-zeroshttps://puzzicle.org/thumbs/x7d23b9c4dae7643b.jpgZeros of polynomials: matching equation to zerosWhen we are given a list of the zeros of a polynomial, we can conclude the polynomial must have certain factors, which gives us information about the equation of the polynomial.https://cdn.kastatic.org/ka-youtube-converted/CtqwiX4wHEY.mp4/CtqwiX4wHEY.mp4231yes https://puzzicle.org/math/algebra2/v/polynomial-equation-from-graphhttps://puzzicle.org/thumbs/x6274b6e74d5b6e77.jpgZeros of polynomials: matching equation to graphWhen we are given the graph of a polynomial, we can deduce what its zeros are, which helps us determine a few factors the polynomial's equation must include.https://cdn.kastatic.org/ka-youtube-converted/f-xluUaAfwo.mp4/f-xluUaAfwo.mp4181yes https://puzzicle.org/math/algebra2/v/polynomial-zeros-groupinghttps://puzzicle.org/thumbs/x394e49d04a8e1912.jpgZeros of polynomials (with factoring): groupingWhen a polynomial is given in factored form, we can quickly find its zeros. When its given in expanded form, we can factor it, and then find the zeros! Here is an example of a 3rd degree polynomial we can factor using the method of grouping.https://cdn.kastatic.org/ka-youtube-converted/w0vTAHIS5B4.mp4/w0vTAHIS5B4.mp4295yes https://puzzicle.org/math/algebra2/v/polynomial-zeros-common-factorhttps://puzzicle.org/thumbs/xdd574d3d4e28f785.jpgZeros of polynomials (with factoring): common factorWhen a polynomial is given in factored form, we can quickly find its zeros. When it's given in expanded form, we can factor it, and then find the zeros! Here is an example of a 3rd degree polynomial we can factor by first taking a common factor and then using the sum-product pattern.https://cdn.kastatic.org/ka-youtube-converted/SONskIMkYdU.mp4/SONskIMkYdU.mp4212yes https://puzzicle.org/math/algebra2/v/polynomial-zero-multiplicityhttps://puzzicle.org/thumbs/x453d8f1967753c96.jpgMultiplicity of zeros of polynomialsThe polynomial p(x)=(x-1)(x-3)² is a 3rd degree polynomial, but it has only 2 distinct zeros. This is because the zero x=3, which is related to the factor (x-3)², repeats twice. This is called multiplicity. It means that x=3 is a zero of multiplicity 2, and x=1 is a zero of multiplicity 1. Multiplicity is a fascinating concept, and it is directly related to graphical behavior of the polynomial around the zero.https://cdn.kastatic.org/ka-youtube-converted/jrFLb9ZoZH0.mp4/jrFLb9ZoZH0.mp4396yes https://puzzicle.org/math/algebra2/v/polynomial-multiplicity-exampleshttps://puzzicle.org/thumbs/xea0e9844108ab8f1.jpgZeros of polynomials (multiplicity)Given the graph of a polynomial and looking at its x-intercepts, we can determine the factors the polynomial must have. Additionally, we can determine whether those factors are raised to an odd power or to an even power (this is called the multiplicity of the factors).https://cdn.kastatic.org/ka-youtube-converted/1ke_xceCevk.mp4/1ke_xceCevk.mp4300yes https://puzzicle.org/math/algebra2/v/solving-exponential-equations-with-exponent-propertieshttps://puzzicle.org/thumbs/x73e71236.jpgSolving exponential equations using exponent propertiesSal solves equations like 26^(9x+5) = 1 and 2^(3x+5) = 64^(x-7).https://cdn.kastatic.org/ka-youtube-converted/etl9KKf6se0.mp4/etl9KKf6se0.mp4296yes https://puzzicle.org/math/algebra2/v/solving-exponential-equations-with-exponent-properties-advancedhttps://puzzicle.org/thumbs/xbe4a3f78.jpgSolving exponential equations using exponent properties (advanced)Sal solves equations like 32^(x/3) = 8^(x-12) and 5^(4x+3) / 25^(9-x) = 5^(2x+5).https://cdn.kastatic.org/ka-youtube-converted/mGSn-qR6k-g.mp4/mGSn-qR6k-g.mp4422yes https://puzzicle.org/math/algebra2/v/constructing-exponential-modelshttps://puzzicle.org/thumbs/xd0ae5341.jpgConstructing exponential modelsSal models the distribution of a chain letter using an exponential function.https://cdn.kastatic.org/ka-youtube-converted/FRW7SINSYco.mp4/FRW7SINSYco.mp4228yes https://puzzicle.org/math/algebra2/v/constructing-exponential-models-half-lifehttps://puzzicle.org/thumbs/x085dfeba.jpgConstructing exponential models: half lifeSal models the decay of a Carbon-14 sample using an exponential function.https://cdn.kastatic.org/ka-youtube-converted/kaxfCiP9d0w.mp4/kaxfCiP9d0w.mp4258yes https://puzzicle.org/math/algebra2/v/constructing-exponential-models-percent-changehttps://puzzicle.org/thumbs/x017b71e8.jpgConstructing exponential models: percent changeSal models a population of narwhals using an exponential function.https://cdn.kastatic.org/ka-youtube-converted/x9a73iHv06U.mp4/x9a73iHv06U.mp4292yes https://puzzicle.org/math/algebra2/v/logarithmshttps://puzzicle.org/thumbs/1242982964.jpgIntro to logarithmsSal explains what logarithms are and gives a few examples of finding logarithms.https://cdn.kastatic.org/ka-youtube-converted/Z5myJ8dg_rM.mp4/Z5myJ8dg_rM.mp4422yes https://puzzicle.org/math/algebra2/v/fancier-logarithm-expressionshttps://puzzicle.org/thumbs/1243193209.jpgEvaluating logarithms (advanced)Sal evaluates log₂(8), log₈(2), log₂(⅛), and log₈(½).https://cdn.kastatic.org/ka-youtube-converted/eTWCARmrzJ0.mp4/eTWCARmrzJ0.mp4260yes https://puzzicle.org/math/algebra2/v/exponential-to-logarithmic-formhttps://puzzicle.org/thumbs/x19c7ee10.jpgRelationship between exponentials & logarithmsSal rewrites 100=10^2 as a logarithmic equation and log_5(1/125)=-3 as an exponential equation.https://cdn.kastatic.org/ka-youtube-converted/Obch1OP5QyA.mp4/Obch1OP5QyA.mp4102yes https://puzzicle.org/math/algebra2/v/plotting-exponential-logarithmhttps://puzzicle.org/thumbs/x07bc9727.jpgRelationship between exponentials & logarithms: graphsGiven a few points on the graph of an exponential function, Sal plots the corresponding points on the graph of the corresponding logarithmic function.https://cdn.kastatic.org/ka-youtube-converted/DhW9pz5Vfwo.mp4/DhW9pz5Vfwo.mp4250yes https://puzzicle.org/math/algebra2/v/logarithm-exponential-deductionshttps://puzzicle.org/thumbs/xce641b55.jpgRelationship between exponentials & logarithms: tablesGiven incomplete tables of values of b^x and its corresponding inverse function, log_b(y), Sal uses the inverse relationship of the functions to fill in the missing values.https://cdn.kastatic.org/ka-youtube-converted/Iz6IVf8frjw.mp4/Iz6IVf8frjw.mp4359yes https://puzzicle.org/math/algebra2/v/natural-logarithm-with-a-calculatorhttps://puzzicle.org/thumbs/367722961.jpgEvaluating natural logarithm with calculatorSal evaluates log_e(67) (which is more commonly written as ln(67) ) using a calculator.https://cdn.kastatic.org/ka-youtube-converted/Dpo_-GrMpNE.mp4/Dpo_-GrMpNE.mp4217yes https://puzzicle.org/math/algebra2/v/introduction-to-logarithm-propertieshttps://puzzicle.org/thumbs/25781.jpgIntro to logarithm properties (1 of 2)Sal introduces the logarithm identities for addition and subtraction of logarithms.https://cdn.kastatic.org/ka-youtube-converted/PupNgv49_WY.mp4/PupNgv49_WY.mp4556yes https://puzzicle.org/math/algebra2/v/introduction-to-logarithm-properties-part-2https://puzzicle.org/thumbs/25783.jpgIntro to logarithm properties (2 of 2)Sal introduces the logarithm identities for multiplication of logarithm by a constant, and the change of base rule.https://cdn.kastatic.org/ka-youtube-converted/TMmxKZaCqe0.mp4/TMmxKZaCqe0.mp4604yes https://puzzicle.org/math/algebra2/v/sum-of-logarithms-with-same-basehttps://puzzicle.org/thumbs/367722960.jpgUsing the logarithmic product ruleSal rewrites log₃(27x) as log₃(27)+log_3(x), which is simplified as 3+log₃(x).https://cdn.kastatic.org/ka-youtube-converted/pkGrXzakRFs.mp4/pkGrXzakRFs.mp4304yes https://puzzicle.org/math/algebra2/v/logarithm-of-a-powerhttps://puzzicle.org/thumbs/367723734.jpgUsing the logarithmic power ruleSal rewrites log₅(x³) as 3log₅(x).https://cdn.kastatic.org/ka-youtube-converted/Pb9V374iOas.mp4/Pb9V374iOas.mp4289yes https://puzzicle.org/math/algebra2/v/using-multiple-logarithm-properties-to-simplifyhttps://puzzicle.org/thumbs/367723732.jpgUsing the properties of logarithms: multiple stepsSal rewrites log_5([25^x]/y) as 2x-log_5(y) by using both the log subtraction property and the log multiplied by a constant property.https://cdn.kastatic.org/ka-youtube-converted/RhzXX5PbsuQ.mp4/RhzXX5PbsuQ.mp4130yes https://puzzicle.org/math/algebra2/v/proof-log-a-log-b-log-abhttps://puzzicle.org/thumbs/23514.jpgProof of the logarithm product ruleSal proves the logarithm addition property, log(a) + log(b) = log(ab).https://cdn.kastatic.org/ka-youtube-converted/FP2arCfAfBY.mp4/FP2arCfAfBY.mp4428yes https://puzzicle.org/math/algebra2/v/proof-a-log-b-log-b-a-log-a-log-b-log-a-bhttps://puzzicle.org/thumbs/23516.jpgProof of the logarithm quotient and power rulesSal proves the logarithm quotient rule, log(a) - log(b) = log(a/b), and the power rule, k⋅log(a) = log(aᵏ).https://cdn.kastatic.org/ka-youtube-converted/yEAxG_D1HDw.mp4/yEAxG_D1HDw.mp4479yes https://puzzicle.org/math/algebra2/v/change-of-base-formulahttps://puzzicle.org/thumbs/367721779.jpgEvaluating logarithms: change of base ruleSal approximates log₅(100) by rewriting it as log(100)/log(5) using the change of base rule, then evaluates with a calculator.https://cdn.kastatic.org/ka-youtube-converted/OkFdDqW9xxM.mp4/OkFdDqW9xxM.mp4452yes https://puzzicle.org/math/algebra2/v/rewriting-logarithmic-expressions-change-of-basehttps://puzzicle.org/thumbs/x97b1cfa7.jpgUsing the logarithm change of base ruleSal rewrites logarithmic expressions like 1/(logₐ4) or logₐ(16)*log₂(a) using the change of base rule.https://cdn.kastatic.org/ka-youtube-converted/qtsMgdZ98Yg.mp4/qtsMgdZ98Yg.mp4403yes https://puzzicle.org/math/algebra2/v/change-of-base-formula-proofhttps://puzzicle.org/thumbs/1335425394.jpgProof of the logarithm change of base ruleSal proves the logarithmic change of base rule, logₐ(b)=logₓ(b)/logₓ(a).https://cdn.kastatic.org/ka-youtube-converted/1reblXFlM6I.mp4/1reblXFlM6I.mp4299yes https://puzzicle.org/math/algebra2/v/exponential-equationhttps://puzzicle.org/thumbs/x1030e61c.jpgSolving exponential equations using logarithms: base-10Sal solves the equation 10^(2t-3)=7.https://cdn.kastatic.org/ka-youtube-converted/R443Db-wJ5o.mp4/R443Db-wJ5o.mp4171yes https://puzzicle.org/math/algebra2/v/shifting-functions-introhttps://puzzicle.org/thumbs/x6e84edb65f8627f7.jpgShifting functions introductionThe graph of y=f(x)+k (where k is a real number) is the same as the graph of y=f(x) only it's shifted up (when k>0) or down (when k<0). Similarly, the graph of y=f(x-h) (where h is a real number) is the same as the graph of y=f(x) only it's shifted to the right (when h>0) or to the left (when h<0)https://cdn.kastatic.org/ka-youtube-converted/RttvubuBhAE.mp4/RttvubuBhAE.mp4338yes https://puzzicle.org/math/algebra2/v/shifting-functions-exampleshttps://puzzicle.org/thumbs/x9389c47e.jpgShifting functions examplesSal analyzes two cases where functions f and g are given graphically, and g is a result of shifting f. He writes formulas for g in terms of f and in terms of x.https://cdn.kastatic.org/ka-youtube-converted/MDav5OMpCto.mp4/MDav5OMpCto.mp4462yes https://puzzicle.org/math/algebra2/v/graphing-shifted-functionshttps://puzzicle.org/thumbs/xc28ac02f.jpgGraphing shifted functionsGiven the graph of f(x)=x², Sal graphs g(x)=(x-2)²-4, which is the graph of f shifted 2 units to the right and 4 units down.https://cdn.kastatic.org/ka-youtube-converted/5DLkB-g8Rr8.mp4/5DLkB-g8Rr8.mp4168yes https://puzzicle.org/math/algebra2/v/reflecting-functions-introhttps://puzzicle.org/thumbs/xa07bf13a87f45a4f.jpgReflecting functions introductionWe can reflect the graph of y=f(x) over the x-axis by graphing y=-f(x) and over the y-axis by graphing y=f(-x). See this in action and understand why it happens.https://cdn.kastatic.org/ka-youtube-converted/mR2y_kohZIw.mp4/mR2y_kohZIw.mp4430yes https://puzzicle.org/math/algebra2/v/reflecting-functions-exampleshttps://puzzicle.org/thumbs/x840d7bd27c963978.jpgReflecting functions: examplesWe can reflect the graph of any function f about the x-axis by graphing y=-f(x) and we can reflect it about the y-axis by graphing y=f(-x). We can even reflect it about both axes by graphing y=-f(-x). See how this is applied to solve various problems.https://cdn.kastatic.org/ka-youtube-converted/0QOnEoQHZ1k.mp4/0QOnEoQHZ1k.mp4393yes https://puzzicle.org/math/algebra2/v/scaling-functions-introhttps://puzzicle.org/thumbs/x3f4f68f86bcdb75a.jpgScaling functions introductionThe graph y=k⋅f(x) (where k is a real number) is similar to the graph y=f(x), but each point's distance from the x-axis is multiplied by k. A similar thing happens when we graph y=f(k⋅x), only now the distance from the y-axis changes. These operations are called "scaling."https://cdn.kastatic.org/ka-youtube-converted/1lt6a77OmLU.mp4/1lt6a77OmLU.mp4363yes https://puzzicle.org/math/algebra2/v/vert-function-scalinghttps://puzzicle.org/thumbs/xb5a485b436f48633.jpgScaling functions vertically: examplesThe function k⋅f(x) is a vertical scaling of f. See multiple examples of how we relate the two functions and their graphs, and determine the value of k.https://cdn.kastatic.org/ka-youtube-converted/tRCEBtKLqEw.mp4/tRCEBtKLqEw.mp4309yes https://puzzicle.org/math/algebra2/v/scaling-functions-horizontallyhttps://puzzicle.org/thumbs/x65945a460335704b.jpgScaling functions horizontally: examplesThe function f(k⋅x) is a horizontal scaling of f. See multiple examples of how we relate the two functions and their graphs, and determine the value of k.https://cdn.kastatic.org/ka-youtube-converted/npMMbXm9sR0.mp4/npMMbXm9sR0.mp4489yes https://puzzicle.org/math/algebra2/v/compressing-functions-examplehttps://puzzicle.org/thumbs/xf63087fc.jpgIdentifying horizontal squash from graphGiven the graphs of functions f and g, where g is the result of compressing f by a factor of 2, Sal finds g(x) in terms of f(x).https://cdn.kastatic.org/ka-youtube-converted/4H5JZnytOfE.mp4/4H5JZnytOfE.mp4196yes https://puzzicle.org/math/algebra2/v/shifting-and-reflecting-functionshttps://puzzicle.org/thumbs/xb05fec98.jpgIdentifying function transformationsSal walks through several examples of how to write g(x) implicitly in terms of f(x) when g(x) is a shift or a reflection of f(x).https://cdn.kastatic.org/ka-youtube-converted/ENFNyNPYfZU.mp4/ENFNyNPYfZU.mp4400yes https://puzzicle.org/math/algebra2/v/graphing-square-and-cube-roothttps://puzzicle.org/thumbs/x9762d161485e0a98.jpgGraphing square and cube root functionsWe can graph various square root and cube root functions by thinking of them as transformations of the parent graphs y=√x and y=∛x.https://cdn.kastatic.org/ka-youtube-converted/PI6IV7K3DbM.mp4/PI6IV7K3DbM.mp4408yes https://puzzicle.org/math/algebra2/v/comparing-exponential-logarithmic-functionshttps://puzzicle.org/thumbs/xd6ac565c.jpgGraphical relationship between 2ˣ and log₂(x)Sal graphs y=2ˣ and y=log₂(x) on the same coordinate plane, showing how they relate as graphs of inverse functions.https://cdn.kastatic.org/ka-youtube-converted/K_PiPfYxtao.mp4/K_PiPfYxtao.mp4342yes https://puzzicle.org/math/algebra2/v/graphing-logarithmic-functions-1https://puzzicle.org/thumbs/x6573016191db84e6.jpgGraphing logarithmic functions (example 1)We can graph y=2log₂(-x-3) by viewing it as a transformation of y=log₂(x).https://cdn.kastatic.org/ka-youtube-converted/GpBSbNoon1E.mp4/GpBSbNoon1E.mp4327yes https://puzzicle.org/math/algebra2/v/graphing-logarithmic-functions-2https://puzzicle.org/thumbs/xf8b2301cb608d79f.jpgGraphing logarithmic functions (example 2)We can graph y=4log₂(x+6)-7 by viewing it as a transformation of y=log₂(x).https://cdn.kastatic.org/ka-youtube-converted/f_BJC6UwDTE.mp4/f_BJC6UwDTE.mp4226yes https://puzzicle.org/math/algebra2/v/rational-equation-introhttps://puzzicle.org/thumbs/1242890488.jpgRational equations introWhen we have an equation where the variable is in the denominator of a quotient, that's a rational equation. We can solve it by multiplying both sides by the denominator, but we have to look out for extraneous solutions in the process.https://cdn.kastatic.org/ka-youtube-converted/3RdNPrNUi4s.mp4/3RdNPrNUi4s.mp4192yes https://puzzicle.org/math/algebra2/v/equations-with-two-rational-expressionshttps://puzzicle.org/thumbs/x62e31296.jpgEquations with rational expressionsSal solves (x²-10x+21)/(3x-12)=(x-5)/(x-4), which has one real solution and one extraneous solution.https://cdn.kastatic.org/ka-youtube-converted/McOMtxI_Jzs.mp4/McOMtxI_Jzs.mp4312yes https://puzzicle.org/math/algebra2/v/equations-with-two-rational-expressions-2https://puzzicle.org/thumbs/x9985bba7.jpgEquations with rational expressions (example 2)Sal solves (-2x+4)/(x-1)=3/(x+1)-1.https://cdn.kastatic.org/ka-youtube-converted/CBlSsy1eslw.mp4/CBlSsy1eslw.mp4432yes https://puzzicle.org/math/algebra2/v/rational-function-inversehttps://puzzicle.org/thumbs/x479fdc23.jpgFinding inverses of rational functionsThe inverse of a function ƒ is a function that maps every output in ƒ's range to its corresponding input in ƒ's domain. We can find an expression for the inverse of ƒ by solving the equation 𝘹=ƒ(𝘺) for the variable 𝘺. See how it's done with a rational function.https://cdn.kastatic.org/ka-youtube-converted/YI30Ecfl4tc.mp4/YI30Ecfl4tc.mp4257yes https://puzzicle.org/math/algebra2/v/extraneous-solutions-to-radical-equationshttps://puzzicle.org/thumbs/23748.jpgIntro to square-root equations & extraneous solutionsSal explains what square-root equations are, and shows an example of solving such an equation and checking for extraneous solutions.https://cdn.kastatic.org/ka-youtube-converted/711pdW8TbbY.mp4/711pdW8TbbY.mp4670yes https://puzzicle.org/math/algebra2/v/extraneous-solutions-of-radical-equationshttps://puzzicle.org/thumbs/xfaa88068.jpgSquare-root equations introSal gives an example of how an extraneous solution arises when solving 2x-1=√(8-x).https://cdn.kastatic.org/ka-youtube-converted/m4eiYHL3PP8.mp4/m4eiYHL3PP8.mp4323yes https://puzzicle.org/math/algebra2/v/solving-radical-equationshttps://puzzicle.org/thumbs/127771048.jpgSolving square-root equations: one solutionSal solves the equation 3+√(5x+6)=12.https://cdn.kastatic.org/ka-youtube-converted/pFFoAGIEyJc.mp4/pFFoAGIEyJc.mp4191yes https://puzzicle.org/math/algebra2/v/solving-square-root-equations-w-two-solutionshttps://puzzicle.org/thumbs/xe1ea042e.jpgSolving square-root equations: two solutionsSal solves the equation 6+3w=√(2w+12)+2w which has two solutions.https://cdn.kastatic.org/ka-youtube-converted/wt6XqG59t5U.mp4/wt6XqG59t5U.mp4329yes https://puzzicle.org/math/algebra2/v/solving-square-root-equations-w-no-solutionhttps://puzzicle.org/thumbs/x91409da4.jpgSolving square-root equations: no solutionSal solves the equation √(3x-7)+√(2x-1)=0, only to find out that the single solution is extraneous, which means the equation has no solution.https://cdn.kastatic.org/ka-youtube-converted/ibeyn2QGjCM.mp4/ibeyn2QGjCM.mp4239yes https://puzzicle.org/math/algebra2/v/extraneous-solutionshttps://puzzicle.org/thumbs/xc87f4bd4f8e58a14.jpgExtraneous solutionsExtraneous solutions are values that we get when solving equations that aren't really solutions to the equation. In this video, we explain how and why we get extraneous solutions, by understanding the logic behind the process of solving equations.https://cdn.kastatic.org/ka-youtube-converted/Hz3vkU3gTeI.mp4/Hz3vkU3gTeI.mp4475yes https://puzzicle.org/math/algebra2/v/finding-radical-equation-with-given-extraneous-solutionhttps://puzzicle.org/thumbs/x194f0296.jpgEquation that has a specific extraneous solutionSal finds the value of d for which √(3x+25)=d+2x has an extraneous solution at x=-3.https://cdn.kastatic.org/ka-youtube-converted/HRVrbJzXxdM.mp4/HRVrbJzXxdM.mp4297yes https://puzzicle.org/math/algebra2/v/unit-circle-definition-of-trig-functions-1https://puzzicle.org/thumbs/1159990979.jpgUnit circleLearn how to use the unit circle to define sine, cosine, and tangent for all real numbers.https://cdn.kastatic.org/ka-youtube-converted/1m9p9iubMLU.mp4/1m9p9iubMLU.mp4544yes https://puzzicle.org/math/algebra2/v/matching-ratios-trig-functionshttps://puzzicle.org/thumbs/xdcacd12a.jpgThe trig functions & right triangle trig ratiosSal shows how, for acute angles, the two different definitions of the trigonometric values (SOH CAH TOA and the unit circle definition) result in the same values.https://cdn.kastatic.org/ka-youtube-converted/WffVdYETdng.mp4/WffVdYETdng.mp4417yes https://puzzicle.org/math/algebra2/v/introduction-to-radianshttps://puzzicle.org/thumbs/1159577071.jpgIntro to radiansSal explains the definition and motivation for radians and the relationship between radians and degrees.https://cdn.kastatic.org/ka-youtube-converted/EnwWxMZVBeg.mp4/EnwWxMZVBeg.mp4651yes https://puzzicle.org/math/algebra2/v/radian-and-degree-conversion-practicehttps://puzzicle.org/thumbs/1159451550.jpgRadians & degreesSal discusses the general approach to converting between radians and degrees and vice versa.https://cdn.kastatic.org/ka-youtube-converted/z8vj8tUCkxY.mp4/z8vj8tUCkxY.mp4432yes https://puzzicle.org/math/algebra2/v/we-converting-degrees-to-radianshttps://puzzicle.org/thumbs/1153849058.jpgDegrees to radiansTo convert from degrees to radians, multiply the number of degrees by π/180. This will give you the measurement in radians. If you have an angle that's 90 degrees, and you want to know what it is in radians, you multiply 90 by π/180. This gives you π/2.https://cdn.kastatic.org/ka-youtube-converted/O3jvUZ8wvZs.mp4/O3jvUZ8wvZs.mp4422yes https://puzzicle.org/math/algebra2/v/we-converting-radians-to-degreeshttps://puzzicle.org/thumbs/1153727649.jpgRadians to degreesTo change from radians to degrees, you need to multiply the number of radians by 180/π. This number will help you switch between the two units. For example, if you multiply π/2 radians by 180/π, you will get 90 degrees.https://cdn.kastatic.org/ka-youtube-converted/z0-1gBy1ykE.mp4/z0-1gBy1ykE.mp4200yes https://puzzicle.org/math/algebra2/v/rotation-by-radians-and-quadrantshttps://puzzicle.org/thumbs/x81df7889.jpgRadian angles & quadrantsSal determines the quadrant at which a ray falls after a rotation by a certain measure of radians.https://cdn.kastatic.org/ka-youtube-converted/fYQ3GRSu4JU.mp4/fYQ3GRSu4JU.mp4231yes https://puzzicle.org/math/algebra2/v/we-graph-domain-and-range-of-sine-functionhttps://puzzicle.org/thumbs/1153456927.jpgGraph of y=sin(x)The graph of y=sin(x) is like a wave that forever oscillates between -1 and 1, in a shape that repeats itself every 2π units. Specifically, this means that the domain of sin(x) is all real numbers, and the range is [-1,1]. See how we find the graph of y=sin(x) using the ​unit-circle definition of sin(x).https://cdn.kastatic.org/ka-youtube-converted/sjUhr0HkLUg.mp4/sjUhr0HkLUg.mp4562yes https://puzzicle.org/math/algebra2/v/we-graphs-of-sine-and-cosine-functionshttps://puzzicle.org/thumbs/1153664550.jpgIntersection points of y=sin(x) and y=cos(x)Sal draws the graphs of the sine and the cosine functions and analyzes their intersection points.https://cdn.kastatic.org/ka-youtube-converted/08zHioOVTd4.mp4/08zHioOVTd4.mp4667yes https://puzzicle.org/math/algebra2/v/tangent-graphhttps://puzzicle.org/thumbs/x9267c937.jpgGraph of y=tan(x)Sal draws the graph of the tangent function based on the unit circle definition of the function.https://cdn.kastatic.org/ka-youtube-converted/FK6-tZ5D7xM.mp4/FK6-tZ5D7xM.mp4612yes https://puzzicle.org/math/algebra2/v/example-graphing-y-3sin-x-2https://puzzicle.org/thumbs/x70280d50ff5ef21c.jpgExample: Graphing y=3⋅sin(½⋅x)-2Sal graphs y=3⋅sin(½⋅x)-2 by thinking about the graph of y=sin(x) and analyzing how the graph (including the midline, amplitude, and period) changes as we perform function transformations to get from y=sin(x) to y=3⋅sin(½⋅x)-2.https://cdn.kastatic.org/ka-youtube-converted/-Si4W4KyeMw.mp4/-Si4W4KyeMw.mp4203yes https://puzzicle.org/math/algebra2/v/example-graphing-y-cos-x-1-5https://puzzicle.org/thumbs/x77b9a0f0a63cfc2e.jpgExample: Graphing y=-cos(π⋅x)+1.5Sal graphs y=-cos(π⋅x)+1.5 by thinking about the graph of y=cos(x) and analyzing how the graph (including the midline, amplitude, and period) changes as we perform function transformations to get from y=cos(x) to y=-cos(π⋅x)+1.5.https://cdn.kastatic.org/ka-youtube-converted/9XtCWJmKF28.mp4/9XtCWJmKF28.mp4190yes https://puzzicle.org/math/algebra2/v/trig-function-equationhttps://puzzicle.org/thumbs/xdaccbc0b.jpgSinusoidal function from graphSal finds the equation of a sinusoidal function from its graph where the minimum point (-2,-5) and the maximum point (2,1) are highlighted.https://cdn.kastatic.org/ka-youtube-converted/yHo0CcDVHsk.mp4/yHo0CcDVHsk.mp4322yes https://puzzicle.org/math/algebra2/v/interpreting-trig-graphs-in-contexthttps://puzzicle.org/thumbs/xc07920a8b52ac0c2.jpgInterpreting trigonometric graphs in contextWhen a trigonometric function models a real-world relationship, we can assign meaning to its midline, amplitude and period.https://cdn.kastatic.org/ka-youtube-converted/7q9eHwP2Smw.mp4/7q9eHwP2Smw.mp4334yes https://puzzicle.org/math/algebra2/v/modeling-with-combined-functionshttps://puzzicle.org/thumbs/x9e853ab5.jpgModeling with function combinationSal models the height of a tree on top of a tower, by adding the functions that model the growth of the tree and the tower separately.https://cdn.kastatic.org/ka-youtube-converted/2IWhknVzeJM.mp4/2IWhknVzeJM.mp4188yes https://puzzicle.org/math/algebra2/v/periodicity-of-algebraic-modelshttps://puzzicle.org/thumbs/x19dc89e1.jpgPeriodicity of algebraic modelsSal analyzes the periodicity of graphs that model real world situations.https://cdn.kastatic.org/ka-youtube-converted/zHS9ZVXw_Z0.mp4/zHS9ZVXw_Z0.mp4201yes https://puzzicle.org/math/algebra2/v/end-behavior-word-problemshttps://puzzicle.org/thumbs/x5ae6fb8c.jpgEnd behavior of algebraic modelsSal analyzes the end behavior of graphs that model real world situations.https://cdn.kastatic.org/ka-youtube-converted/kmzBGnniH2w.mp4/kmzBGnniH2w.mp4180yes https://puzzicle.org/math/algebra2/v/interpreting-features-of-functions-2-example-2https://puzzicle.org/thumbs/x62250c60.jpgSymmetry of algebraic modelsSal interprets the significance of modeling function being even.https://cdn.kastatic.org/ka-youtube-converted/ynIq9IxbVso.mp4/ynIq9IxbVso.mp4198yes https://puzzicle.org/math/algebra2/v/function-symmetryhttps://puzzicle.org/thumbs/62754978.jpgFunction symmetry introductionFunctions can be symmetrical about the y-axis, which means that if we reflect their graph about the y-axis we will get the same graph. There are other functions that we can reflect about both the x- and y-axis and get the same graph. These are two types of symmetry we call even and odd functions.https://cdn.kastatic.org/ka-youtube-converted/4dOQePLgqxE.mp4/4dOQePLgqxE.mp4322yes https://puzzicle.org/math/algebra2/v/recognizing-features-of-functions-2-example-2https://puzzicle.org/thumbs/x09a49d09.jpgEven and odd functions: GraphsSal picks the function that is odd among three functions given by their graphs.https://cdn.kastatic.org/ka-youtube-converted/zltgXTlUVLw.mp4/zltgXTlUVLw.mp4233yes https://puzzicle.org/math/algebra2/v/even-and-odd-functions-tableshttps://puzzicle.org/thumbs/x6219cabb7ec1daa4.jpgEven and odd functions: TablesEven functions are symmetrical about the y-axis: f(x)=f(-x). Odd functions are symmetrical about the x- and y-axis: f(x)=-f(-x). Let's use these definitions to determine if a function given as a table is even, odd, or neither.https://cdn.kastatic.org/ka-youtube-converted/eKwJlJUM_d0.mp4/eKwJlJUM_d0.mp4249yes https://puzzicle.org/math/algebra2/v/even-and-odd-functions-equationshttps://puzzicle.org/thumbs/xc67b21d48f754599.jpgEven and odd functions: EquationsWhen we are given the equation of a function f(x), we can check whether the function is even, odd, or neither by evaluating f(-x). If we get an expression that is equivalent to f(x), we have an even function; if we get an expression that is equivalent to -f(x), we have an odd function; and if neither happens, it is neither!https://cdn.kastatic.org/ka-youtube-converted/fTeO-V_VgEc.mp4/fTeO-V_VgEc.mp4236yes https://puzzicle.org/math/algebra2/v/even-and-odd-functions-find-the-mistakehttps://puzzicle.org/thumbs/xd2801e92243d7dbc.jpgEven and odd functions: Find the mistakeSee another student's work when trying to determine whether a function is even, odd, or either, and decide whether they made a mistake, and if so, where.https://cdn.kastatic.org/ka-youtube-converted/RAksvTJWgxo.mp4/RAksvTJWgxo.mp4215yes https://puzzicle.org/math/algebra-home/v/comparing-features-of-functions-2-example-1https://puzzicle.org/thumbs/x37888c41.jpgComparing functions: x-interceptsGiven several functions represented in different forms (as a formula, as a graph, and as a table of values), Sal finds the one with that has no x-intercepts.https://cdn.kastatic.org/ka-youtube-converted/w2BN2AJ4fOI.mp4/w2BN2AJ4fOI.mp4167yes https://puzzicle.org/math/algebra-home/v/comparing-features-of-functions-2-example-3https://puzzicle.org/thumbs/xbb39e37a.jpgComparing functions: shared featuresSal is given the formula of one function and the graph of the other, and he determines which features are shared by both functions.https://cdn.kastatic.org/ka-youtube-converted/H1H0xmjZMPA.mp4/H1H0xmjZMPA.mp4270yes https://puzzicle.org/math/algebra2/v/equations-by-graphinghttps://puzzicle.org/thumbs/x4b838d2059b027eb.jpgSolving equations by graphingYou probably already solved a system of equations by graphing the equations and looking for intersection points. This method can actually be used to solve (or find an approximate solution to) any single equation, no matter what kind! This is a very exciting tool.https://cdn.kastatic.org/ka-youtube-converted/7JkY19bYF28.mp4/7JkY19bYF28.mp4259yes https://puzzicle.org/math/precalculus/v/complex-conjugateshttps://puzzicle.org/thumbs/8187550.jpgIntro to complex number conjugatesLearn about the conjugate of complex numbers, how to add and divide using conjugates, and the key property that multiplying a complex number by its conjugate produces a real number. Conjugates are very useful for simplifying division of complex numbers.https://cdn.kastatic.org/ka-youtube-converted/BZxZ_eEuJBM.mp4/BZxZ_eEuJBM.mp4483yes https://puzzicle.org/math/precalculus/v/complex-conjugates-examplehttps://puzzicle.org/thumbs/150088078.jpgComplex number conjugatesThrough a guided example with 7 - 5i, this video explains how to find the conjugate of a complex number, which is simply changing the sign of the imaginary part. Multiplying a complex number by its conjugate results in a real number. This is useful for simplifying complex numbers and is similar to the difference of squares.https://cdn.kastatic.org/ka-youtube-converted/dbxJ6LD0344.mp4/dbxJ6LD0344.mp4241yes https://puzzicle.org/math/precalculus/v/dividing-complex-numbershttps://puzzicle.org/thumbs/150088083.jpgDividing complex numbersSal divides (6+3i) by (7-5i).https://cdn.kastatic.org/ka-youtube-converted/Z8j5RDOibV4.mp4/Z8j5RDOibV4.mp4298yes https://puzzicle.org/math/algebra2/v/non-linear-systems-of-equations-3https://puzzicle.org/thumbs/27010.jpgQuadratic system with no solutionsSal solves a system of two quadratic equations algebraically and finds the system has no solutions. He then graphs the equations to show that this is true.https://cdn.kastatic.org/ka-youtube-converted/FksgVpM_iXs.mp4/FksgVpM_iXs.mp4416yes https://puzzicle.org/math/algebra2/v/polynomial-intervalshttps://puzzicle.org/thumbs/x1eca5b5935d679c0.jpgPositive and negative intervals of polynomialsIf we know all the zeros of a polynomial, then we can determine the intervals over which the polynomial is positive and negative. This is because the polynomial has the same sign between consecutive zeros. So all we need to do is check each interval that is between two consecutive zeros (or before the smallest zero and after the largest zero).https://cdn.kastatic.org/ka-youtube-converted/z3faK7OV3Ew.mp4/z3faK7OV3Ew.mp4520yes https://puzzicle.org/math/algebra2/v/polynomial-end-behaviorhttps://puzzicle.org/thumbs/xb79a4102.jpgIntro to end behavior of polynomialsSal explains what "end behavior" is and what affects the end behavior of polynomial functions.https://cdn.kastatic.org/ka-youtube-converted/tZKzaF28sOk.mp4/tZKzaF28sOk.mp4490yes https://puzzicle.org/math/algebra-home/v/example-adding-polynomials-with-multiple-variableshttps://puzzicle.org/thumbs/1159107679.jpgAdding polynomials: two variables (intro)Sal simplifies (4x²y - 3x² - 2y) + (8xy - 3x² + 2x²y + 4).https://cdn.kastatic.org/ka-youtube-converted/jroamh6SIo0.mp4/jroamh6SIo0.mp4150yes https://puzzicle.org/math/algebra-home/v/subtracting-polynomials-with-multiple-variableshttps://puzzicle.org/thumbs/13179529.jpgSubtracting polynomials: two variables (intro)Sal simplifies (4x²y - 3xy + 25) - (9y²x + 7xy - 20).https://cdn.kastatic.org/ka-youtube-converted/AqMT_zB9rP8.mp4/AqMT_zB9rP8.mp4130yes https://puzzicle.org/math/algebra-home/v/example-of-error-when-subtracting-polynomialshttps://puzzicle.org/thumbs/x23cf0821.jpgFinding an error in polynomial subtractionSal analyzes a polynomial subtraction process to find the step that has an error.https://cdn.kastatic.org/ka-youtube-converted/ZBDiMLAQU-8.mp4/ZBDiMLAQU-8.mp4147yes https://puzzicle.org/math/algebra2/v/poly-diff-of-squareshttps://puzzicle.org/thumbs/x08cd42d946762f26.jpgPolynomial special products: difference of squaresDive into the exciting world of special products of polynomials, focusing on the difference of squares. We explore how to expand and simplify algebraic expressions. We also tackle more complex expressions, applying the same principles to make math magic happen!https://cdn.kastatic.org/ka-youtube-converted/uNweU6I4Icw.mp4/uNweU6I4Icw.mp4246yes https://puzzicle.org/math/algebra2/v/poly-perfect-squarehttps://puzzicle.org/thumbs/xce2df43c40bb1be6.jpgPolynomial special products: perfect squareSquaring binomials is a breeze when you recognize patterns! The perfect square pattern tells us that (a+b)²=a²+2ab+b². The video shows how to square more complex binomials. It's all about applying what we know about simple binomials to these trickier ones.https://cdn.kastatic.org/ka-youtube-converted/hw5DE8HSZk0.mp4/hw5DE8HSZk0.mp4310yes https://puzzicle.org/math/precalculus/v/factoring-sums-of-squareshttps://puzzicle.org/thumbs/xe59b0d49.jpgComplex numbers & sum of squares factorizationLearn how expressions of the form x^2+y^2 can be factored into linear factors. This would not be possible without the aid of complex numbers!https://cdn.kastatic.org/ka-youtube-converted/8TPofjGXDR4.mp4/8TPofjGXDR4.mp4291yes https://puzzicle.org/math/precalculus/v/factoring-sum-of-squareshttps://puzzicle.org/thumbs/xbe367cd0.jpgFactoring sum of squaresSal factors 36a^8+2b^6 as (6a^4-i*√2b^3)(6a^4+i*√2b^3).https://cdn.kastatic.org/ka-youtube-converted/C3QPTCwpIZo.mp4/C3QPTCwpIZo.mp4152yes https://puzzicle.org/math/precalculus/v/factoring-polynomials-using-complex-numbershttps://puzzicle.org/thumbs/x1328ad6348f733b1.jpgFactoring polynomials using complex numbersSal shows how to factor a fourth degree polynomial into linear factors using the sum-product rule and the sum of squares identity.https://cdn.kastatic.org/ka-youtube-converted/9NhTVXhfqzU.mp4/9NhTVXhfqzU.mp4190yes https://puzzicle.org/math/algebra-home/v/intro-to-polynomial-synthetic-division-precalculus-khan-academyhttps://puzzicle.org/thumbs/xb2853877f3e1c452.jpgIntro to polynomial synthetic divisionLearn how to use synthetic division as another approach to solve polynomials.https://cdn.kastatic.org/ka-youtube-converted/z7JiQ5etCZs.mp4/z7JiQ5etCZs.mp4299yes https://puzzicle.org/math/algebra-home/v/why-synthetic-division-workshttps://puzzicle.org/thumbs/948944478.jpgWhy synthetic division worksSal explains why synthetic division gives you the same result as traditional algebraic long division.https://cdn.kastatic.org/ka-youtube-converted/-nlMXVrgtjw.mp4/-nlMXVrgtjw.mp4420yes https://puzzicle.org/math/algebra-home/v/factoring-5th-degree-polynomial-to-find-real-zeroshttps://puzzicle.org/thumbs/x2e519607.jpgFactoring higher-degree polynomialsSal factors p(x)=2x^5+x^4-2x-1 as (2x+1)(x^4-1) using grouping. Then he further factors (x^4-1) as (x^2+1)(x+1)(x-1) using the special product form of difference of squares.https://cdn.kastatic.org/ka-youtube-converted/mekBhhFWC9c.mp4/mekBhhFWC9c.mp4482yes https://puzzicle.org/math/algebra-home/v/factoring-sum-of-cubeshttps://puzzicle.org/thumbs/13179584.jpgFactoring sum of cubesSal factors 27x^6+125 as (3x^2+5)(9x^4-15x^2+25) using a special product form for a sum of cubes.https://cdn.kastatic.org/ka-youtube-converted/Touv2APcBRw.mp4/Touv2APcBRw.mp4286yes https://puzzicle.org/math/algebra-home/v/difference-of-cubes-factoringhttps://puzzicle.org/thumbs/466052139.jpgFactoring difference of cubesSal factors 40c^3-5d^3 as 5(2x-d)(4c^2+2cd+d^2) using a special product form for a difference of cubes.https://cdn.kastatic.org/ka-youtube-converted/CxcP4ylUP5w.mp4/CxcP4ylUP5w.mp4252yes https://puzzicle.org/math/precalculus/v/function-compositionhttps://puzzicle.org/thumbs/x2114d642.jpgIntro to composing functionsThis video is about composing functions, which is the process of building up a function by composing it from other functions. It explains how to evaluate the composition of functions step by step, using examples with three different function definitions: f (x), g(t), and h(x).https://cdn.kastatic.org/ka-youtube-converted/wUNWjd4bMmw.mp4/wUNWjd4bMmw.mp4375yes https://puzzicle.org/math/precalculus/v/evaluating-composite-functionshttps://puzzicle.org/thumbs/x10ca4f8b.jpgEvaluating composite functionsSal evaluates (h⚬g)(-6) for g(x)=x²+5x-3 and h(y)=3(y-1)²-5.https://cdn.kastatic.org/ka-youtube-converted/lxtlwnCV-HM.mp4/lxtlwnCV-HM.mp4250yes https://puzzicle.org/math/precalculus/v/evaluating-composite-functions-using-tableshttps://puzzicle.org/thumbs/x19ec1112.jpgEvaluating composite functions: using tablesGiven tables of values of the functions f and g, Sal evaluates f(g(0)) and g(f(0)).https://cdn.kastatic.org/ka-youtube-converted/FR8JH6IldsE.mp4/FR8JH6IldsE.mp4250yes https://puzzicle.org/math/precalculus/v/evaluating-composite-functions-using-graphshttps://puzzicle.org/thumbs/x4dd841a1.jpgEvaluating composite functions: using graphsGiven the graphs of the functions f and g, Sal evaluates g(f(-5)).https://cdn.kastatic.org/ka-youtube-converted/oORnGaJp1pk.mp4/oORnGaJp1pk.mp4182yes https://puzzicle.org/math/precalculus/v/new-function-from-compositionhttps://puzzicle.org/thumbs/xb73f4bfe.jpgFinding composite functionsThrough a worked example involving f(x)=√(x²-1) and g(x)=x/(1+x), learn about function composition: the process of combining two functions to create a new function. This involves replacing the input of one function with the output of another function. Also note that the composition of two functions is typically not the same as their composition in the reverse order.https://cdn.kastatic.org/ka-youtube-converted/_b-2rZpX5z4.mp4/_b-2rZpX5z4.mp4177yes https://puzzicle.org/math/precalculus/v/evaluating-composite-functions-example-1https://puzzicle.org/thumbs/xf6ad0f67.jpgEvaluating composite functions (advanced)Given that h(x)=3x and g(t)=-2t-2-h(t), Sal finds h(g(8)).https://cdn.kastatic.org/ka-youtube-converted/jlID_mIJXi4.mp4/jlID_mIJXi4.mp4251yes https://puzzicle.org/math/precalculus/v/determining-if-a-function-is-invertiblehttps://puzzicle.org/thumbs/x88678122.jpgDetermining if a function is invertibleLearn how to build a mapping diagram for a finite function, and how to use this diagram to determine if the function is invertible. An invertible function has a one-to-one mapping between its domain and range. Functions that map multiple domain elements to the same range element are not invertible.https://cdn.kastatic.org/ka-youtube-converted/dqCTAHHza10.mp4/dqCTAHHza10.mp4244yes https://puzzicle.org/math/precalculus/v/restricting-trig-function-domainhttps://puzzicle.org/thumbs/x5679daee.jpgRestricting domains of functions to make them invertibleSal is given the graph of a trigonometric function, and he discusses ways in which he can change the function to make it invertible.https://cdn.kastatic.org/ka-youtube-converted/tCV9VyIIaw0.mp4/tCV9VyIIaw0.mp4360yes https://puzzicle.org/math/precalculus/v/reading-inverse-values-from-a-graphhttps://puzzicle.org/thumbs/x9cd44c3726ddbc92.jpgReading inverse values from a graphLearn how to find the inverse of a function and how to graph it in this video. The inverse function swaps the x and y values of the original function, and the graph of this inverse function is a reflection of the original function about the line y=x.https://cdn.kastatic.org/ka-youtube-converted/nEWYKpTCX-U.mp4/nEWYKpTCX-U.mp4254yes https://puzzicle.org/math/precalculus/v/reading-inverse-values-from-a-tablehttps://puzzicle.org/thumbs/xd20cb95991e2c2f7.jpgReading inverse values from a tableAn inverse function takes the output from the original function and gives us the input. Given a table of inputs and outputs for a function, evaluate the inverse function for a certain input.https://cdn.kastatic.org/ka-youtube-converted/ygVpyyASavk.mp4/ygVpyyASavk.mp4128yes https://puzzicle.org/math/precalculus/v/verify-inverse-functions-from-tableshttps://puzzicle.org/thumbs/xda04e060cb055b5a.jpgVerifying inverse functions from tablesInverse functions undo each other. Functions s and t are inverses if and only if s(t(x))=x and t(s(x))=x for every x-value in the domains.https://cdn.kastatic.org/ka-youtube-converted/UTUuyCXFAs4.mp4/UTUuyCXFAs4.mp4272yes https://puzzicle.org/math/precalculus/v/using-specific-values-to-test-for-inverseshttps://puzzicle.org/thumbs/x741d01745ab2e2c1.jpgUsing specific values to test for inversesAre two functions inverses? Even one counterexample can show that they are not. How many examples would it take to show that they really are inverses?https://cdn.kastatic.org/ka-youtube-converted/EYv2EoEFRbo.mp4/EYv2EoEFRbo.mp4213yes https://puzzicle.org/math/precalculus/v/verifying-function-inverses-by-compositionhttps://puzzicle.org/thumbs/x365d0a93.jpgVerifying inverse functions by compositionSal composes f(x)=(x+7)³-1 and g(x)=∛(x+1)-7, and finds that f(g(x))=g(f(x))=x, which means the functions are inverses!https://cdn.kastatic.org/ka-youtube-converted/8GEGnSEJA2s.mp4/8GEGnSEJA2s.mp4401yes https://puzzicle.org/math/precalculus/v/composing-non-inverse-functionshttps://puzzicle.org/thumbs/xe665f62f.jpgVerifying inverse functions by composition: not inverseSal composes f(x)=2x-3 and g(x)=½x+3, and finds that f(g(x)) ≠ g(f(x)) ≠ x, which means the functions are not inverses.https://cdn.kastatic.org/ka-youtube-converted/PJIL0kRHND0.mp4/PJIL0kRHND0.mp4252yes https://puzzicle.org/math/algebra2/v/solving-radical-equations-2https://puzzicle.org/thumbs/26908.jpgSolving cube-root equationsSal solves the equation -∛y=4∛y+5.https://cdn.kastatic.org/ka-youtube-converted/b6WtwQddAcY.mp4/b6WtwQddAcY.mp4214yes https://puzzicle.org/math/algebra2/v/solve-exponentialshttps://puzzicle.org/thumbs/xad64718b.jpgSolving exponential equations using logarithms: base-2We can use logarithms to solve *any* exponential equation of the form a⋅bᶜˣ=d. For example, this is how you can solve 3⋅10²ˣ=7: 1. Divide by 3: 10²ˣ=7/3 2. Use the definition of logarithm: 2x=log(7/3) 3. Divide by 2: x=log(7/3)/2 Now you can use a calculator to find the solution of the equation as a rounded decimal number ​https://cdn.kastatic.org/ka-youtube-converted/7Ig6kVZaWoU.mp4/7Ig6kVZaWoU.mp4312yes https://puzzicle.org/math/algebra2/v/solving-exponential-model-word-problems-1https://puzzicle.org/thumbs/x6edba13f.jpgExponential model word problem: medication dissolveSal solves an exponential equation in order to answer a question about an exponential model.https://cdn.kastatic.org/ka-youtube-converted/HnHCQ2X_meg.mp4/HnHCQ2X_meg.mp4245yes https://puzzicle.org/math/algebra2/v/solving-exponential-model-word-problems-2https://puzzicle.org/thumbs/x472204e4.jpgExponential model word problem: bacteria growthSal evaluates an exponential function at a specific value in order to answer a question about an exponential model.https://cdn.kastatic.org/ka-youtube-converted/IpXL9HIkqUQ.mp4/IpXL9HIkqUQ.mp494yes https://puzzicle.org/math/math3/v/simplifying-rational-expressions-2https://puzzicle.org/thumbs/27012.jpgSimplifying rational expressions: groupingSal simplifies and states the domain of (2x^2+13x+20)/(2x^2+17x+30).https://cdn.kastatic.org/ka-youtube-converted/dstNU7It-Ro.mp4/dstNU7It-Ro.mp4402yes https://puzzicle.org/math/math3/v/simplifying-rational-expressions-w-higher-degree-termshttps://puzzicle.org/thumbs/xc35049aa.jpgSimplifying rational expressions: higher degree termsSal simplifies & states the domain of (x⁴+8x²+7)/(3x⁵-3x).https://cdn.kastatic.org/ka-youtube-converted/yPZ7In2tdOQ.mp4/yPZ7In2tdOQ.mp4353yes https://puzzicle.org/math/math3/v/simplifying-rational-expressions-w-two-variableshttps://puzzicle.org/thumbs/x27b04c23.jpgSimplifying rational expressions: two variablesSal simplifies & states the domain of (5x²+20xy+20y²)/(x²-xy-6y²).https://cdn.kastatic.org/ka-youtube-converted/e7vA_S7abSY.mp4/e7vA_S7abSY.mp4371yes https://puzzicle.org/math/precalculus/v/end-behavior-of-rational-functionshttps://puzzicle.org/thumbs/x4f467034.jpgEnd behavior of rational functionsSal analyzes the end behavior of several rational functions, that together cover all cases types of end behavior.https://cdn.kastatic.org/ka-youtube-converted/Vtcmyr5IGYY.mp4/Vtcmyr5IGYY.mp4607yes https://puzzicle.org/math/precalculus/v/discontinuities-of-rational-functionshttps://puzzicle.org/thumbs/xc81ec0b8.jpgDiscontinuities of rational functionsSal analyzes two rational functions to find their vertical asymptotes & removable discontinuities. He distinguishes those from the zeros of the functions.https://cdn.kastatic.org/ka-youtube-converted/qoMimDT7D8w.mp4/qoMimDT7D8w.mp4406yes https://puzzicle.org/math/precalculus/v/finding-asymptotes-examplehttps://puzzicle.org/thumbs/x0149f708.jpgGraphing rational functions according to asymptotesLearn how to find removable discontinuities, horizontal asymptotes, and vertical asymptotes of rational functions. This video explores the specific example f(x)=(3x^2-18x-81)/(6x^2-54) before generalizing findings to all rational functions. Don't forget that not every zero of the denominator is a vertical asymptote!https://cdn.kastatic.org/ka-youtube-converted/P0ZgqB44Do4.mp4/P0ZgqB44Do4.mp4682yes https://puzzicle.org/math/precalculus/v/graphs-of-rational-functions-y-intercepthttps://puzzicle.org/thumbs/xc5366e26.jpgGraphs of rational functions: y-interceptSal picks the graph that matches f(x)=(ax^m+bx+12)/(cx^m+dx+12) based on its y-intercept.https://cdn.kastatic.org/ka-youtube-converted/YElHmwdbkzQ.mp4/YElHmwdbkzQ.mp4188yes https://puzzicle.org/math/precalculus/v/graphs-of-rational-functions-horizontal-asymptotehttps://puzzicle.org/thumbs/x06669d97.jpgGraphs of rational functions: horizontal asymptoteSal picks the graph that matches f(x)=(-x²+ax+b)/(x²+cx+d) based on its horizontal asymptote.https://cdn.kastatic.org/ka-youtube-converted/qAfETdORmcw.mp4/qAfETdORmcw.mp4196yes https://puzzicle.org/math/precalculus/v/graphs-of-rational-functions-vertical-asymptoteshttps://puzzicle.org/thumbs/xcb7ef0f1.jpgGraphs of rational functions: vertical asymptotesSal picks the graph that matches f(x)=g(x)/(x²-x-6) (where g(x) is a polynomial) based on its discontinuities.https://cdn.kastatic.org/ka-youtube-converted/-jXk1gl_sEo.mp4/-jXk1gl_sEo.mp4333yes https://puzzicle.org/math/precalculus/v/graphs-of-rational-functions-zeroshttps://puzzicle.org/thumbs/x70aa392a.jpgGraphs of rational functions: zerosSal picks the graph that matches f(x)=(2x²-18)/g(x) (where g(x) is a polynomial) based on its zeros.https://cdn.kastatic.org/ka-youtube-converted/TX_mx3qULpw.mp4/TX_mx3qULpw.mp4467yes https://puzzicle.org/math/precalculus/v/dogs-cats-and-bears-in-a-pet-store-visual-argumenthttps://puzzicle.org/thumbs/x65947499.jpgAnalyzing structure word problem: pet store (1 of 2)Sal solves a word problem about the unknown number of bears, cats, and dogs in a pet store. This is part 1 where Sal uses a visual reasoning.https://cdn.kastatic.org/ka-youtube-converted/glEP04Pzap8.mp4/glEP04Pzap8.mp4354yes https://puzzicle.org/math/precalculus/v/dogs-cats-and-bears-in-a-pet-store-analytic-argumenthttps://puzzicle.org/thumbs/xd2f9789a.jpgAnalyzing structure word problem: pet store (2 of 2)Sal solves a word problem about the unknown number of bears, cats, and dogs in a pet store. This is part 2 where Sal uses an algebraic reasoning.https://cdn.kastatic.org/ka-youtube-converted/mpHMuaVW0DI.mp4/mpHMuaVW0DI.mp4164yes https://puzzicle.org/math/precalculus/v/combining-mixtures-examplehttps://puzzicle.org/thumbs/x210e0e3e0fe45fe7.jpgCombining mixtures exampleFigure out how much of two gasoline mixtures to combine to get gasoline with a certain concentration of ethanol.https://cdn.kastatic.org/ka-youtube-converted/xMADEddxSEo.mp4/xMADEddxSEo.mp4348yes https://puzzicle.org/math/precalculus/v/applying-rational-equations-1https://puzzicle.org/thumbs/8187347.jpgRational equations word problem: combined ratesSal solves a word problem about the combined leaf-bagging rates of Ian and Kyandre, by creating a rational equation that models the situation.https://cdn.kastatic.org/ka-youtube-converted/gD7A1LA4jO8.mp4/gD7A1LA4jO8.mp4326yes https://puzzicle.org/math/precalculus/v/applying-rational-equations-2https://puzzicle.org/thumbs/27075.jpgRational equations word problem: combined rates (example 2)Sal solves a word problem about the combined deck-staining rates of Anya and Bill, by creating a rational equation that models the situation.https://cdn.kastatic.org/ka-youtube-converted/E1j8W64NQ0Y.mp4/E1j8W64NQ0Y.mp4360yes https://puzzicle.org/math/precalculus/v/applying-rational-equations-3https://puzzicle.org/thumbs/27076.jpgRational equations word problem: eliminating solutionsSal solves a word problem about the combined pool-filling rates of two water hoses, by creating a rational equation that models the situation. The equation has a solution that is eliminated due to the context.https://cdn.kastatic.org/ka-youtube-converted/B0Z4s38YIgQ.mp4/B0Z4s38YIgQ.mp4479yes https://puzzicle.org/math/precalculus/v/reasoning-through-inequality-expressionshttps://puzzicle.org/thumbs/xf89c9064.jpgReasoning about unknown variablesSal solves the following challenge: Given that a>0, b<0, and a/b>a*b, what further can we learn about the variables?https://cdn.kastatic.org/ka-youtube-converted/0_VaUYoNV7Y.mp4/0_VaUYoNV7Y.mp4367yes https://puzzicle.org/math/precalculus/v/using-expressions-to-understand-relationshipshttps://puzzicle.org/thumbs/xec00e95f.jpgReasoning about unknown variables: divisibilitySal solves the following challenge: Given the positive integers a, b, and c, where a is a multiple of c and (a+b)/c is an integer. Is b necessarily a multiple of c?https://cdn.kastatic.org/ka-youtube-converted/zTLxxezKUT8.mp4/zTLxxezKUT8.mp4218yes https://puzzicle.org/math/precalculus/v/structure-in-rational-expressionhttps://puzzicle.org/thumbs/x9a692344.jpgStructure in rational expressionAnalyzing an elaborate rational expression to determine what's bigger: the value of that expression at some unknown c, or the number of times the expression is equal to 0? This is algebraic reasoning at its best!https://cdn.kastatic.org/ka-youtube-converted/mNgW6SAQLhk.mp4/mNgW6SAQLhk.mp4281yes https://puzzicle.org/math/precalculus/v/multiplying-rational-expressionshttps://puzzicle.org/thumbs/x90ca5af3f8374426.jpgMultiplying rational expressionsWhen we multiply rational expressions, we multiply both numerators and multiply both denominators. We can also see if we can reduce the product to lowest terms. This is very similar to multiplying fractions, only we also have to think about the domain while we do it.https://cdn.kastatic.org/ka-youtube-converted/fjJ0C7-N-xU.mp4/fjJ0C7-N-xU.mp4334yes https://puzzicle.org/math/precalculus/v/dividing-rational-expressionshttps://puzzicle.org/thumbs/xb2c6a8969d9d3f67.jpgDividing rational expressionsWhen we divide rational expressions, we multiply the dividend (the first expression) by the reciprocal of the divisor (the second expression). We can also see if we can reduce the quotient to lowest terms. This is very similar to dividing fractions, only we also have to think about the domain while we do it.https://cdn.kastatic.org/ka-youtube-converted/0cmhBgQNGDI.mp4/0cmhBgQNGDI.mp4384yes https://puzzicle.org/math/precalculus/v/least-common-multiples-of-polynomialshttps://puzzicle.org/thumbs/x58140338.jpgLeast common multiple of polynomialsSal finds the least common multiple (LCM) of 3z³-6z²-9z and 7z⁴+21z³+14z².https://cdn.kastatic.org/ka-youtube-converted/Ws03IbNrjfM.mp4/Ws03IbNrjfM.mp4412yes https://puzzicle.org/math/precalculus/v/adding-and-subtracting-rational-expressions-3https://puzzicle.org/thumbs/27019.jpgSubtracting rational expressionsSal subtracts and simplifies (a-2)/(a+2) - (a-3)/(a²+4a+4).https://cdn.kastatic.org/ka-youtube-converted/IKsi-DQU2zo.mp4/IKsi-DQU2zo.mp4276yes https://puzzicle.org/math/algebra-2-tx/v/domain-and-range-of-parent-functionshttps://puzzicle.org/thumbs/xfd6e995c5d95cd0a.jpgDomain and range of parent functionsThis video reviews domain and range in the context of parent functions.https://cdn.kastatic.org/ka-youtube-converted/6Paqbr7CyO8.mp4/6Paqbr7CyO8.mp4195yes https://puzzicle.org/math/algebra-2-tx/v/formulating-absolute-value-linear-equationshttps://puzzicle.org/thumbs/xe2b49e4b9ca02c83.jpgFormulating absolute value linear equationsSal solves an example problem about formulating an absolute value equation that matches a real-world context.https://cdn.kastatic.org/ka-youtube-converted/lLRm1kUgIzc.mp4/lLRm1kUgIzc.mp4281yes https://puzzicle.org/math/algebra-home/v/absolute-inequalities-2https://puzzicle.org/thumbs/26817.jpgSolving absolute value inequalities: fractionsSal solves the inequality |2r-3 1/4| < 2 1/2.https://cdn.kastatic.org/ka-youtube-converted/x5EJG_rAtkY.mp4/x5EJG_rAtkY.mp4450yes https://puzzicle.org/math/algebra-home/v/writing-and-using-inequalitieshttps://puzzicle.org/thumbs/26819.jpgAbsolute value inequalities word problemSal solves a word problem about a carpenter by writing an appropriate absolute value inequality and solving it.https://cdn.kastatic.org/ka-youtube-converted/RHe9X2HDEjA.mp4/RHe9X2HDEjA.mp4278yes https://puzzicle.org/math/precalculus/v/representing-systems-of-equations-with-matriceshttps://puzzicle.org/thumbs/xa9a6ec209a5d8147.jpgRepresenting systems of equations with matricesGiven a system of 3 linear equations with 3 variables, we can represent it with the equation Ax=b, where A is a 3x3 matrix, x is a 3D vector with the variables, and b is a constant 3D vector. Why is this useful? We can solve the system by solving the equation, which in many cases is much more efficient.https://cdn.kastatic.org/ka-youtube-converted/V8lIpRHjthk.mp4/V8lIpRHjthk.mp4410yes https://puzzicle.org/math/precalculus/v/representing-systems-of-any-number-of-equations-with-matriceshttps://puzzicle.org/thumbs/x623f12621cbc424b.jpgRepresenting systems of any number of equations with matricesIn a previous videos, we saw how to represent a system with 3 equations and variables as a single matrix equation. It turns out we can do the same for a system of n equations and variables. The resulting equation will be Ax=b, where A is an nXn matrix, x is an nX1 unknown vector, and b is the nX1 constant vector.https://cdn.kastatic.org/ka-youtube-converted/lVle8hJTZAk.mp4/lVle8hJTZAk.mp4289yes https://puzzicle.org/math/algebra-2-tx/v/solving-systems-of-three-linear-equations-in-three-variables-substitutionhttps://puzzicle.org/thumbs/xb5c5b1e8baed6ca5.jpgSolving systems of three linear equations in three variables: substitutionSal demonstrates how to solve a system of three linear equations in three variables using substitution.https://cdn.kastatic.org/ka-youtube-converted/umCBMJI_nNU.mp4/umCBMJI_nNU.mp4372yes https://puzzicle.org/math/linear-algebra/v/matrices-reduced-row-echelon-form-2https://puzzicle.org/thumbs/25257.jpgSolving linear systems with matricesSal solves a linear system with 3 variables by representing it with an augmented matrix and bringing the matrix to reduced row-echelon form.https://cdn.kastatic.org/ka-youtube-converted/lP1DGtZ8Wys.mp4/lP1DGtZ8Wys.mp4457yes https://puzzicle.org/math/algebra-home/v/synthetic-division-example-2https://puzzicle.org/thumbs/948968139.jpgDividing polynomials: synthetic divisionSal divides (2x^5-x^3+3x^2-2x+7) by (x-3) using synthetic division.https://cdn.kastatic.org/ka-youtube-converted/3Ee_huKclEQ.mp4/3Ee_huKclEQ.mp4300yes https://puzzicle.org/math/algebra-2-tx/v/solving-equations-with-rational-exponentshttps://puzzicle.org/thumbs/xf6c818e2d8c41803.jpgSolving equations with rational exponentsSal solves two equations with rational exponents, one with a decimal exponent and one using radical notation.https://cdn.kastatic.org/ka-youtube-converted/dopR7YdndHE.mp4/dopR7YdndHE.mp4277yes https://puzzicle.org/math/algebra-2-tx/v/writing-quadratic-functions-given-3-pointshttps://puzzicle.org/thumbs/xfbc119fa3f61f75f.jpgWriting quadratic functions given 3 pointsSal shows how to use the general form quadratic equation to write the equation of a parabola given any three points on it.https://cdn.kastatic.org/ka-youtube-converted/tirQoI10a1A.mp4/tirQoI10a1A.mp4427yes https://puzzicle.org/math/geometry/v/equation-for-parabola-from-focus-and-directrixhttps://puzzicle.org/thumbs/xf4347fc5.jpgEquation of a parabola from focus & directrixThe equation of a parabola is derived from the focus and directrix, and then the general formula is used to solve an example.https://cdn.kastatic.org/ka-youtube-converted/okXVhDMuGFg.mp4/okXVhDMuGFg.mp4588yes https://puzzicle.org/math/algebra2/v/quadratic-inequality-word-problemhttps://puzzicle.org/thumbs/x8b9f273d.jpgQuadratic inequality word problemSal models a context that concerns a candy vending machine. The model turns out to be a quadratic inequality.https://cdn.kastatic.org/ka-youtube-converted/GDppV18XDCs.mp4/GDppV18XDCs.mp4600yes https://puzzicle.org/math/algebra2/v/introduction-to-complex-numbershttps://puzzicle.org/thumbs/xc316d3f4.jpgClassifying complex numbersDive into expressions involving the imaginary unit 'i', simplifying them to reveal the real and imaginary components. Learn how real and imaginary numbers are subsets of complex numbers and study how to classify numbers as real, pure imaginary, or complex.https://cdn.kastatic.org/ka-youtube-converted/A_ESfuN1Pkg.mp4/A_ESfuN1Pkg.mp4279yes https://puzzicle.org/math/algebra-2-tx/v/formulating-quadratic-and-square-root-equations-using-graphs-part-1https://puzzicle.org/thumbs/xf497f0f853f8c779.jpgFormulating quadratic and square-root equations using graphs - Part 1Sal demonstrates how to use vertex form to write the equation of a quadratic function given a table of values.https://cdn.kastatic.org/ka-youtube-converted/3Puz8m-vLUc.mp4/3Puz8m-vLUc.mp4275yes https://puzzicle.org/math/algebra-2-tx/v/formulating-quadratic-and-square-root-equations-using-graphs-part-2https://puzzicle.org/thumbs/x6748db709c9b2f74.jpgFormulating quadratic and square-root equations using graphs - Part 2Sal demonstrates how to use vertex form to write the equation of a quadratic function given a table of values, and how to use this equation to write the equation of the inverse square root function.https://cdn.kastatic.org/ka-youtube-converted/r_s4NIjFXmA.mp4/r_s4NIjFXmA.mp4396yes https://puzzicle.org/math/algebra-2-tx/v/transforming-a-cubic-functionhttps://puzzicle.org/thumbs/x5f12a1bf136d6717.jpgGraphing cubic function transformationsSal demonstrates the relationship between changes to the equation of the parent function x^3 and transformations of its original graph.https://cdn.kastatic.org/ka-youtube-converted/gpaRJMGFmyI.mp4/gpaRJMGFmyI.mp4183yes https://puzzicle.org/math/ap-precalculus/v/transforming-rational-functionhttps://puzzicle.org/thumbs/x36793f71389bbd37.jpgGraphing rational function transformationsSal demonstrates the relationship between changes to the equation of the parent function 1/x and transformations of its original graph.https://cdn.kastatic.org/ka-youtube-converted/GsOBAUUNzQg.mp4/GsOBAUUNzQg.mp4277yes https://puzzicle.org/math/algebra2/v/making-more-pizzas-to-spread-cost-per-pizzahttps://puzzicle.org/thumbs/x5d5d623c.jpgRational equation word problemSal models a context that concerns the number and price of pizza slices. The model turns out to be a rational equation.https://cdn.kastatic.org/ka-youtube-converted/jQ15tkoXZoA.mp4/jQ15tkoXZoA.mp4540yes https://puzzicle.org/math/algebra-2-tx/v/intro-to-direct-inverse-variation-indian-accenthttps://puzzicle.org/thumbs/xc2941f2cee8c6163.jpgIntro to direct & inverse variationThis video explains what it means for quantities to vary directly or inversely, and gives many examples of both types of variation.https://cdn.kastatic.org/ka-youtube-converted/OXF_xaAIjf4.mp4/OXF_xaAIjf4.mp4593yes https://puzzicle.org/math/algebra-home/v/recognizing-direct-and-inverse-variationhttps://puzzicle.org/thumbs/62754977.jpgRecognizing direct & inverse variationSal gives many examples of two-variable equations where the variables vary directly, inversely, or neither.https://cdn.kastatic.org/ka-youtube-converted/3_DxJwDTbyQ.mp4/3_DxJwDTbyQ.mp4425yes https://puzzicle.org/math/algebra-home/v/direct-inverse-and-joint-variationhttps://puzzicle.org/thumbs/124291559.jpgRecognizing direct & inverse variation: tableSal is given a table with a few values of the variables x and y, and determines whether the variables vary directly or inversely.https://cdn.kastatic.org/ka-youtube-converted/v-k5L0BPOmc.mp4/v-k5L0BPOmc.mp4330yes https://puzzicle.org/math/algebra-home/v/inverse-variation-applicationhttps://puzzicle.org/thumbs/124291567.jpgInverse variation word problem: string vibrationSal models a context about lengths of strings and the frequency of their vibrations! with an inverse variation equation.https://cdn.kastatic.org/ka-youtube-converted/lkP-E2LUnjA.mp4/lkP-E2LUnjA.mp4282yes https://puzzicle.org/math/algebra2/v/constructing-an-exponential-equation-examplehttps://puzzicle.org/thumbs/x45286516.jpgExponential equation word problemSal models a context that concerns a bank savings account. The model turns out to be an exponential equation.https://cdn.kastatic.org/ka-youtube-converted/IlFD0LzAZeo.mp4/IlFD0LzAZeo.mp4528yes https://puzzicle.org/math/college-algebra/v/solving-logarithmic-equations-2https://puzzicle.org/thumbs/xe2ac1cd0b3c5ed9a.jpgSolving logarithmic equationsSal solves a logarithmic equation with a fractional argument.https://cdn.kastatic.org/ka-youtube-converted/ZQUuhYgjqmw.mp4/ZQUuhYgjqmw.mp4125yes https://puzzicle.org/math/college-algebra/v/solving-real-world-logarithmic-equationshttps://puzzicle.org/thumbs/x44947ec1ef9ad4a3.jpgSolving real world logarithmic equationsSal solves a real world example of a logarithmic equation.https://cdn.kastatic.org/ka-youtube-converted/cZUpKBALvTU.mp4/cZUpKBALvTU.mp4135yes https://puzzicle.org/math/algebra-basics/v/absolute-value-and-number-lineshttps://puzzicle.org/thumbs/2344592.jpgAbsolute value and number linesOne easy way to think of absolute value is the distance it is from zero. To do that, a number line comes in handy. Watch and learn.https://cdn.kastatic.org/ka-youtube-converted/frBJEYvyd-8.mp4/frBJEYvyd-8.mp4339yes https://puzzicle.org/math/algebra-basics/v/understanding-exponents-2https://puzzicle.org/thumbs/2600693.jpgExponent example 1Sal explains the difference between 5 × 3 and 5 to the third powerhttps://cdn.kastatic.org/ka-youtube-converted/5qfOViJda_g.mp4/5qfOViJda_g.mp475yes https://puzzicle.org/math/algebra-basics/v/understanding-exponentshttps://puzzicle.org/thumbs/2600692.jpgExponent example 2We can write a number multiplied by itself multiple times in exponential notation. 6 to the 8th power means we have 8 factors of 6, not that we multiply 6 by 8. 6 to the 8th power is a much larger number than 6 times 8.https://cdn.kastatic.org/ka-youtube-converted/mJ1P4A-KA8k.mp4/mJ1P4A-KA8k.mp4115yes https://puzzicle.org/math/algebra-basics/v/rewriting-square-root-of-fractionhttps://puzzicle.org/thumbs/x1918d185.jpgSimplifying square roots of fractionsWe can simplify the √(1/200) by finding perfect squares that are factors of 200. We could either look directly for the largest perfect square factor or break 200 into smaller factors and find repeated factors. If we have √2 in the denominator, we can multiply by √2/√2 to make an equivalent fraction.https://cdn.kastatic.org/ka-youtube-converted/s9ppnjgmiyk.mp4/s9ppnjgmiyk.mp4281yes https://puzzicle.org/math/algebra-basics/v/converting-decimals-to-fractions-2-ex-1https://puzzicle.org/thumbs/915110987.jpgConverting decimals to fractions 2 (ex 1)To convert a decimal to a fraction, we need to figure out the smallest place value of the digits. In this case, the smallest place is the ten-thousandths place, so the fraction will have a denominator of 10,000. The numerator will be the decimal number without the decimal point. Then we check for common factors to simplify.https://cdn.kastatic.org/ka-youtube-converted/EGr3KC55sfU.mp4/EGr3KC55sfU.mp474yes https://puzzicle.org/math/algebra-basics/v/multiplying-decimalshttps://puzzicle.org/thumbs/1812.jpgMultiplying decimals exampleTo multiply decimals, we multiply them just like whole numbers. We count the number of digits behind the decimal in both numbers we're multiplying, and make sure our answer has the same number of digits after the decimal. - We can check our answer by thinking about what it means to multiply by 0.5 (or one half).https://cdn.kastatic.org/ka-youtube-converted/JEHejQphIYc.mp4/JEHejQphIYc.mp4172yes https://puzzicle.org/math/algebra-basics/v/algebraic-expressions-with-fraction-divisionhttps://puzzicle.org/thumbs/xa376e0ea.jpgNested fractionsSal starts with a simple case of multiplying two fractions and proceeds to simplifying a nested fraction expression.https://cdn.kastatic.org/ka-youtube-converted/_BFaxpf35sY.mp4/_BFaxpf35sY.mp4346yes https://puzzicle.org/math/algebra-basics/v/proportions-2-exercise-exampleshttps://puzzicle.org/thumbs/x8802e219.jpgSolving tricky proportionsSal solves the equation (x-9)/12=2/3.https://cdn.kastatic.org/ka-youtube-converted/qzsR7cChujg.mp4/qzsR7cChujg.mp4264yes https://puzzicle.org/math/algebra-basics/v/slope-and-rate-of-changehttps://puzzicle.org/thumbs/23636.jpgSlope (more examples)Given two points on a line, you can find the slope of the line. Watch Sal doing a bunch of examples.https://cdn.kastatic.org/ka-youtube-converted/Iqws-qzyZwc.mp4/Iqws-qzyZwc.mp4822yes https://puzzicle.org/math/algebra-basics/v/factoring-a-common-binomial-factorhttps://puzzicle.org/thumbs/xb16e9511.jpgFactoring polynomials: common binomial factorSal factors n(n-1)+3(n-1) as (n+3)(n-1) by noticing that (n-1) is a common factor.https://cdn.kastatic.org/ka-youtube-converted/tDlzWNtRzS8.mp4/tDlzWNtRzS8.mp4102yes https://puzzicle.org/math/algebra-basics/v/factoring-difference-of-squares-exercise-examplehttps://puzzicle.org/thumbs/xe909c411.jpgFactoring difference of squares: missing valuesSal analyzes the factorization of 3y^3-100y as 4y(My+g)(My-g) to find the possible values for the missing coefficient g.https://cdn.kastatic.org/ka-youtube-converted/IRmv-Z--3LA.mp4/IRmv-Z--3LA.mp4350yes https://puzzicle.org/math/algebra-basics/v/segment-additionhttps://puzzicle.org/thumbs/xf0270049.jpgEquation practice with segment additionIf we split a line segment into two segments, the length of the total segment equals the sums of the lengths of the two smaller segments. We can make an equation out of the given information. After we solve for the value of x, we can substitute that value back into the expression for the length we are seeking.https://cdn.kastatic.org/ka-youtube-converted/YFqeMzfcyg4.mp4/YFqeMzfcyg4.mp4221yes https://puzzicle.org/math/algebra-basics/v/midpoint-of-a-segmenthttps://puzzicle.org/thumbs/x73a3d40c.jpgEquation practice with midpointsGiven an information about the distance of a segment's midpoint from its endpoints, Sal forms and solves an equation in order to find the length of the segment.https://cdn.kastatic.org/ka-youtube-converted/E0TNh9uWesw.mp4/E0TNh9uWesw.mp4199yes https://puzzicle.org/math/algebra-basics/v/pythagorean-theoremhttps://puzzicle.org/thumbs/23754.jpgIntro to the Pythagorean theorem 2Sal introduces the famous and super important Pythagorean theorem!https://cdn.kastatic.org/ka-youtube-converted/AEIzy1kNRqo.mp4/AEIzy1kNRqo.mp4783yes https://puzzicle.org/math/algebra-home/v/dependent-and-independent-variables-exercise-example-1https://puzzicle.org/thumbs/x063b5c9e.jpgDependent & independent variablesIdentifying the dependent and the independent variables in a word problem.https://cdn.kastatic.org/ka-youtube-converted/i9j_VUMq5yg.mp4/i9j_VUMq5yg.mp4130yes https://puzzicle.org/math/algebra-home/v/square-roots-and-real-numbershttps://puzzicle.org/thumbs/23610.jpgSquare roots and real numbers (old)An old video of Sal where he simplifies square roots in order to determine whether they represent rational or irrational numbers.https://cdn.kastatic.org/ka-youtube-converted/BpBh8gvMifs.mp4/BpBh8gvMifs.mp4619yes https://puzzicle.org/math/algebra-home/v/number-systems-introductionhttps://puzzicle.org/thumbs/x00087af5.jpgIntroduction to number systems and binaryThe base 10 (decimal) system is the most common number system used by humans, but there are other important and useful number systems. For example, base 2, called binary system, is the basis of modern computing. We can convert between the decimal form and binary form of a number to solve different problems.https://cdn.kastatic.org/ka-youtube-converted/ku4KOFQ-bB4.mp4/ku4KOFQ-bB4.mp4600yes https://puzzicle.org/math/algebra-home/v/hexadecimal-number-systemhttps://puzzicle.org/thumbs/x3ba7f09a.jpgHexadecimal number systemThe hexadecimal number system is a way of counting that uses 16 different symbols instead of just the ten we're used to. It uses the numbers 0-9 like we do, but it also uses the letters A-F to represent the numbers 10-15. This system is useful in computer science because it can represent large numbers in a shorter, more compact way.https://cdn.kastatic.org/ka-youtube-converted/4EJay-6Bioo.mp4/4EJay-6Bioo.mp4454yes https://puzzicle.org/math/algebra-home/v/decimal-to-binaryhttps://puzzicle.org/thumbs/x47887da0.jpgConverting from decimal to binaryTo convert a number from decimal to binary, we need to figure out which combination of 0s and 1s will represent the number. Start by finding the largest power of 2 that fits into the number, write down a "1" to represent that power of 2, and subtract it from the number. We keep going until we reach the smallest power of 2, which is 1.https://cdn.kastatic.org/ka-youtube-converted/H4BstqvgBow.mp4/H4BstqvgBow.mp4232yes https://puzzicle.org/math/algebra-home/v/large-number-decimal-to-binaryhttps://puzzicle.org/thumbs/x72738886.jpgConverting larger number from decimal to binaryTo convert a number from decimal to binary, we need to figure out which combination of 0s and 1s will represent the number. Start by finding the largest power of 2 that fits into the number, write down a "1" to represent that power of 2, and subtract it from the number. We keep going until we reach the smallest power of 2, which is 1.https://cdn.kastatic.org/ka-youtube-converted/bvcXEJbEzSs.mp4/bvcXEJbEzSs.mp4252yes https://puzzicle.org/math/algebra-home/v/decimal-to-hexadecimalhttps://puzzicle.org/thumbs/x86cfb88f.jpgConverting from decimal to hexadecimal representationTo convert a decimal number to hexadecimal, you need to break it down into powers of 16. Start by finding the largest power of 16 that is less than or equal to the decimal number. Once the decimal number is expressed as the sum of powers of 16, you can write it in hexadecimal form using place values. Remember that in hexadecimal, the digits A-F represent the numbers 10-15.https://cdn.kastatic.org/ka-youtube-converted/NFmDz1dQyPU.mp4/NFmDz1dQyPU.mp4223yes https://puzzicle.org/math/algebra-home/v/binary-additionhttps://puzzicle.org/thumbs/xfe57d10d.jpgAdding in binaryWhen we add two numbers in binary, the key is that we can only use the digits zero and one. When we add two ones together, the result is two, but we have to write it as "10" in binary. Sometimes we have to "carry" a one over just like in regular addition.https://cdn.kastatic.org/ka-youtube-converted/RgklPQ8rbkg.mp4/RgklPQ8rbkg.mp4168yes https://puzzicle.org/math/algebra-home/v/binary-multiplicationhttps://puzzicle.org/thumbs/xbf936360.jpgMultiplying in binaryTo multiply numbers in base 2, we can use the same process as we would in base 10, but with only two digits: 0 and 1. Start by multiplying the rightmost digit of one number by each digit of the other number, working left. Keep track of any "carries" (when the product is 2, carry over a 1 to the next column). Finally, add up all the partial products.https://cdn.kastatic.org/ka-youtube-converted/xHWKYFhhtJQ.mp4/xHWKYFhhtJQ.mp4260yes https://puzzicle.org/math/algebra-home/v/binary-to-hexadecimalhttps://puzzicle.org/thumbs/x8e9be1be.jpgConverting directly from binary to hexadecimalTo convert from binary to hexadecimal, we can split the binary number into groups of four digits. Each group of four binary digits can be converted into one hexadecimal digit. We can use a table or chart to figure out which hexadecimal digit matches each group of four binary digits. Once we have converted each group, we put the hexadecimal digits together to get the final answer.https://cdn.kastatic.org/ka-youtube-converted/8T4F7WboWPQ.mp4/8T4F7WboWPQ.mp4327yes https://puzzicle.org/math/algebra-home/v/mixture-problems-2https://puzzicle.org/thumbs/26888.jpgLinear equation word problem: salineSal solves the following word problem: You have 50 ounces of a 25% saline solution. How many ounces of a 10% saline solution must you add to make a solution that is 15% saline?https://cdn.kastatic.org/ka-youtube-converted/OBVGQt1Eeug.mp4/OBVGQt1Eeug.mp4403yes https://puzzicle.org/math/algebra-home/v/mixture-problems-3https://puzzicle.org/thumbs/26889.jpgLinear equation word problem: sugary drinksSal solves the following problem: Drink A with 40% sugar is mixed with Drink B to obtain Drink C which has 25% sugar. What is the percentage of sugar of Drink B?https://cdn.kastatic.org/ka-youtube-converted/JVlfQEhzLMM.mp4/JVlfQEhzLMM.mp4247yes https://puzzicle.org/math/algebra-home/v/algebra-linear-equations-1https://puzzicle.org/thumbs/23426.jpgLinear equations 1To solve linear equations, find the value of the variable that makes the equation true. Use the inverse of the number that multiplies the variable, and multiply or divide both sides by it. Simplify the result to get the variable value. Check your answer by plugging it back into the equation.https://cdn.kastatic.org/ka-youtube-converted/bAerID24QJ0.mp4/bAerID24QJ0.mp4447yes https://puzzicle.org/math/algebra-home/v/algebra-linear-equations-2https://puzzicle.org/thumbs/23428.jpgLinear equations 2solving equations of the form AX+B=Chttps://cdn.kastatic.org/ka-youtube-converted/DopnmxeMt-s.mp4/DopnmxeMt-s.mp4367yes https://puzzicle.org/math/algebra-home/v/algebra-linear-equations-3https://puzzicle.org/thumbs/23430.jpgLinear equations 3Linear equations with multiple variable and constant termshttps://cdn.kastatic.org/ka-youtube-converted/Zn-GbH2S0Dk.mp4/Zn-GbH2S0Dk.mp4405yes https://puzzicle.org/math/algebra-home/v/algebra-linear-equations-4https://puzzicle.org/thumbs/23432.jpgLinear equations 4Solving linear equations with variable expressions in the denominators of fractionshttps://cdn.kastatic.org/ka-youtube-converted/9IUEk9fn2Vs.mp4/9IUEk9fn2Vs.mp4458yes https://puzzicle.org/math/algebra-home/v/averageshttps://puzzicle.org/thumbs/23450.jpgAveragesWe can calculate an average by adding up all the numbers in a group and dividing by the number of numbers. We can also use algebra to solve problems involving averages, and figure out how well we need to do on exams to get a good grade in a class.https://cdn.kastatic.org/ka-youtube-converted/9VZsMY15xeU.mp4/9VZsMY15xeU.mp4516yes https://puzzicle.org/math/algebra-home/v/taking-percentageshttps://puzzicle.org/thumbs/23454.jpgTaking percentagesTaking a percentage of a number.https://cdn.kastatic.org/ka-youtube-converted/_SpE4hQ8D_o.mp4/_SpE4hQ8D_o.mp4596yes https://puzzicle.org/math/algebra-home/v/more-percent-problemshttps://puzzicle.org/thumbs/23460.jpgMore percent problemsSlightly harder percent problemshttps://cdn.kastatic.org/ka-youtube-converted/4oeoIOan_h4.mp4/4oeoIOan_h4.mp4545yes https://puzzicle.org/math/algebra-home/v/solving-for-a-variablehttps://puzzicle.org/thumbs/124291556.jpgSolving an equation for a variableThe perimeter of a rectangle is equal to 2 times the length plus 2 times the width. We can solve for the length by isolating it on one side of the equation. We do this by subtracting 2 times the width from both sides, and then dividing both sides by 2. This gives us the formula for the length.https://cdn.kastatic.org/ka-youtube-converted/fnuIT7EhAvs.mp4/fnuIT7EhAvs.mp484yes https://puzzicle.org/math/algebra-home/v/solving-and-graphing-linear-inequalities-in-two-variables-1https://puzzicle.org/thumbs/26822.jpgSolving and graphing linear inequalitiesHow to graph on a number line and coordinate plane.https://cdn.kastatic.org/ka-youtube-converted/EoCeL4SPIcA.mp4/EoCeL4SPIcA.mp4135yes https://puzzicle.org/math/algebra-home/v/ordered-pair-solutions-of-equationshttps://puzzicle.org/thumbs/561901044.jpgSolutions to 2-variable equations: substitution (old)An old video of Sal checking whether (3,-4) is a solution of 5x+2y=7 by substituting x=3 and y=-4.https://cdn.kastatic.org/ka-youtube-converted/HUn4XwV7o9I.mp4/HUn4XwV7o9I.mp489yes https://puzzicle.org/math/algebra-home/v/ordered-pair-solutions-of-equations-2https://puzzicle.org/thumbs/466056216.jpgSolutions to 2-variable equations: graphical (old)An old video of Sal checking whether (3,-4) is a solution of 5x+2y=7 by graphing the equation and checking if (3,-4) is on the line.https://cdn.kastatic.org/ka-youtube-converted/SSNA9gaAOVc.mp4/SSNA9gaAOVc.mp4306yes https://puzzicle.org/math/algebra-home/v/graphing-using-x-and-y-interceptshttps://puzzicle.org/thumbs/33752748.jpgGraphing using intercepts (old)An old video of Sal where he draws the line y=3x-9 by finding its x- and y-intercetps.https://cdn.kastatic.org/ka-youtube-converted/6m642-2D3V4.mp4/6m642-2D3V4.mp4238yes https://puzzicle.org/math/algebra-home/v/equation-of-a-line-2https://puzzicle.org/thumbs/33752755.jpgSlope-intercept equation from slope & point (old)An old video where Sal finds the slope-intercept form of a line that has a slope of 7 and goes through the point (-4,-11).https://cdn.kastatic.org/ka-youtube-converted/MxiqyE2uMCo.mp4/MxiqyE2uMCo.mp4237yes https://puzzicle.org/math/algebra-home/v/equation-of-a-line-hairier-examplehttps://puzzicle.org/thumbs/466052138.jpgSlope-intercept equation from slope & point: fractions (old)An old video where Sal finds the slope-intercept form of a line that has a slope of ⅓ and goes through the point (-12,-14/3).https://cdn.kastatic.org/ka-youtube-converted/k8EfLFvUI5U.mp4/k8EfLFvUI5U.mp4204yes https://puzzicle.org/math/algebra-home/v/finding-y-intercept-given-slope-and-pointhttps://puzzicle.org/thumbs/x81716b8c.jpgFinding y intercept given slope & point (old)An old video where Sal finds the y-intercept of the line y=17/13x+b given that the line passes through (5,8).https://cdn.kastatic.org/ka-youtube-converted/KV_XLL4K2Fw.mp4/KV_XLL4K2Fw.mp4120yes https://puzzicle.org/math/algebra-home/v/interpreting-linear-formulas-1https://puzzicle.org/thumbs/xf811a89f.jpgLinear equations word problem: marblesSal finds the y-intercept and the slope of a linear relationship representing someone filling a glass tank with marbles. He then interprets what the y-intercept and the slope mean in that context.https://cdn.kastatic.org/ka-youtube-converted/deKCm28TkRo.mp4/deKCm28TkRo.mp4160yes https://puzzicle.org/math/algebra-home/v/interpreting-linear-formulas-2https://puzzicle.org/thumbs/x43f08c7b.jpgLinear equations word problem: file transferSal finds the y-intercept and the slope of a linear relationship representing a top secret file transfer. He then interprets what the y-intercept and the slope mean in that context.https://cdn.kastatic.org/ka-youtube-converted/AY8ldIEi3fU.mp4/AY8ldIEi3fU.mp4155yes https://puzzicle.org/math/algebra-home/v/linear-models-1https://puzzicle.org/thumbs/x1b99677b.jpgLinear models word problem: bookSal solves a word problem about a person reading a book. The solution involves the modeling of the situation as a linear function.https://cdn.kastatic.org/ka-youtube-converted/W3flX500w5g.mp4/W3flX500w5g.mp4532yes https://puzzicle.org/math/algebra-home/v/linear-models-2https://puzzicle.org/thumbs/x922ace72.jpgLinear models word problem: marblesSal solves a word problem about a person filling a glass tank with marbles. The solution involves the modeling of the situation as a linear function.https://cdn.kastatic.org/ka-youtube-converted/Rq56-OJR964.mp4/Rq56-OJR964.mp4337yes https://puzzicle.org/math/algebra-home/v/solving-systems-of-equations-by-eliminationhttps://puzzicle.org/thumbs/23678.jpgSolving systems of equations by elimination (old)An old video where Sal introduces the elimination method for systems of linear equations.https://cdn.kastatic.org/ka-youtube-converted/vA-55wZtLeE.mp4/vA-55wZtLeE.mp4764yes https://puzzicle.org/math/algebra-home/v/solving-linear-systems-by-substitutionhttps://puzzicle.org/thumbs/23676.jpgSolving linear systems by substitution (old)An old video where Sal introduces the substitution method for systems of linear equations.https://cdn.kastatic.org/ka-youtube-converted/V7H1oUHXPkg.mp4/V7H1oUHXPkg.mp4561yes https://puzzicle.org/math/algebra-home/v/manipulating-expressions-with-unknown-variables-1https://puzzicle.org/thumbs/xf33e100e.jpgManipulating expressions using structureGiven that a+b=0, Sal finds an expression equivalent to a*b.https://cdn.kastatic.org/ka-youtube-converted/ZYn8MhrwHr4.mp4/ZYn8MhrwHr4.mp495yes https://puzzicle.org/math/algebra-home/v/manipulating-expressions-with-unknown-variables-2https://puzzicle.org/thumbs/xf778691f.jpgManipulating expressions using structure (example 2)Given that a+b=2a, Sal finds an expression equivalent to b-a.https://cdn.kastatic.org/ka-youtube-converted/DF5MkPVjnCc.mp4/DF5MkPVjnCc.mp4161yes https://puzzicle.org/math/algebra-home/v/interpreting-equations-graphically-1https://puzzicle.org/thumbs/x6bfe3d09.jpgInterpreting equations graphicallySal interprets the intersection point of the graphs of functions f and g as the solution of the equation f(x)=g(x).https://cdn.kastatic.org/ka-youtube-converted/LyVqwJkI5jg.mp4/LyVqwJkI5jg.mp4245yes https://puzzicle.org/math/algebra-home/v/interpreting-equations-graphically-2https://puzzicle.org/thumbs/xad6730ce.jpgInterpreting equations graphically (example 2)Sal finds the coordinates of points on a graph that are solutions to a given equation.https://cdn.kastatic.org/ka-youtube-converted/R446VpTeOU8.mp4/R446VpTeOU8.mp4157yes https://puzzicle.org/math/algebra-home/v/estimating-a-solution-to-nonlinear-system-with-calculatorhttps://puzzicle.org/thumbs/x16a2de2b.jpgSolving equations graphically (1 of 2)Sal solves the equation e^x=1/[x(x-1)(x-2)] by considering the graphs of y=e^x and y=1/[x(x-1)(x-2)].https://cdn.kastatic.org/ka-youtube-converted/mXsn-YYUN4Y.mp4/mXsn-YYUN4Y.mp4483yes https://puzzicle.org/math/algebra-home/v/estimating-a-solution-to-nonlinear-system-with-calculator-part-2https://puzzicle.org/thumbs/x364f568c.jpgSolving equations graphically (2 of 2)Sal solves the equation e^x=1/[x(x-1)(x-2)] by considering the graphs of y=e^x and y=1/[x(x-1)(x-2)].https://cdn.kastatic.org/ka-youtube-converted/wUb7tWgv49U.mp4/wUb7tWgv49U.mp4265yes https://puzzicle.org/math/algebra-home/v/systems-of-three-variableshttps://puzzicle.org/thumbs/33752776.jpgIntro to linear systems with 3 variablesSal discusses how we approach solving systems of three variables algebraically, and how we visualize it graphically.https://cdn.kastatic.org/ka-youtube-converted/pANZpI9sOYg.mp4/pANZpI9sOYg.mp4503yes https://puzzicle.org/math/algebra-home/v/systems-of-three-variables-2https://puzzicle.org/thumbs/33752777.jpgSolving linear systems with 3 variablesSal solves a system of three variables using elimination.https://cdn.kastatic.org/ka-youtube-converted/f7cX-Ar2cEM.mp4/f7cX-Ar2cEM.mp4420yes https://puzzicle.org/math/algebra-home/v/solutions-to-three-variable-system-2https://puzzicle.org/thumbs/465991886.jpgSolving linear systems with 3 variables: no solutionSal solves a system with three variables that turns out to have no solution.https://cdn.kastatic.org/ka-youtube-converted/145zOBT0LD4.mp4/145zOBT0LD4.mp4306yes https://puzzicle.org/math/algebra-home/v/three-equation-application-problemhttps://puzzicle.org/thumbs/33752778.jpg3-variable linear system word problemSal solves a word problem about the angles of a given triangle by modeling the given information as a system of three equations and variables.https://cdn.kastatic.org/ka-youtube-converted/TJaHNxJk5-4.mp4/TJaHNxJk5-4.mp4496yes https://puzzicle.org/math/algebra-home/v/constraining-solutions-to-two-variable-linear-inequalitieshttps://puzzicle.org/thumbs/x1849794a.jpgConstraining solutions of two-variable inequalitiesSal determines which x-values make the ordered pair (x,-7) a solution of 2x-7y<25, both algebraically and graphically.https://cdn.kastatic.org/ka-youtube-converted/a5YZ7-JRSUE.mp4/a5YZ7-JRSUE.mp4252yes https://puzzicle.org/math/algebra-home/v/constraining-solutions-of-systems-of-inequalitieshttps://puzzicle.org/thumbs/x66f6081c.jpgConstraining solutions of systems of inequalitiesGiven the graph of a system of inequalities, Sal finds the x-values that make the ordered pair (x,-2) a solution of the system.https://cdn.kastatic.org/ka-youtube-converted/BGyrSuW3GCg.mp4/BGyrSuW3GCg.mp4348yes https://puzzicle.org/math/algebra-home/v/graphing-inequalities-2https://puzzicle.org/thumbs/33752760.jpgGraphing two-variable inequalities (old)An old video where Sal graphs the inequality y-4x<-3.https://cdn.kastatic.org/ka-youtube-converted/-aTy1ED1m5I.mp4/-aTy1ED1m5I.mp4320yes https://puzzicle.org/math/math1/v/analyzing-structure-with-linear-inequalities-1https://puzzicle.org/thumbs/x182dfd3b87d8eecf.jpgAnalyzing structure with linear inequalities: fruitsGiven a real-world context about the number of fruits bought, we find a linear inequality that correctly depicts the situation.https://cdn.kastatic.org/ka-youtube-converted/JeIwsn-EWZU.mp4/JeIwsn-EWZU.mp4233yes https://puzzicle.org/math/math1/v/analyzing-structure-with-linear-inequalities-2https://puzzicle.org/thumbs/xc92fb1665aa8ac6e.jpgAnalyzing structure with linear inequalities: ballsGiven a real-world context about the number of balls in a bag, we find a linear inequality that correctly depicts the situation.https://cdn.kastatic.org/ka-youtube-converted/Pfi_chISQbg.mp4/Pfi_chISQbg.mp4140yes https://puzzicle.org/math/algebra-home/v/domain-of-a-radical-functionhttps://puzzicle.org/thumbs/364967637.jpgDomain of a radical functionFinding the domain of f(x)=√(2x-8).https://cdn.kastatic.org/ka-youtube-converted/4h54s7BBPpA.mp4/4h54s7BBPpA.mp4107yes https://puzzicle.org/math/algebra-home/v/domain-of-algebraic-function-exampleshttps://puzzicle.org/thumbs/x43141392.jpgWorked example: domain of algebraic functionsMany examples of determining the domains of functions according to mathematical limitations.https://cdn.kastatic.org/ka-youtube-converted/n17q8CBiMtQ.mp4/n17q8CBiMtQ.mp4434yes https://puzzicle.org/math/algebra-home/v/average-rate-of-change-when-function-defined-by-equationhttps://puzzicle.org/thumbs/x4f2e6d3a.jpgWorked example: average rate of change from equationFinding the interval where a function has an average rate of change of ½ given its equation.https://cdn.kastatic.org/ka-youtube-converted/6n0yxfWNFR4.mp4/6n0yxfWNFR4.mp4304yes https://puzzicle.org/math/algebra-home/v/average-rate-of-change-with-function-notation-word-problemhttps://puzzicle.org/thumbs/x33e4fdd9.jpgAverage rate of change word problem: equationAverage rate of change tells us how much the function changed per a single time unit, over a specific interval. It has many real-world applications. In this video, we represent the average rate of water draining with an algebraic expression.https://cdn.kastatic.org/ka-youtube-converted/wmQdFLdxLYc.mp4/wmQdFLdxLYc.mp4273yes https://puzzicle.org/math/algebra-home/v/sum-of-functionshttps://puzzicle.org/thumbs/364981332.jpgAdding functionsGiven that f(x)=9-x^2 and g(x)=5x^2+2x+1, Sal finds (f+g)(x).https://cdn.kastatic.org/ka-youtube-converted/n34dqyVCXs4.mp4/n34dqyVCXs4.mp4153yes https://puzzicle.org/math/algebra-home/v/difference-of-functionshttps://puzzicle.org/thumbs/364967636.jpgSubtracting functionsGiven that f(x)=2x√5-4 and g(x)=x^2+2x√5-1, Sal finds (g-f)(x).https://cdn.kastatic.org/ka-youtube-converted/6arI8XhLZ9o.mp4/6arI8XhLZ9o.mp4136yes https://puzzicle.org/math/algebra-home/v/product-of-functionshttps://puzzicle.org/thumbs/364954028.jpgMultiplying functionsSal solves the following problem: given that f(x)=7x-5 and g(x)=x^3+4x, find (f*g)(x). He explains that generally, (f*g)(x)=f(x)*g(x).https://cdn.kastatic.org/ka-youtube-converted/JKvmAexeMgY.mp4/JKvmAexeMgY.mp4180yes https://puzzicle.org/math/algebra-home/v/quotient-of-functionshttps://puzzicle.org/thumbs/364954027.jpgDividing functionsSal solves the following problem: given that f(x)=2x^2+15x-8 and g(x)=x^2+10x+16, find (f/g)(x). He explains that generally, (f/g)(x)=f(x)/g(x).https://cdn.kastatic.org/ka-youtube-converted/u9v_bakOIcU.mp4/u9v_bakOIcU.mp4378yes https://puzzicle.org/math/precalculus/v/modeling-with-composite-functions-exampleshttps://puzzicle.org/thumbs/x11595a4f.jpgModeling with composite functionsSal determines the correct functions to compose (and the correct order) in order to model a given relationship, and vice versa.https://cdn.kastatic.org/ka-youtube-converted/pIMfRbznxKA.mp4/pIMfRbznxKA.mp4403yes https://puzzicle.org/math/precalculus/v/modeling-with-composite-functionshttps://puzzicle.org/thumbs/x7cd69a43.jpgModeling with composite functions: skydivingSal models the maximum speed of a skydiver, by using composing given formulas for the maximum speed as a function of parachute area and parachute area as a function of width.https://cdn.kastatic.org/ka-youtube-converted/UMTeKCLvv9E.mp4/UMTeKCLvv9E.mp4208yes https://puzzicle.org/math/algebra-home/v/function-inverses-example-2https://puzzicle.org/thumbs/26635.jpgFinding inverse functions: quadraticSal finds the inverse of f(x)=(x+2)^2+1.https://cdn.kastatic.org/ka-youtube-converted/aeyFb2eVH1c.mp4/aeyFb2eVH1c.mp4432yes https://puzzicle.org/math/algebra-home/v/function-inverses-example-3https://puzzicle.org/thumbs/26636.jpgFinding inverse functions: quadratic (example 2)Sal finds the inverse of f(x)=(x-1)^2-2.https://cdn.kastatic.org/ka-youtube-converted/Bq9cq9FZuNM.mp4/Bq9cq9FZuNM.mp4455yes https://puzzicle.org/math/algebra-home/v/finding-inverse-functions-radicalhttps://puzzicle.org/thumbs/x8af4cad8.jpgFinding inverse functions: radicalSal finds the inverse of h(x)=-∛(3x-6)+12.https://cdn.kastatic.org/ka-youtube-converted/JBqYTuA_WgE.mp4/JBqYTuA_WgE.mp4277yes https://puzzicle.org/math/algebra-home/v/domain-of-a-functionhttps://puzzicle.org/thumbs/23512.jpgDomain of advanced functionsSal covers many different kinds of functions and shows how to determine their domain.https://cdn.kastatic.org/ka-youtube-converted/U-k5N1WPk4g.mp4/U-k5N1WPk4g.mp4599yes https://puzzicle.org/math/algebra-home/v/finding-a-piecewise-function-definition-from-graphhttps://puzzicle.org/thumbs/x3af6a8cc.jpgGraphs of nonlinear piecewise functionsSal is given the graph of a piecewise function and several possible formulas. He determines which is the correct formula.https://cdn.kastatic.org/ka-youtube-converted/IJWDyPFXGyM.mp4/IJWDyPFXGyM.mp4263yes https://puzzicle.org/math/algebra-home/v/new-operator-definitionshttps://puzzicle.org/thumbs/185730830.jpgNew operator definitions 1Sal solves a few problems where a new operator is defined and Sal evaluates some expression with it.https://cdn.kastatic.org/ka-youtube-converted/ND-Bbp_q46s.mp4/ND-Bbp_q46s.mp4410yes https://puzzicle.org/math/algebra-home/v/new-definitions-2https://puzzicle.org/thumbs/1254312875.jpgNew operator definitions 2Worked examples of working with more newly defined operators.https://cdn.kastatic.org/ka-youtube-converted/soUmhhuF08Y.mp4/soUmhhuF08Y.mp4133yes https://puzzicle.org/math/algebra-home/v/solving-quadratic-equations-by-square-rootshttps://puzzicle.org/thumbs/23720.jpgSolving quadratics by taking square roots: challengeSal solves challenging quadratic equations like (4x+1)²-8=0 by taking the square root of both sides.https://cdn.kastatic.org/ka-youtube-converted/55G8037gsKY.mp4/55G8037gsKY.mp4663yes https://puzzicle.org/math/algebra-home/v/factoring-special-productshttps://puzzicle.org/thumbs/23712.jpgSolving quadratic equations by factoring (old)An old video where Sal solves a bunch of quadratic equations by using factorization methods.https://cdn.kastatic.org/ka-youtube-converted/BI_jmI4xRus.mp4/BI_jmI4xRus.mp4608yes https://puzzicle.org/math/algebra-home/v/quadratic-formula-1https://puzzicle.org/thumbs/27002.jpgUsing the quadratic formulaSal solves the equation -7q^2+2q+9=0 by using the quadratic formula.https://cdn.kastatic.org/ka-youtube-converted/iulx0z1lz8M.mp4/iulx0z1lz8M.mp4334yes https://puzzicle.org/math/algebra-home/v/quadratic-equations-in-standard-formhttps://puzzicle.org/thumbs/150088075.jpgWorked example: quadratic formulaSolving 6x²+3=2x-6 by rewriting in standard form and identifying the parameters a, b, and c, that can be used within the quadratic formula.https://cdn.kastatic.org/ka-youtube-converted/ty4Ohya4hdE.mp4/ty4Ohya4hdE.mp4141yes https://puzzicle.org/math/algebra-home/v/non-linear-systems-of-equations-1https://puzzicle.org/thumbs/27008.jpgQuadratic systems: graphical solutionSal solves a system of a quadratic equation and a linear equation by graphing both equations and looking for their intersections, and then checks the solution algebraically.https://cdn.kastatic.org/ka-youtube-converted/hjigR_rHKDI.mp4/hjigR_rHKDI.mp4344yes https://puzzicle.org/math/algebra-home/v/systems-of-nonlinear-equations-2https://puzzicle.org/thumbs/x0698a8ee.jpgQuadratic systems: both variables are squaredSal solves the system y=0.5x and 2x^2-y^2=7.https://cdn.kastatic.org/ka-youtube-converted/GQf1vjfxuo8.mp4/GQf1vjfxuo8.mp4214yes https://puzzicle.org/math/algebra-home/v/quadratic-inequality-examplehttps://puzzicle.org/thumbs/x63a87fa0.jpgQuadratic inequalitiesSal solves x^2+3x>10.https://cdn.kastatic.org/ka-youtube-converted/OP91XWBRI1w.mp4/OP91XWBRI1w.mp4461yes https://puzzicle.org/math/algebra-home/v/quadratic-inequality-example-2https://puzzicle.org/thumbs/x3de63390.jpgQuadratic inequalities (example 2)Sal solves -x(2x-14)≥24.https://cdn.kastatic.org/ka-youtube-converted/YS_3XRj74oo.mp4/YS_3XRj74oo.mp4474yes https://puzzicle.org/math/algebra-home/v/quadratic-inequalities-visual-explanationhttps://puzzicle.org/thumbs/23502.jpgQuadratic inequalities: graphical approachSal solves a few quadratic inequalities by moving all terms to one side of the inequality and graphing the resulting expression.https://cdn.kastatic.org/ka-youtube-converted/xdiBjypYFRQ.mp4/xdiBjypYFRQ.mp4593yes https://puzzicle.org/math/algebra-home/v/evaluating-a-polynomial-at-a-given-valuehttps://puzzicle.org/thumbs/13179511.jpgEvaluating polynomialsEvaluating 3x²-8x+7 when x=-2.https://cdn.kastatic.org/ka-youtube-converted/03yq7XsErqo.mp4/03yq7XsErqo.mp469yes https://puzzicle.org/math/algebra-home/v/simply-a-polynomialhttps://puzzicle.org/thumbs/13179512.jpgSimplifying polynomialsSimplifying the polynomial 3x²-8x+7+2x³-x²+8x-3 by combining like terms.https://cdn.kastatic.org/ka-youtube-converted/vN0aL-_vIKM.mp4/vN0aL-_vIKM.mp4217yes https://puzzicle.org/math/algebra-home/v/adding-and-subtracting-polynomials-3https://puzzicle.org/thumbs/26922.jpgAdding & subtracting multiple polynomialsSal simplifies (x³ + 3x - 6) + (-2x² + x - 2) - (3x - 4).https://cdn.kastatic.org/ka-youtube-converted/DMyhUb1pZT0.mp4/DMyhUb1pZT0.mp4147yes https://puzzicle.org/math/algebra-home/v/adding-polynomialshttps://puzzicle.org/thumbs/13179513.jpgAdding polynomials (old)An old video where Sal adds (5x²+8x-3) + (2x² - 7x + 13x).https://cdn.kastatic.org/ka-youtube-converted/Oe1PKI_6-38.mp4/Oe1PKI_6-38.mp4130yes https://puzzicle.org/math/algebra-home/v/subtracting-polynomials-in-two-variableshttps://puzzicle.org/thumbs/x206d2842.jpgSubtracting polynomials: two variablesSal subtracts -6x⁴-3x²y²+5y⁴ from 2x⁴-8x²y²-y⁴.https://cdn.kastatic.org/ka-youtube-converted/G9C7uz3XOh0.mp4/G9C7uz3XOh0.mp4224yes https://puzzicle.org/math/algebra-home/v/addition-and-subtraction-of-polynomialshttps://puzzicle.org/thumbs/23702.jpgPolynomials reviewQuickly review what polynomials are, common related terms (e.g. degree, coefficient, binomial, etc.), addition & subtraction of polynomials, and modeling area with polynomials.https://cdn.kastatic.org/ka-youtube-converted/ZgFXL6SEUiI.mp4/ZgFXL6SEUiI.mp4959yes https://puzzicle.org/math/algebra-home/v/special-products-of-polynomials-2https://puzzicle.org/thumbs/8187338.jpgSpecial products of binomials: two variablesSal finds the area of a square with side (6x-5y).https://cdn.kastatic.org/ka-youtube-converted/4fQeHtSdw80.mp4/4fQeHtSdw80.mp4312yes https://puzzicle.org/math/algebra-home/v/special-products-of-binomialshttps://puzzicle.org/thumbs/23706.jpgMore examples of special productsSal gives numerous examples of the two special binomial product forms: perfect squares and the difference of two squares.https://cdn.kastatic.org/ka-youtube-converted/bFtjG45-Udk.mp4/bFtjG45-Udk.mp4636yes https://puzzicle.org/math/algebra-home/v/special-products-of-polynomials-1https://puzzicle.org/thumbs/26976.jpgSquaring a binomial (old)An old video where Sal expands and simplifies (7x+10)².https://cdn.kastatic.org/ka-youtube-converted/4GcNzvILqtM.mp4/4GcNzvILqtM.mp4199yes https://puzzicle.org/math/algebra-home/v/missing-coefficients-in-binomial-producthttps://puzzicle.org/thumbs/xe2571c46.jpgMultiplying binomials by polynomials challengeSal finds the values of coefficients a and b that make (2x+4)(5x-9)=ax²+bx-46 true for all x-values.https://cdn.kastatic.org/ka-youtube-converted/ieC5LgoGJVg.mp4/ieC5LgoGJVg.mp4283yes https://puzzicle.org/math/algebra-home/v/multiplication-of-polynomialshttps://puzzicle.org/thumbs/23704.jpgMultiplying binomials by polynomials (old)An old video where Sal gives several examples of polynomial multiplication.https://cdn.kastatic.org/ka-youtube-converted/fGThIRpWEE4.mp4/fGThIRpWEE4.mp4589yes https://puzzicle.org/math/algebra-home/v/multiplying-binomials-with-radicalshttps://puzzicle.org/thumbs/127771049.jpgMultiplying binomials with radicals (old)An old video Where Sal multiplies and simplifies (x²-√6)(x²+√2).https://cdn.kastatic.org/ka-youtube-converted/yAH3722GrP8.mp4/yAH3722GrP8.mp4364yes https://puzzicle.org/math/algebra-home/v/multiplying-polynomials-3https://puzzicle.org/thumbs/26975.jpgPolynomial multiplication word problemSal finds the volume of a tank whose base has an area of 3x² + 30x + 5 square feet and whose height is 8x - 5 feet.https://cdn.kastatic.org/ka-youtube-converted/bamcYQDzVTw.mp4/bamcYQDzVTw.mp4259yes https://puzzicle.org/math/algebra-home/v/polynomials1https://puzzicle.org/thumbs/8187332.jpgPolynomial word problem: rectangle and circle areaSal writes a binomial to express the difference between the area of a rectangle and the area of a circle.https://cdn.kastatic.org/ka-youtube-converted/EvvxBdNIUeQ.mp4/EvvxBdNIUeQ.mp4277yes https://puzzicle.org/math/algebra-home/v/polynomials-2https://puzzicle.org/thumbs/26919.jpgPolynomial word problem: total value of billsSal writes a polynomial to express the total value of p twenty dollar bills, q ten dollar bills, and r five dollar bills.https://cdn.kastatic.org/ka-youtube-converted/IGs7IB48Fvg.mp4/IGs7IB48Fvg.mp484yes https://puzzicle.org/math/algebra-home/v/multiplying-polynomials-2https://puzzicle.org/thumbs/26974.jpgPolynomial word problem: area of a windowSal writes a polynomial expression that represents the total area of a window made of wood and glass.https://cdn.kastatic.org/ka-youtube-converted/J1HAY8E3gms.mp4/J1HAY8E3gms.mp4288yes https://puzzicle.org/math/algebra2/v/ways-to-factor-a-monomialhttps://puzzicle.org/thumbs/x846ba409.jpgWhich monomial factorization is correct?Factoring monomials involves breaking down an expression into two parts. This process shows that there are multiple correct ways to factor a monomial. By multiplying coefficients and powers, you can verify if the factorization is accurate. It's a fun exploration of the many ways to dissect monomials!https://cdn.kastatic.org/ka-youtube-converted/_MCPV8CQzf8.mp4/_MCPV8CQzf8.mp4126yes https://puzzicle.org/math/algebra2/v/finding-algebraic-factor-of-a-monomialhttps://puzzicle.org/thumbs/xa5ca219a.jpgWorked example: finding the missing monomial factorFollow along as Sal works an example to show you how to factor monomials by isolating variables in equations. Learn how to use exponent properties to rewrite complex expressions and get a firm grasp on the crucial role that coefficients and exponents play in this process. This knowledge will empower you to solve for unknown factors in any monomial equation.https://cdn.kastatic.org/ka-youtube-converted/kkrF9X6Fycg.mp4/kkrF9X6Fycg.mp4199yes https://puzzicle.org/math/algebra2/v/dimensions-of-rectangle-for-algebraically-described-areahttps://puzzicle.org/thumbs/x498ab758.jpgWorked example: finding missing monomial side in area modelLearn how to find the length of a rectangle using algebraic expressions, as Sal finds the length of a rectangle whose area is 42xy^3 and whose height is 14xy. Dive into dividing coefficients and variables to solve for length and uncover how dividing powers of 'y' can simplify your equation.https://cdn.kastatic.org/ka-youtube-converted/UYh7Sj_yR8c.mp4/UYh7Sj_yR8c.mp4129yes https://puzzicle.org/math/algebra2/v/monomial-greatest-common-factorhttps://puzzicle.org/thumbs/13179576.jpgGreatest common factor of monomialsFollow along as Sal finds the greatest common factor of 10cd^2 and 25c^3d^2 and discover the secret to finding the greatest common factor of monomials! Dive into prime factorization and variable parts, and learn how to break down monomials into their simplest forms. Uncover the common factors and master the art of algebraic expressions.https://cdn.kastatic.org/ka-youtube-converted/_01wqwsb66E.mp4/_01wqwsb66E.mp4193yes https://puzzicle.org/math/algebra-home/v/factoring-and-the-distributive-property-2https://puzzicle.org/thumbs/26981.jpgFactoring polynomials: common factor (old)An old video where Sal factors 20u²v - 10uv² as 10uv(2u-v).https://cdn.kastatic.org/ka-youtube-converted/499MvHFrqUU.mp4/499MvHFrqUU.mp4286yes https://puzzicle.org/math/algebra-home/v/factor-polynomials-using-the-gcfhttps://puzzicle.org/thumbs/13179577.jpgFactoring polynomials: common factor (old)An old video where Sal factors 4x⁴y+8x³y as 4x³y(x+2).https://cdn.kastatic.org/ka-youtube-converted/_sIuZHYrdWM.mp4/_sIuZHYrdWM.mp4328yes https://puzzicle.org/math/algebra-home/v/evaluating-expressions-where-individual-variable-values-are-unknownhttps://puzzicle.org/thumbs/1242712428.jpgWorked example: evaluating expressions using structureSal solved a few examples where we need to evaluate an expression, but we don't know the individual variable values.https://cdn.kastatic.org/ka-youtube-converted/TIwGXn4NalM.mp4/TIwGXn4NalM.mp4279yes https://puzzicle.org/math/algebra-home/v/evaluating-expressions-with-unknown-variables-2https://puzzicle.org/thumbs/1242747382.jpgWorked example: evaluating expressions using structure (more examples)A few more examples where Sal evaluates expressions with multiple unknown variables.https://cdn.kastatic.org/ka-youtube-converted/CLQRZ2UbQ4Q.mp4/CLQRZ2UbQ4Q.mp4180yes https://puzzicle.org/math/algebra-home/v/factoring-quadratics-with-two-variableshttps://puzzicle.org/thumbs/xd24800c6.jpgFactoring two-variable quadraticsSal factors x^2+4xy-5y^2 as (x-y)(x+5y).https://cdn.kastatic.org/ka-youtube-converted/DRpdoZQtvOM.mp4/DRpdoZQtvOM.mp4371yes https://puzzicle.org/math/algebra-home/v/factoring-quadratics-with-two-variables-examplehttps://puzzicle.org/thumbs/x97900f0f.jpgFactoring two-variable quadratics: rearrangingSal rearranges 30x^2+11xy+y^2 as y^2+11xy+30x^2 and then factors it as (y+5x)(y+6x).https://cdn.kastatic.org/ka-youtube-converted/0xrvRKHoO2g.mp4/0xrvRKHoO2g.mp4166yes https://puzzicle.org/math/algebra-home/v/factoring-trinomials-by-grouping-1https://puzzicle.org/thumbs/26983.jpgFactoring two-variable quadratics: groupingSal factors 5rs+25r-3s-15 as (s+5)(5r-3).https://cdn.kastatic.org/ka-youtube-converted/HXIj16mjfgk.mp4/HXIj16mjfgk.mp4226yes https://puzzicle.org/math/algebra-home/v/factoring-trinomials-with-a-common-factorhttps://puzzicle.org/thumbs/13179580.jpgFactoring quadratics with common factor (old)An old video where Sal factors 8k²-24k-144 by first taking out a common factor of 8 and then using the sum-product pattern.https://cdn.kastatic.org/ka-youtube-converted/GMoqg_s4Dl4.mp4/GMoqg_s4Dl4.mp4301yes https://puzzicle.org/math/algebra-home/v/u09-l2-t1-we3-factoring-special-products-3https://puzzicle.org/thumbs/26991.jpgFactoring difference of squares: two variables (example 2)Sal factors 49x^2-49y^2 as (7x+7y)(7x-7y) or as 49(x+y)(x-y).https://cdn.kastatic.org/ka-youtube-converted/YahJQvY396o.mp4/YahJQvY396o.mp4150yes https://puzzicle.org/math/algebra-home/v/dividing-polynomials-1https://puzzicle.org/thumbs/13179524.jpgDividing polynomials: long divisionSal divides (x^2-3x+2) by (x-2) and then checks the solution.https://cdn.kastatic.org/ka-youtube-converted/8Wxw9bpKEGQ.mp4/8Wxw9bpKEGQ.mp4200yes https://puzzicle.org/math/algebra-home/v/dividing-polynomials-with-remainders-examplehttps://puzzicle.org/thumbs/x4278b219.jpgDivide polynomials by monomials (with remainders)Sal divides (7x^6+x^3+2x+1) by x^2, and writes the solution as q(x)+r(x)/x^2, where the degree of the remainder, r(x), is less than the degree of x^2.https://cdn.kastatic.org/ka-youtube-converted/WqNc6My1aNU.mp4/WqNc6My1aNU.mp4161yes https://puzzicle.org/math/algebra-home/v/dividing-polynomials-with-remaindershttps://puzzicle.org/thumbs/13179525.jpgDividing polynomials with remaindersSal divides (x^3+5x-4) by (x^2-x+1) using long division.https://cdn.kastatic.org/ka-youtube-converted/pLCmwHsDYqo.mp4/pLCmwHsDYqo.mp4383yes https://puzzicle.org/math/precalculus/v/binomial-theoremhttps://puzzicle.org/thumbs/x143f944b.jpgIntro to the Binomial TheoremThe Binomial theorem tells us how to expand expressions of the form (a+b)ⁿ, for example, (x+y)⁷. The larger the power is, the harder it is to expand expressions like this directly. But with the Binomial theorem, the process is relatively fast!https://cdn.kastatic.org/ka-youtube-converted/iPwrDWQ7hPc.mp4/iPwrDWQ7hPc.mp4795yes https://puzzicle.org/math/precalculus/v/pascals-triangle-binomial-theoremhttps://puzzicle.org/thumbs/x87dcb8c9.jpgPascal's triangle and binomial expansionSal introduces Pascal's triangle, and shows how we can use it to figure out the coefficients in binomial expansions.https://cdn.kastatic.org/ka-youtube-converted/v9Evg2tBdRk.mp4/v9Evg2tBdRk.mp4500yes https://puzzicle.org/math/precalculus/v/coefficient-in-binomial-expansionhttps://puzzicle.org/thumbs/x69d9ce4e.jpgExpanding binomialsSal expands (3y^2+6x^3)^5 using the binomial theorem and Pascal's triangle.https://cdn.kastatic.org/ka-youtube-converted/ojFuf9RYmzI.mp4/ojFuf9RYmzI.mp4521yes https://puzzicle.org/math/precalculus/v/binomial-expansion-algorithmhttps://puzzicle.org/thumbs/x10480fb9.jpgExpanding binomials w/o Pascal's triangleSal gives a "trick" for expanding large powers of binomials, without using Pascal's triangle.https://cdn.kastatic.org/ka-youtube-converted/2IFItASxDVo.mp4/2IFItASxDVo.mp4299yes https://puzzicle.org/math/precalculus/v/binomial-theorem-and-combinatorics-intuitionhttps://puzzicle.org/thumbs/x5fbb948e.jpgBinomial expansion & combinatoricsSal explains why we use the combinatorial formula for (n choose k) to expand binomial expressions.https://cdn.kastatic.org/ka-youtube-converted/_hrN4rVCOfI.mp4/_hrN4rVCOfI.mp4255yes https://puzzicle.org/math/precalculus/v/binomial-theorem-intuitionhttps://puzzicle.org/thumbs/x18e9abfd.jpgPascal's triangle & combinatoricsSal shows how generating the values in Pascal's triangle is related to the combinatorial formula (n choose k).https://cdn.kastatic.org/ka-youtube-converted/x-N76NrMDNE.mp4/x-N76NrMDNE.mp4246yes https://puzzicle.org/math/algebra-home/v/binomial-theorem-part-3https://puzzicle.org/thumbs/25810.jpgBinomial expansion & combinatorics (old)An old video of Sal explaining why we use the combinatorial formula for (n choose k) to expand binomial expressions.https://cdn.kastatic.org/ka-youtube-converted/xF_hJaXUNfE.mp4/xF_hJaXUNfE.mp4719yes https://puzzicle.org/math/algebra2/v/analyzing-polynomial-manipulationshttps://puzzicle.org/thumbs/xa96fc7d0.jpgAnalyzing polynomial identitiesThe video focuses on the critical analysis of polynomial manipulations, emphasizing the importance of checking each step for errors. It provides practice on identifying valid polynomial identities and highlights the significance of correct algebraic multiplication in solving polynomial equations.https://cdn.kastatic.org/ka-youtube-converted/-6qiO49Q180.mp4/-6qiO49Q180.mp4459yes https://puzzicle.org/math/precalculus/v/fundamental-theorem-of-algebra-introhttps://puzzicle.org/thumbs/xd72bdb29.jpgThe Fundamental theorem of AlgebraSal introduces the Fundamental Theorem of Algebra, which, as is reflected in its name, is a very important theorem about polynomials.https://cdn.kastatic.org/ka-youtube-converted/d8-LO6FCna0.mp4/d8-LO6FCna0.mp4334yes https://puzzicle.org/math/precalculus/v/fundamental-theorem-algebra-quadratichttps://puzzicle.org/thumbs/x6d4fe289.jpgQuadratics & the Fundamental Theorem of AlgebraThe proof of the Fundamental Theorem of Algebra for any degree of polynomial is really tough. For now, let's note that it indeed holds for polynomials of the second degree (i.e. quadratics).https://cdn.kastatic.org/ka-youtube-converted/zNxEVaRbdus.mp4/zNxEVaRbdus.mp4395yes https://puzzicle.org/math/precalculus/v/possible-real-rootshttps://puzzicle.org/thumbs/xf74e2e33.jpgNumber of possible real roots of a polynomialThe Fundamental Theorem of Algebra can be used in order to determine how many real roots a given polynomial has. Check it out!https://cdn.kastatic.org/ka-youtube-converted/_HFTRQRpzFM.mp4/_HFTRQRpzFM.mp4226yes https://puzzicle.org/math/algebra-home/v/finding-roots-or-zeros-of-polynomial-1https://puzzicle.org/thumbs/xc9beaa62.jpgFinding zeros of polynomials (1 of 2)Sal finds all the zeros (which is the same as the roots) of p(x)=x⁵+9x³-2x³-18x=0.https://cdn.kastatic.org/ka-youtube-converted/x9lb_frpkH0.mp4/x9lb_frpkH0.mp4549yes https://puzzicle.org/math/algebra-home/v/finding-roots-or-zeros-of-polynomial-2https://puzzicle.org/thumbs/x19e2704e.jpgFinding zeros of polynomials (2 of 2)Sal uses an alternative method to find the zeros of p(x)=x⁵+9x³-2x³-18x=0.https://cdn.kastatic.org/ka-youtube-converted/HJgjAZ7JkHQ.mp4/HJgjAZ7JkHQ.mp4180yes https://puzzicle.org/math/algebra-home/v/finding-roots-or-zeros-of-polynomial-examplehttps://puzzicle.org/thumbs/x0ef1472a.jpgFinding zeros of polynomials (example 2)Sal finds all the zeros (which is the same as the roots) of p(x)=(3x⁴-8x³+15x-40)(3x-8)².https://cdn.kastatic.org/ka-youtube-converted/9zreFD3T0qY.mp4/9zreFD3T0qY.mp4405yes https://puzzicle.org/math/algebra-home/v/identifying-graph-based-on-rootshttps://puzzicle.org/thumbs/x64e707f0.jpgZeros of polynomials & their graphsSal uses the zeros of y=x^3+3x^2+x+3 to determine its corresponding graph.https://cdn.kastatic.org/ka-youtube-converted/uFZvWYPfOmw.mp4/uFZvWYPfOmw.mp4327yes https://puzzicle.org/math/algebra-home/v/recognizing-features-of-functions-example-1https://puzzicle.org/thumbs/x88f83ee2.jpgEnd behavior of functions & their graphsSal picks a function that has a given end behavior based on its graph.https://cdn.kastatic.org/ka-youtube-converted/hpBBuaiIkrg.mp4/hpBBuaiIkrg.mp4143yes https://puzzicle.org/math/algebra-home/v/connection-between-even-and-odd-numbers-and-functionshttps://puzzicle.org/thumbs/62754979.jpgEven/odd functions & numbersThe connection between even and odd functions to even and odd numbers.https://cdn.kastatic.org/ka-youtube-converted/jFazrvLodrA.mp4/jFazrvLodrA.mp4248yes https://puzzicle.org/math/algebra-home/v/identifying-even-and-odd-polynomialshttps://puzzicle.org/thumbs/xd705e9ac.jpgEven & odd polynomialsSal analyzes three different polynomials to see if they are even, odd, or neither.https://cdn.kastatic.org/ka-youtube-converted/bOsNS3FsCPs.mp4/bOsNS3FsCPs.mp4442yes https://puzzicle.org/math/algebra-home/v/radical-expressions-with-higher-rootshttps://puzzicle.org/thumbs/23742.jpgSimplifying higher-index rootsMultiple examples of simplifications of higher-index radicals. For example, simplifying ⁵√96 as 2⁵√3.https://cdn.kastatic.org/ka-youtube-converted/iX7ivCww2ws.mp4/iX7ivCww2ws.mp4526yes https://puzzicle.org/math/algebra-home/v/multiply-and-simplify-a-radical-expression-1https://puzzicle.org/thumbs/127771044.jpgSimplifying cube root expressionsA worked example of simplifying elaborate expressions that contain radicals. In this example, we simplify 5∛(2x²)⋅3∛(4x⁴).https://cdn.kastatic.org/ka-youtube-converted/GEeKOeGHddY.mp4/GEeKOeGHddY.mp4265yes https://puzzicle.org/math/algebra-home/v/rational-exponents-and-exponent-lawshttps://puzzicle.org/thumbs/126711577.jpgSimplifying cube root expressions (two variables)Sal simplifies ∛(125x⁶y³) as 5x²y.https://cdn.kastatic.org/ka-youtube-converted/NuccqpiUHrk.mp4/NuccqpiUHrk.mp4136yes https://puzzicle.org/math/algebra-home/v/radical-equivalent-to-rational-exponents-2https://puzzicle.org/thumbs/126711575.jpgSimplifying higher-index root expressionsHow to rewrite a radical with variables in it as an exponent and then simplify it using exponent properties. In this example, we simplify ∜(5a⁴b¹²).https://cdn.kastatic.org/ka-youtube-converted/zlRKO21qEpQ.mp4/zlRKO21qEpQ.mp4144yes https://puzzicle.org/math/algebra-home/v/simplifying-mixed-exponent-and-radical-expressionshttps://puzzicle.org/thumbs/x3472e447.jpgSimplifying rational exponent expressions: mixed exponents and radicalsSal simplifies v^(-6/5)* ⁵√v as v^(-1).https://cdn.kastatic.org/ka-youtube-converted/95KX0dNZXAA.mp4/95KX0dNZXAA.mp4159yes https://puzzicle.org/math/algebra-home/v/simplifying-square-root-expressions-no-variables-advancedhttps://puzzicle.org/thumbs/x7cc704b6.jpgSimplifying square-root expressions: no variables (advanced)Sal simplifies elaborate expressions with square roots. For example, he simplifies (4√20-3√45)/√35 as -√(1/7).https://cdn.kastatic.org/ka-youtube-converted/suwJmCrSDI8.mp4/suwJmCrSDI8.mp4283yes https://puzzicle.org/math/algebra-home/v/how-to-rationalize-a-denominatorhttps://puzzicle.org/thumbs/23746.jpgIntro to rationalizing the denominatorWhen we have a fraction with a root in the denominator, like 1/√2, it's often desirable to manipulate it so the denominator doesn't have roots. To do that, we can multiply both the numerator and the denominator by the same root, that will get rid of the root in the denominator. For example, we can multiply 1/√2 by √2/√2 to get √2/2https://cdn.kastatic.org/ka-youtube-converted/gY5TvlHg4Vk.mp4/gY5TvlHg4Vk.mp4618yes https://puzzicle.org/math/algebra-home/v/rationalizing-denominators-of-expressionshttps://puzzicle.org/thumbs/127771045.jpgWorked example: rationalizing the denominatorSal rationalizes the denominator of the expression (16+2x²)/(√8).https://cdn.kastatic.org/ka-youtube-converted/Z3db5itCIiQ.mp4/Z3db5itCIiQ.mp4298yes https://puzzicle.org/math/algebra-home/v/subtracting-and-simplifying-radicalshttps://puzzicle.org/thumbs/127771043.jpgSimplifying radical expressions (subtraction)Sal simplifies 4∜(81x⁵)-2∜(81x⁵)-√(x³).https://cdn.kastatic.org/ka-youtube-converted/qH4IQfDD1Nw.mp4/qH4IQfDD1Nw.mp4345yes https://puzzicle.org/math/algebra-home/v/multiply-and-simplify-a-radical-expression-2https://puzzicle.org/thumbs/127771042.jpgSimplifying radical expressions: two variablesA worked example of simplifying elaborate expressions that contain radicals with two variables. In this example, we simplify √(60x²y)/√(48x).https://cdn.kastatic.org/ka-youtube-converted/sBvRJUwXJPo.mp4/sBvRJUwXJPo.mp4187yes https://puzzicle.org/math/algebra-home/v/simplifying-cube-rootshttps://puzzicle.org/thumbs/126711579.jpgSimplifying radical expressions: three variablesA worked example of simplifying the cube root of 27a²b⁵c³ using the properties of exponents.https://cdn.kastatic.org/ka-youtube-converted/c-wtvEdEoVs.mp4/c-wtvEdEoVs.mp4325yes https://puzzicle.org/math/algebra-home/v/simplifying-rational-exponent-expressions-advancedhttps://puzzicle.org/thumbs/x4753d1a9.jpgSimplifying hairy expression with fractional exponentsSal simplifies hairy expressions with rational exponents. For example, he simplifies (125^-⅛*125^⅝)/(5^½) as 5.https://cdn.kastatic.org/ka-youtube-converted/1BRXCIvt9e0.mp4/1BRXCIvt9e0.mp4367yes https://puzzicle.org/math/algebra-home/v/decay-of-cesium-137-examplehttps://puzzicle.org/thumbs/x5621b10d.jpgConstructing exponential models (old example)Sal constructs a function to model the decay of a radioactive element.https://cdn.kastatic.org/ka-youtube-converted/polop-89aIA.mp4/polop-89aIA.mp4553yes https://puzzicle.org/math/algebra-home/v/structure-in-exponential-expressionhttps://puzzicle.org/thumbs/x7de74d32.jpgStructure in exponential expressionAnalyzing an elaborate exponential function to determine its value at t=0 and whether it increases or decreases with t. This is algebraic reasoning at its best!https://cdn.kastatic.org/ka-youtube-converted/gR8-vRg6Yp0.mp4/gR8-vRg6Yp0.mp4439yes https://puzzicle.org/math/algebra-home/v/solving-logarithmic-equationshttps://puzzicle.org/thumbs/370262996.jpgLogarithmic equations: variable in the argumentSal solves the equation log(x)+log(3)=2log(4)-log(2).https://cdn.kastatic.org/ka-youtube-converted/Kv2iHde7Xgw.mp4/Kv2iHde7Xgw.mp4253yes https://puzzicle.org/math/algebra-home/v/solving-logarithmic-equations-dup-1https://puzzicle.org/thumbs/367722962.jpgLogarithmic equations: variable in the baseSal solves the equation 4=log_b(81).https://cdn.kastatic.org/ka-youtube-converted/oqAYqT70EBQ.mp4/oqAYqT70EBQ.mp4208yes https://puzzicle.org/math/algebra-home/v/matching-exponential-functionshttps://puzzicle.org/thumbs/x9061c6c9.jpgGraphs of exponential functions (old example)Sal matches the following four functions to their appropriate graphs: y=2^x, y=-3^x, y=2-(1/3)^x, y=(1/2)^x-2.https://cdn.kastatic.org/ka-youtube-converted/EwEbZI57P1o.mp4/EwEbZI57P1o.mp4442yes https://puzzicle.org/math/algebra-home/v/graphing-logarithmic-functionshttps://puzzicle.org/thumbs/367721780.jpgShape of a logarithmic parent graphSal graphs y=log₅(x).https://cdn.kastatic.org/ka-youtube-converted/DuYgVVU_BwY.mp4/DuYgVVU_BwY.mp4550yes https://puzzicle.org/math/algebra-home/v/logarithmic-function-graphshttps://puzzicle.org/thumbs/x74c0eb39.jpgGraphs of logarithmic functionsThe graph of y=log base 2 of x looks like a curve that increases at an ever-decreasing rate as x gets larger. It becomes very negative as x approaches 0 from the right. The graph of y=-log base 2 of x is the same as the first graph, but flipped over the x-axis. The graph of y=-log base 2 of (x+2) is the same as the first graph, but shifted to the left by 2.https://cdn.kastatic.org/ka-youtube-converted/LqyA96oYtwE.mp4/LqyA96oYtwE.mp4388yes https://puzzicle.org/math/algebra-home/v/vertical-asymptote-natural-loghttps://puzzicle.org/thumbs/xbd675271.jpgVertical asymptote of natural logWhat happens to ln(x) as x approaches 0?https://cdn.kastatic.org/ka-youtube-converted/78tnNYNISOc.mp4/78tnNYNISOc.mp4420yes https://puzzicle.org/math/algebra-home/v/vi-and-sal-explore-how-we-think-about-scalehttps://puzzicle.org/thumbs/214235936.jpgLogarithmic scale (with Vi Hart)Vi Hart and Sal talk about how we humans perceive things nonlinearlyhttps://cdn.kastatic.org/ka-youtube-converted/4xfOq00BzJA.mp4/4xfOq00BzJA.mp4416yes https://puzzicle.org/math/algebra-home/v/logarithmic-scalehttps://puzzicle.org/thumbs/214235939.jpgLogarithmic scaleSal discusses the differences between linear and logarithmic scale.https://cdn.kastatic.org/ka-youtube-converted/sBhEi4L91Sg.mp4/sBhEi4L91Sg.mp4675yes https://puzzicle.org/math/algebra-home/v/richter-scalehttps://puzzicle.org/thumbs/231149553.jpgRichter scaleSal explains the basics of the Richter scale (Moment magnitude scale) and uses this to compare the magnitude of 4 famous earthquakes.https://cdn.kastatic.org/ka-youtube-converted/RFn-IGlayAg.mp4/RFn-IGlayAg.mp4679yes https://puzzicle.org/math/algebra-home/v/vi-and-sal-talk-about-the-mysteries-of-benford-s-lawhttps://puzzicle.org/thumbs/214235938.jpgBenford's law (with Vi Hart, 1 of 2)Vi Hart visits Khan Academy and talks about the mysteries of Benford's Law with Salhttps://cdn.kastatic.org/ka-youtube-converted/6KmeGpjeLZ0.mp4/6KmeGpjeLZ0.mp4485yes https://puzzicle.org/math/algebra-home/v/benford-s-law-explanation-sequel-to-mysteries-of-benford-s-lawhttps://puzzicle.org/thumbs/214235937.jpgBenford's law (with Vi Hart, 2 of 2)Vi Hart visits Khan Academy and talks about the mysteries of Benford's Law with Salhttps://cdn.kastatic.org/ka-youtube-converted/SZUDoEdjTzg.mp4/SZUDoEdjTzg.mp4387yes https://puzzicle.org/math/algebra-home/v/flipping-shifting-radical-functionshttps://puzzicle.org/thumbs/x463eded6.jpgTransforming the square-root functionSal shows various examples of functions and their graphs that are a result of shifting and/or flipping y=√x.https://cdn.kastatic.org/ka-youtube-converted/t5ry2rasWvI.mp4/t5ry2rasWvI.mp4435yes https://puzzicle.org/math/algebra-home/v/graphs-of-square-root-functionshttps://puzzicle.org/thumbs/23740.jpgGraphs of square-root functionsSal graphs y=√x. Then, he shows a few examples of how we can shift and stretch the graphs of y=√x and y=x^2, and how the equations of those graphs look.https://cdn.kastatic.org/ka-youtube-converted/outcfkh69U0.mp4/outcfkh69U0.mp4901yes https://puzzicle.org/math/algebra-home/v/radical-function-graphs-exercisehttps://puzzicle.org/thumbs/xc7be7cc7.jpgSquare-root functions & their graphsSal is given a drawing with four graphs and four formulas of square-root functions. He uses transformations to match each formula with its appropriate graph.https://cdn.kastatic.org/ka-youtube-converted/Uteoh0PtWwI.mp4/Uteoh0PtWwI.mp4222yes https://puzzicle.org/math/math3/v/simplifying-rational-expressions-1https://puzzicle.org/thumbs/8187340.jpgSimplifying rational expressions: common binomial factorsGiven a rectangle with length a²+6a+27 and width a²-9, Sal writes the ratio of the width to the length as a simplified rational expression.https://cdn.kastatic.org/ka-youtube-converted/XChok8XlF90.mp4/XChok8XlF90.mp4251yes https://puzzicle.org/math/math3/v/simplifying-rational-expressions-3https://puzzicle.org/thumbs/27013.jpgSimplifying rational expressions: opposite common binomial factorsSal simplifies and states the domain of (x^2-36)/(6-x).https://cdn.kastatic.org/ka-youtube-converted/ey_b3aPsRl8.mp4/ey_b3aPsRl8.mp4233yes https://puzzicle.org/math/algebra-home/v/solving-rational-equations-3https://puzzicle.org/thumbs/27022.jpgEquations with one rational expression (advanced)Sal solves the following equation by first simplifying the rational expression: x^2-(x^2-4)/(x-2)=4.https://cdn.kastatic.org/ka-youtube-converted/5wUJLMWZ5Fw.mp4/5wUJLMWZ5Fw.mp4272yes https://puzzicle.org/math/algebra-home/v/rational-equationshttps://puzzicle.org/thumbs/124291561.jpgEquation with two rational expressions (old example)Sal solves the equation 4/(p-1)=5/(p+3).https://cdn.kastatic.org/ka-youtube-converted/bRwJ-QCz9XU.mp4/bRwJ-QCz9XU.mp4256yes https://puzzicle.org/math/algebra-home/v/solving-rational-equations-2https://puzzicle.org/thumbs/27021.jpgEquation with two rational expressions (old example 2)Sal solves the equation 5/(2x)-4/(3x)=7/18 by first finding the LCM (least common multiple) of 2x and 3x.https://cdn.kastatic.org/ka-youtube-converted/RdYA8ZpqdJE.mp4/RdYA8ZpqdJE.mp4248yes https://puzzicle.org/math/algebra-home/v/extraneous-solutions-to-rational-equationshttps://puzzicle.org/thumbs/124291568.jpgEquation with two rational expressions (old example 3)Sal solves the following equation and eliminates the resulting extraneous solution: (x^2)/(x+2)=4/(x+2).https://cdn.kastatic.org/ka-youtube-converted/bml74_PsfwA.mp4/bml74_PsfwA.mp4183yes https://puzzicle.org/math/algebra-home/v/direct-variation-applicationhttps://puzzicle.org/thumbs/124291558.jpgDirect variation word problem: space travelSal models a context about space travel with a direct variation equation.https://cdn.kastatic.org/ka-youtube-converted/ZsjfUUJUq9s.mp4/ZsjfUUJUq9s.mp4122yes https://puzzicle.org/math/math3/v/analyzing-vertical-asymptotes-of-rational-functionshttps://puzzicle.org/thumbs/xdf0aa09d.jpgAnalyzing vertical asymptotes of rational functionsSal analyzes the behavior of q(x)=(x²+3x+2)/(x+3) around its vertical asymptote at x=-3.https://cdn.kastatic.org/ka-youtube-converted/X8B4Fi7wadc.mp4/X8B4Fi7wadc.mp4426yes https://puzzicle.org/math/math3/v/example-rational-functions-graphshttps://puzzicle.org/thumbs/x6d96b6fe.jpgGraphs of rational functions (old example)Sal matches three graphs of rational functions to three formulas of such functions by considering asymptotes and intercepts.https://cdn.kastatic.org/ka-youtube-converted/onmNaDrxwmo.mp4/onmNaDrxwmo.mp4367yes https://puzzicle.org/math/math3/v/horizontal-vertical-asymptoteshttps://puzzicle.org/thumbs/xc5653fa0.jpgGraphing rational functions 1Sal graphs f(x)=(2x+10)/(5x-15).https://cdn.kastatic.org/ka-youtube-converted/fvC0dm2wzIo.mp4/fvC0dm2wzIo.mp4664yes https://puzzicle.org/math/math3/v/another-rational-function-graph-examplehttps://puzzicle.org/thumbs/23776.jpgGraphing rational functions 2Sal graphs y=(2x)/(x+1).https://cdn.kastatic.org/ka-youtube-converted/ReEMqdZEEX0.mp4/ReEMqdZEEX0.mp4500yes https://puzzicle.org/math/math3/v/asymptotes-of-rational-functionshttps://puzzicle.org/thumbs/23774.jpgGraphing rational functions 3Sal graphs y=(x^2)/(x^2-16).https://cdn.kastatic.org/ka-youtube-converted/2N62v_63SBo.mp4/2N62v_63SBo.mp41266yes https://puzzicle.org/math/math3/v/a-third-example-of-graphing-a-rational-functionhttps://puzzicle.org/thumbs/23778.jpgGraphing rational functions 4Sal graphs y=(x)/(x^2-x-6).https://cdn.kastatic.org/ka-youtube-converted/p7ycTWq6BFk.mp4/p7ycTWq6BFk.mp4691yes https://puzzicle.org/math/algebra-home/v/rational-inequalitieshttps://puzzicle.org/thumbs/23568.jpgRational inequalities: one side is zeroSal shows two ways to solve the inequality (x-1)/(x+2)>0.https://cdn.kastatic.org/ka-youtube-converted/ZjeMdXV0QMg.mp4/ZjeMdXV0QMg.mp4709yes https://puzzicle.org/math/algebra-home/v/rational-inequalities-2https://puzzicle.org/thumbs/23570.jpgRational inequalities: both sides are not zeroSal solves the rational inequality (x-3)/(x+4)≥2.https://cdn.kastatic.org/ka-youtube-converted/2RnS3fSHVV8.mp4/2RnS3fSHVV8.mp4795yes https://puzzicle.org/math/algebra-home/v/partial-fraction-expansion-1https://puzzicle.org/thumbs/23546.jpgIntro to partial fraction expansionSal explains what partial fraction expansion is by rewriting (x²-2x-37)/(x²-3x-40) as the sum of 1 and two rational expressions with linear denominators.https://cdn.kastatic.org/ka-youtube-converted/S-XKGBesRzk.mp4/S-XKGBesRzk.mp4688yes https://puzzicle.org/math/algebra-home/v/partial-fraction-expansion-2https://puzzicle.org/thumbs/23548.jpgPartial fraction expansionSal performs partial fraction expansion upon (10x²+12x+20)/(x³-8).https://cdn.kastatic.org/ka-youtube-converted/6FrPLJY0rqM.mp4/6FrPLJY0rqM.mp4782yes https://puzzicle.org/math/algebra-home/v/partial-fraction-expansion-3https://puzzicle.org/thumbs/23550.jpgPartial fraction expansion: repeated factorsSal performs partial fraction expansion upon (6x²-19x+15)/(x-1)(x-2)².https://cdn.kastatic.org/ka-youtube-converted/A52fEdPn9lg.mp4/A52fEdPn9lg.mp4635yes https://puzzicle.org/math/precalculus/v/trigonometry-unit-circle-symmetryhttps://puzzicle.org/thumbs/xa88b4a7e.jpgSine & cosine identities: symmetrySal finds several trigonometric identities for sine and cosine by considering horizontal and vertical symmetries of the unit circle.https://cdn.kastatic.org/ka-youtube-converted/tzQ7arA917E.mp4/tzQ7arA917E.mp4478yes https://puzzicle.org/math/precalculus/v/tan-symmetries-unit-circlehttps://puzzicle.org/thumbs/x2cb52cc7.jpgTangent identities: symmetrySal finds several trigonometric identities for tangent by considering horizontal and vertical symmetries of the unit circle.https://cdn.kastatic.org/ka-youtube-converted/k_wJsio68D4.mp4/k_wJsio68D4.mp4435yes https://puzzicle.org/math/precalculus/v/trig-angle-rotationshttps://puzzicle.org/thumbs/x5e52b073.jpgSine & cosine identities: periodicitySal finds trigonometric identities for sine and cosine by considering angle rotations on the unit circle.https://cdn.kastatic.org/ka-youtube-converted/h-TPSylHrvE.mp4/h-TPSylHrvE.mp4373yes https://puzzicle.org/math/precalculus/v/tan-periodicityhttps://puzzicle.org/thumbs/xa5a398fc.jpgTangent identities: periodicitySal solves a problem by considering the periodicity of the tangent function.https://cdn.kastatic.org/ka-youtube-converted/C3HFAyigqoY.mp4/C3HFAyigqoY.mp4242yes https://puzzicle.org/math/algebra2/v/solving-triangle-unit-circlehttps://puzzicle.org/thumbs/x8f71dac4.jpgTrig values of π/4Sal finds the trigonometric values of π/4 using the unit-circle definition.https://cdn.kastatic.org/ka-youtube-converted/KoYZErFpZ5Q.mp4/KoYZErFpZ5Q.mp4474yes https://puzzicle.org/math/algebra2/v/pythagorean-trig-identity-from-unit-circlehttps://puzzicle.org/thumbs/x952a1f00.jpgProof of the Pythagorean trig identityThe Pythagorean identity tells us that no matter what the value of θ is, sin²θ+cos²θ is equal to 1. We can prove this identity using the Pythagorean theorem in the unit circle with x²+y²=1.https://cdn.kastatic.org/ka-youtube-converted/n0DLSIOYBsQ.mp4/n0DLSIOYBsQ.mp4372yes https://puzzicle.org/math/algebra2/v/using-the-pythagorean-trig-identityhttps://puzzicle.org/thumbs/xda6daf69.jpgUsing the Pythagorean trig identityThe Pythagorean identity tells us that no matter what the value of θ is, sin²θ+cos²θ is equal to 1. This follows from the Pythagorean theorem, which is why it's called the Pythagorean identity! We can use this identity to solve various problems.https://cdn.kastatic.org/ka-youtube-converted/soIt2TwV6Xk.mp4/soIt2TwV6Xk.mp4375yes https://puzzicle.org/math/algebra2/v/midline-amplitude-periodhttps://puzzicle.org/thumbs/x2f235e5a.jpgFeatures of sinusoidal functionsSal introduces the main features of sinusoidal functions: midline, amplitude, & period. He shows how these can be found from a sinusoidal function's graph.https://cdn.kastatic.org/ka-youtube-converted/s4cLM0l1gd4.mp4/s4cLM0l1gd4.mp4298yes https://puzzicle.org/math/algebra2/v/we-amplitude-and-periodhttps://puzzicle.org/thumbs/1153684694.jpgAmplitude & period of sinusoidal functions from equationSal finds the amplitude and the period of y=-0.5cos(3x).https://cdn.kastatic.org/ka-youtube-converted/SBqnRja4CW4.mp4/SBqnRja4CW4.mp4501yes https://puzzicle.org/math/algebra2/v/example-amplitude-and-period-transformationshttps://puzzicle.org/thumbs/1153830189.jpgTransforming sinusoidal graphs: vertical stretch & horizontal reflectionSal graphs y=2*sin(-x) by considering it as a vertical stretch and a horizontal reflection of y=sin(x).https://cdn.kastatic.org/ka-youtube-converted/0zCcFSO8ouE.mp4/0zCcFSO8ouE.mp4772yes https://puzzicle.org/math/algebra2/v/amplitude-and-period-cosine-transformationshttps://puzzicle.org/thumbs/1153825235.jpgTransforming sinusoidal graphs: vertical & horizontal stretchesSal graphs y=-2.5*cos(1/3*x) by considering it as a vertical stretch and reflection, and a horizontal stretch, of y=cos(x).https://cdn.kastatic.org/ka-youtube-converted/uBVhtGL9y88.mp4/uBVhtGL9y88.mp4643yes https://puzzicle.org/math/algebra2/v/modeling-with-shifted-trig-functionshttps://puzzicle.org/thumbs/xd667e103.jpgTrig word problem: modeling daily temperatureSal solves a word problem about the daily change in temperature by modeling it with a sinusoidal function.https://cdn.kastatic.org/ka-youtube-converted/RX0DY9eRp8g.mp4/RX0DY9eRp8g.mp4655yes https://puzzicle.org/math/algebra2/v/modeling-temperature-fluxtuationshttps://puzzicle.org/thumbs/x02d93c3e.jpgTrig word problem: modeling annual temperatureSal solves a word problem about the annual change in temperature by modeling it with a sinusoidal function.https://cdn.kastatic.org/ka-youtube-converted/mVlCXkht6hg.mp4/mVlCXkht6hg.mp4406yes https://puzzicle.org/math/precalculus/v/inverse-trig-functions-arcsinhttps://puzzicle.org/thumbs/26269.jpgIntro to arcsineSal introduces arcsine, which is the inverse function of sine, and discusses its principal range.https://cdn.kastatic.org/ka-youtube-converted/JGU74wbZMLg.mp4/JGU74wbZMLg.mp4636yes https://puzzicle.org/math/precalculus/v/inverse-trig-functions-arctanhttps://puzzicle.org/thumbs/26271.jpgIntro to arctangentSal introduces arctangent, which is the inverse function of tangent, and discusses its principal range.https://cdn.kastatic.org/ka-youtube-converted/Idxeo49szW0.mp4/Idxeo49szW0.mp4606yes https://puzzicle.org/math/precalculus/v/inverse-trig-functions-arccoshttps://puzzicle.org/thumbs/26273.jpgIntro to arccosineSal introduces arccosine, which is the inverse function of cosine, and discusses its principal range.https://cdn.kastatic.org/ka-youtube-converted/eTDaJ4ebK28.mp4/eTDaJ4ebK28.mp4818yes https://puzzicle.org/math/precalculus/v/inverse-tan-domainhttps://puzzicle.org/thumbs/x7988ecbb.jpgDomain & range of inverse tangent functionSal finds the formula for the inverse function of g(x)=tan(x-3π/2)+6, and then determines the domain of that inverse function.https://cdn.kastatic.org/ka-youtube-converted/QGfdhqbilY8.mp4/QGfdhqbilY8.mp4584yes https://puzzicle.org/math/precalculus/v/inverse-tan-scenariohttps://puzzicle.org/thumbs/x793351da.jpgUsing inverse trig functions with a calculatorSal discusses the appropriate way to use the calculator in order to find an angle when its tangent value is given.https://cdn.kastatic.org/ka-youtube-converted/MABWdzmZFIQ.mp4/MABWdzmZFIQ.mp4215yes https://puzzicle.org/math/precalculus/v/sine-solutionshttps://puzzicle.org/thumbs/xdbe9f9c2.jpgSolving sinusoidal equations of the form sin(x)=dSal finds the expressions that together represent all possible solutions to the equation sin(x)=1/3.https://cdn.kastatic.org/ka-youtube-converted/NC7iWEQ9Kug.mp4/NC7iWEQ9Kug.mp4370yes https://puzzicle.org/math/precalculus/v/we-graph-of-cosine-functionhttps://puzzicle.org/thumbs/1153795385.jpgSolving cos(θ)=1 and cos(θ)=-1Sal solves the equations cos(θ)=1 and cos(θ)=-1 using the graph of y=cos(θ).https://cdn.kastatic.org/ka-youtube-converted/SdHwokUU8xI.mp4/SdHwokUU8xI.mp4318yes https://puzzicle.org/math/precalculus/v/inverse-trig-with-modelhttps://puzzicle.org/thumbs/x42b7ba00.jpgTrig word problem: solving for temperatureSal solves a word problem about the annual change in temperature by solving a sinusoidal equation.https://cdn.kastatic.org/ka-youtube-converted/2pwnr_soZEU.mp4/2pwnr_soZEU.mp4503yes https://puzzicle.org/math/precalculus/v/trigonometry-identity-review-funhttps://puzzicle.org/thumbs/26275.jpgTrig angle addition identitiesSal reviews 6 related trigonometric angle addition identities: sin(a+b), sin(a-c), cos(a+b), cos(a-b), cos(2a), and sin(2a).https://cdn.kastatic.org/ka-youtube-converted/a70-dYvDJZY.mp4/a70-dYvDJZY.mp4667yes https://puzzicle.org/math/precalculus/v/cosine-addition-identity-examplehttps://puzzicle.org/thumbs/xfb68005e.jpgUsing the cosine angle addition identitySal evaluates the cosine of the sum of 60° and another angle whose right triangle is given. To do this, he must use the cosine angle addition formula.https://cdn.kastatic.org/ka-youtube-converted/h0SNEO25vIw.mp4/h0SNEO25vIw.mp4314yes https://puzzicle.org/math/precalculus/v/double-angle-formula-for-cosine-example-chttps://puzzicle.org/thumbs/x5368b1c4.jpgUsing the cosine double-angle identityThe cosine double angle formula tells us that cos(2θ) is always equal to cos²θ-sin²θ. For example, cos(60) is equal to cos²(30)-sin²(30). We can use this identity to rewrite expressions or solve problems. See some examples in this video.https://cdn.kastatic.org/ka-youtube-converted/D_smr0GBPvA.mp4/D_smr0GBPvA.mp4208yes https://puzzicle.org/math/precalculus/v/proof-angle-addition-sinehttps://puzzicle.org/thumbs/x8c2e47e4.jpgProof of the sine angle addition identitySal proves the identity sin(x+y) = sin(x)*cos(y) + cos(x)*sin(y).https://cdn.kastatic.org/ka-youtube-converted/R0EQg9vgbQw.mp4/R0EQg9vgbQw.mp4506yes https://puzzicle.org/math/precalculus/v/proof-angle-addition-cosinehttps://puzzicle.org/thumbs/x9472d9c4.jpgProof of the cosine angle addition identitySal proves the identity cos(x+y) = cos(x)*cos(y) - sin(x)*sin(y).https://cdn.kastatic.org/ka-youtube-converted/0VBQnR2h8XM.mp4/0VBQnR2h8XM.mp4362yes https://puzzicle.org/math/precalculus/v/sine-angle-addition-2https://puzzicle.org/thumbs/xd0dcedae.jpgFinding trig values using angle addition identitiesSal finds the value of sin(7π/12) by rewriting it as sin(π/3+π/4) and then using the sine angle addition formula.https://cdn.kastatic.org/ka-youtube-converted/2RbKfRfzD-M.mp4/2RbKfRfzD-M.mp4515yes https://puzzicle.org/math/precalculus/v/sin-angle-additionhttps://puzzicle.org/thumbs/xdf09400e.jpgUsing trig angle addition identities: finding side lengthsSal is given a drawing with two triangles and some side lengths, and he uses the sine angle addition identity in order to find a missing side length.https://cdn.kastatic.org/ka-youtube-converted/sI789G6FBb4.mp4/sI789G6FBb4.mp4320yes https://puzzicle.org/math/precalculus/v/cosine-angle-additionhttps://puzzicle.org/thumbs/x3448aa45.jpgUsing trig angle addition identities: manipulating expressionsSal is given that cos(2θ)=C and he uses the cosine double-angle identity in order to find an expression for sin(θ).https://cdn.kastatic.org/ka-youtube-converted/yV4Xa8Xtmrc.mp4/yV4Xa8Xtmrc.mp4372yes https://puzzicle.org/math/precalculus/v/examples-using-pythagorean-identities-to-simplify-trigonometric-expressionshttps://puzzicle.org/thumbs/x756fc164.jpgUsing trigonometric identitiesTrigonometric identities like sin²θ+cos²θ=1 can be used to rewrite expressions in a different, more convenient way. For example, (1-sin²θ)(cos²θ) can be rewritten as (cos²θ)(cos²θ), and then as cos⁴θ.https://cdn.kastatic.org/ka-youtube-converted/4OEeVLo5V1o.mp4/4OEeVLo5V1o.mp4187yes https://puzzicle.org/math/ap-calculus-bc/v/parametric-equations-1https://puzzicle.org/thumbs/25825.jpgParametric equations introIn this video, we learn about parametric equations using the example of a car driving off a cliff. Parametric equations define x and y as functions of a third parameter, t (time). They help us find the path, direction, and position of an object at any given time.https://cdn.kastatic.org/ka-youtube-converted/m6c6dlmUT1c.mp4/m6c6dlmUT1c.mp4668yes https://puzzicle.org/math/algebra-home/v/parametric-equations-2https://puzzicle.org/thumbs/25827.jpgRemoving the parameter in parametric equationsSal starts with parametric equations that give x and y as functions of t, and he manipulates the equations to get y as a function of x.https://cdn.kastatic.org/ka-youtube-converted/wToSIQJ2o_8.mp4/wToSIQJ2o_8.mp4585yes https://puzzicle.org/math/algebra-home/v/parametric-equations-3https://puzzicle.org/thumbs/25829.jpgRemoving the parameter in parametric equations (example 2)Sal is given x=3cost and y=2sint and he finds an equation that gives the relationship between x and y (spoiler: it's an ellipse!).https://cdn.kastatic.org/ka-youtube-converted/57BiI_iD3-U.mp4/57BiI_iD3-U.mp4772yes https://puzzicle.org/math/algebra-home/v/parametric-equations-4https://puzzicle.org/thumbs/25831.jpgParametric equations with the same graphSal shows how different parametric equations can result in the same relationship between x and y (and therefore with the same graph).https://cdn.kastatic.org/ka-youtube-converted/IReD6c_njOY.mp4/IReD6c_njOY.mp4784yes https://puzzicle.org/math/algebra-home/v/can-a-squirrel-avoid-getting-run-overhttps://puzzicle.org/thumbs/xdfd94c1b.jpgWorked example: Rate problemSal solves two related proportion word problems about a squirrel crossing the road and a car that approaches it. He does that using dimensional analysis.https://cdn.kastatic.org/ka-youtube-converted/F0LLR7bs7Qo.mp4/F0LLR7bs7Qo.mp4253yes https://puzzicle.org/math/algebra-home/v/unit-conversion-exercise-example-1https://puzzicle.org/thumbs/xbabf89be.jpgMultiple units word problem: road tripSal solves a word problem where he finds the cost of gas in a road trip given the car's fuel efficiency, length of the trip, and price of gas.https://cdn.kastatic.org/ka-youtube-converted/xpzt0wqMT6Y.mp4/xpzt0wqMT6Y.mp4227yes https://puzzicle.org/math/algebra2/v/plotting-complex-numbers-on-the-complex-planehttps://puzzicle.org/thumbs/x21268ed3.jpgPlotting numbers on the complex planeUnderstand the roles of real and imaginary parts in forming coordinates. Learn how to plot complex numbers on a unique two-dimensional grid known as the complex plane. Discover the real and imaginary parts of complex numbers, and see how they correspond to the horizontal and vertical axes.https://cdn.kastatic.org/ka-youtube-converted/kGzXIbauGQk.mp4/kGzXIbauGQk.mp474yes https://puzzicle.org/math/precalculus/v/absolute-value-of-a-complex-numberhttps://puzzicle.org/thumbs/x49eee15a.jpgAbsolute value of complex numbersSal finds the absolute value of (3-4i).https://cdn.kastatic.org/ka-youtube-converted/yvzyC4VBpUU.mp4/yvzyC4VBpUU.mp4213yes https://puzzicle.org/math/precalculus/v/basic-complex-analysishttps://puzzicle.org/thumbs/8187548.jpgAbsolute value & angle of complex numbersSal finds the modulus (which is the absolute value) and the argument (which is the angle) of √3/2+1/2*i.https://cdn.kastatic.org/ka-youtube-converted/FwuPXchH2rA.mp4/FwuPXchH2rA.mp4784yes https://puzzicle.org/math/precalculus/v/complex-plane-distance-midpointhttps://puzzicle.org/thumbs/xd622f44d.jpgDistance & midpoint of complex numbersSal finds the distance between (2+3i) and (-5-i) and then he finds their midpoint on the complex plane.https://cdn.kastatic.org/ka-youtube-converted/Efoeqb6tC88.mp4/Efoeqb6tC88.mp4371yes https://puzzicle.org/math/precalculus/v/polar-form-complex-numberhttps://puzzicle.org/thumbs/x5cb8c70a.jpgPolar & rectangular forms of complex numbersLearn how to convert a complex number from rectangular form to polar form. This video covers how to find the distance (r) and direction (theta) of the complex number on the complex plane, and how to use trigonometric functions and the Pythagorean theorem to make the conversion.https://cdn.kastatic.org/ka-youtube-converted/8RasCV_Lggg.mp4/8RasCV_Lggg.mp4736yes https://puzzicle.org/math/algebra-home/v/complex-polar-form-intuitionhttps://puzzicle.org/thumbs/x039d8d9ed24bc35c.jpgComplex number polar form intuitionLearn how to convert a complex number from rectangular form to polar form. We find the distance (r) and direction (theta) of the complex number on the complex plane, and use trigonometric functions, special right triangles, and the Pythagorean theorem to make the conversion.https://cdn.kastatic.org/ka-youtube-converted/YZRzt_AsuVs.mp4/YZRzt_AsuVs.mp4361yes https://puzzicle.org/math/algebra-home/v/complex-number-divisionhttps://puzzicle.org/thumbs/x7818ca14.jpgDividing complex numbers: polar & exponential formSal shows how complex division affects the modulus and the argument of the divisor and the dividend.https://cdn.kastatic.org/ka-youtube-converted/xOa-l_tXgWA.mp4/xOa-l_tXgWA.mp4430yes https://puzzicle.org/math/algebra-home/v/powers-of-complex-numbershttps://puzzicle.org/thumbs/xf6a4f54a.jpgPowers of complex numbersSal simplifies the 20th power of a complex number given in polar form.https://cdn.kastatic.org/ka-youtube-converted/dl_9NC_J6yo.mp4/dl_9NC_J6yo.mp4385yes https://puzzicle.org/math/precalculus/v/exponential-form-to-find-complex-rootshttps://puzzicle.org/thumbs/8187549.jpgComplex number equations: x³=1Sal finds all complex solutions to the equation x^3=1.https://cdn.kastatic.org/ka-youtube-converted/N0Y8ia57C24.mp4/N0Y8ia57C24.mp4713yes https://puzzicle.org/math/algebra-home/v/iit-jee-complex-numbers-part-1https://puzzicle.org/thumbs/8187445.jpgChallenging complex numbers problem (1 of 3)Paper 1 Problem 39 from the challenging 2010 IIT JEE exam, about complex numbers.https://cdn.kastatic.org/ka-youtube-converted/_5ei_I02huY.mp4/_5ei_I02huY.mp4441yes https://puzzicle.org/math/algebra-home/v/iit-jee-complex-numbers-part-2https://puzzicle.org/thumbs/8187446.jpgChallenging complex numbers problem (2 of 3)Paper 1 Problem 39 from the challenging 2010 IIT JEE exam, about complex numbers.https://cdn.kastatic.org/ka-youtube-converted/pQwoBOpVoWw.mp4/pQwoBOpVoWw.mp4446yes https://puzzicle.org/math/algebra-home/v/iit-jee-complex-numbers-part-3https://puzzicle.org/thumbs/8187447.jpgChallenging complex numbers problem (3 of 3)Paper 1 Problem 39 from the challenging 2010 IIT JEE exam, about complex numbers.https://cdn.kastatic.org/ka-youtube-converted/FwA_UZkI-JM.mp4/FwA_UZkI-JM.mp4189yes https://puzzicle.org/math/algebra-home/v/complex-determinant-examplehttps://puzzicle.org/thumbs/8187468.jpgChallenging complex numbers problem: complex determinantPaper 1 Problem 53 from the challenging 2010 IIT JEE exam, about determinant of a complex numbers matrix.https://cdn.kastatic.org/ka-youtube-converted/E7OkUomRq1Q.mp4/E7OkUomRq1Q.mp4778yes https://puzzicle.org/math/geometry/v/introduction-to-conic-sectionshttps://puzzicle.org/thumbs/23522.jpgIntro to conic sectionsSal introduces the four conic sections and shows how they are derived by intersecting planes with cones in certain ways.https://cdn.kastatic.org/ka-youtube-converted/0A7RR0oy2ho.mp4/0A7RR0oy2ho.mp4658yes https://puzzicle.org/math/geometry/v/graphing-circles-from-featureshttps://puzzicle.org/thumbs/xa6159ffd.jpgGraphing circles from featuresHow to graph a circle given the center and either the radius or another point on the circle.https://cdn.kastatic.org/ka-youtube-converted/LDjMDIuTWtM.mp4/LDjMDIuTWtM.mp4129yes https://puzzicle.org/math/geometry/v/center-and-radius-of-circle-from-graphhttps://puzzicle.org/thumbs/x48f9959e.jpgFeatures of a circle from its graphGiven a circle on the coordinate plane, Sal finds its center and its radius.https://cdn.kastatic.org/ka-youtube-converted/SLyM_-sQwMM.mp4/SLyM_-sQwMM.mp4230yes https://puzzicle.org/math/geometry/v/radius-and-center-for-a-circle-equation-in-standard-formhttps://puzzicle.org/thumbs/xca06c2ed.jpgFeatures of a circle from its standard equationSal finds the center and the radius of the circle whose equation is (x+3)^2+(y-4)^2=49.https://cdn.kastatic.org/ka-youtube-converted/JvDpYlyKkNU.mp4/JvDpYlyKkNU.mp4231yes https://puzzicle.org/math/geometry/v/graphing-a-circle-from-standard-equationhttps://puzzicle.org/thumbs/x64f04a47.jpgGraphing a circle from its standard equationSal graphs the circle whose equation is (x+5)²+(y-5)²=4.https://cdn.kastatic.org/ka-youtube-converted/thDrJvWNI8M.mp4/thDrJvWNI8M.mp4314yes https://puzzicle.org/math/geometry/v/writing-standard-equation-of-circlehttps://puzzicle.org/thumbs/x6415cfbc.jpgWriting standard equation of a circleGiven a circle on the coordinate plane, Sal finds its standard equation, which is an equation in the form (x-a)²+(y-b)²=r².https://cdn.kastatic.org/ka-youtube-converted/iX5UgArMyiI.mp4/iX5UgArMyiI.mp4340yes https://puzzicle.org/math/geometry/v/completing-the-square-to-write-equation-in-standard-form-of-a-circlehttps://puzzicle.org/thumbs/x577fb2f2.jpgFeatures of a circle from its expanded equationSal finds the center and the radius of a circle whose equation is x^2+y^2+4x-4y-17=0, and then he graphs the circle.https://cdn.kastatic.org/ka-youtube-converted/XyDMsotfJhE.mp4/XyDMsotfJhE.mp4260yes https://puzzicle.org/math/precalculus/v/conic-sections-intro-to-ellipseshttps://puzzicle.org/thumbs/23526.jpgIntro to ellipsesLearn all about ellipses in this video. The standard form for an ellipse centered at the origin is x²/a² + y²/b² = 1. The semi-major axis is the longest radius and the semi-minor axis is the shortest radius. The video also explains how to shift an ellipse.https://cdn.kastatic.org/ka-youtube-converted/lvAYFUIEpFI.mp4/lvAYFUIEpFI.mp4854yes https://puzzicle.org/math/precalculus/v/ellipse-standard-equation-from-graphhttps://puzzicle.org/thumbs/xf0e7990c.jpgEllipse standard equation from graphGiven an ellipse on the coordinate plane, Sal finds its standard equation, which is an equation in the form (x-h)²/a²+(y-k)²/b²=1.https://cdn.kastatic.org/ka-youtube-converted/_JrQF8Rkaio.mp4/_JrQF8Rkaio.mp4215yes https://puzzicle.org/math/precalculus/v/ellipse-graph-from-standard-equationhttps://puzzicle.org/thumbs/xb04ef9af.jpgEllipse graph from standard equationThis video explains how to graph an ellipse when given its equation in standard form. The center of this ellipse is (4,1), the horizontal radius is 4, and the vertical radius is 7. The key definitions are center, horizontal radius, and vertical radius.https://cdn.kastatic.org/ka-youtube-converted/h5dIVNjVjXg.mp4/h5dIVNjVjXg.mp4100yes https://puzzicle.org/math/precalculus/v/foci-of-an-ellipsehttps://puzzicle.org/thumbs/23540.jpgFoci of an ellipse from equationSal explains how the radii and the foci of an ellipse relate to each other, and how we can use this relationship in order to find the foci from the equation of an ellipse.https://cdn.kastatic.org/ka-youtube-converted/QR2vxfwiHAU.mp4/QR2vxfwiHAU.mp4829yes https://puzzicle.org/math/geometry/v/focus-and-directrix-introductionhttps://puzzicle.org/thumbs/x0abb5ff2.jpgIntro to focus & directrixA parabola is the set of all points equidistant from a point (called the "focus") and a line (called the "directrix"). See this video to learn more about this.https://cdn.kastatic.org/ka-youtube-converted/pe8Dm_FUpdU.mp4/pe8Dm_FUpdU.mp4247yes https://puzzicle.org/math/math2/v/finding-focus-and-directrix-from-vertexhttps://puzzicle.org/thumbs/x009570c6.jpgFocus & directrix of a parabola from equationGiven the parabola equation y-23/4=-1/3(x-1)^2, Sal finds the parabola's focus and directrix using the general formula for a parabola whose focus is (a,b) and directrix is y=k.https://cdn.kastatic.org/ka-youtube-converted/w56Vuf9tHfA.mp4/w56Vuf9tHfA.mp4689yes https://puzzicle.org/math/precalculus/v/conic-sections-intro-to-hyperbolashttps://puzzicle.org/thumbs/23528.jpgIntro to hyperbolasSal introduces the standard equation for hyperbolas, and how it can be used in order to determine the direction of the hyperbola and its vertices.https://cdn.kastatic.org/ka-youtube-converted/pzSyOTkAsY4.mp4/pzSyOTkAsY4.mp4814yes https://puzzicle.org/math/precalculus/v/vertices-and-direction-of-a-hyperbolahttps://puzzicle.org/thumbs/x16b23523.jpgVertices & direction of a hyperbolaSal picks the graph of y²/9-x²/4=1 based on the hyperbola's center, direction, & vertices.https://cdn.kastatic.org/ka-youtube-converted/oO3nWnJppqg.mp4/oO3nWnJppqg.mp4315yes https://puzzicle.org/math/precalculus/v/vertices-and-direction-of-a-hyperbola-2https://puzzicle.org/thumbs/xb9aa106e.jpgVertices & direction of a hyperbola (example 2)Sal matches an equation to a given graph of a hyperbola, based on the hyperbola's direction & vertices.https://cdn.kastatic.org/ka-youtube-converted/hnVFThmLW5Q.mp4/hnVFThmLW5Q.mp4141yes https://puzzicle.org/math/precalculus/v/conic-sections-hyperbolas-2https://puzzicle.org/thumbs/23530.jpgGraphing hyperbolas (old example)Given the hyperbola equation y^2/4-x^2/9=1, Sal determines the direction to which it opens and its vertices in order to draw its graph.https://cdn.kastatic.org/ka-youtube-converted/hl58vTCqVIY.mp4/hl58vTCqVIY.mp4450yes https://puzzicle.org/math/precalculus/v/foci-of-a-hyperbolahttps://puzzicle.org/thumbs/23542.jpgFoci of a hyperbola from equationSal discusses the foci of hyperbolas and shows how they relate to hyperbola equations.https://cdn.kastatic.org/ka-youtube-converted/S0Fd2Tg2v7M.mp4/S0Fd2Tg2v7M.mp4924yes https://puzzicle.org/math/precalculus/v/proof-hyperbola-focihttps://puzzicle.org/thumbs/23544.jpgProof of the hyperbola foci formulaSal proves why, for the general hyperbola equation x^2/a^2-y^2/b^2=1, the focal length f forms the equation f^2=a^2+b^2 with the parameters a and b.https://cdn.kastatic.org/ka-youtube-converted/HPRFmu7JsKU.mp4/HPRFmu7JsKU.mp4890yes https://puzzicle.org/math/precalculus/v/conic-sections-hyperbolas-3https://puzzicle.org/thumbs/23532.jpgEquation of a hyperbola not centered at the originSal analyzes the hyperbola whose equation is (x-1)^2/16-(y+1)^2/4=1, and graphs it.https://cdn.kastatic.org/ka-youtube-converted/lGQw-W1PxBE.mp4/lGQw-W1PxBE.mp4629yes https://puzzicle.org/math/algebra-home/v/conic-identification-3https://puzzicle.org/thumbs/23538.jpgConic section from expanded equation: circle & parabolaSal manipulates the equation x^2+y^2-3x+4y=4 in order to find that it represents a circle, and the equation 2x^2+y+12x+16=0 in order to find it represents a parabola.https://cdn.kastatic.org/ka-youtube-converted/Dru0RHgfp2g.mp4/Dru0RHgfp2g.mp4694yes https://puzzicle.org/math/algebra-home/v/identifying-conics-1https://puzzicle.org/thumbs/23534.jpgConic section from expanded equation: ellipseSal manipulates the equation 9x^2+4y^2+54x-8y+49=0 in order to find that it represents an ellipse.https://cdn.kastatic.org/ka-youtube-converted/LSJuu4Qm2qQ.mp4/LSJuu4Qm2qQ.mp4551yes https://puzzicle.org/math/algebra-home/v/identifying-conics-2https://puzzicle.org/thumbs/23536.jpgConic section from expanded equation: hyperbolaSal manipulates the equation 4y^2-50x=25x^2+16y+109 in order to find that it represents a hyperbola.https://cdn.kastatic.org/ka-youtube-converted/cvA4VN1dpuY.mp4/cvA4VN1dpuY.mp4688yes https://puzzicle.org/math/algebra-home/v/tangent-line-hyperbola-relationship-very-optionalhttps://puzzicle.org/thumbs/8187463.jpgRepresenting a line tangent to a hyperbolaHow a tangent line relates to a hyperbola. Might be useful for some competitive exams where there isn't time to derive (like we are doing in this video).https://cdn.kastatic.org/ka-youtube-converted/c_8QQbVQKU0.mp4/c_8QQbVQKU0.mp4604yes https://puzzicle.org/math/algebra-home/v/iit-jee-circle-hyperbola-common-tangent-part-1https://puzzicle.org/thumbs/8187454.jpgCommon tangent of circle & hyperbola (1 of 5)2010 IIT JEE Paper 1 Problem 45: Find the equation of a tangent common a given circle and a given hyperbola.https://cdn.kastatic.org/ka-youtube-converted/MhLfun2Vask.mp4/MhLfun2Vask.mp4376yes https://puzzicle.org/math/algebra-home/v/iit-jee-circle-hyperbola-common-tangent-part-2https://puzzicle.org/thumbs/8187455.jpgCommon tangent of circle & hyperbola (2 of 5)2010 IIT JEE Paper 1 Problem 45 Circle Hyperbola Common Tangent Part 2https://cdn.kastatic.org/ka-youtube-converted/0imeUgSxR10.mp4/0imeUgSxR10.mp4645yes https://puzzicle.org/math/algebra-home/v/iit-jee-circle-hyperbola-common-tangent-part-3https://puzzicle.org/thumbs/8187456.jpgCommon tangent of circle & hyperbola (3 of 5)2010 IIT JEE Paper 1 Problem 45 Circle Hyperbola Common Tangent Part 3https://cdn.kastatic.org/ka-youtube-converted/RomnHMWSLoE.mp4/RomnHMWSLoE.mp4535yes https://puzzicle.org/math/algebra-home/v/iit-jee-circle-hyperbola-common-tangent-part-4https://puzzicle.org/thumbs/8187457.jpgCommon tangent of circle & hyperbola (4 of 5)2010 IIT JEE Paper 1 Problem 45 Circle Hyperbola Common Tangent Part 4https://cdn.kastatic.org/ka-youtube-converted/eGo8C2Jshzs.mp4/eGo8C2Jshzs.mp4636yes https://puzzicle.org/math/algebra-home/v/iit-jee-circle-hyperbola-common-tangent-part-5https://puzzicle.org/thumbs/8187458.jpgCommon tangent of circle & hyperbola (5 of 5)2010 IIT JEE Paper 1 Problem 45 Circle Hyperbola Common Tangent Part 5https://cdn.kastatic.org/ka-youtube-converted/rPoqpQcgNv4.mp4/rPoqpQcgNv4.mp4737yes https://puzzicle.org/math/algebra-home/v/iit-jee-circle-hyperbola-intersectionhttps://puzzicle.org/thumbs/8187453.jpgIntersection of circle & hyperbola2010 IIT JEE Paper 1 Problem 46: Find the equation of a circle whose diameter is defined by the two intersection points of given circle and hyperbola.https://cdn.kastatic.org/ka-youtube-converted/6g3DPg2HqGw.mp4/6g3DPg2HqGw.mp4764yes https://puzzicle.org/math/ap-calculus-ab/v/sigma-notation-sumhttps://puzzicle.org/thumbs/xea358b4a.jpgUnderstanding summation notationSigma notation is the standard notation for writing long sums. Learn how it is used in this video.https://cdn.kastatic.org/ka-youtube-converted/5jwXThH6fg4.mp4/5jwXThH6fg4.mp4267yes https://puzzicle.org/math/precalculus/v/sum-of-arithmetic-sequence-arithmetic-serieshttps://puzzicle.org/thumbs/1393162637.jpgArithmetic series introSal explains the formula for the sum of a finite arithmetic serieshttps://cdn.kastatic.org/ka-youtube-converted/cYw4MFWsB6c.mp4/cYw4MFWsB6c.mp4236yes https://puzzicle.org/math/precalculus/v/evaluating-finite-arithmetic-serieshttps://puzzicle.org/thumbs/xc323fdc4.jpgWorked example: arithmetic series (sigma notation)Sal evaluates the arithmetic series Σ(2k+50) for k=1 to 550. He does that using the arithmetic series formula (a₁+aₙ)*n/2.https://cdn.kastatic.org/ka-youtube-converted/RM644gFKo_g.mp4/RM644gFKo_g.mp4422yes https://puzzicle.org/math/precalculus/v/evaluating-arithmetic-sum-examplehttps://puzzicle.org/thumbs/x75e479bb.jpgWorked example: arithmetic series (sum expression)Sal evaluates the arithmetic sum (-50)+(-44)+(-38)+...+2044. He does that by finding the number of terms and using the arithmetic series formula (a₁+aₙ)*n/2.https://cdn.kastatic.org/ka-youtube-converted/0-wa7voc0uM.mp4/0-wa7voc0uM.mp4407yes https://puzzicle.org/math/precalculus/v/finding-sum-from-recursive-definition-of-sequencehttps://puzzicle.org/thumbs/x779f2dfd.jpgWorked example: arithmetic series (recursive formula)Sal evaluates the sum of the first 650 terms in the sequence defined recursively as {aᵢ=aᵢ₋₁+11, a₁=4}. He does that by finding the 650th term and using the arithmetic series formula (a₁+aₙ)*n/2.https://cdn.kastatic.org/ka-youtube-converted/pIO34mrW_Ro.mp4/pIO34mrW_Ro.mp4321yes https://puzzicle.org/math/precalculus/v/alternate-proof-to-induction-for-integer-sumhttps://puzzicle.org/thumbs/193202563.jpgProof of finite arithmetic series formulaWatch Sal prove the expression for the sum of all positive integers up to and including n.https://cdn.kastatic.org/ka-youtube-converted/1wnIsgUivEQ.mp4/1wnIsgUivEQ.mp4212yes https://puzzicle.org/math/algebra-home/v/geometric-sequenceshttps://puzzicle.org/thumbs/1383732207.jpgIntro to geometric sequences (advanced)Sal introduces geometric sequences and gives a few examples. Notation used in this video is relatively advanced.https://cdn.kastatic.org/ka-youtube-converted/dIGLhLMsy2U.mp4/dIGLhLMsy2U.mp4399yes https://puzzicle.org/math/precalculus/v/series-as-sum-of-sequencehttps://puzzicle.org/thumbs/1393021992.jpgGeometric series introA geometric series is the sum of the terms of a geometric sequence. Learn more about it here.https://cdn.kastatic.org/ka-youtube-converted/rcRg_gO7-7E.mp4/rcRg_gO7-7E.mp4173yes https://puzzicle.org/math/precalculus/v/geometric-series-introductionhttps://puzzicle.org/thumbs/x8bcd35c6.jpgGeometric series with sigma notationA geometric series is the sum of the terms of a geometric sequence. Learn about geometric series and how they can be written in general terms and using sigma notation.https://cdn.kastatic.org/ka-youtube-converted/CecgFWTg9pQ.mp4/CecgFWTg9pQ.mp4268yes https://puzzicle.org/math/precalculus/v/example-finding-sum-of-finite-geometric-serieshttps://puzzicle.org/thumbs/x9a2ad65d.jpgWorked example: finite geometric series (sigma notation)Sal evaluates the geometric series Σ2(3ᵏ) for k=0 to 99 using the finite geometric series formula a(1-rⁿ)/(1-r).https://cdn.kastatic.org/ka-youtube-converted/AXP5PGSaaYk.mp4/AXP5PGSaaYk.mp4168yes https://puzzicle.org/math/precalculus/v/geometric-series-word-problemhttps://puzzicle.org/thumbs/x9ba70b9b.jpgFinite geometric series word problem: social mediaWatch Sal solve an example of using a geometric series to answer a fun word problem.https://cdn.kastatic.org/ka-youtube-converted/aIjzkiijGnA.mp4/aIjzkiijGnA.mp4483yes https://puzzicle.org/math/algebra-home/v/writing-arithmetic-series-in-sigma-notationhttps://puzzicle.org/thumbs/x8ebd01f2.jpgArithmetic series in sigma notationSal writes the arithmetic sum 7+9+11+...+403+405 in sigma notation. There are actually two common ways of doing this.https://cdn.kastatic.org/ka-youtube-converted/XArCfFaOhoo.mp4/XArCfFaOhoo.mp4248yes https://puzzicle.org/math/algebra-home/v/writing-geometric-series-in-sigma-notationhttps://puzzicle.org/thumbs/xeebb3f52.jpgFinite geometric series in sigma notationSal writes the geometric sum 2+6+18+54 in sigma notation. There are actually two common ways of doing this.https://cdn.kastatic.org/ka-youtube-converted/bT_0pn1lHiY.mp4/bT_0pn1lHiY.mp4194yes https://puzzicle.org/math/algebra-home/v/properties-to-find-sumhttps://puzzicle.org/thumbs/x9235e839.jpgEvaluating series using the formula for the sum of n squaresUsing properties of sigma notation to rewrite an elaborate sum as a combination of simpler sums, which we know the formula for.https://cdn.kastatic.org/ka-youtube-converted/LwhJVURumAA.mp4/LwhJVURumAA.mp4414yes https://puzzicle.org/math/ap-calculus-bc/v/partial-sum-notationhttps://puzzicle.org/thumbs/x59079628.jpgPartial sums introThis video introduces partial sums, which are sums of a certain number of terms in an infinite series. It explains the notation S sub n (Sₙ) and shows how to calculate partial sums using a formula.https://cdn.kastatic.org/ka-youtube-converted/k9MEOgcc5KY.mp4/k9MEOgcc5KY.mp4369yes https://puzzicle.org/math/ap-calculus-bc/v/term-from-partial-sumhttps://puzzicle.org/thumbs/xbb49a4841e6d74b5.jpgPartial sums: formula for nth term from partial sumThis video shows how to find a rule for the nth term (aₙ) in a series. We start with the formula for the sum of the first n terms (sₙ). Then, we subtract the sum of the first n-1 terms (sₙ₋₁) from sₙ to get aₙ. Finally, we simplify the expression.https://cdn.kastatic.org/ka-youtube-converted/Ppuj5gw78Vc.mp4/Ppuj5gw78Vc.mp4245yes https://puzzicle.org/math/ap-calculus-bc/v/finding-term-value-from-sum-formulahttps://puzzicle.org/thumbs/x67de3ad3908c900b.jpgPartial sums: term value from partial sumThe partial sum of a sequence gives us the sum of the first n terms in the sequence. If we know the formula for the partial sums of a sequence, we can find the value of any term in the sequence.https://cdn.kastatic.org/ka-youtube-converted/qmWJ3nLII1M.mp4/qmWJ3nLII1M.mp4249yes https://puzzicle.org/math/ap-calculus-bc/v/deriving-geometric-series-sum-formulahttps://puzzicle.org/thumbs/x16f67abe.jpgInfinite geometric series formula intuitionIn the video, we learn about the sum of an infinite geometric series. The sum converges (has a finite value) when the common ratio (r) is between -1 and 1. The formula for the sum is S = a / (1 - r), where a is the first term.https://cdn.kastatic.org/ka-youtube-converted/wqnpSzEzq1w.mp4/wqnpSzEzq1w.mp4474yes https://puzzicle.org/math/ap-calculus-bc/v/infinite-geometric-serieshttps://puzzicle.org/thumbs/xc809ee47.jpgProof of infinite geometric series as a limitThe video provides a proof for the sum of an infinite geometric series using limits. When the absolute value of the common ratio (r) is between 0 and 1, the limit of the series converges to a finite sum. The formula for the sum is a / (1 - r), where a is the first term.https://cdn.kastatic.org/ka-youtube-converted/b-7kCymoUpg.mp4/b-7kCymoUpg.mp4286yes https://puzzicle.org/math/ap-calculus-bc/v/evaluating-infinite-geometric-serieshttps://puzzicle.org/thumbs/x5b3f5531.jpgWorked example: convergent geometric seriesSal evaluates the infinite geometric series 8+8/3+8/9+... Because the common ratio's absolute value is less than 1, the series converges to a finite number.https://cdn.kastatic.org/ka-youtube-converted/Sify7Bp7BmI.mp4/Sify7Bp7BmI.mp4230yes https://puzzicle.org/math/ap-calculus-bc/v/infinite-geometric-series-not-converginghttps://puzzicle.org/thumbs/x6ee026a1.jpgWorked example: divergent geometric seriesSal evaluates the infinite geometric series -0.5+1.5-4.5+... Because the common ratio's absolute value is greater than 1, the series doesn't converge.https://cdn.kastatic.org/ka-youtube-converted/Gk1mWXIpasE.mp4/Gk1mWXIpasE.mp4194yes https://puzzicle.org/math/ap-calculus-bc/v/bouncing-ball-distancehttps://puzzicle.org/thumbs/xb23a1b53.jpgInfinite geometric series word problem: bouncing ballWatch Sal determine the total vertical distance a bouncing ball moves using an infinite geometric series.https://cdn.kastatic.org/ka-youtube-converted/tqTJZEglrvc.mp4/tqTJZEglrvc.mp4310yes https://puzzicle.org/math/ap-calculus-bc/v/repeating-decimal-geometric-serieshttps://puzzicle.org/thumbs/x9a32c336.jpgInfinite geometric series word problem: repeating decimalSee how we can write a repeating decimal as an infinite geometric series.https://cdn.kastatic.org/ka-youtube-converted/2BgWWsypzLA.mp4/2BgWWsypzLA.mp4482yes https://puzzicle.org/math/algebra-home/v/deductive-reasoning-1https://puzzicle.org/thumbs/27088.jpgInductive & deductive reasoningSal discusses the difference between inductive and deductive reasoning by considering a word problem.https://cdn.kastatic.org/ka-youtube-converted/GEId0GonOZM.mp4/GEId0GonOZM.mp4120yes https://puzzicle.org/math/algebra-home/v/deductive-reasoning-2https://puzzicle.org/thumbs/27089.jpgDeductive reasoningSal analyzes a solution of a mathematical problem to determine whether it uses deductive reasoning.https://cdn.kastatic.org/ka-youtube-converted/VMEV__2wW3E.mp4/VMEV__2wW3E.mp4237yes https://puzzicle.org/math/algebra-home/v/deductive-reasoning-3https://puzzicle.org/thumbs/27090.jpgUsing deductive reasoningSal uses deductive reasoning to prove an algebraic identity.https://cdn.kastatic.org/ka-youtube-converted/lYLaaMObgkw.mp4/lYLaaMObgkw.mp4251yes https://puzzicle.org/math/algebra-home/v/inductive-reasoning-2https://puzzicle.org/thumbs/27086.jpgInductive reasoningSal analyzes a solution of a mathematical problem to determine whether it uses inductive reasoning.https://cdn.kastatic.org/ka-youtube-converted/ZSsBh4Ybbj8.mp4/ZSsBh4Ybbj8.mp4141yes https://puzzicle.org/math/algebra-home/v/inductive-reasoning-3https://puzzicle.org/thumbs/27087.jpgInductive reasoning (example 2)Sal analyzes a solution of a mathematical problem to determine whether it uses inductive reasoning.https://cdn.kastatic.org/ka-youtube-converted/pgfWkaySFGY.mp4/pgfWkaySFGY.mp4183yes https://puzzicle.org/math/algebra-home/v/u12-l1-t3-we1-inductive-reasoning-1https://puzzicle.org/thumbs/27085.jpgUsing inductive reasoningSal uses inductive reasoning to find an expression for the nth number in the sequence 6, 9, 12, 15,....https://cdn.kastatic.org/ka-youtube-converted/MnQ7Lizkpqk.mp4/MnQ7Lizkpqk.mp4127yes https://puzzicle.org/math/algebra-home/v/u03-l1-t1-we2-inductive-patternshttps://puzzicle.org/thumbs/2600591.jpgUsing inductive reasoning (example 2)Sal uses inductive reasoning in order to find the 50th element of a pattern toothpick shapes.https://cdn.kastatic.org/ka-youtube-converted/GvbrtnEYRpY.mp4/GvbrtnEYRpY.mp4394yes https://puzzicle.org/math/algebra-home/v/proof-by-inductionhttps://puzzicle.org/thumbs/193202564.jpgProof of finite arithmetic series formula by inductionProving an expression for the sum of all positive integers up to and including n by inductionhttps://cdn.kastatic.org/ka-youtube-converted/wblW_M_HVQ8.mp4/wblW_M_HVQ8.mp4563yes https://puzzicle.org/math/algebra-home/v/sum-of-n-squares-1https://puzzicle.org/thumbs/xe115b9a9.jpgSum of n squares (part 1)What is the sum of the first n squares, 1 + 4 + 9 + 16 + ... + n²? In this video we begin the journey towards finding a formula for this sum.https://cdn.kastatic.org/ka-youtube-converted/i7iKLZQ-vCk.mp4/i7iKLZQ-vCk.mp4376yes https://puzzicle.org/math/algebra-home/v/sum-n-squares-2https://puzzicle.org/thumbs/x4f4861b7.jpgSum of n squares (part 2)What is the sum of the first n squares, 1 + 4 + 9 + 16 + ... + n²? In this video we find one possible formula for this sum.https://cdn.kastatic.org/ka-youtube-converted/MkGXR8umLco.mp4/MkGXR8umLco.mp4312yes https://puzzicle.org/math/algebra-home/v/alternate-sum-of-n-squares-formulahttps://puzzicle.org/thumbs/x463bbd8a.jpgSum of n squares (part 3)The sum of the first n squares, 1 + 4 + 9 + 16 + ... + n², is given by the formula ⅙n(n+1)(2n+1). In this video we factor and rewrite the formula that we found in the previous video and obtain the common formula given above.https://cdn.kastatic.org/ka-youtube-converted/KvvFTPsZxU4.mp4/KvvFTPsZxU4.mp4291yes https://puzzicle.org/math/precalculus/v/introduction-to-vectors-and-scalarshttps://puzzicle.org/thumbs/113633126.jpgIntro to vectors and scalarsScalars and vectors are two kinds of quantities that are used in physics and math. Scalars are quantities that only have magnitude (or size), while vectors have both magnitude and direction. Explore some examples of scalars and vectors, including distance, displacement, speed, and velocity.https://cdn.kastatic.org/ka-youtube-converted/ihNZlp7iUHE.mp4/ihNZlp7iUHE.mp4519yes https://puzzicle.org/math/algebra-home/v/vector-representations-examplehttps://puzzicle.org/thumbs/x88dbd3d7.jpgRecognizing vectorsWatch Sal figure out which of a few options could be represented by a vector. Remember that vectors have both magnitude and direction!https://cdn.kastatic.org/ka-youtube-converted/n8Ic2Oj-zvA.mp4/n8Ic2Oj-zvA.mp4156yes https://puzzicle.org/math/algebra-home/v/equivalent-vectors-exampleshttps://puzzicle.org/thumbs/xf0d96e10.jpgEquivalent vectorsSal determines if two vectors shown on a graph are equivalent by seeing if they have the same magnitude and direction.https://cdn.kastatic.org/ka-youtube-converted/v_W-aaB1irs.mp4/v_W-aaB1irs.mp4397yes https://puzzicle.org/math/precalculus/v/example-finding-components-of-a-vectorhttps://puzzicle.org/thumbs/x014bbaf1.jpgFinding the components of a vectorSal finds the x and y-components of a vector given its graph.https://cdn.kastatic.org/ka-youtube-converted/BmDph4uNfPI.mp4/BmDph4uNfPI.mp4198yes https://puzzicle.org/math/precalculus/v/example-comparing-x-components-of-vectorshttps://puzzicle.org/thumbs/x1e25ccda.jpgComparing the components of vectorsSal figures out which vectors have the same x-component given the graphs of 4 different vectors.https://cdn.kastatic.org/ka-youtube-converted/v7ZZ1ctT1k0.mp4/v7ZZ1ctT1k0.mp4175yes https://puzzicle.org/math/precalculus/v/example-calcuating-magnitude-of-vector-from-graphhttps://puzzicle.org/thumbs/x7e3752a9.jpgVector magnitude from graphSal finds the magnitude of a vector given its graph.https://cdn.kastatic.org/ka-youtube-converted/-U53eHKCLcg.mp4/-U53eHKCLcg.mp4218yes https://puzzicle.org/math/precalculus/v/finding-vector-magnitude-from-componentshttps://puzzicle.org/thumbs/x79bdd17f.jpgVector magnitude from componentsSal finds the magnitude of a vector given its components of (5, -3).https://cdn.kastatic.org/ka-youtube-converted/qMZnsj4ZO9g.mp4/qMZnsj4ZO9g.mp4186yes https://puzzicle.org/math/precalculus/v/vector-magnitude-from-initial-and-terminal-pointshttps://puzzicle.org/thumbs/x904d03e1b40a61f2.jpgVector magnitude from initial & terminal pointsExample of finding the magnitude of a vector when what we're given about the vector is where it starts and where it ends.https://cdn.kastatic.org/ka-youtube-converted/jfaSZhNFEOE.mp4/jfaSZhNFEOE.mp4323yes https://puzzicle.org/math/precalculus/v/understanding-multiplying-vectors-by-scalarshttps://puzzicle.org/thumbs/xd8f11403.jpgScalar multiplication: component formSal explains what happens graphically and to the components of a vector when we multiply it by a scalar.https://cdn.kastatic.org/ka-youtube-converted/JuTvjQgpUMQ.mp4/JuTvjQgpUMQ.mp4342yes https://puzzicle.org/math/precalculus/v/examples-from-understanding-scalar-multiplication-exercisehttps://puzzicle.org/thumbs/xa8a0effa.jpgScalar multiplication: magnitude and directionSal analyzes the magnitude and direction of vectors that are a result of scalar multiplication of a vector whose magnitude is given.https://cdn.kastatic.org/ka-youtube-converted/kLLsJXc1E1E.mp4/kLLsJXc1E1E.mp4293yes https://puzzicle.org/math/precalculus/v/adding-and-subtracting-vectorshttps://puzzicle.org/thumbs/xa1f61db1.jpgAdding & subtracting vectorsSal shows how to add vectors by adding their components, then explains the intuition behind adding vectors using a graph.https://cdn.kastatic.org/ka-youtube-converted/VgqsM-XdBD0.mp4/VgqsM-XdBD0.mp4462yes https://puzzicle.org/math/precalculus/v/visually-adding-and-subtracting-vectorshttps://puzzicle.org/thumbs/xdccf8d66.jpgAdding & subtracting vectors end-to-endBuild intuition behind adding and subtracting vectors visually and the "end-to-end" method.https://cdn.kastatic.org/ka-youtube-converted/BsBH8nAv5l4.mp4/BsBH8nAv5l4.mp4382yes https://puzzicle.org/math/precalculus/v/subtracting-vectors-exercise-examplehttps://puzzicle.org/thumbs/x12b923fb.jpgSubtracting vectors end-to-endWatch Sal determine which diagram represents vector subtraction graphically.https://cdn.kastatic.org/ka-youtube-converted/coDfeoi2zcQ.mp4/coDfeoi2zcQ.mp4104yes https://puzzicle.org/math/linear-algebra/v/adding-vectorshttps://puzzicle.org/thumbs/x6bc2f1ae.jpgAdding vectors algebraically & graphicallyTo add the vectors (x₁,y₁) and (x₂,y₂), we add the corresponding components from each vector: (x₁+x₂,y₁+y₂). Here's a concrete example: the sum of (2,4) and (1,5) is (2+1,4+5), which is (3,9). There's also a nice graphical way to add vectors, and the two ways will always result in the same vector.​​https://cdn.kastatic.org/ka-youtube-converted/8QihetGj3pg.mp4/8QihetGj3pg.mp4432yes https://puzzicle.org/math/algebra-home/v/combined-vector-operations-examplehttps://puzzicle.org/thumbs/xea5f084d.jpgCombined vector operationsWatch Sal find new vector 3u + 1/5w when u = (2, -1) and w = (-5, 5).https://cdn.kastatic.org/ka-youtube-converted/TdhbEonaHrc.mp4/TdhbEonaHrc.mp4359yes https://puzzicle.org/math/algebra-home/v/unit-vector-introhttps://puzzicle.org/thumbs/xb1784845.jpgWorked example: finding unit vector with given directionLearn what a unit vector is and how to find a unit vector in the direction of a given vector.https://cdn.kastatic.org/ka-youtube-converted/jCkhbKFZgLk.mp4/jCkhbKFZgLk.mp4302yes https://puzzicle.org/math/linear-algebra/v/intro-unit-vector-notationhttps://puzzicle.org/thumbs/xad013660.jpgUnit vectors introUnit vectors are vectors whose magnitude is exactly 1 unit. They are very useful for different reasons. Specifically, the unit vectors [0,1] and [1,0] can form together any other vector.https://cdn.kastatic.org/ka-youtube-converted/9ylUcCOTH8Y.mp4/9ylUcCOTH8Y.mp4479yes https://puzzicle.org/math/algebra-home/v/scaling-unit-vectorhttps://puzzicle.org/thumbs/x95ab94f3.jpgWorked example: Scaling unit vectorsWatch Sal scale up a unit vector to have a magnitude greater than 1.https://cdn.kastatic.org/ka-youtube-converted/OFBI_rNDZE0.mp4/OFBI_rNDZE0.mp4173yes https://puzzicle.org/math/precalculus/v/angles-of-vectors-from-componentshttps://puzzicle.org/thumbs/xadedc828.jpgDirection of vectors from components: 1st & 2nd quadrantsSal first finds the direction angle of a vector in the first quadrant, then moves onto a trickier one in the second quadrant.https://cdn.kastatic.org/ka-youtube-converted/A3OuFLHum6w.mp4/A3OuFLHum6w.mp4541yes https://puzzicle.org/math/precalculus/v/more-examples-finding-vector-angleshttps://puzzicle.org/thumbs/x06688cbf.jpgDirection of vectors from components: 3rd & 4th quadrantsSal finds the direction angle of a vector in the third quadrant and a vector in the fourth quadrant.https://cdn.kastatic.org/ka-youtube-converted/EX0Ha42WQ24.mp4/EX0Ha42WQ24.mp4402yes https://puzzicle.org/math/precalculus/v/vector-components-from-magnitude-and-directionhttps://puzzicle.org/thumbs/xa3355657.jpgVector components from magnitude & directionSal finds the components of a couple of vectors given in magnitude and direction form.https://cdn.kastatic.org/ka-youtube-converted/MkFbisCyjFc.mp4/MkFbisCyjFc.mp4616yes https://puzzicle.org/math/algebra-home/v/mag-dir-vec-sumshttps://puzzicle.org/thumbs/xb581cb7c.jpgAdding vectors in magnitude & direction form (1 of 2)Watch Sal add two vectors given in magnitude and direction form by breaking them down into components first.https://cdn.kastatic.org/ka-youtube-converted/6Kw2nIwWYL0.mp4/6Kw2nIwWYL0.mp4562yes https://puzzicle.org/math/algebra-home/v/mag-dir-vec-sums-2https://puzzicle.org/thumbs/x6d0446b8.jpgAdding vectors in magnitude & direction form (2 of 2)Watch Sal finish the problem he started in part 1 by converting the sum back to magnitude and direction form.https://cdn.kastatic.org/ka-youtube-converted/SjY5UytlTTs.mp4/SjY5UytlTTs.mp4487yes https://puzzicle.org/math/precalculus/v/mag-vec-sumshttps://puzzicle.org/thumbs/x5a2e610c.jpgVector addition & magnitudeSal explains some interesting properties of the magnitude of vector sums.https://cdn.kastatic.org/ka-youtube-converted/0t8W4JFpP2M.mp4/0t8W4JFpP2M.mp4330yes https://puzzicle.org/math/precalculus/v/vector-component-in-directionhttps://puzzicle.org/thumbs/xff91329c.jpgVector components from magnitude & direction: word problemGiven the magnitudes and directions of the forces two people are applying to a box, find the overall force that is applied in the direction of the target.https://cdn.kastatic.org/ka-youtube-converted/9A8wM1UMLyo.mp4/9A8wM1UMLyo.mp4536yes https://puzzicle.org/math/algebra-home/v/adding-three-vectorshttps://puzzicle.org/thumbs/xf3e77364.jpgVectors word problem: tug of warSal solves a word problem where three vector forces act upon a ring in different magnitudes & directions.https://cdn.kastatic.org/ka-youtube-converted/KNklNvGof8o.mp4/KNklNvGof8o.mp4524yes https://puzzicle.org/math/precalculus/v/adding-displacement-vectorshttps://puzzicle.org/thumbs/x8e6a40ba.jpgVector word problem: hikingSal solves a word problem with vectors where he finds the total straight-line distance traveled over a few days.https://cdn.kastatic.org/ka-youtube-converted/cf1-CwrDwrA.mp4/cf1-CwrDwrA.mp4416yes https://puzzicle.org/math/precalculus/v/introduction-to-the-matrixhttps://puzzicle.org/thumbs/1400915680.jpgIntro to matricesTake the red pill and enter the Matrix!https://cdn.kastatic.org/ka-youtube-converted/0oGJTQCy4cQ.mp4/0oGJTQCy4cQ.mp4269yes https://puzzicle.org/math/precalculus/v/matrix-addition-and-subtraction-1https://puzzicle.org/thumbs/1401149467.jpgAdding matrices (with -1 scalar also)Sal defines what it means to add or subtract matrices. He shows a few examples and discusses some important properties of matrix addition and subtraction.https://cdn.kastatic.org/ka-youtube-converted/WR9qCSXJlyY.mp4/WR9qCSXJlyY.mp4335yes https://puzzicle.org/math/precalculus/v/scalar-multiplicationhttps://puzzicle.org/thumbs/1400835599.jpgMultiplying matrices by scalarsSal defines what it means to multiply a matrix by a scalar (in the world of matrices, a scalar is simply a regular number).https://cdn.kastatic.org/ka-youtube-converted/TbaltFbJ3wE.mp4/TbaltFbJ3wE.mp4138yes https://puzzicle.org/math/precalculus/v/matrix-multiplication-introhttps://puzzicle.org/thumbs/xe9bce422.jpgIntro to matrix multiplicationSal explains what it means to multiply two matrices, and gives an example.https://cdn.kastatic.org/ka-youtube-converted/kT4Mp9EdVqs.mp4/kT4Mp9EdVqs.mp4386yes https://puzzicle.org/math/precalculus/v/multiplying-a-matrix-by-a-matrixhttps://puzzicle.org/thumbs/x07e01747.jpgMultiplying matricesSal gives an example of a multiplication of two matrices that don't have the same dimensions.https://cdn.kastatic.org/ka-youtube-converted/OMA2Mwo0aZg.mp4/OMA2Mwo0aZg.mp4330yes https://puzzicle.org/math/precalculus/v/defined-and-undefined-matrix-operationshttps://puzzicle.org/thumbs/x044ecd2f.jpgDefined matrix operationsSal discusses the conditions of matrix dimensions for which addition or multiplication are defined.https://cdn.kastatic.org/ka-youtube-converted/O1-9f1g0OsI.mp4/O1-9f1g0OsI.mp4212yes https://puzzicle.org/math/precalculus/v/identity-matrixhttps://puzzicle.org/thumbs/x45df6c9a.jpgIntro to identity matrixJust as any number remains the same when multiplied by 1, any matrix remains the same when multiplied by the identity matrix. Learn more from Sal.https://cdn.kastatic.org/ka-youtube-converted/3cnIa0fYJkY.mp4/3cnIa0fYJkY.mp4479yes https://puzzicle.org/math/precalculus/v/identity-matrix-dimensionshttps://puzzicle.org/thumbs/xcb707bb1.jpgDimensions of identity matrixSal explains why the identity matrix is always a square matrix, even though it works with non-square matrices.https://cdn.kastatic.org/ka-youtube-converted/l7p1X5pdDoc.mp4/l7p1X5pdDoc.mp4224yes https://puzzicle.org/math/precalculus/v/commutative-property-matrix-multiplicationhttps://puzzicle.org/thumbs/xe5c8c1bb.jpgIs matrix multiplication commutative?Sal checks whether the commutative property applies for matrix multiplication. In other words, he checks whether for any two matrices A and B, A*B=B*A (the answer is NO, by the way).https://cdn.kastatic.org/ka-youtube-converted/z-SU7P-gIoQ.mp4/z-SU7P-gIoQ.mp4452yes https://puzzicle.org/math/precalculus/v/associative-property-matrix-multiplicationhttps://puzzicle.org/thumbs/xa56c25c1.jpgAssociative property of matrix multiplicationSal shows that matrix multiplication is associative. Mathematically, this means that for any three matrices A, B, and C, (A*B)*C=A*(B*C).https://cdn.kastatic.org/ka-youtube-converted/8Ryfe82DTcM.mp4/8Ryfe82DTcM.mp4517yes https://puzzicle.org/math/precalculus/v/zero-matrixhttps://puzzicle.org/thumbs/x04b88473.jpgZero matrix & matrix multiplicationJust as any number multiplied by zero is zero, there is a zero matrix such that any matrix multiplied by it results in that zero matrix. Learn more from Sal.https://cdn.kastatic.org/ka-youtube-converted/LOf8bfjiLow.mp4/LOf8bfjiLow.mp4282yes https://puzzicle.org/math/precalculus/v/matrix-expressionshttps://puzzicle.org/thumbs/x6ed8fca0.jpgUsing properties of matrix operationsSal determines which of a few optional matrix expressions is equivalent to the matrix expression A*B*C. This is done using what we know about the properties of matrix addition and multiplication.https://cdn.kastatic.org/ka-youtube-converted/OjF765iVuF8.mp4/OjF765iVuF8.mp4210yes https://puzzicle.org/math/precalculus/v/identity-zero-matrix-equationhttps://puzzicle.org/thumbs/x620f2504.jpgUsing identity & zero matricesSal solves a problem where he has to determine whether unknown matrices are zero or identity to make an equation true.https://cdn.kastatic.org/ka-youtube-converted/ikadoCpDNT8.mp4/ikadoCpDNT8.mp4238yes https://puzzicle.org/math/algebra-home/v/transforming-position-vectorhttps://puzzicle.org/thumbs/x181bbe4e.jpgTransforming vectors using matricesSal transforms a 2-dimensional vector using a 2x2 matrix, and draws the original vector and its image on the plane.https://cdn.kastatic.org/ka-youtube-converted/Kh8HKAxdEyw.mp4/Kh8HKAxdEyw.mp4283yes https://puzzicle.org/math/algebra-home/v/matrix-transformation-trianglehttps://puzzicle.org/thumbs/xe4b9c0d5.jpgTransforming polygons using matricesSal transforms a triangle using a 2x2 matrix.https://cdn.kastatic.org/ka-youtube-converted/RKBSX-6pKgY.mp4/RKBSX-6pKgY.mp4435yes https://puzzicle.org/math/algebra-home/v/matrix-transformation-quadrilateralhttps://puzzicle.org/thumbs/x723c8dfd.jpgVisual representation of transformation from matrixSal finds the drawing that appropriately represents the effect of a given 2x2 transformation matrix on the plane.https://cdn.kastatic.org/ka-youtube-converted/3tyM_M3FQNM.mp4/3tyM_M3FQNM.mp4368yes https://puzzicle.org/math/algebra-home/v/finding-the-determinant-of-a-2x2-matrixhttps://puzzicle.org/thumbs/1128184016.jpgDeterminant of a 2x2 matrixSal shows how to find the determinant of a 2x2 matrix.https://cdn.kastatic.org/ka-youtube-converted/OU9sWHk_dlw.mp4/OU9sWHk_dlw.mp470yes https://puzzicle.org/math/algebra-home/v/inverse-matrix-part-1https://puzzicle.org/thumbs/25203.jpgIntro to matrix inversesSal introduces the concept of an inverse matrix.https://cdn.kastatic.org/ka-youtube-converted/iUQR0enP7RQ.mp4/iUQR0enP7RQ.mp4854yes https://puzzicle.org/math/algebra-home/v/singular-matriceshttps://puzzicle.org/thumbs/25213.jpgDetermining invertible matricesSal shows why a matrix is invertible if and only if its determinant is not 0.https://cdn.kastatic.org/ka-youtube-converted/UqyN7-tRS00.mp4/UqyN7-tRS00.mp4867yes https://puzzicle.org/math/precalculus/v/inverse-of-a-2x2-matrixhttps://puzzicle.org/thumbs/1128298896.jpgFinding inverses of 2x2 matricesSal gives an example of how to find the inverse of a given 2x2 matrix.https://cdn.kastatic.org/ka-youtube-converted/01c12NaUQDw.mp4/01c12NaUQDw.mp4168yes https://puzzicle.org/math/algebra-home/v/finding-the-determinant-of-a-3x3-matrix-method-2https://puzzicle.org/thumbs/1128472045.jpgDeterminant of a 3x3 matrix: standard method (1 of 2)Sal shows the standard method for finding the determinant of a 3x3 matrix.https://cdn.kastatic.org/ka-youtube-converted/u00I3MCrspU.mp4/u00I3MCrspU.mp4237yes https://puzzicle.org/math/algebra-home/v/finding-the-determinant-of-a-3x3-matrix-method-1https://puzzicle.org/thumbs/1128416416.jpgDeterminant of a 3x3 matrix: shortcut method (2 of 2)Sal shows a "shortcut" method for finding the determinant of a 3x3 matrix.https://cdn.kastatic.org/ka-youtube-converted/v4MenooI1J0.mp4/v4MenooI1J0.mp4159yes https://puzzicle.org/math/algebra-home/v/inverting-matrices-part-3https://puzzicle.org/thumbs/25207.jpgInverting a 3x3 matrix using Gaussian eliminationSal explains how we can find the inverse of a 3x3 matrix using Gaussian elimination.https://cdn.kastatic.org/ka-youtube-converted/obts_JDS6_Q.mp4/obts_JDS6_Q.mp4816yes https://puzzicle.org/math/algebra-home/v/inverting-3x3-part-1-calculating-matrix-of-minors-and-cofactor-matrixhttps://puzzicle.org/thumbs/1128367509.jpgInverting a 3x3 matrix using determinants Part 1: Matrix of minors and cofactor matrixSal shows how to find the inverse of a 3x3 matrix using its determinant. In Part 1 we learn how to find the matrix of minors of a 3x3 matrix and its cofactor matrix.https://cdn.kastatic.org/ka-youtube-converted/xZBbfLLfVV4.mp4/xZBbfLLfVV4.mp4527yes https://puzzicle.org/math/algebra-home/v/inverting-3x3-part-2-determinant-and-adjugate-of-a-matrixhttps://puzzicle.org/thumbs/1128279750.jpgInverting a 3x3 matrix using determinants Part 2: Adjugate matrixSal shows how to find the inverse of a 3x3 matrix using its determinant. In Part 2 we complete the process by finding the determinant of the matrix and its adjugate matrix.https://cdn.kastatic.org/ka-youtube-converted/ArcrdMkEmKo.mp4/ArcrdMkEmKo.mp4383yes https://puzzicle.org/math/algebra-home/v/matrix-equations-systemshttps://puzzicle.org/thumbs/x41cbb463.jpgRepresenting linear systems with matrix equationsSal shows how a system of two linear equations can be represented with the equation A*x=b where A is the coefficient matrix, x is the variable vector, and b is the constant vector.https://cdn.kastatic.org/ka-youtube-converted/EC2mgUZyzoA.mp4/EC2mgUZyzoA.mp4595yes https://puzzicle.org/math/precalculus/v/solving-matrix-equationhttps://puzzicle.org/thumbs/x516a2c36.jpgSolving linear systems with matricesSal solves that matrix equation using the inverse of the coefficient matrix.https://cdn.kastatic.org/ka-youtube-converted/0_DYEFtlCiM.mp4/0_DYEFtlCiM.mp4398yes https://puzzicle.org/math/algebra-home/v/matrices-to-solve-a-vector-combination-problemhttps://puzzicle.org/thumbs/25211.jpgMatrix word problem: vector combinationSal finds the appropriate combination of two given vectors in order to obtain a third given vector. This is done by representing the problem with a single matrix equation and solving that equation.https://cdn.kastatic.org/ka-youtube-converted/gsNgdVdAT1o.mp4/gsNgdVdAT1o.mp4860yes https://puzzicle.org/math/algebra-home/v/data-in-matriceshttps://puzzicle.org/thumbs/xf6a9994c.jpgMatrix word problem: pricesSal shows how matrices can be used to efficiently represent data about the prices of toilet paper and toothpaste in two different cities.https://cdn.kastatic.org/ka-youtube-converted/PPOIlLhsT6s.mp4/PPOIlLhsT6s.mp4343yes https://puzzicle.org/math/algebra2/v/factor-high-deg-monomialshttps://puzzicle.org/thumbs/x7af1edb2cc92a08b.jpgIntroduction to factoring higher degree monomialsFactoring monomials involves breaking down a single term into the product of other terms. This video explores how to factor monomials, like 6x to the 7th power, into simpler parts. It also introduces the concept of prime factorization, which is useful for factoring higher degree expressions.https://cdn.kastatic.org/ka-youtube-converted/Kze7cw30F60.mp4/Kze7cw30F60.mp4283yes https://puzzicle.org/math/algebra2/v/polynomial-identities-introhttps://puzzicle.org/thumbs/x1534feedace1666e.jpgPolynomial identities introductionPolynomial identities are equal expressions involving polynomials. We can prove these identities through algebraic manipulation, like expanding and simplifying. This helps us see if two expressions are the same for all values of a variable, making them true polynomial identities.https://cdn.kastatic.org/ka-youtube-converted/EvNKKyhLSpQ.mp4/EvNKKyhLSpQ.mp4296yes https://puzzicle.org/math/algebra2/v/relationships-w-poly-identitieshttps://puzzicle.org/thumbs/xbc8b185cf7a1a65b.jpgDescribing numerical relationships with polynomial identitiesExplore numerical relationships in polynomial identities, demonstrating how they can reveal patterns in sequences of integer squares. Learn how algebra can prove that the difference between successive terms in a sequence of integer squares is always an increasing odd number.https://cdn.kastatic.org/ka-youtube-converted/uJwq67y5wAY.mp4/uJwq67y5wAY.mp4274yes https://puzzicle.org/math/algebra2/v/geo-series-word-problem-swinghttps://puzzicle.org/thumbs/xabd79ca6fc794e36.jpgGeometric series word problems: swingThe length a monkey passes when swinging from a tree can be modeled by a geometric series! We learn that each swing's length is half the previous one, forming a finite geometric series. By applying the sum formula, we calculate the total distance the monkey swings.https://cdn.kastatic.org/ka-youtube-converted/NlzK79p3QDQ.mp4/NlzK79p3QDQ.mp4352yes https://puzzicle.org/math/algebra2/v/geo-series-word-problem-hikehttps://puzzicle.org/thumbs/xe4bdeef0a16ce982.jpgGeometric series word problems: hikeCan a finite geometric series help us plan a hiking trip? Sloan embarks on a four-day hiking trip, increasing her distance by 20% each day. To figure out her first day's distance, we use the finite geometric series formula. This math tool helps us solve Sloan's hiking puzzle.https://cdn.kastatic.org/ka-youtube-converted/9jxd1EUfQnc.mp4/9jxd1EUfQnc.mp4390yes https://puzzicle.org/math/algebra2/v/modeling-periodic-function-with-shifthttps://puzzicle.org/thumbs/x049abc7f.jpgTrig word problem: length of day (phase shift)Sal solves a word problem about the annual change in the length of day by modeling it with a sinusoidal function that has a phase shift.https://cdn.kastatic.org/ka-youtube-converted/MJBjGnR6vlk.mp4/MJBjGnR6vlk.mp4463yes https://puzzicle.org/math/algebra2/v/graph-labels-and-scaleshttps://puzzicle.org/thumbs/x338c6e3dc24cbc11.jpgGraph labels and scalesWhen graphing a real-world relationship, we need to pick labels and axis scales that are appropriate for the purpose of our model.https://cdn.kastatic.org/ka-youtube-converted/IoPQtvXkde0.mp4/IoPQtvXkde0.mp4273yes https://puzzicle.org/math/algebra2/v/modeling-with-multiple-variables-pancakeshttps://puzzicle.org/thumbs/x4bde56bfed975b68.jpgModeling with multiple variables: PancakesModeling the relationship between three quantities (or more) isn't that different from modeling the relationship between two quantities. Here is an example of a model that relates different quantities in the context of making pancakes.https://cdn.kastatic.org/ka-youtube-converted/bp29Cf4xCqU.mp4/bp29Cf4xCqU.mp4145yes https://puzzicle.org/math/algebra2/v/modeling-with-multiple-variables-roller-coasterhttps://puzzicle.org/thumbs/x056b41d61dd7068a.jpgModeling with multiple variables: Roller coasterModeling the relationship between three quantities (or more) isn't that different from modeling the relationship between two quantities. Here is an example of a model that relates different quantities in the context of roller coaster rides.https://cdn.kastatic.org/ka-youtube-converted/hANm5-51ywk.mp4/hANm5-51ywk.mp4140yes https://puzzicle.org/math/algebra2/v/modeling-with-multiple-variables-taco-standhttps://puzzicle.org/thumbs/xb64563073b5a7e70.jpgModeling with multiple variables: Taco standModeling the relationship between three quantities (or more) isn't that different from modeling the relationship between two quantities. Here is an example of a model that relates different quantities related to the daily profits of a taco stand.https://cdn.kastatic.org/ka-youtube-converted/YvuW40fpptI.mp4/YvuW40fpptI.mp4196yes https://puzzicle.org/math/algebra2/v/modeling-with-multiple-variables-ice-creamhttps://puzzicle.org/thumbs/xaa312f986d971d09.jpgModeling with multiple variables: Ice creamModeling the relationship between three quantities (or more) isn't that different from modeling the relationship between two quantities. Here is an example of a model that relates different quantities related to two people walking to the same ice cream shop.https://cdn.kastatic.org/ka-youtube-converted/_auAZcmJF3M.mp4/_auAZcmJF3M.mp4208yes https://puzzicle.org/math/ap-calculus-ab/v/khan-academy-in-the-classroom-bill-scotthttps://puzzicle.org/thumbs/xec0c91221a4cd607.jpgKhan Academy in the classroomBill Scott uses Khan Academy to teach AP Calculus at Phillips Academy in Andover, Massachusetts, and he's part of the teaching team that helped develop Khan Academy's AP lessons. Phillips Academy was one of the first schools to teach AP nearly 60 years ago.https://cdn.kastatic.org/ka-youtube-converted/Tx-AtZOJYfE.mp4/Tx-AtZOJYfE.mp4206yes https://puzzicle.org/math/ap-calculus-ab/v/sal-interviews-the-ap-calculus-lead-at-college-boardhttps://puzzicle.org/thumbs/xc8a53bbe.jpgSal interviews the AP Calculus Lead at College BoardIn this video, Sal interviews Ben Hedrick, the AP Calculus Lead at College Board. They discuss the AP Calculus exam's focus on conceptual understanding, the importance of mastering core topics like derivatives and integrals, and the value of practicing with past exams. Ben also advises students to relax and approach the exam as just another test.https://cdn.kastatic.org/ka-youtube-converted/_3VKtPGbSzk.mp4/_3VKtPGbSzk.mp4597yes https://puzzicle.org/math/ap-calculus-ab/v/introduction-to-limits-hdhttps://puzzicle.org/thumbs/91852634.jpgIntroducing limitsIn this video, we learn about limits, a fundamental concept in calculus. Limits help us understand what a function approaches as the input gets closer to a certain value, even when the function is undefined at that point. The video demonstrates this concept using two examples with different functions.https://cdn.kastatic.org/ka-youtube-converted/riXcZT2ICjA.mp4/riXcZT2ICjA.mp4692yes https://puzzicle.org/math/ap-calculus-ab/v/limits-from-graphshttps://puzzicle.org/thumbs/xbb039ef35482481d.jpgEstimating limit values from graphsIn this video, we learn to estimate limit values from graphs by observing the function's behavior as x approaches a value from both left and right sides. If the function approaches the same value from both sides, the limit exists. If it approaches different values or is unbounded, the limit doesn't exist.https://cdn.kastatic.org/ka-youtube-converted/mols6pMKrto.mp4/mols6pMKrto.mp4361yes https://puzzicle.org/math/ap-calculus-ab/v/unbounded-limitshttps://puzzicle.org/thumbs/xedd08d5c288f2e21.jpgExamining unbounded limitsThis video discusses estimating limit values from graphs, focusing on two functions: y = 1/x² and y = 1/x. For y = 1/x², the limit is unbounded as x approaches 0, since the function increases without bound. For y = 1/x, the limit doesn't exist as x approaches 0, since it's unbounded in opposite directions.https://cdn.kastatic.org/ka-youtube-converted/56qtGRCd8bE.mp4/56qtGRCd8bE.mp4152yes https://puzzicle.org/math/ap-calculus-ab/v/one-sided-limits-from-graphshttps://puzzicle.org/thumbs/x0aee8ec3.jpgEstimating one-sided limits from graphsA one-sided limit is the value the function approaches as the x-values approach the limit from *one side only*. For example, f(x)=|x|/x returns -1 for negative numbers, 1 for positive numbers, and isn't defined for 0. The one-sided *right* limit of f at x=0 is 1, and the one-sided *left* limit at x=0 is -1.https://cdn.kastatic.org/ka-youtube-converted/nOnd3SiYZqM.mp4/nOnd3SiYZqM.mp4551yes https://puzzicle.org/math/ap-calculus-ab/v/one-sided-limits-from-graphs-asymptotehttps://puzzicle.org/thumbs/xc8975421.jpgEstimating one-sided limits from graphs: asymptoteThis video explores estimating one-sided limit values from graphs. As x approaches 6 from the left, the function becomes unbounded with an asymptote, making the left-sided limit nonexistent. However, when approaching 6 from the right, the function approaches -3, indicating that the right-handed limit exists. Sal's analysis highlights the importance of understanding limits from both sides.https://cdn.kastatic.org/ka-youtube-converted/5f1-Rg3MmKs.mp4/5f1-Rg3MmKs.mp4147yes https://puzzicle.org/math/ap-calculus-ab/v/connecting-limits-and-graphical-behaviorhttps://puzzicle.org/thumbs/x38f5a88c7c08b6ed.jpgConnecting limits and graphical behaviorUsually when we analyze a function's limits from its graph, we are looking at the more "interesting" points. It's important to remember that you can talk about the function's value at any point. Also, a description of a limit can apply to multiple different functions.https://cdn.kastatic.org/ka-youtube-converted/l6FX_r_Tkls.mp4/l6FX_r_Tkls.mp4303yes https://puzzicle.org/math/ap-calculus-ab/v/missing-function-detail-when-zoomed-outhttps://puzzicle.org/thumbs/x07fe160200ec1a22.jpgMissing function detail when zoomed outUnderstand because of issues of scale, graphical representations of functions may miss important function behavior.https://cdn.kastatic.org/ka-youtube-converted/oCfQz8BcJN4.mp4/oCfQz8BcJN4.mp4209yes https://puzzicle.org/math/ap-calculus-ab/v/missing-function-features-when-zoomed-inhttps://puzzicle.org/thumbs/x35f7d4dfe4c31d89.jpgMissing function features when zoomed inUnderstand because of issues of scale, graphical representations of functions may miss important function behavior.https://cdn.kastatic.org/ka-youtube-converted/Nfdu6x4vJuQ.mp4/Nfdu6x4vJuQ.mp4198yes https://puzzicle.org/math/ap-calculus-ab/v/finding-limit-of-an-oscillating-functionhttps://puzzicle.org/thumbs/xbaf15f030d45a687.jpgFinding limit of an oscillating functionA limit might not exist if the function is oscillating near a value of x.https://cdn.kastatic.org/ka-youtube-converted/vyWni0VxLao.mp4/vyWni0VxLao.mp475yes https://puzzicle.org/math/ap-calculus-ab/v/approximating-limit-from-tablehttps://puzzicle.org/thumbs/x91d9e677c75da969.jpgApproximating limits using tablesIn this video, we learn about estimating limit values from tables. The main points are approximating the limit from the left (values less than the target) and the right (values greater than the target). By getting closer to the target value from both sides, we can estimate the limit even if the expression is not defined at the target value.https://cdn.kastatic.org/ka-youtube-converted/1tn0vd2835k.mp4/1tn0vd2835k.mp4267yes https://puzzicle.org/math/ap-calculus-ab/v/estimating-limit-from-tablehttps://puzzicle.org/thumbs/xba5a348d6c1f4a1e.jpgEstimating limits from tablesWhen given a table of values for a function, we can estimate the limit at a certain point by observing the values the function approaches from both sides. The limit is the value the function converges to, even if the function's value at that point is different.https://cdn.kastatic.org/ka-youtube-converted/zX49LocrXbg.mp4/zX49LocrXbg.mp4204yes https://puzzicle.org/math/ap-calculus-ab/v/one-sided-limits-from-tableshttps://puzzicle.org/thumbs/x618f6823de234d0f.jpgEstimating one-sided limits from tablesIn this video, we explore one-sided limits using tables, focusing on estimating the value a function approaches from either the left or right. We emphasize the importance of understanding the negative and positive superscripts, which indicate the direction we're approaching the limit from.https://cdn.kastatic.org/ka-youtube-converted/9geXUOXTJJQ.mp4/9geXUOXTJJQ.mp4380yes https://puzzicle.org/math/ap-calculus-ab/v/limit-propertieshttps://puzzicle.org/thumbs/1313230441.jpgIntroducing properties of limitsThis video introduces limit properties, which are intuitive rules that help simplify limit problems. The main properties covered are the sum, difference, product, quotient, and exponent rules. These properties allow you to break down complex limits into simpler components, making it easier to find the limit of a function.https://cdn.kastatic.org/ka-youtube-converted/lSwsAFgWqR8.mp4/lSwsAFgWqR8.mp4308yes https://puzzicle.org/math/ap-calculus-ab/v/limits-of-combined-functionshttps://puzzicle.org/thumbs/xf598b75a.jpgDetermining limits of combined functionsIn this video, we learn how to find the limit of combined functions using algebraic properties of limits. The main ideas are that the limit of a product is the product of the limits, and that the limit of a quotient is the quotient of the limits, provided the denominator's limit isn't zero.https://cdn.kastatic.org/ka-youtube-converted/AhnJtTI_DMM.mp4/AhnJtTI_DMM.mp4249yes https://puzzicle.org/math/ap-calculus-ab/v/limits-of-combined-functions-piecewisehttps://puzzicle.org/thumbs/xb8e7508a99eac58f.jpgDetermining limits of combined functions: piecewise functionsThis video demonstrates that even when individual limits of functions f(x) and g(x) don't exist, the limit of their sum or product might still exist. By analyzing left and right-hand limits, we can determine if the limit of the combined functions exists and find its value.https://cdn.kastatic.org/ka-youtube-converted/Xo49w4DDEfQ.mp4/Xo49w4DDEfQ.mp4253yes https://puzzicle.org/math/ap-calculus-ab/v/limits-of-composite-functionshttps://puzzicle.org/thumbs/xce616e87.jpgExamining the theorem for limits of composite functionsThis video focuses on finding the limit of composite functions, specifically the limit as 'x' approaches 'a' of f(g(x)). It explains that this limit equals f(limit as 'x' approaches 'a' of g(x)) if two conditions are met: the limit of g(x) exists, and f(x) is continuous at that limit.https://cdn.kastatic.org/ka-youtube-converted/9Z3ngNi-12k.mp4/9Z3ngNi-12k.mp4193yes https://puzzicle.org/math/ap-calculus-ab/v/theorem-for-limits-of-composite-functions-when-conditions-aren-t-methttps://puzzicle.org/thumbs/x747e37f92542c9b6.jpgExamining the theorem for limits of composite functions: when conditions aren't metThis video focuses on determining limits of composite functions using algebraic properties of limits. It explains that if the theorem for limits of composite functions doesn't apply, the limit might still exist. The video demonstrates this through examples and analyzing graphs of functions.https://cdn.kastatic.org/ka-youtube-converted/eK8T3CcSvwE.mp4/eK8T3CcSvwE.mp4202yes https://puzzicle.org/math/ap-calculus-ab/v/limits-of-composite-functions-internal-limit-doesn-t-existhttps://puzzicle.org/thumbs/x73e09706aab6477d.jpgDetermining limits of composite functions: internal limit doesn't existThis video focuses on finding the limit of composite functions, even when the limit of the internal function does not exist. It demonstrates how to use right-handed and left-handed limits to determine if the composite function's limit exists by analyzing the graphs of the functions involved.https://cdn.kastatic.org/ka-youtube-converted/XYaZgRQdV3g.mp4/XYaZgRQdV3g.mp4257yes https://puzzicle.org/math/ap-calculus-ab/v/limits-of-composite-functions-external-limit-doesn-t-existhttps://puzzicle.org/thumbs/x6300390872f3a878.jpgDetermining limits of composite functions: external limit doesn't existFinding the limit of g(h(x)) at x=1 when the limit of h(x) at x=1 is 2 and the limit of g(x) at x=2 doesn't exist. Does it mean that the composite limit doesn't exist? Not necessarily! See how we analyze it.https://cdn.kastatic.org/ka-youtube-converted/VAk50-zIes8.mp4/VAk50-zIes8.mp4205yes https://puzzicle.org/math/ap-calculus-ab/v/limit-by-substitutionhttps://puzzicle.org/thumbs/xc6cdef6f.jpgFinding limits by direct substitutionSal explains how you can easily find limits of functions at points where the functions are continuous: simply plug in the x-value into the function! Later we will learn how to find limits even when the function isn't continuous.https://cdn.kastatic.org/ka-youtube-converted/cfOcOgr0E7U.mp4/cfOcOgr0E7U.mp4127yes https://puzzicle.org/math/ap-calculus-ab/v/undefined-limit-by-substitutionhttps://puzzicle.org/thumbs/xcb791a87.jpgExamining undefined limits by direct substitutionSal gives an example of a limit where direct substitution ends in a quotient with 0 in the denominator and non-0 in the numerator. Such limits are undefined. What about limits where substitution ends in 0/0? Keep going and you'll see!https://cdn.kastatic.org/ka-youtube-converted/JlOyb_RTejo.mp4/JlOyb_RTejo.mp4134yes https://puzzicle.org/math/ap-calculus-ab/v/limits-of-trigonometric-functionshttps://puzzicle.org/thumbs/x2e2de15b2bd817a2.jpgDetermining limits of trigonometric functionsThis video covers limits of trigonometric functions, focusing on sine, cosine, and tangent. It emphasizes that sine and cosine are continuous and defined for all real numbers, so their limits can be found using direct substitution. For tangent and cotangent, limits depend on whether the point is in their domain.https://cdn.kastatic.org/ka-youtube-converted/5_zxtSyTfOY.mp4/5_zxtSyTfOY.mp4366yes https://puzzicle.org/math/ap-calculus-ab/v/limits-of-piecewise-functionshttps://puzzicle.org/thumbs/xc5697551683ca362.jpgDetermining limits of piecewise functionsIn this video, we explore limits of piecewise functions using algebraic properties of limits and direct substitution. We learn that to find one-sided and two-sided limits, we need to consider the function definition for the specific interval we're approaching and substitute the value of x accordingly.https://cdn.kastatic.org/ka-youtube-converted/2xdh0yKopB8.mp4/2xdh0yKopB8.mp4229yes https://puzzicle.org/math/ap-calculus-ab/v/limit-at-a-point-of-discontinuityhttps://puzzicle.org/thumbs/1313337273.jpgDetermining limits of piecewise functions: absolute valueThis video focuses on finding the limit of |x-3|/(x-3) at x=3 by rewriting it and examining it as a piecewise function. This approach helps us understand the behavior of the function for x values greater or less than 3, revealing that the limit doesn't exist.https://cdn.kastatic.org/ka-youtube-converted/Y7sqB1e4RBI.mp4/Y7sqB1e4RBI.mp4380yes https://puzzicle.org/math/ap-calculus-ab/v/limit-example-1https://puzzicle.org/thumbs/1313161995.jpgDetermining limits by factoringIn this video, we explore the limit of (x²+x-6)/(x-2) as x approaches 2. By factoring and simplifying the expression, we discover that the function is undefined at x = 2, but its limit from both sides as x approaches 2 is in fact 5.https://cdn.kastatic.org/ka-youtube-converted/EAa3J_nDkoI.mp4/EAa3J_nDkoI.mp4344yes https://puzzicle.org/math/ap-calculus-ab/v/limits-by-rationalizinghttps://puzzicle.org/thumbs/x626f2630.jpgDetermining limits by rationalizingIn this video, we explore how to find the limit of a function as x approaches -1. The function is (x+1)/(√(x+5)-2). To tackle the indeterminate form 0/0, we "rationalize the denominator" by multiplying the numerator and denominator by the conjugate of the denominator. This simplifies the expression, allowing us to evaluate the limit.https://cdn.kastatic.org/ka-youtube-converted/yVWxxQ7SMOw.mp4/yVWxxQ7SMOw.mp4572yes https://puzzicle.org/math/ap-calculus-ab/v/trig-limit-using-pythagorean-identityhttps://puzzicle.org/thumbs/xb48ac8d7.jpgUsing the Pythagorean identity to determine trig limitsIn this video, we explore finding the limit as θ approaches 0 for the expression (1-cosθ)/(2sin²θ). By using the Pythagorean identity, we rewrite the expression to simplify it and avoid the indeterminate form 0/0. This allows us to evaluate the limit and find the answer, 1/4.https://cdn.kastatic.org/ka-youtube-converted/d8qtbGMB2gI.mp4/d8qtbGMB2gI.mp4422yes https://puzzicle.org/math/ap-calculus-ab/v/trig-limit-using-double-angle-identityhttps://puzzicle.org/thumbs/x44aa2481.jpgUsing double angle identities to determine trig limitsIn this video, we dive into finding the limit at θ=-π/4 of (1+√2sinθ)/(cos2θ) by employing trigonometric identities. We use the cosine double angle identity to rewrite the expression, allowing us to simplify and cancel terms. This approach helps us overcome the indeterminate form and find the limit, showcasing the power of trig identities in solving limit problems.https://cdn.kastatic.org/ka-youtube-converted/lXShNH1G6Pk.mp4/lXShNH1G6Pk.mp4631yes https://puzzicle.org/math/ap-calculus-ab/v/flow-chart-of-limit-strategieshttps://puzzicle.org/thumbs/x32b0695e0ad9fefa.jpgExploring strategies to find limitsIn this video we explore strategies for determining which technique to use when finding limits. We also highlight the importance of understanding various methods, such as direct substitution, factoring, multiplying by conjugates, and using trig identities. By knowing when to apply each technique, we can effectively tackle different types of limit problems and enhance our problem-solving skills.https://cdn.kastatic.org/ka-youtube-converted/ZaLw1cunN3s.mp4/ZaLw1cunN3s.mp4751yes https://puzzicle.org/math/ap-calculus-ab/v/squeeze-sandwich-theoremhttps://puzzicle.org/thumbs/x1006b0c1.jpgIntroducing the squeeze theoremThe squeeze (or sandwich) theorem states that if f(x)≤g(x)≤h(x) for all numbers, and at some point x=k we have f(k)=h(k), then g(k) must also be equal to them. We can use the theorem to find tricky limits like sin(x)/x at x=0, by "squeezing" sin(x)/x between two nicer functions and ​using them to find the limit at x=0.https://cdn.kastatic.org/ka-youtube-converted/WvxKwRcHGHg.mp4/WvxKwRcHGHg.mp4431yes https://puzzicle.org/math/ap-calculus-ab/v/sinx-over-x-as-x-approaches-0https://puzzicle.org/thumbs/x55458aa544b67d20.jpgDetermining the limit of sin(x)/x as x approaches 0In this video, we prove that the limit of sin(θ)/θ as θ approaches 0 is equal to 1. We use a geometric construction involving a unit circle, triangles, and trigonometric functions. By comparing the areas of these triangles and applying the squeeze theorem, we demonstrate that the limit is indeed 1. This proof helps clarify a fundamental concept in calculus.https://cdn.kastatic.org/ka-youtube-converted/5xitzTutKqM.mp4/5xitzTutKqM.mp4556yes https://puzzicle.org/math/ap-calculus-ab/v/1-cosx-over-x-as-x-approaches-0https://puzzicle.org/thumbs/x6b5ae9affbc7334e.jpgDetermining the limit of (1-cos(x))/x as x approaches 0In this video, we explore the limit of (1-cos(x))/x as x approaches 0 and show that it equals 0. We use the Pythagorean trigonometric identity, algebraic manipulation, and the known limit of sin(x)/x as x approaches 0 to prove this result. This concept is helpful for understanding the derivative of sin(x).https://cdn.kastatic.org/ka-youtube-converted/Ojtwh8XDF8M.mp4/Ojtwh8XDF8M.mp4245yes https://puzzicle.org/math/ap-calculus-ab/v/types-of-discontinuitieshttps://puzzicle.org/thumbs/xe1171210a532febe.jpgExploring types of discontinuitiesThis video discusses three discontinuity types: point/removable, jump, and asymptotic/infinite. Point/removable occurs when the two-sided limit exists but differs from the function's value. Jump arises when one-sided limits aren't equal, and asymptotic/infinite happens when the two-sided limit is unbounded. Continuity requires the two-sided limit to match the function's value.https://cdn.kastatic.org/ka-youtube-converted/NVGBq_V-F6U.mp4/NVGBq_V-F6U.mp4436yes https://puzzicle.org/math/ap-calculus-ab/v/continuity-at-a-pointhttps://puzzicle.org/thumbs/xc3f8ab9b4556a5fd.jpgExploring continuity at a pointA function f is continuous at a point x=c if we can draw its graph at that point without lifting our pencil. To define this rigorously, f is continuous at x=c if the two-sided limit of f(x) as x approaches c exists and matches the value of f(c).https://cdn.kastatic.org/ka-youtube-converted/3KMqU5j7irw.mp4/3KMqU5j7irw.mp4496yes https://puzzicle.org/math/ap-calculus-ab/v/continuity-at-a-point-graphicallyhttps://puzzicle.org/thumbs/x0db3f6a2.jpgWorking an example: Continuity at a point (graphical)In this video, we explore the necessary conditions for continuity at a point using graphical representations of functions. We analyze two examples to determine if the left-hand and right-hand limits exist, if the function is defined at the point, and then we use these observations to determine if the function is continuous at that point.https://cdn.kastatic.org/ka-youtube-converted/jzn4rTPajT0.mp4/jzn4rTPajT0.mp4439yes https://puzzicle.org/math/ap-calculus-ab/v/limit-of-piecewise-function-that-is-definedhttps://puzzicle.org/thumbs/x05e1dd0f.jpgWorking an example: point where a function is continuousIn this video, we explore the limit of a piecewise function at the point where two cases of the function meet. By finding the left-hand and right-hand limits, we can determine if they're equal. If so, the limit exists at that point, and we've successfully analyzed the function's behavior.https://cdn.kastatic.org/ka-youtube-converted/TdVDqWOR3dU.mp4/TdVDqWOR3dU.mp4239yes https://puzzicle.org/math/ap-calculus-ab/v/limit-of-piecewise-function-that-is-undefinedhttps://puzzicle.org/thumbs/x42e9592b.jpgWorking an example: point where a function isn't continuousDiscover how to determine the limit of a piecewise function at the boundary between two distinct cases. By examining one-sided limits approaching from the left and right, we reveal that unequal values result in a non-existent limit for this particular function.https://cdn.kastatic.org/ka-youtube-converted/LVqv8PBR1is.mp4/LVqv8PBR1is.mp4243yes https://puzzicle.org/math/ap-calculus-ab/v/continuity-over-an-intervalhttps://puzzicle.org/thumbs/xedde6ebaf1ab032d.jpgExploring continuity over an intervalA function ƒ is continuous over the open interval (a,b) if and only if it's continuous on every point in (a,b). ƒ is continuous over the closed interval [a,b] if and only if it's continuous on (a,b), the right-sided limit of ƒ at x=a is ƒ(a) and the left-sided limit of ƒ at x=b is ƒ(b).https://cdn.kastatic.org/ka-youtube-converted/ENIQiqxFpBc.mp4/ENIQiqxFpBc.mp4540yes https://puzzicle.org/math/ap-calculus-ab/v/functions-continuous-on-all-numbershttps://puzzicle.org/thumbs/x83a5cb78.jpgDetermining if functions are continuous on all real numbersDiscover how to determine if a function is continuous on all real numbers by examining two examples: eˣ and √x. Generally, common functions exhibit continuity within their domain. Explore the concept of continuity, including asymptotic and jump discontinuities, and learn how to identify continuous functions in various scenarios.https://cdn.kastatic.org/ka-youtube-converted/DCWeH62w-NA.mp4/DCWeH62w-NA.mp4197yes https://puzzicle.org/math/ap-calculus-ab/v/functions-continuous-on-specific-numbershttps://puzzicle.org/thumbs/x27857f00.jpgDetermining if functions are continuous at specific x-valuesDetermine the continuity of two functions, ln(x-3) and e^(x-3), at x=3. Explore the concept of continuity, highlighting that common functions are continuous within their domain. Discover that ln(x-3) is not continuous at x=3, while e^(x-3) is continuous for all real numbers, including x=3.https://cdn.kastatic.org/ka-youtube-converted/X51CaI4tfQg.mp4/X51CaI4tfQg.mp4262yes https://puzzicle.org/math/ap-calculus-ab/v/defining-a-function-at-a-point-to-make-it-continuoushttps://puzzicle.org/thumbs/xf8a1e5c8.jpgRemoving discontinuities (factoring)Discover how to make a function continuous by determining the appropriate value for a discontinuous point. In this example, we explore the function f(x) = (6x² + 18x + 12) / (x² - 4) and find the value it should have at x = -2 to ensure continuity.https://cdn.kastatic.org/ka-youtube-converted/oUgDaEwMbiU.mp4/oUgDaEwMbiU.mp4297yes https://puzzicle.org/math/ap-calculus-ab/v/fancy-algebra-to-find-a-limit-and-make-a-function-continuoushttps://puzzicle.org/thumbs/x0f1bf7aa.jpgRemoving discontinuities (rationalization)Discover how to determine the value of the function f(x) = (√(x+4) - 3) / (x - 5) at x = 5 to ensure continuity. By employing algebraic techniques like rationalization and factoring, we can find the limit as x approaches 5 and identify the value that makes the function continuous.https://cdn.kastatic.org/ka-youtube-converted/P1DJxuG7U9A.mp4/P1DJxuG7U9A.mp4320yes https://puzzicle.org/math/ap-calculus-ab/v/introduction-to-infinite-limitshttps://puzzicle.org/thumbs/x03359d4283dfc6fd.jpgIntroducing infinite limitsWe will now explore infinite limits in calculus, focusing on the notation used for unbounded limits. Learn how to determine if a limit approaches positive or negative infinity, or if it doesn't exist. Dive into one-sided limits and practice with examples using graphs of rational functions.https://cdn.kastatic.org/ka-youtube-converted/WS-76fFFICY.mp4/WS-76fFFICY.mp4263yes https://puzzicle.org/math/ap-calculus-ab/v/infinite-limits-and-asymptoteshttps://puzzicle.org/thumbs/x2a99dc091edd63c8.jpgExploring infinite limits and asymptotesExploring unbounded limits and limits at infinity, this video delves into the relationship between vertical and horizontal asymptotes. Vertical asymptotes signify undefined limits, while horizontal asymptotes can have existing limits as x approaches infinity or negative infinity. The video highlights key differences and demonstrates how functions can cross horizontal asymptotes.https://cdn.kastatic.org/ka-youtube-converted/6BwDYdfw9NQ.mp4/6BwDYdfw9NQ.mp4254yes https://puzzicle.org/math/ap-calculus-ab/v/unbounded-limits-algebraic-1https://puzzicle.org/thumbs/x3b574f8c.jpgAnalyzing unbounded limits: rational functionExplore one-sided limits of the function f(x) = -1/(x-1)² as it approaches x=1. By evaluating f(x) from both the right and left sides, we discover that the function tends towards negative infinity in both cases. This analysis helps us understand the behavior of functions around asymptotes.https://cdn.kastatic.org/ka-youtube-converted/_ZGHGDKWpp8.mp4/_ZGHGDKWpp8.mp4303yes https://puzzicle.org/math/ap-calculus-ab/v/unbounded-limits-algebraic-2https://puzzicle.org/thumbs/x650d767e.jpgAnalyzing unbounded limits: mixed functionExplore the fascinating behavior of the function f(x) = x / (1 - cos(x - 2)) around its asymptote at x = 2. Dive into one-sided limits and discover how the function approaches positive infinity from both the left and right sides. Unravel the properties of the cosine function and learn to evaluate limits using algebraic and graphical methods.https://cdn.kastatic.org/ka-youtube-converted/AI3aOkafYFw.mp4/AI3aOkafYFw.mp4377yes https://puzzicle.org/math/ap-calculus-ab/v/introduction-to-limits-at-infinityhttps://puzzicle.org/thumbs/x3e64d63d151a2eb6.jpgIntroducing limits at infinityDiscover the concept of limits at infinity in calculus, where functions approach a finite value as x approaches positive or negative infinity. This notation helps identify horizontal asymptotes and understand the behavior of functions as x gets larger or smaller, providing insight into the graph's structure.https://cdn.kastatic.org/ka-youtube-converted/eh_ATp0hbB0.mp4/eh_ATp0hbB0.mp4166yes https://puzzicle.org/math/ap-calculus-ab/v/functions-with-same-limit-at-inifinityhttps://puzzicle.org/thumbs/x417fbfe644109072.jpgExploring functions with same limit at infinityExploring limits at infinity reveals that various functions can share the same limit as x approaches infinity. By examining diverse functions, including those with oscillations and natural logs, we can appreciate the infinite possibilities of functions converging to a common limit as x grows infinitely large.https://cdn.kastatic.org/ka-youtube-converted/H_Nm3qGE65s.mp4/H_Nm3qGE65s.mp4226yes https://puzzicle.org/math/ap-calculus-ab/v/limits-at-positive-and-negative-infinityhttps://puzzicle.org/thumbs/1313454812.jpgDetermining limits at infinity of quotients (Part 1)Discover how to determine the limits at positive and negative infinity for rational functions like (4x⁵-3x²+3)/(6x⁵-100x²-10). By focusing on the dominant terms and their growth rates, you'll learn to identify horizontal asymptotes and understand the function's behavior as x approaches infinity.https://cdn.kastatic.org/ka-youtube-converted/gv9ogppphso.mp4/gv9ogppphso.mp4247yes https://puzzicle.org/math/ap-calculus-ab/v/more-limits-at-infinityhttps://puzzicle.org/thumbs/1313658038.jpgDetermining limits at infinity of quotients (Part 2)In this exploration of limits at infinity for rational functions, we examine three general cases of limit behavior. By identifying dominating terms and simplifying expressions, we can determine horizontal asymptotes and unbounded limits, ultimately deepening our understanding these essential calculus concepts.https://cdn.kastatic.org/ka-youtube-converted/KcqO1fX9b_I.mp4/KcqO1fX9b_I.mp4273yes https://puzzicle.org/math/ap-calculus-ab/v/limits-with-two-horizontal-asymptoteshttps://puzzicle.org/thumbs/1315254826.jpgDetermining limits at infinity of quotients with square roots (odd power)Discover how to find the limits at positive and negative infinity for the function x/√(x²+1). Learn how the leading term's odd power (1) results in different limits at positive and negative infinity, and understand how this impacts horizontal asymptotes and end behavior.https://cdn.kastatic.org/ka-youtube-converted/xks4cETlN58.mp4/xks4cETlN58.mp4314yes https://puzzicle.org/math/ap-calculus-ab/v/limit-at-infinity-of-rational-expression-with-radical-evenhttps://puzzicle.org/thumbs/x0d461730.jpgDetermining limits at infinity of quotients with square roots (even power)Discover how to find the limit at infinity of rational expressions with radicals by dividing the numerator and denominator by the highest degree term. Mastering this technique will help us to analyze end behavior and to evaluate other limits involving infinity.https://cdn.kastatic.org/ka-youtube-converted/uPksX_O9ARo.mp4/uPksX_O9ARo.mp4217yes https://puzzicle.org/math/ap-calculus-ab/v/intermediate-value-theoremhttps://puzzicle.org/thumbs/xf5e9f146.jpgDiscovering the intermediate value theoremDiscover the Intermediate Value Theorem, a fundamental concept in calculus that states if a function is continuous over a closed interval [a, b], it encompasses every value between f(a) and f(b) within that range. Dive into this foundational theorem and explore its connection to continuous functions and their behavior on intervals.https://cdn.kastatic.org/ka-youtube-converted/9xgO-EJ3sr0.mp4/9xgO-EJ3sr0.mp4485yes https://puzzicle.org/math/ap-calculus-ab/v/intermediate-value-theorem-examplehttps://puzzicle.org/thumbs/x9233a0b3.jpgWorking an example: using the intermediate value theoremDiscover how the Intermediate Value Theorem guarantees specific outcomes for continuous functions. With a given function f, where f(-2) = 3 and f(1) = 6, learn to identify the correct statement that aligns with the theorem's principles and understand its real-life applications.https://cdn.kastatic.org/ka-youtube-converted/TrQTK-B4bzM.mp4/TrQTK-B4bzM.mp4403yes https://puzzicle.org/math/ap-calculus-ab/v/justification-with-the-intermediate-value-theorem-tablehttps://puzzicle.org/thumbs/x04398cbad4cdf87c.jpgApplying the intermediate value theorem: tableDiscover how to apply the Intermediate Value Theorem with a table-defined function. Learn to determine if a continuous function has a solution within a specified interval, and justify your reasoning using the theorem's principles. Explore real-life applications and enhance your understanding of calculus concepts.https://cdn.kastatic.org/ka-youtube-converted/B0A2lDzn3yw.mp4/B0A2lDzn3yw.mp4329yes https://puzzicle.org/math/ap-calculus-ab/v/justification-with-the-intermediate-value-theorem-equationhttps://puzzicle.org/thumbs/xbd6d29e4792ae6b7.jpgApplying the intermediate value theorem: equationDiscover how to apply the Intermediate Value Theorem to determine if a function has a solution within a specific interval. This engaging lesson explores the theorem's conditions, continuity, and justification using real-world examples and equations. Master the art of working with rational functions and their domains to enhance your calculus skills.https://cdn.kastatic.org/ka-youtube-converted/cSw5R-jdMiI.mp4/cSw5R-jdMiI.mp4226yes https://puzzicle.org/math/ap-calculus-ab/v/limit-intuition-reviewhttps://puzzicle.org/thumbs/1312376615.jpgExploring the formal definition of limits Part 1: intuition reviewDiscover the essence of limits in calculus as we prepare to dive into the formal definition. Enhance your understanding of this fundamental concept by reviewing how function values approach a specific limit as the input variable gets closer to a certain point. Get ready to explore the mathematical rigor behind limits!https://cdn.kastatic.org/ka-youtube-converted/5i8HLmVTcRQ.mp4/5i8HLmVTcRQ.mp4216yes https://puzzicle.org/math/ap-calculus-ab/v/building-the-idea-of-epsilon-delta-definitionhttps://puzzicle.org/thumbs/1312579492.jpgExploring the formal definition of limits Part 2: building the ideaExplore the rigorous mathematical definition of a limit as x approaches c, and understand how to get f(x) as close to L as desired by finding a range around c. Dive into the epsilon-delta definition and its application in proving limits for various functions.https://cdn.kastatic.org/ka-youtube-converted/ljzn4SADuZc.mp4/ljzn4SADuZc.mp4366yes https://puzzicle.org/math/ap-calculus-ab/v/epsilon-delta-definition-of-limitshttps://puzzicle.org/thumbs/1312571391.jpgExploring the formal definition of limits Part 3: the definitionExplore the epsilon-delta definition of limits, which states that the limit of f(x) at x=c equals L if, for any ε>0, there's a δ>0 ensuring that when the distance between x and c is less than δ, the distance between f(x) and L is less than ε. This concept captures the idea of getting arbitrarily close to L.https://cdn.kastatic.org/ka-youtube-converted/w70af5Ou70M.mp4/w70af5Ou70M.mp4419yes https://puzzicle.org/math/ap-calculus-ab/v/proving-a-limit-using-epsilon-delta-definitionhttps://puzzicle.org/thumbs/1312638166.jpgExploring the formal definition of limits Part 4: using the definitionExplore the epsilon-delta definition of limits in calculus, as we rigorously prove a limit exists for a piecewise function. Dive into the process of defining delta as a function of epsilon, and learn how to apply this concept to validate limits with precision.https://cdn.kastatic.org/ka-youtube-converted/0sCttufU-jQ.mp4/0sCttufU-jQ.mp4510yes https://puzzicle.org/math/ap-calculus-ab/v/2011-calculus-ab-free-response-6ahttps://puzzicle.org/thumbs/244051677.jpgDetermining continuity: 2011 Calculus AB free response #6aContinuity of piecewise-defined functionshttps://cdn.kastatic.org/ka-youtube-converted/fr-8tjLoeDw.mp4/fr-8tjLoeDw.mp4263yes https://puzzicle.org/math/ap-calculus-ab/v/newton-leibniz-and-usain-bolthttps://puzzicle.org/thumbs/581541547.jpgConnecting Newton, Leibniz, and Usain BoltWhy do we study differential calculus? Dive into the core concepts of instantaneous rates of change and their practical applications. Uncover the connection between Newton and Leibniz's pioneering work and Usain Bolt's impressive sprinting speeds, while grasping the significance of determining instantaneous velocity in diverse situations.https://cdn.kastatic.org/ka-youtube-converted/EKvHQc3QEow.mp4/EKvHQc3QEow.mp4547yes https://puzzicle.org/math/ap-calculus-ab/v/derivative-as-a-concepthttps://puzzicle.org/thumbs/xfea4e4d112667558.jpgExploring derivatives as a conceptLearn about derivatives as the instantaneous rate of change and the slope of the tangent line. This video introduces key concepts, including the difference between average and instantaneous rates of change, and how derivatives are central to differential calculus. Master various notations used to represent derivatives, such as Leibniz's, Lagrange's, and Newton's notations.https://cdn.kastatic.org/ka-youtube-converted/N2PpRnFqnqY.mp4/N2PpRnFqnqY.mp4436yes https://puzzicle.org/math/ap-calculus-ab/v/secant-lines-and-average-rate-of-changehttps://puzzicle.org/thumbs/x9c4c1968ea8e3c9f.jpgConnecting secant lines and average rate of changeLearn how to calculate the average rate of change for a function and its connection to the slope of a secant line. Grasp the concept of instantaneous rate of change and its significance in calculus, leading to the idea of the derivative.https://cdn.kastatic.org/ka-youtube-converted/0z_MDIWMBwU.mp4/0z_MDIWMBwU.mp4317yes https://puzzicle.org/math/ap-calculus-ab/v/derivative-as-slope-of-curvehttps://puzzicle.org/thumbs/xa17015bd.jpgExploring the derivative as slope of a curveExplore how to interpret the derivative of a function at a specific point as the curve's slope or the tangent line's slope at that point. Dive into problem-solving examples that demonstrate this concept and strengthen your understanding of derivatives in calculus.https://cdn.kastatic.org/ka-youtube-converted/S-dcMvJlMJs.mp4/S-dcMvJlMJs.mp4370yes https://puzzicle.org/math/ap-calculus-ab/v/derivative-as-slope-of-tangent-linehttps://puzzicle.org/thumbs/xad889a94.jpgConnecting the derivative and tangent line equationsDiscover how the derivative of a function reveals the slope of the tangent line at any point on the graph. We'll explore how to use this powerful tool to determine the equation of the tangent line, enhancing our understanding of instantaneous rates of change.https://cdn.kastatic.org/ka-youtube-converted/fqQ6sslzyhY.mp4/fqQ6sslzyhY.mp4453yes https://puzzicle.org/math/ap-calculus-ab/v/calculus-derivatives-1-new-hd-versionhttps://puzzicle.org/thumbs/24100.jpgExploring the formal definition of the derivative as a limitDiscover how to define the derivative of a function at a specific point using the limit of the slope of the secant line. We'll explore the concept of finding the slope as the difference in function values approaches zero, represented by the limit of [f(c)-f(c+h)]/h as h→0.https://cdn.kastatic.org/ka-youtube-converted/ANyVpMS3HL4.mp4/ANyVpMS3HL4.mp4943yes https://puzzicle.org/math/ap-calculus-ab/v/alternate-form-of-the-derivativehttps://puzzicle.org/thumbs/x2c6e38f4.jpgUnderstanding the formal and alternate form of the derivativeDiscover the two methods for defining the derivative of a function at a specific point using limit expressions. In this video we'll explore the standard form and alternate form, enhancing our understanding of tangent line slopes and secant line approximations.https://cdn.kastatic.org/ka-youtube-converted/Df2escG-Vu0.mp4/Df2escG-Vu0.mp4293yes https://puzzicle.org/math/ap-calculus-ab/v/formal-and-alternate-form-of-the-derivative-for-ln-xhttps://puzzicle.org/thumbs/x86a51654.jpgWorking an example: Derivative as a limitDiscover how to apply the formal and alternate forms of the derivative in real-world scenarios. We'll explore the process of finding the slope of tangent lines using both methods and compare their effectiveness in solving calculus problems. Let's dive into the practical side of derivatives to deepen our understanding.https://cdn.kastatic.org/ka-youtube-converted/m8yC7kR5Fuk.mp4/m8yC7kR5Fuk.mp4346yes https://puzzicle.org/math/ap-calculus-ab/v/formal-and-alternate-form-of-the-derivative-example-1https://puzzicle.org/thumbs/x04bbe3f0.jpgWorking an example: Derivative from limit expressionWe explore a limit expression and discover that it represents the derivative of the function f(x) = x³ at the point x = 5. By analyzing the alternate form of the derivative, we gain a deeper understanding of tangent lines and their slopes.https://cdn.kastatic.org/ka-youtube-converted/ePh8iCbcXfA.mp4/ePh8iCbcXfA.mp4317yes https://puzzicle.org/math/ap-calculus-ab/v/calculus-derivatives-2-new-hd-versionhttps://puzzicle.org/thumbs/24102.jpgFinding the derivative of x² at x=3 using the formal definitionDiscover how to find the derivative of x² at x=3 using the formal definition of a derivative. Learn to calculate the slope of the tangent line at a specific point on the curve y=x² by applying the limit as the change in x approaches zero. This method helps determine the instantaneous rate of change for the function.https://cdn.kastatic.org/ka-youtube-converted/IePCHjMeFkE.mp4/IePCHjMeFkE.mp4508yes https://puzzicle.org/math/ap-calculus-ab/v/calculus-derivatives-2-5-new-hd-versionhttps://puzzicle.org/thumbs/24104.jpgFinding the derivative of x² at any point using the formal definitionLet's find the derivative of x² at any point using the formal definition of a derivative. We will learn to apply the limit as h approaches 0 to determine the slope of the tangent line at a given point on the curve y = x². This powerful concept leads to a general formula for the derivative: f'(x) = 2x.https://cdn.kastatic.org/ka-youtube-converted/HEH_oKNLgUU.mp4/HEH_oKNLgUU.mp4665yes https://puzzicle.org/math/ap-calculus-ab/v/estimating-derivative-at-a-pointhttps://puzzicle.org/thumbs/xd63a671d93acf351.jpgEstimating derivativesLet's explore how to estimate the derivative of a function at a specific point using the slope of a secant line connecting nearby points. By finding the average rate of change between these points, we can make our best guess for the instantaneous rate of change at the desired point.https://cdn.kastatic.org/ka-youtube-converted/jvYZNp5myXg.mp4/jvYZNp5myXg.mp4356yes https://puzzicle.org/math/ap-calculus-ab/v/differentiabilityhttps://puzzicle.org/thumbs/x2eefa38e99f06afd.jpgExploring differentiability and continuityWe explore the notion of differentiability at a point and its relationship with continuity, using the definition of a derivative. If a function is differentiable at x = c, then it is continuous at x = c. However, a function being continuous does not guarantee that it is differentiable, as demonstrated with an absolute value function.https://cdn.kastatic.org/ka-youtube-converted/xuAiQOzIkWY.mp4/xuAiQOzIkWY.mp4578yes https://puzzicle.org/math/ap-calculus-ab/v/differentiability-at-a-point-graphicalhttps://puzzicle.org/thumbs/xcec6c3f0.jpgUnderstanding differentiability at a point: graphicalLet's dive into examples of functions and their graphs, focusing on finding points where the function isn't differentiable. By examining various cases such as vertical tangents, discontinuities, and sharp turns, we gain a deeper understanding of the conditions that make a function non-differentiable. We then determine the x-values where the derivative doesn't exist, further solidifying our knowledge of differentiability.https://cdn.kastatic.org/ka-youtube-converted/pwh1dK3vTkM.mp4/pwh1dK3vTkM.mp4339yes https://puzzicle.org/math/ap-calculus-ab/v/differentiability-at-a-point-algebraic-is-differentiablehttps://puzzicle.org/thumbs/x75490a85.jpgExamining differentiability at a point: algebraic (function is differentiable)We examine a piecewise function to determine its continuity and differentiability at an edge point. By analyzing left and right hand limits, we establish continuity. Checking the limit of the difference quotient confirms both left and right hand limits are equal, making the function continuous and differentiable at the edge point.https://cdn.kastatic.org/ka-youtube-converted/_DkttKctXFY.mp4/_DkttKctXFY.mp4302yes https://puzzicle.org/math/ap-calculus-ab/v/differentiability-at-a-point-algebraic-not-differentiablehttps://puzzicle.org/thumbs/xac6f5146.jpgExamining differentiability at a point: algebraic (function isn't differentiable)We examine a piecewise function to determine its differentiability and continuity at an edge point. Through analyzing left and right-hand limits, we find that the function is continuous at the point. However, due to differing slopes from the left and right, the function is not differentiable at the edge point.https://cdn.kastatic.org/ka-youtube-converted/BpPHve1cCCs.mp4/BpPHve1cCCs.mp4382yes https://puzzicle.org/math/ap-calculus-ab/v/power-rulehttps://puzzicle.org/thumbs/1280002940.jpgExploring the power ruleLet's dive into the power rule, a handy tool for finding the derivative of xⁿ. This rule simplifies the process of taking derivatives, especially for polynomials, by bringing the exponent out front and decrementing the power. We explore examples with positive, negative, and fractional exponents.https://cdn.kastatic.org/ka-youtube-converted/bRZmfc1YFsQ.mp4/bRZmfc1YFsQ.mp4234yes https://puzzicle.org/math/ap-calculus-ab/v/power-rule-with-rewritinghttps://puzzicle.org/thumbs/x13db1ccef0efc252.jpgApplying the power rule (with rewriting the expression)Discover how the power rule helps us find derivatives of functions like 1/x, ∛x, or ∛x². By rewriting these functions as xⁿ, where n is a negative or fractional exponent, we can apply the power rule to calculate their derivatives with ease.https://cdn.kastatic.org/ka-youtube-converted/H-v4oraDjuM.mp4/H-v4oraDjuM.mp4264yes https://puzzicle.org/math/ap-calculus-ab/v/derivative-properties-and-polynomial-derivativeshttps://puzzicle.org/thumbs/1280224216.jpgExploring basic derivative rulesLet's explore how to find the derivative of any polynomial using the power rule and additional properties. The derivative of a constant is always 0, and we can pull out a scalar constant when taking the derivative. Furthermore, the derivative of a sum of two functions is simply the sum of their derivatives.https://cdn.kastatic.org/ka-youtube-converted/mzOBlH32qdk.mp4/mzOBlH32qdk.mp4593yes https://puzzicle.org/math/ap-calculus-ab/v/differentiating-linear-functionshttps://puzzicle.org/thumbs/x949b135c.jpgApplying basic derivative rules: find the errorWe analyze two students' attempts to differentiate linear functions using basic rules. One student forgets a negative sign, while another incorrectly assumes the derivative of a product equals the product of derivatives. By identifying common mistakes, we help improve our understanding of differentiation.https://cdn.kastatic.org/ka-youtube-converted/4l_IvxXD9o8.mp4/4l_IvxXD9o8.mp4286yes https://puzzicle.org/math/ap-calculus-ab/v/derivative-properties-examplehttps://puzzicle.org/thumbs/x6d547a43.jpgApplying basic derivative rules: tableLet's explore a problem involving two functions, f and g, and their derivatives at specific points. Our goal is to find the derivative of a new function, h(x), which is a combination of these functions: 3f(x)+2g(x). By applying basic derivative rules, we determine the derivative—and thus the slope of the tangent line—of h(x) at x = 9.https://cdn.kastatic.org/ka-youtube-converted/rOftmuhGLjY.mp4/rOftmuhGLjY.mp4540yes https://puzzicle.org/math/ap-calculus-ab/v/differentiating-polynomials-examplehttps://puzzicle.org/thumbs/xc6fb9f2b.jpgDifferentiating polynomialsLet's explore how to differentiate polynomials using the power rule and derivative properties. We work with the function f(x)=x⁵+2x³-x² and apply the power rule to find its derivative, f'(x)=5x⁴+6x²-2x. Next, we evaluate f'(x) at x=2, determining that f'(2)=100, which represents the rate of change or slope of the tangent line at the specified point.https://cdn.kastatic.org/ka-youtube-converted/-CTaxKTzbEI.mp4/-CTaxKTzbEI.mp4399yes https://puzzicle.org/math/ap-calculus-ab/v/negative-powers-differentiationhttps://puzzicle.org/thumbs/xb74318e1.jpgDifferentiating integer powers (mixed positive and negative)Let's explore how to differentiate the function g(x) = 2/(x³) - 1/(x²) and evaluate the derivative at x = 2. By applying the Power Rule and simplifying, we efficiently find the slope of the tangent line at the given point.https://cdn.kastatic.org/ka-youtube-converted/W233qRK6HPs.mp4/W233qRK6HPs.mp4289yes https://puzzicle.org/math/ap-calculus-ab/v/tangents-of-polynomialshttps://puzzicle.org/thumbs/x0cea991e.jpgFinding equations of tangents to polynomialsLet's explore the process of finding the equation of the tangent line to the graph of f(x) = x³ - 6x² + x - 5 at x = 1. By calculating the derivative and evaluating it at x = 1, we determine the slope and use point-slope form to obtain the tangent line equation.https://cdn.kastatic.org/ka-youtube-converted/j9FDoYNxZlw.mp4/j9FDoYNxZlw.mp4282yes https://puzzicle.org/math/ap-calculus-ab/v/derivatives-of-sinx-and-cosxhttps://puzzicle.org/thumbs/xe7075be468fa015c.jpgExploring derivatives of sin(x) and cos(x)Now we explore the intuition behind the derivatives of trigonometric functions, discovering that the derivative of sin(x) is cos(x) and the derivative of cos(x) is -sin(x). By analyzing tangent line slopes, we gain a deeper understanding of these fundamental relationships.https://cdn.kastatic.org/ka-youtube-converted/UwFlrPNf5ZE.mp4/UwFlrPNf5ZE.mp4221yes https://puzzicle.org/math/ap-calculus-ab/v/sine-and-cosine-differentiationhttps://puzzicle.org/thumbs/x0015df60.jpgWorking an example: Derivatives of sin(x) and cos(x)Dive into the derivative of the function g(x) = 7sin(x) - 3cos(x) - (π/∛x)². By applying the power rule and the derivatives of sine and cosine functions, we efficiently determine the derivative g'(x) = 7cos(x) + 3sin(x) + 2π²/3 * x^(-5/3). Through algebraic manipulation and careful attention to detail, we tackle the problem's initially intimidating appearance.https://cdn.kastatic.org/ka-youtube-converted/Iur13MNO0Ro.mp4/Iur13MNO0Ro.mp4314yes https://puzzicle.org/math/ap-calculus-ab/v/derivative-of-exhttps://puzzicle.org/thumbs/x707a8c1550464685.jpgFinding the derivative of 𝑒ˣDiscover the fascinating property of the derivative of 𝑒ˣ, which remains 𝑒ˣ after differentiation. This unique characteristic is central to our understanding and exploration of exponential functions, showcasing the remarkable nature of the number 𝑒.https://cdn.kastatic.org/ka-youtube-converted/W_gNAjWWvBg.mp4/W_gNAjWWvBg.mp4182yes https://puzzicle.org/math/ap-calculus-ab/v/derivative-of-lnxhttps://puzzicle.org/thumbs/xd9664db59c1550c6.jpgFinding the derivative of ln(x)Let's explore the concept that the derivative of ln(x) is 1/x by examining the slopes of tangent lines at various points on the graph of y=ln(x). We'll observe that the slopes match the values of 1/x, reinforcing the statement. A future video will provide a proof for this concept.https://cdn.kastatic.org/ka-youtube-converted/Qm1TDobNrns.mp4/Qm1TDobNrns.mp4123yes https://puzzicle.org/math/ap-calculus-ab/v/finding-limits-with-derivativeshttps://puzzicle.org/thumbs/x0ec8624760fbf06b.jpgFinding limits with derivativesUse the definition of the derivative of a function whose derivative is known to determine a limit.https://cdn.kastatic.org/ka-youtube-converted/aJ55LrZ_Vqw.mp4/aJ55LrZ_Vqw.mp4133yes https://puzzicle.org/math/ap-calculus-ab/v/applying-the-product-rule-for-derivativeshttps://puzzicle.org/thumbs/1332054864.jpgDiscovering the product ruleDiscover the product rule, a fundamental technique for finding the derivative of a function expressed as a product of two functions. We'll learn how to apply this rule to simplify differentiation and enhance our understanding of calculus.https://cdn.kastatic.org/ka-youtube-converted/79ngr0Bur38.mp4/79ngr0Bur38.mp4160yes https://puzzicle.org/math/ap-calculus-ab/v/differentiating-productshttps://puzzicle.org/thumbs/xc926838b.jpgDifferentiating productsWe explore the product rule by finding the derivative of eˣcos(x). We identify eˣ as the first function and cos(x) as the second, then apply the product rule to calculate the derivative, simplifying our result for a clearer understanding.https://cdn.kastatic.org/ka-youtube-converted/WxTrxxW0qeM.mp4/WxTrxxW0qeM.mp4268yes https://puzzicle.org/math/ap-calculus-ab/v/product-rule-example-implicithttps://puzzicle.org/thumbs/xf0f4ce72.jpgWorking an example: Product rule with tableWe explore how to evaluate the derivative of the product of two functions, f(x) and h(x), at x=3, using given values for f, h, and their derivatives. Applying the product rule, we calculate the derivative of f(x)⋅h(x) at x=3 with ease.https://cdn.kastatic.org/ka-youtube-converted/SQUSh1LNjIo.mp4/SQUSh1LNjIo.mp4189yes https://puzzicle.org/math/ap-calculus-ab/v/product-rule-example-mixed-implicit-explicithttps://puzzicle.org/thumbs/x80b91d3f.jpgWorking an example: Product rule with mixed implicit and explicitWe explore how to calculate the derivative of F(x) = f(x)⋅g(x) at x = -1, given the values of f and f' at x = -1 and g(x) = 1/x. By applying the product rule, we efficiently determine F'(x) and evaluate it at the specified point.https://cdn.kastatic.org/ka-youtube-converted/xioPD_ldZNY.mp4/xioPD_ldZNY.mp4190yes https://puzzicle.org/math/ap-calculus-ab/v/quotient-rulehttps://puzzicle.org/thumbs/xef64cebda9ec86f4.jpgDiscovering the quotient ruleDiscover the quotient rule, a powerful technique for finding the derivative of a function expressed as a quotient. We'll explore how to apply this rule by differentiating the numerator and denominator functions, and then combining them to simplify the result.https://cdn.kastatic.org/ka-youtube-converted/WqzY3xibFL8.mp4/WqzY3xibFL8.mp4233yes https://puzzicle.org/math/ap-calculus-ab/v/quotient-rule-examplehttps://puzzicle.org/thumbs/xc807f948.jpgWorking an example: Quotient rule with tableLet's explore how to find the derivative of F(x) = f(x)/g(x) at x = -1, given the values of f and f' at x = -1 and g(x) = 2x³. By applying the quotient rule, we efficiently calculate the derivative, F'(x), and evaluate it at the specified point.https://cdn.kastatic.org/ka-youtube-converted/ZAElYVh5DXU.mp4/ZAElYVh5DXU.mp4351yes https://puzzicle.org/math/ap-calculus-ab/v/rational-functions-differentiationhttps://puzzicle.org/thumbs/x6df23c44.jpgDifferentiating rational functionsLet's dive into the differentiation of the rational function (5-3x)/(x²+3x) using the Quotient Rule. By identifying the numerator and denominator as separate functions, we apply the Quotient Rule to find the derivative, simplifying the expression for a clear understanding of the process. This approach can be applied to differentiate any other rational function as well.https://cdn.kastatic.org/ka-youtube-converted/wjm5ZYZ4hj8.mp4/wjm5ZYZ4hj8.mp4404yes https://puzzicle.org/math/ap-calculus-ab/v/derivatives-of-tanx-and-cotxhttps://puzzicle.org/thumbs/x19c6a074.jpgFinding derivatives of tan(x) and cot(x)We find the derivatives of tan(x) and cot(x) by rewriting them as quotients of sin(x) and cos(x). Using the quotient rule, we determine that the derivative of tan(x) is sec^2(x) and the derivative of cot(x) is -csc^2(x). This process involves applying the Pythagorean identity to simplify final results.https://cdn.kastatic.org/ka-youtube-converted/Rr_1GQyiRYs.mp4/Rr_1GQyiRYs.mp4278yes https://puzzicle.org/math/ap-calculus-ab/v/derivatives-of-secx-and-cscxhttps://puzzicle.org/thumbs/x813a440b.jpgFinding derivatives of sec(x) and csc(x)Let's explore the derivatives of sec(x) and csc(x) by expressing them as 1/cos(x) and 1/sin(x), respectively, and applying the quotient rule. We discover that the derivative of sec(x) can be written as sin(x)/cos²(x) or tan(x)sec(x), and the derivative of csc(x) can be expressed as -cos(x)/sin²(x) or -cot(x)csc(x).https://cdn.kastatic.org/ka-youtube-converted/TDJ5nXWEkWM.mp4/TDJ5nXWEkWM.mp4268yes https://puzzicle.org/math/ap-calculus-ab/v/differentiability-implies-continuityhttps://puzzicle.org/thumbs/x428a8acc.jpgExploring the proof: Differentiability implies continuityWe explore the connection between differentiability and continuity, demonstrating that if a function is differentiable at a point, it must also be continuous at that point. By examining the properties of limits and using algebra, we prove this relationship for a function at a given point.https://cdn.kastatic.org/ka-youtube-converted/eytyVWA5ZQs.mp4/eytyVWA5ZQs.mp4700yes https://puzzicle.org/math/ap-calculus-ab/v/is-the-power-rule-reasonablehttps://puzzicle.org/thumbs/1280226135.jpgJustifying the power ruleLet's explore the power rule's validity by examining the derivatives of x¹ and x². We'll analyze the slopes of tangent lines for these functions and then see how the power rule provides reasonable results, building our confidence in its usefulness.https://cdn.kastatic.org/ka-youtube-converted/BYTfCnR9Sl0.mp4/BYTfCnR9Sl0.mp4402yes https://puzzicle.org/math/ap-calculus-ab/v/proof-d-dx-x-nhttps://puzzicle.org/thumbs/24124.jpgProving the power rule for positive integer powersWe dive into proving the formula for the derivative of x^n by skillfully applying the binomial theorem. Together, we expand (x + Δx)^n, simplify the expression, and take the limit as Δx approaches zero to reveal the power rule for derivatives.https://cdn.kastatic.org/ka-youtube-converted/dZnc3PtNaN4.mp4/dZnc3PtNaN4.mp4423yes https://puzzicle.org/math/ap-calculus-ab/v/proof-d-dx-sqrt-xhttps://puzzicle.org/thumbs/24126.jpgProving the power rule for the square root functionWe explore a proof of the power rule for the special case when n=½, focusing on the derivative of √x. By applying the definition of a derivative and utilizing the conjugate, we demonstrate that the power rule holds true for this specific case.https://cdn.kastatic.org/ka-youtube-converted/789aMeepbxI.mp4/789aMeepbxI.mp4307yes https://puzzicle.org/math/ap-calculus-ab/v/derivative-of-sin-xhttps://puzzicle.org/thumbs/x58015703874630dc.jpgProving the derivative of sin(x)Let's dive into the proof that the derivative of sin(x) equals cos(x). By applying angle addition formulas, the squeeze theorem, and exploring the concept of limits, we unravel this interesting proof. This exploration not only solidifies our understanding, but also equips us to handle more complex derivatives in the future.https://cdn.kastatic.org/ka-youtube-converted/HVvCbnrUxek.mp4/HVvCbnrUxek.mp4352yes https://puzzicle.org/math/ap-calculus-ab/v/derivative-of-cos-xhttps://puzzicle.org/thumbs/x15e2d5aca402cdb0.jpgProving the derivative of cos(x)Let's leverage our understanding that the derivative of sin(x) equals cos(x) to visually demonstrate that the derivative of cos(x) equals -sin(x). By strategically shifting graphs and applying trigonometric identities, we'll establish a strong visual argument, deepening our comprehension of these key calculus concepts.https://cdn.kastatic.org/ka-youtube-converted/PszFmdiMUcs.mp4/PszFmdiMUcs.mp4198yes https://puzzicle.org/math/ap-calculus-ab/v/product-rule-proofhttps://puzzicle.org/thumbs/xd4dbda2d.jpgProving the product ruleLet's delve into the proof of the product rule, a key concept in calculus. We apply the definition of a derivative to the product of two functions, making sense of this rule. Through smart algebraic manipulation, we arrive at the classic product rule formula.https://cdn.kastatic.org/ka-youtube-converted/L5ErlC0COxI.mp4/L5ErlC0COxI.mp4566yes https://puzzicle.org/math/ap-calculus-ab/v/estimating-rate-of-change-2011-ap-calculus-ab-free-response-2ahttps://puzzicle.org/thumbs/xe1fed4290b5bf938.jpgEstimating rate of change 2011 AP Calculus AB free response 2aEstimating rate of change using a table.https://cdn.kastatic.org/ka-youtube-converted/gXVKbTwricA.mp4/gXVKbTwricA.mp4216yes https://puzzicle.org/math/ap-calculus-ab/v/writing-equation-of-tangent-line-ap-calculus-ab-free-response-2011-3ahttps://puzzicle.org/thumbs/x3e1d1ce7afa6ae82.jpgWriting equation of tangent line AP Calculus AB free response 2011 3aWriting the equation of a tangent line given a function and an x-value.https://cdn.kastatic.org/ka-youtube-converted/-KaohGMliAU.mp4/-KaohGMliAU.mp4179yes https://puzzicle.org/math/ap-calculus-ab/v/2015-ap-calculus-abbc-3ahttps://puzzicle.org/thumbs/x5356f90d.jpgEstimating rate of change: 2015 AP Calculus AB/BC 3aEstimating accelerationhttps://cdn.kastatic.org/ka-youtube-converted/sdFT_8AJsy0.mp4/sdFT_8AJsy0.mp4349yes https://puzzicle.org/math/ap-calculus-ab/v/2015-ap-calculus-ab-6ahttps://puzzicle.org/thumbs/x3c9aaf26.jpgWriting equation of tangent line: 2015 AP Calculus AB 6aEquation for tangent linehttps://cdn.kastatic.org/ka-youtube-converted/4ThUMRAvUkY.mp4/4ThUMRAvUkY.mp4148yes https://puzzicle.org/math/ap-calculus-ab/v/using-the-quotient-rule-2016-calculus-ab-free-response-6bhttps://puzzicle.org/thumbs/xf48db77409954f3c.jpgUsing the quotient rule, 2016 Calculus AB free response 6bDemonstrate how to apply the quotient rule to find the first derivative of a function. All the values necessary to find the value of the derivative are found in the table that is given in the stem, 2016 Calculus AB free response 6b.https://cdn.kastatic.org/ka-youtube-converted/5RJZhyeUuHk.mp4/5RJZhyeUuHk.mp4173yes https://puzzicle.org/math/ap-calculus-ab/v/using-average-rate-of-change-and-first-derivative-2018-calculus-ab-free-responsehttps://puzzicle.org/thumbs/xaabc4b1438ae3762.jpgUsing average rate of change and first derivative, 2018 Calculus AB free response 5a,bCalculate the average rate of change on an interval, determine the equation for the first derivative using the product rule, and use the first derivative to determine the slope of a line tangent to a graph at a given point, 2018 Calculus AB free response 5a,b.https://cdn.kastatic.org/ka-youtube-converted/zXfGK-Eh4c0.mp4/zXfGK-Eh4c0.mp4233yes https://puzzicle.org/math/ap-calculus-ab/v/relating-derivative-to-tangent-line-2019-calculus-ab-free-response-6abhttps://puzzicle.org/thumbs/x701e098ed14f048f.jpgRelating derivative to tangent line, 2019 Calculus AB free response 6a,bThe relationship between the value of the first derivative of a function at a point and the slope of the tangent line, and how to use the product rule, 2019 Calculus AB free response 6a,b.https://cdn.kastatic.org/ka-youtube-converted/w_hXTuSwNJ4.mp4/w_hXTuSwNJ4.mp4252yes https://puzzicle.org/math/ap-calculus-ab/v/determining-when-first-derivative-equals-average-rate-of-change-2025-calculus-abhttps://puzzicle.org/thumbs/x255cc3edab9361b3.jpgDetermining when first derivative equals average rate of change, 2025 Calculus AB free response 1bCalculate the average rate of change, and the relationship between the first derivative and instantaneous rate of change, 2025 Calculus AB free response 1b.https://cdn.kastatic.org/ka-youtube-converted/HDwfKnZ5A2U.mp4/HDwfKnZ5A2U.mp4427yes https://puzzicle.org/math/ap-calculus-ab/v/chain-rule-introductionhttps://puzzicle.org/thumbs/1332165556.jpgUsing the chain ruleThe chain rule states that the derivative of f(g(x)) is f'(g(x))⋅g'(x). In other words, it helps us differentiate *composite functions*. For example, sin(x²) is a composite function because it can be constructed as f(g(x)) for f(x)=sin(x) and g(x)=x². Using the chain rule and the derivatives of sin(x) and x², we can then find the derivative of sin(x²).https://cdn.kastatic.org/ka-youtube-converted/0T0QrHO56qg.mp4/0T0QrHO56qg.mp4307yes https://puzzicle.org/math/ap-calculus-ab/v/common-chain-rule-misunderstandingshttps://puzzicle.org/thumbs/xd1df901f5054119f.jpgExamining common chain rule misunderstandingsWe tackle common misconceptions about the Chain Rule, ensuring we distinguish between product and composition of functions. We also emphasize the importance of fully applying the Chain Rule, and avoid the pitfall of taking the derivative of the outer function with respect to the derivative of the inner function.https://cdn.kastatic.org/ka-youtube-converted/WVdPekbXSfE.mp4/WVdPekbXSfE.mp4408yes https://puzzicle.org/math/ap-calculus-ab/v/recognizing-compositions-of-functionshttps://puzzicle.org/thumbs/xec328d89b4ff5a8f.jpgIdentifying composite functionsNow let's explore the world of composite functions, honing our skills at recognizing them in various forms. This understanding is crucial to get the most out of the Chain Rule. We'll explore how functions can be composed, and even how they can be expressed as products, setting a strong foundation for advanced calculus concepts.https://cdn.kastatic.org/ka-youtube-converted/MT1-bqKpal8.mp4/MT1-bqKpal8.mp4493yes https://puzzicle.org/math/ap-calculus-ab/v/differentiating-composite-functions-1https://puzzicle.org/thumbs/x1c1ae57d.jpgWorking an example: Derivative of cos³(x) using the chain ruleNow let's tackle a worked example of composite functions, using the example f(x) = cos³(x). By applying the chain rule, we differentiate this function, breaking it down into its components x³ and cos(x). This helps us understand how to handle other complex derivatives with ease.https://cdn.kastatic.org/ka-youtube-converted/yAFphxolU5w.mp4/yAFphxolU5w.mp4557yes https://puzzicle.org/math/ap-calculus-ab/v/chain-rule-definition-and-examplehttps://puzzicle.org/thumbs/1331367901.jpgWorking an example: Derivative of √(3x²-x) using the chain ruleLet's dive into the process of differentiating a composite function, specifically f(x)=sqrt(3x^2-x), using the chain rule. By breaking down the function into its components, sqrt(x) and 3x^2-x, we demonstrate how their derivatives work together to make differentiation easier.https://cdn.kastatic.org/ka-youtube-converted/IiBC4ngwH6E.mp4/IiBC4ngwH6E.mp4331yes https://puzzicle.org/math/ap-calculus-ab/v/differentiating-composite-functions-2https://puzzicle.org/thumbs/xd528cea6.jpgWorking an example: Derivative of ln(√x) using the chain ruleIn this worked example, we dissect the composite function f(x)=ln(√x) into its parts, ln(x) and √x. By applying the chain rule, we successfully differentiate this function, providing a clear step-by-step process for finding the derivative of similar composite functions.https://cdn.kastatic.org/ka-youtube-converted/G2ZXpbMukEQ.mp4/G2ZXpbMukEQ.mp4405yes https://puzzicle.org/math/ap-calculus-ab/v/chain-rule-example-implicithttps://puzzicle.org/thumbs/x17495d13.jpgWorking an example: Chain rule with tableThrough a worked example, we explore the Chain rule with a table. Using specific x-values for functions f and g, and their derivatives, we collaboratively evaluate the derivative of a composite function F(x) = f(g(x)). By applying the chain rule, we illuminate the process, making it easy to understand.https://cdn.kastatic.org/ka-youtube-converted/bLKTkM-w5ps.mp4/bLKTkM-w5ps.mp4147yes https://puzzicle.org/math/ap-calculus-ab/v/exponential-functions-differentiation-introhttps://puzzicle.org/thumbs/x43b56dc0.jpgDetermining the derivative of aˣ (for any positive base a)Let's dive deeper into the fascinating world of derivatives, specifically focusing on the derivative of aˣ for any positive base a. Using the derivative of eˣ and the chain rule, we unravel the mystery behind differentiating exponential functions. We then apply our newfound knowledge to differentiate the expression 8⋅3ˣ.https://cdn.kastatic.org/ka-youtube-converted/gHzLHknEk1M.mp4/gHzLHknEk1M.mp4325yes https://puzzicle.org/math/ap-calculus-ab/v/logarithmic-functions-differentiation-introhttps://puzzicle.org/thumbs/x3d849d5f.jpgDetermining the derivative of logₐx (for any positive base a≠1)Now let's look into the fascinating world of logarithms, exploring how to find the derivative of logₐx for any positive base a≠1. Leveraging the derivative of ln(x) and the change of base rule, we successfully differentiate log₇x and -3log_π(x). Join us on this mathematical journey!https://cdn.kastatic.org/ka-youtube-converted/2XB4K6T-yKo.mp4/2XB4K6T-yKo.mp4288yes https://puzzicle.org/math/ap-calculus-ab/v/exponential-functions-differentiationhttps://puzzicle.org/thumbs/x6bc656f4.jpgWorking an example: Derivative of 7^(x²-x) using the chain ruleIn this worked example we explore the process of differentiating the exponential function 7^(x²-x). We Leverage our previous understanding of the derivative of aˣ and the chain rule to unravel the complexities of this composite function to find its derivative.https://cdn.kastatic.org/ka-youtube-converted/fqYds-feWcw.mp4/fqYds-feWcw.mp4220yes https://puzzicle.org/math/ap-calculus-ab/v/log-functions-differentiationhttps://puzzicle.org/thumbs/xfad2980a.jpgWorking an example: Derivative of log₄(x²+x) using the chain ruleLet's explore a worked example of differentiating the logarithmic function log₄(x²+x) using the chain rule. Leveraging our understanding of the derivative of logₐ(x), we simplify the complexities of composite functions, making differentiation more approachable and fun!https://cdn.kastatic.org/ka-youtube-converted/WZPpbkVESVk.mp4/WZPpbkVESVk.mp4250yes https://puzzicle.org/math/ap-calculus-ab/v/trig-functions-differentiation-sechttps://puzzicle.org/thumbs/x08b4707d.jpgWorking an example: Derivative of sec(3π/2-x) using the chain ruleWe explore the fascinating process of differentiating the function sec(3π/2-x) in this worked example. Using the chain rule and trigonometric identities, we calculate the derivative and evaluate it at x=π/4. This problem illuminates the beauty of composite functions and their derivatives.https://cdn.kastatic.org/ka-youtube-converted/6QRtLh_THmo.mp4/6QRtLh_THmo.mp4431yes https://puzzicle.org/math/ap-calculus-ab/v/radical-functions-differentiationhttps://puzzicle.org/thumbs/xec617875.jpgWorking an example: Derivative of ∜(x³+4x²+7) using the chain ruleLet's tackle the challenge of differentiating the fourth root of a polynomial expression in this worked example. Using the chain rule and the power rule, we simplify the complex function and evaluate the derivative at x=-3. Join us on this mathematical journey!https://cdn.kastatic.org/ka-youtube-converted/-_7td0mDbQs.mp4/-_7td0mDbQs.mp4355yes https://puzzicle.org/math/ap-calculus-ab/v/implicit-differentiation-1https://puzzicle.org/thumbs/1335926801.jpgExploring implicit differentiationSome relationships cannot be represented by an explicit function. For example, x²+y²=1. Implicit differentiation helps us find ​dy/dx even for relationships like that. This is done using the chain ​rule, and viewing y as an implicit function of x. For example, according to the chain rule, the derivative of y² would be 2y⋅(dy/dx).https://cdn.kastatic.org/ka-youtube-converted/mSVrqKZDRF4.mp4/mSVrqKZDRF4.mp4482yes https://puzzicle.org/math/ap-calculus-ab/v/implicit-derivative-of-x-y-2-x-y-1https://puzzicle.org/thumbs/1309724635.jpgWorking an example: Implicit differentiationUsing implicit differentiation, let's take on the challenge of the equation (x-y)² = x + y - 1 in this worked example. We utilize the chain rule and algebraic techniques to find the derivative of y with respect to x. This adventure deepens our grasp of how variables interact within intricate equations.https://cdn.kastatic.org/ka-youtube-converted/9uxvm-USEYE.mp4/9uxvm-USEYE.mp4296yes https://puzzicle.org/math/ap-calculus-ab/v/finding-slope-of-tangent-line-with-implicit-differentiationhttps://puzzicle.org/thumbs/1336152047.jpgWorking an example: Evaluating derivative with implicit differentiationUncover the process of calculating the slope of a tangent line at a specific point on a curve using implicit differentiation. We navigate through the steps of finding the derivative, substituting values, and simplifying to reveal the slope at x=1 for the curve x²+(y-x)³=28.https://cdn.kastatic.org/ka-youtube-converted/KyYC8XzKsHU.mp4/KyYC8XzKsHU.mp4325yes https://puzzicle.org/math/ap-calculus-ab/v/showing-explicit-and-implicit-differentiation-give-same-resulthttps://puzzicle.org/thumbs/1336144047.jpgShowing explicit and implicit differentiation give same resultDiscover the fascinating connection between implicit and explicit differentiation! In this video we'll explore a simple equation, unravel it using both methods, and find that they both lead us to the same derivative. This engaging journey demonstrates the versatility and consistency of calculus.https://cdn.kastatic.org/ka-youtube-converted/2CsQ_l1S2_Y.mp4/2CsQ_l1S2_Y.mp4468yes https://puzzicle.org/math/ap-calculus-ab/v/derivatives-of-inverse-functionshttps://puzzicle.org/thumbs/xc6bf99923afd4174.jpgExploring the derivatives of inverse functionsExplore the intriguing connection between inverse functions and their derivatives. We delve into the unique relationship that exists when f(g(x))=x=g(f(x)), using the functions 𝑒ˣ and ln(x) as examples. Uncover how this principle can help us understand the derivatives of these inverses in a new light.https://cdn.kastatic.org/ka-youtube-converted/XOs9vVmzE70.mp4/XOs9vVmzE70.mp4286yes https://puzzicle.org/math/ap-calculus-ab/v/derivatives-of-inverse-functions-implicithttps://puzzicle.org/thumbs/x60332421.jpgExploring the derivatives of inverse functions: from equationsLet's explore the intriguing relationship between a function and its inverse, focusing on the function f(x)=½x³+3x-4. We delve into the derivative of the inverse of f, applying the chain rule and the power rule to evaluate it at x=-14. Join us as we unravel this complex calculus concept.https://cdn.kastatic.org/ka-youtube-converted/UBctoCirje0.mp4/UBctoCirje0.mp4304yes https://puzzicle.org/math/ap-calculus-ab/v/derivatives-of-inverse-functions-tablehttps://puzzicle.org/thumbs/x21b6e9c4.jpgExploring the derivatives of inverse functions: from tablesLet's delve into the fascinating realm of inverse functions, exploring how to evaluate the derivative of an inverse function, h', at a specific x-value. Using a provided table of values for function g, its inverse h, and its derivative g', we unravel the mystery of h' using the chain rule and the concept of inverse functions.https://cdn.kastatic.org/ka-youtube-converted/9Eg97Rtg-pE.mp4/9Eg97Rtg-pE.mp4312yes https://puzzicle.org/math/ap-calculus-ab/v/derivative-inverse-sinehttps://puzzicle.org/thumbs/x27ff4cd3.jpgDetermining the derivative of inverse sineUnlock the mystery of the derivative of inverse sine! Let's dive into the world of calculus, rearranging equations and applying implicit differentiation to find the derivative of y with respect to x. Using trigonometric identities, we transform the derivative into a function of x, revealing a fascinating relationship.https://cdn.kastatic.org/ka-youtube-converted/yIQUhXa-n-M.mp4/yIQUhXa-n-M.mp4296yes https://puzzicle.org/math/ap-calculus-ab/v/derivative-inverse-cosinehttps://puzzicle.org/thumbs/xb6e07b84.jpgDetermining the derivative of inverse cosineUnraveling the mystery of the inverse cosine function, we find its derivative equals -1/(sqrt(1 - x^2)). This step-by-step proof guides us to a fascinating comparison with the derivative of inverse sine, revealing a captivating connection between these two trigonometric functions.https://cdn.kastatic.org/ka-youtube-converted/v_OfFmMRvOc.mp4/v_OfFmMRvOc.mp4224yes https://puzzicle.org/math/ap-calculus-ab/v/derivative-inverse-tangenthttps://puzzicle.org/thumbs/x44778b77.jpgDetermining the derivative of inverse tangentNow let's explore the derivative of the inverse tangent function. Starting with the derivative of tangent, we use the chain rule and trigonometric identities to find the derivative of its inverse. Join us as we investigate this fascinating mathematical process!https://cdn.kastatic.org/ka-youtube-converted/G7WyEp8gHs0.mp4/G7WyEp8gHs0.mp4362yes https://puzzicle.org/math/ap-calculus-ab/v/correcting-work-on-derivative-strategieshttps://puzzicle.org/thumbs/x837229133733f991.jpgDifferentiating functions: Find the errorLet's delve into the process of differentiating functions, analyzing student work for potential errors. Our focus is on the correct application of the product, chain, and quotient rules. This exploration underscores the crucial role these rules play in calculus, enhancing our grasp of differentiation.https://cdn.kastatic.org/ka-youtube-converted/1l4mmTSIbvU.mp4/1l4mmTSIbvU.mp4639yes https://puzzicle.org/math/ap-calculus-ab/v/strategies-applying-derivative-ruleshttps://puzzicle.org/thumbs/x4a3a5f3dfd06dfa8.jpgManipulating functions before differentiationLet's explore how manipulating functions before differentiating can make our calculus journey smoother and quicker. By tweaking our functions a bit, we can make differentiation less complex. We'll walk through several examples, showing how a touch of algebra can go a long way!https://cdn.kastatic.org/ka-youtube-converted/AntQBDFOjUw.mp4/AntQBDFOjUw.mp4680yes https://puzzicle.org/math/ap-calculus-ab/v/disguised-derivativeshttps://puzzicle.org/thumbs/x3b58cdbf.jpgExploring disguised derivativesWe tackle a complex limit expression, viewing it as 5 times the derivative of log(x) when x=2. This perspective enables us to evaluate the limit using differentiation rules, providing a simpler alternative to direct limit analysis. Together, let's navigate this challenging problem.https://cdn.kastatic.org/ka-youtube-converted/fUWn_SJSLBk.mp4/fUWn_SJSLBk.mp4273yes https://puzzicle.org/math/ap-calculus-ab/v/differentiating-using-multiple-rules-strategyhttps://puzzicle.org/thumbs/x34eb537c5c891863.jpgDifferentiating using multiple rules: strategyLet's explore how to break down complex expressions and determine the right derivative rules to apply. We'll learn how to identify the structure of these expressions and decide the order of operations, using the chain rule and product rule. This strategy will help us tackle even the most elaborate expressions with confidence.https://cdn.kastatic.org/ka-youtube-converted/Qgj2_4r3VBs.mp4/Qgj2_4r3VBs.mp4347yes https://puzzicle.org/math/ap-calculus-ab/v/applying-the-chain-rule-and-product-rulehttps://puzzicle.org/thumbs/x1f82ad74663177f8.jpgApplying the chain rule and product ruleLet's explore multiple strategies to tackle derivatives involving both the product and chain rules. We start by applying the chain rule first, then the product rule. Next, we switch it up, using the product rule first, then the chain rule. Discover how these different approaches lead us to the same answer, demonstrating the flexibility and interconnectedness of these mathematical rules.https://cdn.kastatic.org/ka-youtube-converted/QqAPCIIdx-E.mp4/QqAPCIIdx-E.mp4354yes https://puzzicle.org/math/ap-calculus-ab/v/applying-chain-rule-twicehttps://puzzicle.org/thumbs/x8f0090e095853d20.jpgApplying the chain rule twiceUnravel the intricacies of applying the chain rule twice in a single problem. We'll dissect the process of finding the derivative of a function like sin(x^2)^3, demonstrating the power and adaptability of the chain rule when used not just once, but twice!https://cdn.kastatic.org/ka-youtube-converted/eY1LuoSfiys.mp4/eY1LuoSfiys.mp4190yes https://puzzicle.org/math/ap-calculus-ab/v/using-the-product-rule-and-the-chain-rulehttps://puzzicle.org/thumbs/1334798762.jpgDetermining the derivative of eᶜᵒˢˣ⋅cos(eˣ)By applying both the product and chain rule, we tackle the task of differentiating the function eᶜᵒˢˣ⋅cos(eˣ). This exercise not only provides us with a solution to this complex problem, but also deepens our understanding of these fundamental calculus techniques, reinforcing the importance of these rules in the broader context of calculus.https://cdn.kastatic.org/ka-youtube-converted/UXQGzgPf1LE.mp4/UXQGzgPf1LE.mp4352yes https://puzzicle.org/math/ap-calculus-ab/v/chain-rule-with-triple-compositionhttps://puzzicle.org/thumbs/1332001812.jpgDetermining the derivative of sin(ln(x²))We apply the chain rule twice to differentiate sin(ln(x²)), breaking down this composite function into simpler elements. This double application allows us to handle the complexity of the function, turning it into manageable parts. Together, we'll tackle this mathematical task with precision and clarity.https://cdn.kastatic.org/ka-youtube-converted/1015d63VKh4.mp4/1015d63VKh4.mp4292yes https://puzzicle.org/math/ap-calculus-ab/v/second-derivativeshttps://puzzicle.org/thumbs/xf0d302c5.jpgExploring second derivativesWe dive into the fascinating realm of second derivatives, starting with the function y=6/x². Together, we apply the power rule to find the first derivative, then repeat the process to reveal the second derivative. This journey illuminates how we can use mathematical tools to uncover deeper layers of a function's behavior.https://cdn.kastatic.org/ka-youtube-converted/WC5VYKI807Q.mp4/WC5VYKI807Q.mp4147yes https://puzzicle.org/math/ap-calculus-ab/v/finding-second-derivative-implicithttps://puzzicle.org/thumbs/xbcaed91db72d24e2.jpgExploring second derivatives (implicit equations): find expressionLet's explore the process of determining the second derivative of y with respect to x from an implicit equation. Using implicit differentiation, we apply the chain rule and quotient rule. This leads us to a simplified expression for the second derivative in terms of x and y.https://cdn.kastatic.org/ka-youtube-converted/oPijG5Bfemg.mp4/oPijG5Bfemg.mp4287yes https://puzzicle.org/math/ap-calculus-ab/v/evaluating-second-derivative-implicithttps://puzzicle.org/thumbs/x6c67e58001ee1c64.jpgExploring second derivatives (implicit equations): evaluate derivativeWe tackle the challenge of implicit differentiation, using an equation in x and y to find the first derivative. Then, we step it up a notch by evaluating the second derivative at a specific point. All this is based on a problem from the 2015 AP Calculus AB exam.https://cdn.kastatic.org/ka-youtube-converted/komjMq5jm9k.mp4/komjMq5jm9k.mp4290yes https://puzzicle.org/math/ap-calculus-ab/v/change-in-continuous-function-approaches-0https://puzzicle.org/thumbs/xdadf5f63.jpgExploring if function u is continuous at x, then Δu→0 as Δx→0Let's explore the fascinating relationship between changes in function values and x-values. When we have a continuous function, we discover that as the change in x-values (Δx) nears zero, the change in the function's values (Δu) also tends towards zero. This insight offers us a unique way to understand continuity.https://cdn.kastatic.org/ka-youtube-converted/l6T4RhlgkG0.mp4/l6T4RhlgkG0.mp4385yes https://puzzicle.org/math/ap-calculus-ab/v/chain-rule-proofhttps://puzzicle.org/thumbs/xb6e914af.jpgProving the chain ruleLet's explore the proof of the chain rule, using the formal properties of continuity and differentiability. This proof helps us to more deeply understand this fundamental concept. In making sense of the chain rule, we turn it into a powerful tool in our mathematical toolkit.https://cdn.kastatic.org/ka-youtube-converted/FKraGDm2fUY.mp4/FKraGDm2fUY.mp4340yes https://puzzicle.org/math/ap-calculus-ab/v/quotient-rule-from-product-rulehttps://puzzicle.org/thumbs/1332076874.jpgExploring the quotient rule from product and chain rulesWe explore the connection between the quotient rule, product rule, and chain rule in calculus. Rather than memorizing another rule, we see how the quotient rule naturally emerges from applying the product and chain rules. This approach simplifies our understanding of these fundamental calculus tools.https://cdn.kastatic.org/ka-youtube-converted/ho87DN9wO70.mp4/ho87DN9wO70.mp4315yes https://puzzicle.org/math/ap-calculus-ab/v/2015-ap-calculus-ab-6chttps://puzzicle.org/thumbs/x82672503.jpgDifferentiating implicitly: 2015 AP Calculus AB 6cFinding second derivative through implicit differentiationhttps://cdn.kastatic.org/ka-youtube-converted/qSb78ptLZU0.mp4/qSb78ptLZU0.mp4377yes https://puzzicle.org/math/ap-calculus-ab/v/using-the-chain-rule-2017-calculus-ab-free-response-6abchttps://puzzicle.org/thumbs/x71a2abdd5772cc84.jpgUsing the chain rule, 2017 Calculus AB free response 6a,b,cDetermine the slope of a line tangent to the graph of a function (which involves using the chain rule), and evaluate the first derivative of a function using the chain rule, 2017 Calculus AB free response 6a,b,c.https://cdn.kastatic.org/ka-youtube-converted/IeCRLXKu2Lk.mp4/IeCRLXKu2Lk.mp4552yes https://puzzicle.org/math/ap-calculus-ab/v/applying-implicit-differentiation-2023-calculus-ab-free-response-6ahttps://puzzicle.org/thumbs/xe260ce5c53b4a4a2.jpgApplying implicit differentiation, 2023 Calculus AB free response 6aUse implicit differentiation to find the first derivative of an equation, 2023 Calculus AB free response 6a.https://cdn.kastatic.org/ka-youtube-converted/13kB8wPSiuo.mp4/13kB8wPSiuo.mp4202yes https://puzzicle.org/math/ap-calculus-ab/v/interpreting-the-meaning-of-the-derivative-in-contexthttps://puzzicle.org/thumbs/x926ec566b9e26d0d.jpgInterpreting the meaning of the derivative in contextLet's get hands-on with the concept of derivatives! We'll learn how to interpret the meaning of a derivative within a real-world context, turning complex calculus into practical applications. We'll see how derivatives can effectively express rates of change and motion in our everyday lives.https://cdn.kastatic.org/ka-youtube-converted/IElkjzC9YhQ.mp4/IElkjzC9YhQ.mp4293yes https://puzzicle.org/math/ap-calculus-ab/v/one-dimensional-motion-with-calculushttps://puzzicle.org/thumbs/x36586540e4f670e8.jpgIntroducing one-dimensional motion with calculusLet's explore one-dimensional motion! We'll use calculus to connect position, velocity, speed, and acceleration. By understanding how position changes over time, we'll gain insights into the intricate dynamics of straight-line motion. It's all about making sense of movement in a one-dimensional world!https://cdn.kastatic.org/ka-youtube-converted/MFrpe4Wm8-g.mp4/MFrpe4Wm8-g.mp4391yes https://puzzicle.org/math/ap-calculus-ab/v/interpreting-direction-from-position-timehttps://puzzicle.org/thumbs/x4c422f7b71d9d21d.jpgInterpreting direction of motion from position-time graphGiven the position-time graph of a linear motion, we can interpret whether the moving object moves forward or backward (or right or left, depending on how we define the motion!).https://cdn.kastatic.org/ka-youtube-converted/PNBJQ2OtS_4.mp4/PNBJQ2OtS_4.mp4159yes https://puzzicle.org/math/ap-calculus-ab/v/interpreting-direction-from-velocity-timehttps://puzzicle.org/thumbs/x9c2767f4bc513a73.jpgInterpreting direction of motion from velocity-time graphLet's explore how a velocity-time graph can reveal the direction of an object's linear motion. We'll decipher whether the object is moving forward or backward, or even right or left, based on the graph's details. Together, we'll make sense of these intriguing mathematical representations of motion.https://cdn.kastatic.org/ka-youtube-converted/lsufpQEVHZ8.mp4/lsufpQEVHZ8.mp4106yes https://puzzicle.org/math/ap-calculus-ab/v/interpreting-speed-from-velocity-timehttps://puzzicle.org/thumbs/x2b01a75d8d96b72c.jpgInterpreting change in speed from velocity-time graphLet's examine a velocity-time graph to understand an object's motion. By distinguishing between positive and negative slopes, we discern whether the object is speeding up, slowing down, or maintaining its speed. It's a practical application of calculus concepts to real-world motion scenarios!https://cdn.kastatic.org/ka-youtube-converted/pp08hpkTN-o.mp4/pp08hpkTN-o.mp4237yes https://puzzicle.org/math/ap-calculus-ab/v/motion-problems-with-derivativeshttps://puzzicle.org/thumbs/xf18565f55a7b2a76.jpgWorking an example: Motion problems with derivativesJoin us as we explore the relationship between position, velocity, and acceleration for a particle moving along the x-axis. We'll calculate velocity and acceleration from a given position function, and interpret these values to understand the particle's direction of motion and changes in speed. It's another practical application of calculus to real-world motion problems!https://cdn.kastatic.org/ka-youtube-converted/b4w0xmuOiQo.mp4/b4w0xmuOiQo.mp4298yes https://puzzicle.org/math/ap-calculus-ab/v/modeling-a-forgetting-curvehttps://puzzicle.org/thumbs/x1c23041a.jpgExploring applied rate of change: forgetfulnessLet's face it—sometimes we forget things! Ever wonder about the dynamics underpinning this phenomenon? Let's explore this by tracking the number of vocabulary words remembered over time. Together, we'll use a mathematical model to calculate the rate of change in our memory, revealing the fascinating dynamics of knowledge retention and loss.https://cdn.kastatic.org/ka-youtube-converted/WbyODj_OiZU.mp4/WbyODj_OiZU.mp4300yes https://puzzicle.org/math/ap-calculus-ab/v/derivative-and-marginal-costhttps://puzzicle.org/thumbs/xa93afa81.jpgExamining marginal cost and differential calculusThrough the lens of economics, we examine how the derivative can be used as a tool to understand marginal cost. We'll use a factory scenario to see how cost changes with quantity, and how this insight informs production decisions. Calculus—in real life!https://cdn.kastatic.org/ka-youtube-converted/Wkjz1LlX1CQ.mp4/Wkjz1LlX1CQ.mp4280yes https://puzzicle.org/math/ap-calculus-ab/v/rates-of-change-between-radius-and-area-of-circlehttps://puzzicle.org/thumbs/1339382000.jpgIntroducing related ratesJoin us as we explore the intriguing relationship between the rate at which a circle's radius expands and the corresponding rate of area growth. Using the tools of calculus, specifically derivatives, we'll tackle this concept, providing a practical application to real-world problems involving rates of change. This understanding forms a key part of mastering calculus and its applications.https://cdn.kastatic.org/ka-youtube-converted/kQF9pOqmS0U.mp4/kQF9pOqmS0U.mp4463yes https://puzzicle.org/math/ap-calculus-ab/v/analyzing-related-rates-problems-expressionshttps://puzzicle.org/thumbs/x54c311a6a4dfe567.jpgAnalyzing related rates problems: expressionsGrasping related rates problems starts with understanding the quantities involved. In this exploration, we'll identify these quantities, match each expression with its units and values, and use this information to confidently tackle these problems. It's all about making sense of the relationships and changes over time.https://cdn.kastatic.org/ka-youtube-converted/TX69hjJpBFk.mp4/TX69hjJpBFk.mp4487yes https://puzzicle.org/math/ap-calculus-ab/v/analyzing-related-rates-problems-equationshttps://puzzicle.org/thumbs/x8b79a8b9220004e6.jpgAnalyzing related rates problems: equations (Pythagoras)Mastering related rates problems hinges on the careful selection of an equation that accurately links the quantities. We'll illustrate this with a diagram, emphasizing the role of the right equation in solving these problems and understanding the interplay of quantities over time.https://cdn.kastatic.org/ka-youtube-converted/hFRQBhHdINM.mp4/hFRQBhHdINM.mp4215yes https://puzzicle.org/math/ap-calculus-ab/v/related-rates-example-with-trigonometryhttps://puzzicle.org/thumbs/x16eb78b269660673.jpgAnalyzing related rates problems: equations (trig)Let's tackle a practical problem involving a ladder sliding down a wall. We'll examine how the changing position of the ladder influences the angle it makes with the ground. We'll use calculus to calculate the rate of change of this angle. It's a hands-on way to see calculus in action!https://cdn.kastatic.org/ka-youtube-converted/XwEGKNPLQfQ.mp4/XwEGKNPLQfQ.mp4481yes https://puzzicle.org/math/ap-calculus-ab/v/differentiating-related-functions-introhttps://puzzicle.org/thumbs/x6086e3c067b3b3de.jpgDifferentiating related functions introIn this video, we explore how to differentiate related functions. We'll use the chain rule to find the derivatives of these functions, all linked to the same variable. It's a cool way to see how changes in one function can affect another.https://cdn.kastatic.org/ka-youtube-converted/Mj_2fVKWa0k.mp4/Mj_2fVKWa0k.mp4204yes https://puzzicle.org/math/ap-calculus-ab/v/differentiating-related-functionshttps://puzzicle.org/thumbs/x322654ab7f8eebbd.jpgWorking an example: Differentiating related functionsIn this video, we tackle a fascinating problem involving related functions. We'll explore how to use implicit differentiation to find the derivatives of these functions with respect to a shared variable. By the end, we'll have a clear understanding of how these functions interact and change together.https://cdn.kastatic.org/ka-youtube-converted/C3MI_mspN3o.mp4/C3MI_mspN3o.mp4472yes https://puzzicle.org/math/ap-calculus-ab/v/rate-of-change-of-distance-between-approaching-carshttps://puzzicle.org/thumbs/1353379359.jpgExploring related rates: Approaching carsIn this video, we explore the fascinating world of related rates with two cars approaching an intersection. We'll figure out how the rate of change of the distance between the two cars changes as they move. It's a real-world application of math that shows how calculus helps us understand motion and rates of change.https://cdn.kastatic.org/ka-youtube-converted/wV59XWLmv0c.mp4/wV59XWLmv0c.mp4413yes https://puzzicle.org/math/ap-calculus-ab/v/falling-ladder-related-rateshttps://puzzicle.org/thumbs/1339617514.jpgExploring related rates: Falling ladderLet's explore a thrilling real-world scenario in this video: a ladder slipping away from a wall! We'll use related rates to calculate how fast the top of the ladder falls. It's a fun and practical application of calculus that'll keep us on our toes.https://cdn.kastatic.org/ka-youtube-converted/kBVDSu7v8os.mp4/kBVDSu7v8os.mp4349yes https://puzzicle.org/math/ap-calculus-ab/v/related-rates-of-water-pouring-into-conehttps://puzzicle.org/thumbs/1339646438.jpgExploring related rates: water pouring into a coneIn this video, we explore an intriguing scenario where we pour water into a cone-shaped cup at a constant rate. We'll discover how the rate of change in the water's depth connects to the rate of change in volume, all with the help of our new related rates tools.https://cdn.kastatic.org/ka-youtube-converted/Xe6YlrCgkIo.mp4/Xe6YlrCgkIo.mp4692yes https://puzzicle.org/math/ap-calculus-ab/v/speed-of-shadow-of-diving-birdhttps://puzzicle.org/thumbs/1353395672.jpgExploring related rates: shadowLet's take a look at a problem involving an owl hunting a mouse and the shadow it casts. We can use calculus to determine how fast the shadow moves as the owl dives towards its prey. It's a real-world application of related rates that brings the concept to life!https://cdn.kastatic.org/ka-youtube-converted/cPOjnyjii-4.mp4/cPOjnyjii-4.mp4639yes https://puzzicle.org/math/ap-calculus-ab/v/rate-of-change-of-balloon-heighthttps://puzzicle.org/thumbs/1341818796.jpgExploring related rates: balloonLet's imagine we're at a hot air balloon show, and we're tracking a balloon's ascent. Using our knowledge of angles, distances, and related rates we can apply trigonometry and calculus to find the rate of change in the balloon's height.https://cdn.kastatic.org/ka-youtube-converted/_kbd6troMgA.mp4/_kbd6troMgA.mp4497yes https://puzzicle.org/math/ap-calculus-ab/v/local-linearization-introhttps://puzzicle.org/thumbs/x1822071e.jpgIntroducing local linearitySal introduces the idea of approximating curves using their tangent line equations. This is also called "local linearization."https://cdn.kastatic.org/ka-youtube-converted/u7dhn-hBHzQ.mp4/u7dhn-hBHzQ.mp4578yes https://puzzicle.org/math/ap-calculus-ab/v/local-linearity-and-differentiabilityhttps://puzzicle.org/thumbs/xfb771e9ced51a63c.jpgRelating local linearity and differentiabilityIntuition for how local linearity relates to differentiability using the Desmos graphing calculator.https://cdn.kastatic.org/ka-youtube-converted/h2lARdo7Xko.mp4/h2lARdo7Xko.mp4390yes https://puzzicle.org/math/ap-calculus-ab/v/approximation-with-local-linearityhttps://puzzicle.org/thumbs/x35355b305a3b40be.jpgWorking an example: Approximation with local linearityFinding the equation of a tangent line at a point of a curve by knowing the derivative at that point. Then using that equation to approximate the value of the function at close by x-values.https://cdn.kastatic.org/ka-youtube-converted/wdQRY3Er7PE.mp4/wdQRY3Er7PE.mp4215yes https://puzzicle.org/math/ap-calculus-ab/v/linear-approximation-examplehttps://puzzicle.org/thumbs/xac8728bc.jpgUsing linear approximation of a rational functionSal finds a linear expression that approximates y=1/(x-1) around x=-1. This is done by finding the equation of the line tangent to the graph at x=-1, a process called "linear approximation."https://cdn.kastatic.org/ka-youtube-converted/vRw4ovZK1CQ.mp4/vRw4ovZK1CQ.mp4431yes https://puzzicle.org/math/ap-calculus-ab/v/introduction-to-l-hopital-s-rulehttps://puzzicle.org/thumbs/27376.jpgIntroducing L'Hôpital's ruleWhen you are solving a limit, and get 0/0 or ∞/∞, L'Hôpital's rule is the tool you need.https://cdn.kastatic.org/ka-youtube-converted/PdSzruR5OeE.mp4/PdSzruR5OeE.mp4531yes https://puzzicle.org/math/ap-calculus-ab/v/l-hopital-s-rule-example-1https://puzzicle.org/thumbs/27377.jpgApplying L'Hôpital's rule: limit at 0 exampleSal uses L'Hôpital's rule to find the limit at 0 of (2sin(x)-sin(2x))/(x-sin(x)).https://cdn.kastatic.org/ka-youtube-converted/BiVOC3WocXs.mp4/BiVOC3WocXs.mp4463yes https://puzzicle.org/math/ap-calculus-ab/v/l-hopital-s-rule-example-2https://puzzicle.org/thumbs/27378.jpgApplying L'Hôpital's rule: limit at infinity exampleSal uses L'Hôpital's rule to find the limit at infinity of (4x²-5x)/(1-3x²).https://cdn.kastatic.org/ka-youtube-converted/FJo18AwLfuI.mp4/FJo18AwLfuI.mp4315yes https://puzzicle.org/math/ap-calculus-ab/v/proof-of-special-case-of-l-hopital-s-rulehttps://puzzicle.org/thumbs/1383606650.jpgProving a special case of l'Hôpital's ruleThis isn't a full proof of L'Hopital's rule, but it should give some intuition for why it works.https://cdn.kastatic.org/ka-youtube-converted/SIB4WDYF5DQ.mp4/SIB4WDYF5DQ.mp4321yes https://puzzicle.org/math/ap-calculus-ab/v/2011-calculus-ab-free-response-1ahttps://puzzicle.org/thumbs/244051678.jpgAnalyzing speed of a particle: 2011 Calculus AB free response #1aDetermining whether speed is increasing. Difference between speed and accelerationhttps://cdn.kastatic.org/ka-youtube-converted/ppBJWf_Wdmc.mp4/ppBJWf_Wdmc.mp4448yes https://puzzicle.org/math/ap-calculus-ab/v/2011-calculus-ab-free-response-5ahttps://puzzicle.org/thumbs/238520370.jpgUsing tangent line for approximation: 2011 Calculus AB free response #5aUsing a tangent line to extrapolate a function from a known initial conditionhttps://cdn.kastatic.org/ka-youtube-converted/n9UvuzI2fq0.mp4/n9UvuzI2fq0.mp4419yes https://puzzicle.org/math/ap-calculus-ab/v/estimating-rate-of-change-2015-ap-calculus-ab-bc-1bhttps://puzzicle.org/thumbs/x12330977f2e83320.jpgEstimating rate of change 2015 AP Calculus AB BC 1bEstimating a rate of change based on given functions.https://cdn.kastatic.org/ka-youtube-converted/PTWT41LmPl8.mp4/PTWT41LmPl8.mp4228yes https://puzzicle.org/math/ap-calculus-ab/v/2015-ap-calculus-2chttps://puzzicle.org/thumbs/xb51b7830.jpgFinding rate of change: 2015 AP Calculus 2cRate of change of difference between functionshttps://cdn.kastatic.org/ka-youtube-converted/Ncv6gt2Luew.mp4/Ncv6gt2Luew.mp4225yes https://puzzicle.org/math/ap-calculus-ab/v/calculating-acceleration-ap-calculus-ab-bc-2015-3chttps://puzzicle.org/thumbs/x7d3d1a48e3de2b9c.jpgCalculating acceleration AP Calculus AB BC 2015 3cCalculating acceleration from a velocity function with context.https://cdn.kastatic.org/ka-youtube-converted/bt1eSGeC3ak.mp4/bt1eSGeC3ak.mp4106yes https://puzzicle.org/math/ap-calculus-ab/v/2017-ap-calculus-abbc-4ahttps://puzzicle.org/thumbs/x1b6d8fe24f533104.jpgWriting and using equation of tangent line: 2017 AP Calculus AB/BC 4aPotato problem from 2017 AP exam (Question 4). Finding and using the equation of a tangent line to approximate solution to differential equation.https://cdn.kastatic.org/ka-youtube-converted/tMA17GzO0TY.mp4/tMA17GzO0TY.mp4368yes https://puzzicle.org/math/ap-calculus-ab/v/mean-value-theorem-1https://puzzicle.org/thumbs/x497fcea6.jpgIntroducing the mean value theoremThe Mean Value Theorem states that if a function f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a point c in the interval (a,b) such that f'(c) is equal to the function's average rate of change over [a,b]. In other words, the graph has a tangent somewhere in (a,b) that is parallel to the secant line over [a,b].https://cdn.kastatic.org/ka-youtube-converted/6hri9k_2R8o.mp4/6hri9k_2R8o.mp4397yes https://puzzicle.org/math/ap-calculus-ab/v/finding-where-the-derivative-is-equal-to-the-average-changehttps://puzzicle.org/thumbs/xb8ad8f0a.jpgApplying the mean value theorem example: polynomialSal finds the number that satisfies the Mean value theorem for f(x)=x²-6x+8 over the interval [2,5].https://cdn.kastatic.org/ka-youtube-converted/S_YIUXy-WFM.mp4/S_YIUXy-WFM.mp4290yes https://puzzicle.org/math/ap-calculus-ab/v/mean-value-theorem-example-square-roothttps://puzzicle.org/thumbs/x0c6ab8b7.jpgApplying the mean value theorem example: square root functionSal finds the number that satisfies the Mean value theorem for f(x)=√(4x-3) over the interval [1,3].https://cdn.kastatic.org/ka-youtube-converted/Xg24JhQHVCM.mp4/Xg24JhQHVCM.mp4384yes https://puzzicle.org/math/ap-calculus-ab/v/justification-with-the-mean-value-theorem-tablehttps://puzzicle.org/thumbs/xa788d96f0783b454.jpgJustifying use of the mean value theorem: tableExample justifying use of mean value theorem (where function is defined with a table).https://cdn.kastatic.org/ka-youtube-converted/PE8NNZG9IYw.mp4/PE8NNZG9IYw.mp4221yes https://puzzicle.org/math/ap-calculus-ab/v/justification-with-the-mean-value-theorem-equationhttps://puzzicle.org/thumbs/xfa2d1f5961783950.jpgJustifying use of the mean value theorem: equationExample justifying use of mean value theorem (where function is defined with an equation).https://cdn.kastatic.org/ka-youtube-converted/IJ-obdnR_j8.mp4/IJ-obdnR_j8.mp4174yes https://puzzicle.org/math/ap-calculus-ab/v/getting-a-ticket-because-of-the-mean-value-theoremhttps://puzzicle.org/thumbs/xe252a20d.jpgApplying the mean value theoremEven if a cop never spots you while you are speeding, he can still infer when you must have been speeding...https://cdn.kastatic.org/ka-youtube-converted/InrLtU3k3Y8.mp4/InrLtU3k3Y8.mp4432yes https://puzzicle.org/math/ap-calculus-ab/v/extreme-value-theoremhttps://puzzicle.org/thumbs/xbf69f66b.jpgIntroducing the extreme value theoremThe Extreme value theorem states that if a function is continuous on a closed interval [a,b], then the function must have a maximum and a minimum on the interval. This makes sense: when a function is continuous you can draw its graph without lifting the pencil, so you must hit a high point and a low point on that interval.https://cdn.kastatic.org/ka-youtube-converted/bZYTDst1MOo.mp4/bZYTDst1MOo.mp4478yes https://puzzicle.org/math/ap-calculus-ab/v/minima-maxima-and-critical-pointshttps://puzzicle.org/thumbs/1337154539.jpgIntroducing critical pointsSal introduces the "critical points" of a function and discusses their relationship with the extremum points of the function.https://cdn.kastatic.org/ka-youtube-converted/lDY9JcFaRd4.mp4/lDY9JcFaRd4.mp4473yes https://puzzicle.org/math/ap-calculus-ab/v/finding-critical-numbershttps://puzzicle.org/thumbs/x13fa2a4c.jpgFinding critical pointsSal finds the critical points of f(x)=xe^(-2x²).https://cdn.kastatic.org/ka-youtube-converted/MUQfl385Yug.mp4/MUQfl385Yug.mp4351yes https://puzzicle.org/math/ap-calculus-ab/v/increasing-decreasing-intervals-given-the-functionhttps://puzzicle.org/thumbs/xa3538e83.jpgFinding decreasing interval given the functionSal finds the intervals where the function f(x)=x⁶-3x⁵ is decreasing by analyzing the intervals where f' is positive or negative.https://cdn.kastatic.org/ka-youtube-converted/Fk4pn8ZOy1w.mp4/Fk4pn8ZOy1w.mp4318yes https://puzzicle.org/math/ap-calculus-ab/v/finding-increasing-interval-given-derivativehttps://puzzicle.org/thumbs/x69b9761a.jpgFinding increasing interval given the derivativeSal is given that the derivative of function g, is g'(x)=x²/(x-2)³. He uses that to find the intervals where g is increasing, by looking for the intervals where g' is positive.https://cdn.kastatic.org/ka-youtube-converted/sOluh1zX5tE.mp4/sOluh1zX5tE.mp4360yes https://puzzicle.org/math/ap-calculus-ab/v/testing-critical-points-for-local-extremahttps://puzzicle.org/thumbs/1337206217.jpgFinding relative extrema (first derivative test)Once you find a critical point, how can you tell if it is a minimum, maximum or neither?https://cdn.kastatic.org/ka-youtube-converted/lFQ4kMcODzU.mp4/lFQ4kMcODzU.mp4325yes https://puzzicle.org/math/ap-calculus-ab/v/finding-relative-maximum-examplehttps://puzzicle.org/thumbs/xb5d57cc3.jpgWorking an example: finding relative extremaSal finds the relative maximum point of g(x)=x⁴-x⁵ by analyzing the intervals where its derivative, g', is negative or positive.https://cdn.kastatic.org/ka-youtube-converted/x09FpMmGB4A.mp4/x09FpMmGB4A.mp4469yes https://puzzicle.org/math/ap-calculus-ab/v/analyzing-mistakes-when-finding-extrema-1https://puzzicle.org/thumbs/xac26df03383cb341.jpgAnalyzing mistakes when finding extrema (example 1)Analyzing the work of someone who tried to find extrema of a function, to see whether they made mistakes.https://cdn.kastatic.org/ka-youtube-converted/-ihDprWkcY8.mp4/-ihDprWkcY8.mp4273yes https://puzzicle.org/math/ap-calculus-ab/v/analyzing-mistakes-when-finding-extrema-2https://puzzicle.org/thumbs/x82b0ec2f6975a56d.jpgAnalyzing mistakes when finding extrema (example 2)Analyzing the work of someone who tried to find extrema of a function, to see whether they made mistakes.https://cdn.kastatic.org/ka-youtube-converted/Qcn8A5Ll7g4.mp4/Qcn8A5Ll7g4.mp4282yes https://puzzicle.org/math/ap-calculus-ab/v/using-extreme-value-theoremhttps://puzzicle.org/thumbs/x1efba280.jpgFinding absolute extrema on a closed intervalSal finds the absolute maximum value of f(x)=8ln(x)-x² over the interval [1,4].https://cdn.kastatic.org/ka-youtube-converted/JXVGPEOQCb8.mp4/JXVGPEOQCb8.mp4416yes https://puzzicle.org/math/ap-calculus-ab/v/absolute-extrema-on-entire-domainhttps://puzzicle.org/thumbs/x0caa19a6.jpgAnalyzing absolute minima and maxima (entire domain)Sal analyzes the absolute minimum and maximum points of g(x)=x²ln(x) over its entire domain.https://cdn.kastatic.org/ka-youtube-converted/Xhc7Hens0f8.mp4/Xhc7Hens0f8.mp4564yes https://puzzicle.org/math/ap-calculus-ab/v/concavity-concave-upwards-and-concave-downwards-intervalshttps://puzzicle.org/thumbs/1337193144.jpgIntroducing concavitySal introduces the concept of concavity, what it means for a graph to be "concave up" or "concave down," and how this relates to the second derivative of a function.https://cdn.kastatic.org/ka-youtube-converted/LcEqOzNov4E.mp4/LcEqOzNov4E.mp4594yes https://puzzicle.org/math/ap-calculus-ab/v/recognizing-concavity-exercisehttps://puzzicle.org/thumbs/x22647994.jpgAnalyzing concavity (graphical)Sal walks through an exercise where you are asked to recognize the concavity of a function in certain regions.https://cdn.kastatic.org/ka-youtube-converted/ulyyusmpc9w.mp4/ulyyusmpc9w.mp4143yes https://puzzicle.org/math/ap-calculus-ab/v/inflection-pointshttps://puzzicle.org/thumbs/1336987809.jpgIntroducing inflection pointsInflection points are points where the function changes concavity, i.e. from being "concave up" to being "concave down" or vice versa. They can be found by considering where the second derivative changes signs. In similar to critical points in the first derivative, inflection points will occur when the second derivative is either zero or undefined.https://cdn.kastatic.org/ka-youtube-converted/UK2shgCXALo.mp4/UK2shgCXALo.mp4154yes https://puzzicle.org/math/ap-calculus-ab/v/inflection-points-graphically-given-functionhttps://puzzicle.org/thumbs/x70e8e6ed.jpgAnalyzing inflection points (graphical)Sal analyzes the graph of a function g to find all the inflection points of g.https://cdn.kastatic.org/ka-youtube-converted/iWgW2nD4wGQ.mp4/iWgW2nD4wGQ.mp4200yes https://puzzicle.org/math/ap-calculus-ab/v/analyzing-concavity-algebraicallyhttps://puzzicle.org/thumbs/x8e1390d7.jpgAnalyzing concavity (algebraic)Sal finds the intervals where g(x)=-x⁴+6x²-2x-3 is concave down/up by finding where its second derivative, g'', is positive/negative.https://cdn.kastatic.org/ka-youtube-converted/72L_N7AtCRE.mp4/72L_N7AtCRE.mp4556yes https://puzzicle.org/math/ap-calculus-ab/v/inflection-points-algebraicallyhttps://puzzicle.org/thumbs/xcc3df6cb.jpgAnalyzing inflection points (algebraic)Sal analyzes the points of inflection of g(x)=¼x⁴-4x³+24x² by looking for values where the second derivative g'' changes signs.https://cdn.kastatic.org/ka-youtube-converted/RoxefQ_Qgm8.mp4/RoxefQ_Qgm8.mp4335yes https://puzzicle.org/math/ap-calculus-ab/v/mistakes-when-finding-inflection-points-second-derivative-undefinedhttps://puzzicle.org/thumbs/x23301524ce5be9eb.jpgExploring mistakes when finding inflection points: second derivative undefinedCandidates for inflection points are points where the second derivative is zero *and* points where the second derivative is undefined. It's important not to overlook any candidate.https://cdn.kastatic.org/ka-youtube-converted/TaRPDFqEd-c.mp4/TaRPDFqEd-c.mp4371yes https://puzzicle.org/math/ap-calculus-ab/v/mistakes-when-finding-inflection-points-not-checking-candidateshttps://puzzicle.org/thumbs/xda174a6c45f7203f.jpgExploring mistakes when finding inflection points: not checking candidatesCandidates for inflection points are x-values where the second derivative is equal to zero or undefined. But they are only candidates! Once we've found them, we need to test them and see if they really are inflection points.https://cdn.kastatic.org/ka-youtube-converted/1lA1hWdw62Q.mp4/1lA1hWdw62Q.mp4242yes https://puzzicle.org/math/ap-calculus-ab/v/second-derivative-testhttps://puzzicle.org/thumbs/x404646fd.jpgApplying the second derivative testSal justifies the second derivative test, which is a way of determining relative minima & maxima, and gives an example.https://cdn.kastatic.org/ka-youtube-converted/-cW5hCsc9Yc.mp4/-cW5hCsc9Yc.mp4373yes https://puzzicle.org/math/ap-calculus-ab/v/calculus-graphing-using-derivativeshttps://puzzicle.org/thumbs/355054.jpgSketching a curve with calculus: polynomialSal sketches a graph of f(x)=3x⁴-4x³+2 including extremum and inflection points.https://cdn.kastatic.org/ka-youtube-converted/hIgnece9ins.mp4/hIgnece9ins.mp41231yes https://puzzicle.org/math/ap-calculus-ab/v/calculus-graphing-with-derivatives-examplehttps://puzzicle.org/thumbs/366056.jpgSketching a curve with calculus: logarithmSal sketches a graph of f(x)=ln(x⁴+27) including extremum and inflection points.https://cdn.kastatic.org/ka-youtube-converted/zC_dTaEY2AY.mp4/zC_dTaEY2AY.mp41508yes https://puzzicle.org/math/ap-calculus-ab/v/analyzing-a-function-with-its-derivativehttps://puzzicle.org/thumbs/1337114573.jpgAnalyzing a function with its derivativeTaking the derivative of f(x)=x³-12x+2 and graphing the derivative, so we can tell when f is increasing or decreasing, and where is its relative extremum points.https://cdn.kastatic.org/ka-youtube-converted/pInFesXIfg8.mp4/pInFesXIfg8.mp4582yes https://puzzicle.org/math/ap-calculus-ab/v/calculus-based-justification-for-function-increasinghttps://puzzicle.org/thumbs/xc63dbafee7c67364.jpgEstablishing a calculus-based justification for a function increasingA calculus-based justification is when we explain a property of a function f based on its derivative f'. See a good example (and a few wrong ones) for how to do this when explaining why a function increases.https://cdn.kastatic.org/ka-youtube-converted/qLhZdSNDQ0M.mp4/qLhZdSNDQ0M.mp4192yes https://puzzicle.org/math/ap-calculus-ab/v/justification-using-first-derivativehttps://puzzicle.org/thumbs/x559e4677ad53fe94.jpgJustifying a function's features using the first derivativeWe can explain why a function is increasing, decreasing, or having a relative maximum using information about its first derivative. This is called calculus-based justification.https://cdn.kastatic.org/ka-youtube-converted/t-RXwwBjxsc.mp4/t-RXwwBjxsc.mp4353yes https://puzzicle.org/math/ap-calculus-ab/v/identifying-inflection-points-from-graphs-of-function-and-derivativeshttps://puzzicle.org/thumbs/x5875debbcd4d8e9e.jpgIdentifying inflection points from graphs of function and derivativesIdentifying inflection points from graphs of function, first derivative and second derivative. Graphs are done using Desmos.com.https://cdn.kastatic.org/ka-youtube-converted/YBQUw0NhqgU.mp4/YBQUw0NhqgU.mp4306yes https://puzzicle.org/math/ap-calculus-ab/v/justification-for-inflection-pointhttps://puzzicle.org/thumbs/x69ba0a53e7397e88.jpgJustifying a function's features using the second derivative: inflection pointWe can justify whether a function has an inflection point by looking at where the second derivative changes its sign.https://cdn.kastatic.org/ka-youtube-converted/QgpjYgAAW7o.mp4/QgpjYgAAW7o.mp4227yes https://puzzicle.org/math/ap-calculus-ab/v/calculus-based-justification-using-second-derivative-maxhttps://puzzicle.org/thumbs/xcf9943a1a5d9101e.jpgJustifying a function's features using the second derivative: maximum pointGiven that the derivative of a function is zero, we can justify whether the function has a relative maximum point by looking at the second derivative.https://cdn.kastatic.org/ka-youtube-converted/PvZaQZBJzGU.mp4/PvZaQZBJzGU.mp4254yes https://puzzicle.org/math/ap-calculus-ab/v/connecting-function-and-derivatives-graphicallyhttps://puzzicle.org/thumbs/x3ef49788b2269af6.jpgConnecting f, f', and f'' graphicallyAnalyzing three graphs to see which describes the derivative of which other graph.https://cdn.kastatic.org/ka-youtube-converted/elugjT_Np3M.mp4/elugjT_Np3M.mp4336yes https://puzzicle.org/math/ap-calculus-ab/v/connecting-function-and-derivatives-graphically-exphttps://puzzicle.org/thumbs/xc58a981587b5fb54.jpgConnecting f, f', and f'' graphically (another example)Analyzing three graphs to see which describes the derivative of which other graph.https://cdn.kastatic.org/ka-youtube-converted/G6ReTLn9vLc.mp4/G6ReTLn9vLc.mp4261yes https://puzzicle.org/math/ap-calculus-ab/v/minimizing-sum-of-squareshttps://puzzicle.org/thumbs/1339427794.jpgApplying optimization: sum of squaresWhat is the minimum possible value of x^2+y^2 given that their product has to be fixed at xy = -16https://cdn.kastatic.org/ka-youtube-converted/1TK8V_qmqrk.mp4/1TK8V_qmqrk.mp4455yes https://puzzicle.org/math/ap-calculus-ab/v/optimizing-box-volume-graphicallyhttps://puzzicle.org/thumbs/1338874025.jpgApplying optimization: box volume (Part 1)If you are making a box out of a flat piece of cardboard, how do you maximize the volume of that box?https://cdn.kastatic.org/ka-youtube-converted/MC0tq6fNRwU.mp4/MC0tq6fNRwU.mp4590yes https://puzzicle.org/math/ap-calculus-ab/v/optimizing-box-volume-analyticallyhttps://puzzicle.org/thumbs/1339153034.jpgApplying optimization: box volume (Part 2)Finishing up the last video by working through the formulas.https://cdn.kastatic.org/ka-youtube-converted/cRboY08YG8g.mp4/cRboY08YG8g.mp4540yes https://puzzicle.org/math/ap-calculus-ab/v/optimizing-profit-at-a-shoe-factoryhttps://puzzicle.org/thumbs/1339412561.jpgApplying optimization: profitWho knows, you may end up running a shoe factory one day. So it might not be a bad idea to know how to maximize profits.https://cdn.kastatic.org/ka-youtube-converted/dam16G6cZ8k.mp4/dam16G6cZ8k.mp4687yes https://puzzicle.org/math/ap-calculus-ab/v/minimizing-the-cost-of-a-storage-containerhttps://puzzicle.org/thumbs/1339500281.jpgApplying optimization: cost of materialsWith all the storage you might have to handle for your shoe factory, I bet you'd also like to be able to minize the cost of storage.https://cdn.kastatic.org/ka-youtube-converted/tSMuKcN-RKM.mp4/tSMuKcN-RKM.mp4760yes https://puzzicle.org/math/ap-calculus-ab/v/expression-for-combined-area-of-triangle-and-squarehttps://puzzicle.org/thumbs/1393028914.jpgApplying optimization: area of triangle & square (Part 1)Sal constructs an equilateral triangle & a square whose bases are 100m together, such that their area is the smallest possible.https://cdn.kastatic.org/ka-youtube-converted/IFU7Go6Qg6E.mp4/IFU7Go6Qg6E.mp4379yes https://puzzicle.org/math/ap-calculus-ab/v/minimizing-combined-areahttps://puzzicle.org/thumbs/1392948670.jpgApplying optimization: area of triangle & square (Part 2)Sal constructs an equilateral triangle & a square whose bases are 100m together, such that their area is the smallest possible.https://cdn.kastatic.org/ka-youtube-converted/eS-_ZFzHjYA.mp4/eS-_ZFzHjYA.mp4331yes https://puzzicle.org/math/ap-calculus-ab/v/rectilinear-motion-example-maximum-accelerationhttps://puzzicle.org/thumbs/x4a5d6352.jpgAnalyzing motion problems: finding the maximum accelerationThe speed of a particle moving along the x-axis is given by v(t)=-t³+6t²+2t. Sal analyzes it to find the time when the particle's acceleration attains its maximum value.https://cdn.kastatic.org/ka-youtube-converted/5P1O6tBiTlQ.mp4/5P1O6tBiTlQ.mp4265yes https://puzzicle.org/math/ap-calculus-ab/v/implicit-curve-horizontal-tangenthttps://puzzicle.org/thumbs/x27943efb79d490b1.jpgDetermining horizontal tangent to implicit curveFinding the equation of a horizontal tangent to a curve that is defined implicitly as an equation in x and y.https://cdn.kastatic.org/ka-youtube-converted/ZhOojHeslhs.mp4/ZhOojHeslhs.mp4175yes https://puzzicle.org/math/ap-calculus-ab/v/determining-equation-of-vertical-line-tangent-to-curvehttps://puzzicle.org/thumbs/x19165ce49a3ca1a8.jpgDetermining equation of vertical line tangent to curveDetermine the equation of a vertical tangent line to the graph of an implicit equation.https://cdn.kastatic.org/ka-youtube-converted/CQt9SZzArY8.mp4/CQt9SZzArY8.mp4297yes https://puzzicle.org/math/ap-calculus-ab/v/2011-calculus-ab-free-response-4dhttps://puzzicle.org/thumbs/238520373.jpgUsing the Mean Value Theorem: 2011 Calculus AB free response #4dMean Value Theorem and differentiabilityhttps://cdn.kastatic.org/ka-youtube-converted/yX3JpmEgNHo.mp4/yX3JpmEgNHo.mp4346yes https://puzzicle.org/math/ap-calculus-ab/v/2015-ap-calculus-ab-5-ahttps://puzzicle.org/thumbs/x269df2e9.jpgFinding relative maximum: 2015 AP Calculus AB 5aMaxima for f from graph of f'https://cdn.kastatic.org/ka-youtube-converted/j_kI7OcaLKI.mp4/j_kI7OcaLKI.mp4275yes https://puzzicle.org/math/ap-calculus-ab/v/2015-ap-calculus-ab-5bhttps://puzzicle.org/thumbs/x29c15a44.jpgAnalyzing function features: 2015 AP Calculus AB 5bIntervals where f is concave downwards and decreasing.https://cdn.kastatic.org/ka-youtube-converted/bba11I93w8o.mp4/bba11I93w8o.mp4181yes https://puzzicle.org/math/ap-calculus-ab/v/2015-ap-calculus-ab-5chttps://puzzicle.org/thumbs/x05f1c7c9.jpgFinding inflection points: 2015 AP Calculus AB 5cInflection points for a functionhttps://cdn.kastatic.org/ka-youtube-converted/ol4bW-3qs_s.mp4/ol4bW-3qs_s.mp4196yes https://puzzicle.org/math/ap-calculus-ab/v/2015-ap-calculus-ab-6bhttps://puzzicle.org/thumbs/xdf20cb71.jpgFinding vertical tangent lines: 2015 AP Calculus AB 6bPoint on curve where tangent is verticalhttps://cdn.kastatic.org/ka-youtube-converted/s3jvXHQlO-c.mp4/s3jvXHQlO-c.mp4282yes https://puzzicle.org/math/ap-calculus-ab/v/2017-ap-calculus-abbc-4bhttps://puzzicle.org/thumbs/x8c6b96ca66873089.jpgUsing concavity for reasoning: 2017 AP Calculus AB/BC 4bPotato problem from 2017 AP exam (Question 4, part b). Taking the second derivative using the chain rule to think about concavity.https://cdn.kastatic.org/ka-youtube-converted/OzhjlMnQ114.mp4/OzhjlMnQ114.mp4312yes https://puzzicle.org/math/ap-calculus-ab/v/introduction-to-integral-calculushttps://puzzicle.org/thumbs/x26ac85dafbab3184.jpgIntroducing integral calculusThe basic idea of Integral calculus is finding the area under a curve. To find it exactly, we can divide the area into infinite rectangles of infinitely small width and sum their areas—calculus is great for working with infinite things! This idea is actually quite rich, and it's also tightly related to Differential calculus, as you will see in the upcoming videos.https://cdn.kastatic.org/ka-youtube-converted/__Uw1SXPW7s.mp4/__Uw1SXPW7s.mp4292yes https://puzzicle.org/math/ap-calculus-ab/v/definite-integrals-introhttps://puzzicle.org/thumbs/x5d1b85e9cd11f651.jpgIntroducing definite integralsDefinite integrals represent the area under the curve of a function and above the 𝘹-axis. Learn about the notation we use to write them and see some introductory examples.https://cdn.kastatic.org/ka-youtube-converted/0dDIPzqKgYk.mp4/0dDIPzqKgYk.mp4148yes https://puzzicle.org/math/ap-calculus-ab/v/area-under-rate-function-examplehttps://puzzicle.org/thumbs/x4d22ba5a.jpgWorking an example: accumulation of changeAn example relating rates of change with a leaky bathtub.https://cdn.kastatic.org/ka-youtube-converted/okjt-OZY1JU.mp4/okjt-OZY1JU.mp4415yes https://puzzicle.org/math/ap-calculus-ab/v/simple-riemann-approximation-using-rectangleshttps://puzzicle.org/thumbs/1319674060.jpgIntroducing Riemann approximationsApproximating the area under a curve using some rectangles. This is called a Riemann sum.https://cdn.kastatic.org/ka-youtube-converted/dEAk0BHBYCM.mp4/dEAk0BHBYCM.mp4405yes https://puzzicle.org/math/ap-calculus-ab/v/riemann-sums-over-and-under-estimationhttps://puzzicle.org/thumbs/x2dd8112342f80589.jpgOver- and under-estimating of Riemann sumsRiemann sums are approximations of area, so usually they aren't equal to the exact area. Sometimes they are larger than the exact area (this is called overestimation) and sometimes they are smaller (this is called underestimation).https://cdn.kastatic.org/ka-youtube-converted/ER5eoDkjsrc.mp4/ER5eoDkjsrc.mp4241yes https://puzzicle.org/math/ap-calculus-ab/v/riemann-sum-from-tablehttps://puzzicle.org/thumbs/xdeb8581d02120d29.jpgWorking an example: finding a Riemann sum using a tableWhen we have a table of values of a function, we can use it to find a Riemann sum approximation of that function.https://cdn.kastatic.org/ka-youtube-converted/dki2K3Z-1kI.mp4/dki2K3Z-1kI.mp4399yes https://puzzicle.org/math/ap-calculus-ab/v/riemann-sums-over-under-examplehttps://puzzicle.org/thumbs/x2449dade551c2767.jpgWorking an example: over- and under-estimation of Riemann sumsOrdering different areas from least to greatest.https://cdn.kastatic.org/ka-youtube-converted/XmI80eJZAWM.mp4/XmI80eJZAWM.mp4282yes https://puzzicle.org/math/ap-calculus-ab/v/midpoint-sumshttps://puzzicle.org/thumbs/x3c51cdc21a21f6f6.jpgApproximating area using midpoint sumsApproximating area under a curve using rectangle where the heights are the value of the function at the midpoint of each interval.https://cdn.kastatic.org/ka-youtube-converted/kJ57S0Z4Vp8.mp4/kJ57S0Z4Vp8.mp4329yes https://puzzicle.org/math/ap-calculus-ab/v/trapezoidal-approximation-of-area-under-curvehttps://puzzicle.org/thumbs/1319516835.jpgApproximating area using trapezoidal sumsThe area under a curve is commonly approximated using rectangles (e.g. left, right, and midpoint Riemann sums), but it can also be approximated by trapezoids. Trapezoidal sums actually give a better approximation, in general, than rectangular sums that use the same number of subdivisions.https://cdn.kastatic.org/ka-youtube-converted/1p0NHR5w0Lc.mp4/1p0NHR5w0Lc.mp4507yes https://puzzicle.org/math/ap-calculus-ab/v/summation-notation-worked-exampleshttps://puzzicle.org/thumbs/x6cf630144647fe01.jpgWorking examples: Summation notationSummation notation uses the sigma Σ symbol to represent sums with multiple terms. See some more involved examples of how we read expressions in summation notation.https://cdn.kastatic.org/ka-youtube-converted/luSh3hz_b18.mp4/luSh3hz_b18.mp4321yes https://puzzicle.org/math/ap-calculus-ab/v/generalizing-a-left-riemann-sum-with-equally-spaced-rectangleshttps://puzzicle.org/thumbs/1319350944.jpgUsing Riemann sums in summation notationGeneralizing the technique of approximating area under a curve with rectangles.https://cdn.kastatic.org/ka-youtube-converted/ViqrHGae7FA.mp4/ViqrHGae7FA.mp4511yes https://puzzicle.org/math/ap-calculus-ab/v/practice-approximating-area-under-curvehttps://puzzicle.org/thumbs/xf873a305.jpgWorking an example: Riemann sums in summation notationHere we express the approximation of the area under a curve in sigma notation.https://cdn.kastatic.org/ka-youtube-converted/3ZhUdleD1lQ.mp4/3ZhUdleD1lQ.mp4427yes https://puzzicle.org/math/ap-calculus-ab/v/riemann-sums-and-integralshttps://puzzicle.org/thumbs/1319594360.jpgRepresenting a definite integral as the limit of a Riemann sumDefinite integrals represent the exact area under a given curve, and Riemann sums are used to approximate those areas. However, if we take Riemann sums with infinite rectangles of infinitely small width (using limits), we get the exact area, i.e. the definite integral!https://cdn.kastatic.org/ka-youtube-converted/CXCtqBlEZ7g.mp4/CXCtqBlEZ7g.mp4266yes https://puzzicle.org/math/ap-calculus-ab/v/rewriting-definite-integral-as-limit-of-riemann-sumhttps://puzzicle.org/thumbs/xc09fd81209309140.jpgWorking an example: Rewriting definite integral as limit of Riemann sumGiven a definite integral expression, we can write the corresponding limit of a Riemann sum with infinite rectangles.https://cdn.kastatic.org/ka-youtube-converted/eidemLqiPyQ.mp4/eidemLqiPyQ.mp4334yes https://puzzicle.org/math/ap-calculus-ab/v/writing-riemann-sum-limit-as-definite-integralhttps://puzzicle.org/thumbs/x6ba31798dda20c91.jpgWorking an example: Rewriting limit of Riemann sum as definite integralWhen given a limit of Riemann sum with infinite rectangles, we can analyze the expression to find the corresponding definite integral.https://cdn.kastatic.org/ka-youtube-converted/Himr2l8Rd18.mp4/Himr2l8Rd18.mp4395yes https://puzzicle.org/math/ap-calculus-ab/v/fundamental-theorem-of-calculushttps://puzzicle.org/thumbs/1353533334.jpgIntroducing the fundamental theorem of calculus and accumulation functionsThe fundamental theorem of calculus shows how, in some sense, integration is the opposite of differentiation.https://cdn.kastatic.org/ka-youtube-converted/C7ducZoLKgw.mp4/C7ducZoLKgw.mp4483yes https://puzzicle.org/math/ap-calculus-ab/v/functions-defined-by-definite-integralshttps://puzzicle.org/thumbs/x5d0078e4e21024fd.jpgUnderstanding functions defined by definite integrals (accumulation functions)Understanding that a function can be defined using a definite integral. Thinking about how to evaluate functions defined this way.https://cdn.kastatic.org/ka-youtube-converted/qJMbAfbb_Ek.mp4/qJMbAfbb_Ek.mp4257yes https://puzzicle.org/math/ap-calculus-ab/v/applying-the-fundamental-theorem-of-calculushttps://puzzicle.org/thumbs/1362669934.jpgFinding derivative with fundamental theorem of calculusThe Fundamental Theorem of Calculus tells us that the derivative of the definite integral from 𝘢 to 𝘹 of ƒ(𝑡)𝘥𝑡 is ƒ(𝘹), provided that ƒ is continuous. See how this can be used to evaluate the derivative of accumulation functions.https://cdn.kastatic.org/ka-youtube-converted/WapSrovs7yM.mp4/WapSrovs7yM.mp4204yes https://puzzicle.org/math/ap-calculus-ab/v/derivative-with-ftc-and-chain-rulehttps://puzzicle.org/thumbs/x56ccc0203d1640e5.jpgFinding derivative with fundamental theorem of calculus: chain ruleThe Fundamental Theorem of Calculus tells us how to find the derivative of the integral from 𝘢 to 𝘹 of a certain function. But what if instead of 𝘹 we have a function of 𝘹, for example sin(𝘹)? Then we need to also use the chain rule.https://cdn.kastatic.org/ka-youtube-converted/dptES2sb1oc.mp4/dptES2sb1oc.mp4187yes https://puzzicle.org/math/ap-calculus-ab/v/interpreting-behavior-of-antiderivativehttps://puzzicle.org/thumbs/xe75754b91c3ca2f4.jpgInterpreting the behavior of accumulation functionsWhen given the graph of function ƒ, we can reason about the graph of its antiderivative 𝑔 (so 𝑔'=ƒ).https://cdn.kastatic.org/ka-youtube-converted/51JgTZKnytA.mp4/51JgTZKnytA.mp4502yes https://puzzicle.org/math/ap-calculus-ab/v/negative-definite-integralshttps://puzzicle.org/thumbs/xe6d840e1958cb41c.jpgUnderstanding definite integrals with a negative valueWe learned that definite integrals give us the area under the curve and above the x-axis. But what if the curve itself is below the x-axis? In this case, the definite integral is still related to area, but it's negative. See how this works and get some intuition for why this is so.https://cdn.kastatic.org/ka-youtube-converted/TtRAY6Eyp4w.mp4/TtRAY6Eyp4w.mp4324yes https://puzzicle.org/math/ap-calculus-ab/v/definite-integrals-with-area-formulashttps://puzzicle.org/thumbs/xce6edc1757a2754a.jpgFinding definite integrals using area formulasSince definite integrals are the net area between a curve and the x-axis, we can sometimes use geometric area formulas to find definite integrals. See how it's done.https://cdn.kastatic.org/ka-youtube-converted/7R_DTzVHw5Y.mp4/7R_DTzVHw5Y.mp4257yes https://puzzicle.org/math/ap-calculus-ab/v/calculating-net-area-using-geometryhttps://puzzicle.org/thumbs/x5c0aa1089f1709ae.jpgCalculating net area using geometryThere are definite integrals that can be evaluated using geometry.https://cdn.kastatic.org/ka-youtube-converted/077LCl3IX7M.mp4/077LCl3IX7M.mp4242yes https://puzzicle.org/math/ap-calculus-ab/v/same-integration-boundshttps://puzzicle.org/thumbs/x931cc126.jpgEvaluating a definite integral over a single pointWhat happens when the bounds of your integral are the same?https://cdn.kastatic.org/ka-youtube-converted/q3qvkZxhchE.mp4/q3qvkZxhchE.mp4151yes https://puzzicle.org/math/ap-calculus-ab/v/integrating-scaled-functionhttps://puzzicle.org/thumbs/xdfa7ddb8.jpgIntegrating scaled version of functionSal uses a graph to explain why we can take a constant out of a definite integral.https://cdn.kastatic.org/ka-youtube-converted/JyArK4jw3XU.mp4/JyArK4jw3XU.mp4321yes https://puzzicle.org/math/ap-calculus-ab/v/switching-integral-boundshttps://puzzicle.org/thumbs/x87d0c9ae.jpgSwitching bounds of definite integralWhat happens when you swap the bounds on an integral?https://cdn.kastatic.org/ka-youtube-converted/L5eBJjx2FP8.mp4/L5eBJjx2FP8.mp4277yes https://puzzicle.org/math/ap-calculus-ab/v/integrating-function-sumshttps://puzzicle.org/thumbs/x78778540.jpgIntegrating sums of functionsIf you know the integrals of two functions, what is the integral of their sum?https://cdn.kastatic.org/ka-youtube-converted/JUouea9XoF8.mp4/JUouea9XoF8.mp4406yes https://puzzicle.org/math/ap-calculus-ab/v/properties-of-definite-integrals-2https://puzzicle.org/thumbs/xb67ea312.jpgWorking examples: Finding definite integrals using algebraic propertiesSal evaluates definite integrals of functions given their graphs. He does so using various properties of integrals.https://cdn.kastatic.org/ka-youtube-converted/pRZzfoM-9lc.mp4/pRZzfoM-9lc.mp4139yes https://puzzicle.org/math/ap-calculus-ab/v/breaking-integral-intervalhttps://puzzicle.org/thumbs/x99ada59b.jpgEvaluating definite integrals on adjacent intervalsBy subdividing the interval of numbers where you are integrating, you can break up an integral into parts.https://cdn.kastatic.org/ka-youtube-converted/BE1FK7vDVq0.mp4/BE1FK7vDVq0.mp4186yes https://puzzicle.org/math/ap-calculus-ab/v/evaluating-function-defined-by-integralhttps://puzzicle.org/thumbs/x1decd87a.jpgWorking an example: Breaking up the integral's intervalFinding a definite integral by breaking it down to smaller intervals that are adjacent to each other.https://cdn.kastatic.org/ka-youtube-converted/h6mZLTpTHcU.mp4/h6mZLTpTHcU.mp4445yes https://puzzicle.org/math/ap-calculus-ab/v/merging-definite-integrals-over-adjacent-intervalshttps://puzzicle.org/thumbs/xa3d0a3c8c0bf97f6.jpgWorking an example: Merging definite integrals over adjacent intervalsThinking about how to evaluate the sum of definite integrals over adjacent intervals.https://cdn.kastatic.org/ka-youtube-converted/w00eCt3i9KM.mp4/w00eCt3i9KM.mp4242yes https://puzzicle.org/math/ap-calculus-ab/v/evaluating-function-defined-by-integral-switchhttps://puzzicle.org/thumbs/x44acaab0.jpgEvaluating functions defined by integrals: switched intervalSal evaluates a function defined by the integral of a graphed function. In order to evaluate he must switch the sides of the interval.https://cdn.kastatic.org/ka-youtube-converted/Kq5hqFg17bg.mp4/Kq5hqFg17bg.mp4174yes https://puzzicle.org/math/ap-calculus-ab/v/swapping-the-bounds-for-definite-integralhttps://puzzicle.org/thumbs/1363004264.jpgFinding derivative with fundamental theorem of calculus: x is on lower boundSometimes you need to swap the bounds of integration before applying the fundamental theorem of calculus.https://cdn.kastatic.org/ka-youtube-converted/TqGCNNlx6pU.mp4/TqGCNNlx6pU.mp4180yes https://puzzicle.org/math/ap-calculus-ab/v/both-bounds-being-a-function-of-xhttps://puzzicle.org/thumbs/1362611642.jpgFinding derivative with fundamental theorem of calculus: x is on both boundsHow do you apply the fundamental theorem of calculus when both integral bounds are a function of x.https://cdn.kastatic.org/ka-youtube-converted/0z52CLjC2C0.mp4/0z52CLjC2C0.mp4304yes https://puzzicle.org/math/ap-calculus-ab/v/connecting-the-first-and-second-fundamental-theorems-of-calculushttps://puzzicle.org/thumbs/1362356573.jpgExploring the fundamental theorem of calculus and definite integralsThere are really two versions of the fundamental theorem of calculus, and we go through the connection here.https://cdn.kastatic.org/ka-youtube-converted/JbfVrwxuPxM.mp4/JbfVrwxuPxM.mp4285yes https://puzzicle.org/math/ap-calculus-ab/v/antiderivatives-and-indefinite-integralshttps://puzzicle.org/thumbs/1328880986.jpgExploring antiderivatives and indefinite integralsWhat's the opposite of a derivative? It's something called the indefinite integral.https://cdn.kastatic.org/ka-youtube-converted/MMv-027KEqU.mp4/MMv-027KEqU.mp4223yes https://puzzicle.org/math/ap-calculus-ab/v/indefinite-integrals-of-x-raised-to-a-powerhttps://puzzicle.org/thumbs/1328847625.jpgDetermining the reverse power ruleCan you find a function whose derivative is x^n?https://cdn.kastatic.org/ka-youtube-converted/QxbJsg-Vdms.mp4/QxbJsg-Vdms.mp4348yes https://puzzicle.org/math/ap-calculus-ab/v/indefinite-integral-propertieshttps://puzzicle.org/thumbs/x178e6d86e8b0bec8.jpgExploring indefinite integrals: sums and multiplesAn indefinite integral of a sum is the same as the sum of the integrals of the component parts. Constants can be taken out of integrals.https://cdn.kastatic.org/ka-youtube-converted/YwR_82jgJ_Q.mp4/YwR_82jgJ_Q.mp4278yes https://puzzicle.org/math/ap-calculus-ab/v/rewriting-integrand-before-integratinghttps://puzzicle.org/thumbs/xc8d3f667a3156637.jpgRewriting before integratingSome indefinite integrals are much simpler to integrate by algebraically rewriting the integrand first.https://cdn.kastatic.org/ka-youtube-converted/hzSef_HVqz4.mp4/hzSef_HVqz4.mp4317yes https://puzzicle.org/math/ap-calculus-ab/v/antiderivative-of-x-1https://puzzicle.org/thumbs/1328998382.jpgDetermining the indefinite integral of 1/xIn differential ​calculus we learned that the derivative of ln(x) is 1/x. Integration goes the other way: the integral (or antiderivative) of 1/x should be a function whose derivative is 1/x. As we just saw, this is ln(x). However, if x is negative then ln(x) is undefined! The solution is quite simple: the antiderivative of 1/x is ln(|x|).https://cdn.kastatic.org/ka-youtube-converted/sPPjk4aXzmQ.mp4/sPPjk4aXzmQ.mp4455yes https://puzzicle.org/math/ap-calculus-ab/v/basic-trig-and-exponential-antiderivativeshttps://puzzicle.org/thumbs/1329132099.jpgDetermining the indefinite integrals of sin(x), cos(x), and eˣIntegral of sin(x)dx is -cos(x)+C, integral of cos(x)dx is sin(x)+C, and integral of eˣdx is eˣ+C. Learn why and see worked examples.https://cdn.kastatic.org/ka-youtube-converted/hXg-6YgAARk.mp4/hXg-6YgAARk.mp4244yes https://puzzicle.org/math/ap-calculus-ab/v/reverse-power-rule-for-definite-integralshttps://puzzicle.org/thumbs/x057a358cb0b0afa3.jpgEvaluating definite integrals: reverse power ruleExamples of calculating definite integrals of polynomials using the fundamental theorem of calculus and the reverse power rule.https://cdn.kastatic.org/ka-youtube-converted/auOcNZFKfo0.mp4/auOcNZFKfo0.mp4254yes https://puzzicle.org/math/ap-calculus-ab/v/evaluating-definite-integral-with-power-rulehttps://puzzicle.org/thumbs/xa03481ca.jpgEvaluating definite integral of a rational functionSal finds the definite integral of (16-x^3)/x^3; between -1 and -2 using the reverse power rule.https://cdn.kastatic.org/ka-youtube-converted/4WJUEXIksH0.mp4/4WJUEXIksH0.mp4305yes https://puzzicle.org/math/ap-calculus-ab/v/definite-integral-of-cube-roothttps://puzzicle.org/thumbs/x89fe4f56.jpgEvaluating definite integral of a radical functionSal finds the definite integral of 12∛x between -1 and 8 using the reverse power rule.https://cdn.kastatic.org/ka-youtube-converted/5UWdzPq8_Qs.mp4/5UWdzPq8_Qs.mp4231yes https://puzzicle.org/math/ap-calculus-ab/v/definite-integral-of-sine-based-functionhttps://puzzicle.org/thumbs/x41874f6b.jpgEvaluating definite integral of a trig functionSal finds the definite integral of 9sin(x) between 11π/2 and 6π.https://cdn.kastatic.org/ka-youtube-converted/ldLdWj6DLTw.mp4/ldLdWj6DLTw.mp4300yes https://puzzicle.org/math/ap-calculus-ab/v/definite-integral-involving-natural-loghttps://puzzicle.org/thumbs/x651ef59b.jpgEvaluating definite integral involving a natural log functionSal finds the definite integral of (6+x^2;)/x^3; between 2 and 4. To do that, he has to use the integral of 1/x, which is ln(x).https://cdn.kastatic.org/ka-youtube-converted/JMqKtEC2bbY.mp4/JMqKtEC2bbY.mp4447yes https://puzzicle.org/math/ap-calculus-ab/v/definite-integrals-of-piecewise-functionshttps://puzzicle.org/thumbs/x3953c53a.jpgEvaluating definite integral of a piecewise functionSal evaluates definite integral of a piecewise function over an interval that goes through the two cases of the function.https://cdn.kastatic.org/ka-youtube-converted/f5qr1lH6ViU.mp4/f5qr1lH6ViU.mp4374yes https://puzzicle.org/math/ap-calculus-ab/v/definite-integral-of-absolute-valuehttps://puzzicle.org/thumbs/x3b98bdf7.jpgEvaluating definite integral of an absolute value functionSal evaluates the definite integral of f(x)=|x+2| between -4 and 0.https://cdn.kastatic.org/ka-youtube-converted/PWcwhA2BwVk.mp4/PWcwhA2BwVk.mp4413yes https://puzzicle.org/math/ap-calculus-ab/v/evaluating-definite-integral-of-function-with-removable-and-jump-discontinuitieshttps://puzzicle.org/thumbs/xe89d09dff1fe417d.jpgEvaluating definite integral of function with removable and jump discontinuitiesEvaluate definite integrals of functions with removable or jump discontinuities.https://cdn.kastatic.org/ka-youtube-converted/cnw9lhxQGn4.mp4/cnw9lhxQGn4.mp4430yes https://puzzicle.org/math/ap-calculus-ab/v/u-substitutionhttps://puzzicle.org/thumbs/1254417751.jpgIntroducing 𝘶-substitutionUsing u-substitution to find the anti-derivative of a function. Seeing that u-substitution is the inverse of the chain rule.https://cdn.kastatic.org/ka-youtube-converted/b76wePnIBdU.mp4/b76wePnIBdU.mp4311yes https://puzzicle.org/math/ap-calculus-ab/v/u-substitution-example-3https://puzzicle.org/thumbs/1254165861.jpgUsing 𝘶-substitution: multiplying by a constantManipulating the expression to make u-substitution a little more obvious.https://cdn.kastatic.org/ka-youtube-converted/oqCfqIcbE10.mp4/oqCfqIcbE10.mp4333yes https://puzzicle.org/math/ap-calculus-ab/v/u-substitution-defining-uhttps://puzzicle.org/thumbs/xcef304a893f93d7a.jpgUsing 𝘶-substitution: defining 𝘶A common challenge when performing 𝘶-substitution is to realize which part should be our 𝘶.https://cdn.kastatic.org/ka-youtube-converted/r5XXDSOh5Nk.mp4/r5XXDSOh5Nk.mp4215yes https://puzzicle.org/math/ap-calculus-ab/v/u-substitution-defining-u-exphttps://puzzicle.org/thumbs/xf4e4b2d05feeefa8.jpgUsing 𝘶-substitution: defining 𝘶 (more examples)A common challenge when performing 𝘶-substitution is to realize which part should be our 𝘶.https://cdn.kastatic.org/ka-youtube-converted/rsBALP8QNns.mp4/rsBALP8QNns.mp4274yes https://puzzicle.org/math/ap-calculus-ab/v/u-substitution-example-2https://puzzicle.org/thumbs/1254490089.jpgUsing 𝘶-substitution: rational functionAnother example of using u-substitution with a rational function.https://cdn.kastatic.org/ka-youtube-converted/Zp5z0wa0kgo.mp4/Zp5z0wa0kgo.mp4244yes https://puzzicle.org/math/ap-calculus-ab/v/u-substitution-with-ln-xhttps://puzzicle.org/thumbs/1254468209.jpgUsing 𝘶-substitution: logarithmic functionDoing u-substitution with ln(x)https://cdn.kastatic.org/ka-youtube-converted/OLO64d4Y1qI.mp4/OLO64d4Y1qI.mp4219yes https://puzzicle.org/math/ap-calculus-ab/v/u-substitution-definite-integralshttps://puzzicle.org/thumbs/xf96099c4eaa7e1db.jpgUsing 𝘶-substitution: definite integralsWhen using 𝘶-substitution in definite integrals, we must make sure we take care of the boundaries of integration.https://cdn.kastatic.org/ka-youtube-converted/paNqHnR3-aM.mp4/paNqHnR3-aM.mp4353yes https://puzzicle.org/math/ap-calculus-ab/v/u-substitution-exponentialhttps://puzzicle.org/thumbs/xadef2972.jpgUsing 𝘶-substitution: definite integral of exponential functionFinding the definite integral from 0 to 1 of x^2(2^(x^3)).https://cdn.kastatic.org/ka-youtube-converted/1ct7LUx23io.mp4/1ct7LUx23io.mp4473yes https://puzzicle.org/math/ap-calculus-ab/v/integral-partial-fractionhttps://puzzicle.org/thumbs/xa5f8f98b.jpgIntegrating using long divisionHere we do polynomial long division to make an integral easier to evaluate.https://cdn.kastatic.org/ka-youtube-converted/5j81gyHn9i0.mp4/5j81gyHn9i0.mp4318yes https://puzzicle.org/math/ap-calculus-ab/v/integration-using-completing-the-square-arctanhttps://puzzicle.org/thumbs/x1ff0a76ebf38d991.jpgIntegrating using completing the square and the derivative of arctan(x)Sometimes we can integrate rational functions by using the method of completing the square in the denominator and then integrating using u-substitution and our knowledge about the derivative of arctan(x).https://cdn.kastatic.org/ka-youtube-converted/1nXSO3POyZ8.mp4/1nXSO3POyZ8.mp4328yes https://puzzicle.org/math/ap-calculus-ab/v/proof-of-fundamental-theorem-of-calculushttps://puzzicle.org/thumbs/1363005217.jpgProving the fundamental theorem of calculusThe first part of the fundamental theorem of calculus tells us that if we define 𝘍(𝘹) to be the definite integral of function ƒ from some constant 𝘢 to 𝘹, then 𝘍 is an antiderivative of ƒ. In other words, 𝘍'(𝘹)=ƒ(𝘹). See why this is so.https://cdn.kastatic.org/ka-youtube-converted/pWtt0AvU0KA.mp4/pWtt0AvU0KA.mp4840yes https://puzzicle.org/math/ap-calculus-ab/v/intuition-for-second-fundamental-theorem-of-calculushttps://puzzicle.org/thumbs/1319534849.jpgProving the second part of fundamental theorem of calculusThe second part of the fundamental theorem of calculus tells us that to find the definite integral of a function f; from 𝘢 to 𝘣, we need to take an antiderivative of f, call it 𝘍, and calculate 𝘍(𝘣)-𝘍(𝘢). Get some intuition into why this is true.https://cdn.kastatic.org/ka-youtube-converted/Cz_GWNdf_68.mp4/Cz_GWNdf_68.mp4744yes https://puzzicle.org/math/ap-calculus-ab/v/2011-calculus-ab-free-response-4ahttps://puzzicle.org/thumbs/237109444.jpgUsing equations involving integrals: 2011 Calculus AB free response #4aTaking derivatives and integrals of strangely defined functionshttps://cdn.kastatic.org/ka-youtube-converted/tnoPnyvVjyc.mp4/tnoPnyvVjyc.mp4288yes https://puzzicle.org/math/ap-calculus-ab/v/differential-equation-introductionhttps://puzzicle.org/thumbs/x64571eb4.jpgIntroducing differential equationsDifferential equations are equations that relate a function with one or more of its derivatives. This means their solution is a function! Learn more in this video.https://cdn.kastatic.org/ka-youtube-converted/6o7b9yyhH7k.mp4/6o7b9yyhH7k.mp4470yes https://puzzicle.org/math/ap-calculus-ab/v/writing-a-differential-equationhttps://puzzicle.org/thumbs/xc721bc33777fe0a5.jpgWriting a differential equationDifferential equations describe relationships that involve quantities and their rates of change. See how we write the equation for such a relationship.https://cdn.kastatic.org/ka-youtube-converted/H9h-Xg7zjPU.mp4/H9h-Xg7zjPU.mp4139yes https://puzzicle.org/math/ap-calculus-ab/v/verifying-solutions-to-differential-equationshttps://puzzicle.org/thumbs/x38bbf6ba25cf4539.jpgVerifying solutions to differential equationsWe can check whether a potential solution to a differential equation is indeed a solution. What we need to do is differentiate and substitute both the solution and the derivative into the equation.https://cdn.kastatic.org/ka-youtube-converted/KZqg4L--YC0.mp4/KZqg4L--YC0.mp4352yes https://puzzicle.org/math/ap-calculus-ab/v/creating-a-slope-fieldhttps://puzzicle.org/thumbs/xc8a02f3a.jpgIntroducing slope fieldsSlope fields allow us to analyze differential equations graphically. Learn how to draw them and use them to find particular solutions.https://cdn.kastatic.org/ka-youtube-converted/8Amgakx5aII.mp4/8Amgakx5aII.mp4352yes https://puzzicle.org/math/ap-calculus-ab/v/differential-equation-from-slope-fieldhttps://puzzicle.org/thumbs/xb57e783b.jpgWorking an example: equation from slope fieldGiven a slope field and a few differential equations, we can determine which equation corresponds to the slope field by considering specific slopes.https://cdn.kastatic.org/ka-youtube-converted/sU5MCM3npXU.mp4/sU5MCM3npXU.mp4371yes https://puzzicle.org/math/ap-calculus-ab/v/identifying-slope-field-examplehttps://puzzicle.org/thumbs/xe8ae4b4059518a88.jpgWorking an example: slope field from equationGiven a differential equation in x and y, we can draw a segment with dy/dx as slope at any point (x,y). That's the slope field of the equation. See how we match an equation to its slope field by considering the various slopes in the diagram.https://cdn.kastatic.org/ka-youtube-converted/sPe6qoQFFD8.mp4/sPe6qoQFFD8.mp4183yes https://puzzicle.org/math/ap-calculus-ab/v/example-of-steps-for-constructing-slope-fieldhttps://puzzicle.org/thumbs/x5d6cbc82222694cb.jpgWorking an example: forming a slope fieldGiven a differential equation in x and y, we can draw a segment with dy/dx as slope at any point (x,y). That's the slope field of the equation. See how we determine the slopes of a few segments in the slope field of an equation.https://cdn.kastatic.org/ka-youtube-converted/tgS5kgkZNGw.mp4/tgS5kgkZNGw.mp4211yes https://puzzicle.org/math/ap-calculus-ab/v/slope-field-to-visualize-solutionshttps://puzzicle.org/thumbs/x8ea5d072.jpgApproximating solution curves in slope fieldsGiven the slope field of a differential equation, we can sketch various solutions to the equation.https://cdn.kastatic.org/ka-youtube-converted/LoaagZPWvpM.mp4/LoaagZPWvpM.mp4416yes https://puzzicle.org/math/ap-calculus-ab/v/range-of-solution-curve-from-slope-fieldhttps://puzzicle.org/thumbs/xfcaa1cd3c0b4ece0.jpgWorking an example: range of solution curve from slope fieldGiven the slope field of a differential equation, we can sketch various solutions to the equation. In this example, we analyze the range of a specific solution.https://cdn.kastatic.org/ka-youtube-converted/8EA5wv-kQPw.mp4/8EA5wv-kQPw.mp4129yes https://puzzicle.org/math/ap-calculus-ab/v/separable-differential-equations-introductionhttps://puzzicle.org/thumbs/x4bbc820f.jpgIntroducing separable equationsSeparation of variables allows us to rewrite differential equations so we obtain an equality between two integrals we can evaluate. Separable equations are the class of differential equations that can be solved using this method.https://cdn.kastatic.org/ka-youtube-converted/DL-ozRGDlkY.mp4/DL-ozRGDlkY.mp4546yes https://puzzicle.org/math/ap-calculus-ab/v/addressing-treating-differentials-algebraicallyhttps://puzzicle.org/thumbs/x31a949b063c9d896.jpgAddressing treating differentials algebraicallyAddressing treating differentials algebraicallyhttps://cdn.kastatic.org/ka-youtube-converted/KQ7Y31QgX_w.mp4/KQ7Y31QgX_w.mp4284yes https://puzzicle.org/math/ap-calculus-ab/v/separable-differential-equations-exampleshttps://puzzicle.org/thumbs/xcd8d49b4a7193569.jpgWorking examples: separable differential equationsTwo worked examples of finding general solutions to separable differential equations.https://cdn.kastatic.org/ka-youtube-converted/UAcRCt9pEDI.mp4/UAcRCt9pEDI.mp4264yes https://puzzicle.org/math/ap-calculus-ab/v/identifying-separable-equationshttps://puzzicle.org/thumbs/x4d1f7dd1f9ad4a51.jpgWorking an example: identifying separable equationsSeparable equations can be written in the form dy/dx=f(x)g(y). See how we analyze various differential equations to see if they are separable.https://cdn.kastatic.org/ka-youtube-converted/6vUjGgI8Dso.mp4/6vUjGgI8Dso.mp4297yes https://puzzicle.org/math/ap-calculus-ab/v/finding-constant-of-integration-rationalhttps://puzzicle.org/thumbs/x5e2c88b0.jpgFinding particular solutions to differential equations: rational functionSal finds f(-1) given that f(x)=24/x^3; and f(2)=12.https://cdn.kastatic.org/ka-youtube-converted/bjdOWH43-9s.mp4/bjdOWH43-9s.mp4264yes https://puzzicle.org/math/ap-calculus-ab/v/finding-constant-of-integration-exponentialhttps://puzzicle.org/thumbs/x5b20ed82.jpgFinding particular solutions to differential equations: exponential functionSal finds f(0) given that f(x)=5eˣ and f(7)=40+5e⁷.https://cdn.kastatic.org/ka-youtube-converted/T8rbpI4OZCc.mp4/T8rbpI4OZCc.mp4188yes https://puzzicle.org/math/ap-calculus-ab/v/particular-solution-to-differential-equation-examplehttps://puzzicle.org/thumbs/xc7b88d99.jpgWorking an example: finding a specific solution to a separable equationSolving a separable differential equation given initial conditions. In this video, the equation is dy/dx=2y^2; with y(1)=1.https://cdn.kastatic.org/ka-youtube-converted/E444KhRcWSk.mp4/E444KhRcWSk.mp4338yes https://puzzicle.org/math/ap-calculus-ab/v/using-particular-solution-to-separable-differential-equationhttps://puzzicle.org/thumbs/x2db6052e61b90ad0.jpgWorking an example: separable equation with an implicit solutionSometimes the solution of a separable differential equation can't be written as an explicit function. This doesn't mean we can't use it!https://cdn.kastatic.org/ka-youtube-converted/72xChzYjJdM.mp4/72xChzYjJdM.mp4198yes https://puzzicle.org/math/ap-calculus-ab/v/determining-the-domain-of-particular-solution-to-differential-equationhttps://puzzicle.org/thumbs/x1719599353ddb91e.jpgDetermining the domain of particular solution to differential equationDetermine any restrictions on the domain of a particular solution to a differential equation.https://cdn.kastatic.org/ka-youtube-converted/R_tIbSCqioU.mp4/R_tIbSCqioU.mp4228yes https://puzzicle.org/math/ap-calculus-ab/v/modeling-population-with-simple-differential-equationhttps://puzzicle.org/thumbs/x71b1c9d7.jpgExploring exponential models and differential equations (Part 1)Assuming a quantity grows proportionally to its size results in the general equation dy/dx=ky. Solving it with separation of variables results in the general exponential function y=Ceᵏˣ.https://cdn.kastatic.org/ka-youtube-converted/IYFkXWlgC_w.mp4/IYFkXWlgC_w.mp4461yes https://puzzicle.org/math/ap-calculus-ab/v/particular-solution-given-initial-conditions-for-populationhttps://puzzicle.org/thumbs/xb0ac5177.jpgExploring exponential models and differential equations (Part 2)Given the general solution P=Ceᵏᵗ and the conditions P(0)=100 and P(50)=200, we find the solution to an exponential modeling problem.https://cdn.kastatic.org/ka-youtube-converted/6jqx5uEBs5U.mp4/6jqx5uEBs5U.mp4293yes https://puzzicle.org/math/ap-calculus-ab/v/exponential-solution-to-differential-equationhttps://puzzicle.org/thumbs/xeaa1614aea9948f2.jpgWorking an example: exponential solution to differential equationThe solution of the general differential equation dy/dx=ky (for some k) is Ceᵏˣ (for some C). See how this is derived and used for finding a particular solution to a differential equation.https://cdn.kastatic.org/ka-youtube-converted/pRuwcphjJeg.mp4/pRuwcphjJeg.mp4267yes https://puzzicle.org/math/ap-calculus-ab/v/2011-calculus-ab-free-response-5chttps://puzzicle.org/thumbs/238520371.jpgFinding particular solution to differential equation: 2011 Calculus AB free response #5cSolving a differential equation using separation of variableshttps://cdn.kastatic.org/ka-youtube-converted/R-2Uw10QgIo.mp4/R-2Uw10QgIo.mp4555yes https://puzzicle.org/math/ap-calculus-ab/v/sketching-a-slope-field-ap-calculus-ab-bc-2015-4ahttps://puzzicle.org/thumbs/xe106de13958d1a93.jpgSketching a slope field AP Calculus AB BC 2015 4aSketching a slope field given a differential equationhttps://cdn.kastatic.org/ka-youtube-converted/cbkrtV5Qajs.mp4/cbkrtV5Qajs.mp4494yes https://puzzicle.org/math/ap-calculus-ab/v/2017-ap-calculus-abbc-4chttps://puzzicle.org/thumbs/x8a58bdc13f086d4e.jpgFinding particular solution to differential equation: 2017 AP Calculus AB/BC 4cPotato problem from 2017 AP exam (Question 4, part c). Finding a particular solution for a separable differential equation.https://cdn.kastatic.org/ka-youtube-converted/0DGrdJA5Mx4.mp4/0DGrdJA5Mx4.mp4424yes https://puzzicle.org/math/ap-calculus-ab/v/average-function-value-closed-intervalhttps://puzzicle.org/thumbs/x32f2123f.jpgCalculating average value over a closed intervalAverage value of a function over a closed intervalhttps://cdn.kastatic.org/ka-youtube-converted/NCF4m8BDs7w.mp4/NCF4m8BDs7w.mp4283yes https://puzzicle.org/math/ap-calculus-ab/v/calculating-function-average-over-intervalhttps://puzzicle.org/thumbs/x24937653.jpgCalculating average value of function over intervalHere we find the average value of x^2+1 on the interval between 0 and 3.https://cdn.kastatic.org/ka-youtube-converted/0rzL08BHr5c.mp4/0rzL08BHr5c.mp4308yes https://puzzicle.org/math/ap-calculus-ab/v/mean-value-theorem-integralshttps://puzzicle.org/thumbs/x5ca5c5c1.jpgExploring the mean value theorem for integralsHere Sal goes through the connection between the mean value theorem and integration.https://cdn.kastatic.org/ka-youtube-converted/59UeshgSuEE.mp4/59UeshgSuEE.mp4547yes https://puzzicle.org/math/ap-calculus-ab/v/motion-problems-with-integralshttps://puzzicle.org/thumbs/x3722df7eacb6e00b.jpgSolving motion problems with integrals: displacement vs. distanceThe definite integral of a velocity function gives us the displacement. To find the actual distance traveled, we need to use the speed function, which is the absolute value of the velocity.https://cdn.kastatic.org/ka-youtube-converted/Sy_7PkoTCtA.mp4/Sy_7PkoTCtA.mp4484yes https://puzzicle.org/math/ap-calculus-ab/v/analyzing-motion-problems-positionhttps://puzzicle.org/thumbs/x08f18c2e8eb0e883.jpgAnalyzing motion problems: positionFinding the appropriate expression to use when looking for the position at a certain time.https://cdn.kastatic.org/ka-youtube-converted/M7IZClYqdEo.mp4/M7IZClYqdEo.mp4176yes https://puzzicle.org/math/ap-calculus-ab/v/analyzing-motion-problems-total-dist-traveledhttps://puzzicle.org/thumbs/x6334a37724249bbe.jpgAnalyzing motion problems: total distance traveledFinding the appropriate expression to use when looking for the total distance traveled over a certain time interval.https://cdn.kastatic.org/ka-youtube-converted/Ih-NVgXuTnQ.mp4/Ih-NVgXuTnQ.mp4173yes https://puzzicle.org/math/ap-calculus-ab/v/antiderivative-accelerationhttps://puzzicle.org/thumbs/xecee68b6.jpgWorking an example: motion problems (with definite integrals)What can you say about the velocity and position of a particle given its acceleration.https://cdn.kastatic.org/ka-youtube-converted/sLw902V40L0.mp4/sLw902V40L0.mp4512yes https://puzzicle.org/math/ap-calculus-ab/v/average-acceleration-over-intervalhttps://puzzicle.org/thumbs/xeac69708.jpgFinding the average acceleration over intervalAs an example of finding a mean value, we find an average acceleration.https://cdn.kastatic.org/ka-youtube-converted/wdAgEzdkFZI.mp4/wdAgEzdkFZI.mp4307yes https://puzzicle.org/math/ap-calculus-ab/v/area-under-rate-net-changehttps://puzzicle.org/thumbs/x195c42e1.jpgDetermining area under rate function gives the net changeIf you have a function representing rate, what does the area under its curve represent?https://cdn.kastatic.org/ka-youtube-converted/6FTiHeius1c.mp4/6FTiHeius1c.mp4447yes https://puzzicle.org/math/ap-calculus-ab/v/interpreting-definite-integral-as-net-changehttps://puzzicle.org/thumbs/xb67abbc5a7b138fa.jpgInterpreting definite integral as net changeThe definite integral of a rate function gives us the net change in the quantity described by the rate. See how we interpret definite integrals in a real-world context.https://cdn.kastatic.org/ka-youtube-converted/5-qpfFTaY8c.mp4/5-qpfFTaY8c.mp4316yes https://puzzicle.org/math/ap-calculus-ab/v/interpreting-definite-integrals-in-contexthttps://puzzicle.org/thumbs/x5ad32ab30fabf1f0.jpgWorking examples: interpreting definite integrals in contextInterpreting expressions involving definite integrals in a real-world context.https://cdn.kastatic.org/ka-youtube-converted/FWcXFU8Zj8U.mp4/FWcXFU8Zj8U.mp4261yes https://puzzicle.org/math/ap-calculus-ab/v/analyzing-problems-involving-definite-integralshttps://puzzicle.org/thumbs/xf42789b7e0119925.jpgAnalyzing problems involving definite integralsSee worked example of how to find the appropriate expression to use in order to solve word problems using definite integrals.https://cdn.kastatic.org/ka-youtube-converted/U4dtruMdQ7Y.mp4/U4dtruMdQ7Y.mp4352yes https://puzzicle.org/math/ap-calculus-ab/v/definite-integral-word-problemhttps://puzzicle.org/thumbs/x98122687f9058cd3.jpgWorking an example: problem involving definite integral (algebraic)Using definite integral to solve a word problem about the growth in the population of a town.https://cdn.kastatic.org/ka-youtube-converted/vYix-RPhT64.mp4/vYix-RPhT64.mp4470yes https://puzzicle.org/math/ap-calculus-ab/v/evaluating-simple-definite-integralhttps://puzzicle.org/thumbs/1319502914.jpgCalculating area between a curve and the x-axisBased on the fundamental theorem of calculus, we can use antiderivatives to compute integrals.https://cdn.kastatic.org/ka-youtube-converted/xR4AnXDBnsw.mp4/xR4AnXDBnsw.mp4354yes https://puzzicle.org/math/ap-calculus-ab/v/definite-integrals-and-negative-areahttps://puzzicle.org/thumbs/1329076337.jpgCalculating area between a curve and the x-axis: negative areaWith integrals, it can be helpful to introduce the notion of negative area. See why!https://cdn.kastatic.org/ka-youtube-converted/MM0FTzvedH4.mp4/MM0FTzvedH4.mp4448yes https://puzzicle.org/math/ap-calculus-ab/v/area-between-curveshttps://puzzicle.org/thumbs/1337260457.jpgCalculating the area between curvesBy integrating the difference of two functions, you can find the area between them.https://cdn.kastatic.org/ka-youtube-converted/wQXYtsyfbqg.mp4/wQXYtsyfbqg.mp4410yes https://puzzicle.org/math/ap-calculus-ab/v/area-between-curves-examplehttps://puzzicle.org/thumbs/x731d2664722c6dbc.jpgWorking an example: area between curvesThe area between the graphs of functions ƒ and 𝑔 can be found by taking the definite integral of ƒ-𝑔 (suppose ƒ is above 𝑔).https://cdn.kastatic.org/ka-youtube-converted/aaCRnT8o-Ng.mp4/aaCRnT8o-Ng.mp4325yes https://puzzicle.org/math/ap-calculus-ab/v/area-between-curves-with-multiple-boundarieshttps://puzzicle.org/thumbs/1337371091.jpgCalculating the composite area between curvesSometimes we need to break the area into multiple different areas, because the enclosing functions change.https://cdn.kastatic.org/ka-youtube-converted/LbTH7MGMNjk.mp4/LbTH7MGMNjk.mp4414yes https://puzzicle.org/math/ap-calculus-ab/v/area-between-curve-and-y-axishttps://puzzicle.org/thumbs/x0ed00c813377b4ac.jpgCalculating area between a curve and the 𝘺-axisWe can use a definite integral in terms of 𝘺 to find the area between a curve and the 𝘺-axis.https://cdn.kastatic.org/ka-youtube-converted/6XqsLv0QDUE.mp4/6XqsLv0QDUE.mp4230yes https://puzzicle.org/math/ap-calculus-ab/v/area-between-two-functions-of-yhttps://puzzicle.org/thumbs/x8dcff5d39bb68119.jpgCalculating horizontal area between curvesWe can use a definite integral in terms of 𝘺 to find the horizontal area between curves of two functions of 𝘺.https://cdn.kastatic.org/ka-youtube-converted/B5441_DREY0.mp4/B5441_DREY0.mp4483yes https://puzzicle.org/math/ap-calculus-ab/v/calculating-areas-between-curves-with-calculatorhttps://puzzicle.org/thumbs/xfe150f653277effb.jpgCalculating areas between curves with calculatorCalculate the area between curves that intersect in 3 points using a sum of two or more definite integrals.https://cdn.kastatic.org/ka-youtube-converted/OzTI5d5VKkM.mp4/OzTI5d5VKkM.mp4251yes https://puzzicle.org/math/ap-calculus-ab/v/volume-with-cross-sections-introhttps://puzzicle.org/thumbs/xa76e90a6f6f16638.jpgIntroducing volume with cross sectionsUsing definite integration to find volume of a solid whose base is given as a region between function and whose cross sections are squares.https://cdn.kastatic.org/ka-youtube-converted/4vLy5VoUcQE.mp4/4vLy5VoUcQE.mp4330yes https://puzzicle.org/math/ap-calculus-ab/v/determining-integral-for-volume-with-rectangular-cross-sectionshttps://puzzicle.org/thumbs/x3f3ce37d6316a884.jpgDetermining integral for volume with rectangular cross sectionsDetermine the integral used to calculate the volume of a solid with a known base and a cross section that is a rectangle.https://cdn.kastatic.org/ka-youtube-converted/Cm-sRrHY-HM.mp4/Cm-sRrHY-HM.mp4348yes https://puzzicle.org/math/ap-calculus-ab/v/volume-from-cross-sections-without-graph-shownhttps://puzzicle.org/thumbs/x94793087a3fe91f9.jpgExploring volume with cross sections: squares and rectangles (no graph)Expressing the volume of a 3-D figure as a definite integral (where the area of cross sections are functions of x). The problem doesn't come with a graph but that doesn't mean we shouldn't sketch one!https://cdn.kastatic.org/ka-youtube-converted/_y6zKNoWiKI.mp4/_y6zKNoWiKI.mp4357yes https://puzzicle.org/math/ap-calculus-ab/v/volume-with-cross-sections-perpendicular-to-y-axishttps://puzzicle.org/thumbs/x9f023c02bceb6cae.jpgExploring volume with cross sections perpendicular to y-axisWorked example expressing the volume of a figure based on cross sections perpendicular to the y-axis as a definite integral (integrating with respect to y).https://cdn.kastatic.org/ka-youtube-converted/aYT0Gm-v5T8.mp4/aYT0Gm-v5T8.mp4265yes https://puzzicle.org/math/ap-calculus-ab/v/volume-solid-semicircle-cross-sectionhttps://puzzicle.org/thumbs/x50d51a99.jpgExploring volume with cross sections: semicircleThe volume of a solid with semi-circular cross sections and a triangular base.https://cdn.kastatic.org/ka-youtube-converted/tf4C8x8e7HQ.mp4/tf4C8x8e7HQ.mp4497yes https://puzzicle.org/math/ap-calculus-ab/v/volume-triangular-cross-sectionhttps://puzzicle.org/thumbs/x2605e92e.jpgExploring volume with cross sections: triangleThis time, the cross section of our solid is given as the area between two curves.https://cdn.kastatic.org/ka-youtube-converted/S-agS4YaxxU.mp4/S-agS4YaxxU.mp4514yes https://puzzicle.org/math/ap-calculus-ab/v/disk-method-around-x-axishttps://puzzicle.org/thumbs/1276987869.jpgUsing the disc method around x-axisFinding the solid of revolution (constructed by revolving around the x-axis) using the disc method.https://cdn.kastatic.org/ka-youtube-converted/btGaOTXxXs8.mp4/btGaOTXxXs8.mp4541yes https://puzzicle.org/math/ap-calculus-ab/v/generalizing-disc-method-around-x-axishttps://puzzicle.org/thumbs/1276878010.jpgGeneralizing the disc method around x-axisGeneralizing what we did in the last video for f(x) to get the "formula" for using the disc method around the x-axishttps://cdn.kastatic.org/ka-youtube-converted/vmS3cTaxKdA.mp4/vmS3cTaxKdA.mp4155yes https://puzzicle.org/math/ap-calculus-ab/v/disc-method-around-y-axishttps://puzzicle.org/thumbs/1277026678.jpgUsing the disc method around y-axisFinding the volume of a figure that is rotated around the y-axis using the disc methodhttps://cdn.kastatic.org/ka-youtube-converted/43AS7bPUORc.mp4/43AS7bPUORc.mp4453yes https://puzzicle.org/math/ap-calculus-ab/v/disc-method-rotation-around-horizontal-linehttps://puzzicle.org/thumbs/1276975478.jpgUsing the disc method: rotation around horizontal lineSolid of revolution constructing by rotating around line that is not an axishttps://cdn.kastatic.org/ka-youtube-converted/XdzcU5JbVcA.mp4/XdzcU5JbVcA.mp4594yes https://puzzicle.org/math/ap-calculus-ab/v/disc-method-rotating-around-vertical-linehttps://puzzicle.org/thumbs/1276946626.jpgUsing the disc method: rotation around vertical lineVolume of solid created by rotating around vertical line that is not the y-axis using the disc method.https://cdn.kastatic.org/ka-youtube-converted/jxf7XqvZWWg.mp4/jxf7XqvZWWg.mp4291yes https://puzzicle.org/math/ap-calculus-ab/v/calculating-integral-disc-method-around-vertical-linehttps://puzzicle.org/thumbs/1277077144.jpgCalculating integral: disc around vertical lineCalculate the integral from the last video.https://cdn.kastatic.org/ka-youtube-converted/eLdhifUUMmk.mp4/eLdhifUUMmk.mp4339yes https://puzzicle.org/math/ap-calculus-ab/v/disc-method-washer-method-for-rotation-around-x-axishttps://puzzicle.org/thumbs/1277082263.jpgFinding the solid of revolution between two functions (leading up to the washer method)Finding the volume of a solid of revolution that is defined between two functionshttps://cdn.kastatic.org/ka-youtube-converted/vhMl755vR5Q.mp4/vhMl755vR5Q.mp4547yes https://puzzicle.org/math/ap-calculus-ab/v/generalizing-the-washer-methodhttps://puzzicle.org/thumbs/1276911918.jpgGeneralizing the washer methodLooking at the example from the last video in a more generalized wayhttps://cdn.kastatic.org/ka-youtube-converted/Thvc2s9aUP4.mp4/Thvc2s9aUP4.mp4511yes https://puzzicle.org/math/ap-calculus-ab/v/washer-method-rotating-around-non-axishttps://puzzicle.org/thumbs/1277007629.jpgUsing the washer method: rotating around horizontal line (not x-axis), part 1Washer method when rotating around a horizontal line that is not the x-axishttps://cdn.kastatic.org/ka-youtube-converted/OFNGpKGg9IQ.mp4/OFNGpKGg9IQ.mp4398yes https://puzzicle.org/math/ap-calculus-ab/v/part-2-of-washer-for-not-axis-rotationhttps://puzzicle.org/thumbs/1276843891.jpgUsing the washer method: rotating around horizontal line (not x-axis), part 2Doing some hairy algebra and arithmetic to evaluate the definite integral from the last videohttps://cdn.kastatic.org/ka-youtube-converted/LKzpw_HUKNQ.mp4/LKzpw_HUKNQ.mp4494yes https://puzzicle.org/math/ap-calculus-ab/v/washer-or-ring-method-for-vertical-line-rotationhttps://puzzicle.org/thumbs/1278890062.jpgUsing the washer method: rotating around vertical line (not y-axis), part 1Setting up the definite integral for the volume of a solid of revolution around a vertical line using the washer or ring method.https://cdn.kastatic.org/ka-youtube-converted/WAPZihVUmzE.mp4/WAPZihVUmzE.mp4470yes https://puzzicle.org/math/ap-calculus-ab/v/evaluating-integral-for-washer-method-around-vertical-linehttps://puzzicle.org/thumbs/1278699793.jpgUsing the washer method: rotating around vertical line (not y-axis), part 2Evaluating the integral set up in the last video using the washer method.https://cdn.kastatic.org/ka-youtube-converted/i-Rb4_n929k.mp4/i-Rb4_n929k.mp4335yes https://puzzicle.org/math/ap-calculus-ab/v/2011-calculus-ab-free-response-1-parts-b-c-dhttps://puzzicle.org/thumbs/237109442.jpgUsing integrals to analyze particle motion: 2011 Calculus AB Free Response #1 (b, c, & d)Integration to find average value of a function. Using a graphing calculator to calculate definite integrals.https://cdn.kastatic.org/ka-youtube-converted/OSipkz8vDjE.mp4/OSipkz8vDjE.mp4484yes https://puzzicle.org/math/ap-calculus-ab/v/2011-calculus-ab-free-response-2-c-dhttps://puzzicle.org/thumbs/235193472.jpgApplying second fundamental theorem of calculus: 2011 Calculus AB free response #2 (c & d)Second fundamental theorem of calculus applicationhttps://cdn.kastatic.org/ka-youtube-converted/2axaRJQkfVk.mp4/2axaRJQkfVk.mp4447yes https://puzzicle.org/math/ap-calculus-ab/v/calculating-area-between-curves-ap-calculus-ab-free-response-2011-3bhttps://puzzicle.org/thumbs/x1f2813e75d8391c4.jpgCalculating area between curves AP Calculus AB free response 2011 3bCalculating the area between curves.https://cdn.kastatic.org/ka-youtube-converted/0QsFXGAqx9c.mp4/0QsFXGAqx9c.mp4315yes https://puzzicle.org/math/ap-calculus-ab/v/2011-calculus-ab-free-response-3-chttps://puzzicle.org/thumbs/235193473.jpgWriting integral for solid of revolution: 2011 Calculus AB free response #3 (c)Disk method to find the volume of a solid generated by rotation of a functionhttps://cdn.kastatic.org/ka-youtube-converted/v-H_7o7EMoU.mp4/v-H_7o7EMoU.mp4399yes https://puzzicle.org/math/ap-calculus-ab/v/2011-calculus-ab-free-response-6chttps://puzzicle.org/thumbs/244051676.jpgFinding average value: 2011 Calculus AB free response #6cAverage value of a piecewise-defined function on an intervalhttps://cdn.kastatic.org/ka-youtube-converted/ucghWyc2_qs.mp4/ucghWyc2_qs.mp4449yes https://puzzicle.org/math/ap-calculus-ab/v/2015-ap-calculus-ab-bc-1ahttps://puzzicle.org/thumbs/xb6e61bac6ad2b410.jpgUsing accumulation functions 2015 AP Calculus AB BC 1aUsing accumulation functions to estimate the amount of water in a pipe.https://cdn.kastatic.org/ka-youtube-converted/DQsYrGDDs5I.mp4/DQsYrGDDs5I.mp4285yes https://puzzicle.org/math/ap-calculus-ab/v/2015-ap-calculus-abbc-1dhttps://puzzicle.org/thumbs/x44d517e4.jpgWriting integral to solve problem in context: 2015 AP Calculus AB/BC 1dWhen the pipe overflowshttps://cdn.kastatic.org/ka-youtube-converted/9Fqy_stBdFk.mp4/9Fqy_stBdFk.mp4199yes https://puzzicle.org/math/ap-calculus-ab/v/2015-ap-calculus-ap-2ahttps://puzzicle.org/thumbs/xe129d8d1.jpgFinding area between curves: 2015 AP Calculus AB 2aAreas between curveshttps://cdn.kastatic.org/ka-youtube-converted/5QY8yvZViO0.mp4/5QY8yvZViO0.mp4295yes https://puzzicle.org/math/ap-calculus-ab/v/2015-ap-calculus-ap-2bhttps://puzzicle.org/thumbs/x59861801.jpgFinding volume of solid with known cross section: 2015 AP Calculus AB 2bVolume of solid with known cross sectionhttps://cdn.kastatic.org/ka-youtube-converted/xmVBv7vk3Gg.mp4/xmVBv7vk3Gg.mp4387yes https://puzzicle.org/math/ap-calculus-ab/v/finding-average-velocity-2015-ap-calculus-ab-bc-3dhttps://puzzicle.org/thumbs/x6cb1a7b4c4bcfbf9.jpgFinding average velocity 2015 AP Calculus AB BC 3dFinding the average velocity over a given time interval in context.https://cdn.kastatic.org/ka-youtube-converted/31pieUEqLWw.mp4/31pieUEqLWw.mp4200yes https://puzzicle.org/math/ap-calculus-ab/v/2015-ap-calculus-ab-5dhttps://puzzicle.org/thumbs/x5c1c07cc.jpgUsing function that involves an integral: 2015 AP Calculus AB 5dConstructing expression for f with integral.https://cdn.kastatic.org/ka-youtube-converted/a601bUPEYUs.mp4/a601bUPEYUs.mp4329yes https://puzzicle.org/math/ap-calculus-ab/v/estimating-average-value-with-trapezoidal-sum-ap-calculus-ab-free-response-2011-https://puzzicle.org/thumbs/x54712dc49943244a.jpgEstimating average value with trapezoidal sum AP Calculus AB free response 2011 2bEstimating the average value of a function using a trapezoidal sum.https://cdn.kastatic.org/ka-youtube-converted/qS7P8pOgYTI.mp4/qS7P8pOgYTI.mp4338yes https://puzzicle.org/math/ap-calculus-ab/v/differentiating-implicitly-and-determining-concavity-ap-calculus-ab-bc-2015-4bhttps://puzzicle.org/thumbs/xf2606ff017f925c1.jpgDifferentiating implicitly and determining concavity AP Calculus AB BC 2015 4bDifferentiating implicitly and determining concavity of a given differential equationhttps://cdn.kastatic.org/ka-youtube-converted/gPEgwN4ZHCA.mp4/gPEgwN4ZHCA.mp4494yes https://puzzicle.org/math/ap-calculus-bc/v/deriving-integration-by-parts-formulahttps://puzzicle.org/thumbs/1329232699.jpgIntegration by parts introThis video explains integration by parts, a technique for finding antiderivatives. It starts with the product rule for derivatives, then takes the antiderivative of both sides. By rearranging the equation, we get the formula for integration by parts. It helps simplify complex antiderivatives.https://cdn.kastatic.org/ka-youtube-converted/dh__n9FVKA0.mp4/dh__n9FVKA0.mp4232yes https://puzzicle.org/math/ap-calculus-bc/v/antiderivative-of-xcosx-using-integration-by-partshttps://puzzicle.org/thumbs/1329293927.jpgIntegration by parts: ∫x⋅cos(x)dxThis video shows how to find the antiderivative of x*cos(x) using integration by parts. It assigns f(x)=x and g'(x)=cos(x), making f'(x)=1 and g(x)=sin(x). The formula becomes x*sin(x) - ∫sin(x)dx, which simplifies to x*sin(x) + cos(x) + C.https://cdn.kastatic.org/ka-youtube-converted/bZ8YAHDTFJ8.mp4/bZ8YAHDTFJ8.mp4232yes https://puzzicle.org/math/ap-calculus-bc/v/integral-of-ln-xhttps://puzzicle.org/thumbs/1377705570.jpgIntegration by parts: ∫ln(x)dxThis video shows how to find the antiderivative of the natural log of x using integration by parts. We rewrite the integral as ln(x) times 1dx, then choose f(x) = ln(x) and g'(x) = 1. The antiderivative is xln(x) - x + C.https://cdn.kastatic.org/ka-youtube-converted/iw5eLJV0Sj4.mp4/iw5eLJV0Sj4.mp4230yes https://puzzicle.org/math/ap-calculus-bc/v/integration-by-parts-twice-for-antiderivative-of-x-2-e-xhttps://puzzicle.org/thumbs/1329248194.jpgIntegration by parts: ∫x²⋅𝑒ˣdxIntegration by parts helps find antiderivatives of products of functions. We assign f(x) and g'(x) to parts of the product. Then, we find f'(x) and g(x). The formula is ∫f(x)g'(x)dx = f(x)g(x) - ∫f'(x)g(x)dx. Sometimes, we use integration by parts twice!https://cdn.kastatic.org/ka-youtube-converted/n-iEqLhGfd4.mp4/n-iEqLhGfd4.mp4406yes https://puzzicle.org/math/ap-calculus-bc/v/integration-by-parts-of-e-x-cos-x-1https://puzzicle.org/thumbs/1369495388.jpgIntegration by parts: ∫𝑒ˣ⋅cos(x)dxIn the video, we learn about integration by parts to find the antiderivative of e^x * cos(x). We assign f(x) = e^x and g'(x) = cos(x), then apply integration by parts twice. The result is the antiderivative e^x * sin(x) + e^x * cos(x) / 2 + C.https://cdn.kastatic.org/ka-youtube-converted/LJqNdG6Y2cM.mp4/LJqNdG6Y2cM.mp4437yes https://puzzicle.org/math/ap-calculus-bc/v/integration-by-parts-of-definite-integralhttps://puzzicle.org/thumbs/x72ccc1ab3a4944c6.jpgIntegration by parts: definite integralsWhen finding a definite integral using integration by parts, we should first find the antiderivative (as we do with indefinite integrals), but then we should also evaluate the antiderivative at the boundaries and subtract.https://cdn.kastatic.org/ka-youtube-converted/vS3bpx5ubBI.mp4/vS3bpx5ubBI.mp4343yes https://puzzicle.org/math/ap-calculus-bc/v/integration-with-partial-fractionshttps://puzzicle.org/thumbs/xcf3c678b37108e75.jpgIntegration with partial fractionsFinding the integral of a rational function using linear partial fraction decomposition.https://cdn.kastatic.org/ka-youtube-converted/Mt92Ozs7aJg.mp4/Mt92Ozs7aJg.mp4439yes https://puzzicle.org/math/ap-calculus-bc/v/introduction-to-improper-integralshttps://puzzicle.org/thumbs/1377828561.jpgIntroduction to improper integralsImproper integrals are definite integrals where one or both of the ​boundaries is at infinity, or where the integrand has a vertical asymptote in the interval of integration. As crazy as it may sound, we can actually calculate some improper integrals using some clever methods that involve limits.https://cdn.kastatic.org/ka-youtube-converted/qv7DM5Ph0vU.mp4/qv7DM5Ph0vU.mp4232yes https://puzzicle.org/math/ap-calculus-bc/v/divergent-improper-integralhttps://puzzicle.org/thumbs/1378156106.jpgDivergent improper integralSometimes the value of an infinite integral is, well, infinite.https://cdn.kastatic.org/ka-youtube-converted/9VqL6POWIoc.mp4/9VqL6POWIoc.mp4162yes https://puzzicle.org/math/ap-calculus-bc/v/eulers-methodhttps://puzzicle.org/thumbs/x5c5705d2.jpgEuler's methodThis video introduces Euler's Method, a numerical way to approximate solutions to differential equations when analytical methods don't work. Using a table with x, y, and dy/dx values, we start with an initial condition and increment x by a chosen delta x to estimate y values: smaller delta x gives better approximations.https://cdn.kastatic.org/ka-youtube-converted/q87L9R9v274.mp4/q87L9R9v274.mp4608yes https://puzzicle.org/math/ap-calculus-bc/v/example-eulers-method-exercisehttps://puzzicle.org/thumbs/x7ac26ce4.jpgWorked example: Euler's methodFinding the initial condition based on the result of approximating with Euler's method.https://cdn.kastatic.org/ka-youtube-converted/_-a-yNubpk8.mp4/_-a-yNubpk8.mp4275yes https://puzzicle.org/math/ap-calculus-bc/v/modeling-population-with-differential-equationshttps://puzzicle.org/thumbs/xff25b528.jpgGrowth models: introductionPopulation should grow proportionally to its size, but it can't keep growing forever! Learn more about this problem, posed by Malthus, and embark on a journey towards its mathematical solution.https://cdn.kastatic.org/ka-youtube-converted/_JpS8k1a9yE.mp4/_JpS8k1a9yE.mp4577yes https://puzzicle.org/math/ap-calculus-bc/v/logistic-differential-equation-intuitionhttps://puzzicle.org/thumbs/xb9599623.jpgThe logistic growth modelThe logistic differential equation dN/dt=rN(1-N/K) describes the situation where a population grows proportionally to its size, but stops growing when it reaches the size of K.https://cdn.kastatic.org/ka-youtube-converted/oiDvNs15tkE.mp4/oiDvNs15tkE.mp4649yes https://puzzicle.org/math/ap-calculus-bc/v/analyzing-logistic-differential-equationshttps://puzzicle.org/thumbs/xe60ace5dccb9437f.jpgWorked example: Logistic model word problemFinding the carrying capacity of a population that grows logistically. Also finding the population's size when it's growing the fastest.https://cdn.kastatic.org/ka-youtube-converted/kDZlMFkKhWA.mp4/kDZlMFkKhWA.mp4527yes https://puzzicle.org/math/ap-calculus-bc/v/solving-logistic-differential-equation-part-1https://puzzicle.org/thumbs/x152ee0ed.jpgLogistic equations (Part 1)Finding the general solution of the general logistic equation dN/dt=rN(1-N/K). The solution is kind of hairy, but it's worth bearing with us!https://cdn.kastatic.org/ka-youtube-converted/j_Taf2Tgggo.mp4/j_Taf2Tgggo.mp4818yes https://puzzicle.org/math/ap-calculus-bc/v/solving-logistic-differential-equation-part2https://puzzicle.org/thumbs/xd9cb2b32.jpgLogistic equations (Part 2)Finding the general solution of the general logistic equation dN/dt=rN(1-N/K). The solution is kind of hairy, but it's worth bearing with us!https://cdn.kastatic.org/ka-youtube-converted/NU1v-8VRirU.mp4/NU1v-8VRirU.mp4598yes https://puzzicle.org/math/ap-calculus-bc/v/arc-length-formulahttps://puzzicle.org/thumbs/x28571cbc.jpgArc length introIn this video, learn how to find the arc length of a curve using definite integrals. To find arc length, take the derivative of the function, square it, add 1, find the square root, and integrate from a to b.https://cdn.kastatic.org/ka-youtube-converted/8Y-snjheI9M.mp4/8Y-snjheI9M.mp4381yes https://puzzicle.org/math/ap-calculus-bc/v/arc-length-examplehttps://puzzicle.org/thumbs/x35f6ea19.jpgWorked example: arc lengthThis video shows how to find the arc length of a curve using the arc length formula. It uses the function y = x^(3/2) and finds the arc length from x = 0 to x = 32/9. The main steps are: find the derivative, square it, add 1, take the square root, and integrate.https://cdn.kastatic.org/ka-youtube-converted/OhISsmqv4_8.mp4/OhISsmqv4_8.mp4423yes https://puzzicle.org/math/ap-calculus-bc/v/parametric-function-differentiationhttps://puzzicle.org/thumbs/x037d4a44.jpgParametric equations differentiationSal finds the derivative of the function defined by the parametric equations x=sin(1+3t) and y=2t³, and evaluates it at t=-⅓.https://cdn.kastatic.org/ka-youtube-converted/AXasQLqazWU.mp4/AXasQLqazWU.mp4335yes https://puzzicle.org/math/ap-calculus-bc/v/second-derivative-of-parametric-functionshttps://puzzicle.org/thumbs/x32dbbbf7.jpgSecond derivatives (parametric functions)Sal finds the second derivative of the function defined by the parametric equations x=3e²ᵗ and y=3³ᵗ-1.https://cdn.kastatic.org/ka-youtube-converted/iHNa1fhV7IA.mp4/iHNa1fhV7IA.mp4390yes https://puzzicle.org/math/ap-calculus-bc/v/parametric-curve-arc-lengthhttps://puzzicle.org/thumbs/xf846ca3307ce3b26.jpgParametric curve arc lengthConceptual introduction to the formula for arc length of a parametric curve.https://cdn.kastatic.org/ka-youtube-converted/DXHM7LYGcbI.mp4/DXHM7LYGcbI.mp4365yes https://puzzicle.org/math/ap-calculus-bc/v/parametric-arc-length-examplehttps://puzzicle.org/thumbs/x1cd2aab57f889617.jpgWorked example: Parametric arc lengthFinding the length of the parametric curve 𝘹=cos(𝑡), 𝘺=sin(𝑡) from 𝑡=0 to 𝑡=π/2, using the formula for arc length of a parametric curve.https://cdn.kastatic.org/ka-youtube-converted/F3KyoI26fKI.mp4/F3KyoI26fKI.mp4288yes https://puzzicle.org/math/ap-calculus-bc/v/position-vector-valued-functionshttps://puzzicle.org/thumbs/24316.jpgVector-valued functions introUsing a position vector valued function to describe a curve or pathhttps://cdn.kastatic.org/ka-youtube-converted/7fYDCUIvZnM.mp4/7fYDCUIvZnM.mp4425yes https://puzzicle.org/math/ap-calculus-bc/v/derivative-of-a-position-vector-valued-functionhttps://puzzicle.org/thumbs/24318.jpgVector-valued functions differentiationVisualizing the derivative of a position vector valued functionhttps://cdn.kastatic.org/ka-youtube-converted/E9Q_Lc0g1xE.mp4/E9Q_Lc0g1xE.mp4885yes https://puzzicle.org/math/ap-calculus-bc/v/second-derivatives-of-vector-valued-functionshttps://puzzicle.org/thumbs/xc0b3f860.jpgSecond derivatives (vector-valued functions)Sal finds the first and the second derivatives of the vector-valued function h(t)=(-t⁵-6,4t⁴+2t+1).https://cdn.kastatic.org/ka-youtube-converted/-vv63FS__bY.mp4/-vv63FS__bY.mp4222yes https://puzzicle.org/math/ap-calculus-bc/v/planar-motion-example-acceleration-vectorhttps://puzzicle.org/thumbs/x9cae018f.jpgPlanar motion example: acceleration vectorThe position of a particle moving in the xy-plane is given by the position vector (-3t³+4t²,t³+2). Sal analyzes it to find the acceleration vector of the particle at time t=3.https://cdn.kastatic.org/ka-youtube-converted/HFdeHX_wA4k.mp4/HFdeHX_wA4k.mp4275yes https://puzzicle.org/math/ap-calculus-bc/v/motion-along-a-curve-example-1https://puzzicle.org/thumbs/xc751f553.jpgMotion along a curve: finding rate of changeGiven that a particle moves along the implicit curve x²y²=16 and given the rate of change of x with respect to t at some point, Sal finds the particle's rate of change with respect to y using implicit differentiation.https://cdn.kastatic.org/ka-youtube-converted/Sv3Y0F0ASZ0.mp4/Sv3Y0F0ASZ0.mp4352yes https://puzzicle.org/math/ap-calculus-bc/v/motion-along-a-curve-example-2https://puzzicle.org/thumbs/x123d8b65.jpgMotion along a curve: finding velocity magnitudeGiven that a particle moves along the implicit curve xy=16 and given its rate of change with respect to y at some point, Sal finds the magnitude of the particle's velocity vector using implicit differentiation.https://cdn.kastatic.org/ka-youtube-converted/nC5UgHDjbcg.mp4/nC5UgHDjbcg.mp4382yes https://puzzicle.org/math/ap-calculus-bc/v/planar-motion-with-integralshttps://puzzicle.org/thumbs/x2729df269f69de0e.jpgPlanar motion (with integrals)To analyze planar motion where the rate vector is given, we need to find the displacement in each direction separately. Then we will either use that to find the new position, or to find the magnitude of the displacement using the Pythagorean theorem.https://cdn.kastatic.org/ka-youtube-converted/EZbgMfuh9C8.mp4/EZbgMfuh9C8.mp4428yes https://puzzicle.org/math/ap-calculus-bc/v/derivatives-in-polar-coordinateshttps://puzzicle.org/thumbs/x6005fd8e4a8e9fcd.jpgPolar functions derivativesFinding derivatives of 𝑟, 𝘹, and 𝘺 of a function given in polar coordinates.https://cdn.kastatic.org/ka-youtube-converted/8mS9eDHf0fQ.mp4/8mS9eDHf0fQ.mp4565yes https://puzzicle.org/math/ap-calculus-bc/v/differentiating-polar-functions-examplehttps://puzzicle.org/thumbs/x58a369d140415dc6.jpgWorked example: differentiating polar functionsAn AP Calculus sample item where we find the rate of change of 𝘹 with respect to θ.https://cdn.kastatic.org/ka-youtube-converted/ConsnUZGfmw.mp4/ConsnUZGfmw.mp4357yes https://puzzicle.org/math/ap-calculus-bc/v/formula-area-polar-graphhttps://puzzicle.org/thumbs/xb44c8c08.jpgArea bounded by polar curvesDevelop intuition for the area enclosed by polar graph formulahttps://cdn.kastatic.org/ka-youtube-converted/qVn_Lfec-Ac.mp4/qVn_Lfec-Ac.mp4391yes https://puzzicle.org/math/ap-calculus-bc/v/cardioid-areahttps://puzzicle.org/thumbs/x8a47b52d.jpgWorked example: Area enclosed by cardioidArea enclosed by cardioidhttps://cdn.kastatic.org/ka-youtube-converted/-cR6FzM1zNE.mp4/-cR6FzM1zNE.mp4386yes https://puzzicle.org/math/ap-calculus-bc/v/area-defined-by-two-polar-graphshttps://puzzicle.org/thumbs/xd198d3cc.jpgWorked example: Area between two polar graphsMore practice with the area encloised by polar functions, this time with two graphs.https://cdn.kastatic.org/ka-youtube-converted/KwR_ysrv3sg.mp4/KwR_ysrv3sg.mp4369yes https://puzzicle.org/math/ap-calculus-bc/v/calculator-integralhttps://puzzicle.org/thumbs/xb21bd894.jpgEvaluating definite integral with calculatorThis video shows how to find the overlapping area between two circles using definite integrals and a graphing calculator. It demonstrates entering the integral function, specifying the variable, and setting the bounds of integration. The result matches the hand-calculated answer.https://cdn.kastatic.org/ka-youtube-converted/4tWFdI7DkJ4.mp4/4tWFdI7DkJ4.mp4179yes https://puzzicle.org/math/ap-calculus-bc/v/convergent-and-divergent-sequenceshttps://puzzicle.org/thumbs/1383707269.jpgConvergent and divergent sequencesThis video talks about a sequence that alternates between positive and negative values. It shows how to find the limit of the sequence as n approaches infinity. If the limit exists, the sequence converges; if not, it diverges.https://cdn.kastatic.org/ka-youtube-converted/lfZGtjSWcQs.mp4/lfZGtjSWcQs.mp4299yes https://puzzicle.org/math/ap-calculus-bc/v/identifying-sequence-convergence-divergencehttps://puzzicle.org/thumbs/x1a7b5891.jpgWorked example: sequence convergence/divergenceHow can we tell if a sequence converges or diverges? See Sal in action, determining the convergence/divergence of several sequences.https://cdn.kastatic.org/ka-youtube-converted/muqyereWEh4.mp4/muqyereWEh4.mp4233yes https://puzzicle.org/math/ap-calculus-bc/v/infinite-series-as-limit-of-partial-sumshttps://puzzicle.org/thumbs/xb8d9548c.jpgInfinite series as limit of partial sumsInfinite series are defined as the limit of the infinite sequence of partial sums. Since we already know how to work with limits of sequences, this definition is really useful.https://cdn.kastatic.org/ka-youtube-converted/bh1UrCGWz1k.mp4/bh1UrCGWz1k.mp4288yes https://puzzicle.org/math/ap-calculus-bc/v/divergence-testhttps://puzzicle.org/thumbs/x1404a71d.jpgnth term divergence testThe video introduces the divergence test, a tool to check if a series diverges. If the limit of a sequence aₙ as n approaches infinity isn't zero, the series will diverge. The test can't prove convergence, but it helps identify divergence.https://cdn.kastatic.org/ka-youtube-converted/1yInzfzDDKY.mp4/1yInzfzDDKY.mp4476yes https://puzzicle.org/math/ap-calculus-bc/v/integral-test-intuitionhttps://puzzicle.org/thumbs/x05f544f8.jpgIntegral testThe integral test helps us determine a series convergence by comparing it to an improper integral, which is something we already know how to find. Learn how it works in this video.https://cdn.kastatic.org/ka-youtube-converted/u1UKIljUWuc.mp4/u1UKIljUWuc.mp4526yes https://puzzicle.org/math/ap-calculus-bc/v/integral-test-divergencehttps://puzzicle.org/thumbs/xe42fe616.jpgWorked example: Integral testSee how the integral test is put to use in determining whether a sequence converges or diverges.https://cdn.kastatic.org/ka-youtube-converted/xRyXz_UZ14Q.mp4/xRyXz_UZ14Q.mp4545yes https://puzzicle.org/math/ap-calculus-bc/v/harmonic-and-p-serieshttps://puzzicle.org/thumbs/xa3a79760191bd066.jpgHarmonic series and 𝑝-series𝑝-series is a family of series where the terms are of the form 1/(nᵖ) for some value of 𝑝. The Harmonic series is the special case where 𝑝=1. These series are very interesting and useful.https://cdn.kastatic.org/ka-youtube-converted/m6ByHObCs6Y.mp4/m6ByHObCs6Y.mp4310yes https://puzzicle.org/math/ap-calculus-bc/v/p-series-convergencehttps://puzzicle.org/thumbs/x71d725aa03d31325.jpgWorked example: p-seriesIn this video, we'll explore p-series, which are infinite sums written as Σ(1/xᵖ) with a positive p value. Through engaging examples, we'll learn how to identify if a p-series converges (has a finite sum) or diverges (doesn't have a finite sum) by examining the value of p. Get ready for a fun and informative journey into the world of p-series!https://cdn.kastatic.org/ka-youtube-converted/vspcKkcLuuA.mp4/vspcKkcLuuA.mp4175yes https://puzzicle.org/math/ap-calculus-bc/v/comparison-testhttps://puzzicle.org/thumbs/xaf5004b1.jpgDirect comparison testThe comparison test helps us decide if a series converges (sums to a finite value) or diverges (goes to infinity). If a smaller series converges, the larger one does too. If a smaller series diverges, the larger one also diverges. This test works for series with non-negative terms.https://cdn.kastatic.org/ka-youtube-converted/0tXxFPHzFFI.mp4/0tXxFPHzFFI.mp4270yes https://puzzicle.org/math/ap-calculus-bc/v/comparison-test-convergencehttps://puzzicle.org/thumbs/x8d385a4e.jpgWorked example: direct comparison testUsing the direct comparison test to determine that the infinite sum of 1/(2ⁿ+n) converges by comparing it to the infinite sum 1/2ⁿ.https://cdn.kastatic.org/ka-youtube-converted/XoBlfbrdBpQ.mp4/XoBlfbrdBpQ.mp4295yes https://puzzicle.org/math/ap-calculus-bc/v/limit-comparison-test-corhttps://puzzicle.org/thumbs/x5b63d7f9.jpgLimit comparison testThe video explains the Comparison Test and Limit Comparison Test for infinite series. The Comparison Test checks if a series converges by comparing it to another series with known convergence. The Limit Comparison Test compares the behavior of two series as n approaches infinity. If their limit is positive and finite, both series either converge or diverge together.https://cdn.kastatic.org/ka-youtube-converted/AtbZZiSLemQ.mp4/AtbZZiSLemQ.mp4413yes https://puzzicle.org/math/ap-calculus-bc/v/limit-comparison-test-examplehttps://puzzicle.org/thumbs/xa4936ff42719938e.jpgWorked example: limit comparison testTo use the limit comparison test for a series S₁, we need to find another series S₂ that is similar in structure (so the infinite limit of S₁/S₂ is finite) and whose convergence is already determined. See a worked example of using the test in this video.https://cdn.kastatic.org/ka-youtube-converted/0dWF35WPKfM.mp4/0dWF35WPKfM.mp4290yes https://puzzicle.org/math/ap-calculus-bc/v/harmonic-series-divergenthttps://puzzicle.org/thumbs/x0039c7d4.jpgProof: harmonic series divergesShowing that the harmonic series 1 + ½ + ⅓ + ¼ + ... actually diverges, using the direct comparison test. This proof is famous for its clever use of algebraic manipulation!https://cdn.kastatic.org/ka-youtube-converted/4yyLfrsSXQQ.mp4/4yyLfrsSXQQ.mp4393yes https://puzzicle.org/math/ap-calculus-bc/v/alternating-series-testhttps://puzzicle.org/thumbs/x30ab4c93.jpgAlternating series testWhen a series alternates (plus, minus, plus, minus,...) there's a fairly simple way to determine whether it converges or diverges: see if the terms of the series approach 0.https://cdn.kastatic.org/ka-youtube-converted/UsMom3yNsX8.mp4/UsMom3yNsX8.mp4348yes https://puzzicle.org/math/ap-calculus-bc/v/worked-example-alternating-serieshttps://puzzicle.org/thumbs/xfd653f28e36ff338.jpgWorked example: alternating seriesExample using the alternating series test to determine which values of a variable will make the series converge.https://cdn.kastatic.org/ka-youtube-converted/DPxf4-ta4PM.mp4/DPxf4-ta4PM.mp4177yes https://puzzicle.org/math/ap-calculus-bc/v/ratio-test-convergencehttps://puzzicle.org/thumbs/x18608b0a.jpgRatio testThe ratio test is a most useful test for series convergence. It caries over intuition from geometric series to more general series. Learn more about it here.https://cdn.kastatic.org/ka-youtube-converted/av947KCWf2U.mp4/av947KCWf2U.mp4539yes https://puzzicle.org/math/ap-calculus-bc/v/conditional-and-absolute-convergencehttps://puzzicle.org/thumbs/xf35d880b.jpgConditional & absolute convergenceThis video talks about two types of convergence: conditional and absolute. A series converges conditionally if it converges, but taking the absolute value of its terms makes it diverge. A series converges absolutely if it converges even when taking the absolute value of its terms.https://cdn.kastatic.org/ka-youtube-converted/ORTI4kk1okM.mp4/ORTI4kk1okM.mp4301yes https://puzzicle.org/math/ap-calculus-bc/v/alternating-series-error-estimationhttps://puzzicle.org/thumbs/x37b67eba.jpgAlternating series remainderThis video shows how to estimate the sum of an alternating series using partial sums. It demonstrates that the error (remainder) is between 0 and the first term not included. This helps find a range for the actual sum.https://cdn.kastatic.org/ka-youtube-converted/gs9O255gdEY.mp4/gs9O255gdEY.mp4558yes https://puzzicle.org/math/ap-calculus-bc/v/alternating-series-remainderhttps://puzzicle.org/thumbs/xef940f21.jpgWorked example: alternating series remainderUsing the alternating series remainder to approximating the sum of an alternating series to a given error bound.https://cdn.kastatic.org/ka-youtube-converted/vTYxY4IpmF8.mp4/vTYxY4IpmF8.mp4621yes https://puzzicle.org/math/ap-calculus-bc/v/maclaurin-and-taylor-series-intuitionhttps://puzzicle.org/thumbs/89857202.jpgTaylor & Maclaurin polynomials intro (part 1)This video introduces the Maclaurin series, a way to approximate functions using polynomials. It shows how to create better approximations by adding more terms, making the function and its derivatives match at x=0. The general formula for the n-th term is the n-th derivative of the function at 0 times x^n divided by n factorial.https://cdn.kastatic.org/ka-youtube-converted/epgwuzzDHsQ.mp4/epgwuzzDHsQ.mp4779yes https://puzzicle.org/math/ap-calculus-bc/v/generalized-taylor-series-approximationhttps://puzzicle.org/thumbs/90906157.jpgTaylor & Maclaurin polynomials intro (part 2)This video explains Taylor series, a way to approximate functions around any point x=c. It starts with a constant approximation (f(c)), then adds terms for matching the slope (f'(c)(x-c)) and higher derivatives. More terms give better approximations.https://cdn.kastatic.org/ka-youtube-converted/1LxhXqD3_CE.mp4/1LxhXqD3_CE.mp4447yes https://puzzicle.org/math/ap-calculus-bc/v/maclaurin-polynomial-examplehttps://puzzicle.org/thumbs/xf501bdc865844281.jpgWorked example: Maclaurin polynomialFinding the second degree Maclaurin polynomial of 1/√(x+1).https://cdn.kastatic.org/ka-youtube-converted/tWNxdsWoAd4.mp4/tWNxdsWoAd4.mp4283yes https://puzzicle.org/math/ap-calculus-bc/v/finding-coefficient-of-maclaurin-polynomialhttps://puzzicle.org/thumbs/x4edb6df3557032e7.jpgWorked example: coefficient in Maclaurin polynomialFinding the coefficient of the x² term in a Maclaurin polynomial, given the formula for the value of any derivative at x=0.https://cdn.kastatic.org/ka-youtube-converted/RRYh4cqv6ZQ.mp4/RRYh4cqv6ZQ.mp4185yes https://puzzicle.org/math/ap-calculus-bc/v/fourth-degree-coefficient-for-taylor-polynomialhttps://puzzicle.org/thumbs/x3c942abd69be14fd.jpgWorked example: coefficient in Taylor polynomialFinding the coefficient of the term containing (x+2)⁴ in the Taylor polynomial centered at x=-2 of x⁶-x³.https://cdn.kastatic.org/ka-youtube-converted/eo9dra_yIbc.mp4/eo9dra_yIbc.mp4233yes https://puzzicle.org/math/ap-calculus-bc/v/visualizing-taylor-series-for-e-xhttps://puzzicle.org/thumbs/90906158.jpgVisualizing Taylor polynomial approximationsApproximating eˣ with a Taylor polynomial centered at x=3. In the video we find the first few terms of such a polynomial and graph it to see how close it gets to eˣ.https://cdn.kastatic.org/ka-youtube-converted/AFMXixBVP-0.mp4/AFMXixBVP-0.mp4398yes https://puzzicle.org/math/ap-calculus-bc/v/error-or-remainder-of-a-taylor-polynomial-approximationhttps://puzzicle.org/thumbs/250668427.jpgTaylor polynomial remainder (part 1)The more terms we have in a Taylor polynomial approximation of a function, the closer we get to the function. But HOW close? Let's embark on a journey to find a bound for the error of a Taylor polynomial approximation.https://cdn.kastatic.org/ka-youtube-converted/wgkRH5Uoavk.mp4/wgkRH5Uoavk.mp4687yes https://puzzicle.org/math/ap-calculus-bc/v/proof-bounding-the-error-or-remainder-of-a-taylor-polynomial-approximationhttps://puzzicle.org/thumbs/250668428.jpgTaylor polynomial remainder (part 2)The more terms we have in a Taylor polynomial approximation of a function, the closer we get to the function. But HOW close? In this video, we prove the Lagrange error bound for Taylor polynomials.https://cdn.kastatic.org/ka-youtube-converted/yUUPP70Fhpo.mp4/yUUPP70Fhpo.mp4908yes https://puzzicle.org/math/ap-calculus-bc/v/lagrange-error-bound-for-sine-functionhttps://puzzicle.org/thumbs/x7a80166ea896e1ec.jpgWorked example: estimating sin(0.4) using Lagrange error boundLagrange error bound (also called Taylor remainder theorem) can help us determine the degree of Taylor/Maclaurin polynomial to use to approximate a function to a given error bound. See how it's done when approximating the sine function.https://cdn.kastatic.org/ka-youtube-converted/2KWifFPOy5s.mp4/2KWifFPOy5s.mp4527yes https://puzzicle.org/math/ap-calculus-bc/v/lagrange-error-bound-exponential-examplehttps://puzzicle.org/thumbs/xd240c30ca59e3c2d.jpgWorked example: estimating eˣ using Lagrange error boundThis video explains how to find the least degree of a Taylor polynomial to estimate e^x with an error smaller than 0.001. It uses the LaGrange error bound and Taylor's remainder theorem to find the smallest n (degree) that satisfies the error condition.https://cdn.kastatic.org/ka-youtube-converted/IeHlXGpfaeA.mp4/IeHlXGpfaeA.mp4560yes https://puzzicle.org/math/ap-calculus-bc/v/power-series-radius-interval-convergencehttps://puzzicle.org/thumbs/xf2f4de02.jpgPower series introPower series define functions using infinite sums. A geometric series is a special case of a power series with a constant coefficient. Convergence happens when the common ratio's absolute value is less than 1. The interval of convergence is the range of x values where the series converges. The radius of convergence is half the interval.https://cdn.kastatic.org/ka-youtube-converted/DlBQcj_zQk0.mp4/DlBQcj_zQk0.mp4409yes https://puzzicle.org/math/ap-calculus-bc/v/radius-convergence-ratio-testhttps://puzzicle.org/thumbs/x1f1a1ec842581890.jpgWorked example: interval of convergenceThis video focuses on finding the interval of convergence for an infinite series using the ratio test. It shows how to calculate the limit of the ratio of consecutive terms, then determine when the series converges, diverges, or is inconclusive. Finally, it tests the inconclusive cases.https://cdn.kastatic.org/ka-youtube-converted/oXUJXsoCtDs.mp4/oXUJXsoCtDs.mp4506yes https://puzzicle.org/math/ap-calculus-bc/v/function-as-a-geometric-serieshttps://puzzicle.org/thumbs/x43a07b696ac7d30f.jpgFunction as a geometric seriesExpressions of the form a/(1-r) represent the infinite sum of a geometric series whose initial term is a and constant ratio is r, which is written as Σa(r)ⁿ. Since geometric series are a class of power series, we obtained the power series representation of a/(1-r) very quickly.https://cdn.kastatic.org/ka-youtube-converted/EXIUYib9d8M.mp4/EXIUYib9d8M.mp4241yes https://puzzicle.org/math/ap-calculus-bc/v/function-as-geometric-serieshttps://puzzicle.org/thumbs/x4776b79b.jpgGeometric series as a functionPower series of the form Σk(x-a)ⁿ (where k is constant) are a geometric series with initial term k and common ratio (x-a). Since we have an expression for the sum of a geometric series, we can rewrite such power series as a finite expression.https://cdn.kastatic.org/ka-youtube-converted/OzH735zyTWs.mp4/OzH735zyTWs.mp4298yes https://puzzicle.org/math/ap-calculus-bc/v/finding-power-series-through-integrationhttps://puzzicle.org/thumbs/xc5188ec7.jpgPower series of arctan(2x)We can represent arctan(2x) with a power series by representing its derivative as a power series and then integrating that series. You have to admit this is pretty neat.https://cdn.kastatic.org/ka-youtube-converted/NgYrsqoKXpM.mp4/NgYrsqoKXpM.mp4666yes https://puzzicle.org/math/ap-calculus-bc/v/power-series-using-integrationhttps://puzzicle.org/thumbs/x420cdc1d.jpgPower series of ln(1+x³)We can represent ln(1+x³) with a power series by representing its derivative as a power series and then integrating that series. You have to admit this is pretty neat.https://cdn.kastatic.org/ka-youtube-converted/znT4W6n3WkI.mp4/znT4W6n3WkI.mp4800yes https://puzzicle.org/math/ap-calculus-bc/v/cosine-taylor-series-at-0-maclaurinhttps://puzzicle.org/thumbs/89857203.jpgMaclaurin series of cos(x)This video explains the Maclaurin series, a special case of the Taylor series. It shows how to find the Maclaurin series for cosine of x by taking derivatives and evaluating them at 0. The resulting pattern creates a polynomial representation of cos(x).https://cdn.kastatic.org/ka-youtube-converted/WWe7pZjc4s8.mp4/WWe7pZjc4s8.mp4337yes https://puzzicle.org/math/ap-calculus-bc/v/sine-taylor-series-at-0-maclaurinhttps://puzzicle.org/thumbs/89860962.jpgMaclaurin series of sin(x)We can approximate sin(x) using a Maclaurin series, which is like a Taylor polynomial centered at x=0 with infinite terms. Surprisingly, this series is the same as the sin(x) function itself! The series includes odd powers of x divided by their factorials, with alternating signs. This pattern helps us understand the relationship between sin(x) and cos(x).https://cdn.kastatic.org/ka-youtube-converted/LlKvubIqHc8.mp4/LlKvubIqHc8.mp4393yes https://puzzicle.org/math/ap-calculus-bc/v/taylor-series-at-0-maclaurin-for-e-to-the-xhttps://puzzicle.org/thumbs/89858536.jpgMaclaurin series of eˣWe can estimate eˣ using a Maclaurin series, which is an infinite Taylor polynomial centered at x=0. Surprisingly, this series is identical to the original function, eˣ! This means the approximation gets closer to eˣ as we add more terms.https://cdn.kastatic.org/ka-youtube-converted/JYQqml4-4q4.mp4/JYQqml4-4q4.mp4370yes https://puzzicle.org/math/ap-calculus-bc/v/maclaurin-series-based-on-cos-xhttps://puzzicle.org/thumbs/x1aeaa19d.jpgWorked example: power series from cos(x)Finding a power series to represent x³cos(x²) using the Maclaurin series of cos(x).https://cdn.kastatic.org/ka-youtube-converted/kXULrUxGzmU.mp4/kXULrUxGzmU.mp4481yes https://puzzicle.org/math/ap-calculus-bc/v/evaluating-power-series-for-mystery-functionhttps://puzzicle.org/thumbs/x13b06289.jpgWorked example: cosine function from power seriesGiven a power series, we recognize it as the Maclaurin series of cos(x³) and evaluate it at a given x-value.https://cdn.kastatic.org/ka-youtube-converted/BAFOOQxiwkU.mp4/BAFOOQxiwkU.mp4369yes https://puzzicle.org/math/ap-calculus-bc/v/recognizing-function-from-taylor-serieshttps://puzzicle.org/thumbs/xcf549f34c730725c.jpgWorked example: recognizing function from Taylor seriesSample AP Calculus question asking to recognize a function from its Taylor series.https://cdn.kastatic.org/ka-youtube-converted/0BdZZCwDE-I.mp4/0BdZZCwDE-I.mp4266yes https://puzzicle.org/math/ap-calculus-bc/v/visualizing-taylor-series-approximationshttps://puzzicle.org/thumbs/90906156.jpgVisualizing Taylor series approximationsThe larger the degree of a Taylor polynomial, the better it approximates the function. See that in action with sin(x) and its Taylor polynomials.https://cdn.kastatic.org/ka-youtube-converted/8dMLK2Wueaw.mp4/8dMLK2Wueaw.mp4361yes https://puzzicle.org/math/ap-calculus-bc/v/euler-s-formula-and-euler-s-identityhttps://puzzicle.org/thumbs/90906155.jpgEuler's formula & Euler's identityEuler's Formula, e^(ix) = cos(x) + i*sin(x), and Euler's Identity, e^(iπ) + 1 = 0, are incredible math discoveries. They connect e, i, and pi through Maclaurin series of cos(x), sin(x), and e^x. These formulas reveal a deep, mysterious connection between important numbers in math.https://cdn.kastatic.org/ka-youtube-converted/mgNtPOgFje0.mp4/mgNtPOgFje0.mp4687yes https://puzzicle.org/math/ap-calculus-bc/v/rep-function-with-geometric-serieshttps://puzzicle.org/thumbs/x38e71a1b.jpgGeometric series interval of convergenceWhen a power series is a geometric series, we can find its interval of convergence without using the ratio test!https://cdn.kastatic.org/ka-youtube-converted/aiwy2fNF_ZQ.mp4/aiwy2fNF_ZQ.mp4332yes https://puzzicle.org/math/ap-calculus-bc/v/integrating-power-serieshttps://puzzicle.org/thumbs/xbf473cee828d6007.jpgIntegrating power seriesWithin its interval of convergence, the integral of a power series is the sum of integrals of individual terms: ∫Σf(x)dx=Σ∫f(x)dx. See how this is used to find the integral of a power series.https://cdn.kastatic.org/ka-youtube-converted/dEHZV7q5VV0.mp4/dEHZV7q5VV0.mp4365yes https://puzzicle.org/math/ap-calculus-bc/v/differentiating-power-serieshttps://puzzicle.org/thumbs/xe25757f0bfe90db1.jpgDifferentiating power seriesWithin its interval of convergence, the derivative of a power series is the sum of derivatives of individual terms: [Σf(x)]'=Σf'(x). See how this is used to find the derivative of a power series.https://cdn.kastatic.org/ka-youtube-converted/904esvqoJns.mp4/904esvqoJns.mp4482yes https://puzzicle.org/math/ap-calculus-bc/v/finding-function-from-series-by-integratinghttps://puzzicle.org/thumbs/x263746f1c1fe3e3e.jpgFinding function from power series by integratingWhen a power series S₁ is an antiderivative of a geometric series S₂, we can find the function represented by S₁ by integrating the expression for S₂.https://cdn.kastatic.org/ka-youtube-converted/ixFgd1lFe9g.mp4/ixFgd1lFe9g.mp4347yes https://puzzicle.org/math/ap-calculus-bc/v/interval-of-convergence-for-derivative-and-integralhttps://puzzicle.org/thumbs/x1449caf36ce2ba6a.jpgInterval of convergence for derivative and integralIntegrating or differentiating a power series term-by-term can only work within the interval of convergence. The interval of convergence of the integral/derivative will be the same, except maybe for the endpoints. See an example here.https://cdn.kastatic.org/ka-youtube-converted/TeIgljoLIxA.mp4/TeIgljoLIxA.mp4515yes https://puzzicle.org/math/ap-calculus-bc/v/definition-of-limit-of-a-sequence-and-sequence-convergencehttps://puzzicle.org/thumbs/1392656058.jpgFormal definition for limit of a sequenceA sequence is "converging" if its terms approach a specific value at infinity. This video is a more formal definition of what it means for a sequence to converge.https://cdn.kastatic.org/ka-youtube-converted/wzw9ll80Zbc.mp4/wzw9ll80Zbc.mp4289yes https://puzzicle.org/math/ap-calculus-bc/v/proving-a-sequence-convergeshttps://puzzicle.org/thumbs/1392561319.jpgProving a sequence converges using the formal definitionApplying the formal definition of the limit of a sequence to prove that a sequence converges.https://cdn.kastatic.org/ka-youtube-converted/-XRQovTI04Q.mp4/-XRQovTI04Q.mp4331yes https://puzzicle.org/math/ap-calculus-bc/v/proving-p-series-convergence-criteriahttps://puzzicle.org/thumbs/x530a4495c588ee72.jpgProof of p-series convergence criteriaA p-series converges for p>1 and diverges for 0https://cdn.kastatic.org/ka-youtube-converted/4WChwIGKdaA.mp4/4WChwIGKdaA.mp4577yes https://puzzicle.org/math/ap-calculus-bc/v/2015-ap-calculus-bc-2ahttps://puzzicle.org/thumbs/x48730aff.jpg2015 AP Calculus BC 2ax coordinate of particle at a certain timehttps://cdn.kastatic.org/ka-youtube-converted/UmU0L6j-48s.mp4/UmU0L6j-48s.mp4251yes https://puzzicle.org/math/ap-calculus-bc/v/2015-ap-calculus-bc-2bhttps://puzzicle.org/thumbs/x67e9485a.jpg2015 AP Calculus BC 2bPoint on curve where tangent line has a certain slopehttps://cdn.kastatic.org/ka-youtube-converted/ZFBwkvVa6NE.mp4/ZFBwkvVa6NE.mp4303yes https://puzzicle.org/math/ap-calculus-bc/v/2015-ap-calculus-bc-2chttps://puzzicle.org/thumbs/x592677be.jpg2015 AP Calculus BC 2cTime when particle reaches a certain speedhttps://cdn.kastatic.org/ka-youtube-converted/eU7MCBQTvoY.mp4/eU7MCBQTvoY.mp4280yes https://puzzicle.org/math/ap-calculus-bc/v/2015-ap-calculus-bc-2dhttps://puzzicle.org/thumbs/xfaa31c60.jpg2015 AP Calculus BC 2dFinding distance travelledhttps://cdn.kastatic.org/ka-youtube-converted/zhNP1LM479g.mp4/zhNP1LM479g.mp4285yes https://puzzicle.org/math/ap-calculus-bc/v/2015-ap-calculus-bc-5ahttps://puzzicle.org/thumbs/x831577c3.jpg2015 AP Calculus BC 5aEquation of tangent linehttps://cdn.kastatic.org/ka-youtube-converted/1RGnb-QvQzU.mp4/1RGnb-QvQzU.mp4276yes https://puzzicle.org/math/ap-calculus-bc/v/2015-ap-calculus-bc-5bhttps://puzzicle.org/thumbs/x5cb99082.jpg2015 AP Calculus BC 5bRelative minima and maximahttps://cdn.kastatic.org/ka-youtube-converted/kWQzA37MqAk.mp4/kWQzA37MqAk.mp4293yes https://puzzicle.org/math/ap-calculus-bc/v/2015-ap-calculus-bc-5chttps://puzzicle.org/thumbs/x294f1da5.jpg2015 AP Calculus BC 5cEngineering expression for critical pointhttps://cdn.kastatic.org/ka-youtube-converted/DYCwLAtNsjQ.mp4/DYCwLAtNsjQ.mp4267yes https://puzzicle.org/math/ap-calculus-bc/v/2015-ap-calculus-bc-5dhttps://puzzicle.org/thumbs/x60b91981.jpg2015 AP Calculus BC 5dPartial fraction expansionhttps://cdn.kastatic.org/ka-youtube-converted/wwc5yi3c4gY.mp4/wwc5yi3c4gY.mp4302yes https://puzzicle.org/math/ap-calculus-bc/v/2015-ap-calculus-bc-6ahttps://puzzicle.org/thumbs/xb9c4aefa.jpg2015 AP Calculus BC 6aRatio testhttps://cdn.kastatic.org/ka-youtube-converted/AZ6B7J8K1VU.mp4/AZ6B7J8K1VU.mp4395yes https://puzzicle.org/math/ap-calculus-bc/v/2015-ap-calculus-bc-6bhttps://puzzicle.org/thumbs/xd7ef5b25.jpg2015 AP Calculus BC 6bMaclaurin series for derivative.https://cdn.kastatic.org/ka-youtube-converted/59Jop9bZ-mw.mp4/59Jop9bZ-mw.mp4376yes https://puzzicle.org/math/ap-calculus-bc/v/2015-ap-calculus-bc-6chttps://puzzicle.org/thumbs/xc6263210.jpg2015 AP Calculus BC 6cMultiplying two Taylor polynomialshttps://cdn.kastatic.org/ka-youtube-converted/IAJHoGeevSQ.mp4/IAJHoGeevSQ.mp4478yes https://puzzicle.org/math/ap-calculus-bc/v/2011-calculus-bc-free-response-1ahttps://puzzicle.org/thumbs/244051675.jpg2011 Calculus BC free response #1aVelocity and acceleration vectors for particlehttps://cdn.kastatic.org/ka-youtube-converted/wtLTb_VaI-k.mp4/wtLTb_VaI-k.mp4403yes https://puzzicle.org/math/ap-calculus-bc/v/2011-calculus-bc-free-response-1-b-chttps://puzzicle.org/thumbs/244051674.jpg2011 Calculus BC free response #1 (b & c)Position and slope of particle's path at a given timehttps://cdn.kastatic.org/ka-youtube-converted/73eSiVcg4tQ.mp4/73eSiVcg4tQ.mp4459yes https://puzzicle.org/math/ap-calculus-bc/v/2011-calculus-bc-free-response-1dhttps://puzzicle.org/thumbs/246408189.jpg2011 Calculus BC free response #1dArc length for a curvehttps://cdn.kastatic.org/ka-youtube-converted/v-4QScXlN0o.mp4/v-4QScXlN0o.mp4514yes https://puzzicle.org/math/ap-calculus-bc/v/2011-calculus-bc-free-response-3ahttps://puzzicle.org/thumbs/246408188.jpg2011 Calculus BC free response #3aArc length of a functionhttps://cdn.kastatic.org/ka-youtube-converted/_G9JwTwjulA.mp4/_G9JwTwjulA.mp4341yes https://puzzicle.org/math/ap-calculus-bc/v/2011-calculus-bc-free-response-3-b-chttps://puzzicle.org/thumbs/246408186.jpg2011 Calculus BC free response #3 (b & c)Volume of a solid of rotation and Chain Rule for rates of change.https://cdn.kastatic.org/ka-youtube-converted/lpXCvX6ZenM.mp4/lpXCvX6ZenM.mp4506yes https://puzzicle.org/math/ap-calculus-bc/v/2011-calculus-bc-free-response-6ahttps://puzzicle.org/thumbs/246408187.jpg2011 Calculus BC free response #6aTaylor Series approximation of sin(x)https://cdn.kastatic.org/ka-youtube-converted/1BGlbx67B6s.mp4/1BGlbx67B6s.mp4462yes https://puzzicle.org/math/ap-calculus-bc/v/2011-calculus-bc-free-response-6bhttps://puzzicle.org/thumbs/248599678.jpg2011 Calculus BC free response #6bTaylor Series for cos x at x=0https://cdn.kastatic.org/ka-youtube-converted/NQazgcu8c7s.mp4/NQazgcu8c7s.mp4599yes https://puzzicle.org/math/ap-calculus-bc/v/2011-calculus-bc-free-response-6chttps://puzzicle.org/thumbs/248599679.jpg2011 Calculus BC free response #6cCalculating the 6th derivative at 0 from the Taylor Series approximationhttps://cdn.kastatic.org/ka-youtube-converted/l1hJABkcuYI.mp4/l1hJABkcuYI.mp4178yes https://puzzicle.org/math/ap-calculus-bc/v/2011-calculus-bc-free-response-6dhttps://puzzicle.org/thumbs/250668426.jpg2011 Calculus BC free response #6dLagrange error bound for Taylor Polynomial approximationhttps://cdn.kastatic.org/ka-youtube-converted/m6oMgtGsAa4.mp4/m6oMgtGsAa4.mp4712yes https://puzzicle.org/math/ap-calculus-bc/v/ap-calculus-bc-exams-2008-1-ahttps://puzzicle.org/thumbs/24254.jpgAP Calculus BC exams: 2008 1 aPart 1a of the 2008 BC free responsehttps://cdn.kastatic.org/ka-youtube-converted/upO6Mh862PI.mp4/upO6Mh862PI.mp4534yes https://puzzicle.org/math/ap-calculus-bc/v/ap-calculus-bc-exams-2008-1-b-chttps://puzzicle.org/thumbs/24256.jpgAP Calculus BC exams: 2008 1 b&cParts b and c of problem 1 (free response)https://cdn.kastatic.org/ka-youtube-converted/xPb6HLM3xEQ.mp4/xPb6HLM3xEQ.mp4574yes https://puzzicle.org/math/ap-calculus-bc/v/ap-calculus-bc-exams-2008-1-c-dhttps://puzzicle.org/thumbs/24258.jpgAP Calculus BC exams: 2008 1 c&dparts c and d of problem 1 in the 2008 AP Calculus BC free responsehttps://cdn.kastatic.org/ka-youtube-converted/_l0Mfsu__gU.mp4/_l0Mfsu__gU.mp4560yes https://puzzicle.org/math/ap-calculus-bc/v/ap-calculus-bc-exams-2008-1-dhttps://puzzicle.org/thumbs/24260.jpgAP Calculus BC exams: 2008 1 dPart 1d of the 2008 AP Calculus BC exam (free response)https://cdn.kastatic.org/ka-youtube-converted/sPTuCE5zd3s.mp4/sPTuCE5zd3s.mp4322yes https://puzzicle.org/math/ap-calculus-bc/v/calculus-bc-2008-2-ahttps://puzzicle.org/thumbs/24262.jpgCalculus BC 2008 2 a2a of 2008 Calculus BC exam (free response)https://cdn.kastatic.org/ka-youtube-converted/xvvI_QRYxBY.mp4/xvvI_QRYxBY.mp4525yes https://puzzicle.org/math/ap-calculus-bc/v/calculus-bc-2008-2-b-chttps://puzzicle.org/thumbs/24264.jpgCalculus BC 2008 2 b &cParts 2b and 2c of the 2008 BC exam (free response)https://cdn.kastatic.org/ka-youtube-converted/S4oOSgTj9C8.mp4/S4oOSgTj9C8.mp4642yes https://puzzicle.org/math/ap-calculus-bc/v/calculus-bc-2008-2dhttps://puzzicle.org/thumbs/24266.jpgCalculus BC 2008 2dPart 2d of the 2008 Calculus BC exam free-response sectionhttps://cdn.kastatic.org/ka-youtube-converted/o_vMb655dFk.mp4/o_vMb655dFk.mp4268yes https://puzzicle.org/math/ap-precalculus/v/interpreting-key-features-of-polynomial-graph-applications-precalculus-khan-acadhttps://puzzicle.org/thumbs/xae934b315abdabe4.jpgInterpreting key features of polynomial graph applicationsInterpret the meaning of key features on a transformed graph in a given context.https://cdn.kastatic.org/ka-youtube-converted/tPiJEFFHtg0.mp4/tPiJEFFHtg0.mp4122yes https://puzzicle.org/math/ap-precalculus/v/connecting-average-rates-of-change-with-the-concavity-of-a-graphhttps://puzzicle.org/thumbs/x53d5a518cdac6322.jpgConnecting average rates of change with the concavity of a graphLearn how to use a table of values of a graph to identify the concavity over a certain interval.https://cdn.kastatic.org/ka-youtube-converted/cGRJ55Q-9ks.mp4/cGRJ55Q-9ks.mp4218yes https://puzzicle.org/math/ap-precalculus/v/describing-the-average-rate-of-change-using-the-concavity-of-the-graphhttps://puzzicle.org/thumbs/x720d3877fc524354.jpgDescribing the average rate of change using the concavity of the graphLearn how to describe the average rate of change on a graph by using its concavity.https://cdn.kastatic.org/ka-youtube-converted/kQf0gmw-6Po.mp4/kQf0gmw-6Po.mp4120yes https://puzzicle.org/math/ap-precalculus/v/applying-the-rational-root-theoremhttps://puzzicle.org/thumbs/xa07226b73c6b277c.jpgApplying the rational root theoremLearn how to apply the rational root theorem to polynomial functions in order to solve for their zeros.https://cdn.kastatic.org/ka-youtube-converted/Y9fwl_qocJc.mp4/Y9fwl_qocJc.mp4455yes https://puzzicle.org/math/ap-precalculus/v/analyzing-horizontal-asymptotes-of-rational-functions-precalculus-khan-academyhttps://puzzicle.org/thumbs/xfcc2ec588b19cdc5.jpgAnalyzing horizontal asymptotes of rational functionsIdentify the horizontal asymptote of different rational functions by understanding their end behavior as the function approaches infinity and negative infinity.https://cdn.kastatic.org/ka-youtube-converted/UfY72qdxnvk.mp4/UfY72qdxnvk.mp4351yes https://puzzicle.org/math/ap-precalculus/v/solving-rational-inequalities-from-a-graphhttps://puzzicle.org/thumbs/xec60105a144afcba.jpgSolving rational inequalities from a graphIdentify when an inequality is true for a rational function when provided its graph.https://cdn.kastatic.org/ka-youtube-converted/04lcQT1QfA0.mp4/04lcQT1QfA0.mp4117yes https://puzzicle.org/math/ap-precalculus/v/solving-rational-inequalities-by-testing-intervalshttps://puzzicle.org/thumbs/xffd54bc646df185a.jpgSolving rational inequalities by testing intervals in terms of a constantLearn how to solve rational inequalities when the inequality is in the form of a rational expression on one side and a constant on the other.https://cdn.kastatic.org/ka-youtube-converted/JfXMsEi2S-s.mp4/JfXMsEi2S-s.mp4617yes https://puzzicle.org/math/ap-precalculus/v/understanding-equivalent-forms-of-polynomial-functions-to-answer-questions-in-cohttps://puzzicle.org/thumbs/x66cca8437b339792.jpgUnderstanding equivalent forms of polynomial functions to answer questions in contextLearn how different forms of the same polynomial function can help you answer questions in context.https://cdn.kastatic.org/ka-youtube-converted/YL73k-e8Qs4.mp4/YL73k-e8Qs4.mp4313yes https://puzzicle.org/math/ap-precalculus/v/understanding-equivalent-forms-of-rational-functions-to-answer-questions-in-conthttps://puzzicle.org/thumbs/x7d6f76c2728513de.jpgUnderstanding equivalent forms of rational functions to answer questions in contextUnderstanding what different rational forms support finding specific key features.https://cdn.kastatic.org/ka-youtube-converted/Si1TCvVJ-mk.mp4/Si1TCvVJ-mk.mp4219yes https://puzzicle.org/math/ap-precalculus/v/connecting-equivalent-forms-of-rational-functionshttps://puzzicle.org/thumbs/x374a53eaa72145b1.jpgConnecting equivalent forms of rational functionsLearn how to connect different forms of equivalent rational equations and understand what key features each form provides.https://cdn.kastatic.org/ka-youtube-converted/uNFTcN4ZZMo.mp4/uNFTcN4ZZMo.mp4541yes https://puzzicle.org/math/ap-precalculus/v/estimating-an-oblique-asymptote-graphicallyhttps://puzzicle.org/thumbs/xff1dda67966aab1d.jpgEstimating an oblique asymptote graphicallyUse a rational graph and your knowledge of oblique asymptotes to estimate its equation.https://cdn.kastatic.org/ka-youtube-converted/l_Z8UNIshw0.mp4/l_Z8UNIshw0.mp4109yes https://puzzicle.org/math/ap-precalculus/v/finding-oblique-asymptotes-of-a-rational-functionhttps://puzzicle.org/thumbs/x445eebc639bc674a.jpgFinding oblique asymptotes of a rational functionUse polynomial division to algebraically identify the oblique asymptote of a rational function if it exists.https://cdn.kastatic.org/ka-youtube-converted/aYRquz4FsKk.mp4/aYRquz4FsKk.mp4316yes https://puzzicle.org/math/ap-precalculus/v/graph-a-polynomial-with-transformations-precalculus-khan-academyhttps://puzzicle.org/thumbs/x2ac85c746e0fd37c.jpgGraph a polynomial with transformationsGraph a polynomial function with transformations by identifying the translations, dilations and reflections.https://cdn.kastatic.org/ka-youtube-converted/W21OQ7qBBZs.mp4/W21OQ7qBBZs.mp4246yes https://puzzicle.org/math/ap-precalculus/v/writing-transformation-form-equations-of-polynomials-from-a-graphhttps://puzzicle.org/thumbs/x1aadcdc37c594477.jpgWriting transformation form equations of polynomials from a graphWrite an equation for a polynomial function in transformation form when provided with a graph of the function.https://cdn.kastatic.org/ka-youtube-converted/6lM0zam_sHs.mp4/6lM0zam_sHs.mp4213yes https://puzzicle.org/math/ap-precalculus/v/constructing-rational-function-modelshttps://puzzicle.org/thumbs/xafb07bdef26a9716.jpgConstructing rational function modelsConstruct a rational function model based off of inversely proportional relationships.https://cdn.kastatic.org/ka-youtube-converted/in8QJym_rzM.mp4/in8QJym_rzM.mp4137yes https://puzzicle.org/math/ap-precalculus/v/constructing-models-of-data-sets-using-technology-and-regressionshttps://puzzicle.org/thumbs/x46ae4fda9af9dcf1.jpgConstructing models of data sets using technology and regressionsUsing a TI-84 to construct models of data sets.https://cdn.kastatic.org/ka-youtube-converted/WDO-eotwPP8.mp4/WDO-eotwPP8.mp4377yes https://puzzicle.org/math/ap-precalculus/v/constructing-exponential-functions-from-tableshttps://puzzicle.org/thumbs/x9234532a3a1c3a60.jpgConstructing exponential functions from tablesConstruct an exponential function from a table where you need to find a constant k.https://cdn.kastatic.org/ka-youtube-converted/3byo_eDa4aU.mp4/3byo_eDa4aU.mp4541yes https://puzzicle.org/math/ap-precalculus/v/constructing-logarithmic-functions-from-tableshttps://puzzicle.org/thumbs/x635c5ace0e2535c0.jpgConstructing logarithmic functions from tablesConstruct a logarithmic function when provided a table of values.https://cdn.kastatic.org/ka-youtube-converted/CIJXML8n6mI.mp4/CIJXML8n6mI.mp4241yes https://puzzicle.org/math/ap-precalculus/v/exploring-the-characteristics-of-periodic-functionshttps://puzzicle.org/thumbs/x21a0e90230791bbe.jpgExploring the characteristics of periodic functionsExplore characteristics of periodic functions and identify patterns to fill in a table of values.https://cdn.kastatic.org/ka-youtube-converted/GIhvaBecuO8.mp4/GIhvaBecuO8.mp4101yes https://puzzicle.org/math/ap-precalculus/v/identifying-reference-angles-of-positive-angles-in-standard-positionhttps://puzzicle.org/thumbs/x2d92b89cf1d866a7.jpgIdentifying reference angles of positive angles in standard positionFind the reference angle of a positive angle drawn in standard position in either radians or degrees.https://cdn.kastatic.org/ka-youtube-converted/DUNQsJAJZOU.mp4/DUNQsJAJZOU.mp4276yes https://puzzicle.org/math/ap-precalculus/v/identifying-reference-angles-for-negative-angles-in-standard-positionhttps://puzzicle.org/thumbs/xddc4ac5183c57437.jpgIdentifying reference angles for negative angles in standard positionIdentifying the reference angle of a negative angle drawn in standard position in either radians or degrees.https://cdn.kastatic.org/ka-youtube-converted/ZJk9y3sdZYE.mp4/ZJk9y3sdZYE.mp4206yes https://puzzicle.org/math/ap-precalculus/v/evaluating-trigonometric-angles-in-standard-positionhttps://puzzicle.org/thumbs/x9673ffc4a1049883.jpgEvaluating trigonometric angles in standard positionUse your knowledge of the unit circle and right triangle trigonometry to evaluate angles in standard position.https://cdn.kastatic.org/ka-youtube-converted/ROG5LyuP-yo.mp4/ROG5LyuP-yo.mp4216yes https://puzzicle.org/math/ap-precalculus/v/graphing-sinusoidal-functions-with-phase-shifts-ap-precalculus-khan-academyhttps://puzzicle.org/thumbs/xf9279539f0fc8bcc.jpgGraphing sinusoidal functions with transformationsGraph a sine function that has multiple transformations, including a phase shift.https://cdn.kastatic.org/ka-youtube-converted/QnuCfZD2Xyw.mp4/QnuCfZD2Xyw.mp4290yes https://puzzicle.org/math/ap-precalculus/v/understanding-the-tangent-functionhttps://puzzicle.org/thumbs/x7bb00dd00a9038a3.jpgUnderstanding the tangent functionUnderstand how the tangent function performs on a graph and how different transformations change it.https://cdn.kastatic.org/ka-youtube-converted/XwyawGBo3Lw.mp4/XwyawGBo3Lw.mp4554yes https://puzzicle.org/math/ap-precalculus/v/graphs-of-arccos-and-arcsinhttps://puzzicle.org/thumbs/xae03ba5dd585fc45.jpgGraphs of arccosine and arcsineGraph the parent functions of the inverse of cosine and the inverse of sine.https://cdn.kastatic.org/ka-youtube-converted/UinjqPjVrWU.mp4/UinjqPjVrWU.mp4217yes https://puzzicle.org/math/precalculus/v/interpreting-solutions-of-trigonometric-equationshttps://puzzicle.org/thumbs/xfe047acd4c891e24.jpgInterpreting solutions of trigonometric equationsStarting from a context represented by a trigonometric function, interpret equations based on the function.https://cdn.kastatic.org/ka-youtube-converted/c-VkOXIGTqU.mp4/c-VkOXIGTqU.mp4249yes https://puzzicle.org/math/ap-precalculus/v/identifying-the-asymptote-equations-of-transformed-reciprocal-trigonometric-funchttps://puzzicle.org/thumbs/x35b7d540d79b555c.jpgIdentifying the asymptote equations of transformed reciprocal trigonometric functionsLearn how to identify the asymptote equations of transformed reciprocal trigonometric functions.https://cdn.kastatic.org/ka-youtube-converted/fiH1jEPP3I0.mp4/fiH1jEPP3I0.mp4371yes https://puzzicle.org/math/ap-precalculus/v/graphing-polar-coordinateshttps://puzzicle.org/thumbs/x815c23cc43edd979.jpgGraphing polar coordinatesGraph polar coordinates with radii that may be positive or negative, using angles measured in either degrees or radians.https://cdn.kastatic.org/ka-youtube-converted/MiIC7zvBbOI.mp4/MiIC7zvBbOI.mp4246yes https://puzzicle.org/math/ap-precalculus/v/converting-polar-and-rectangular-coordinates-precalculus-khan-academyhttps://puzzicle.org/thumbs/xaab37e5499c392b7.jpgConverting polar and rectangular coordinatesConvert polar coordinates to rectangular, and vice versa, using conversion formulas.https://cdn.kastatic.org/ka-youtube-converted/LWje358QRkU.mp4/LWje358QRkU.mp4363yes https://puzzicle.org/math/precalculus/v/convert-complex-polar-to-rectangularhttps://puzzicle.org/thumbs/x3528768eebf3981a.jpgConverting a complex number from polar to rectangular formGiven a complex number in polar form, we can convert that number to rectangular form and plot it on the complex plane.https://cdn.kastatic.org/ka-youtube-converted/auywa7dydAk.mp4/auywa7dydAk.mp4159yes https://puzzicle.org/math/ap-precalculus/v/graphing-polar-functionshttps://puzzicle.org/thumbs/x0586475f6f8ca1fc.jpgGraphing polar functionsGraph polar functions by using a table of values.https://cdn.kastatic.org/ka-youtube-converted/KbPj4xXrBrk.mp4/KbPj4xXrBrk.mp4508yes https://puzzicle.org/math/ap-precalculus/v/describing-characteristics-of-polar-function-graphshttps://puzzicle.org/thumbs/x9a971e698ab3ec6d.jpgDescribing characteristics of polar function graphsUse your understanding of increasing and decreasing functions to identify how the distance a polar graph is from the origin changes.https://cdn.kastatic.org/ka-youtube-converted/oDKomV5oLrI.mp4/oDKomV5oLrI.mp4307yes https://puzzicle.org/math/ap-precalculus/v/estimating-the-value-of-a-polar-function-with-linear-approximation-ap-precalculuhttps://puzzicle.org/thumbs/x7b6dac6d5d69d1d9.jpgEstimating the value of a polar function with linear approximationUse linear approximation to estimate the value of a point on a polar function graph.https://cdn.kastatic.org/ka-youtube-converted/Z7492K-qa_c.mp4/Z7492K-qa_c.mp4397yes https://puzzicle.org/math/ap-precalculus/v/identifying-vertical-extremum-of-parametric-planar-motion-functionshttps://puzzicle.org/thumbs/xf73f86224186f66f.jpgIdentifying vertical extremum of parametric planar motion functionsLearn how to identify extremum of parametric planar motion functions.https://cdn.kastatic.org/ka-youtube-converted/rEuOOY7PO-Q.mp4/rEuOOY7PO-Q.mp4266yes https://puzzicle.org/math/ap-precalculus/v/identifying-intercepts-of-parametric-planar-motion-functionshttps://puzzicle.org/thumbs/x78510ff84da6372f.jpgIdentifying intercepts of parametric planar motion functionsWhen given a parametric planar motion function, Sal shows how to identify a specified intercept.https://cdn.kastatic.org/ka-youtube-converted/tcbD-QLtB3o.mp4/tcbD-QLtB3o.mp4188yes https://puzzicle.org/math/ap-precalculus/v/determining-the-average-rate-of-change-of-parametric-planar-motion-functions-ovehttps://puzzicle.org/thumbs/x7ebab024bd36c5e9.jpgDetermining the average rate of change of parametric planar motion functions over a given intervalLearn how to determine the average rate of change of parametric planar motion functions over a given interval.https://cdn.kastatic.org/ka-youtube-converted/s0C08510fHs.mp4/s0C08510fHs.mp4311yes https://puzzicle.org/math/ap-precalculus/v/parametric-equations-for-projectile-motionhttps://puzzicle.org/thumbs/x61ba17ed7ef6518e.jpgParametric equations for projectile motionUse parametric equations to solve applications of projectile motion.https://cdn.kastatic.org/ka-youtube-converted/pNiydPVY6Z8.mp4/pNiydPVY6Z8.mp4552yes https://puzzicle.org/math/ap-precalculus/v/writing-the-equation-of-an-ellipse-from-graphhttps://puzzicle.org/thumbs/xd4725704bcee78db.jpgWriting the equation of an ellipse from graphWrite the equation of an ellipse when provided a graph by finding the radii of the major and minor axes.https://cdn.kastatic.org/ka-youtube-converted/pae1hExrc5o.mp4/pae1hExrc5o.mp4205yes https://puzzicle.org/math/ap-precalculus/v/parameterizing-implicitly-defined-functionshttps://puzzicle.org/thumbs/x4c2f2aef5c412b1f.jpgParameterizing implicitly defined functionsLearn how to identify a parameterization that satisfies a given equation for all real values.https://cdn.kastatic.org/ka-youtube-converted/eKygWXbAZRI.mp4/eKygWXbAZRI.mp4256yes https://puzzicle.org/math/precalculus/v/representing-quantities-with-vectorshttps://puzzicle.org/thumbs/x2b6cd2a4f67ecbb0.jpgRepresenting quantities with vectorsVectors can represent real-world quantities where it's important to have both a magnitude and a direction. One example for that is force: you want to represent a force's magnitude (or strength), but it's also important to indicate in what direction the force is applied.https://cdn.kastatic.org/ka-youtube-converted/TCOQzIONaz8.mp4/TCOQzIONaz8.mp4166yes https://puzzicle.org/math/precalculus/v/interpreting-statements-about-vectorshttps://puzzicle.org/thumbs/x5ce6dc5dc16cfb72.jpgInterpreting statements about vectorsVector quantities have both a magnitude and a direction. In this example, we interpret a mathematical statement about two vector quantities in terms of the real-world quantities they represent.https://cdn.kastatic.org/ka-youtube-converted/eA2AW5F4N_M.mp4/eA2AW5F4N_M.mp4158yes https://puzzicle.org/math/precalculus/v/introduction-to-vector-componentshttps://puzzicle.org/thumbs/x4a572c4edae38f8e.jpgIntroduction to vector componentsVectors are quantities that have a magnitude and a direction. In the two-dimensional plane, we can describe them in an equivalent way, by thinking about the changes in x and y from the vector's tail to its head.https://cdn.kastatic.org/ka-youtube-converted/hJkKADcQWj0.mp4/hJkKADcQWj0.mp4316yes https://puzzicle.org/math/precalculus/v/adding-vectors-in-magnitude-and-direction-formhttps://puzzicle.org/thumbs/x429b31be7da5c379.jpgAdding vectors in magnitude and direction formIn this example, Sal takes two vectors given by magnitude and direction, and finds the magnitude and direction of their sum.https://cdn.kastatic.org/ka-youtube-converted/yFPfO_eHJdY.mp4/yFPfO_eHJdY.mp4501yes https://puzzicle.org/math/linear-algebra/v/matrix-vector-products-as-linear-transformationshttps://puzzicle.org/thumbs/25291.jpgMatrix vector products as linear transformationsMatrix Vector Products as Linear Transformationshttps://cdn.kastatic.org/ka-youtube-converted/ondmopWLiEg.mp4/ondmopWLiEg.mp41024yes https://puzzicle.org/math/precalculus/v/using-matrices-to-manipulate-data-game-showhttps://puzzicle.org/thumbs/x56b27d8d6e4815b6.jpgUsing matrices to manipulate data: Game showIn other videos, we saw how we can use matrices to represent real-world situations. Once we do that, we can also manipulate the matrices (using common matrix operations like addition, subtraction, and scalar multiplication) to reveal more information about the context. Here, we manipulate matrices that represent the possible prizes in a game show.https://cdn.kastatic.org/ka-youtube-converted/NKG6DifB3OA.mp4/NKG6DifB3OA.mp4216yes https://puzzicle.org/math/precalculus/v/using-matrices-to-manipulate-data-pet-storehttps://puzzicle.org/thumbs/x8fb8a59d203b117c.jpgUsing matrices to manipulate data: Pet storeIn other videos, we saw how we can use matrices to represent real-world situations. Once we do that, we can also manipulate the matrices (using common matrix operations like addition, subtraction, and scalar multiplication) to reveal more information about the context. Here, we manipulate matrices that represent the inventory of dog food in a pet store chain.https://cdn.kastatic.org/ka-youtube-converted/g-ljbR62EAQ.mp4/g-ljbR62EAQ.mp4246yes https://puzzicle.org/math/ap-statistics/v/identifying-a-sample-and-populationhttps://puzzicle.org/thumbs/x8f6cbec077980a24.jpgIdentifying a sample and populationIdentifying a sample and populationhttps://cdn.kastatic.org/ka-youtube-converted/VPM84_yfx5Q.mp4/VPM84_yfx5Q.mp4132yes https://puzzicle.org/math/ap-statistics/v/statistical-questionshttps://puzzicle.org/thumbs/x875b9f8c.jpgStatistical and non statistical questionsA statistical question is one that can be answered by collecting data and where there will be variability in that data. For example, there will likely be variability in the data collected to answer the question, "How much do the animals at Fancy Farm weigh?" but not to answer, "What color hat is Sara wearing?".https://cdn.kastatic.org/ka-youtube-converted/qyYSQDcSNlY.mp4/qyYSQDcSNlY.mp4471yes https://puzzicle.org/math/ap-statistics/v/classifying-discrete-and-continuous-variableshttps://puzzicle.org/thumbs/xa50293e781216840.jpgClassifying discrete and continuous variablesSal shows how to classify a quantitative variable as discrete or continuous, and give the reason for the classification.https://cdn.kastatic.org/ka-youtube-converted/OnnWywg0q3Y.mp4/OnnWywg0q3Y.mp4269yes https://puzzicle.org/math/ap-statistics/v/creating-a-relative-frequency-table-for-one-categorical-variablehttps://puzzicle.org/thumbs/x988d2d17a6a0e115.jpgCreating a relative frequency table for one categorical variableSal shows how to turn counts for a single categorical variable into proportions in a relative frequency table.https://cdn.kastatic.org/ka-youtube-converted/xQSTwehQdCc.mp4/xQSTwehQdCc.mp4113yes https://puzzicle.org/math/ap-statistics/v/reading-bar-charts-2https://puzzicle.org/thumbs/x4f968262.jpgReading bar charts: comparing two sets of dataA bar chart is especially useful with comparing two sets of data. The difference in the bars give us a quick snapshot that allows us to draw some conclusions.https://cdn.kastatic.org/ka-youtube-converted/gnyHsgTFXIY.mp4/gnyHsgTFXIY.mp490yes https://puzzicle.org/math/ap-statistics/v/u08-l1-t2-we3-stem-and-leaf-plotshttps://puzzicle.org/thumbs/380855565.jpgStem-and-leaf plotsWe learn how to interpret a stem-and-leaf plot using basketball scores as an example. The video demonstrates how to read the plot, translate it into individual scores, and then use those scores to calculate the team's total points. This method provides a clear view of data distribution and makes it easier to understand the range of scores.https://cdn.kastatic.org/ka-youtube-converted/LEFE1km5ROY.mp4/LEFE1km5ROY.mp4358yes https://puzzicle.org/math/ap-statistics/v/reading-stem-and-leaf-plotshttps://puzzicle.org/thumbs/xb424e537.jpgReading stem and leaf plotsA stem and leaf plot displays data by breaking each data point into two parts: the "stem" and the "leaf." Each stem represents a range of values (e.g. if the stem is 2, the corresponding leaves might be data points in the 20s). Each leaf typically represents the last digit of a data point (e.g. if the leaf is 6 next to a stem of 2, it could represent a data point whose value is 26).https://cdn.kastatic.org/ka-youtube-converted/PXKHyT__B2k.mp4/PXKHyT__B2k.mp4188yes https://puzzicle.org/math/ap-statistics/v/comparing-distributions-using-summary-statisticshttps://puzzicle.org/thumbs/x29d4a4cd6116d063.jpgComparing distributions using summary statisticsIn this video, Sal shows how to compare the center, shape, and variability of two distributions, and determine whether summary statistics support claims about the corresponding populations.https://cdn.kastatic.org/ka-youtube-converted/7zK6OejZhjk.mp4/7zK6OejZhjk.mp4219yes https://puzzicle.org/math/ap-statistics/v/z-score-introductionhttps://puzzicle.org/thumbs/xdc0f07c2c91917f7.jpgZ-score introductionA z-score is an example of a standardized score. A z-score measures how many standard deviations a data point is from the mean in a distribution.https://cdn.kastatic.org/ka-youtube-converted/5S-Zfa-vOXs.mp4/5S-Zfa-vOXs.mp4305yes https://puzzicle.org/math/ap-statistics/v/comparing-with-z-scoreshttps://puzzicle.org/thumbs/x250dec1c92a744df.jpgComparing with z-scoresUse standardized scores—also called z-scores— to compare data points from different distributions.https://cdn.kastatic.org/ka-youtube-converted/_rtQGAX5wsQ.mp4/_rtQGAX5wsQ.mp4210yes https://puzzicle.org/math/ap-statistics/v/identifying-a-censushttps://puzzicle.org/thumbs/x30b62db229d3940f.jpgIdentifying a censusSal shows how to decide whether a described data collection method is a census, by naming the population and checking that every individual in it was recorded.https://cdn.kastatic.org/ka-youtube-converted/apc8jmzOqzE.mp4/apc8jmzOqzE.mp4151yes https://puzzicle.org/math/ap-statistics/v/worked-example-identifying-experimenthttps://puzzicle.org/thumbs/xfb74d7129be01707.jpgWorked example identifying experimentWorked example identifying experimenthttps://cdn.kastatic.org/ka-youtube-converted/28lzQ_OhBHY.mp4/28lzQ_OhBHY.mp4276yes https://puzzicle.org/math/ap-statistics/v/generalizabilty-of-survey-results-examplehttps://puzzicle.org/thumbs/x166bd2fcd09e99a0.jpgGeneralizability of survey results examplePractice identifying to which population the results of a survey can be generalized.https://cdn.kastatic.org/ka-youtube-converted/2FXwGI_debU.mp4/2FXwGI_debU.mp4150yes https://puzzicle.org/math/ap-statistics/v/sampling-with-and-without-replacementhttps://puzzicle.org/thumbs/x98ba54da73335b8d.jpgSampling with and without replacementSal shows how to classify a described sampling procedure as with or without replacement.https://cdn.kastatic.org/ka-youtube-converted/ZU7lZAaNgVA.mp4/ZU7lZAaNgVA.mp4195yes https://puzzicle.org/math/ap-statistics/v/techniques-for-generating-a-simple-random-samplehttps://puzzicle.org/thumbs/xe7433b050109a6da.jpgTechniques for generating a simple random sampleTechniques for generating a simple random samplehttps://cdn.kastatic.org/ka-youtube-converted/acfjqWTwee0.mp4/acfjqWTwee0.mp4386yes https://puzzicle.org/math/ap-statistics/v/techniques-for-random-sampling-and-avoiding-biashttps://puzzicle.org/thumbs/x9db3cdd29f45dfba.jpgTechniques for random sampling and avoiding biasTechniques for random sampling and avoiding biashttps://cdn.kastatic.org/ka-youtube-converted/PdXDLNNXPik.mp4/PdXDLNNXPik.mp4553yes https://puzzicle.org/math/ap-statistics/v/systematic-random-samplinghttps://puzzicle.org/thumbs/x0a7759e472f9c753.jpgSystematic random samplingIn a systematic random sample, we arrange members of a population in some order, pick a random starting point, and select every member in a set interval.https://cdn.kastatic.org/ka-youtube-converted/TrOS8OnM1u4.mp4/TrOS8OnM1u4.mp4288yes https://puzzicle.org/math/ap-statistics/v/examples-of-bias-in-surveyshttps://puzzicle.org/thumbs/x69eb52cf5567f908.jpgExamples of bias in surveysExamples of bias in surveyshttps://cdn.kastatic.org/ka-youtube-converted/r1R00l8Z5lg.mp4/r1R00l8Z5lg.mp4342yes https://puzzicle.org/math/ap-statistics/v/example-of-under-coverage-introducing-biashttps://puzzicle.org/thumbs/x9e8eb23cad527582.jpgExample of undercoverage introducing biasNonresponse, undercoverage, and voluntary responses can all introduce bias when we sample a population for a study. Given the description of a study, we can think about potential sources of bias, and how they may have impacted the results of the study.https://cdn.kastatic.org/ka-youtube-converted/D9f7dCMxu2Y.mp4/D9f7dCMxu2Y.mp4261yes https://puzzicle.org/math/ap-statistics/v/introduction-to-experiment-designhttps://puzzicle.org/thumbs/xaf6650305a49db77.jpgIntroduction to experiment designIntroduction to experiment design. Explanatory and response variables. Control and treatment groups.https://cdn.kastatic.org/ka-youtube-converted/DaBq0naj0YY.mp4/DaBq0naj0YY.mp4628yes https://puzzicle.org/math/ap-statistics/v/matched-pairs-experiment-designhttps://puzzicle.org/thumbs/x20a1993463318741.jpgMatched pairs experiment designThe video presents an in-depth exploration of experimental design in statistics, focusing on the use of control and treatment groups, block design, and matched pairs design. It emphasizes the importance of random assignment to mitigate lurking variables and bias, and the value of double-blind experiments. The video also discusses the potential for imbalance in experimental groups and how matched pairs design can help address this issue.https://cdn.kastatic.org/ka-youtube-converted/UEFBGewP2ik.mp4/UEFBGewP2ik.mp4337yes https://puzzicle.org/math/ap-statistics/v/mosaic-plots-introhttps://puzzicle.org/thumbs/xb767da60dfdca857.jpgMosaic plots and segmented bar chartsWe can display categorical data from a two-way table with a segmented bar chart or a mosaic plot.https://cdn.kastatic.org/ka-youtube-converted/b9G5pWuxb1Y.mp4/b9G5pWuxb1Y.mp4346yes https://puzzicle.org/math/ap-statistics/v/analyzing-mosaic-plotshttps://puzzicle.org/thumbs/x9d2347a8565b6267.jpgAnalyzing mosaic plotsWe can use mosaic plots to draw conclusions about relationships between two categorical variables.https://cdn.kastatic.org/ka-youtube-converted/2sHkluggZp8.mp4/2sHkluggZp8.mp4160yes https://puzzicle.org/math/ap-statistics/v/experimental-versus-theoretical-probability-simulationhttps://puzzicle.org/thumbs/xdd932b06d7d9ffac.jpgExperimental versus theoretical probability simulationExperimental versus theoretical probability simulationhttps://cdn.kastatic.org/ka-youtube-converted/Nos-xOCpQqg.mp4/Nos-xOCpQqg.mp4297yes https://puzzicle.org/math/ap-statistics/v/coin-flipping-examplehttps://puzzicle.org/thumbs/380860449.jpgExample: All the ways you can flip a coinManually going through the combinatorics to determine the probability of an event occuringhttps://cdn.kastatic.org/ka-youtube-converted/3UlE8gyKbkU.mp4/3UlE8gyKbkU.mp4134yes https://puzzicle.org/math/ap-statistics/v/testing-independence-from-experimental-datahttps://puzzicle.org/thumbs/x484ec0da2dc9ed37.jpgConditional probability and independenceUse conditional probability to see if events are independent or not.https://cdn.kastatic.org/ka-youtube-converted/pIfpHdGVwLU.mp4/pIfpHdGVwLU.mp4247yes https://puzzicle.org/math/ap-statistics/v/conditional-probability-tree-diagram-examplehttps://puzzicle.org/thumbs/xf4d467541e2b366a.jpgConditional probability tree diagram exampleUsing a tree diagram to work out a conditional probability question. If someone fails a drug test, what is the probability that they actually are taking drugs?https://cdn.kastatic.org/ka-youtube-converted/hxEdXUB_IdQ.mp4/hxEdXUB_IdQ.mp4644yes https://puzzicle.org/math/precalculus/v/general-multiplication-example-independenthttps://puzzicle.org/thumbs/x9b4022f8d54d2ee0.jpgGeneral multiplication rule example: independent eventsWe can use the general multiplication rule to find the probability that two events both occur when the events are independent.https://cdn.kastatic.org/ka-youtube-converted/OqbkCYy37hI.mp4/OqbkCYy37hI.mp4149yes https://puzzicle.org/math/precalculus/v/introduction-to-dependent-probabilityhttps://puzzicle.org/thumbs/1200026245.jpgDependent probability introductionLet's get you started with a great explanation of dependent probability using a scenario involving a casino game.https://cdn.kastatic.org/ka-youtube-converted/VjLEoo3hIoM.mp4/VjLEoo3hIoM.mp4398yes https://puzzicle.org/math/precalculus/v/general-multiplication-rule-example-dependenthttps://puzzicle.org/thumbs/xd746275804f8a727.jpgGeneral multiplication rule example: dependent eventsWe can use the general multiplication rule to find the probability that two events both occur when the events are not independent.https://cdn.kastatic.org/ka-youtube-converted/iyQNk090FGM.mp4/iyQNk090FGM.mp4159yes https://puzzicle.org/math/precalculus/v/discrete-probability-distributionhttps://puzzicle.org/thumbs/x970b9601.jpgConstructing a probability distribution for random variableSal breaks down how to create the probability distribution of the number of "heads" after 3 flips of a fair coin.https://cdn.kastatic.org/ka-youtube-converted/cqK3uRoPtk0.mp4/cqK3uRoPtk0.mp4408yes https://puzzicle.org/math/precalculus/v/valid-discrete-probability-distribution-exampleshttps://puzzicle.org/thumbs/x32040e0dffffea40.jpgValid discrete probability distribution examplesWorked examples on identifying valid discrete probability distributions.https://cdn.kastatic.org/ka-youtube-converted/uXtxKxYl1Iw.mp4/uXtxKxYl1Iw.mp4240yes https://puzzicle.org/math/precalculus/v/example-analyzing-discrete-probability-distributionhttps://puzzicle.org/thumbs/x344d97846e9fa2e4.jpgProbability with discrete random variable exampleExample analyzing discrete probability distributionhttps://cdn.kastatic.org/ka-youtube-converted/d2STHFVHGAg.mp4/d2STHFVHGAg.mp4230yes https://puzzicle.org/math/precalculus/v/theoretical-probability-distribution-example-tableshttps://puzzicle.org/thumbs/x6eea606716b37dba.jpgTheoretical probability distribution example: tablesWe can create the probability distribution for rolling two dice by creating a table to represent the events that make up the sample space.https://cdn.kastatic.org/ka-youtube-converted/hA8VmhkKEJo.mp4/hA8VmhkKEJo.mp4195yes https://puzzicle.org/math/precalculus/v/theoretical-probability-distribution-example-multiplicationhttps://puzzicle.org/thumbs/xa325f9755680187e.jpgTheoretical probability distribution example: multiplicationWe can create a probability distribution for the number of times someone wins a prize using the multiplication rule.https://cdn.kastatic.org/ka-youtube-converted/2jExPaoTrQE.mp4/2jExPaoTrQE.mp4205yes https://puzzicle.org/math/precalculus/v/expected-value-of-a-discrete-random-variablehttps://puzzicle.org/thumbs/x9996bd6c7bed2406.jpgMean (expected value) of a discrete random variableWe can calculate the mean (or expected value) of a discrete random variable as the weighted average of all the outcomes of that random variable based on their probabilities. We interpret expected value as the predicted average outcome if we looked at that random variable over an infinite number of trials.https://cdn.kastatic.org/ka-youtube-converted/qafPcWNUiM8.mp4/qafPcWNUiM8.mp4272yes https://puzzicle.org/math/precalculus/v/interpreting-expected-valuehttps://puzzicle.org/thumbs/x6bac398edf6f3339.jpgInterpreting expected valueWe can interpret expected value as a long term average outcome. This example looks at expected value in the context of a lottery ticket.https://cdn.kastatic.org/ka-youtube-converted/SNIW7MmCdhA.mp4/SNIW7MmCdhA.mp4174yes https://puzzicle.org/math/ap-statistics/v/variance-and-standard-deviation-of-a-discrete-random-variablehttps://puzzicle.org/thumbs/x74e7ad7c9d4508ca.jpgVariance and standard deviation of a discrete random variableWe learn how to calculate the mean and standard deviation of a discrete random variable. The concept of a random variable is explained, along with methods to calculate its expected value (mean) and measure its spread (variance and standard deviation). A practical example makes the concept easier to understand.https://cdn.kastatic.org/ka-youtube-converted/2egl_5c8i-g.mp4/2egl_5c8i-g.mp4386yes https://puzzicle.org/math/ap-statistics/v/binomial-variableshttps://puzzicle.org/thumbs/x04c2ddde6bb2c5b3.jpgBinomial variablesAn introduction to a special class of random variables called binomial random variableshttps://cdn.kastatic.org/ka-youtube-converted/yUDFir8oYl8.mp4/yUDFir8oYl8.mp4451yes https://puzzicle.org/math/ap-statistics/v/recognizing-binomial-variableshttps://puzzicle.org/thumbs/x5a9f0870c146fc09.jpgRecognizing binomial variablesSome examples and counterexamples of binomial variableshttps://cdn.kastatic.org/ka-youtube-converted/ITtFw5PH-hs.mp4/ITtFw5PH-hs.mp4402yes https://puzzicle.org/math/ap-statistics/v/expected-value-of-binomial-variablehttps://puzzicle.org/thumbs/x6349c94e04e93b4b.jpgExpected value of a binomial variableDeriving and using the expected value (mean) formula for binomial random variableshttps://cdn.kastatic.org/ka-youtube-converted/fjjhAm5trSM.mp4/fjjhAm5trSM.mp4393yes https://puzzicle.org/math/ap-statistics/v/variance-of-binomial-variablehttps://puzzicle.org/thumbs/x1062c356a16065ba.jpgVariance of a binomial variableWe can derive the variance of a binomial variable to be p(1-p), and the standard deviation is the square root of the variance.https://cdn.kastatic.org/ka-youtube-converted/uudB5XiAYOw.mp4/uudB5XiAYOw.mp4411yes https://puzzicle.org/math/ap-statistics/v/finding-the-mean-and-standard-deviation-of-a-binomial-random-variablehttps://puzzicle.org/thumbs/x15bc0464cc477e7a.jpgFinding the mean and standard deviation of a binomial random variableFor a binomal random variable, the mean is n times p (np), where n is the sample size and p is the probability of success. The standard deviation is the square root of np(1-p). We can use them to make predictions in a binomial setting. In this example, we look at how many defective chips we expect, on average, in a sample.https://cdn.kastatic.org/ka-youtube-converted/gro5b3PZ-Bw.mp4/gro5b3PZ-Bw.mp4302yes https://puzzicle.org/math/ap-statistics/v/probability-of-making-2-shots-in-6-attemptshttps://puzzicle.org/thumbs/x88ef095e.jpgBinomial probability exampleWe can use the binomial distribution to find the probability of getting a certain number of successes, like successful basketball shots, out of a fixed number of trials. We use the binomial distribution to find discrete probabilities.https://cdn.kastatic.org/ka-youtube-converted/Ctytn4a6zjw.mp4/Ctytn4a6zjw.mp4624yes https://puzzicle.org/math/ap-statistics/v/ck12-org-normal-distribution-problems-qualitative-sense-of-normal-distributionshttps://puzzicle.org/thumbs/26198.jpgQualitative sense of normal distributionsDiscussion of how "normal" a distribution might be.https://cdn.kastatic.org/ka-youtube-converted/79duxPXpyKQ.mp4/79duxPXpyKQ.mp4653yes https://puzzicle.org/math/ap-statistics/v/ck12-org-normal-distribution-problems-empirical-rulehttps://puzzicle.org/thumbs/26202.jpgNormal distribution problems: Empirical ruleThe empirical rule (also called the "68-95-99.7 rule") is a guideline for how data is distributed in a normal distribution. The rule states that (approximately): - 68% of the data points will fall within one standard deviation of the mean. - 95% of the data points will fall within two standard deviations of the mean. - 99.7% of the data points will fall within three standard deviations of the mean.https://cdn.kastatic.org/ka-youtube-converted/OhRr26AfFBU.mp4/OhRr26AfFBU.mp4625yes https://puzzicle.org/math/ap-statistics/v/z-table-for-proportion-belowhttps://puzzicle.org/thumbs/xd7aeb1e8fab6fb00.jpgStandard normal table for proportion belowFinding the proportion of a normal distribution that is below a value by calculating a z-score and using a z-table.https://cdn.kastatic.org/ka-youtube-converted/Fo4kitkFB3I.mp4/Fo4kitkFB3I.mp4260yes https://puzzicle.org/math/ap-statistics/v/z-table-for-proportion-abovehttps://puzzicle.org/thumbs/xaff92a3b77dfe7d7.jpgStandard normal table for proportion aboveFinding the proportion of a normal distribution that is above a value by calculating a z-score and using a z-table.https://cdn.kastatic.org/ka-youtube-converted/i9FzFfv1rQg.mp4/i9FzFfv1rQg.mp4307yes https://puzzicle.org/math/ap-statistics/v/z-table-for-proportion-between-valueshttps://puzzicle.org/thumbs/xd01bbf159422bc87.jpgStandard normal table for proportion between valuesFinding the proportion of a normal distribution that is between two values by calculating z-scores and using a z-table.https://cdn.kastatic.org/ka-youtube-converted/uwhV0TAPmWc.mp4/uwhV0TAPmWc.mp4430yes https://puzzicle.org/math/ap-statistics/v/threshold-for-low-percentilehttps://puzzicle.org/thumbs/xe17171ee612b4210.jpgThreshold for low percentileFind the cutoff for a given lower percentile in a normal distributionhttps://cdn.kastatic.org/ka-youtube-converted/umYWWMfxUCI.mp4/umYWWMfxUCI.mp4360yes https://puzzicle.org/math/ap-statistics/v/introduction-to-sampling-distributionshttps://puzzicle.org/thumbs/x52e03bfc13da458b.jpgIntroduction to sampling distributionsIntroduction to sampling distributionshttps://cdn.kastatic.org/ka-youtube-converted/z0Ry_3_qhDw.mp4/z0Ry_3_qhDw.mp4438yes https://puzzicle.org/math/ap-statistics/v/central-limit-theoremhttps://puzzicle.org/thumbs/371055.jpgCentral limit theoremIntroduction to the central limit theorem and the sampling distribution of the meanhttps://cdn.kastatic.org/ka-youtube-converted/JNm3M9cqWyc.mp4/JNm3M9cqWyc.mp4589yes https://puzzicle.org/math/ap-statistics/v/sampling-distribution-of-the-sample-meanhttps://puzzicle.org/thumbs/411022.jpgSampling distribution of the sample meanTake a sample from a population, calculate the mean of that sample, put everything back, and do it over and over. No matter what the population looks like, those sample means will be roughly normally distributed given a reasonably large sample size (at least 30). This is the main idea of the Central Limit Theorem — the sampling distribution of the sample mean is approximately normal for "large" samples.https://cdn.kastatic.org/ka-youtube-converted/FXZ2O1Lv-KE.mp4/FXZ2O1Lv-KE.mp4652yes https://puzzicle.org/math/ap-statistics/v/sampling-distribution-of-the-sample-mean-2https://puzzicle.org/thumbs/378042.jpgSampling distribution of the sample mean (part 2)More on the Central Limit Theorem and the Sampling Distribution of the Sample Meanhttps://cdn.kastatic.org/ka-youtube-converted/NYd6wzYkQIM.mp4/NYd6wzYkQIM.mp4800yes https://puzzicle.org/math/ap-statistics/v/identifying-statistics-and-parametershttps://puzzicle.org/thumbs/xd0d35ef2986c09dc.jpgIdentifying statistics and parametersIn this video, Sal explains how to identify sample statistics and population parameters in context, and recognize the appropriate reasoning for using each term.https://cdn.kastatic.org/ka-youtube-converted/_WGYJrZlUPk.mp4/_WGYJrZlUPk.mp4223yes https://puzzicle.org/math/ap-statistics/v/sample-statistic-bias-worked-examplehttps://puzzicle.org/thumbs/x41b8a947b56549fd.jpgSample statistic bias worked exampleDetermining bias of sample statistics based on approximate sampling distributions examplehttps://cdn.kastatic.org/ka-youtube-converted/SRwMfEmKx3A.mp4/SRwMfEmKx3A.mp4171yes https://puzzicle.org/math/ap-statistics/v/sampling-distribution-of-sample-proportion-part-1https://puzzicle.org/thumbs/xf3a96d17db7a0a33.jpgSampling distribution of sample proportion part 1Formulas for the mean and standard deviation of a sampling distribution of sample proportions.https://cdn.kastatic.org/ka-youtube-converted/Br067hrasc8.mp4/Br067hrasc8.mp4597yes https://puzzicle.org/math/ap-statistics/v/sampling-distribution-of-sample-proportion-part-2https://puzzicle.org/thumbs/xe64811e724d79e4e.jpgSampling distribution of sample proportion part 2Building intuition for the sampling distribution of sample proportions using a simulation.https://cdn.kastatic.org/ka-youtube-converted/ftYKMNfFcI8.mp4/ftYKMNfFcI8.mp4274yes https://puzzicle.org/math/ap-statistics/v/normal-conditions-for-sampling-distributions-of-sample-proportionshttps://puzzicle.org/thumbs/x04a6f94b827f8df4.jpgNormal conditions for sampling distributions of sample proportionsConditions for roughly normal sampling distribution of sample proportionshttps://cdn.kastatic.org/ka-youtube-converted/L7AX2RcbqCg.mp4/L7AX2RcbqCg.mp4377yes https://puzzicle.org/math/ap-statistics/v/probability-of-sample-proportions-examplehttps://puzzicle.org/thumbs/x70dec620eba28cbd.jpgProbability of sample proportions exampleProbability of sample proportions examplehttps://cdn.kastatic.org/ka-youtube-converted/BkRJJsHoRdw.mp4/BkRJJsHoRdw.mp4455yes https://puzzicle.org/math/ap-statistics/v/conditions-for-valid-confidence-intervalshttps://puzzicle.org/thumbs/xafe586eaa18b21b5.jpgConditions for valid confidence intervals for a proportionThere are three conditions we need to satisfy before we make a one-sample z-interval to estimate a population proportion. We need to satisfy the random, normal, and independence conditions for these confidence intervals to be valid.https://cdn.kastatic.org/ka-youtube-converted/8E8YQY5qE3s.mp4/8E8YQY5qE3s.mp4404yes https://puzzicle.org/math/ap-statistics/v/conditions-for-confidence-intervals-worked-exampleshttps://puzzicle.org/thumbs/x59be24b1707f5dff.jpgConditions for confidence interval for a proportion worked examplesConditions for confidence intervals for a population proportion worked examples--random condition, independence condition and normal condition.https://cdn.kastatic.org/ka-youtube-converted/PwMqupyG4D0.mp4/PwMqupyG4D0.mp4294yes https://puzzicle.org/math/ap-statistics/v/critical-value-for-a-given-confidence-levelhttps://puzzicle.org/thumbs/x07560ab28415eebe.jpgCritical value (z*) for a given confidence levelIf we want to be 95% confident, we need to build a confidence interval that extends about 2 standard errors above and below our estimate. More precisely, it's actually 1.96 standard errors. This is called a critical value (z*). We can calculate a critical value z* for any given confidence level using normal distribution calculations.https://cdn.kastatic.org/ka-youtube-converted/VILsYH84HJE.mp4/VILsYH84HJE.mp4382yes https://puzzicle.org/math/ap-statistics/v/example-calculating-confidence-intervalhttps://puzzicle.org/thumbs/xfa0cc7216c02b4bd.jpgExample constructing and interpreting a confidence interval for pCheck conditions, calculate, and interpret a confidence interval to estimate a population proportionhttps://cdn.kastatic.org/ka-youtube-converted/sjG9Ns_8x6c.mp4/sjG9Ns_8x6c.mp4395yes https://puzzicle.org/math/ap-statistics/v/determining-sample-size-based-on-confidence-and-margin-of-errorhttps://puzzicle.org/thumbs/xeeac6e2fe9157c97.jpgDetermining sample size based on confidence and margin of errorDetermining sample size based on confidence level and margin of errorhttps://cdn.kastatic.org/ka-youtube-converted/VyFs7fsWE6w.mp4/VyFs7fsWE6w.mp4481yes https://puzzicle.org/math/ap-statistics/v/idea-behind-hypothesis-testinghttps://puzzicle.org/thumbs/xe84952a8.jpgIdea behind hypothesis testingThe lesson explores hypothesis testing in statistics, demonstrating how varied outcomes from a test with a known accuracy rate can influence the acceptance or rejection of a hypothesis. This understanding is key to evaluating the reliability of statistical tests.https://cdn.kastatic.org/ka-youtube-converted/dpGmVV0-4jc.mp4/dpGmVV0-4jc.mp4469yes https://puzzicle.org/math/ap-statistics/v/examples-of-null-and-alternative-hypotheseshttps://puzzicle.org/thumbs/x1e4199a69e22e6db.jpgExamples of null and alternative hypothesesThe null and alternative hypotheses are both statements about the population that you are studying. The null hypothesis is often stated as the assumption that there is no change, no difference between two groups, or no relationship between two variables. The alternative hypothesis, on the other hand, is the statement that there is a change, difference, or relationship.https://cdn.kastatic.org/ka-youtube-converted/_3_6wjycJdk.mp4/_3_6wjycJdk.mp4342yes https://puzzicle.org/math/ap-statistics/v/constructing-hypotheses-for-a-significance-testhttps://puzzicle.org/thumbs/x4d18b4c2c1fb3da4.jpgConstructing hypotheses for a significance test about a proportionConstructing hypotheses for a significance test about a proportionhttps://cdn.kastatic.org/ka-youtube-converted/8MJvK26PwSo.mp4/8MJvK26PwSo.mp4256yes https://puzzicle.org/math/ap-statistics/v/conditions-for-a-z-test-about-a-proportionhttps://puzzicle.org/thumbs/xb2824472b0e22385.jpgConditions for a z test about a proportionExamples showing how to check whether or not the conditions have been met for doing a z test about a proportion.https://cdn.kastatic.org/ka-youtube-converted/5JlJ4Qfk8g4.mp4/5JlJ4Qfk8g4.mp4333yes https://puzzicle.org/math/ap-statistics/v/p-values-and-significance-testshttps://puzzicle.org/thumbs/x05c8d25c91aab4bb.jpgP-values and significance testsWe compare a P-value to a significance level to make a conclusion in a significance test. Given the null hypothesis is true, a p-value is the probability of getting a result as or more extreme than the sample result by random chance alone. If a p-value is lower than our significance level, we reject the null hypothesis. If not, we fail to reject the null hypothesis.https://cdn.kastatic.org/ka-youtube-converted/KS6KEWaoOOE.mp4/KS6KEWaoOOE.mp4478yes https://puzzicle.org/math/ap-statistics/v/comparing-p-value-diff-significance-levelshttps://puzzicle.org/thumbs/x9672ca0f07f2f79d.jpgComparing P-values to different significance levelsExample comparing P-values to different significance levels, and why it's important to set the significance level before a test.https://cdn.kastatic.org/ka-youtube-converted/DQCF4kTXf9Q.mp4/DQCF4kTXf9Q.mp4281yes https://puzzicle.org/math/ap-statistics/v/estimating-p-value-from-simulationhttps://puzzicle.org/thumbs/x830e99a9880651fa.jpgEstimating a P-value from a simulationExample of estimating a P-value based on a simulation to approximate a sampling distribution assuming the null hypothesis is true.https://cdn.kastatic.org/ka-youtube-converted/hNDlqAYRDyg.mp4/hNDlqAYRDyg.mp4285yes https://puzzicle.org/math/ap-statistics/v/calculating-a-z-statistic-in-a-significance-testhttps://puzzicle.org/thumbs/x083d0a99edd2451d.jpgCalculating a z statistic in a test about a proportionCalculating a z statistic in a one-sample z test about a proportionhttps://cdn.kastatic.org/ka-youtube-converted/DfpFfIdwcIo.mp4/DfpFfIdwcIo.mp4291yes https://puzzicle.org/math/ap-statistics/v/calculating-p-value-from-z-tablehttps://puzzicle.org/thumbs/xb627473a87be1a83.jpgCalculating a P-value given a z statisticIn a significance test about a population proportion, we first calculate a test statistic based on our sample results. We then calculate a p-value based on that test statistic using a normal distribution.https://cdn.kastatic.org/ka-youtube-converted/NNpv-n_Hbvc.mp4/NNpv-n_Hbvc.mp4327yes https://puzzicle.org/math/ap-statistics/v/introduction-to-type-i-and-type-ii-errorshttps://puzzicle.org/thumbs/xf25154435cc3c778.jpgIntroduction to Type I and Type II errorsBoth type 1 and type 2 errors are mistakes made when testing a hypothesis. A type 1 error occurs when you wrongly reject the null hypothesis (i.e. you think you found a significant effect when there really isn't one). A type 2 error occurs when you wrongly fail to reject the null hypothesis (i.e. you miss a significant effect that is really there).https://cdn.kastatic.org/ka-youtube-converted/Hdbbx7DIweQ.mp4/Hdbbx7DIweQ.mp4303yes https://puzzicle.org/math/ap-statistics/v/examples-identifying-type-i-and-type-ii-errorshttps://puzzicle.org/thumbs/x458609e30135e0d0.jpgExamples identifying Type I and Type II errorsExamples identifying Type I and Type II errorshttps://cdn.kastatic.org/ka-youtube-converted/JNOWWJeZBTc.mp4/JNOWWJeZBTc.mp4344yes https://puzzicle.org/math/ap-statistics/v/introduction-to-power-in-significance-testshttps://puzzicle.org/thumbs/x69796141f98e1fbb.jpgIntroduction to power in significance testsIntroduction to power in significance testshttps://cdn.kastatic.org/ka-youtube-converted/6_Cuz0QqRWc.mp4/6_Cuz0QqRWc.mp4585yes https://puzzicle.org/math/ap-statistics/v/examples-thinking-about-power-in-significance-testshttps://puzzicle.org/thumbs/x5ddacb57b3192721.jpgExamples thinking about power in significance testsExamples thinking about power in significance testshttps://cdn.kastatic.org/ka-youtube-converted/XnYUFgfbvm4.mp4/XnYUFgfbvm4.mp4262yes https://puzzicle.org/math/ap-statistics/v/sampling-dist-diff-proportionshttps://puzzicle.org/thumbs/x779db425586e4212.jpgSampling distribution of the difference in sample proportionsWe can calculate the mean and standard deviation for the sampling distribution of the difference in sample proportions. Also, we can tell if the shape of that sampling distribution is approximately normal.https://cdn.kastatic.org/ka-youtube-converted/oThlns-qBaI.mp4/oThlns-qBaI.mp4538yes https://puzzicle.org/math/ap-statistics/v/sampling-dist-diff-proportion-probabilityhttps://puzzicle.org/thumbs/x0192a06875173c7b.jpgSampling distribution of the difference in sample proportions: Probability exampleWe can use the mean and standard deviation and normal shape to calculate probability in a sampling distribution of the difference in sample proportions.https://cdn.kastatic.org/ka-youtube-converted/-Sg3F-vK_cY.mp4/-Sg3F-vK_cY.mp4232yes https://puzzicle.org/math/ap-statistics/v/confidence-intervals-for-the-difference-between-two-proportionshttps://puzzicle.org/thumbs/x18d0aa6bccac869c.jpgConfidence intervals for the difference between two proportionsIntroducing two-sample z intervals to estimate the difference between two population proportions.https://cdn.kastatic.org/ka-youtube-converted/hcUxFwUjCxM.mp4/hcUxFwUjCxM.mp4523yes https://puzzicle.org/math/ap-statistics/v/examples-identifying-conditions-for-two-sample-z-tests-and-z-intervalshttps://puzzicle.org/thumbs/x442bd7f649a8f16c.jpgExamples identifying conditions for inference on two proportionsExamples identifying conditions for confidence intervals and tests about two proportions.https://cdn.kastatic.org/ka-youtube-converted/DR5ZytN7Pj0.mp4/DR5ZytN7Pj0.mp4276yes https://puzzicle.org/math/ap-statistics/v/calculating-two-sample-z-interval-confidence-interval-for-difference-of-proportihttps://puzzicle.org/thumbs/xb316c2bc7a3ece27.jpgCalculating a confidence interval for the difference of proportionsCalculating two-sample z interval to estimate the difference between two population proportions.https://cdn.kastatic.org/ka-youtube-converted/h_ZIMHgKTP0.mp4/h_ZIMHgKTP0.mp4341yes https://puzzicle.org/math/ap-statistics/v/hypothesis-test-for-difference-in-proportionshttps://puzzicle.org/thumbs/x906625cbaa9c085c.jpgHypothesis test for difference in proportionsHypothesis test for difference in proportionshttps://cdn.kastatic.org/ka-youtube-converted/7iTWX8bfBv0.mp4/7iTWX8bfBv0.mp4559yes https://puzzicle.org/math/ap-statistics/v/constructing-hypotheses-for-two-proportionshttps://puzzicle.org/thumbs/x9c00eba8ee3befe5.jpgConstructing hypotheses for two proportionsConstructing hypotheses for two proportionshttps://cdn.kastatic.org/ka-youtube-converted/NCENjXTMp9I.mp4/NCENjXTMp9I.mp4291yes https://puzzicle.org/math/ap-statistics/v/hypothesis-test-for-difference-in-proportions-examplehttps://puzzicle.org/thumbs/xf576d78506270a34.jpgHypothesis test for difference in proportions exampleHypothesis test for difference in proportions examplehttps://cdn.kastatic.org/ka-youtube-converted/HOyNEU94xJQ.mp4/HOyNEU94xJQ.mp4628yes https://puzzicle.org/math/ap-statistics/v/p-value-conclusions-difference-proportionshttps://puzzicle.org/thumbs/x83803c85b979bb7d.jpgComparing P value to significance level for test involving difference of proportionsComparing P value to significance level for test involving difference of proportionshttps://cdn.kastatic.org/ka-youtube-converted/Y650kaYNlJU.mp4/Y650kaYNlJU.mp4150yes https://puzzicle.org/math/ap-statistics/v/chi-square-test-homogeneityhttps://puzzicle.org/thumbs/xa927b364259449d6.jpgIntroduction to the chi-square test for homogeneityWe investigate a scenario that includes applying the chi-squared statistic for a test of homogeneity, setting up null and alternative hypotheses, calculating expected values, verifying conditions for conducting a chi-squared test, and determining degrees of freedom based on the structure of a contingency table.https://cdn.kastatic.org/ka-youtube-converted/t_jfTOE44YQ.mp4/t_jfTOE44YQ.mp4476yes https://puzzicle.org/math/ap-statistics/v/chi-square-test-association-independencehttps://puzzicle.org/thumbs/x3fa714d0d3bae06f.jpgChi-square test for association (independence)Chi-square test for association/independence.https://cdn.kastatic.org/ka-youtube-converted/zOvUQWOzTlc.mp4/zOvUQWOzTlc.mp4628yes https://puzzicle.org/math/ap-statistics/v/standard-error-of-the-meanhttps://puzzicle.org/thumbs/397052.jpgStandard error of the meanTake a sample from a population, calculate the mean of that sample, put everything back, and do it over and over. How much do those sample means tend to vary from the "average" sample mean? This is what the standard error of the mean measures. Its longer name is the standard deviation of the sampling distribution of the sample mean.https://cdn.kastatic.org/ka-youtube-converted/J1twbrHel3o.mp4/J1twbrHel3o.mp4915yes https://puzzicle.org/math/ap-statistics/v/sampling-distribution-example-problemhttps://puzzicle.org/thumbs/6022796.jpgExample: Probability of sample mean exceeding a valueEstimating the probability that the sample mean exceeds a given value in the sampling distribution of the sample mean.https://cdn.kastatic.org/ka-youtube-converted/0ZstEh_8bYc.mp4/0ZstEh_8bYc.mp4868yes https://puzzicle.org/math/ap-statistics/v/introduction-to-t-statisticshttps://puzzicle.org/thumbs/x8c7a88d2d7e2e655.jpgIntroduction to t statisticsAn introduction to why we use t statistics.https://cdn.kastatic.org/ka-youtube-converted/a2rd4Qy8yNI.mp4/a2rd4Qy8yNI.mp4266yes https://puzzicle.org/math/ap-statistics/v/simulation-showing-value-of-t-statistichttps://puzzicle.org/thumbs/xd476a6729ce90085.jpgSimulation showing value of t statisticSee why we use t statistics when building confidence intervals for a mean using the sample standard deviation in place of the population standard deviation.https://cdn.kastatic.org/ka-youtube-converted/gLE6y_NwmhQ.mp4/gLE6y_NwmhQ.mp4194yes https://puzzicle.org/math/ap-statistics/v/conditions-for-valid-t-intervalshttps://puzzicle.org/thumbs/x76eae7745fbf1dc1.jpgConditions for valid t intervalsExamples showing how to determine if the conditions have been met for making a t interval to estimate a mean.https://cdn.kastatic.org/ka-youtube-converted/WPxnNjD_yoU.mp4/WPxnNjD_yoU.mp4294yes https://puzzicle.org/math/ap-statistics/v/example-finding-critical-t-valuehttps://puzzicle.org/thumbs/x4d48d6d578ca9c06.jpgExample finding critical t valueGiven a confidence level, we can calculate a critical value (t*) in a t-distribution with n-1 degrees of freedom.https://cdn.kastatic.org/ka-youtube-converted/mLh_FPwMdcY.mp4/mLh_FPwMdcY.mp4239yes https://puzzicle.org/math/ap-statistics/v/calculating-a-one-sample-t-interval-for-a-meanhttps://puzzicle.org/thumbs/x7d96e465ca35a4b0.jpgExample constructing a t interval for a meanExample showing how to calculate a one-sample t interval for a mean.https://cdn.kastatic.org/ka-youtube-converted/6G3jMjAqNqM.mp4/6G3jMjAqNqM.mp4358yes https://puzzicle.org/math/ap-statistics/v/confidence-interval-for-a-mean-differencehttps://puzzicle.org/thumbs/x06d4814f0bdba692.jpgConfidence interval for a mean with paired dataConfidence interval for a mean difference with paired datahttps://cdn.kastatic.org/ka-youtube-converted/gWaA8mVBZQk.mp4/gWaA8mVBZQk.mp4586yes https://puzzicle.org/math/ap-statistics/v/writing-hypotheses-for-significance-test-about-meanshttps://puzzicle.org/thumbs/xd988b0853e5e8e20.jpgWriting hypotheses for a significance test about a meanExample of constructing hypotheses for a test about a meanhttps://cdn.kastatic.org/ka-youtube-converted/89Dp9Xq4eaA.mp4/89Dp9Xq4eaA.mp4154yes https://puzzicle.org/math/ap-statistics/v/conditions-t-test-meanhttps://puzzicle.org/thumbs/x946974efaad1ff8e.jpgConditions for a t test about a meanExample showing how to check which conditions have been met for a t test about a meanhttps://cdn.kastatic.org/ka-youtube-converted/GtokpL4f32s.mp4/GtokpL4f32s.mp4348yes https://puzzicle.org/math/ap-statistics/v/when-to-use-z-or-t-statistics-in-significance-testshttps://puzzicle.org/thumbs/x587b2b5cf9c4884a.jpgWhen to use z or t statistics in significance testsWhen to use z or t statistics in significance testshttps://cdn.kastatic.org/ka-youtube-converted/-bhfcBvyWoc.mp4/-bhfcBvyWoc.mp4291yes https://puzzicle.org/math/ap-statistics/v/example-calculating-t-statistic-for-signficance-testhttps://puzzicle.org/thumbs/xed967efdbf796cc7.jpgExample calculating t statistic for a test about a meanThe lesson demonstrates conducting a t-test for a population mean with a teacher experience sample. It explains the t-statistic calculation and the p-value interpretation to test a null hypothesis. This process helps ascertain if the true mean is less than the hypothesized mean.https://cdn.kastatic.org/ka-youtube-converted/BJl9WA787-Q.mp4/BJl9WA787-Q.mp4302yes https://puzzicle.org/math/ap-statistics/v/using-ti-calculator-for-p-value-from-t-statistichttps://puzzicle.org/thumbs/x7905126b8d0a8152.jpgUsing TI calculator for P-value from t statisticIn a significance test about a population mean, we first calculate a test statistic based on our sample results. We can then use technology to calculate the p-value based on that test statistic using a t distribution with n-1 degrees of freedom.https://cdn.kastatic.org/ka-youtube-converted/M2P96TJMEaM.mp4/M2P96TJMEaM.mp4199yes https://puzzicle.org/math/ap-statistics/v/calculating-p-value-from-t-statistichttps://puzzicle.org/thumbs/x23ff937dfc559767.jpgUsing a table to estimate P-value from t statisticIn a significance test about a population mean, we first calculate a test statistic based on our sample results. We can then use a table to estimate the p-value based on that test statistic using a t distribution with n-1 degrees of freedom.https://cdn.kastatic.org/ka-youtube-converted/z-HSsVARNnk.mp4/z-HSsVARNnk.mp4293yes https://puzzicle.org/math/ap-statistics/v/comparing-p-value-from-t-statistic-to-significance-levelhttps://puzzicle.org/thumbs/x1bbbd6bc7ce03c14.jpgComparing P-value from t statistic to significance levelExample showing how to compare the P-value to a significance level to make a conclusion in a t test.https://cdn.kastatic.org/ka-youtube-converted/I5WSC_iRtwE.mp4/I5WSC_iRtwE.mp4219yes https://puzzicle.org/math/ap-statistics/v/ap-statistics-free-response-on-significance-test-for-meanhttps://puzzicle.org/thumbs/xd576802c969703a3.jpgFree response example: Significance test for a meanAP Statistics free response on significance test for a meanhttps://cdn.kastatic.org/ka-youtube-converted/8nCsnsD8Fnw.mp4/8nCsnsD8Fnw.mp4713yes https://puzzicle.org/math/ap-statistics/v/sampling-distribution-diff-meanshttps://puzzicle.org/thumbs/x2fac87e4484b6025.jpgSampling distribution of the difference in sample meansWe can calculate the mean and standard deviation for the sampling distribution of the difference in sample means. And we can tell if the shape of that sampling distribution is approximately normal.https://cdn.kastatic.org/ka-youtube-converted/p24UTvbKZog.mp4/p24UTvbKZog.mp4487yes https://puzzicle.org/math/ap-statistics/v/sampling-dist-diff-means-probabilityhttps://puzzicle.org/thumbs/x8e08ca1374efe3a6.jpgSampling distribution of the difference in sample means: Probability exampleWe can use the mean and standard deviation and normal shape to calculate probability in a sampling distribution of the difference in sample means.https://cdn.kastatic.org/ka-youtube-converted/5OYtncxCR50.mp4/5OYtncxCR50.mp4223yes https://puzzicle.org/math/ap-statistics/v/conditions-for-inference-for-difference-of-meanshttps://puzzicle.org/thumbs/x332a11a1b1aa7c98.jpgConditions for inference for difference of meansConditions for inference for difference of meanshttps://cdn.kastatic.org/ka-youtube-converted/Yadshgsx6FQ.mp4/Yadshgsx6FQ.mp4203yes https://puzzicle.org/math/ap-statistics/v/constructing-t-interval-for-difference-of-meanshttps://puzzicle.org/thumbs/x55b4bb1b6469ce0b.jpgConstructing t interval for difference of meansConstructing t interval for difference of meanshttps://cdn.kastatic.org/ka-youtube-converted/Lh1TPIFH7iI.mp4/Lh1TPIFH7iI.mp4393yes https://puzzicle.org/math/ap-statistics/v/calculating-confidence-interval-for-difference-of-meanshttps://puzzicle.org/thumbs/xf4da59b85aaf41fc.jpgCalculating confidence interval for difference of meansCalculating confidence interval for difference of meanshttps://cdn.kastatic.org/ka-youtube-converted/pVPPDbZGd04.mp4/pVPPDbZGd04.mp4290yes https://puzzicle.org/math/ap-statistics/v/alternative-hypothesis-for-2-sample-t-testhttps://puzzicle.org/thumbs/x28325b281c9a31f9.jpgHypotheses for a two-sample t testExample of how to write hypotheses for a two-sample t test, and how to distinguish between paired and two-sample comparisons between means.https://cdn.kastatic.org/ka-youtube-converted/8CJ-RsoYbg0.mp4/8CJ-RsoYbg0.mp4240yes https://puzzicle.org/math/ap-statistics/v/example-of-hypotheses-for-paired-and-2-sample-t-testshttps://puzzicle.org/thumbs/xfea83edcce1cd42b.jpgExample of hypotheses for paired and two-sample t testsExample of hypotheses for paired and two-sample t testshttps://cdn.kastatic.org/ka-youtube-converted/X0gIJUXz6jc.mp4/X0gIJUXz6jc.mp4229yes https://puzzicle.org/math/ap-statistics/v/two-sample-t-test-for-difference-of-meanshttps://puzzicle.org/thumbs/x5108846a61b527e8.jpgTwo-sample t test for difference of meansGiven data from two samples, we can do a signficance test to compare the sample means with a test statistic and p-value, and determine if there is enough evidence to suggest a difference between the two population means.https://cdn.kastatic.org/ka-youtube-converted/NkGvw18zlGQ.mp4/NkGvw18zlGQ.mp4416yes https://puzzicle.org/math/ap-statistics/v/conclusion-for-2-sample-t-test-from-p-valuehttps://puzzicle.org/thumbs/xd4c85a8cd12c1924.jpgConclusion for a two-sample t test using a P-valueGiven results of a two-sample t test, compare the P-value to the significance level to make a conclusion in context about the difference between two means.https://cdn.kastatic.org/ka-youtube-converted/ogFLbvKru8A.mp4/ogFLbvKru8A.mp4149yes https://puzzicle.org/math/ap-statistics/v/calculating-the-equation-of-a-regression-linehttps://puzzicle.org/thumbs/x3acdc86c38b8c2f6.jpgCalculating the equation of a regression lineWe determine the correlation coefficient for bivariate data, which helps understand the relationship between variables. We then build the equation for the least squares line, using standard deviations and the correlation coefficient. The regression line equation y hat = mx + b is calculated.https://cdn.kastatic.org/ka-youtube-converted/FGesqq22TCM.mp4/FGesqq22TCM.mp4491yes https://puzzicle.org/math/ap-statistics/v/using-least-squares-regression-outputhttps://puzzicle.org/thumbs/x143f9602822e8462.jpgUsing least squares regression outputWorked example using least squares regression outputhttps://cdn.kastatic.org/ka-youtube-converted/vEon1weoA3o.mp4/vEon1weoA3o.mp4214yes https://puzzicle.org/math/statistics-probability/v/review-and-intuition-why-we-divide-by-n-1-for-the-unbiased-sample-variancehttps://puzzicle.org/thumbs/1164004886.jpgReview and intuition why we divide by n-1 for the unbiased sample varianceReviewing the population mean, sample mean, population variance, sample variance and building an intuition for why we divide by n-1 for the unbiased sample variancehttps://cdn.kastatic.org/ka-youtube-converted/KkaU2ur3Ymw.mp4/KkaU2ur3Ymw.mp4584yes https://puzzicle.org/math/statistics-probability/v/another-simulation-giving-evidence-that-n-1-gives-us-an-unbiased-estimate-of-varhttps://puzzicle.org/thumbs/1174713420.jpgWhy we divide by n - 1 in varianceAnother visualization providing evidence that dividing by n-1 truly gives an unbiased estimate of population variance. Simulation at: http://www.khanacademy.org/cs/unbiased-variance-visualization/1167453164https://cdn.kastatic.org/ka-youtube-converted/F2mfEldxsPI.mp4/F2mfEldxsPI.mp4286yes https://puzzicle.org/math/statistics-probability/v/simulation-showing-bias-in-sample-variancehttps://puzzicle.org/thumbs/1174647817.jpgSimulation showing bias in sample varianceSimulation by Peter Collingridge giving us a better understanding of why we divide by (n-1) when calculating the unbiased sample variance. Simulation available at: http://www.khanacademy.org/cs/challenge-unbiased-estimate-of-population-variance/1169428428https://cdn.kastatic.org/ka-youtube-converted/Cn0skMJ2F3c.mp4/Cn0skMJ2F3c.mp4384yes https://puzzicle.org/math/statistics-probability/v/simulation-providing-evidence-that-n-1-gives-us-unbiased-estimatehttps://puzzicle.org/thumbs/1174782081.jpgSimulation providing evidence that (n-1) gives us unbiased estimateSimulation by KA user tetef showing that dividing by (n-1) gives us an unbiased estimate of population variance. Simulation at: http://www.khanacademy.org/cs/will-it-converge-towards-1/1167579097https://cdn.kastatic.org/ka-youtube-converted/0QS4VHPV4JA.mp4/0QS4VHPV4JA.mp4270yes https://puzzicle.org/math/probability/v/calculating-percentilehttps://puzzicle.org/thumbs/x14d5c1380b3b7971.jpgCalculating percentileLearn how to calculate the percentile rank for a given data point.https://cdn.kastatic.org/ka-youtube-converted/Ngyt8Q5tWkU.mp4/Ngyt8Q5tWkU.mp4218yes https://puzzicle.org/math/statistics-probability/v/analyzing-a-cumulative-relative-frequency-graphhttps://puzzicle.org/thumbs/x6970bb2f4549cd5e.jpgAnalyzing a cumulative relative frequency graphWe investigate a scenario that includes interpreting cumulative relative frequency graphs, using percentiles and quartiles to estimate central tendency and spread within a distribution, and calculating the interquartile range based on the interpretation of the graph.https://cdn.kastatic.org/ka-youtube-converted/TwGYLQ-DNdc.mp4/TwGYLQ-DNdc.mp4355yes https://puzzicle.org/math/statistics-probability/v/density-curveshttps://puzzicle.org/thumbs/x38d7e38a58c80236.jpgDensity CurvesAn introduction to density curves for visualizing distributions. A brief review of frequency histograms and relative frequency histograms as well.https://cdn.kastatic.org/ka-youtube-converted/PUvUQMQ7xQk.mp4/PUvUQMQ7xQk.mp4574yes https://puzzicle.org/math/statistics-probability/v/median-mean-and-skew-from-density-curveshttps://puzzicle.org/thumbs/xb4e2aa7a83b8779f.jpgMedian, mean and skew from density curvesHigh level analysis of density curves. A focus on median, mean, left-skew and right-skew.https://cdn.kastatic.org/ka-youtube-converted/JFesFhraX2M.mp4/JFesFhraX2M.mp4365yes https://puzzicle.org/math/probability/v/finding-z-score-for-a-percentilehttps://puzzicle.org/thumbs/x2f6934241a8b25ce.jpgFinding z-score for a percentileIn any normal distribution, we can find the z-score that corresponds to some percentile rank. If we're given a particular normal distribution with some mean and standard deviation, we can use that z-score to find the actual cutoff for that percentile. In this example, we find what pulse rate represents the top 30% of all pulse rates in a population.https://cdn.kastatic.org/ka-youtube-converted/S5_5KyCVjrU.mp4/S5_5KyCVjrU.mp4254yes https://puzzicle.org/math/statistics-probability/v/r-squared-or-coefficient-of-determinationhttps://puzzicle.org/thumbs/8187298.jpgR-squared or coefficient of determinationIn linear regression, r-squared (also called the coefficient of determination) is the proportion of variation in the response variable that is explained by the explanatory variable in the model.https://cdn.kastatic.org/ka-youtube-converted/lng4ZgConCM.mp4/lng4ZgConCM.mp4761yes https://puzzicle.org/math/ap-statistics/v/standard-dev-residualshttps://puzzicle.org/thumbs/xaecf749b9a5db4fb.jpgStandard deviation of residuals or root mean square deviation (RMSD)Standard deviation of the residuals are a measure of how well a regression line fits the data. It is also known as root mean square deviation or root mean square error.https://cdn.kastatic.org/ka-youtube-converted/hSP_rmNHU_U.mp4/hSP_rmNHU_U.mp4426yes https://puzzicle.org/math/statistics-probability/v/interpreting-computer-regression-datahttps://puzzicle.org/thumbs/xde97338b3dce169e.jpgInterpreting computer regression dataInterpreting computer generated regression data to find the equation of a least-squares regression line. Predictors and coefficients. S and R-squared.https://cdn.kastatic.org/ka-youtube-converted/sIJj7Q77SVI.mp4/sIJj7Q77SVI.mp4313yes https://puzzicle.org/math/statistics-probability/v/impact-of-removing-outliers-on-regression-lineshttps://puzzicle.org/thumbs/xc6423bf0474428d0.jpgImpact of removing outliers on regression linesImpact of removing outliers on slope, y-intercept and r of least-squares regression lines.https://cdn.kastatic.org/ka-youtube-converted/TbS7S7lb9Es.mp4/TbS7S7lb9Es.mp4344yes https://puzzicle.org/math/ap-statistics/v/influential-points-regressionhttps://puzzicle.org/thumbs/x69913ce3b3f4e63e.jpgInfluential points in regressionOutliers and high-leverage points can be influential to different measurements in least-squares regression like the slope, y-intercept, and correlation coefficient (r).https://cdn.kastatic.org/ka-youtube-converted/jHHPGhrcMOY.mp4/jHHPGhrcMOY.mp4255yes https://puzzicle.org/math/statistics-probability/v/transforming-nonlinear-datahttps://puzzicle.org/thumbs/x62fff31072ed0c1c.jpgTransforming nonlinear dataUse logarithms to transform nonlinear data into a linear relationship so we can use least-squares regression methods.https://cdn.kastatic.org/ka-youtube-converted/A1H4j97paI4.mp4/A1H4j97paI4.mp4175yes https://puzzicle.org/math/statistics-probability/v/transformed-linear-regression-examplehttps://puzzicle.org/thumbs/x27e206519815626b.jpgWorked example of linear regression using transformed dataWorked example of linear regression using transformed data. Adapted from 2007 AP Statistics free response, form b, question 6, part d.https://cdn.kastatic.org/ka-youtube-converted/dEVGfxHF1yo.mp4/dEVGfxHF1yo.mp4460yes https://puzzicle.org/math/probability/v/intro-experiment-designhttps://puzzicle.org/thumbs/x2d7343379f74c4c7.jpgIntroduction to experiment designExplanatory and response variables. Random sampling and random assignment (including block design). Placebos and blind/double-blind experiments.https://cdn.kastatic.org/ka-youtube-converted/Fw8qsuZNzH4.mp4/Fw8qsuZNzH4.mp4400yes https://puzzicle.org/math/probability/v/bayes-theorem-visualizedhttps://puzzicle.org/thumbs/1317894497.jpgConditional probability with Bayes' TheoremConditional probability visualized using trees.https://cdn.kastatic.org/ka-youtube-converted/Zxm4Xxvzohk.mp4/Zxm4Xxvzohk.mp4306yes https://puzzicle.org/math/statistics-probability/v/getting-at-least-one-headshttps://puzzicle.org/thumbs/191769946.jpg"At least one" probability with coin flippingIn this video, we 'll explore the probability of getting at least one heads in multiple flips of a fair coin.https://cdn.kastatic.org/ka-youtube-converted/mkyZ45KQYi4.mp4/mkyZ45KQYi4.mp4560yes https://puzzicle.org/math/statistics-probability/v/impact-of-scaling-and-shifting-random-variableshttps://puzzicle.org/thumbs/x39496cee07d4d44f.jpgImpact of transforming (scaling and shifting) random variablesLinear transformations (addition and multiplication of a constant) and their impacts on center (mean) and spread (standard deviation) of a distribution.https://cdn.kastatic.org/ka-youtube-converted/fx3TtExi9e8.mp4/fx3TtExi9e8.mp4386yes https://puzzicle.org/math/statistics-probability/v/example-transforming-a-discrete-random-variablehttps://puzzicle.org/thumbs/x09997886d004552f.jpgExample: Transforming a discrete random variableExample of transforming a discrete random variablehttps://cdn.kastatic.org/ka-youtube-converted/nb3m_Q7rfT4.mp4/nb3m_Q7rfT4.mp4250yes https://puzzicle.org/math/statistics-probability/v/mean-of-sum-and-difference-of-random-variableshttps://puzzicle.org/thumbs/x0961ac866dd6c24a.jpgMean of sum and difference of random variablesMean of sum and difference of random variableshttps://cdn.kastatic.org/ka-youtube-converted/RYSN1CARHtE.mp4/RYSN1CARHtE.mp4202yes https://puzzicle.org/math/statistics-probability/v/variance-of-sum-and-difference-of-random-variableshttps://puzzicle.org/thumbs/xbed7d77d46342fbc.jpgVariance of sum and difference of random variablesIntuition for why the variance of both the sum and difference of two independent random variables is equal to the sum of their variances.https://cdn.kastatic.org/ka-youtube-converted/3VXhi_0Mclg.mp4/3VXhi_0Mclg.mp4487yes https://puzzicle.org/math/statistics-probability/v/intuition-for-why-independence-matters-for-variance-of-sumhttps://puzzicle.org/thumbs/xcd147b9df75754e0.jpgIntuition for why independence matters for variance of sumIntuition for why independence matters for variance of sumhttps://cdn.kastatic.org/ka-youtube-converted/UAOJgqdsK9w.mp4/UAOJgqdsK9w.mp4263yes https://puzzicle.org/math/statistics-probability/v/variance-of-differences-of-random-variableshttps://puzzicle.org/thumbs/6022811.jpgDeriving the variance of the difference of random variablesSal derives the variance of the difference of random variableshttps://cdn.kastatic.org/ka-youtube-converted/rLdoKZ7w0xI.mp4/rLdoKZ7w0xI.mp4647yes https://puzzicle.org/math/statistics-probability/v/analyzing-distribution-of-sum-of-two-normally-distributed-random-variableshttps://puzzicle.org/thumbs/x874938306036423b.jpgExample: Analyzing distribution of sum of two normally distributed random variablesFinding the probability that the total of some random variables exceeds an amount by understanding the distribution of the sum of normally distributed variables.https://cdn.kastatic.org/ka-youtube-converted/Z6vANdo1L68.mp4/Z6vANdo1L68.mp4351yes https://puzzicle.org/math/statistics-probability/v/analyzing-the-difference-in-distributionshttps://puzzicle.org/thumbs/x45f991a6e84e9da8.jpgExample: Analyzing the difference in distributionsFinding the probability that a randomly selected woman is taller than a randomly selected man by understanding the distribution of the difference of normally distributed variables.https://cdn.kastatic.org/ka-youtube-converted/_Pq547nYFaE.mp4/_Pq547nYFaE.mp4585yes https://puzzicle.org/math/statistics-probability/v/10-percent-rule-independencehttps://puzzicle.org/thumbs/xee11001de8e951c9.jpg10% Rule of assuming "independence" between trials10% Rule of assuming "independence" between trialshttps://cdn.kastatic.org/ka-youtube-converted/elcLysy7exM.mp4/elcLysy7exM.mp4403yes https://puzzicle.org/math/statistics-probability/v/generalizing-k-scores-in-n-attemptshttps://puzzicle.org/thumbs/x0a129ac6.jpgGeneralizing k scores in n attemptsSal generalizes 2 scores in 6 attempts to k scores in n attempts.https://cdn.kastatic.org/ka-youtube-converted/efE8xuvUjAo.mp4/efE8xuvUjAo.mp4253yes https://puzzicle.org/math/statistics-probability/v/free-throw-binomial-probability-distributionhttps://puzzicle.org/thumbs/xead3220b.jpgFree throw binomial probability distributionSal uses the binomial distribution to calculate the probability of making different number of free throws.https://cdn.kastatic.org/ka-youtube-converted/eL965_Lscb8.mp4/eL965_Lscb8.mp4519yes https://puzzicle.org/math/statistics-probability/v/graphing-basketball-binomial-distributionhttps://puzzicle.org/thumbs/xba5fcceb.jpgGraphing basketball binomial distributionSal graphs the results of using the binomial distribution to find the probabilities of making different numbers of free throws.https://cdn.kastatic.org/ka-youtube-converted/y2G03Lumhe0.mp4/y2G03Lumhe0.mp4262yes https://puzzicle.org/math/statistics-probability/v/ti-84-binompdf-and-binomcdf-functionshttps://puzzicle.org/thumbs/x1138367bae9683e9.jpgBinompdf and binomcdf functionsThis tutorial demonstrates how to use a graphing calculator to calculate probabilities for a binomial distribution. It covers key concepts like binomial random variables, binomial probability distribution function, and binomial cumulative distribution function, using the example of making free throws in basketball.https://cdn.kastatic.org/ka-youtube-converted/IngLJs6T1Qw.mp4/IngLJs6T1Qw.mp4327yes https://puzzicle.org/math/statistics-probability/v/geometric-random-variables-introductionhttps://puzzicle.org/thumbs/x19fae2466b67f624.jpgGeometric random variables introductionDistinguishing between geometric and binomial random variableshttps://cdn.kastatic.org/ka-youtube-converted/IVlzp8g1McE.mp4/IVlzp8g1McE.mp4375yes https://puzzicle.org/math/ap-statistics/v/geometric-mean-std-deviationhttps://puzzicle.org/thumbs/xee486f6ef9da8309.jpgGeometric distribution mean and standard deviationWe can calculate and interpret the mean and standard deviation for the distribution of a geometric random variable, and describe its shape.https://cdn.kastatic.org/ka-youtube-converted/XKCeLA4UsXw.mp4/XKCeLA4UsXw.mp4386yes https://puzzicle.org/math/statistics-probability/v/probability-for-a-geometric-random-variablehttps://puzzicle.org/thumbs/x372e97d6330a7fe3.jpgProbability for a geometric random variableFinding the probability for a single outcome of a geometric random variablehttps://cdn.kastatic.org/ka-youtube-converted/jKlgEvRFItE.mp4/jKlgEvRFItE.mp4171yes https://puzzicle.org/math/statistics-probability/v/cumulative-geometric-probability-greater-than-a-valuehttps://puzzicle.org/thumbs/xfdbbc68b8f5b3a7f.jpgCumulative geometric probability (greater than a value)Probability for a geometric random variable being greater than a certain value.https://cdn.kastatic.org/ka-youtube-converted/7TYMKKQK8s8.mp4/7TYMKKQK8s8.mp4303yes https://puzzicle.org/math/statistics-probability/v/cumulative-geometric-probability-less-than-a-valuehttps://puzzicle.org/thumbs/xaacc5e5c87f80355.jpgCumulative geometric probability (less than a value)Probability for a geometric random variable being less than a certain value.https://cdn.kastatic.org/ka-youtube-converted/nxUP4lJ_hbU.mp4/nxUP4lJ_hbU.mp4247yes https://puzzicle.org/math/statistics-probability/v/ti-84-geometpdf-and-geometcdf-functionshttps://puzzicle.org/thumbs/x120d1a2744a614d3.jpgTI-84 geometpdf and geometcdf functionsUsing a TI-84 (very similar for TI-85 or TI-89) calculator for making calculations regarding geometric random variables.https://cdn.kastatic.org/ka-youtube-converted/RMf0F72I7dY.mp4/RMf0F72I7dY.mp4348yes https://puzzicle.org/math/statistics-probability/v/proof-of-expected-value-of-geometric-random-variablehttps://puzzicle.org/thumbs/x609952a24493dc6d.jpgProof of expected value of geometric random variableProof of expected value of geometric random variablehttps://cdn.kastatic.org/ka-youtube-converted/7br_-emGNec.mp4/7br_-emGNec.mp4439yes https://puzzicle.org/math/statistics-probability/v/probabilities-from-density-curveshttps://puzzicle.org/thumbs/xcf52ad8fc5bff8ac.jpgProbabilities from density curvesExamples finding probabilities from probability distributions for continuous random variables.https://cdn.kastatic.org/ka-youtube-converted/3E8BO7VRMEA.mp4/3E8BO7VRMEA.mp4293yes https://puzzicle.org/math/statistics-probability/v/confidence-intervals-and-margin-of-errorhttps://puzzicle.org/thumbs/x6f4d752a0bd525e7.jpgConfidence intervals and margin of errorIf we poll 100 people, and 56% of them support a candidate, we can use what we know about sampling distributions and margin of error to build a confidence interval to estimate the true value of the percentage in the population.https://cdn.kastatic.org/ka-youtube-converted/hlM7zdf7zwU.mp4/hlM7zdf7zwU.mp4705yes https://puzzicle.org/math/statistics-probability/v/confidence-interval-simulationhttps://puzzicle.org/thumbs/x3514d7ca10c2a248.jpgConfidence interval simulationConfidence interval simulationhttps://cdn.kastatic.org/ka-youtube-converted/bGALoCckICI.mp4/bGALoCckICI.mp4265yes https://puzzicle.org/math/statistics-probability/v/interpreting-confidence-intervals-examplehttps://puzzicle.org/thumbs/x2136930aa89bfa56.jpgInterpreting confidence level exampleInterpreting confidence level examplehttps://cdn.kastatic.org/ka-youtube-converted/XZAGtQt-lQo.mp4/XZAGtQt-lQo.mp4300yes https://puzzicle.org/math/statistics-probability/v/comparing-p-value-to-significance-level-examplehttps://puzzicle.org/thumbs/xff94fdc886b2120a.jpgMaking conclusions in a test about a proportionExample of how to make a conclusion in a test about a proportionhttps://cdn.kastatic.org/ka-youtube-converted/RcjZwZJO5qo.mp4/RcjZwZJO5qo.mp4314yes https://puzzicle.org/math/ap-statistics/v/full-significance-test-example-from-ap-testhttps://puzzicle.org/thumbs/x9b9b2ac10698f2ee.jpgSignificance test for a proportion free response exampleCarrying out every step of a significance test on a proportion.https://cdn.kastatic.org/ka-youtube-converted/sqRQSPwNE-4.mp4/sqRQSPwNE-4.mp4782yes https://puzzicle.org/math/ap-statistics/v/full-significance-test-example-from-ap-test-part-2-with-correctionhttps://puzzicle.org/thumbs/x3cd99e7c7d0dd3f2.jpgSignificance test for a proportion free response (part 2 with correction)Full significance test example from AP test part 2 with correction for maximum points on AP exam.https://cdn.kastatic.org/ka-youtube-converted/l5uCPu2t7W4.mp4/l5uCPu2t7W4.mp4170yes https://puzzicle.org/math/statistics-tx/v/confidence-interval-for-hypothesis-test-for-difference-in-proportionshttps://puzzicle.org/thumbs/x35d1563873c98288.jpgConfidence interval for hypothesis test for difference in proportionsConfidence interval for hypothesis test for difference in proportionshttps://cdn.kastatic.org/ka-youtube-converted/-VZUijm02h0.mp4/-VZUijm02h0.mp4330yes https://puzzicle.org/math/statistics-tx/v/conclusion-for-2-sample-t-test-from-confidence-intervalhttps://puzzicle.org/thumbs/x23ef855ee4d013f8.jpgConclusion for a two-sample t test using a confidence intervalGiven a confidence interval estimating the difference between two means, make a conclusion in a significance test about the difference between those means.https://cdn.kastatic.org/ka-youtube-converted/iJr9PpY3PjM.mp4/iJr9PpY3PjM.mp4226yes https://puzzicle.org/math/statistics-probability/v/chi-square-statistichttps://puzzicle.org/thumbs/x842fa2bbaaeadaef.jpgChi-square statistic for hypothesis testingChi-square statistic for hypothesis testing (chi-square goodness-of-fit test)https://cdn.kastatic.org/ka-youtube-converted/jABsbNBPXIk.mp4/jABsbNBPXIk.mp4506yes https://puzzicle.org/math/statistics-probability/v/goodness-of-fit-examplehttps://puzzicle.org/thumbs/x202e0eed5e346fd9.jpgChi-square goodness-of-fit exampleChi-square goodness-of-fit examplehttps://cdn.kastatic.org/ka-youtube-converted/VjO9hWVlOBo.mp4/VjO9hWVlOBo.mp4409yes https://puzzicle.org/math/statistics-probability/v/intro-inference-slopehttps://puzzicle.org/thumbs/xa84bb36461ffc2b3.jpgIntroduction to inference about slope in linear regressionIntroduction to sample slopes and using them to make confidence intervals or do a test about the population slope in least-squares regressionhttps://cdn.kastatic.org/ka-youtube-converted/Pzglbt9_xCI.mp4/Pzglbt9_xCI.mp4432yes https://puzzicle.org/math/statistics-probability/v/conditions-inference-slopehttps://puzzicle.org/thumbs/x997bc64c21b63d1f.jpgConditions for inference on slopeIntroducing the conditions for making a confidence interval or doing a test about slope in least-squares regression.https://cdn.kastatic.org/ka-youtube-converted/kOQx3LtWHEQ.mp4/kOQx3LtWHEQ.mp4291yes https://puzzicle.org/math/statistics-probability/v/confidence-interval-slopehttps://puzzicle.org/thumbs/xd40810aa6b9f90e4.jpgConfidence interval for the slope of a regression lineConfidence interval for the slope of a regression linehttps://cdn.kastatic.org/ka-youtube-converted/8w6EPyEqE9M.mp4/8w6EPyEqE9M.mp4415yes https://puzzicle.org/math/statistics-probability/v/t-statistic-slopehttps://puzzicle.org/thumbs/x8e50c43032661b0c.jpgCalculating t statistic for slope of regression lineCalculate the test statistic in a test about the slope of a regression linehttps://cdn.kastatic.org/ka-youtube-converted/7MAuojBTF-g.mp4/7MAuojBTF-g.mp4436yes https://puzzicle.org/math/statistics-probability/v/making-conclusions-for-regression-slope-hypothesis-testhttps://puzzicle.org/thumbs/xe4ba7721c7cc54d9.jpgUsing a P-value to make conclusions in a test about slopeMaking conclusions in a hypothesis test about the slope of a least-squares regression line.https://cdn.kastatic.org/ka-youtube-converted/Mpd83AuDTrU.mp4/Mpd83AuDTrU.mp4385yes https://puzzicle.org/math/statistics-probability/v/interval-test-slopehttps://puzzicle.org/thumbs/x40f4765e2292a3e6.jpgUsing a confidence interval to test slopeUsing confidence interval for hypothesis test on regression slopehttps://cdn.kastatic.org/ka-youtube-converted/rL9MPL3wqsk.mp4/rL9MPL3wqsk.mp4234yes https://puzzicle.org/math/cc-2nd-grade-math/v/expanded-form-of-numbershttps://puzzicle.org/thumbs/x1e06e0d4e43f89e1.jpgExpanded form of numbersSal shows how to turn numbers from expanded form to number form.https://cdn.kastatic.org/ka-youtube-converted/drZjIlmHdME.mp4/drZjIlmHdME.mp4177yes https://puzzicle.org/math/cc-1st-grade-math/v/number-gridhttps://puzzicle.org/thumbs/xb19b2406.jpgNumber gridSal goes through all the numbers from 0 to 100 and shows some interesting patterns.https://cdn.kastatic.org/ka-youtube-converted/9XZypM2Z3Ro.mp4/9XZypM2Z3Ro.mp4298yes https://puzzicle.org/math/cc-1st-grade-math/v/numbers-to-120https://puzzicle.org/thumbs/xf78091c1.jpgMissing numbers between 0 and 120Sal finds missing numbers between 0 and 120.https://cdn.kastatic.org/ka-youtube-converted/1AqkBdCBm9o.mp4/1AqkBdCBm9o.mp4114yes https://puzzicle.org/math/cc-1st-grade-math/v/place-value-examplehttps://puzzicle.org/thumbs/x426bd66b.jpgPlace value example: 25Sal shows 25 as 2 tens and 5 ones.https://cdn.kastatic.org/ka-youtube-converted/X_PnRFAKbkg.mp4/X_PnRFAKbkg.mp4175yes https://puzzicle.org/math/cc-1st-grade-math/v/rep-quantity-with-digitshttps://puzzicle.org/thumbs/x8a682fbc.jpgPlace value example: 42Sal shows 42 as 4 tens and 2 ones.https://cdn.kastatic.org/ka-youtube-converted/-Zlq5tNl94M.mp4/-Zlq5tNl94M.mp4259yes https://puzzicle.org/math/cc-2nd-grade-math/v/greater-than-and-less-than-symbolshttps://puzzicle.org/thumbs/x48102d13.jpgGreater than and less than symbolsThis video teaches comparing numbers using equal, greater than, and less than symbols. It clarifies the equal sign's meaning and demonstrates using comparison symbols with numbers and expressions.https://cdn.kastatic.org/ka-youtube-converted/nFsQA2Zvy1o.mp4/nFsQA2Zvy1o.mp4304yes https://puzzicle.org/math/cc-2nd-grade-math/v/number-and-word-form-of-a-numberhttps://puzzicle.org/thumbs/x2ca722b3f06e68c6.jpgNumber and word form of a numberSal shows how to turn a number from number form into word form.https://cdn.kastatic.org/ka-youtube-converted/hBCz0VDt2cI.mp4/hBCz0VDt2cI.mp4137yes https://puzzicle.org/math/cc-2nd-grade-math/v/word-form-and-expanded-formhttps://puzzicle.org/thumbs/x10d92a563c3a621a.jpgWord form and expanded formSal shows how to write numbers from word form to expanded form.https://cdn.kastatic.org/ka-youtube-converted/83Tji22tYGY.mp4/83Tji22tYGY.mp486yes https://puzzicle.org/math/arithmetic-home/v/place-value-2https://puzzicle.org/thumbs/2600658.jpgWriting a number in standard formThis video demonstrates converting large numbers from expanded and written form to standard form. Using the example "six hundred forty-five million five hundred eighty-four thousand four hundred sixty-two," it explains the process and arrives at 645,584,462, emphasizing place value understanding.https://cdn.kastatic.org/ka-youtube-converted/wd4cRAoBOiE.mp4/wd4cRAoBOiE.mp4249yes https://puzzicle.org/math/arithmetic-home/v/intro-to-roman-numeralshttps://puzzicle.org/thumbs/x496ef053298635d7.jpgIntro to Roman numeralsLet's get a quick introduction to what Roman numerals are why we learn about them.https://cdn.kastatic.org/ka-youtube-converted/Q3X6Coa2FWo.mp4/Q3X6Coa2FWo.mp4267yes https://puzzicle.org/math/arithmetic-home/v/how-to-read-and-write-roman-numerals-class-6-india-khan-academyhttps://puzzicle.org/thumbs/xba705027c2eec9e8.jpgReading and writing Roman numeralsLet's learn how to read and write Roman numerals.https://cdn.kastatic.org/ka-youtube-converted/W_NymI3Aamo.mp4/W_NymI3Aamo.mp4454yes https://puzzicle.org/math/arithmetic-home/v/roman-numerals-worked-example-class-6-india-khan-academyhttps://puzzicle.org/thumbs/x5d30f29cd339dbbb.jpgRoman numerals worked exampleLet's look at an example of how to convert a number from the usual Hindu-Arabic numerals to Roman numerals.https://cdn.kastatic.org/ka-youtube-converted/dGZTQxyfwPI.mp4/dGZTQxyfwPI.mp4205yes https://puzzicle.org/math/arithmetic-home/v/basic-additionhttps://puzzicle.org/thumbs/23798.jpgBasic additionLet's learn about basic addition by starting with simple examples and moving on to more difficult problems. Two methods for solving these problems are demonstrated: drawing circles to represent each quantity, or using a number line. Practice is emphasized as the key to mastering this skill.https://cdn.kastatic.org/ka-youtube-converted/AuX7nPBqDts.mp4/AuX7nPBqDts.mp4462yes https://puzzicle.org/math/arithmetic-home/v/basic-subtractionhttps://puzzicle.org/thumbs/23808.jpgBasic subtractionIntroduction to subtraction.https://cdn.kastatic.org/ka-youtube-converted/aNqG4ChKShI.mp4/aNqG4ChKShI.mp4512yes https://puzzicle.org/math/cc-kindergarten-math/v/adding-fruithttps://puzzicle.org/thumbs/x041e77f7.jpgAdd and subtract: pieces of fruitSal adds fruit and subtracts fruit. The examples he uses are 5+3 and 5-3.https://cdn.kastatic.org/ka-youtube-converted/ie0waMJxnTs.mp4/ie0waMJxnTs.mp4137yes https://puzzicle.org/math/cc-1st-grade-math/v/relating-addition-and-subtractionhttps://puzzicle.org/thumbs/xa7a18531.jpgRelating addition and subtractionSal shows the relationship between 2+3=5, 5-2=3, and 5-3=2.https://cdn.kastatic.org/ka-youtube-converted/zVLjWIftX_o.mp4/zVLjWIftX_o.mp4125yes https://puzzicle.org/math/cc-kindergarten-math/v/getting-to-10-by-filling-boxeshttps://puzzicle.org/thumbs/x5d9fb90d.jpgGetting to 10 by filling boxesSal shows how to add two numbers to make 10. He uses 3 + ___ = 10 and 6 + ___ = 10 as examples.https://cdn.kastatic.org/ka-youtube-converted/An46SYAxhtc.mp4/An46SYAxhtc.mp4114yes https://puzzicle.org/math/cc-kindergarten-math/v/adding-to-10-examplehttps://puzzicle.org/thumbs/xeebd76e8.jpgAdding to 10Sal finds out how many bananas we need to add to 3 bananas to make 10 bananas.https://cdn.kastatic.org/ka-youtube-converted/9FC0WT186aY.mp4/9FC0WT186aY.mp4120yes https://puzzicle.org/math/cc-kindergarten-math/v/addition-word-problems-within-10https://puzzicle.org/thumbs/xd3451833.jpgAddition word problems within 10Sal solves word problems by adding small numbers (numbers 10 or less).https://cdn.kastatic.org/ka-youtube-converted/-3DFzxbP9Fk.mp4/-3DFzxbP9Fk.mp4102yes https://puzzicle.org/math/cc-kindergarten-math/v/subtraction-word-problems-within-10https://puzzicle.org/thumbs/x9854c66d.jpgSubtraction word problems within 10Sal solves word problems by subtracting small numbers (numbers 10 or less).https://cdn.kastatic.org/ka-youtube-converted/qSkpZswoZTc.mp4/qSkpZswoZTc.mp463yes https://puzzicle.org/math/cc-2nd-grade-math/v/adding-within-20-using-place-value-blockshttps://puzzicle.org/thumbs/x8cc10311ecdb3895.jpgAdding within 20 using place value blocksSal adds 8 + 6 using place value blocks.https://cdn.kastatic.org/ka-youtube-converted/dDv4FTqKBmY.mp4/dDv4FTqKBmY.mp484yes https://puzzicle.org/math/cc-2nd-grade-math/v/adding-within-20-using-ten-frameshttps://puzzicle.org/thumbs/xfa530001a926b085.jpgAdding within 20 using ten framesSal adds 8 + 7 using ten frames.https://cdn.kastatic.org/ka-youtube-converted/UzQnoXrJg1o.mp4/UzQnoXrJg1o.mp4158yes https://puzzicle.org/math/cc-2nd-grade-math/v/adding-within-20https://puzzicle.org/thumbs/x22d1c9bd.jpgAdding 7 + 6Sal adds 7 + 6 using the number line.https://cdn.kastatic.org/ka-youtube-converted/ZgzpTx-s9Zo.mp4/ZgzpTx-s9Zo.mp4233yes https://puzzicle.org/math/cc-2nd-grade-math/v/adding-within-20-examplehttps://puzzicle.org/thumbs/x515583ed.jpgAdding 8 + 7In the video it demonstrates adding 8 + 7 using a visual approach. Adding emphasizes the concept of place value, showing that 15 is made up of 1 group of 10 and 5 ones. This method helps students understand addition within 20 in a more concrete way.https://cdn.kastatic.org/ka-youtube-converted/AK0NZITv5Ns.mp4/AK0NZITv5Ns.mp4199yes https://puzzicle.org/math/cc-2nd-grade-math/v/adding-with-arrayshttps://puzzicle.org/thumbs/x5f1f4fa5b655b190.jpgAdding with arraysSal shows how to relate repeated addition to an array.https://cdn.kastatic.org/ka-youtube-converted/IRjLEJWVtLQ.mp4/IRjLEJWVtLQ.mp4114yes https://puzzicle.org/math/arithmetic-home/v/subtraction-2https://puzzicle.org/thumbs/23810.jpgSubtracting different waysSal subtracts numbers using models and number lines.https://cdn.kastatic.org/ka-youtube-converted/incKJchBCLo.mp4/incKJchBCLo.mp4752yes https://puzzicle.org/math/cc-2nd-grade-math/v/subtract-within-20-using-a-number-linehttps://puzzicle.org/thumbs/x1ed4febfd0a96ed1.jpgSubtract within 20 using a number lineSal subtracts 17 - 9 using a number line.https://cdn.kastatic.org/ka-youtube-converted/HjRcLJMHuVA.mp4/HjRcLJMHuVA.mp480yes https://puzzicle.org/math/cc-2nd-grade-math/v/subtract-within-20-using-place-value-blockshttps://puzzicle.org/thumbs/x4412c816995f85ee.jpgSubtract within 20 using place value blocksSal subtracts 13-4 using place value blocks.https://cdn.kastatic.org/ka-youtube-converted/jhkYFCMUYwA.mp4/jhkYFCMUYwA.mp479yes https://puzzicle.org/math/cc-2nd-grade-math/v/subtract-within-20-using-ten-frameshttps://puzzicle.org/thumbs/x02a60b8a9b838251.jpgSubtract within 20 using ten framesSal subtracts 13-6 using ten frames.https://cdn.kastatic.org/ka-youtube-converted/r8HsSXMn-4E.mp4/r8HsSXMn-4E.mp456yes https://puzzicle.org/math/cc-2nd-grade-math/v/subtracting-within-20https://puzzicle.org/thumbs/x616fafa9.jpgSubtracting 14 - 6Sal subtracts 14 - 6 by first thinking about subtracting 2 and 4.https://cdn.kastatic.org/ka-youtube-converted/9McJ3GobPaY.mp4/9McJ3GobPaY.mp4243yes https://puzzicle.org/math/cc-2nd-grade-math/v/sea-monsters-and-superheroeshttps://puzzicle.org/thumbs/x2e0bef65.jpgAddition and subtraction word problems: superheroesSal solves 1 addition and 1 subtraction word problem.https://cdn.kastatic.org/ka-youtube-converted/fGok2nHOjnI.mp4/fGok2nHOjnI.mp4218yes https://puzzicle.org/math/cc-2nd-grade-math/v/exercising-gorillashttps://puzzicle.org/thumbs/xb8abeb4c.jpgAddition and subtraction word problems: gorillasSal solves addition and subtraction word problems.https://cdn.kastatic.org/ka-youtube-converted/SfgD7Sm08ns.mp4/SfgD7Sm08ns.mp4332yes https://puzzicle.org/math/cc-1st-grade-math/v/comparison-word-problemshttps://puzzicle.org/thumbs/xa765b7ce.jpgComparison word problems: marblesSal solves word problems comparing 2 numbers. Numbers in the problems are less than 10.https://cdn.kastatic.org/ka-youtube-converted/osIFa9zcI-w.mp4/osIFa9zcI-w.mp4203yes https://puzzicle.org/math/cc-1st-grade-math/v/more-comparison-word-problemshttps://puzzicle.org/thumbs/x2a46299a.jpgComparison word problems: roly-poliesSal solves word problems comparing 2 numbers. Numbers in the problems are less than 10.https://cdn.kastatic.org/ka-youtube-converted/3rs7sFPoiZ4.mp4/3rs7sFPoiZ4.mp4194yes https://puzzicle.org/math/cc-1st-grade-math/v/adding-two-digit-numbers-no-regroupinghttps://puzzicle.org/thumbs/x6c1f38f1.jpgAdding 2-digit numbers without regrouping 1Sal adds 71 + 24.https://cdn.kastatic.org/ka-youtube-converted/EV38zfiY6Vs.mp4/EV38zfiY6Vs.mp4113yes https://puzzicle.org/math/arithmetic-home/v/adding-whole-numbers-and-applications-1https://puzzicle.org/thumbs/2600666.jpgExample: Adding 2-digit numbers (no carrying)Adding Whole Numbers and Applications 1https://cdn.kastatic.org/ka-youtube-converted/b22tMEc6Kko.mp4/b22tMEc6Kko.mp492yes https://puzzicle.org/math/cc-2nd-grade-math/v/breaking-apart-two-digit-addition-problemshttps://puzzicle.org/thumbs/xbbe54e50.jpgBreaking apart 2-digit addition problemsSal thinks about different ways to break up addition problems.https://cdn.kastatic.org/ka-youtube-converted/UjfmOmPARyk.mp4/UjfmOmPARyk.mp4288yes https://puzzicle.org/math/cc-1st-grade-math/v/regrouping-when-adding-two-digit-and-one-digit-numberhttps://puzzicle.org/thumbs/xe227d7c8.jpgRegrouping to add 1-digit numberSal adds 35 + 6.https://cdn.kastatic.org/ka-youtube-converted/G1cNKc3PD74.mp4/G1cNKc3PD74.mp4253yes https://puzzicle.org/math/cc-2nd-grade-math/v/adding-by-getting-to-group-of-10-firsthttps://puzzicle.org/thumbs/x03b57659.jpgAdding by making a group of 10Sal adds 5+68 by breaking the 5 into a 2 and a 3.https://cdn.kastatic.org/ka-youtube-converted/DzJvR56Suss.mp4/DzJvR56Suss.mp4170yes https://puzzicle.org/math/cc-2nd-grade-math/v/example-exercises-on-ways-to-add-two-digit-numbershttps://puzzicle.org/thumbs/x9d9dccf4.jpgStrategies for adding 2-digit numbersSal goes through multiple strategies for adding 2-digit numbers.https://cdn.kastatic.org/ka-youtube-converted/kMHFY6ZrKbU.mp4/kMHFY6ZrKbU.mp4196yes https://puzzicle.org/math/cc-2nd-grade-math/v/correct-number-lines-for-addition-and-subtractionhttps://puzzicle.org/thumbs/x39288e47.jpgAddition and subtraction with number linesSal figures out which number line matches up with a 2-digit addition or subtraction problem.https://cdn.kastatic.org/ka-youtube-converted/I5xcZgyY4ag.mp4/I5xcZgyY4ag.mp4117yes https://puzzicle.org/math/cc-2nd-grade-math/v/understanding-place-value-when-adding-oneshttps://puzzicle.org/thumbs/x971fdd55.jpgUnderstanding place value when adding onesSal adds 37 + 2 by thinking about place value.https://cdn.kastatic.org/ka-youtube-converted/7_QPAdHILzw.mp4/7_QPAdHILzw.mp4156yes https://puzzicle.org/math/cc-2nd-grade-math/v/understanding-place-value-while-adding-tenshttps://puzzicle.org/thumbs/xb228ca07.jpgUnderstanding place value when adding tensSal adds 23 + 30 by thinking about place value.https://cdn.kastatic.org/ka-youtube-converted/Dy5uDkOoMNc.mp4/Dy5uDkOoMNc.mp4196yes https://puzzicle.org/math/early-math/v/introduction-to-regrouping-borrowinghttps://puzzicle.org/thumbs/x340aa2ea.jpgSubtracting with regrouping (borrowing)Sal subtracts 25 from 83 by thinking about tens and ones.https://cdn.kastatic.org/ka-youtube-converted/egjDLFX9VHg.mp4/egjDLFX9VHg.mp4327yes https://puzzicle.org/math/cc-2nd-grade-math/v/adding-two-digit-numbers-on-a-number-linehttps://puzzicle.org/thumbs/x5d5dfc84de48cb7a.jpgAdding two digit numbers on a number lineSal uses a number line to add two digit numbers.https://cdn.kastatic.org/ka-youtube-converted/qHbw1_9HstU.mp4/qHbw1_9HstU.mp4216yes https://puzzicle.org/math/early-math/v/snow-on-fridayhttps://puzzicle.org/thumbs/xfce06433.jpgSubtraction word problem: snowSal solves a subtraction word problem with numbers less than 100.https://cdn.kastatic.org/ka-youtube-converted/wyWy4PouQxQ.mp4/wyWy4PouQxQ.mp4259yes https://puzzicle.org/math/cc-2nd-grade-math/v/multi-step-addition-word-problemhttps://puzzicle.org/thumbs/xeb55304ed2298655.jpgMulti step addition word problemSal solves a multi-step addition word problems using a tape diagram.https://cdn.kastatic.org/ka-youtube-converted/dbCub716oSg.mp4/dbCub716oSg.mp4251yes https://puzzicle.org/math/cc-2nd-grade-math/v/multi-step-subtraction-word-problemhttps://puzzicle.org/thumbs/x94e9c6f7067f562a.jpgMulti-step subtraction word problemSal solves a multi-step subtraction word problem using a tape diagram.https://cdn.kastatic.org/ka-youtube-converted/qAUnDB0LZL4.mp4/qAUnDB0LZL4.mp4257yes https://puzzicle.org/math/early-math/v/rethinking-three-digit-addition-for-mental-calculationhttps://puzzicle.org/thumbs/xc14c8a26.jpgSolving 3-digit addition in your headSal shows some techniques for doing mental addition, such as making groups of 10 and 100.https://cdn.kastatic.org/ka-youtube-converted/jAfJcgPGqgI.mp4/jAfJcgPGqgI.mp4448yes https://puzzicle.org/math/arithmetic-home/v/mental-technique-for-subtraction-without-regroupinghttps://puzzicle.org/thumbs/xc1b529a7.jpgSubtracting in your head (no regrouping)Sal uses an understanding of place value to subtract.https://cdn.kastatic.org/ka-youtube-converted/iivtjjdSu9I.mp4/iivtjjdSu9I.mp4157yes https://puzzicle.org/math/cc-2nd-grade-math/v/missing-numbers-in-addition-and-subtraction-within-100https://puzzicle.org/thumbs/x3266cde9.jpgMissing numbers in addition and subtractionSal solves missing number in problems like 73 = ___ + 57.https://cdn.kastatic.org/ka-youtube-converted/zTK6XUF0HAs.mp4/zTK6XUF0HAs.mp4344yes https://puzzicle.org/math/arithmetic-home/v/addition-4https://puzzicle.org/thumbs/23806.jpgMulti-digit addition with regroupingSal adds multi-digit numbers like 9367 and 2459 using an understanding of place value.https://cdn.kastatic.org/ka-youtube-converted/fOXo4p4WDKM.mp4/fOXo4p4WDKM.mp4529yes https://puzzicle.org/math/arithmetic-home/v/alternate-mental-subtraction-methodhttps://puzzicle.org/thumbs/23814.jpgAlternate mental subtraction methodSal explains how he subtracts numbers like 9456 and 7589 in his head.https://cdn.kastatic.org/ka-youtube-converted/GBtcGO44e-A.mp4/GBtcGO44e-A.mp4310yes https://puzzicle.org/math/arithmetic-home/v/basic-multiplicationhttps://puzzicle.org/thumbs/23824.jpgBasic multiplicationIntroduction to multiplication.https://cdn.kastatic.org/ka-youtube-converted/mvOkMYCygps.mp4/mvOkMYCygps.mp4800yes https://puzzicle.org/math/arithmetic-home/v/number-line-1https://puzzicle.org/thumbs/x5c700875.jpgWorked example: Whole numbers on the number lineWorked example: Whole numbers on the number line from Arithmetic (all content)https://cdn.kastatic.org/ka-youtube-converted/ez1MIB8_2O0.mp4/ez1MIB8_2O0.mp480yes https://puzzicle.org/math/arithmetic-home/v/multiplying-whole-numbers-and-applications-1https://puzzicle.org/thumbs/2600673.jpgWays to represent multiplicationSal explains different ways to represent 7 times 5.https://cdn.kastatic.org/ka-youtube-converted/Yw3EoxC_GXU.mp4/Yw3EoxC_GXU.mp4183yes https://puzzicle.org/math/arithmetic-home/v/multiplication-3-10-11-12-times-tableshttps://puzzicle.org/thumbs/23828.jpgMultiplication tables for 10, 11, 12Learn to multiply 10, 11, and 12 using 'times tables.'https://cdn.kastatic.org/ka-youtube-converted/qihoczo1Ujk.mp4/qihoczo1Ujk.mp4731yes https://puzzicle.org/math/arithmetic-home/v/lattice-multiplicationhttps://puzzicle.org/thumbs/23840.jpgLattice multiplicationSal introduces lattice multiplication.https://cdn.kastatic.org/ka-youtube-converted/gS6TfWUv97I.mp4/gS6TfWUv97I.mp4435yes https://puzzicle.org/math/arithmetic-home/v/why-lattice-multiplication-workshttps://puzzicle.org/thumbs/23842.jpgWhy lattice multiplication worksSal explains why lattice multiplication works.https://cdn.kastatic.org/ka-youtube-converted/S3z4XqC_YSc.mp4/S3z4XqC_YSc.mp4501yes https://puzzicle.org/math/arithmetic-home/v/2-digit-times-1-digit-example-no-carryinghttps://puzzicle.org/thumbs/x82d7bb46.jpgMultiplying 2-digit by 1-digitLearn to multiply a 2-digit number by a 1-digit number without regrouping. In this video, we will multiply 32x3.https://cdn.kastatic.org/ka-youtube-converted/k68CPfcehTE.mp4/k68CPfcehTE.mp4111yes https://puzzicle.org/math/arithmetic-home/v/multiplication-5-2-digit-times-a-2-digit-numberhttps://puzzicle.org/thumbs/23832.jpgMultiplying 2-digit by 2-digit: 36x23Learn to multiply two-digit numbers. In this video, we will multiply 36 times 23.https://cdn.kastatic.org/ka-youtube-converted/t8m0NalQtEk.mp4/t8m0NalQtEk.mp4675yes https://puzzicle.org/math/arithmetic-home/v/multiplying-whole-numbers-and-applications-2https://puzzicle.org/thumbs/2600674.jpgMultiplying 2-digit by 2-digit: 23x44Learn to multiply two-digit numbers. In this video, we will multiply 23 times 44.https://cdn.kastatic.org/ka-youtube-converted/0WUXQNjdRvM.mp4/0WUXQNjdRvM.mp4198yes https://puzzicle.org/math/arithmetic-home/v/division-1https://puzzicle.org/thumbs/23844.jpgIntro to divisionIntroduction to division.https://cdn.kastatic.org/ka-youtube-converted/MTzTqvzWzm8.mp4/MTzTqvzWzm8.mp41022yes https://puzzicle.org/math/arithmetic-home/v/dividing-whole-numbers-and-applications-1https://puzzicle.org/thumbs/2600678.jpgWays to represent divisionSal explains different ways to represent 18 divided by 2.https://cdn.kastatic.org/ka-youtube-converted/4I9iibPLdBw.mp4/4I9iibPLdBw.mp4148yes https://puzzicle.org/math/arithmetic-home/v/canceling-zeros-when-dividinghttps://puzzicle.org/thumbs/x3ad9bd47.jpgCanceling zeros when dividingLindsay breaks down division problems involving multiples of 10.https://cdn.kastatic.org/ka-youtube-converted/ccS5Fy5yLjk.mp4/ccS5Fy5yLjk.mp4459yes https://puzzicle.org/math/arithmetic-home/v/division-2https://puzzicle.org/thumbs/23846.jpgIntro to long division (remainders)Introduction to long division with remaindershttps://cdn.kastatic.org/ka-youtube-converted/8Ft5iHhauJ0.mp4/8Ft5iHhauJ0.mp4699yes https://puzzicle.org/math/arithmetic-home/v/level-4-divisionhttps://puzzicle.org/thumbs/23850.jpgDividing by 2-digits: 6250÷25Learn to divide a 2-digit number into a larger number.https://cdn.kastatic.org/ka-youtube-converted/gHTH6PKfpMc.mp4/gHTH6PKfpMc.mp4575yes https://puzzicle.org/math/arithmetic-home/v/multi-step-word-problems-with-whole-numbers-exercise-3https://puzzicle.org/thumbs/x17e8cc3e.jpgMultiplication, division word problem: pedalingUsing multiplication and division to make estimates. Easy? Maybe, but what's more important is that you try.https://cdn.kastatic.org/ka-youtube-converted/l7h8A6SX5kY.mp4/l7h8A6SX5kY.mp4130yes https://puzzicle.org/math/arithmetic-home/v/absolute-value-1https://puzzicle.org/thumbs/26784.jpgIdentify and order absolute valuesUse a number line to find absolute value and then order absolute values from least to greatest.https://cdn.kastatic.org/ka-youtube-converted/NvGTCzAfvr0.mp4/NvGTCzAfvr0.mp4202yes https://puzzicle.org/math/arithmetic-home/v/absolute-value-word-problems-exercisehttps://puzzicle.org/thumbs/x7fd5c099.jpgAbsolute value word problemsLearn to solve word problems that ask us to apply our new absolute value knowledge.https://cdn.kastatic.org/ka-youtube-converted/Ig7RSc-93Bs.mp4/Ig7RSc-93Bs.mp4232yes https://puzzicle.org/math/arithmetic-home/v/multiplying-negative-real-numbershttps://puzzicle.org/thumbs/13179320.jpgMultiplying numbers with different signsFind the product of numbers with different signs.https://cdn.kastatic.org/ka-youtube-converted/4ve7OaCnDLs.mp4/4ve7OaCnDLs.mp4146yes https://puzzicle.org/math/arithmetic-home/v/multiplying-and-dividing-negative-numbershttps://puzzicle.org/thumbs/25749.jpgMultiplying and dividing negative numbersSal explains multiplying and dividing positive and negative numbers.https://cdn.kastatic.org/ka-youtube-converted/d8lP5tR2R3Q.mp4/d8lP5tR2R3Q.mp4510yes https://puzzicle.org/math/arithmetic-home/v/introduction-to-fractionshttps://puzzicle.org/thumbs/x4b75a153.jpgMore about fractionsSal shows fractions as parts of a whole and parts of a group.https://cdn.kastatic.org/ka-youtube-converted/kZzoVCmUyKg.mp4/kZzoVCmUyKg.mp4341yes https://puzzicle.org/math/arithmetic-home/v/recognizing-fractions-exercisehttps://puzzicle.org/thumbs/x0b18ce2b.jpgWorked example: Recognizing fractionsWorked example: Recognizing fractions from Arithmetic (all content)https://cdn.kastatic.org/ka-youtube-converted/dXHqO955HxE.mp4/dXHqO955HxE.mp4117yes https://puzzicle.org/math/arithmetic-home/v/fractions-with-like-denominators-numeratorshttps://puzzicle.org/thumbs/x8fe87fd9.jpgComparing fractions with like numerators and denominatorsComparing fractions with like numerators and denominators from Arithmetic (all content)https://cdn.kastatic.org/ka-youtube-converted/wbAxarp_Ug4.mp4/wbAxarp_Ug4.mp4288yes https://puzzicle.org/math/arithmetic-home/v/comparing-fractionshttps://puzzicle.org/thumbs/2600757.jpgComparing fractionsComparing Fractionshttps://cdn.kastatic.org/ka-youtube-converted/Io9i1JkKgN4.mp4/Io9i1JkKgN4.mp4150yes https://puzzicle.org/math/arithmetic-home/v/ordering-fractionshttps://puzzicle.org/thumbs/211177354.jpgComparing and ordering fractionsFinding common denominators of multiple fractions to order themhttps://cdn.kastatic.org/ka-youtube-converted/N8dIOmk_lHs.mp4/N8dIOmk_lHs.mp4468yes https://puzzicle.org/math/arithmetic-home/v/equivalent-fractionshttps://puzzicle.org/thumbs/4802211.jpgEquivalent fractionsLearn how to write equivalent fractions.https://cdn.kastatic.org/ka-youtube-converted/U2ovEuEUxXQ.mp4/U2ovEuEUxXQ.mp4351yes https://puzzicle.org/math/arithmetic-home/v/comparing-improper-fractions-and-mixed-numbershttps://puzzicle.org/thumbs/915154307.jpgComparing improper fractions and mixed numbersWorked examples comparing improper fractions and mixed numbershttps://cdn.kastatic.org/ka-youtube-converted/R8YKuGJ0plI.mp4/R8YKuGJ0plI.mp4294yes https://puzzicle.org/math/arithmetic-home/v/postive-improper-fractions-on-the-number-linehttps://puzzicle.org/thumbs/xf78b285e.jpgMixed number or improper fraction on a number lineFraction or mixed number...either way you should end up on the same spot on the number line. Give it a try with us.https://cdn.kastatic.org/ka-youtube-converted/g-SfwjbpA4U.mp4/g-SfwjbpA4U.mp4132yes https://puzzicle.org/math/arithmetic-home/v/multiplying-a-fraction-by-a-fractionhttps://puzzicle.org/thumbs/x369b9278.jpgIntro to multiplying 2 fractionsSal introduces multiplying 2 fractions.https://cdn.kastatic.org/ka-youtube-converted/x6xtezhuCZ4.mp4/x6xtezhuCZ4.mp4306yes https://puzzicle.org/math/arithmetic-home/v/multiplying-fractions-word-problem-1https://puzzicle.org/thumbs/x683d1719.jpgMultiplying fractions word problem: pumpkin pieOops! Someone tried to solve this problem and made a mistake. See if you can spot the error.https://cdn.kastatic.org/ka-youtube-converted/XDjbNykJ28E.mp4/XDjbNykJ28E.mp4276yes https://puzzicle.org/math/arithmetic-home/v/decimal-place-valuehttps://puzzicle.org/thumbs/1804.jpgDecimal place valueIntroduction to decimal place values and the value of decimal digitshttps://cdn.kastatic.org/ka-youtube-converted/x-Dqe5U1TXA.mp4/x-Dqe5U1TXA.mp4240yes https://puzzicle.org/math/arithmetic-home/v/converting-fractions-to-decimalshttps://puzzicle.org/thumbs/23852.jpgRewriting tricky fractions to decimalsLearn to convert the fraction, 17/93 to a decimal.https://cdn.kastatic.org/ka-youtube-converted/Gn2pdkvdbGQ.mp4/Gn2pdkvdbGQ.mp4562yes https://puzzicle.org/math/arithmetic-home/v/significant-figureshttps://puzzicle.org/thumbs/67068552.jpgIntro to significant figuresSignificant figures are the number of digits in a value, often a measurement, that contribute to the degree of accuracy of the value. We start counting significant figures at the first non-zero digit. Calculate the number of significant figures for an assortment of numbers.https://cdn.kastatic.org/ka-youtube-converted/eCJ76hz7jPM.mp4/eCJ76hz7jPM.mp4303yes https://puzzicle.org/math/arithmetic-home/v/more-on-significant-figureshttps://puzzicle.org/thumbs/67068553.jpgRules of significant figuresThis video teaches significant figures rules, crucial for measurements and calculations. It covers identifying significant digits, including non-zero digits, zeros in between, leading zeros, and trailing zeros. Additionally, it introduces scientific notation for clearer representation of significant figures.https://cdn.kastatic.org/ka-youtube-converted/eMl2z3ezlrQ.mp4/eMl2z3ezlrQ.mp4340yes https://puzzicle.org/math/arithmetic-home/v/multiplying-and-dividing-with-significant-figureshttps://puzzicle.org/thumbs/67068554.jpgMultiplying and dividing with significant figuresLearn to multiply and divide with significant figures.https://cdn.kastatic.org/ka-youtube-converted/iorZdz4dsBU.mp4/iorZdz4dsBU.mp4571yes https://puzzicle.org/math/arithmetic-home/v/addition-and-subtraction-with-significant-figureshttps://puzzicle.org/thumbs/67861020.jpgAddition and subtraction with significant figuresThis video teaches addition and subtraction with significant figures, emphasizing that the result should match the least precise measurement. It offers examples and explains the real-world importance of maintaining precision in calculations.https://cdn.kastatic.org/ka-youtube-converted/xHgPtFUbAeU.mp4/xHgPtFUbAeU.mp4539yes https://puzzicle.org/math/differential-calculus/v/determining-which-limit-statements-are-truehttps://puzzicle.org/thumbs/1313140631.jpgConnecting limits and graphical behavior (more examples)Sal analyzes various 1- and 2-sided limits of a function given graphically.https://cdn.kastatic.org/ka-youtube-converted/_WOr9-_HbAM.mp4/_WOr9-_HbAM.mp4316yes https://puzzicle.org/math/differential-calculus/v/unbounded-limits-graphicalhttps://puzzicle.org/thumbs/x39979770.jpgConnecting limits at infinity notation and graphSal analyzes a function with an asymptote and finds the correct description of the two one-sided limits of the function at that asymptote.https://cdn.kastatic.org/ka-youtube-converted/TlWvQvMB9eI.mp4/TlWvQvMB9eI.mp4118yes https://puzzicle.org/math/differential-calculus/v/limit-at-infinity-involving-trig-definedhttps://puzzicle.org/thumbs/x60640c11.jpgLimits at infinity of quotients with trigSal finds the limit of cosx/(x²-1) at infinity, by putting it between two limits of rational functions, 1/(x²-1) and -1/(x²-1).https://cdn.kastatic.org/ka-youtube-converted/oAQnDsmu41E.mp4/oAQnDsmu41E.mp4184yes https://puzzicle.org/math/differential-calculus/v/limit-at-infinity-involving-trig-undefinedhttps://puzzicle.org/thumbs/x531d1f13.jpgLimits at infinity of quotients with trig (limit undefined)Sal analyzes the limit of (x²+1)/sin(x) at infinity. It turns out this limit doesn't exist, as the function keeps oscillating between positive and negative infinity.https://cdn.kastatic.org/ka-youtube-converted/waWYIKAfkWw.mp4/waWYIKAfkWw.mp4141yes https://puzzicle.org/math/differential-calculus/v/limits-infinity-algebrahttps://puzzicle.org/thumbs/xb85faf80.jpgLimit at infinity of a difference of functionsSal finds the limit at infinity of √(100+x)-√(x).https://cdn.kastatic.org/ka-youtube-converted/nDXFgexOM5c.mp4/nDXFgexOM5c.mp4326yes https://puzzicle.org/math/differential-calculus/v/slope-of-a-line-secant-to-a-curvehttps://puzzicle.org/thumbs/xed57a1a5.jpgSlope of a line secant to a curveWhat is the slope of a line between two points on a curve? This is called a "secant" line.https://cdn.kastatic.org/ka-youtube-converted/8r8Vp_1iB4k.mp4/8r8Vp_1iB4k.mp4437yes https://puzzicle.org/math/differential-calculus/v/slope-of-a-secant-line-arbitrary-differencehttps://puzzicle.org/thumbs/x7f146c13.jpgSecant line with arbitrary differenceSal finds the slope of the secant line on the graph of ln(x) between the points (2,ln2) and (2+h,ln(2+h)).https://cdn.kastatic.org/ka-youtube-converted/RaNeCD8t01I.mp4/RaNeCD8t01I.mp4259yes https://puzzicle.org/math/differential-calculus/v/slope-of-a-secant-linehttps://puzzicle.org/thumbs/xdb7e3de5.jpgSecant line with arbitrary pointSal finds the slope of the secant line on the graph of ln(x) between the points (e,1) and (x,lnx).https://cdn.kastatic.org/ka-youtube-converted/0zExhHh7_Ic.mp4/0zExhHh7_Ic.mp4259yes https://puzzicle.org/math/differential-calculus/v/simplified-secant-line-arbitrary-differencehttps://puzzicle.org/thumbs/x48d3cd9a.jpgSecant line with arbitrary difference (with simplification)Sal finds and simplifies the expression for the slope of the secant line between x=4 and x=4+h on the graph of y=2x²+1.https://cdn.kastatic.org/ka-youtube-converted/l1xdKkIGxOc.mp4/l1xdKkIGxOc.mp4234yes https://puzzicle.org/math/differential-calculus/v/simplified-secant-line-arbitrary-pointhttps://puzzicle.org/thumbs/x4d713576.jpgSecant line with arbitrary point (with simplification)Sal finds and simplifies the expression for the slope of the secant line between x=3 and x=t on the graph of y=2x²+5x.https://cdn.kastatic.org/ka-youtube-converted/T3nVKbxEQdo.mp4/T3nVKbxEQdo.mp4232yes https://puzzicle.org/math/differential-calculus/v/slope-of-a-secant-line-example-3https://puzzicle.org/thumbs/xec2af377.jpgSecant lines: challenging problem 1Sal solves a challenging problem involving slopes of secant lines to a curve.https://cdn.kastatic.org/ka-youtube-converted/H6ZNLD1AeM8.mp4/H6ZNLD1AeM8.mp4378yes https://puzzicle.org/math/differential-calculus/v/graphs-of-functions-and-their-derivatives-example-1https://puzzicle.org/thumbs/x31dde586.jpgSecant lines: challenging problem 2Sal interprets an expression as the slope of a secant line between a specific point on a graph and any other point on that graph.https://cdn.kastatic.org/ka-youtube-converted/P3IlneCNm8A.mp4/P3IlneCNm8A.mp4357yes https://puzzicle.org/math/differential-calculus/v/tangent-slope-as-limiting-value-of-secant-slope-example-3https://puzzicle.org/thumbs/x22e79f7d.jpgLimit expression for the derivative of function (graphical)Sal interprets limit expressions as the derivatives of a function given graphically, and evaluates them.https://cdn.kastatic.org/ka-youtube-converted/hoRISaqp1Po.mp4/hoRISaqp1Po.mp4226yes https://puzzicle.org/math/differential-calculus/v/tangent-line-equation-example-1https://puzzicle.org/thumbs/x29623156.jpgTangent to y=𝑒ˣ/(2+x³)Sal finds the equation of the line tangent to the curve y=eˣ/(2+x³) at the point (1,e/3).https://cdn.kastatic.org/ka-youtube-converted/GH8-URjRQpQ.mp4/GH8-URjRQpQ.mp4437yes https://puzzicle.org/math/differential-calculus/v/normal-line-equationhttps://puzzicle.org/thumbs/x8626cdc3.jpgNormal to y=𝑒ˣ/x²Sal finds the equation of the line normal to the curve y=eˣ/x² at the point (1,e).https://cdn.kastatic.org/ka-youtube-converted/C5G2Js2_Ep4.mp4/C5G2Js2_Ep4.mp4327yes https://puzzicle.org/math/differential-calculus/v/implicit-derivative-of-y-cos-5x-3yhttps://puzzicle.org/thumbs/1336092146.jpgImplicit differentiation (advanced example)Implicit differentiation of y = cos(5x - 3y)https://cdn.kastatic.org/ka-youtube-converted/-EG10aI0rt0.mp4/-EG10aI0rt0.mp4248yes https://puzzicle.org/math/differential-calculus/v/implicit-derivative-of-x-2-y-2-3-5x-2y-2https://puzzicle.org/thumbs/1335955875.jpgImplicit differentiation (advanced example)Implicit derivative of (x²+y²)³=5x²y².https://cdn.kastatic.org/ka-youtube-converted/1DcsREjyoiM.mp4/1DcsREjyoiM.mp4430yes https://puzzicle.org/math/differential-calculus/v/implicit-derivative-of-e-xy-2-x-yhttps://puzzicle.org/thumbs/1335852020.jpgImplicit differentiation (advanced example)Sal finds dy/dx for e^(xy²)=x-y using implicit differentiation.https://cdn.kastatic.org/ka-youtube-converted/ZtI94pI4Uzc.mp4/ZtI94pI4Uzc.mp4358yes https://puzzicle.org/math/differential-calculus/v/derivative-natural-logarithmhttps://puzzicle.org/thumbs/x03a2d28c.jpgDerivative of ln(x) from derivative of 𝑒ˣ and implicit differentiationHow can you find the derivative of ln(x) by viewing it as the inverse of e^x?https://cdn.kastatic.org/ka-youtube-converted/765X_PAxhAw.mp4/765X_PAxhAw.mp4188yes https://puzzicle.org/math/differential-calculus/v/product-rule-for-more-than-two-functionshttps://puzzicle.org/thumbs/1335339467.jpgProduct rule to find derivative of product of three functionsSal differentiates the product of three different functions, and generalizes for the derivative of the product of any number of functions.https://cdn.kastatic.org/ka-youtube-converted/Bkkk0RLSEy8.mp4/Bkkk0RLSEy8.mp4214yes https://puzzicle.org/math/differential-calculus/v/calculus-derivative-of-x-x-xhttps://puzzicle.org/thumbs/24134.jpgComposite exponential function differentiationSal finds the derivatives of xˣ and x^(xˣ). They are surprisingly fun to do!https://cdn.kastatic.org/ka-youtube-converted/N5kkwVoAtkc.mp4/N5kkwVoAtkc.mp4542yes https://puzzicle.org/math/differential-calculus/v/composite-exponential-function-differentiationhttps://puzzicle.org/thumbs/xf03668ef.jpgWorked example: Composite exponential function differentiationSal differentiates the composite exponential function [ln(x)]ˣ and evaluates the derivative at x=e. Composite exponential functions are functions where the variable is in both the base and the exponent.https://cdn.kastatic.org/ka-youtube-converted/BSrJi9uEJp0.mp4/BSrJi9uEJp0.mp4524yes https://puzzicle.org/math/calculus-1/v/analyzing-particle-movement-based-on-graphshttps://puzzicle.org/thumbs/xc4d31415.jpgAnalyzing straight-line motion graphicallyLearn how to analyze a particles motion given the graph of its position over time.https://cdn.kastatic.org/ka-youtube-converted/wKPyzrEVqaQ.mp4/wKPyzrEVqaQ.mp4320yes https://puzzicle.org/math/differential-calculus/v/total-distance-traveled-by-a-particlehttps://puzzicle.org/thumbs/x6ae031ba.jpgTotal distance traveled with derivativesGiven a function representing the position of a particle over time, how can you find the total distance traveled?https://cdn.kastatic.org/ka-youtube-converted/HDntI7zfBNs.mp4/HDntI7zfBNs.mp4609yes https://puzzicle.org/math/differential-calculus/v/l-hopital-s-rule-example-3https://puzzicle.org/thumbs/27379.jpgL'Hôpital's rule: challenging problemSal uses L'Hôpital's rule to find the limit at 1 of x/(x-1)-1/lnx.https://cdn.kastatic.org/ka-youtube-converted/MeVFZjT-ABM.mp4/MeVFZjT-ABM.mp4470yes https://puzzicle.org/math/differential-calculus/v/lhopitals-rule-to-solve-for-variablehttps://puzzicle.org/thumbs/x831dfb4a.jpgL'Hôpital's rule: solve for a variableSal uses L'Hôpital's rule to find the value of a in (√(4+a)-√(4-ax))/x so the expression's limit at 0 is ¾.https://cdn.kastatic.org/ka-youtube-converted/Ot0FxL8Vl7I.mp4/Ot0FxL8Vl7I.mp4250yes https://puzzicle.org/math/differential-calculus/v/tricky-lhopitals-rule-problemhttps://puzzicle.org/thumbs/xdb5ee8fb.jpgL’Hôpital’s rule (composite exponential functions)Sal uses L'Hôpital's rule to find the limit at 0 of (sinx)^(1/lnx).https://cdn.kastatic.org/ka-youtube-converted/CDf_aE5yg3A.mp4/CDf_aE5yg3A.mp4791yes https://puzzicle.org/math/differential-calculus/v/extreme-derivative-word-problem-advancedhttps://puzzicle.org/thumbs/24136.jpgOptimization problem: extreme normaline to y=x²A difficult but interesting derivative word problemhttps://cdn.kastatic.org/ka-youtube-converted/viaPc8zDcRI.mp4/viaPc8zDcRI.mp41393yes https://puzzicle.org/math/integral-calculus/v/approximating-distance-with-rectangleshttps://puzzicle.org/thumbs/xde408b5b.jpgMotion problem with Riemann sum approximationHere we see how estimating the area under a curve representing velocity estimates distance traveled.https://cdn.kastatic.org/ka-youtube-converted/v13fraCiLEg.mp4/v13fraCiLEg.mp4600yes https://puzzicle.org/math/integral-calculus/v/rectangular-and-trapezoidal-riemann-approximationshttps://puzzicle.org/thumbs/1319439886.jpgMidpoint and trapezoidal sums in summation notationEstimating the area under a curve with trapazoids instead of rectangles can give a closer approximation.https://cdn.kastatic.org/ka-youtube-converted/naRUWPvsQbQ.mp4/naRUWPvsQbQ.mp4493yes https://puzzicle.org/math/integral-calculus/v/riemann-sum-negative-functionhttps://puzzicle.org/thumbs/x9cd535ed.jpgRiemann sums in summation notation: challenge problemWhen a function is negative, Riemann sums seem to treat it as having "negative area".https://cdn.kastatic.org/ka-youtube-converted/WeVWv_OEJsY.mp4/WeVWv_OEJsY.mp4396yes https://puzzicle.org/math/integral-calculus/v/integral-defined-function-0https://puzzicle.org/thumbs/x346b5c30.jpgFunctions defined by integrals: challenge problemSolving for where a function defined by an integral equals 0.https://cdn.kastatic.org/ka-youtube-converted/W20SoZiJOHU.mp4/W20SoZiJOHU.mp4585yes https://puzzicle.org/math/integral-calculus/v/antiderivative-of-hairier-expressionhttps://puzzicle.org/thumbs/1328896635.jpgRewriting before integrating: challenge problemIn this example, we find the antiderivative an expression which is not so simple.https://cdn.kastatic.org/ka-youtube-converted/cBi4a1iSaPk.mp4/cBi4a1iSaPk.mp4376yes https://puzzicle.org/math/integral-calculus/v/u-substitution-and-back-substitutionhttps://puzzicle.org/thumbs/1254533049.jpg𝘶-substitution: special applicationUsing 𝘶-substitution in a situation that is a bit different than "classic" 𝘶-substitution. In this case, the substitution helps us take a hairy expression and make it easier to expand and integrate.https://cdn.kastatic.org/ka-youtube-converted/J-NC1M6obKo.mp4/J-NC1M6obKo.mp4301yes https://puzzicle.org/math/integral-calculus/v/doing-u-substitution-twice-second-time-with-whttps://puzzicle.org/thumbs/1254240553.jpg𝘶-substitution: double substitutionFinding the indefinite integral of cos(5x)/e^[sin(5x)]. To do that, we need to perform 𝘶-substitution twice.https://cdn.kastatic.org/ka-youtube-converted/ShpI3gPgLBA.mp4/ShpI3gPgLBA.mp4537yes https://puzzicle.org/math/integral-calculus/v/2-ln-x-x-antiderivative-examplehttps://puzzicle.org/thumbs/24306.jpg𝘶-substitution: challenging applicationFinding ∫(2^ln x)/x dxhttps://cdn.kastatic.org/ka-youtube-converted/C5Lbjbyr1t4.mp4/C5Lbjbyr1t4.mp4520yes https://puzzicle.org/math/integral-calculus/v/using-trig-identity-to-use-u-substitutionhttps://puzzicle.org/thumbs/xc397250c.jpgIntegral of cos^3(x)A specialized form of u-substitution involves taking advantage of trigonometric identities.https://cdn.kastatic.org/ka-youtube-converted/rElAJA9GyL4.mp4/rElAJA9GyL4.mp4366yes https://puzzicle.org/math/integral-calculus/v/integral-of-odd-powered-trig-function-with-u-substitutionhttps://puzzicle.org/thumbs/x6b973043.jpgIntegral of sin^2(x) cos^3(x)Another example where u substitution combined with certain trigonometric identities can be used.https://cdn.kastatic.org/ka-youtube-converted/xSeYS3V5di8.mp4/xSeYS3V5di8.mp4273yes https://puzzicle.org/math/integral-calculus/v/trig-identies-for-trig-integralhttps://puzzicle.org/thumbs/xc716abcc.jpgIntegral of sin^4(x)It is a bit involved, but we can use u-substitution to find the integral of sin^4(x)https://cdn.kastatic.org/ka-youtube-converted/n34jx1FIN8M.mp4/n34jx1FIN8M.mp4333yes https://puzzicle.org/math/integral-calculus/v/introduction-to-trigonometric-substitutionhttps://puzzicle.org/thumbs/x7161c395.jpgIntroduction to trigonometric substitutionIntroduction to trigonometric substitutionhttps://cdn.kastatic.org/ka-youtube-converted/EV5dhv0A2wU.mp4/EV5dhv0A2wU.mp4532yes https://puzzicle.org/math/integral-calculus/v/another-substitution-with-x-sin-thetahttps://puzzicle.org/thumbs/1373246393.jpgSubstitution with x=sin(theta)When you are integrating something that has the expression (1-x^2), try substituting sin(theta) for x.https://cdn.kastatic.org/ka-youtube-converted/8Yl_u_Otcjg.mp4/8Yl_u_Otcjg.mp4308yes https://puzzicle.org/math/integral-calculus/v/integrals-trig-substitution-1https://puzzicle.org/thumbs/24208.jpgMore trig sub practiceExample of using trig substitution to solve an indefinite integralhttps://cdn.kastatic.org/ka-youtube-converted/n4EK92CSuBE.mp4/n4EK92CSuBE.mp4458yes https://puzzicle.org/math/integral-calculus/v/trig-and-u-substitution-together-part-1https://puzzicle.org/thumbs/1373141693.jpgTrig and u substitution together (part 1)All of the substitution!https://cdn.kastatic.org/ka-youtube-converted/nMrJ6nbOQhQ.mp4/nMrJ6nbOQhQ.mp4535yes https://puzzicle.org/math/integral-calculus/v/trig-and-u-substitution-together-part-2https://puzzicle.org/thumbs/1373336307.jpgTrig and u substitution together (part 2)More of all of the substitution!https://cdn.kastatic.org/ka-youtube-converted/WAoaBTWKLoI.mp4/WAoaBTWKLoI.mp4345yes https://puzzicle.org/math/integral-calculus/v/trig-substitution-with-tangenthttps://puzzicle.org/thumbs/1373225524.jpgTrig substitution with tangentWhen you are integrating something which looks like 1+(x^2), try replacing x with tan(theta)https://cdn.kastatic.org/ka-youtube-converted/CYjAgOTVfZk.mp4/CYjAgOTVfZk.mp4286yes https://puzzicle.org/math/integral-calculus/v/integrals-trig-substitution-2https://puzzicle.org/thumbs/24210.jpgMore trig substitution with tangentAnother practice problem replacing x with tan(theta) in an integral.https://cdn.kastatic.org/ka-youtube-converted/fD7MbnXbTls.mp4/fD7MbnXbTls.mp4491yes https://puzzicle.org/math/integral-calculus/v/integrals-trig-substitution-3-long-problemhttps://puzzicle.org/thumbs/24212.jpgLong trig sub problemMore practice with a hairy trig sub problem.https://cdn.kastatic.org/ka-youtube-converted/sw2p2tUIFpc.mp4/sw2p2tUIFpc.mp41072yes https://puzzicle.org/math/integral-calculus/v/improper-integral-with-two-infinite-boundshttps://puzzicle.org/thumbs/1378097320.jpgImproper integral with two infinite boundsA worked example of a challenging improper integral that involves two infinite bounds and an inverse trig substitution.https://cdn.kastatic.org/ka-youtube-converted/9JX2s90_RNQ.mp4/9JX2s90_RNQ.mp4687yes https://puzzicle.org/math/integral-calculus/v/logistic-function-applicationhttps://puzzicle.org/thumbs/x0c018898.jpgWorked example: logistic model equationsThe general logistic function is N(t)=(N₀K)/(N₀+(K-N₀)e⁻ʳᵗ). In this video, we solve a real-world word problem about logistic growth.https://cdn.kastatic.org/ka-youtube-converted/-fIsaqN-aaQ.mp4/-fIsaqN-aaQ.mp4574yes https://puzzicle.org/math/integral-calculus/v/arc-length-example-2https://puzzicle.org/thumbs/x985d984a.jpgWorked example: arc length (advanced)A worked example of finding arc length using definite integrals. This example involves some challenging algebra.https://cdn.kastatic.org/ka-youtube-converted/MtRXjXdXDow.mp4/MtRXjXdXDow.mp4558yes https://puzzicle.org/math/integral-calculus/v/polar-arc-length-formulahttps://puzzicle.org/thumbs/x307ed42c.jpgArc length of polar curvesSal shows the polar arc length formula, and explains why it is true.https://cdn.kastatic.org/ka-youtube-converted/CEPuNmVKnZM.mp4/CEPuNmVKnZM.mp4706yes https://puzzicle.org/math/integral-calculus/v/polar-arc-length-examplehttps://puzzicle.org/thumbs/x749aa43b.jpgWorked example: Arc length of polar curvesWhat is the arc length of the polar curve r = 4sin(theta)?https://cdn.kastatic.org/ka-youtube-converted/sFlH7V7QDbQ.mp4/sFlH7V7QDbQ.mp4383yes https://puzzicle.org/math/integral-calculus/v/geometric-series-convergence-divergencehttps://puzzicle.org/thumbs/x9a18b2d7.jpgConvergent & divergent geometric series (with manipulation)Sal looks at examples of three infinite geometric series and determines if each of them converges or diverges. To do that, he needs to manipulate the expressions to find the common ratio.https://cdn.kastatic.org/ka-youtube-converted/HP8ZTo2iDtw.mp4/HP8ZTo2iDtw.mp4428yes https://puzzicle.org/math/integral-calculus/v/writing-series-sigma-notationhttps://puzzicle.org/thumbs/xc5145c94.jpgConverting explicit series terms to summation notationSome sums are really long, but we can use sigma notation to write them in a shorthand way. See such an example here.https://cdn.kastatic.org/ka-youtube-converted/0DBDuNYsciY.mp4/0DBDuNYsciY.mp4196yes https://puzzicle.org/math/integral-calculus/v/explicitly-defining-serieshttps://puzzicle.org/thumbs/x1985e33e.jpgConverting explicit series terms to summation notation (n ≥ 2)Usually, when writing a series in sigma notation, we start summing at n=1 or n=0. Sometimes though, we would like to start at n=2 or larger values. See an example here.https://cdn.kastatic.org/ka-youtube-converted/yvddTWa9ptU.mp4/yvddTWa9ptU.mp4265yes https://puzzicle.org/math/integral-calculus/v/telescoping-serieshttps://puzzicle.org/thumbs/xf1ee6394.jpgTelescoping seriesTelescoping series is a series where all terms cancel out except for the first and last one. This makes such series easy to analyze. In this video, we use partial fraction decomposition to find sum of telescoping series.https://cdn.kastatic.org/ka-youtube-converted/qUNGPqCPzMg.mp4/qUNGPqCPzMg.mp4751yes https://puzzicle.org/math/integral-calculus/v/divergence-telescoping-serieshttps://puzzicle.org/thumbs/x6f205945.jpgDivergent telescoping seriesTelescoping series is a series where all terms cancel out except for the first and last one. This makes such series easy to analyze. In this video we take a close look at the series 1-1+1-1+1-...https://cdn.kastatic.org/ka-youtube-converted/5LMzbgfZ8cA.mp4/5LMzbgfZ8cA.mp4293yes https://puzzicle.org/math/cc-1st-grade-math/v/adding-3-numbershttps://puzzicle.org/thumbs/x5ef91da0.jpgAdding 5 + 3 + 6Sal adds 5 + 3 + 6 two ways.https://cdn.kastatic.org/ka-youtube-converted/Utm875JRi-o.mp4/Utm875JRi-o.mp4158yes https://puzzicle.org/math/cc-1st-grade-math/v/equal-signhttps://puzzicle.org/thumbs/xfa19f713.jpgEqual signLearn what the equals sign means.https://cdn.kastatic.org/ka-youtube-converted/gEGKvx1wlFg.mp4/gEGKvx1wlFg.mp4296yes https://puzzicle.org/math/cc-1st-grade-math/v/comparing-adding-1-and-10https://puzzicle.org/thumbs/xfb4db072.jpgAdding 1 vs. adding 10Sal talks about the difference between adding 1 and adding 10.https://cdn.kastatic.org/ka-youtube-converted/mKsKU0BAiRM.mp4/mKsKU0BAiRM.mp4137yes https://puzzicle.org/math/cc-1st-grade-math/v/order-by-lengthhttps://puzzicle.org/thumbs/xfc9c120b.jpgOrdering by lengthSal orders 3 objects by length and compares the lengths of 2 objects indirectly by using a third object.https://cdn.kastatic.org/ka-youtube-converted/SQmyTmO8OKw.mp4/SQmyTmO8OKw.mp4157yes https://puzzicle.org/math/cc-1st-grade-math/v/reading-bar-graph-exampleshttps://puzzicle.org/thumbs/xc2bbf559.jpgReading bar graphs: dog bonesSal reads and interprets bar graphs.https://cdn.kastatic.org/ka-youtube-converted/OmLl6pkvV-I.mp4/OmLl6pkvV-I.mp4177yes https://puzzicle.org/math/cc-1st-grade-math/v/halves-and-fourthshttps://puzzicle.org/thumbs/x54bcdbc6.jpgHalves and fourthsWhen a whole is divided into equal size pieces, the pieces are fractions of the whole. If the whole is divided into 2 equal sized pieces, each piece is ​one half of the whole. If the whole is divided into 4 equal sized pieces, each piece is one fourth of the whole.https://cdn.kastatic.org/ka-youtube-converted/0lSTXtwPuOU.mp4/0lSTXtwPuOU.mp4145yes https://puzzicle.org/math/cc-2nd-grade-math/v/using-arrays-in-word-problemshttps://puzzicle.org/thumbs/xf64adba19592ff9f.jpgUsing arrays in word problemsSal uses addition to solve array word problems.https://cdn.kastatic.org/ka-youtube-converted/2ey3UYi8VOY.mp4/2ey3UYi8VOY.mp4101yes https://puzzicle.org/math/cc-2nd-grade-math/v/creating-the-largest-numberhttps://puzzicle.org/thumbs/x76acec6d3e49525a.jpgCreating the largest numberSal uses 3 numbers to show how to create the largest 3 digit number possible.https://cdn.kastatic.org/ka-youtube-converted/1guztZj-RSU.mp4/1guztZj-RSU.mp4188yes https://puzzicle.org/math/cc-2nd-grade-math/v/understanding-place-value-while-subtracting-oneshttps://puzzicle.org/thumbs/x535bc6da.jpgSubtracting 1s using place valueSal subtracts 4 from 46 by thinking about place value.https://cdn.kastatic.org/ka-youtube-converted/FWEqB0J6mgA.mp4/FWEqB0J6mgA.mp4165yes https://puzzicle.org/math/cc-2nd-grade-math/v/understanding-place-value-when-subtracting-tenshttps://puzzicle.org/thumbs/x93cf889d.jpgSubtracting 10s using place valueSal subtracts 40 from 65 by thinking about place value.https://cdn.kastatic.org/ka-youtube-converted/bNNrlzel4Gw.mp4/bNNrlzel4Gw.mp4157yes https://puzzicle.org/math/cc-2nd-grade-math/v/fewer-word-problemshttps://puzzicle.org/thumbs/x469b5449.jpgSubtraction word problem: basketballSal solves a word problem with the term "fewer".https://cdn.kastatic.org/ka-youtube-converted/qyvwN-2PoXA.mp4/qyvwN-2PoXA.mp4132yes https://puzzicle.org/math/cc-2nd-grade-math/v/adding-four-two-digit-numbershttps://puzzicle.org/thumbs/xde376408da2d1e66.jpgAdding four two digit numbersSal uses place value to add four two digit numbers.https://cdn.kastatic.org/ka-youtube-converted/7Y_wVnAHajg.mp4/7Y_wVnAHajg.mp4195yes https://puzzicle.org/math/cc-2nd-grade-math/v/adding-multiple-2-digit-numbers-word-problemshttps://puzzicle.org/thumbs/x4f69fd4e4442fd11.jpgAdding multiple 2-digit numbers word problemsSal solve word problems that involve adding multiple two digit numbers by using place value.https://cdn.kastatic.org/ka-youtube-converted/WJNOtHFN-mM.mp4/WJNOtHFN-mM.mp4210yes https://puzzicle.org/math/cc-2nd-grade-math/v/time-on-a-number-linehttps://puzzicle.org/thumbs/x3389df4fee7131c5.jpgTime on a number lineSal tells time on a number line.https://cdn.kastatic.org/ka-youtube-converted/Rc2v8JdebwQ.mp4/Rc2v8JdebwQ.mp4117yes https://puzzicle.org/math/cc-2nd-grade-math/v/getting-a-sense-of-meters-and-centimetershttps://puzzicle.org/thumbs/x429912e303017554.jpgGetting a sense of meters and centimetersSal explains how to get a sense of the size of meters and centimeters and applies it to measuring objects.https://cdn.kastatic.org/ka-youtube-converted/pyx-6vfCuW0.mp4/pyx-6vfCuW0.mp4388yes https://puzzicle.org/math/cc-2nd-grade-math/v/getting-a-sense-of-inches-and-feethttps://puzzicle.org/thumbs/xa86f39f2288d7ef7.jpgGetting a sense of inches and feetExplore inches and feet as units of length. Learn to visualize an inch with a quarter or thumb and understand a foot equals 12 inches. The video includes examples of when to use inches or feet for measuring various objects.https://cdn.kastatic.org/ka-youtube-converted/o31t2ta-EtQ.mp4/o31t2ta-EtQ.mp4350yes https://puzzicle.org/math/cc-2nd-grade-math/v/measuring-lengths-in-different-unitshttps://puzzicle.org/thumbs/x4e8d34b713688739.jpgMeasuring lengths in different unitsSal measures the same rectangle in different units and then compares the size of the different units.https://cdn.kastatic.org/ka-youtube-converted/9dur4s-4lhk.mp4/9dur4s-4lhk.mp4168yes https://puzzicle.org/math/cc-2nd-grade-math/v/length-word-problem-example-us-customaryhttps://puzzicle.org/thumbs/xf9a16e0a1728e9fe.jpgLength word problem example (US Customary)Sal shows how to solve a length word problem example.https://cdn.kastatic.org/ka-youtube-converted/h43MrQ5G_2k.mp4/h43MrQ5G_2k.mp4223yes https://puzzicle.org/math/cc-2nd-grade-math/v/partition-rectangleshttps://puzzicle.org/thumbs/x7e440e59031fe1cc.jpgPartitioning rectanglesSal explains how to partition a rectangle into 2 rows and 5 columns.https://cdn.kastatic.org/ka-youtube-converted/8AdcPD50aTQ.mp4/8AdcPD50aTQ.mp4193yes https://puzzicle.org/math/cc-kindergarten-math/v/counting-with-small-numbershttps://puzzicle.org/thumbs/x9b4a5e7a.jpgCounting with small numbersSal counts squirrels and horses.https://cdn.kastatic.org/ka-youtube-converted/y2-uaPiyoxc.mp4/y2-uaPiyoxc.mp456yes https://puzzicle.org/math/cc-kindergarten-math/v/counting-in-orderhttps://puzzicle.org/thumbs/x1a52b63e.jpgCounting in orderLearn how to count without making mistakes.https://cdn.kastatic.org/ka-youtube-converted/PEeUTQ0Gri8.mp4/PEeUTQ0Gri8.mp4118yes https://puzzicle.org/math/cc-kindergarten-math/v/counting-by-tens-example-problemshttps://puzzicle.org/thumbs/x591925fe29fb20c1.jpgCounting by tensPractice counting by tens using numbers and ten frames.https://cdn.kastatic.org/ka-youtube-converted/UnPpFw3natI.mp4/UnPpFw3natI.mp495yes https://puzzicle.org/math/cc-kindergarten-math/v/counting-in-sceneshttps://puzzicle.org/thumbs/x37933822.jpgCounting in picturesSal counts the number of objects in pictures.https://cdn.kastatic.org/ka-youtube-converted/leDYnoNSvD4.mp4/leDYnoNSvD4.mp473yes https://puzzicle.org/math/cc-kindergarten-math/v/how-many-objects-1https://puzzicle.org/thumbs/xaa31d6e5.jpgCounting objects 1Learn to count groups of whales, sheep, and flowers.https://cdn.kastatic.org/ka-youtube-converted/I9S5CvSqb5A.mp4/I9S5CvSqb5A.mp463yes https://puzzicle.org/math/cc-kindergarten-math/v/counting-objects-2https://puzzicle.org/thumbs/x4a6f7890.jpgCounting objects 2Learn to count dogs, mice, and cookies in random patterns.https://cdn.kastatic.org/ka-youtube-converted/EUqhLxFccbM.mp4/EUqhLxFccbM.mp4104yes https://puzzicle.org/math/cc-kindergarten-math/v/comparing-groups-through-10-examplehttps://puzzicle.org/thumbs/x1fef85b3.jpgComparing numbers of objectsSal talks about what "more than" and "less than" mean.https://cdn.kastatic.org/ka-youtube-converted/__nkbr6DeTg.mp4/__nkbr6DeTg.mp4123yes https://puzzicle.org/math/cc-kindergarten-math/v/comparing-numbers-through-10-examplehttps://puzzicle.org/thumbs/x546ea510.jpgComparing numbers on the number lineSal uses number line to compare numbers less than or equal to 10.https://cdn.kastatic.org/ka-youtube-converted/tJrSILRXOUc.mp4/tJrSILRXOUc.mp491yes https://puzzicle.org/math/cc-kindergarten-math/v/count-by-categoryhttps://puzzicle.org/thumbs/x21faccb5.jpgCounting by categorySal categorizes objects then count the number of things in each category.https://cdn.kastatic.org/ka-youtube-converted/UA975j_qsTQ.mp4/UA975j_qsTQ.mp444yes https://puzzicle.org/math/cc-kindergarten-math/v/teens-and-tenhttps://puzzicle.org/thumbs/x79debcfe.jpgTeens as sums with 10Sal looks at the 1 in each teen number and thinks about what it really means.https://cdn.kastatic.org/ka-youtube-converted/ourH3ueWNmA.mp4/ourH3ueWNmA.mp4180yes https://puzzicle.org/math/cc-kindergarten-math/v/monkeys-for-partyhttps://puzzicle.org/thumbs/x04d6c646.jpgTeen numbers: monkeysSal breaks the number 18 into 10 and 8.https://cdn.kastatic.org/ka-youtube-converted/zqwVKhQV_2w.mp4/zqwVKhQV_2w.mp4178yes https://puzzicle.org/math/cc-kindergarten-math/v/addition-introductionhttps://puzzicle.org/thumbs/x7d5effb1.jpgIntro to additionLearn what it means to add. The examples used are 1+1 and 2+3.https://cdn.kastatic.org/ka-youtube-converted/fsTD_jqseBA.mp4/fsTD_jqseBA.mp4191yes https://puzzicle.org/math/cc-kindergarten-math/v/subtraction-introductionhttps://puzzicle.org/thumbs/x2fefc435.jpgIntro to subtractionSal talks about what it means to subtract. The examples he uses are 4-3 and 5-2.https://cdn.kastatic.org/ka-youtube-converted/AO9bHbUdg-M.mp4/AO9bHbUdg-M.mp4219yes https://puzzicle.org/math/cc-kindergarten-math/v/making-5https://puzzicle.org/thumbs/x65eda1aa.jpgMaking 5Learn how to add numbers to make the number five.https://cdn.kastatic.org/ka-youtube-converted/A-ykhY_IoaU.mp4/A-ykhY_IoaU.mp4246yes https://puzzicle.org/math/cc-kindergarten-math/v/addition-and-subtraction-within-10https://puzzicle.org/thumbs/xacce5bfc.jpgAddition and subtraction within 10Sal solves addition and subtraction problems with number 10 or less.https://cdn.kastatic.org/ka-youtube-converted/kpEJwpemL2Q.mp4/kpEJwpemL2Q.mp4135yes https://puzzicle.org/math/cc-kindergarten-math/v/relative-positionhttps://puzzicle.org/thumbs/x99affdbee2d6c2d8.jpgRelative positionSal decides if objects are above, below, beside, in front of, or behind other objects.https://cdn.kastatic.org/ka-youtube-converted/uikYPJw0tnE.mp4/uikYPJw0tnE.mp465yes https://puzzicle.org/math/cc-kindergarten-math/v/compose-shapeshttps://puzzicle.org/thumbs/x8b5b5806.jpgComposing shapesSal combines shapes to make other shapes.https://cdn.kastatic.org/ka-youtube-converted/K3GV13uokbk.mp4/K3GV13uokbk.mp4124yes https://puzzicle.org/math/differential-equations/v/finding-particular-linear-solution-to-differential-equationhttps://puzzicle.org/thumbs/x6f28d28d.jpgWorked example: linear solution to differential equationIf a particular solution to a differential equation is linear, y=mx+b, we can set up a system of equations to find m and b. See how it works in this video.https://cdn.kastatic.org/ka-youtube-converted/zid7J4EhZN8.mp4/zid7J4EhZN8.mp4391yes https://puzzicle.org/math/differential-equations/v/separable-differential-equationshttps://puzzicle.org/thumbs/24710.jpgSeparable equations (old)An old introduction video to separable differential equations.https://cdn.kastatic.org/ka-youtube-converted/C5-lz0hcqsE.mp4/C5-lz0hcqsE.mp4720yes https://puzzicle.org/math/differential-equations/v/separable-differential-equations-2https://puzzicle.org/thumbs/24712.jpgSeparable equations example (old)An old worked example video of solving separable equations.https://cdn.kastatic.org/ka-youtube-converted/xVWCfMe97ws.mp4/xVWCfMe97ws.mp4337yes https://puzzicle.org/math/differential-equations/v/newtons-law-of-coolinghttps://puzzicle.org/thumbs/xb63a9856.jpgNewton's Law of CoolingNewton's law of cooling can be modeled with the general equation dT/dt=-k(T-Tₐ), whose solutions are T=Ce⁻ᵏᵗ+Tₐ (for cooling) and T=Tₐ-Ce⁻ᵏᵗ (for heating).https://cdn.kastatic.org/ka-youtube-converted/IICR-w1jYcA.mp4/IICR-w1jYcA.mp4602yes https://puzzicle.org/math/differential-equations/v/applying-newtons-law-of-cooling-to-warm-oatmealhttps://puzzicle.org/thumbs/x97e7847e.jpgWorked example: Newton's law of coolingThe general function for Newton's law of cooling is T=Ce⁻ᵏᵗ+Tₐ. In this video, we solve a word problem that involves the cooling of a freshly baked cookie!https://cdn.kastatic.org/ka-youtube-converted/jH5qflAe3C8.mp4/jH5qflAe3C8.mp4743yes https://puzzicle.org/math/differential-equations/v/exact-equations-intuition-1-proofyhttps://puzzicle.org/thumbs/24714.jpgExact equations intuition 1 (proofy)Chain rule using partial derivatives (not a proof; more intuition).https://cdn.kastatic.org/ka-youtube-converted/iEpqcdaJNTQ.mp4/iEpqcdaJNTQ.mp4594yes https://puzzicle.org/math/differential-equations/v/exact-equations-intuition-2-proofyhttps://puzzicle.org/thumbs/24716.jpgExact equations intuition 2 (proofy)More intuitive building blocks for exact equations.https://cdn.kastatic.org/ka-youtube-converted/a7wYAtMjORQ.mp4/a7wYAtMjORQ.mp4651yes https://puzzicle.org/math/differential-equations/v/exact-equations-example-1https://puzzicle.org/thumbs/24718.jpgExact equations example 1First example of solving an exact differential equation.https://cdn.kastatic.org/ka-youtube-converted/Pb04ntcDJcQ.mp4/Pb04ntcDJcQ.mp4729yes https://puzzicle.org/math/differential-equations/v/exact-equations-example-2https://puzzicle.org/thumbs/24720.jpgExact equations example 2Some more exact equation exampleshttps://cdn.kastatic.org/ka-youtube-converted/utQi1ZhF__Q.mp4/utQi1ZhF__Q.mp4481yes https://puzzicle.org/math/differential-equations/v/exact-equations-example-3https://puzzicle.org/thumbs/24722.jpgExact equations example 3One more exact equation examplehttps://cdn.kastatic.org/ka-youtube-converted/eu_GFuU7tLI.mp4/eu_GFuU7tLI.mp4599yes https://puzzicle.org/math/differential-equations/v/integrating-factors-1https://puzzicle.org/thumbs/24724.jpgIntegrating factors 1Using an integrating factor to make a differential equation exacthttps://cdn.kastatic.org/ka-youtube-converted/j511hg7Hlbg.mp4/j511hg7Hlbg.mp4616yes https://puzzicle.org/math/differential-equations/v/integrating-factors-2https://puzzicle.org/thumbs/24726.jpgIntegrating factors 2Now that we've made the equation exact, let's solve it!https://cdn.kastatic.org/ka-youtube-converted/0NyeDUhKwBE.mp4/0NyeDUhKwBE.mp4506yes https://puzzicle.org/math/differential-equations/v/first-order-homegenous-equationshttps://puzzicle.org/thumbs/24728.jpgFirst order homogenous equationsIntroduction to first order homogenous equations.https://cdn.kastatic.org/ka-youtube-converted/9Wfn-WWV1aY.mp4/9Wfn-WWV1aY.mp4441yes https://puzzicle.org/math/differential-equations/v/first-order-homogenous-equations-2https://puzzicle.org/thumbs/24730.jpgFirst order homogeneous equations 2Another example of using substitution to solve a first order homogeneous differential equations.https://cdn.kastatic.org/ka-youtube-converted/6YRGEsQWZzY.mp4/6YRGEsQWZzY.mp4502yes https://puzzicle.org/math/differential-equations/v/2nd-order-linear-homogeneous-differential-equations-1https://puzzicle.org/thumbs/24732.jpg2nd order linear homogeneous differential equations 1Introduction to 2nd order, linear, homogeneous differential equations with constant coefficients.https://cdn.kastatic.org/ka-youtube-converted/UFWAu8Ptth0.mp4/UFWAu8Ptth0.mp4584yes https://puzzicle.org/math/differential-equations/v/2nd-order-linear-homogeneous-differential-equations-2https://puzzicle.org/thumbs/24734.jpg2nd order linear homogeneous differential equations 2Let's find the general solution!https://cdn.kastatic.org/ka-youtube-converted/SPVqgkOZMAc.mp4/SPVqgkOZMAc.mp4508yes https://puzzicle.org/math/differential-equations/v/2nd-order-linear-homogeneous-differential-equations-3https://puzzicle.org/thumbs/24736.jpg2nd order linear homogeneous differential equations 3Let's use some initial conditions to solve for the particular solutionhttps://cdn.kastatic.org/ka-youtube-converted/UyCwAFQt4v0.mp4/UyCwAFQt4v0.mp4359yes https://puzzicle.org/math/differential-equations/v/2nd-order-linear-homogeneous-differential-equations-4https://puzzicle.org/thumbs/24738.jpg2nd order linear homogeneous differential equations 4Another example with initial conditions!https://cdn.kastatic.org/ka-youtube-converted/3uO_uPb9H8w.mp4/3uO_uPb9H8w.mp4550yes https://puzzicle.org/math/differential-equations/v/complex-roots-of-the-characteristic-equations-1https://puzzicle.org/thumbs/24740.jpgComplex roots of the characteristic equations 1What happens when the characteristic equations has complex roots?!https://cdn.kastatic.org/ka-youtube-converted/6xEO4BeawzA.mp4/6xEO4BeawzA.mp4627yes https://puzzicle.org/math/differential-equations/v/complex-roots-of-the-characteristic-equations-2https://puzzicle.org/thumbs/24742.jpgComplex roots of the characteristic equations 2What happens when the characteristic equation has complex roots?https://cdn.kastatic.org/ka-youtube-converted/jJyRrIZ595c.mp4/jJyRrIZ595c.mp4623yes https://puzzicle.org/math/differential-equations/v/complex-roots-of-the-characteristic-equations-3https://puzzicle.org/thumbs/24744.jpgComplex roots of the characteristic equations 3Lets do an example with initial conditions!https://cdn.kastatic.org/ka-youtube-converted/rGaM6pwqhB0.mp4/rGaM6pwqhB0.mp4612yes https://puzzicle.org/math/differential-equations/v/repeated-roots-of-the-characteristic-equationhttps://puzzicle.org/thumbs/24746.jpgRepeated roots of the characteristic equationWhat happens when the characteristic equation only has 1 repeated root?https://cdn.kastatic.org/ka-youtube-converted/NW9JfMvIsxw.mp4/NW9JfMvIsxw.mp4718yes https://puzzicle.org/math/differential-equations/v/repeated-roots-of-the-characterisitic-equations-part-2https://puzzicle.org/thumbs/24748.jpgRepeated roots of the characteristic equations part 2An example where we use initial conditions to solve a repeated-roots differential equation.https://cdn.kastatic.org/ka-youtube-converted/YezsgIlcb68.mp4/YezsgIlcb68.mp4532yes https://puzzicle.org/math/differential-equations/v/undetermined-coefficients-1https://puzzicle.org/thumbs/24750.jpgUndetermined coefficients 1Using the method of undetermined coefficients to solve nonhomogeneous linear differential equations.https://cdn.kastatic.org/ka-youtube-converted/znE4Nq9NJCQ.mp4/znE4Nq9NJCQ.mp4611yes https://puzzicle.org/math/differential-equations/v/undetermined-coefficients-2https://puzzicle.org/thumbs/24752.jpgUndetermined coefficients 2Another example using undetermined coefficients.https://cdn.kastatic.org/ka-youtube-converted/hbJ2o9EUmJ0.mp4/hbJ2o9EUmJ0.mp4660yes https://puzzicle.org/math/differential-equations/v/undetermined-coefficients-3https://puzzicle.org/thumbs/24754.jpgUndetermined coefficients 3Another example where the nonhomogeneous part is a polynomialhttps://cdn.kastatic.org/ka-youtube-converted/I3vIAzMcm4Y.mp4/I3vIAzMcm4Y.mp4489yes https://puzzicle.org/math/differential-equations/v/undetermined-coefficients-4https://puzzicle.org/thumbs/24756.jpgUndetermined coefficients 4Putting it all together!https://cdn.kastatic.org/ka-youtube-converted/vAepSNDLZRM.mp4/vAepSNDLZRM.mp4354yes https://puzzicle.org/math/differential-equations/v/laplace-transform-1https://puzzicle.org/thumbs/24758.jpgLaplace transform 1Introduction to the Laplace Transformhttps://cdn.kastatic.org/ka-youtube-converted/OiNh2DswFt4.mp4/OiNh2DswFt4.mp4481yes https://puzzicle.org/math/differential-equations/v/laplace-transform-2https://puzzicle.org/thumbs/24760.jpgLaplace transform 2Laplace transform of e^athttps://cdn.kastatic.org/ka-youtube-converted/33TYoybjqPg.mp4/33TYoybjqPg.mp4454yes https://puzzicle.org/math/differential-equations/v/laplace-transform-3-l-sin-athttps://puzzicle.org/thumbs/24762.jpgL{sin(at)} - transform of sin(at)Laplace Transform of sin(at) (part 1)https://cdn.kastatic.org/ka-youtube-converted/gMcs6RF_LrQ.mp4/gMcs6RF_LrQ.mp4644yes https://puzzicle.org/math/differential-equations/v/laplace-transform-4https://puzzicle.org/thumbs/24764.jpgPart 2 of the transform of the sin(at)Part 2 of getting the Laplace transform of sin(at)https://cdn.kastatic.org/ka-youtube-converted/-cApVwKR1Ps.mp4/-cApVwKR1Ps.mp4553yes https://puzzicle.org/math/differential-equations/v/laplace-transform-5https://puzzicle.org/thumbs/24766.jpgLaplace as linear operator and Laplace of derivativesUseful properties of the Laplace Transformhttps://cdn.kastatic.org/ka-youtube-converted/E2VxbufzuWQ.mp4/E2VxbufzuWQ.mp4696yes https://puzzicle.org/math/differential-equations/v/laplace-transform-6https://puzzicle.org/thumbs/24768.jpgLaplace transform of cos t and polynomialsLaplace transform of cosine and polynomials!https://cdn.kastatic.org/ka-youtube-converted/fyJkXBvcA2Q.mp4/fyJkXBvcA2Q.mp4585yes https://puzzicle.org/math/differential-equations/v/more-laplace-transform-toolshttps://puzzicle.org/thumbs/24774.jpg"Shifting" transform by multiplying function by exponentialA grab bag of things to know about the Laplace Transform.https://cdn.kastatic.org/ka-youtube-converted/_X_QwpXsdOs.mp4/_X_QwpXsdOs.mp4677yes https://puzzicle.org/math/differential-equations/v/laplace-transform-of-l-thttps://puzzicle.org/thumbs/24778.jpgLaplace transform of t: L{t}Determining the Laplace Transform of thttps://cdn.kastatic.org/ka-youtube-converted/qhKJ1DG9kYY.mp4/qhKJ1DG9kYY.mp4546yes https://puzzicle.org/math/differential-equations/v/laplace-transform-of-t-n-l-t-nhttps://puzzicle.org/thumbs/24780.jpgLaplace transform of t^n: L{t^n}Laplace Transform of t^n: L{t^n}https://cdn.kastatic.org/ka-youtube-converted/pZIaUx3UQoA.mp4/pZIaUx3UQoA.mp4616yes https://puzzicle.org/math/differential-equations/v/laplace-transform-of-the-unit-step-functionhttps://puzzicle.org/thumbs/24782.jpgLaplace transform of the unit step functionIntroduction to the unit step function and its Laplace Transformhttps://cdn.kastatic.org/ka-youtube-converted/4Mr7aEHQr8E.mp4/4Mr7aEHQr8E.mp41455yes https://puzzicle.org/math/differential-equations/v/inverse-laplace-exampleshttps://puzzicle.org/thumbs/24784.jpgInverse Laplace examplesUsing our toolkit to take some inverse Laplace Transformshttps://cdn.kastatic.org/ka-youtube-converted/BbqbvSig6Mc.mp4/BbqbvSig6Mc.mp41155yes https://puzzicle.org/math/differential-equations/v/dirac-delta-functionhttps://puzzicle.org/thumbs/24788.jpgDirac delta functionIntroduction to the Dirac Delta Functionhttps://cdn.kastatic.org/ka-youtube-converted/4qfdCwys2ew.mp4/4qfdCwys2ew.mp41068yes https://puzzicle.org/math/differential-equations/v/laplace-transform-of-the-dirac-delta-functionhttps://puzzicle.org/thumbs/24790.jpgLaplace transform of the dirac delta functionFiguring out the Laplace Transform of the Dirac Delta Functionhttps://cdn.kastatic.org/ka-youtube-converted/vhfjEpQWWeE.mp4/vhfjEpQWWeE.mp4733yes https://puzzicle.org/math/differential-equations/v/laplace-transform-to-solve-an-equationhttps://puzzicle.org/thumbs/24770.jpgLaplace transform to solve an equationUsing the Laplace Transform to solve an equation we already knew how to solve.https://cdn.kastatic.org/ka-youtube-converted/3uYb-RhM7lU.mp4/3uYb-RhM7lU.mp4651yes https://puzzicle.org/math/differential-equations/v/laplace-transform-solves-an-equation-2https://puzzicle.org/thumbs/24772.jpgLaplace transform solves an equation 2Second part of using the Laplace Transform to solve a differential equation.https://cdn.kastatic.org/ka-youtube-converted/EdQ7Q9VoF44.mp4/EdQ7Q9VoF44.mp4646yes https://puzzicle.org/math/differential-equations/v/using-the-laplace-transform-to-solve-a-nonhomogenous-eqhttps://puzzicle.org/thumbs/24776.jpgUsing the Laplace transform to solve a nonhomogeneous eqSolving a non-homogeneous differential equation using the Laplace Transformhttps://cdn.kastatic.org/ka-youtube-converted/nlUG4OKN1IY.mp4/nlUG4OKN1IY.mp41128yes https://puzzicle.org/math/differential-equations/v/laplace-step-function-differential-equationhttps://puzzicle.org/thumbs/24786.jpgLaplace/step function differential equationHairy differential equation involving a step function that we use the Laplace Transform to solve.https://cdn.kastatic.org/ka-youtube-converted/R0JY8oWsuOY.mp4/R0JY8oWsuOY.mp41152yes https://puzzicle.org/math/differential-equations/v/introduction-to-the-convolutionhttps://puzzicle.org/thumbs/24792.jpgIntroduction to the convolutionIntroduction to the Convolutionhttps://cdn.kastatic.org/ka-youtube-converted/IW4Reburjpc.mp4/IW4Reburjpc.mp41139yes https://puzzicle.org/math/differential-equations/v/the-convolution-and-the-laplace-transformhttps://puzzicle.org/thumbs/24794.jpgThe convolution and the Laplace transformUnderstanding how the product of the Transforms of two functions relates to their convolution.https://cdn.kastatic.org/ka-youtube-converted/TJgBEI3drUc.mp4/TJgBEI3drUc.mp4826yes https://puzzicle.org/math/differential-equations/v/using-the-convolution-theorem-to-solve-an-initial-value-probhttps://puzzicle.org/thumbs/24796.jpgUsing the convolution theorem to solve an initial value probUsing the Convolution Theorem to solve an initial value problemhttps://cdn.kastatic.org/ka-youtube-converted/rfyq32mHcYs.mp4/rfyq32mHcYs.mp4734yes https://puzzicle.org/math/early-math/v/comparing-subtracting-1-and-10https://puzzicle.org/thumbs/xa493150b.jpgSubtracting 1 vs. subtracting 10Sal talks about the difference between subtracting 1 and subtracting 10.https://cdn.kastatic.org/ka-youtube-converted/8YR1E7XWPz8.mp4/8YR1E7XWPz8.mp4152yes https://puzzicle.org/math/early-math/v/starfish-rulershttps://puzzicle.org/thumbs/x0ca5caaa.jpgAddition word problem: starfishSal solves a word problem with the term "fewer".https://cdn.kastatic.org/ka-youtube-converted/QYOieJuzDr4.mp4/QYOieJuzDr4.mp4217yes https://puzzicle.org/math/early-math/v/spots-on-a-diehttps://puzzicle.org/thumbs/xc13e6bcb.jpgAddition word problem: spotsSal solves a word problem with the term "more".https://cdn.kastatic.org/ka-youtube-converted/pvyqucC-0NU.mp4/pvyqucC-0NU.mp4203yes https://puzzicle.org/math/statistics-probability/v/comparing-distribution-meanshttps://puzzicle.org/thumbs/x7c6716a3.jpgComparing means of distributionsSal compares the means of two different distributions given as dot plots.https://cdn.kastatic.org/ka-youtube-converted/pPnxPrhf6Ww.mp4/pPnxPrhf6Ww.mp4469yes https://puzzicle.org/math/statistics-probability/v/comparing-means-and-medianshttps://puzzicle.org/thumbs/x3cb6dff6.jpgMeans and medians of different distributionsSal compares the mean and median based on a few different distributions.https://cdn.kastatic.org/ka-youtube-converted/eLyLbaXfJXo.mp4/eLyLbaXfJXo.mp4389yes https://puzzicle.org/math/trigonometry/v/example-the-six-trig-ratioshttps://puzzicle.org/thumbs/1153576581.jpgFinding reciprocal trig ratiosSal finds all six trigonometric ratios (sine, cosine, tangent, secant, cosecant, and cotangent) of an angle in a given right triangle.https://cdn.kastatic.org/ka-youtube-converted/Q7htxHDN8LE.mp4/Q7htxHDN8LE.mp4285yes https://puzzicle.org/math/trigonometry/v/example-using-trig-to-solve-for-missing-informationhttps://puzzicle.org/thumbs/1153530471.jpgUsing reciprocal trig ratiosSal is given two sides in a right triangle and the cotangent of one of the angles, and he uses this information to find the missing side.https://cdn.kastatic.org/ka-youtube-converted/rufFQZDDXCE.mp4/rufFQZDDXCE.mp4372yes https://puzzicle.org/math/precalculus/v/formula-for-arithmetic-serieshttps://puzzicle.org/thumbs/1392989924.jpgArithmetic series formulaLearn how to write an arithmetic sequence in general terms, using a common difference d and a first term a. Next, we'll see that this formula is equivalent to multiplying the average of the first and last terms by the number of terms.https://cdn.kastatic.org/ka-youtube-converted/Uy_L8tnihDM.mp4/Uy_L8tnihDM.mp4467yes https://puzzicle.org/math/statistics-probability/v/calculating-conditional-probabilityhttps://puzzicle.org/thumbs/xcbc59718.jpgCalculating conditional probabilityConditional probabilities are written like P(A|B), which can be read to mean, "the probability that A happens GIVEN B has happened." If we know probabilities like P(A), P(B), and P(A|B), we can solve for other probabilties like P(B|A).https://cdn.kastatic.org/ka-youtube-converted/6xPkG2pA-TU.mp4/6xPkG2pA-TU.mp4403yes https://puzzicle.org/math/statistics-probability/v/independent-dependent-probabilityhttps://puzzicle.org/thumbs/xfd725922.jpgAnalyzing event probability for independenceSal uses an example about shirts, scarves, hats, and pants to explain how to use probabilities to figure out if two events are independent.https://cdn.kastatic.org/ka-youtube-converted/R-NeYKSEqns.mp4/R-NeYKSEqns.mp4408yes https://puzzicle.org/math/statistics-probability/v/intersection-and-union-of-setshttps://puzzicle.org/thumbs/x99efa12b.jpgIntersection and union of setsTo find the intersection of two or more sets, you look for elements that are contained in all of the sets. To find the union of two or more sets, you combine all the elements from each set together, making sure to remove any duplicates.https://cdn.kastatic.org/ka-youtube-converted/jAfNg3ylZAI.mp4/jAfNg3ylZAI.mp4464yes https://puzzicle.org/math/statistics-probability/v/density-curve-worked-examplehttps://puzzicle.org/thumbs/x9e13b7bc0c3ce4b4.jpgDensity curve worked exampleAnalyzing skew, median, mean and height of a density curve.https://cdn.kastatic.org/ka-youtube-converted/P5y83-ZnQeo.mp4/P5y83-ZnQeo.mp4232yes https://puzzicle.org/math/statistics-probability/v/worked-example-finding-area-under-density-curveshttps://puzzicle.org/thumbs/x947652f549308b0a.jpgWorked example finding area under density curvesWorked example finding area under density curveshttps://cdn.kastatic.org/ka-youtube-converted/Tu3O9tC3Eh8.mp4/Tu3O9tC3Eh8.mp4280yes https://puzzicle.org/math/statistics-probability/v/ck12-org-normal-distribution-problems-z-scorehttps://puzzicle.org/thumbs/26200.jpgNormal distribution problem: z-scores (from ck12.org)Z-score practicehttps://cdn.kastatic.org/ka-youtube-converted/Wp2nVIzBsE8.mp4/Wp2nVIzBsE8.mp4468yes https://puzzicle.org/math/statistics-probability/v/types-of-statistical-studieshttps://puzzicle.org/thumbs/xf6f54d0cbcdb49b8.jpgTypes of statistical studiesTypes of statistical studieshttps://cdn.kastatic.org/ka-youtube-converted/SaP1O0i1bdc.mp4/SaP1O0i1bdc.mp4632yes https://puzzicle.org/math/statistics-probability/v/worked-example-identifying-sample-studyhttps://puzzicle.org/thumbs/x80ec3f947438bdda.jpgWorked example identifying sample studyWorked example identifying sample studyhttps://cdn.kastatic.org/ka-youtube-converted/QSHMOWQUZ54.mp4/QSHMOWQUZ54.mp4181yes https://puzzicle.org/math/statistics-probability/v/statistics-sample-vs-population-meanhttps://puzzicle.org/thumbs/26173.jpgInferring population mean from sample meanMuch of statistics is based upon using data from a random sample that is representative of the population at large. From that sample mean, we can infer things about the greater population mean. We'll explain.https://cdn.kastatic.org/ka-youtube-converted/k5EbijWu-Ss.mp4/k5EbijWu-Ss.mp4603yes https://puzzicle.org/math/geometry/v/constructing-regular-hexagon-inscribed-in-circlehttps://puzzicle.org/thumbs/x3e59c6d1.jpgGeometric constructions: circle-inscribed regular hexagonSal constructs a regular hexagon that is inscribed inside a given circle using compass and straightedge.https://cdn.kastatic.org/ka-youtube-converted/Yu4eqwGFJK8.mp4/Yu4eqwGFJK8.mp4221yes https://puzzicle.org/math/geometry/v/constructing-an-angle-bisector-using-a-compass-and-straightedgehttps://puzzicle.org/thumbs/x999a7ffd.jpgGeometric constructions: angle bisectorSal constructs a line that bisects a given angle using compass and straightedge.https://cdn.kastatic.org/ka-youtube-converted/KNr52iFk7n4.mp4/KNr52iFk7n4.mp4137yes https://puzzicle.org/math/geometry/v/constructing-a-perpendicular-line-using-a-compass-and-straightedgehttps://puzzicle.org/thumbs/xb27e19ab.jpgGeometric constructions: perpendicular line through a point on the lineSal constructs a line perpendicular to a given line through a point on the line using compass and straightedge.https://cdn.kastatic.org/ka-youtube-converted/nM-O8llUdwg.mp4/nM-O8llUdwg.mp472yes https://puzzicle.org/math/geometry/v/constructing-a-perpendicular-bisector-using-a-compass-and-straightedgehttps://puzzicle.org/thumbs/x14f9639b.jpgGeometric constructions: perpendicular bisectorSal constructs a perpendicular bisector to a given line segment using compass and straightedge.https://cdn.kastatic.org/ka-youtube-converted/WR2R_wrdEVY.mp4/WR2R_wrdEVY.mp486yes https://puzzicle.org/math/geometry-home/v/incenter-and-incircles-of-a-trianglehttps://puzzicle.org/thumbs/309318058.jpgIncenter and incircles of a triangleThe incenter of a triangle is the point at which the three angle bisectors intersect. To locate the incenter, one can draw each of the three angle bisectors, and then determine the point at which they all intersect. The incenter is also notable for being the center of the largest possible inscribed circle within the triangle.https://cdn.kastatic.org/ka-youtube-converted/21vbBiCVijE.mp4/21vbBiCVijE.mp4478yes https://puzzicle.org/math/geometry/v/constructing-circle-inscribing-trianglehttps://puzzicle.org/thumbs/x22e61e65.jpgGeometric constructions: triangle-inscribing circleSal constructs a circle that inscribes a given triangle using compass and straightedge.https://cdn.kastatic.org/ka-youtube-converted/lmm767eIsGU.mp4/lmm767eIsGU.mp4153yes https://puzzicle.org/math/geometry/v/constructing-circumscribing-circlehttps://puzzicle.org/thumbs/xba7a49ce.jpgGeometric constructions: triangle-circumscribing circleSal constructs a circle that circumscribes a given triangle using compass and straightedge.https://cdn.kastatic.org/ka-youtube-converted/eSjbEWb4Qp4.mp4/eSjbEWb4Qp4.mp4142yes https://puzzicle.org/math/geometry-home/v/proof-triangle-altitudes-are-concurrent-orthocenterhttps://puzzicle.org/thumbs/319066154.jpgProof: Triangle altitudes are concurrent (orthocenter)Showing that any triangle can be the medial triangle for some larger triangle. Using this to show that the altitudes of a triangle are concurrent (at the orthocenter).https://cdn.kastatic.org/ka-youtube-converted/aGwT2-RERXY.mp4/aGwT2-RERXY.mp4601yes https://puzzicle.org/math/geometry-home/v/showing-angles-have-same-measurehttps://puzzicle.org/thumbs/x07781167.jpgProving angles are congruentSal proves that two angles are congruent in a really interesting triangle like figure.https://cdn.kastatic.org/ka-youtube-converted/2SOK_IrpHeQ.mp4/2SOK_IrpHeQ.mp4293yes https://puzzicle.org/math/geometry/v/points-after-rotationhttps://puzzicle.org/thumbs/xd0c722ed.jpgRotating shapesSal is given a triangle on the coordinate plane and the definition of a rotation about the origin, and he manually draws the image of that rotation.https://cdn.kastatic.org/ka-youtube-converted/6nUMiJfHLSA.mp4/6nUMiJfHLSA.mp4492yes https://puzzicle.org/math/geometry/v/reflecting-segments-over-linehttps://puzzicle.org/thumbs/x135a8040.jpgReflecting shapes: diagonal line of reflectionSal is given two line segments on the coordinate plane and the definition of a reflection, and he draws the image of the segments under that reflection.https://cdn.kastatic.org/ka-youtube-converted/kj3ZfOQGKdE.mp4/kj3ZfOQGKdE.mp4439yes https://puzzicle.org/math/geometry/v/points-on-line-of-reflectionhttps://puzzicle.org/thumbs/x2d79cf1a.jpgDetermining reflections (advanced)Sal is given two line segments on the coordinate plane, and determines the reflection that maps one of them into the other.https://cdn.kastatic.org/ka-youtube-converted/63mWxNXQQAk.mp4/63mWxNXQQAk.mp4341yes https://puzzicle.org/math/geometry/v/example-rotating-polygonshttps://puzzicle.org/thumbs/x6918834b.jpgIntro to rotational symmetrySal checks whether various figure are symmetrical under a 180 degrees rotation.https://cdn.kastatic.org/ka-youtube-converted/P35LyN9g0oI.mp4/P35LyN9g0oI.mp4248yes https://puzzicle.org/math/geometry/v/constructing-quad-based-on-symmetryhttps://puzzicle.org/thumbs/x2bf83c62.jpgFinding a quadrilateral from its symmetriesTwo of the points that define a certain quadrilateral are (0,9) and (3,4). The quadrilateral has reflective symmetry over the line y=3-x. Draw and classify the quadrilateral.https://cdn.kastatic.org/ka-youtube-converted/-nufZ41Kg5c.mp4/-nufZ41Kg5c.mp4187yes https://puzzicle.org/math/geometry/v/reflecting-across-two-lineshttps://puzzicle.org/thumbs/x16ace6c3.jpgFinding a quadrilateral from its symmetries (example 2)Two of the points that define a certain quadrilateral are (-4,-2) and (0,5). The quadrilateral has a reflective symmetry over the lines y=x/2 and y=-2x+5. Draw and classify the quadrilateral.https://cdn.kastatic.org/ka-youtube-converted/ANnuQZyEKVo.mp4/ANnuQZyEKVo.mp4440yes https://puzzicle.org/math/geometry/v/line-and-angle-proofs-exercisehttps://puzzicle.org/thumbs/xab39abd2.jpgProofs with transformationsProofs with transformations from High school geometryhttps://cdn.kastatic.org/ka-youtube-converted/uYXhga17q1g.mp4/uYXhga17q1g.mp4525yes https://puzzicle.org/math/geometry/v/congruent-triangles-and-ssshttps://puzzicle.org/thumbs/267245528.jpgCorresponding parts of congruent triangles are congruentWhen two triangles are congruent, we can know that all of their corresponding sides and angles are congruent too!https://cdn.kastatic.org/ka-youtube-converted/yp49bc4TT-s.mp4/yp49bc4TT-s.mp4363yes https://puzzicle.org/math/geometry/v/other-triangle-congruence-postulateshttps://puzzicle.org/thumbs/267245529.jpgTriangle congruence postulates/criteriaSal introduces and justifies the SSS, SAS, ASA and AAS postulates for congruent triangles. He also shows that AAA is only good for similarity. For SSA, better to watch next video.https://cdn.kastatic.org/ka-youtube-converted/8Ld8Csu4sEs.mp4/8Ld8Csu4sEs.mp4808yes https://puzzicle.org/math/geometry/v/more-on-why-ssa-is-not-a-postulatehttps://puzzicle.org/thumbs/274218421.jpgWhy SSA isn't a congruence postulate/criterionThere are some cases when SSA can imply triangle congruence, but not always. This is why it's not like the other triangle congruence postulates/criteria.https://cdn.kastatic.org/ka-youtube-converted/f8svAm237xM.mp4/f8svAm237xM.mp4444yes https://puzzicle.org/math/geometry/v/finding-congruent-triangleshttps://puzzicle.org/thumbs/274218420.jpgDetermining congruent trianglesSal uses the SSS, ASA, SAS, and AAS postulates to find congruent triangles.https://cdn.kastatic.org/ka-youtube-converted/d5UCZ9hO8X4.mp4/d5UCZ9hO8X4.mp4448yes https://puzzicle.org/math/geometry/v/congruent-legs-and-base-angles-of-isosceles-triangleshttps://puzzicle.org/thumbs/281570417.jpgProofs concerning isosceles trianglesSal proves that the base angles in isosceles triangles are congruent, and conversely, that triangles with congruent base angles are isosceles. He also proves that the perpendicular to the base of an isosceles triangle bisects it.https://cdn.kastatic.org/ka-youtube-converted/7UISwx2Mr4c.mp4/7UISwx2Mr4c.mp4543yes https://puzzicle.org/math/geometry/v/equilateral-triangle-sides-and-angles-congruenthttps://puzzicle.org/thumbs/281570421.jpgProofs concerning equilateral trianglesSal proves that the angles of an equilateral triangle are all congruent (and therefore they all measure 60°), and conversely, that triangles with all congruent angles are equilateral.https://cdn.kastatic.org/ka-youtube-converted/CYNUwiUzlPk.mp4/CYNUwiUzlPk.mp4389yes https://puzzicle.org/math/geometry/v/figuring-out-all-the-angles-for-congruent-triangles-examplehttps://puzzicle.org/thumbs/281570422.jpgCorresponding angles in congruent trianglesWe write the letters of congruent triangles so their order tells us which parts are corresponding. In this video, Sal uses this notation to solve some angles.https://cdn.kastatic.org/ka-youtube-converted/aDCXPdzyS0s.mp4/aDCXPdzyS0s.mp4275yes https://puzzicle.org/math/geometry/v/congruent-triangle-proof-examplehttps://puzzicle.org/thumbs/274218423.jpgGeometry proof problem: midpointSal proves that a point is the midpoint of a segment using triangle congruence.https://cdn.kastatic.org/ka-youtube-converted/RFesGHsuFZw.mp4/RFesGHsuFZw.mp4346yes https://puzzicle.org/math/geometry/v/congruent-triangle-example-2https://puzzicle.org/thumbs/274218422.jpgGeometry proof problem: congruent segmentsSal proves that two pairs of segments are congruent using the ASA and AAS congruence criteria.https://cdn.kastatic.org/ka-youtube-converted/Xc3oHzKXVh8.mp4/Xc3oHzKXVh8.mp4734yes https://puzzicle.org/math/geometry/v/problem-involving-angle-derived-from-square-and-circlehttps://puzzicle.org/thumbs/353965912.jpgGeometry proof problem: squared circleSal finds a missing angle using triangle congruence in a diagram that contains sector of a circle inscribed within a square.https://cdn.kastatic.org/ka-youtube-converted/3n0LvI99-KM.mp4/3n0LvI99-KM.mp4495yes https://puzzicle.org/math/geometry/v/proof-opposite-sides-of-parallelogram-congruenthttps://puzzicle.org/thumbs/348974659.jpgProof: Opposite sides of a parallelogramSal proves that a figure is a parallelogram if and only if opposite sides are congruent.https://cdn.kastatic.org/ka-youtube-converted/LhrGS4-Dd9I.mp4/LhrGS4-Dd9I.mp4510yes https://puzzicle.org/math/geometry/v/proof-diagonals-of-a-parallelogram-bisect-each-otherhttps://puzzicle.org/thumbs/348974660.jpgProof: Diagonals of a parallelogramSal proves that a quadrilateral is a parallelogram if and only if its diagonals bisect each other.https://cdn.kastatic.org/ka-youtube-converted/TErJ-Yr67BI.mp4/TErJ-Yr67BI.mp4546yes https://puzzicle.org/math/geometry/v/proof-opposite-angles-of-parallelogram-congruenthttps://puzzicle.org/thumbs/348974657.jpgProof: Opposite angles of a parallelogramSal proves that opposite angles of a parallelogram are congruent.https://cdn.kastatic.org/ka-youtube-converted/oIV1zM8qlpk.mp4/oIV1zM8qlpk.mp4248yes https://puzzicle.org/math/geometry/v/two-column-proof-showing-segments-are-perpendicularhttps://puzzicle.org/thumbs/267245527.jpgProof: The diagonals of a kite are perpendicularSal proves that the diagonals of a kite are perpendicular, by using the SSS and SAS triangle congruence criteria.https://cdn.kastatic.org/ka-youtube-converted/fSu1LKnhM5Q.mp4/fSu1LKnhM5Q.mp4666yes https://puzzicle.org/math/geometry/v/proof-rhombus-diagonals-are-perpendicular-bisectorshttps://puzzicle.org/thumbs/348974658.jpgProof: Rhombus diagonals are perpendicular bisectorsSal proves that the diagonals of a rhombus are perpendicular, and that they intersect at the midpoints of both.https://cdn.kastatic.org/ka-youtube-converted/GDcVdBAnBdU.mp4/GDcVdBAnBdU.mp4244yes https://puzzicle.org/math/geometry/v/proof-rhombus-area-half-product-of-diagonal-lengthhttps://puzzicle.org/thumbs/351468822.jpgProof: Rhombus areaSal proves that we can find the area of a rhombus by taking half the product of the lengths of the diagonals.https://cdn.kastatic.org/ka-youtube-converted/3FManXv4mZM.mp4/3FManXv4mZM.mp4307yes https://puzzicle.org/math/geometry-home/v/triangle-medians-and-centroidshttps://puzzicle.org/thumbs/13179266.jpgTriangle medians & centroidsThe centroid of a triangle is the point at which the three medians intersect. To locate the centroid, draw each of the three medians (which connect the vertices of the triangle to the midpoints of the opposite sides). It is referred to as the "center of mass" or "balance point" of the triangle. Centroid divides medians in a ratio 2:3.https://cdn.kastatic.org/ka-youtube-converted/GiGLhXFBtRg.mp4/GiGLhXFBtRg.mp4539yes https://puzzicle.org/math/geometry-home/v/triangle-medians-and-centroids-2d-proofhttps://puzzicle.org/thumbs/13179267.jpgTriangle medians and centroids (2D proof)Showing that the centroid is 2/3 of the way along a medianhttps://cdn.kastatic.org/ka-youtube-converted/Vc09LURoMQ8.mp4/Vc09LURoMQ8.mp4915yes https://puzzicle.org/math/geometry-home/v/medians-divide-into-smaller-triangles-of-equal-areahttps://puzzicle.org/thumbs/311468473.jpgDividing triangles with mediansShowing that the three medians of a triangle divide it into six smaller triangles of equal area. Brief discussion of the centroid as wellhttps://cdn.kastatic.org/ka-youtube-converted/Xt4sT0fV9Pw.mp4/Xt4sT0fV9Pw.mp4466yes https://puzzicle.org/math/geometry/v/exploring-medial-triangleshttps://puzzicle.org/thumbs/311469473.jpgExploring medial trianglesThe three midsegments (segments joining the midpoints of the sides) of a triangle form a medial triangle. Find out the properties of the midsegments, the medial triangle and the other 3 triangles formed in this way.https://cdn.kastatic.org/ka-youtube-converted/Rsi5mzEaCPw.mp4/Rsi5mzEaCPw.mp4693yes https://puzzicle.org/math/geometry-home/v/proving-that-the-centroid-is-2-3rds-along-the-medianhttps://puzzicle.org/thumbs/319053662.jpgCentroid & median proofShowing that the centroid divides each median into segments with a 2:1 ratio (or that the centroid is 2/3 along the median)https://cdn.kastatic.org/ka-youtube-converted/i0VS3eEGjiQ.mp4/i0VS3eEGjiQ.mp4416yes https://puzzicle.org/math/geometry-home/v/median-centroid-right-triangle-examplehttps://puzzicle.org/thumbs/319066153.jpgMedian, centroid exampleExample involving properties of medianshttps://cdn.kastatic.org/ka-youtube-converted/k45QTFCHSVs.mp4/k45QTFCHSVs.mp4540yes https://puzzicle.org/math/geometry-home/v/dilating-from-an-arbitrary-point-examplehttps://puzzicle.org/thumbs/xd85b2957.jpgPerforming dilationsTo dilate a point on a coordinate plane, you multiply its coordinates by a scale factor. The scale factor is a positive number that tells you how much to enlarge or shrink the point. If the scale factor is greater than 1, the point moves away from the origin. If the scale factor is less than 1, the point moves closer to the origin.https://cdn.kastatic.org/ka-youtube-converted/j5xis6Hlnds.mp4/j5xis6Hlnds.mp4166yes https://puzzicle.org/math/geometry/v/seeing-circle-similarity-through-translation-and-dilationhttps://puzzicle.org/thumbs/xc9e9791e.jpgProof: all circles are similarWatch Sal informally prove that all circles are similar by showing how we can translate then dilate any circle onto another.https://cdn.kastatic.org/ka-youtube-converted/tcQjJssEMgo.mp4/tcQjJssEMgo.mp495yes https://puzzicle.org/math/geometry/v/angle-bisector-theorem-proofhttps://puzzicle.org/thumbs/309317056.jpgIntro to angle bisector theoremThe Angle Bisector Theorem states that when an angle in a triangle is split into two equal angles, it divides the opposite side into two parts. The ratio of these parts will be the same as the ratio of the sides next to the angle.https://cdn.kastatic.org/ka-youtube-converted/TpIBLnRAslI.mp4/TpIBLnRAslI.mp4515yes https://puzzicle.org/math/geometry/v/angle-bisector-theorem-exampleshttps://puzzicle.org/thumbs/311469472.jpgUsing the angle bisector theoremSal uses the angle bisector theorem to solve for sides of a triangle.https://cdn.kastatic.org/ka-youtube-converted/PlY3e_-9JUA.mp4/PlY3e_-9JUA.mp4184yes https://puzzicle.org/math/geometry/v/finding-area-using-similarity-and-congruencehttps://puzzicle.org/thumbs/290279019.jpgUsing similar & congruent trianglesSal uses the similarity of triangles and the congruence of others in this multi-step problem to find the area of a polygon.https://cdn.kastatic.org/ka-youtube-converted/EqNzr56h1Ic.mp4/EqNzr56h1Ic.mp4607yes https://puzzicle.org/math/geometry/v/challenging-similarity-problemhttps://puzzicle.org/thumbs/289578996.jpgChallenging similarity problemInteresting similarity problem where we don't have a lot of information to work withhttps://cdn.kastatic.org/ka-youtube-converted/7aGEvpHaNJ8.mp4/7aGEvpHaNJ8.mp4593yes https://puzzicle.org/math/geometry/v/goldren-ratio-and-rembrandt-s-self-portraithttps://puzzicle.org/thumbs/xeed4a573.jpgGeometry word problem: the golden ratioTriangle similarity, the Golden ratio, and art, all converge in this inspiring video!https://cdn.kastatic.org/ka-youtube-converted/v5SAMuRanGM.mp4/v5SAMuRanGM.mp4503yes https://puzzicle.org/math/geometry/v/golden-ratio-to-find-radius-of-moonhttps://puzzicle.org/thumbs/xbb6e0621.jpgGeometry word problem: Earth & Moon radiiSimilarity and the Golden ratio join forces again, this time to save the Earth! (Or maybe just finding its radius)https://cdn.kastatic.org/ka-youtube-converted/cnCQGKx6LhE.mp4/cnCQGKx6LhE.mp4383yes https://puzzicle.org/math/geometry/v/triangle-similarity-in-poolhttps://puzzicle.org/thumbs/xd0f9f2e0.jpgGeometry word problem: a perfect pool shotSal uses triangle similarity to plan the perfect shot in a pool game.https://cdn.kastatic.org/ka-youtube-converted/MiyFReTAdCA.mp4/MiyFReTAdCA.mp4409yes https://puzzicle.org/math/geometry-home/v/basic-trigonometryhttps://puzzicle.org/thumbs/19647490.jpgIntro to the trigonometric ratiosSin, cos, and tan are trigonometric ratios that relate the angles and sides of right triangles. Sin is the ratio of the opposite side to the hypotenuse, cos is the ratio of the adjacent side to the hypotenuse, and tan is the ratio of the opposite side to the adjacent side. They are often written as sin(x), cos(x), and tan(x), where x is an angle in radians or degrees.https://cdn.kastatic.org/ka-youtube-converted/Jsiy4TxgIME.mp4/Jsiy4TxgIME.mp4557yes https://puzzicle.org/math/geometry/v/basic-trigonometry-iihttps://puzzicle.org/thumbs/19647491.jpgTrigonometric ratios in right trianglesSal shows a few examples where he starts with the two legs of a right triangle and he finds the trig ratios of one of the acute angles.https://cdn.kastatic.org/ka-youtube-converted/G-T_6hCdMQc.mp4/G-T_6hCdMQc.mp4731yes https://puzzicle.org/math/geometry/v/similarity-to-define-sine-cosine-and-tangenthttps://puzzicle.org/thumbs/xc0627a6a.jpgTriangle similarity & the trigonometric ratiosSal explains how the trigonometric ratios are derived from triangle similarity considerations.https://cdn.kastatic.org/ka-youtube-converted/QuZMXVJNLCo.mp4/QuZMXVJNLCo.mp4536yes https://puzzicle.org/math/geometry/v/example-trig-to-solve-the-sides-and-angles-of-a-right-trianglehttps://puzzicle.org/thumbs/1153614407.jpgSolving for a side in right triangles with trigonometrySal is given a right triangle with an acute angle of 65° and a leg of 5 units, and he uses trigonometry to find the two missing sides.https://cdn.kastatic.org/ka-youtube-converted/l5VbdqRjTXc.mp4/l5VbdqRjTXc.mp4424yes https://puzzicle.org/math/geometry/v/showing-relationship-between-cosine-and-sine-of-complementshttps://puzzicle.org/thumbs/x71f71130.jpgSine & cosine of complementary anglesSal shows that the sine of any angle is equal to the cosine of its complementary angle.https://cdn.kastatic.org/ka-youtube-converted/BLQNL_UGONg.mp4/BLQNL_UGONg.mp4255yes https://puzzicle.org/math/geometry/v/sine-and-cosine-of-complements-examplehttps://puzzicle.org/thumbs/xee258334.jpgUsing complementary anglesSal solved the following problem: Given that cos(58°)=0.53, find sin(32°).https://cdn.kastatic.org/ka-youtube-converted/yiH6GoscimY.mp4/yiH6GoscimY.mp4245yes https://puzzicle.org/math/geometry/v/how-much-of-a-pyramid-is-submergedhttps://puzzicle.org/thumbs/xf177c90b.jpgTrig word problem: complementary anglesSal solves a problem about a submerged pyramid by finding congruent angles in the diagram and using the fact that the cosine of a given angle is always the same, no matter how large the triangle it appears in.https://cdn.kastatic.org/ka-youtube-converted/Z5EnuVJawmY.mp4/Z5EnuVJawmY.mp4344yes https://puzzicle.org/math/geometry-home/v/45-45-90-triangleshttps://puzzicle.org/thumbs/24965.jpgSpecial right triangles intro (part 1)A 45-45-90 triangle is a special type of right triangle, where the ratio of the lengths of the sides of a 45-45-90 triangle is always 1:1:√2, meaning that if one leg is x units long, then the other leg is also x units long, and the hypotenuse is x√2 units long.https://cdn.kastatic.org/ka-youtube-converted/tSHitjFIjd8.mp4/tSHitjFIjd8.mp4570yes https://puzzicle.org/math/geometry-home/v/intro-to-30-60-90-triangleshttps://puzzicle.org/thumbs/24967.jpgSpecial right triangles intro (part 2)A 30-60-90 triangle is a special right triangle with angles of 30, 60, and 90 degrees. It has properties similar to the 45-45-90 triangle. The side opposite the 30-degree angle is half the length of the hypotenuse, and the side opposite the 60-degree angle is the length of the short leg times the square root of three. Alternatively, the ratio of the sides is 1 : √3 : 2.https://cdn.kastatic.org/ka-youtube-converted/Qwet4cIpnCM.mp4/Qwet4cIpnCM.mp4579yes https://puzzicle.org/math/geometry/v/30-60-90-triangle-example-problemhttps://puzzicle.org/thumbs/294677485.jpg30-60-90 triangle example problemUsing what we know about 30-60-90 triangles to solve what at first seems to be a challenging problemhttps://cdn.kastatic.org/ka-youtube-converted/dgHksfBFbjk.mp4/dgHksfBFbjk.mp4400yes https://puzzicle.org/math/geometry/v/30-60-90-triangle-side-ratios-proofhttps://puzzicle.org/thumbs/292160871.jpgSpecial right triangles proof (part 1)Learn how to prove the ratios between the sides of a 30-60-90 triangle.https://cdn.kastatic.org/ka-youtube-converted/SFL4stapeUs.mp4/SFL4stapeUs.mp4419yes https://puzzicle.org/math/geometry/v/45-45-90-triangle-side-ratioshttps://puzzicle.org/thumbs/292160870.jpgSpecial right triangles proof (part 2)Showing the ratios of the sides of a 45-45-90 triangle are 1:1:sqrt(2)https://cdn.kastatic.org/ka-youtube-converted/McINBOFCGH8.mp4/McINBOFCGH8.mp4342yes https://puzzicle.org/math/geometry/v/angle-to-aim-to-get-alienhttps://puzzicle.org/thumbs/x9a39e2a2.jpgRight triangle word problemSal solves a modeling problem where he finds the necessary angle to shoot at a vicious alien.https://cdn.kastatic.org/ka-youtube-converted/aHzd-u35LuA.mp4/aHzd-u35LuA.mp4275yes https://puzzicle.org/math/precalculus/v/law-of-sineshttps://puzzicle.org/thumbs/x29e70bc4.jpgSolving for a side with the law of sinesSal is given a triangle with two angle measures and one side length, and he finds all the missing side lengths and angle measures using the law of sines.https://cdn.kastatic.org/ka-youtube-converted/VjmFKle7xIw.mp4/VjmFKle7xIw.mp4358yes https://puzzicle.org/math/precalculus/v/law-of-sines-examplehttps://puzzicle.org/thumbs/x955c68f1.jpgSolving for an angle with the law of sinesSal is given a triangle with two side lengths and one angle measure, and he finds all the missing angle measures using the law of sines.https://cdn.kastatic.org/ka-youtube-converted/IJySBMtFlnQ.mp4/IJySBMtFlnQ.mp4334yes https://puzzicle.org/math/precalculus/v/proof-law-of-sineshttps://puzzicle.org/thumbs/26258.jpgProof of the law of sinesSal gives a simple proof of the Law of sines.https://cdn.kastatic.org/ka-youtube-converted/APNkWrD-U1k.mp4/APNkWrD-U1k.mp4394yes https://puzzicle.org/math/precalculus/v/law-of-cosines-examplehttps://puzzicle.org/thumbs/x17ddf742.jpgSolving for a side with the law of cosinesLearn how to use the law of cosines to find the missing side length of a triangle when given two side lengths and the contained angle measure.https://cdn.kastatic.org/ka-youtube-converted/ZElOxG7_m3c.mp4/ZElOxG7_m3c.mp4278yes https://puzzicle.org/math/precalculus/v/law-of-cosines-missing-anglehttps://puzzicle.org/thumbs/xb7d3bb05.jpgSolving for an angle with the law of cosinesSal is given a triangle with all side lengths but no angle measure, and he finds one of the angle measures using the law of cosines.https://cdn.kastatic.org/ka-youtube-converted/Ei54NnQ0FKs.mp4/Ei54NnQ0FKs.mp4401yes https://puzzicle.org/math/precalculus/v/law-of-cosineshttps://puzzicle.org/thumbs/26254.jpgProof of the law of cosinesSal gives a simple proof of the Law of cosines.https://cdn.kastatic.org/ka-youtube-converted/pGaDcOMdw48.mp4/pGaDcOMdw48.mp4566yes https://puzzicle.org/math/precalculus/v/law-of-cosines-word-problemhttps://puzzicle.org/thumbs/x461dc00c.jpgTrig word problem: starsSal solves a word problem about the distance between stars using the law of cosines.https://cdn.kastatic.org/ka-youtube-converted/6kGi1dvGZNY.mp4/6kGi1dvGZNY.mp4360yes https://puzzicle.org/math/geometry-home/v/specifying-planes-in-three-dimensionshttps://puzzicle.org/thumbs/x9922e4bf.jpgSpecifying planes in three dimensionsIn a three-dimensional space, a plane can be defined by three points it contains, as long as those points are not on the same line. Learn more about it in this video.https://cdn.kastatic.org/ka-youtube-converted/J2Qz-7ZWDAE.mp4/J2Qz-7ZWDAE.mp4252yes https://puzzicle.org/math/geometry/v/cavalieris-principle-in-3dhttps://puzzicle.org/thumbs/x8f8fedc99264b1ee.jpgCavalieri's principle in 3DCavalieri's principle tells us that if 2 figures have the same height and the same cross-sectional area at every point along that height, they have the same volume.https://cdn.kastatic.org/ka-youtube-converted/2TDFtCKMoKk.mp4/2TDFtCKMoKk.mp4296yes https://puzzicle.org/math/geometry/v/using-related-volumeshttps://puzzicle.org/thumbs/x676e3e2d4c24bfe0.jpgUsing related volumesUse related volumes (using the volume of one figure to determine the volume of another).https://cdn.kastatic.org/ka-youtube-converted/knNLHsjIcl8.mp4/knNLHsjIcl8.mp4376yes https://puzzicle.org/math/geometry/v/classify-lineshttps://puzzicle.org/thumbs/x36e2a7ae.jpgParallel & perpendicular lines from graphThe slopes of parallel lines are equal, and the slopes of perpendicular lines are opposite reciprocals. This is a worked example of determining whether given lines are parallel or perpendicular.https://cdn.kastatic.org/ka-youtube-converted/6riBTnlI4fw.mp4/6riBTnlI4fw.mp4446yes https://puzzicle.org/math/geometry/v/area-of-trapezoid-on-coordinate-planehttps://puzzicle.org/thumbs/x07456d4c.jpgArea of trapezoid on the coordinate planeSal finds the area of a trapezoid using the distance formula and the trapezoid area formula.https://cdn.kastatic.org/ka-youtube-converted/3AxyrK2jYc8.mp4/3AxyrK2jYc8.mp4414yes https://puzzicle.org/math/geometry/v/minions-within-reach-of-wizardhttps://puzzicle.org/thumbs/x63e2adb6.jpgCoordinate plane word problemWatch Sal solve an example where has to determine which of the Minions a wizard can reach using the coordinate plane.https://cdn.kastatic.org/ka-youtube-converted/gS6s3MGZlRc.mp4/gS6s3MGZlRc.mp4352yes https://puzzicle.org/math/geometry/v/finding-a-point-part-way-between-two-pointshttps://puzzicle.org/thumbs/x967aacb2.jpgDividing line segments: graphicalWatch Sal figure out the coordinates of a point between two other points that give a certain ratio. A graph is given to make it easier to visualize the problem.https://cdn.kastatic.org/ka-youtube-converted/0rlNHYHhrWs.mp4/0rlNHYHhrWs.mp4280yes https://puzzicle.org/math/geometry/v/ratios-of-distances-between-colinear-pointshttps://puzzicle.org/thumbs/xbc774557.jpgDividing line segmentsWatch Sal figure out the coordinates of a point between two other points that give a certain ratio.https://cdn.kastatic.org/ka-youtube-converted/Cx-_PxD4DJM.mp4/Cx-_PxD4DJM.mp4237yes https://puzzicle.org/math/geometry/v/proving-triangle-medians-intersect-at-a-pointhttps://puzzicle.org/thumbs/x0f74f8a208da028f.jpgProving triangle medians intersect at a pointProve that for any triangle, the medians intersect at a point 2/3 of the way from each vertex to the midpoint of the opposite side.https://cdn.kastatic.org/ka-youtube-converted/yfcEgqPZBSk.mp4/yfcEgqPZBSk.mp4442yes https://puzzicle.org/math/geometry-home/v/distance-between-a-point-and-a-linehttps://puzzicle.org/thumbs/xb8d341b5.jpgDistance between point & lineThe distance from a point to a line is the shortest distance between the point and any point on the line. This can be done with a variety of tools like slope-intercept form and the Pythagorean Theorem.https://cdn.kastatic.org/ka-youtube-converted/iATjsfAX8yc.mp4/iATjsfAX8yc.mp4332yes https://puzzicle.org/math/geometry/v/inscribed-angles-exercise-examplehttps://puzzicle.org/thumbs/x481d7c00.jpgInscribed anglesSal finds a missing inscribed angle using the inscribed angle theorem.https://cdn.kastatic.org/ka-youtube-converted/aNi1-zeEps8.mp4/aNi1-zeEps8.mp480yes https://puzzicle.org/math/geometry/v/inscribed-and-central-angleshttps://puzzicle.org/thumbs/24977.jpgInscribed angle theorem proofProving that an inscribed angle is half of a central angle that subtends the same arc.https://cdn.kastatic.org/ka-youtube-converted/MyzGVbCHh5M.mp4/MyzGVbCHh5M.mp4856yes https://puzzicle.org/math/geometry/v/right-triangles-inscribed-in-circles-proofhttps://puzzicle.org/thumbs/24981.jpgProof: Right triangles inscribed in circlesProof showing that a triangle inscribed in a circle having a diameter as one side is a right trianglehttps://cdn.kastatic.org/ka-youtube-converted/b0U1NxbRU4w.mp4/b0U1NxbRU4w.mp4335yes https://puzzicle.org/math/geometry/v/hypotenuse-of-right-triangle-inscribed-in-circlehttps://puzzicle.org/thumbs/x2218bca9.jpgInscribed shapes: find diameterFind the diameter of a circle using an inscribed right triangle.https://cdn.kastatic.org/ka-youtube-converted/T971zHhZ3S4.mp4/T971zHhZ3S4.mp4153yes https://puzzicle.org/math/geometry/v/solving-for-unknown-anglehttps://puzzicle.org/thumbs/x6f35e673.jpgInscribed shapes: angle subtended by diameterOne method using an inscribed angle intercepting a diameter of a circle.https://cdn.kastatic.org/ka-youtube-converted/pCLGpRaJfJ8.mp4/pCLGpRaJfJ8.mp4349yes https://puzzicle.org/math/geometry/v/finding-inscribed-angle-measurehttps://puzzicle.org/thumbs/x0594ad71.jpgInscribed shapes: find inscribed angleUsing either the inscribed angle theorem or the fact that two inscribed angles that intercept the same arc must be congruent.https://cdn.kastatic.org/ka-youtube-converted/h-_BDon5oes.mp4/h-_BDon5oes.mp4130yes https://puzzicle.org/math/geometry/v/proving-the-opposite-angles-of-inscribed-quadrilateral-are-supplementaryhttps://puzzicle.org/thumbs/xaaf3d462.jpgInscribed quadrilaterals proofSal uses the inscribed angle theorem and some algebra to prove that opposite angles of an inscribed quadrilateral are supplementary.https://cdn.kastatic.org/ka-youtube-converted/LGnL17ESs3Q.mp4/LGnL17ESs3Q.mp4236yes https://puzzicle.org/math/geometry/v/example-showing-supplementary-oppositie-angles-in-inscribed-quadrilateralhttps://puzzicle.org/thumbs/x265f991c.jpgSolving inscribed quadrilateralsExample showing supplementary opposite angles in inscribed quadrilateral.https://cdn.kastatic.org/ka-youtube-converted/Y9cznjBl0Hk.mp4/Y9cznjBl0Hk.mp4298yes https://puzzicle.org/math/geometry/v/constructing-square-inscribed-in-circlehttps://puzzicle.org/thumbs/xe85c0014.jpgGeometric constructions: circle-inscribed squareSal constructs a square that is inscribed inside a given circle using compass and straightedge.https://cdn.kastatic.org/ka-youtube-converted/-gWtl6mdpeY.mp4/-gWtl6mdpeY.mp4167yes https://puzzicle.org/math/geometry/v/constructing-equilateral-triangle-inscribed-in-circlehttps://puzzicle.org/thumbs/xb4a12e50.jpgGeometric constructions: circle-inscribed equilateral triangleSal constructs an equilateral triangle that is inscribed inside a given circle using compass and straightedge.https://cdn.kastatic.org/ka-youtube-converted/gWMTTP58_J0.mp4/gWMTTP58_J0.mp4208yes https://puzzicle.org/math/geometry/v/sss-to-show-a-radius-is-perpendicular-to-a-chord-that-it-bisectshttps://puzzicle.org/thumbs/267245530.jpgProof: radius is perpendicular to a chord it bisectsSal proves that if a radius in a circle is drawn so it bisects a chord, then the radius is also perpendicular to that chord. The proof uses SSS congruence.https://cdn.kastatic.org/ka-youtube-converted/TgDk06Qayxw.mp4/TgDk06Qayxw.mp4499yes https://puzzicle.org/math/geometry/v/perpendicular-radius-bisects-chordhttps://puzzicle.org/thumbs/298420732.jpgProof: perpendicular radius bisects chordSimple proof using right triangle-side-hypotenuse (RSH) congruence criterion to show that a radius perpendicular to a chord bisects ithttps://cdn.kastatic.org/ka-youtube-converted/2NgzruZ4_5c.mp4/2NgzruZ4_5c.mp4233yes https://puzzicle.org/math/geometry/v/intro-arc-measurehttps://puzzicle.org/thumbs/xa3a0dfdf.jpgIntro to arc measureSal introduces arc measure including that they must sum to 360, conventions of minor and major arcs, and a comparison to arc length.https://cdn.kastatic.org/ka-youtube-converted/_H4jllna_ec.mp4/_H4jllna_ec.mp4327yes https://puzzicle.org/math/geometry/v/arc-measure-example-problemshttps://puzzicle.org/thumbs/xda646acf.jpgFinding arc measuresWatch Sal solve a few problems where he finds a missing arc measure.https://cdn.kastatic.org/ka-youtube-converted/GOA9XWEo7QI.mp4/GOA9XWEo7QI.mp4396yes https://puzzicle.org/math/geometry/v/arc-measure-with-equations-exampleshttps://puzzicle.org/thumbs/x2061f76a.jpgFinding arc measures with equationsSal solves a few items where arc measures are given in equations, we have to find a variable, then use it to find an arc measure.https://cdn.kastatic.org/ka-youtube-converted/bsDu8VQKwxA.mp4/bsDu8VQKwxA.mp4418yes https://puzzicle.org/math/geometry/v/length-of-an-arc-that-subtends-a-central-anglehttps://puzzicle.org/thumbs/xb22c18d7.jpgArc length from subtended angleFinding the length of an arc using the degree of the angle subtended by the arc and the perimeter of the circle.https://cdn.kastatic.org/ka-youtube-converted/tVcasOt55Lc.mp4/tVcasOt55Lc.mp4298yes https://puzzicle.org/math/geometry/v/finding-central-angle-measure-given-arc-lengthhttps://puzzicle.org/thumbs/x99baffff.jpgSubtended angle from arc lengthWatch Sal solve an example where he finds the central angle given arc length.https://cdn.kastatic.org/ka-youtube-converted/1BH2TNzAAik.mp4/1BH2TNzAAik.mp4138yes https://puzzicle.org/math/geometry/v/arc-length-as-fraction-of-circumferencehttps://puzzicle.org/thumbs/xc3486592.jpgArc length as fraction of circumferenceSal finds the fraction of an arc length out of the entire circumference using the radian measure of the central angle subtended by the arc.https://cdn.kastatic.org/ka-youtube-converted/XFI7GK4Nv-s.mp4/XFI7GK4Nv-s.mp4179yes https://puzzicle.org/math/geometry/v/arc-length-from-angle-measurehttps://puzzicle.org/thumbs/x056f0d59.jpgArc length from subtended angle: radiansSal finds the length of an arc using the radius and the radian measure of the angle subtended by the arc.https://cdn.kastatic.org/ka-youtube-converted/dtQyiDMoub4.mp4/dtQyiDMoub4.mp4159yes https://puzzicle.org/math/geometry/v/area-of-a-sector-given-a-central-anglehttps://puzzicle.org/thumbs/x44915f06.jpgArea of a sectorA worked example of finding the area of a circle's sector using the area of the circle and the central angle of the sector.https://cdn.kastatic.org/ka-youtube-converted/u8JFdwmBvvQ.mp4/u8JFdwmBvvQ.mp4146yes https://puzzicle.org/math/geometry/v/proving-radius-is-perpendicular-to-tangent-linehttps://puzzicle.org/thumbs/x5cef9421.jpgProof: Radius is perpendicular to tangent lineSal proves that the radius that connects the intersection point of a tangent line with the circle is perpendicular to the tangent line.https://cdn.kastatic.org/ka-youtube-converted/St3MU5i3chc.mp4/St3MU5i3chc.mp4521yes https://puzzicle.org/math/geometry/v/segments-tangent-to-circle-from-outside-point-congruenthttps://puzzicle.org/thumbs/x7d734c90.jpgProof: Segments tangent to circle from outside point are congruentSal proves that two tangent segments to a circle that are drawn from the same outside point are congruent.https://cdn.kastatic.org/ka-youtube-converted/pJABplruJIU.mp4/pJABplruJIU.mp4236yes https://puzzicle.org/math/geometry/v/tangents-of-circles-exercise-exampleshttps://puzzicle.org/thumbs/x1e3a5b56.jpgTangents of circles problem (example 1)Sal finds missing angles using the property that tangents are perpendicular to the radius.https://cdn.kastatic.org/ka-youtube-converted/8vFhNhL-zm8.mp4/8vFhNhL-zm8.mp4277yes https://puzzicle.org/math/geometry/v/measure-of-circumscribed-anglehttps://puzzicle.org/thumbs/xedc9cfc6.jpgTangents of circles problem (example 2)Sal finds a missing angle using the property that tangents are perpendicular to the radius.https://cdn.kastatic.org/ka-youtube-converted/ZiqHJwzv_HI.mp4/ZiqHJwzv_HI.mp4227yes https://puzzicle.org/math/geometry/v/example-with-tangent-and-radiushttps://puzzicle.org/thumbs/x7b77bf0f.jpgTangents of circles problem (example 3)Sal finds a missing length using the property that tangents are perpendicular to the radius.https://cdn.kastatic.org/ka-youtube-converted/KjQ1KN5GgoE.mp4/KjQ1KN5GgoE.mp4154yes https://puzzicle.org/math/geometry/v/constructing-a-tangent-line-using-compass-and-straightedgehttps://puzzicle.org/thumbs/xd131cdd3.jpgGeometric constructions: circle tangentSal constructs a line tangent to a circle using compass and straightedge.https://cdn.kastatic.org/ka-youtube-converted/n5cKWjZqpIE.mp4/n5cKWjZqpIE.mp4350yes https://puzzicle.org/math/geometry/v/another-example-using-compass-and-straightedge-for-tangent-linehttps://puzzicle.org/thumbs/x6b83da1c.jpgGeometric constructions: circle tangent (example 2)Sal constructs a line tangent to a circle using compass and straightedge.https://cdn.kastatic.org/ka-youtube-converted/g_dStt4st2I.mp4/g_dStt4st2I.mp4335yes https://puzzicle.org/math/precalculus/v/multiply-complex-graphically-3ihttps://puzzicle.org/thumbs/xf2a26e18a5a55bd3.jpgMultiplying complex numbers graphically example: -3iWe can multiply complex numbers graphically on the complex plane by rotating and scaling. Multiplying a complex number z by -3i rotates and scales z.https://cdn.kastatic.org/ka-youtube-converted/fqwR6RNPJgc.mp4/fqwR6RNPJgc.mp4122yes https://puzzicle.org/math/precalculus/v/multiply-complex-graphically-1-ihttps://puzzicle.org/thumbs/xec05d579abc39d4c.jpgMultiplying complex numbers graphically example: -1-iWe can multiply complex numbers graphically on the complex plane. We rotate an amount equal to the argument and scale by the modulus of the complex number by which we're multiplying.https://cdn.kastatic.org/ka-youtube-converted/ebEwF4kb6pI.mp4/ebEwF4kb6pI.mp4217yes https://puzzicle.org/math/precalculus/v/using-matrices-to-transform-the-plane-mapping-a-vectorhttps://puzzicle.org/thumbs/x2abc8976bce511cf.jpgUsing matrices to transform the plane: Mapping a vector2X2 matrices can define transformations for the entire plane. In this worked example, we see how to find the image of a given vector under the transformation defined by a given matrix.https://cdn.kastatic.org/ka-youtube-converted/TuDYc-8L2Bw.mp4/TuDYc-8L2Bw.mp4255yes https://puzzicle.org/math/precalculus/v/using-matrices-to-transform-the-plane-composing-matriceshttps://puzzicle.org/thumbs/xdb226d4911f54a41.jpgUsing matrices to transform the plane: Composing matrices2X2 matrices can define transformations for the entire plane. In this worked example, we see how to find a single matrix that defines the same transformation as the composition of two other matrices.https://cdn.kastatic.org/ka-youtube-converted/OzoEqs1f1S8.mp4/OzoEqs1f1S8.mp4317yes https://puzzicle.org/math/precalculus/v/using-matrices-to-represent-data-networkshttps://puzzicle.org/thumbs/x8a19e3afa1ae7a58.jpgUsing matrices to represent data: NetworksMatrices are basically tables of numeric values. But you may be surprised at how many real-world situations you can model with this structure. Here, for example, we represent a network of bus routes between cities.https://cdn.kastatic.org/ka-youtube-converted/IxmFLAPSx5I.mp4/IxmFLAPSx5I.mp4340yes https://puzzicle.org/math/precalculus/v/using-matrices-to-represent-data-payoffshttps://puzzicle.org/thumbs/xfd2ba4c6357cffcf.jpgUsing matrices to represent data: PayoffsMatrices are basically tables of numeric values. But you may be surprised at how many real-world situations you can model with this structure. Here, for example, we represent a game's payoffs: we represent the points each player gets in an elaborate version of rock, paper, scissors.https://cdn.kastatic.org/ka-youtube-converted/2ObBIZoZnJ0.mp4/2ObBIZoZnJ0.mp4403yes https://puzzicle.org/math/engageny-precalc/v/rendering-5https://puzzicle.org/thumbs/x894807a8.jpg1. Ray tracing intuitionFirst, let's look at ray tracing in 2D using a simple example. In 2D, an image line is formed from the camera location and the viewing direction. When a pixel's color is being determined, a ray is built from the camera, through a point on the image line, and off into the scene. Let's look at the algebra of intersecting a ray with a line segment!https://cdn.kastatic.org/ka-youtube-converted/-4s4vYJnIW4.mp4/-4s4vYJnIW4.mp4148yes https://puzzicle.org/math/engageny-precalc/v/rendering-6https://puzzicle.org/thumbs/x03b6b9b8.jpg2. Parametric form of a rayThis video introduces the parametric form of a ray in 2D. To find the intersection of a two-dimensional ray and a line segment, we use slope intercept form. But this method doesn't work in three dimensions! In 3D, we can represent a ray using a parametric function. This function, R(t), is a weighted average of two points, and it's super handy for naming points on the ray. Click here to review weighted averages of two points. Click here to review slope-intercept form of a line.https://cdn.kastatic.org/ka-youtube-converted/oPTRwYdOEAM.mp4/oPTRwYdOEAM.mp4183yes https://puzzicle.org/math/engageny-precalc/v/rendering-7https://puzzicle.org/thumbs/xdd89aa6a.jpg3. Calculate intersection pointNow we are ready to calculate an intersection point using our ray in parametric form and our line in slope-intercept form. Different values of a parameter t locate different points on the ray. The intersection point we're looking for is one such point, so there's a value of t, t*, that gives us the intersection point! How do solve for t*?https://cdn.kastatic.org/ka-youtube-converted/Mm7ZmL9zfpQ.mp4/Mm7ZmL9zfpQ.mp4162yes https://puzzicle.org/math/engageny-precalc/v/rendering-8https://puzzicle.org/thumbs/x637f7234.jpg4. Using the line equationVertical lines have an undefined slope, so how can we solve for the intersection point? Instead of using slope-intercept form, we need to use the line equation, or implicit form. Click here to see why dividing by zero is bad. Click here to try the interactive program shown in the video.https://cdn.kastatic.org/ka-youtube-converted/b0cgLILc2Eo.mp4/b0cgLILc2Eo.mp4154yes https://puzzicle.org/math/engageny-precalc/v/rendering-9https://puzzicle.org/thumbs/xb97c6b84.jpg5. 3D ray tracing part 1Now we are ready to ray trace in 3D! We'll look at the problem of ray triangle intersection. We can break down complex shapes, like characters, into smaller four-sided polygons, which we can then convert into triangles. Click here for more detail on the plane equation.https://cdn.kastatic.org/ka-youtube-converted/KCSGeTdjWiI.mp4/KCSGeTdjWiI.mp4251yes https://puzzicle.org/math/engageny-precalc/v/half-lifehttps://puzzicle.org/thumbs/423007.jpgHalf-life and carbon datingCarbon dating is a real-life example of a first-order reaction. This video explains half-life in the context of radioactive decay.https://cdn.kastatic.org/ka-youtube-converted/9REPnibO4IQ.mp4/9REPnibO4IQ.mp4751yes https://puzzicle.org/science/physics/v/wave-interference-pulseshttps://puzzicle.org/thumbs/x8a690f5a.jpgWave interferenceLet's explore wave interference and solve a few examples to find the value of the total wave when two wave pulses overlap.https://cdn.kastatic.org/ka-youtube-converted/lpee2Qddqjo.mp4/lpee2Qddqjo.mp4444yes https://puzzicle.org/math/precalculus/v/permutation-formulahttps://puzzicle.org/thumbs/x26c39251.jpgPermutation formulaWant to learn about the permutation formula and how to apply it to tricky problems? Explore this useful technique by solving seating arrangement problems with factorial notation and a general formula. This video also demonstrates the benefits of deductive reasoning over memorization.https://cdn.kastatic.org/ka-youtube-converted/DROZVHObeko.mp4/DROZVHObeko.mp4455yes https://puzzicle.org/math/precalculus/v/zero-factorial-or-0https://puzzicle.org/thumbs/xdb036a16.jpgZero factorial or 0!Learn all about factorials! Factorials are a quick way to represent multiplying a number by all the smaller positive integers down to one. This video also shows why mathematicians have defined zero factorial as one, instead of zero, to make the formula for permutations work in all cases.https://cdn.kastatic.org/ka-youtube-converted/HGoZfzz6dU0.mp4/HGoZfzz6dU0.mp4290yes https://puzzicle.org/math/precalculus/v/possible-three-letter-wordshttps://puzzicle.org/thumbs/x66bfedb8.jpgPossible three letter wordsSal explains how to find all of the possible three letter words when we can use each letter as many times as we want, and when each letter can only be used once.https://cdn.kastatic.org/ka-youtube-converted/VYbqG2NuOo8.mp4/VYbqG2NuOo8.mp4339yes https://puzzicle.org/math/precalculus/v/permutations-and-combinations-1https://puzzicle.org/thumbs/27097.jpgWays to arrange colorsThinking about how many ways you can pick four colors from a group of 6https://cdn.kastatic.org/ka-youtube-converted/oQpKtm5TtxU.mp4/oQpKtm5TtxU.mp4225yes https://puzzicle.org/math/precalculus/v/permutations-and-combinations-2https://puzzicle.org/thumbs/27098.jpgWays to pick officersHow many ways can we pick officers for our organization?https://cdn.kastatic.org/ka-youtube-converted/v9NLtiVt3XY.mp4/v9NLtiVt3XY.mp4139yes https://puzzicle.org/math/precalculus/v/introduction-to-combinationshttps://puzzicle.org/thumbs/xe7fa4091.jpgIntro to combinationsLearn the difference between permutations and combinations, using the example of seating six people in three chairs. Permutations count the different arrangements of people in specific chairs, while combinations count the different groups of people, regardless of order or chair.https://cdn.kastatic.org/ka-youtube-converted/iKy-d5_erhI.mp4/iKy-d5_erhI.mp4378yes https://puzzicle.org/math/precalculus/v/combination-formulahttps://puzzicle.org/thumbs/x4a534332.jpgCombination formulaLearn the difference between permutations and combinations, and how to calculate them using factorials. This video also discusses binomial coefficients and the formula for combinations, nCk. Solidify your understanding with an example: how would one seat six people in four chairs?https://cdn.kastatic.org/ka-youtube-converted/p8vIcmr_Pqo.mp4/p8vIcmr_Pqo.mp4677yes https://puzzicle.org/math/precalculus/v/handshaking-combinationshttps://puzzicle.org/thumbs/x70c33285.jpgHandshaking combinationsSal figures out how different combinations of people can shake hands.https://cdn.kastatic.org/ka-youtube-converted/boH4l1SgJbM.mp4/boH4l1SgJbM.mp4449yes https://puzzicle.org/math/precalculus/v/permutations-and-combinations-3https://puzzicle.org/thumbs/27099.jpgCombination example: 9 card handsThinking about how many ways we can construct a hand of 9 cardshttps://cdn.kastatic.org/ka-youtube-converted/SbpoyXTpC84.mp4/SbpoyXTpC84.mp4462yes https://puzzicle.org/math/statistics-probability/v/getting-exactly-two-heads-combinatoricshttps://puzzicle.org/thumbs/193210641.jpgGetting exactly two heads (combinatorics)A different way to think about the probability of getting 2 heads in 4 flipshttps://cdn.kastatic.org/ka-youtube-converted/8TIben0bJpU.mp4/8TIben0bJpU.mp4600yes https://puzzicle.org/math/statistics-probability/v/exactly-three-heads-in-five-flipshttps://puzzicle.org/thumbs/193210640.jpgExactly three heads in five flipsProbability of exactly 3 heads in 5 flips using combinationshttps://cdn.kastatic.org/ka-youtube-converted/udG9KhNMKJw.mp4/udG9KhNMKJw.mp4411yes https://puzzicle.org/math/precalculus/v/probability-of-dependent-eventshttps://puzzicle.org/thumbs/27101.jpgExample: Different ways to pick officersThinking about the different ways we can pick officers in order to find the probability of one situation in particular.https://cdn.kastatic.org/ka-youtube-converted/l9ft9jpriNA.mp4/l9ft9jpriNA.mp4143yes https://puzzicle.org/math/statistics-probability/v/probability-of-dependent-events-2https://puzzicle.org/thumbs/27102.jpgExample: Combinatorics and probabilityProbability of getting a set of cardshttps://cdn.kastatic.org/ka-youtube-converted/ccrYD6iX_SY.mp4/ccrYD6iX_SY.mp4651yes https://puzzicle.org/math/precalculus/v/permutations-and-combinations-4https://puzzicle.org/thumbs/27100.jpgExample: Lottery probabilityWhat is the probability of winning a 4-number lottery?https://cdn.kastatic.org/ka-youtube-converted/DIjlllgq3dc.mp4/DIjlllgq3dc.mp4310yes https://puzzicle.org/math/precalculus/v/mega-millions-jackpot-probabilityhttps://puzzicle.org/thumbs/690344673.jpgMega millions jackpot probabilityProbability of winning the Mega Millions jackpothttps://cdn.kastatic.org/ka-youtube-converted/gyqodNhM3EU.mp4/gyqodNhM3EU.mp4330yes https://puzzicle.org/math/statistics-probability/v/random-variableshttps://puzzicle.org/thumbs/1203203255.jpgRandom variablesBasic idea and definitions of random variableshttps://cdn.kastatic.org/ka-youtube-converted/3v9w79NhsfI.mp4/3v9w79NhsfI.mp4332yes https://puzzicle.org/math/statistics-probability/v/discrete-and-continuous-random-variableshttps://puzzicle.org/thumbs/1202908818.jpgDiscrete and continuous random variablesDiscrete random variables can only take on a finite number of values. For example, the outcome of rolling a die is a discrete random variable, as it can only land on one of six possible numbers. Continuous random variables, on the other hand, can take on any value in a given interval. For example, the mass of an animal would be a continuous random variable, as it could theoretically be any non-negative number.https://cdn.kastatic.org/ka-youtube-converted/dOr0NKyD31Q.mp4/dOr0NKyD31Q.mp4717yes https://puzzicle.org/math/statistics-probability/v/expected-value-profit-lottery-tickethttps://puzzicle.org/thumbs/xf4ac78a7.jpgExpected profit from lottery ticketSal multiplies outcomes by probabilities to find the expected value of a lottery ticket.https://cdn.kastatic.org/ka-youtube-converted/6vlBOHckmzU.mp4/6vlBOHckmzU.mp4558yes https://puzzicle.org/math/statistics-probability/v/expected-value-wagerhttps://puzzicle.org/thumbs/xfd69eb4f.jpgExpected value while fishingSal walks through an example where he multiplies probability by values and sums to find expected value.https://cdn.kastatic.org/ka-youtube-converted/86nb02Bx_5w.mp4/86nb02Bx_5w.mp4504yes https://puzzicle.org/math/statistics-probability/v/expected-value-insurancehttps://puzzicle.org/thumbs/xb959145f.jpgComparing insurance with expected valueSal uses expected value to compare a couple of different insurance policies.https://cdn.kastatic.org/ka-youtube-converted/BveA7JiRGS8.mp4/BveA7JiRGS8.mp4556yes https://puzzicle.org/math/financial-mathematics-tx-teks/v/estimating-total-postsecondary-costshttps://puzzicle.org/thumbs/xd20e8eb13d5a526a.jpgEstimating total postsecondary costsCalculate the total funding required to complete a desired postsecondary education program, accounting for increasing and fixed costs.https://cdn.kastatic.org/ka-youtube-converted/wE_EKGUiJt0.mp4/wE_EKGUiJt0.mp4205yes https://puzzicle.org/math/financial-mathematics-tx-teks/v/roi-of-educationhttps://puzzicle.org/thumbs/x400018a305393ce1.jpgROI of educationCalculating education return on investment (ROI) involves dividing the net benefit of a degree by the total investment required. One approach focuses strictly on cash flows, while a more comprehensive method incorporates opportunity costs like foregone wages during years of study. Accurate long-term evaluations should further consider factors such as wage growth, loan interest, and the time value of money.https://cdn.kastatic.org/ka-youtube-converted/OJfPASqhLUo.mp4/OJfPASqhLUo.mp4386yes https://puzzicle.org/math/financial-mathematics-tx-teks/v/cost-vs-career-comparisonhttps://puzzicle.org/thumbs/x0f211796d101043c.jpgCost vs career comparisonTo determine if a four-year degree is financially worthwhile, one can use algebra to compare immediate employment earnings against the costs of education followed by a higher salary. This calculation accounts for both tuition expenses and the opportunity cost of lost wages during the years of study.https://cdn.kastatic.org/ka-youtube-converted/dGaILHaNPbc.mp4/dGaILHaNPbc.mp4285yes https://puzzicle.org/math/financial-mathematics-tx-teks/v/employment-vocabularyhttps://puzzicle.org/thumbs/x8edcfd078dd0b822.jpgEmployment vocabularyWhen it comes to employment, the main ideas are understanding your wage rights, like minimum wage and overtime pay, and knowing the difference between a W2 employee and a 1099 contractor, which affects how you handle taxes.https://cdn.kastatic.org/ka-youtube-converted/LHwIGwc82K0.mp4/LHwIGwc82K0.mp4372yes https://puzzicle.org/math/financial-mathematics-tx-teks/v/calculating-gross-pay-with-overtimehttps://puzzicle.org/thumbs/x4900282326ac4b7e.jpgCalculating gross pay with overtimeGross pay with overtime is calculated by paying regular hours at the normal hourly rate and any hours over 40 in a week at 1.5 times that rate. The total gross pay is the sum of regular pay and overtime pay.https://cdn.kastatic.org/ka-youtube-converted/qg-8X4JW5uk.mp4/qg-8X4JW5uk.mp4148yes https://puzzicle.org/math/financial-mathematics-tx-teks/v/calculating-pay-for-exempt-vs-non-exempt-employeeshttps://puzzicle.org/thumbs/x628e22ee67fd141f.jpgCalculating pay for exempt vs non-exempt employeesSometimes non-exempt employees can earn more than exempt employees because they are paid overtime for extra hours worked. Exempt employees usually earn a fixed salary, so their pay doesn’t increase when they work longer hours.https://cdn.kastatic.org/ka-youtube-converted/HxNI_2lfhJE.mp4/HxNI_2lfhJE.mp4222yes https://puzzicle.org/math/financial-mathematics-tx-teks/v/calculating-backpayhttps://puzzicle.org/thumbs/x7de0e7a14c8b2235.jpgCalculating back payEven when an employee is paid a flat amount, like $200 per week, the employer must ensure that the pay still meets minimum wage requirements based on hours worked. If it doesn’t, the employer is required to provide back pay to cover the shortfall.https://cdn.kastatic.org/ka-youtube-converted/LI2o3MNwEJ4.mp4/LI2o3MNwEJ4.mp478yes https://puzzicle.org/math/financial-mathematics-tx-teks/v/calculating-wages-when-earning-tipshttps://puzzicle.org/thumbs/x25213ccf4c424716.jpgCalculating wages when earning tipsWhen you earn tips, your employer is allowed to pay a lower base hourly wage, assuming tips will make up the difference. However, if your tips plus base pay don’t add up to at least the state minimum wage, your employer is required to cover the gap.https://cdn.kastatic.org/ka-youtube-converted/OrLn73I_pI0.mp4/OrLn73I_pI0.mp4181yes https://puzzicle.org/math/financial-mathematics-tx-teks/v/cost-of-using-credit-cardshttps://puzzicle.org/thumbs/xe7d9c3de014c87f7.jpgCost of using credit cardsThe cost of using a credit card includes more than just interest; it also includes various fees. Some fees are charged simply for having the card, such as an annual fee, while others are triggered by specific actions, such as making a foreign purchase or missing a payment.https://cdn.kastatic.org/ka-youtube-converted/pBM7Mf65axo.mp4/pBM7Mf65axo.mp4232yes https://puzzicle.org/math/financial-mathematics-tx-teks/v/estimating-life-insurance-needshttps://puzzicle.org/thumbs/x6eab452a49dd7cae.jpgEstimating life insurance needsLife insurance is essential for supporting your family financially and covering expenses like mortgages, debts, and education in the event of your death. You can estimate your coverage needs by simply multiplying your annual income by 10 to 15, or by precisely calculating your future financial obligations minus current assets.https://cdn.kastatic.org/ka-youtube-converted/U5wgzYwQ6Os.mp4/U5wgzYwQ6Os.mp4319yes https://puzzicle.org/math/financial-mathematics-tx-teks/v/marginal-and-effective-tax-rateshttps://puzzicle.org/thumbs/x6c4d5767b4f3a2da.jpgMarginal and effective tax ratesMany people think moving into a higher tax bracket means all their income is taxed more. That myth comes from misunderstanding marginal vs. effective tax rates.https://cdn.kastatic.org/ka-youtube-converted/5YJcyw3r6Ww.mp4/5YJcyw3r6Ww.mp4208yes https://puzzicle.org/math/financial-mathematics-tx-teks/v/capital-gains-taxhttps://puzzicle.org/thumbs/xd0ef5b111b83e1ab.jpgCalculating capital gains taxCapital gains tax is the tax you pay on the profit from selling an asset, such as stocks or real estate. The tax you pay depends on how long you’ve owned the asset. If you hold the asset for one year or less, the gain is short-term, while holding it for more than one year makes it a long-term gain, which is usually taxed at a lower rate.https://cdn.kastatic.org/ka-youtube-converted/ZYioN1rIHZE.mp4/ZYioN1rIHZE.mp4273yes https://puzzicle.org/math/financial-mathematics-tx-teks/v/calculating-tax-on-interesthttps://puzzicle.org/thumbs/x1ba0da35634a5737.jpgCalculating tax on interestYou pay taxes on interest income you earn from sources like savings accounts, bonds, or certificates of deposit. The government includes this interest in your taxable income and applies your regular income tax rate to it.https://cdn.kastatic.org/ka-youtube-converted/xrnsCEuEHsI.mp4/xrnsCEuEHsI.mp4274yes https://puzzicle.org/math/financial-mathematics-tx-teks/v/calculating-savings-when-paying-pre-taxhttps://puzzicle.org/thumbs/xf55d03f83e4f2765.jpgCalculating savings when paying pre-taxUsing pre-tax dollars to pay for expenses allows individuals to avoid paying income tax on that money, resulting in significant savings. This approach is more efficient than using after-tax dollars because it reduces the total amount of gross income needed to cover the same cost.https://cdn.kastatic.org/ka-youtube-converted/QxVKAu-SbrE.mp4/QxVKAu-SbrE.mp4150yes https://puzzicle.org/math/financial-mathematics-tx-teks/v/what-determines-if-you-are-getting-a-tax-refund-or-billhttps://puzzicle.org/thumbs/x843aabca48892ee2.jpgWhat determines if you are getting a tax refund or bill?Your employer takes federal income tax out of your paycheck, and that amount appears on your W-2. If they took out more than you actually owe, you get a refund; if they took out less, you have to pay the difference.https://cdn.kastatic.org/ka-youtube-converted/oOwZygUZIN8.mp4/oOwZygUZIN8.mp4229yes https://puzzicle.org/math/financial-mathematics-tx-teks/v/charitable-contributions-as-tax-deductionshttps://puzzicle.org/thumbs/xc3a550762b90ed60.jpgCharitable contributions as tax deductionsYou can deduct charitable contributions you make to registered nonprofits or qualified charities, whether you take the standard deduction or itemize. To claim the deduction, add up your eligible donations, make sure you have a receipt for each one, and apply the applicable tax rules on your tax return.https://cdn.kastatic.org/ka-youtube-converted/BfKDwIcFp-4.mp4/BfKDwIcFp-4.mp4338yes https://puzzicle.org/math/financial-mathematics-tx-teks/v/calculating-total-purchase-price-of-bondshttps://puzzicle.org/thumbs/xa20073ec40d06ded.jpgCalculating total purchase price of bondsBuying bonds typically involves more than just their market value, as investors must also account for commissions and accrued interest. This walkthrough explains how to factor in these additional costs to determine the total purchase price for a bulk investment.https://cdn.kastatic.org/ka-youtube-converted/7mSGhHarTFw.mp4/7mSGhHarTFw.mp4283yes https://puzzicle.org/math/financial-mathematics-tx-teks/v/calculating-proceeds-from-selling-bondshttps://puzzicle.org/thumbs/x145f36160e36f363.jpgCalculating proceeds from selling bondsSelling bonds requires factoring in market price, broker commissions, and accrued interest to determine total proceeds. This video guides you through these calculations for corporate bonds, highlighting the industry's standard 30/360 day-count convention for determining interest earned between payment dates.https://cdn.kastatic.org/ka-youtube-converted/5tguTHQReQc.mp4/5tguTHQReQc.mp4273yes https://puzzicle.org/math/financial-mathematics-tx-teks/v/calculating-savings-for-financial-independencehttps://puzzicle.org/thumbs/x87c806ba9b358940.jpgCalculating savings for financial independenceTo reach financial independence, you calculate a total savings goal based on how much you plan to spend during retirement while accounting for inflation. Once you have that goal, you determine the yearly savings needed by factoring in the growth of your current assets and a steady investment return.https://cdn.kastatic.org/ka-youtube-converted/htZjYFhfI9A.mp4/htZjYFhfI9A.mp4462yes https://puzzicle.org/math/financial-mathematics-tx-teks/v/calculating-savings-for-financial-independence-using-tvm-solverhttps://puzzicle.org/thumbs/x6b6618a68d4c4037.jpgCalculating savings for financial independence (using TVM solver)To calculate savings needed for financial independence we can use a TVM (Time Value of Money) solver, available in many financial calculators, including TI-84.https://cdn.kastatic.org/ka-youtube-converted/Ytc9sPvgv-o.mp4/Ytc9sPvgv-o.mp4327yes https://puzzicle.org/math/financial-mathematics-tx-teks/v/calculating-and-graphing-exponential-functions-for-investinghttps://puzzicle.org/thumbs/x854de0c8e9c62571.jpgCalculating and graphing exponential functions for investingStarting to save early for retirement provides a significant long-term financial advantage over waiting to start later, even if a person contributes less money per month.https://cdn.kastatic.org/ka-youtube-converted/xHfwJ0ycE8s.mp4/xHfwJ0ycE8s.mp4379yes https://puzzicle.org/math/financial-mathematics-tx-teks/v/calculating-and-graphing-exponential-functions-for-investing-part-2https://puzzicle.org/thumbs/x5259e9adb8784c10.jpgCalculating and graphing exponential functions for investing (part 2)Graphical analysis of exponential functions when it comes to investing.https://cdn.kastatic.org/ka-youtube-converted/q6Hw5vsSTD4.mp4/q6Hw5vsSTD4.mp4172yes https://puzzicle.org/math/financial-mathematics-tx-teks/v/calculating-future-value-of-annuityhttps://puzzicle.org/thumbs/xb39e9ccb46201c9a.jpgCalculating future value of annuityAn ordinary annuity involves recurring payments made at the end of each compounding period, while an annuity due involves payments made at the beginning of each period. Because every payment in an annuity due has an extra period to compound interest, its total future value is higher and its growth graph will lie above that of an ordinary annuity.https://cdn.kastatic.org/ka-youtube-converted/4oulW0biqqg.mp4/4oulW0biqqg.mp4348yes https://puzzicle.org/math/financial-mathematics-tx-teks/v/early-withdrawals-and-penaltieshttps://puzzicle.org/thumbs/x4e65d489a40eca25.jpgEarly withdrawals and penaltiesEarly withdrawals from a Roth IRA only incur penalties and federal income taxes on the investment gains because the principal contributions consist of post-tax dollars. Conversely, an early withdrawal from a traditional 401(k) requires paying both penalties and income taxes on the entire amount withdrawn since it consists entirely of pre-tax money.https://cdn.kastatic.org/ka-youtube-converted/AbNev7JKgKg.mp4/AbNev7JKgKg.mp4221yes https://puzzicle.org/math/geometry/v/euclid-as-the-father-of-geometryhttps://puzzicle.org/thumbs/611293511.jpgEuclid as the father of geometryEuclid, often called the father of geometry, changed the way we learn about shapes with his 13-book series, Euclid's Elements. He used basic ideas called axioms or postulates to create solid proofs and figure out new ideas called theorems and propositions. Studying Euclidean geometry helps us think better and solve problems more effectively.https://cdn.kastatic.org/ka-youtube-converted/WqzK3UAXaHs.mp4/WqzK3UAXaHs.mp4503yes https://puzzicle.org/math/geometry/v/geometric-precision-practicehttps://puzzicle.org/thumbs/xf56c1db9.jpgGeometric definitions exampleUsing precise language is important in geometry, as it allows for clear communication and understanding of geometric concepts. This video provides an interactive exercise that emphasizes the significance of accurate definitions, helping learners develop their mathematical communication skills and enhance their grasp of geometry.https://cdn.kastatic.org/ka-youtube-converted/iDQ1foxYf0o.mp4/iDQ1foxYf0o.mp4526yes https://puzzicle.org/math/geometry/v/determing-a-translation-between-pointshttps://puzzicle.org/thumbs/x4435dc5a.jpgTranslation challenge problemA translation acting on the coordinate plane takes the point (-169,434) to point (-203, -68). What are the coordinates of the image of point (31, -529) under this translation?https://cdn.kastatic.org/ka-youtube-converted/nV8taxEW8-0.mp4/nV8taxEW8-0.mp4194yes https://puzzicle.org/math/geometry/v/dilating-shapes-shrinkinghttps://puzzicle.org/thumbs/x7315a4e59ce93066.jpgDilating shapes: shrinkingDraw the image of the triangle under a dilation with scale factor 1/4 about the center of dilation on the coordinate plane.https://cdn.kastatic.org/ka-youtube-converted/xAxgxUHzgaY.mp4/xAxgxUHzgaY.mp4146yes https://puzzicle.org/math/geometry/v/dilating-triangles-find-the-errorhttps://puzzicle.org/thumbs/xd4418350026eb760.jpgDilating triangles: find the errorDilating doesn't just shrink or grow a figure. The scale factor also changes the distance from each point to the center of dilation.https://cdn.kastatic.org/ka-youtube-converted/Kry9Bhp4PQg.mp4/Kry9Bhp4PQg.mp4161yes https://puzzicle.org/math/geometry/v/sequences-of-transformationshttps://puzzicle.org/thumbs/x682a7badb97ae737.jpgSequences of transformationsDetermining whether segment lengths and angle measures are preserved under a given sequence of transformations.https://cdn.kastatic.org/ka-youtube-converted/KFSBop9TZyI.mp4/KFSBop9TZyI.mp4279yes https://puzzicle.org/math/geometry/v/defining-transformationshttps://puzzicle.org/thumbs/x47fa972dba5c8314.jpgDefining transformationsGiven a description of the effect of a certain transformation, we determine whether that transformation is a translation, a rotation, or a reflection.https://cdn.kastatic.org/ka-youtube-converted/4zvwNsou9vU.mp4/4zvwNsou9vU.mp4385yes https://puzzicle.org/math/geometry/v/possible-transformations-examplehttps://puzzicle.org/thumbs/x70ee9a30.jpgIdentifying type of transformationSal is given information about a transformation in terms of a few pairs of points and their corresponding images, and he determines what kind of transformation it can be.https://cdn.kastatic.org/ka-youtube-converted/_eAWDuLYVfg.mp4/_eAWDuLYVfg.mp4281yes https://puzzicle.org/math/geometry/v/segment-congruence-equivalent-to-having-same-lengthhttps://puzzicle.org/thumbs/xb9c821ecc3070866.jpgSegment congruence equivalent to having same lengthTwo line segments are congruent if and only if they have equal lengthshttps://cdn.kastatic.org/ka-youtube-converted/WDHW-VZK3YI.mp4/WDHW-VZK3YI.mp4405yes https://puzzicle.org/math/geometry/v/angle-congruence-equivalent-to-having-same-measurehttps://puzzicle.org/thumbs/x74b58778e74808bf.jpgAngle congruence equivalent to having same measureTwo angles are congruent if and only if they have the same measure.https://cdn.kastatic.org/ka-youtube-converted/t41aV0XkYPQ.mp4/t41aV0XkYPQ.mp4323yes https://puzzicle.org/math/geometry/v/proving-the-sss-triangle-congruence-criterion-using-transformationshttps://puzzicle.org/thumbs/xa38f9384e43310a2.jpgProving the SSS triangle congruence criterion using transformationsWe can prove the side-side-side (SSS) triangle congruence criterion using the rigid transformation definition of congruence.https://cdn.kastatic.org/ka-youtube-converted/OIRZd5mDu_o.mp4/OIRZd5mDu_o.mp4347yes https://puzzicle.org/math/geometry/v/proving-the-sas-triangle-congruence-criterion-using-transformationshttps://puzzicle.org/thumbs/x0c3f4d86589e9208.jpgProving the SAS triangle congruence criterion using transformationsWe can prove the side-angle-side (SAS) triangle congruence criterion using the rigid transformation definition of congruence.https://cdn.kastatic.org/ka-youtube-converted/CMTeCNILzwU.mp4/CMTeCNILzwU.mp4271yes https://puzzicle.org/math/geometry/v/proving-asa-and-aas-triangle-congruencehttps://puzzicle.org/thumbs/xd4ce5ee228f7781f.jpgProving the ASA and AAS triangle congruence criteria using transformationsWe can prove the angle-side-angle (ASA) and angle-angle-side (AAS) triangle congruence criteria using the rigid transformation definition of congruence.https://cdn.kastatic.org/ka-youtube-converted/5EX7tYQEB_M.mp4/5EX7tYQEB_M.mp4444yes https://puzzicle.org/math/geometry/v/calculating-angle-measures-to-verify-congruencehttps://puzzicle.org/thumbs/xab0f842981801d8c.jpgCalculating angle measures to verify congruenceCalculate missing angle measures to determine which triangles must be congruent.https://cdn.kastatic.org/ka-youtube-converted/i63Ruo2jj1E.mp4/i63Ruo2jj1E.mp4269yes https://puzzicle.org/math/geometry/v/proving-triangle-congruencehttps://puzzicle.org/thumbs/xecf8b47700b9c9a3.jpgProving triangle congruenceGiven a figure composed of 2 triangles, prove that the triangles are congruent or determine that there's not enough information to tell.https://cdn.kastatic.org/ka-youtube-converted/Q8EtrFIHJuc.mp4/Q8EtrFIHJuc.mp4258yes https://puzzicle.org/math/geometry/v/another-isosceles-example-problemhttps://puzzicle.org/thumbs/281570420.jpgFinding angles in isosceles trianglesThe measures of two angles of an isosceles triangle are 3x+5 and x+16. Find all possible values of x.https://cdn.kastatic.org/ka-youtube-converted/ceDV0QBpcMA.mp4/ceDV0QBpcMA.mp4334yes https://puzzicle.org/math/geometry/v/example-involving-an-isosceles-triangle-and-parallel-lineshttps://puzzicle.org/thumbs/281570419.jpgFinding angles in isosceles triangles (example 2)Sal combines what we know about isosceles triangles and parallel lines with the power of algebra to solve the angles of an isosceles triangle.https://cdn.kastatic.org/ka-youtube-converted/CVKAro3HUxQ.mp4/CVKAro3HUxQ.mp4195yes https://puzzicle.org/math/geometry/v/geometric-constructions-congruent-angleshttps://puzzicle.org/thumbs/xdb60efded6941322.jpgGeometric constructions: congruent anglesWe can construct congruent angles with a compass and straight edge. They're congruent corresponding angles of congruent triangles.https://cdn.kastatic.org/ka-youtube-converted/D9UXfNc1ePs.mp4/D9UXfNc1ePs.mp4249yes https://puzzicle.org/math/geometry/v/constructing-parallel-lineshttps://puzzicle.org/thumbs/xca8f02960808bf73.jpgGeometric constructions: parallel lineWe can parallel lines with compass and straightedge by creating a pair of congruent corresponding angles on a transversal.https://cdn.kastatic.org/ka-youtube-converted/BHLre3UiIho.mp4/BHLre3UiIho.mp4123yes https://puzzicle.org/math/geometry/v/geometric-constructions-perpendicular-line-through-a-point-not-on-the-linehttps://puzzicle.org/thumbs/xd06437d8694742d8.jpgGeometric constructions: perpendicular line through a point not on the lineWe can construct a perpendicular line through a point off the line. The lines must be perpendicular because a rhombus's diagonals are perpendicular.https://cdn.kastatic.org/ka-youtube-converted/-ltjjKgGY7o.mp4/-ltjjKgGY7o.mp4162yes https://puzzicle.org/math/geometry/v/proof-parallel-lines-divide-triangle-sides-proportionallyhttps://puzzicle.org/thumbs/x7d6e8a3bde72266c.jpgProof: Parallel lines divide triangle sides proportionallyProve that a line parallel to one triangle side divides other sides proportionately.https://cdn.kastatic.org/ka-youtube-converted/L1c0SP4W4ro.mp4/L1c0SP4W4ro.mp4392yes https://puzzicle.org/math/geometry/v/multi-step-word-problem-with-pythagorean-theoremhttps://puzzicle.org/thumbs/x6a2de3960f599c04.jpgMulti-step word problem with Pythagorean theoremUse the Pythagorean theorem and proportional reasoning in context.https://cdn.kastatic.org/ka-youtube-converted/jVvpQAIrJ-o.mp4/jVvpQAIrJ-o.mp4275yes https://puzzicle.org/math/geometry/v/area-of-a-regular-hexagonhttps://puzzicle.org/thumbs/353965911.jpgArea of a regular hexagonUsing what we know about triangles to find the area of a regular hexagonhttps://cdn.kastatic.org/ka-youtube-converted/LrxZMdQ6tiw.mp4/LrxZMdQ6tiw.mp4471yes https://puzzicle.org/math/geometry/v/using-similarity-to-estimate-ratio-between-side-lengthshttps://puzzicle.org/thumbs/x5dc8769cc2300a48.jpgUsing similarity to estimate ratio between side lengthsWhen two right triangles share an acute angle measure, the ratios of the corresponding side lengths within the triangles are equal.https://cdn.kastatic.org/ka-youtube-converted/M-jQMTWBuzQ.mp4/M-jQMTWBuzQ.mp4275yes https://puzzicle.org/math/geometry/v/using-right-triangle-ratios-to-approximate-angle-measurehttps://puzzicle.org/thumbs/xe4df20f25da310a0.jpgUsing right triangle ratios to approximate angle measureBecause of similarity, all right triangles with a given acute angle measure have equal ratios between their side lengths. So if we know two of the side lengths of a right triangle, we can figure out the angle measures, too!https://cdn.kastatic.org/ka-youtube-converted/fPXQTEYjF-o.mp4/fPXQTEYjF-o.mp4186yes https://puzzicle.org/math/geometry/v/example-relating-trig-function-to-side-ratioshttps://puzzicle.org/thumbs/x98eb8563.jpgTrig challenge problem: trig values & side ratiosSal is given a diagram with multiple right triangles and is asked to match different expressions with different trig values.https://cdn.kastatic.org/ka-youtube-converted/EBKNtjZAjXg.mp4/EBKNtjZAjXg.mp4564yes https://puzzicle.org/math/geometry/v/point-relative-to-circlehttps://puzzicle.org/thumbs/x4dbb5644.jpgPoints inside/outside/on a circleSal uses the distance formula to determine whether the point (-6,-6) is inside, outside, or on the circle centered at (-1,-3) with radius 6.https://cdn.kastatic.org/ka-youtube-converted/S6BHQMk8C_A.mp4/S6BHQMk8C_A.mp4233yes https://puzzicle.org/math/geometry/v/recognizing-points-on-a-circlehttps://puzzicle.org/thumbs/x93881235.jpgChallenge problem: Points on two circlesWatch Sal solve a challenging problem where he has to determine if points are on both, one, or neither of two circles.https://cdn.kastatic.org/ka-youtube-converted/9ASWQi14FlE.mp4/9ASWQi14FlE.mp4621yes https://puzzicle.org/math/geometry/v/classfying-a-quadrilateral-on-the-coordinate-planehttps://puzzicle.org/thumbs/x4f7e87ad.jpgClassifying quadrilaterals on the coordinate planeWatch Sal classify a quadrilateral as a square, rhombus, rectangle, parallelogram, or trapezoid given its four vertices.https://cdn.kastatic.org/ka-youtube-converted/vsgrWDLEzcQ.mp4/vsgrWDLEzcQ.mp4308yes https://puzzicle.org/math/geometry/v/classifying-figures-with-coordinateshttps://puzzicle.org/thumbs/x5bf0f7821a0a4ca2.jpgClassifying figures with coordinatesUse coordinates to determine the slopes of sides of a figure in order to classify it.https://cdn.kastatic.org/ka-youtube-converted/2B5SQba9xSo.mp4/2B5SQba9xSo.mp4417yes https://puzzicle.org/math/geometry/v/language-and-notation-of-the-circlehttps://puzzicle.org/thumbs/261106564.jpgCircles glossaryFormal definition of a circle. Tangent and secant lines. Diameters and radii. major and minor arcshttps://cdn.kastatic.org/ka-youtube-converted/U2W7HPyC0cM.mp4/U2W7HPyC0cM.mp4672yes https://puzzicle.org/math/geometry/v/radians-as-ratio-of-arc-length-to-radiushttps://puzzicle.org/thumbs/x7740664e9e4a71b9.jpgRadians as ratio of arc length to radiusDerive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality.https://cdn.kastatic.org/ka-youtube-converted/9qrx0aqyhS8.mp4/9qrx0aqyhS8.mp4400yes https://puzzicle.org/math/geometry/v/dilating-in-3dhttps://puzzicle.org/thumbs/xd997f0a2af72c819.jpgDilating in 3DThe cross sections of 3D shapes are dilations of the original shape, centered at a specific point. The scale factor of the dilation depends on the height of the cross-section or the distance from the point on the base.https://cdn.kastatic.org/ka-youtube-converted/3wTy5aI9AF4.mp4/3wTy5aI9AF4.mp4273yes https://puzzicle.org/math/geometry/v/rotating-2d-shapes-in-3dhttps://puzzicle.org/thumbs/x0c885529.jpgRotating 2D shapes in 3DIf you rotate a 2D shape about an axis, the shape will define a 3D object. Watch Sal rotating various 2D shapes and see what 3D objects he gets!https://cdn.kastatic.org/ka-youtube-converted/vdpyWeiHXmU.mp4/vdpyWeiHXmU.mp4252yes https://puzzicle.org/math/geometry/v/area-densityhttps://puzzicle.org/thumbs/x878b740b72678000.jpgArea densityArea or surface density is the amount of a quantity (often mass) per unit of area. Density=Quantity/Areahttps://cdn.kastatic.org/ka-youtube-converted/lqXzyybVh6o.mp4/lqXzyybVh6o.mp4248yes https://puzzicle.org/math/geometry/v/volume-densityhttps://puzzicle.org/thumbs/x2830fa495b885384.jpgVolume densityVolume density is the amount of a quantity (often mass) per unit of volume. Density=Quantity/Volumehttps://cdn.kastatic.org/ka-youtube-converted/QoKiCWCsSCo.mp4/QoKiCWCsSCo.mp4243yes https://puzzicle.org/math/geometry-fl-best/v/conditional-statements-and-logical-equivalencehttps://puzzicle.org/thumbs/x9279e643d902029b.jpgConditional statements and logical equivalenceWe assert that a conditional statement is true. Let's identify the converse, inverse, and contrapositive and determine whether each of them must be true based on the original statement.https://cdn.kastatic.org/ka-youtube-converted/5kojy1CHuyI.mp4/5kojy1CHuyI.mp4344yes https://puzzicle.org/math/geometry-fl-best/v/biconditional-statementshttps://puzzicle.org/thumbs/xb3e5f22c8d89d1a3.jpgBiconditional statementsFor most conditional statements, swapping the "if" and the "then" parts creates a new statement, and the new statement might not have the same truth value as the original. Biconditional statements are conditional statements that are true regardless of which clause is the "if" and which is the "then" part.https://cdn.kastatic.org/ka-youtube-converted/bs5YLyGHa60.mp4/bs5YLyGHa60.mp4221yes https://puzzicle.org/math/geometry-fl-best/v/weighted-average-two-pointshttps://puzzicle.org/thumbs/x0ff0352e.jpgWeighted average of two pointsHow can we calculate a weighted average between two points? Instead of treating both points equally, we give more weight to one point than the other. For example, if we have two points A and B, and we want to find a point M that is closer to B, we might give B twice as much weight as A. In computer graphics, weighted averages help us find points along a line or curve exactly. This helps animators draw shapes smoothly, rather than guessing points, which might make the shape look jagged.https://cdn.kastatic.org/ka-youtube-converted/SkdYqyCIh5g.mp4/SkdYqyCIh5g.mp4237yes https://puzzicle.org/math/geometry-fl-best/v/charactermodeling5https://puzzicle.org/thumbs/x9f3605b8.jpgWeighted average of three pointsFirst, we'll review weighted averages of two points and extend the idea to three points. Computer graphics use weighted averages to help determine the shape and smoothness of curves. Each point's weight can influence the curve's direction and form. It's like each point is pulling the curve towards itself with a certain strength!https://cdn.kastatic.org/ka-youtube-converted/LzkL0Iik2B8.mp4/LzkL0Iik2B8.mp4222yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/geometric-constructions-congruent-segmentshttps://puzzicle.org/thumbs/x884f0022ddc02a30.jpgGeometric constructions congruent segmentsLet's construct a congruent line segment using a compass and a straightedge. All radii of a given circle are congruent, so we can use the compass to mark possible endpoints for a congruent line segment.https://cdn.kastatic.org/ka-youtube-converted/DyzeQO8M9z4.mp4/DyzeQO8M9z4.mp490yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/translations-graph-to-algebraic-rulehttps://puzzicle.org/thumbs/xc05f117decdb7326.jpgTranslations: graph to algebraic ruleA worked example of going from a graph of a translation to an algebraic rule.https://cdn.kastatic.org/ka-youtube-converted/se4o4QzKkTk.mp4/se4o4QzKkTk.mp4141yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/translations-verbal-to-algebraic-rulehttps://puzzicle.org/thumbs/xd1beace66e3332dd.jpgTranslations: description to algebraic ruleA worked example of going from a verbal description of a translation to an algebraic rule.https://cdn.kastatic.org/ka-youtube-converted/vO5d9iXcYtU.mp4/vO5d9iXcYtU.mp489yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/rotations-graph-to-algebraic-rulehttps://puzzicle.org/thumbs/x3576e424277c48f8.jpgRotations: graph to algebraic ruleA worked example of going from a graph of a rotation to an algebraic rule.https://cdn.kastatic.org/ka-youtube-converted/aZnEcR6ZRSw.mp4/aZnEcR6ZRSw.mp4247yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/rotations-description-to-algebraic-rulehttps://puzzicle.org/thumbs/x127a8adb0622871e.jpgRotations: description to algebraic ruleA worked example of going from a verbal description of a rotation to an algebraic rule.https://cdn.kastatic.org/ka-youtube-converted/JIFGwyphKXo.mp4/JIFGwyphKXo.mp4255yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/reflections-graph-to-algebraic-rulehttps://puzzicle.org/thumbs/x886cc4ee1c9af919.jpgReflections: graph to algebraic ruleA worked example of going from a graph of a reflection to an algebraic rule.https://cdn.kastatic.org/ka-youtube-converted/dXxfHjzql2M.mp4/dXxfHjzql2M.mp4148yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/reflections-description-to-algebraic-rulehttps://puzzicle.org/thumbs/x9b532ca8cda073a2.jpgReflections: description to algebraic ruleA worked example of going from a verbal description of a reflection to an algebraic rule.https://cdn.kastatic.org/ka-youtube-converted/sJ8XrA4Vb_k.mp4/sJ8XrA4Vb_k.mp4159yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/representing-dilations-algebraically-k-greater-than-1https://puzzicle.org/thumbs/xff9c6418c253afd7.jpgRepresenting dilations algebraically, k greater than 1Use an algebraic rule to represent a dilation with k > 1.https://cdn.kastatic.org/ka-youtube-converted/OuBo1EomJJ8.mp4/OuBo1EomJJ8.mp4236yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/representing-dilations-algebraically-k-less-than-1https://puzzicle.org/thumbs/xa20b4a4ff0df7bfa.jpgRepresenting dilations algebraically, k less than 1Use an algebraic rule to represent a dilation with k less than 1.https://cdn.kastatic.org/ka-youtube-converted/E5ScRFT4zl8.mp4/E5ScRFT4zl8.mp4208yes https://puzzicle.org/math/geometry-tx/v/rotating-algebraically-about-a-pointhttps://puzzicle.org/thumbs/x972c01e6d63db705.jpgRotating algebraically about a pointRotate a figure about a point other than the origin on the plane and with algebraic coordinate notation. We can do this by translating the plane so the center of rotation is at the origin, rotating about the origin, and then translating the plane back.https://cdn.kastatic.org/ka-youtube-converted/c1qJuIxXOMY.mp4/c1qJuIxXOMY.mp4455yes https://puzzicle.org/math/geometry-tx/v/dilating-algebraically-about-a-pointhttps://puzzicle.org/thumbs/x0ced15701e1c51cc.jpgDilating algebraically about a pointDilate a figure about a point other than the origin on the plane and in algebraic coordinate notation. We can do this by translating the plane so the center of dilation is at the origin, dilating about the origin, and then translating the plane back.https://cdn.kastatic.org/ka-youtube-converted/CySwZBgM_j4.mp4/CySwZBgM_j4.mp4315yes https://puzzicle.org/math/geometry-tx/v/reflecting-over-yxhttps://puzzicle.org/thumbs/xde030f04555404de.jpgReflecting over y=xReflect a figure over the line y = x in the plane and with algebraic coordinate notation. This common transformation swaps the x- and y-coordinates of every point in the figure.https://cdn.kastatic.org/ka-youtube-converted/-47SHD2cPAY.mp4/-47SHD2cPAY.mp4179yes https://puzzicle.org/math/geometry-tx/v/reflecting-over-y-negative-xhttps://puzzicle.org/thumbs/x2e314c47dd967bdc.jpgReflecting over y=-xReflect a figure over the line y = -x in the plane and in algebraic coordinate notation.https://cdn.kastatic.org/ka-youtube-converted/EydeoSJ3248.mp4/EydeoSJ3248.mp4244yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/composing-transformationshttps://puzzicle.org/thumbs/x2c374acbd5a18fc0.jpgComposing transformationsGraph the image of a ray after the composition of a dilation and a rotation.https://cdn.kastatic.org/ka-youtube-converted/gYIBbDLYFY0.mp4/gYIBbDLYFY0.mp4317yes https://puzzicle.org/math/geometry-fl-best/v/trapezoid-midsegment-lengthhttps://puzzicle.org/thumbs/x22628c7130e06c36.jpgTrapezoid midsegment lengthGiven the lengths of the 2 bases of a trapezoid (a quadrilateral with a pair of parallel sides), let's find the length of its midsegment.https://cdn.kastatic.org/ka-youtube-converted/Vc7s7Q7yeMg.mp4/Vc7s7Q7yeMg.mp493yes https://puzzicle.org/math/geometry-fl-best/v/proof-trapezoid-midsegment-lengthhttps://puzzicle.org/thumbs/x44b15056e7527831.jpgProof: Trapezoid midsegment lengthLet's prove that the midsegment of a trapezoid (the segment that joins the midpoints of the 2 legs) has a length equal to the average of the lengths of the two bases. We can use the properties of congruent and similar triangles to support our reasoning. This property holds true whether we define trapezoids has having at least one or exactly one pair of parallel sides.https://cdn.kastatic.org/ka-youtube-converted/_HEF49sraFg.mp4/_HEF49sraFg.mp4419yes https://puzzicle.org/math/geometry-fl-best/v/proof-base-angles-of-isosceles-trapezoidshttps://puzzicle.org/thumbs/x14a9e4aee2dc304d.jpgProof: Base angles of isosceles trapezoidsLet's prove that for isosceles trapezoids (quadrilaterals with one pair of parallel sides and one pair of congruent, non-parallel sides), the base angles are congruent. We can use triangle similarity and the properties of isosceles triangles to help us out.https://cdn.kastatic.org/ka-youtube-converted/xmdgp7LskQI.mp4/xmdgp7LskQI.mp4399yes https://puzzicle.org/math/geometry-fl-best/v/opposite-angles-of-isosceles-trapezoidhttps://puzzicle.org/thumbs/x34cbb4f8c50283b7.jpgOpposite angles of isosceles trapezoidGiven one angle measure in an isosceles trapezoid (a quadrilateral with one pair of parallel sides and one pair of congruent, non-parallel sides), find the angle measure of the opposite angle.https://cdn.kastatic.org/ka-youtube-converted/63kIeQgjOIM.mp4/63kIeQgjOIM.mp469yes https://puzzicle.org/math/geometry-fl-best/v/trapezoid-diagonals-conjecturehttps://puzzicle.org/thumbs/xc658430fb5ef1b4c.jpgTrapezoid diagonals conjectureWhat do you notice when you observe the diagonals in several different trapezoids (quadrilaterals with at least one pair of parallel sides)? Let's form a conjecture.https://cdn.kastatic.org/ka-youtube-converted/uKuzHyUytfg.mp4/uKuzHyUytfg.mp4261yes https://puzzicle.org/math/geometry-fl-best/v/proof-diagonals-of-isosceles-trapezoidhttps://puzzicle.org/thumbs/x68ea5409cd076556.jpgProof: Diagonals of isosceles trapezoidLet's prove that for isosceles trapezoids (quadrilaterals with exactly one pair of parallel sides and exactly one pair of congruent sides), the diagonals are congruent. We'll explore how our assumptions affect the generalizability of our proof.https://cdn.kastatic.org/ka-youtube-converted/amfaIdh_QzM.mp4/amfaIdh_QzM.mp4159yes https://puzzicle.org/math/geometry-tx/v/applying-the-triangle-proportionality-theoremhttps://puzzicle.org/thumbs/x11032f25388fe977.jpgApplying the triangle proportionality theoremApply the triangle proportionality theorem to determine distances along roads split up by parallel roads.https://cdn.kastatic.org/ka-youtube-converted/Fc8JVfHAcQU.mp4/Fc8JVfHAcQU.mp4278yes https://puzzicle.org/math/geometry-tx/v/applying-similar-triangleshttps://puzzicle.org/thumbs/x3138d200955b8fc0.jpgApplying similar trianglesRecognize similar triangles in context and apply them to find a distance. Handle units appropriately.https://cdn.kastatic.org/ka-youtube-converted/MlGOhzYn-QQ.mp4/MlGOhzYn-QQ.mp4362yes https://puzzicle.org/math/geometry-tx/v/finding-perimeter-of-shape-with-arcshttps://puzzicle.org/thumbs/x7a59ff15ddb11307.jpgFinding perimeter of shape with arcsGiven a figure composed of parts of circles, find the exact perimeter in terms of pi. We can reason about the arc lengths based on the central angles of the arcs.https://cdn.kastatic.org/ka-youtube-converted/ybqHUv8cLVk.mp4/ybqHUv8cLVk.mp4245yes https://puzzicle.org/math/geometry-tx/v/finding-perimeter-of-shape-with-semicircleshttps://puzzicle.org/thumbs/xd6940d97dd122186.jpgFinding perimeter of shape with semicirclesGiven a figure composed of semicircles, find the exact perimeter in terms of pi. The figure may look unusual, but we can break down the perimeter into very familiar shapes.https://cdn.kastatic.org/ka-youtube-converted/nh56seaS6iA.mp4/nh56seaS6iA.mp4119yes https://puzzicle.org/math/geometry-tx/v/proof-the-product-of-the-segments-of-intersecting-chordshttps://puzzicle.org/thumbs/xda57be7098fc4d4b.jpgProof The product of the segments of intersecting chordsLet's prove the general case that the product of the segments of intersecting chords are equal. We'll create similar triangles and use the properties of inscribed angles and vertical angles to convince ourselves that this is always true.https://cdn.kastatic.org/ka-youtube-converted/3hV7ZXxal7k.mp4/3hV7ZXxal7k.mp4284yes https://puzzicle.org/math/geometry-tx/v/proof-angle-formed-by-intersecting-chordshttps://puzzicle.org/thumbs/x07af2979a99b576d.jpgProof: Angle formed by intersecting chordsLet's prove that the angle formed by intersecting chords is half the sum of the measures of the intercepted arcs. We'll construct a triangle, then use the properties of inscribed angles to convince ourselves that this is always true.https://cdn.kastatic.org/ka-youtube-converted/nxHUx1o8-sA.mp4/nxHUx1o8-sA.mp4270yes https://puzzicle.org/math/geometry-fl-best/v/proof-angle-between-a-tangent-and-a-chordhttps://puzzicle.org/thumbs/xc7c4467ee35f0c1d.jpgProof: Angle between a tangent and a chordLet's prove that the measure of the angle formed by the intersection of a tangent and a chord is half the measure of the intercepted arc. We'll construct an isosceles triangle and use the relationship between radii and tangents of a circle to convince ourselves that this is true for both the major and minor arcs formed.https://cdn.kastatic.org/ka-youtube-converted/hbdvuygTlMA.mp4/hbdvuygTlMA.mp4328yes https://puzzicle.org/math/geometry-tx/v/proof-diameter-that-bisects-a-chordhttps://puzzicle.org/thumbs/x7949a19d6e45cb88.jpgProof: Diameter that bisects a chordLet's prove that if a diameter of a circle bisects a chord, it is perpendicular to the chord. We'll construct congruent triangles to convince ourselves that this is always true.https://cdn.kastatic.org/ka-youtube-converted/diQqp7pWxZY.mp4/diQqp7pWxZY.mp4124yes https://puzzicle.org/math/geometry-tx/v/proof-the-product-of-segments-of-intersecting-secantshttps://puzzicle.org/thumbs/x4bb776e0c2237cb0.jpgProof: The product of segments of intersecting secantsLet's prove that the product of the length of a secant segment and its external segment is equal to the product of the other secant segment and its external segment from the same point outside the circle. We'll construct similar triangles and use the properties of inscribed triangles to convince ourselves that this is always true.https://cdn.kastatic.org/ka-youtube-converted/J3FpOlIkH3Q.mp4/J3FpOlIkH3Q.mp4190yes https://puzzicle.org/math/geometry-tx/v/proof-angle-formed-by-intersecting-secantshttps://puzzicle.org/thumbs/x67af77fb0561fcdf.jpgProof: Angle formed by intersecting secantsLet's prove that the measure of the angle formed by 2 secants intersecting at a point outside the circle is half the difference of the measures of the intercepted arcs. We'll construct a triangle and use the properties of inscribed angles to convince ourselves that this is always true.https://cdn.kastatic.org/ka-youtube-converted/dHP6PxXlrV0.mp4/dHP6PxXlrV0.mp4272yes https://puzzicle.org/math/geometry-tx/v/proof-the-product-of-segments-of-intersecting-secant-and-tangenthttps://puzzicle.org/thumbs/x75ffbc79aaf64530.jpgProof: The product of segments of intersecting secant and tangentLet's prove that the product of the length of a secant segment and its external segment is equal to the square of the length of the tangent segment intersecting the secant segment at a point outside the circle. We'll construct a pair of similar triangles and use the properties of inscribed angles, tangents, and chords to convince ourselves that this is always true.https://cdn.kastatic.org/ka-youtube-converted/BnpRSGP4TTE.mp4/BnpRSGP4TTE.mp4246yes https://puzzicle.org/math/geometry-tx/v/proof-angle-between-a-tangent-and-a-secanthttps://puzzicle.org/thumbs/x2f6a05c9eb504b22.jpgProof: Angle between a tangent and a secantProve that the measure of the angle formed by a secant and a tangent intersecting at a point outside the circle is half the difference of the measures of the intercepted arcs. We'll construct a triangle, then use the properties of inscribed angles, tangents, and chords to convince ourselves that this is always true.https://cdn.kastatic.org/ka-youtube-converted/OzeHAwjaG60.mp4/OzeHAwjaG60.mp4208yes https://puzzicle.org/math/geometry-tx/v/area-between-sector-and-trianglehttps://puzzicle.org/thumbs/x93ae9db2045a25b5.jpgArea between sector and triangleFind the area of a figure composed subtractively of a sector and special right triangles.https://cdn.kastatic.org/ka-youtube-converted/RlOXnhTujDQ.mp4/RlOXnhTujDQ.mp4235yes https://puzzicle.org/math/geometry-tx/v/composite-area-related-to-circleshttps://puzzicle.org/thumbs/x98800953c7f04271.jpgComposite area related to circlesLet's decompose a figure into a square and quarter circles to find its total area.https://cdn.kastatic.org/ka-youtube-converted/_cF1cwv2EQU.mp4/_cF1cwv2EQU.mp4150yes https://puzzicle.org/math/geometry-home/v/area-of-an-equilateral-trianglehttps://puzzicle.org/thumbs/294658865.jpgArea of equilateral triangleFinding the formula for the area of an equilateral triangle with side shttps://cdn.kastatic.org/ka-youtube-converted/UKQ65tiIQ6o.mp4/UKQ65tiIQ6o.mp4222yes https://puzzicle.org/math/geometry-tx/v/area-of-regular-polygon-given-side-lengthhttps://puzzicle.org/thumbs/xb114845fdb9afb49.jpgArea of regular polygon given side lengthGiven the side length of a regular polygon, find its area. The area of a regular polygon is 1/2 * (apothem length) * perimeter. We can split the polygon into congruent right triangles. Then trigonometric ratios help us find the apothem length based on the side length and angle measures.https://cdn.kastatic.org/ka-youtube-converted/a6dbxqVWOf4.mp4/a6dbxqVWOf4.mp4345yes https://puzzicle.org/math/geometry-tx/v/area-of-regular-polygon-given-radiushttps://puzzicle.org/thumbs/x7e3d93cf201c061d.jpgArea of regular polygon given radiusGiven the radius of a regular polygon, find its area. The area of a regular polygon is 1/2 * (apothem length) * perimeter. We can split the polygon into congruent right triangles. Then trigonometric ratios help us find the apothem length and side length based on the radius length and angle measures.https://cdn.kastatic.org/ka-youtube-converted/9LlgG6l5eXU.mp4/9LlgG6l5eXU.mp4333yes https://puzzicle.org/math/geometry-home/v/area-of-shaded-region-made-from-equilateral-triangleshttps://puzzicle.org/thumbs/xd65354b0.jpgArea of equilateral triangle (advanced)Sal finds a shaded area defined by two equilateral triangles whose side lengths are given.https://cdn.kastatic.org/ka-youtube-converted/QVxqgxVtKbs.mp4/QVxqgxVtKbs.mp4337yes https://puzzicle.org/math/geometry-fl-best/v/surface-area-of-a-rectangular-pyramidhttps://puzzicle.org/thumbs/xce7d93d890d7a558.jpgSurface area of a rectangular pyramidGiven a right rectangular (non-square) pyramid with the dimensions of the base and the height of the pyramid, let's find the lateral and total surface area. We'll use the Pythagorean theorem to find the slant heights of the lateral faces.https://cdn.kastatic.org/ka-youtube-converted/zNL3W-rXbY0.mp4/zNL3W-rXbY0.mp4338yes https://puzzicle.org/math/geometry-fl-best/v/surface-area-of-a-conehttps://puzzicle.org/thumbs/x516666bbb90f1329.jpgSurface area of a coneTo find the surface area of a right circular cone, we can draw its net, which looks like a circle and a sector. Let's build intuition for where the general formulas for lateral surface area and total surface area of a cone come from, then apply them to a specific cone.https://cdn.kastatic.org/ka-youtube-converted/9w6qYXatwkM.mp4/9w6qYXatwkM.mp4376yes https://puzzicle.org/math/geometry-fl-best/v/surface-area-of-an-octagonal-pyramidhttps://puzzicle.org/thumbs/xb145ced3d8e9fb95.jpgSurface area of an octagonal pyramidGiven a right pyramid with a regular octagon as the base, let's find the lateral and total surface area. Since the base is a regular octagon, we can use trigonometric ratios to help find the areas of the base and the lateral faces.https://cdn.kastatic.org/ka-youtube-converted/Pfoy2IW7oHw.mp4/Pfoy2IW7oHw.mp4442yes https://puzzicle.org/math/geometry-fl-best/v/surface-area-of-sphere-computation-and-sense-of-formulahttps://puzzicle.org/thumbs/xdb17394461c959ae.jpgSurface area of sphere: computation and sense of formulaWhy Is the formula for surface area of sphere equal to 4pir^2? And applying its formula to find the surface area.https://cdn.kastatic.org/ka-youtube-converted/IZ4DpWYO3kY.mp4/IZ4DpWYO3kY.mp4270yes https://puzzicle.org/math/grade-8-math-tx/v/scaling-perimeter-and-area-example-1https://puzzicle.org/thumbs/xaa96b0e015db384b.jpgScaling perimeter and area example 1Given a dilation and its scale factor, predict the effect of that dilation on the perimeter and area of a figure.https://cdn.kastatic.org/ka-youtube-converted/DAt886IH4Cw.mp4/DAt886IH4Cw.mp4204yes https://puzzicle.org/math/grade-8-math-tx/v/scaling-perimeter-and-area-example-2https://puzzicle.org/thumbs/x069e25bd8241d871.jpgScaling perimeter and area example 2Given a dilation and its scale factor, predict the effect of that dilation on the perimeter and area of a figure.https://cdn.kastatic.org/ka-youtube-converted/bp8DSaYrvF0.mp4/bp8DSaYrvF0.mp4163yes https://puzzicle.org/math/geometry-home/v/identifying-rayshttps://puzzicle.org/thumbs/372432665.jpgIdentifying raysA ray is a shape that starts at one point and extends forever in one direction. To identify a ray in a picture, look for a line that has one endpoint (the point where the ray starts) and an arrow on the other end (to show that it keeps going).https://cdn.kastatic.org/ka-youtube-converted/w9jEq6dmqPg.mp4/w9jEq6dmqPg.mp4340yes https://puzzicle.org/math/geometry-home/v/measuring-segmentshttps://puzzicle.org/thumbs/xc6f1d2f0.jpgMeasuring Line SegmentsIn this example of measuring a line segment, the numbers span across the positive and negative. Remember, line segments and points are the foundations of geometry, so this is an important concept.https://cdn.kastatic.org/ka-youtube-converted/y-O_B-wWivQ.mp4/y-O_B-wWivQ.mp489yes https://puzzicle.org/math/geometry-home/v/congruent-segmentshttps://puzzicle.org/thumbs/xa1695c23.jpgCongruent line segmentsTwo line segments on a number line are congruent if they have the same length. To check, you can count the number of units between the two endpoints of each line segment. If the counts are the same, the line segments are congruent.https://cdn.kastatic.org/ka-youtube-converted/bJF9R8_-0O0.mp4/bJF9R8_-0O0.mp461yes https://puzzicle.org/math/geometry-home/v/the-golden-ratiohttps://puzzicle.org/thumbs/294676192.jpgThe golden ratioAn introduction to one of the most amazing ideas/numbers in mathematicshttps://cdn.kastatic.org/ka-youtube-converted/5zosU6XTgSY.mp4/5zosU6XTgSY.mp4882yes https://puzzicle.org/math/geometry-home/v/vertical-adjacent-and-linearly-paired-angleshttps://puzzicle.org/thumbs/x40ad80ea.jpgAngle relationships exampleSal finds a linear pair, vertical angles, and adjacent angles from a diagram.https://cdn.kastatic.org/ka-youtube-converted/vAlazPPFlyY.mp4/vAlazPPFlyY.mp4283yes https://puzzicle.org/math/geometry-home/v/proof-corresponding-angle-equivalence-implies-parallel-lineshttps://puzzicle.org/thumbs/265540641.jpgParallel lines & corresponding angles proofProof by contradiction that corresponding angle equivalence implies parallel lineshttps://cdn.kastatic.org/ka-youtube-converted/9_3OxtdqmqE.mp4/9_3OxtdqmqE.mp4318yes https://puzzicle.org/math/geometry-home/v/ca-geometry-deducing-angle-measureshttps://puzzicle.org/thumbs/24456.jpgMissing angles (CA geometry)46-50, deducing the measure of angles.https://cdn.kastatic.org/ka-youtube-converted/_HJljJuVHLw.mp4/_HJljJuVHLw.mp4658yes https://puzzicle.org/math/geometry-home/v/introduction-to-angles-oldhttps://puzzicle.org/thumbs/24939.jpgIntro to angles (old)What an angle is. Angles in a circle. Complementary and supplementary angles.https://cdn.kastatic.org/ka-youtube-converted/2439OIVBgPg.mp4/2439OIVBgPg.mp4595yes https://puzzicle.org/math/geometry-home/v/angles-part-2https://puzzicle.org/thumbs/24941.jpgAngles (part 2)More on complementary and supplementary angles. Introduction to opposite angles.https://cdn.kastatic.org/ka-youtube-converted/zrqzG6xKa1A.mp4/zrqzG6xKa1A.mp4544yes https://puzzicle.org/math/geometry-home/v/angles-part-3https://puzzicle.org/thumbs/24943.jpgAngles (part 3)Angles formed when a transversal intersects parallel lines.https://cdn.kastatic.org/ka-youtube-converted/NLg6hfoKKlE.mp4/NLg6hfoKKlE.mp4587yes https://puzzicle.org/math/geometry-home/v/angles-formed-between-transversals-and-parallel-lineshttps://puzzicle.org/thumbs/2600797.jpgAngles formed between transversals and parallel linesAngles of parallel lineshttps://cdn.kastatic.org/ka-youtube-converted/gRKZaojKeP0.mp4/gRKZaojKeP0.mp4473yes https://puzzicle.org/math/geometry-home/v/angles-of-parallel-lines-2https://puzzicle.org/thumbs/2600798.jpgAngles of parallel lines 2Angles of parallel lines exampleshttps://cdn.kastatic.org/ka-youtube-converted/0eDwckZOffc.mp4/0eDwckZOffc.mp4315yes https://puzzicle.org/math/geometry-home/v/the-angle-gamehttps://puzzicle.org/thumbs/24947.jpgThe angle gameUsing what we know to solve for angles in the Angle Game.https://cdn.kastatic.org/ka-youtube-converted/kqU_ymV581c.mp4/kqU_ymV581c.mp4541yes https://puzzicle.org/math/geometry-home/v/angle-game-part-2https://puzzicle.org/thumbs/24953.jpgThe angle game (part 2)More examples of the Angle Game.https://cdn.kastatic.org/ka-youtube-converted/ko-cYG3d6ec.mp4/ko-cYG3d6ec.mp4589yes https://puzzicle.org/math/geometry-home/v/intro-to-curves-basic-geometrical-ideas-class-6-india-khan-academyhttps://puzzicle.org/thumbs/x73b10c58108327fa.jpgIntro to curvesIn daily language, we all use the word curve to mean "not straight" but does the word curve have the same meaning in math? Can a straight line be a curve? Also, what makes a curve a 'simple curve'?https://cdn.kastatic.org/ka-youtube-converted/EhmPK6iLz0Q.mp4/EhmPK6iLz0Q.mp4412yes https://puzzicle.org/math/geometry-home/v/polygons-as-special-curves-basic-geometrical-ideas-class-6-india-khan-academyhttps://puzzicle.org/thumbs/x50a7930413204e35.jpgPolygons as special curvesYou may have learnt that triangles, rectangles etc. are all examples of polygons but that circles are not. What exactly makes a curve a polygon?https://cdn.kastatic.org/ka-youtube-converted/hGRu2Pyl92w.mp4/hGRu2Pyl92w.mp4193yes https://puzzicle.org/math/geometry-home/v/sum-of-interior-angles-of-a-polygonhttps://puzzicle.org/thumbs/287107579.jpgSum of interior angles of a polygonTo find the interior angle sum of a polygon, we can use a formula: interior angle sum = (n - 2) x 180°, where n is the number of sides. For example, a pentagon has 5 sides, so its interior angle sum is (5 - 2) x 180° = 3 x 180° = 540°.https://cdn.kastatic.org/ka-youtube-converted/qG3HnRccrQU.mp4/qG3HnRccrQU.mp4550yes https://puzzicle.org/math/geometry-home/v/sum-of-the-exterior-angles-of-convex-polygonhttps://puzzicle.org/thumbs/272207189.jpgSum of the exterior angles of a polygonLearn a simple and elegant way to find the sum of the exterior angles of any convex polygon. You will see how to redraw the angles adjacent to each other and form a circle. Then you will discover that the sum of the exterior angles is always 360 degrees. Watch this video to master this important skill in geometry.https://cdn.kastatic.org/ka-youtube-converted/W9B3VYdC5T8.mp4/W9B3VYdC5T8.mp4324yes https://puzzicle.org/math/geometry-home/v/circumcenter-of-a-trianglehttps://puzzicle.org/thumbs/298421783.jpgCircumcenter of a triangleTo prove that any point on the perpendicular bisector of a segment is equidistant from the endpoints, draw the segment and bisector, and note that the bisector creates two right triangles. By the definition of a perpendicular bisector, the hypotenuses of these triangles are congruent. Using the Pythagorean theorem, show that the two right triangles are congruent, and thus the point on the bisector is equidistant from the two endpoints. Other proofs are also included in the video.https://cdn.kastatic.org/ka-youtube-converted/KXZ6w91DioU.mp4/KXZ6w91DioU.mp4749yes https://puzzicle.org/math/geometry-home/v/circumcenter-of-a-right-trianglehttps://puzzicle.org/thumbs/298435073.jpgCircumcenter of a right triangleShowing that the midpoint of the hypotenuse is the circumcenterhttps://cdn.kastatic.org/ka-youtube-converted/VejCw2NlE60.mp4/VejCw2NlE60.mp4342yes https://puzzicle.org/math/geometry-home/v/three-points-defining-a-circlehttps://puzzicle.org/thumbs/309312179.jpgThree points defining a circleThree points uniquely define a circle. If you circumscribe a circle around a triangle, the circumcenter of that triangle will also be the center of that circle.https://cdn.kastatic.org/ka-youtube-converted/4_xhiP6g2ow.mp4/4_xhiP6g2ow.mp4608yes https://puzzicle.org/math/geometry-home/v/area-circumradius-formula-proofhttps://puzzicle.org/thumbs/13179257.jpgArea circumradius formula proofProof of the formula relating the area of a triangle to its circumradiushttps://cdn.kastatic.org/ka-youtube-converted/x_aax6LSRtE.mp4/x_aax6LSRtE.mp4546yes https://puzzicle.org/math/geometry-home/v/2003-aime-ii-problem-7https://puzzicle.org/thumbs/13179269.jpg2003 AIME II problem 7Find the area of rhombus ABCD given that the radii of the circles circumscribed about triangles ABD and ACDhttps://cdn.kastatic.org/ka-youtube-converted/jpKjXtywTlQ.mp4/jpKjXtywTlQ.mp4793yes https://puzzicle.org/math/geometry-home/v/point-line-distance-and-angle-bisectorshttps://puzzicle.org/thumbs/309312178.jpgDistance between a point & lineThinking about the distance between a point and a line. Proof that a point on an angle bisector is equidistant to the sides of the angle and a point equidistant to the sides is on an angle bisector.https://cdn.kastatic.org/ka-youtube-converted/yj4oS-27Q3k.mp4/yj4oS-27Q3k.mp4657yes https://puzzicle.org/math/geometry-home/v/inradius-perimeter-and-areahttps://puzzicle.org/thumbs/311447418.jpgInradius, perimeter, & areaShowing that area is equal to inradius times semiperimeterhttps://cdn.kastatic.org/ka-youtube-converted/5NjgcPri07E.mp4/5NjgcPri07E.mp4449yes https://puzzicle.org/math/geometry-home/v/common-orthocenter-and-centroidhttps://puzzicle.org/thumbs/337196124.jpgCommon orthocenter and centroidIf the orthocenter and centroid are the same point, then the triangle is equilateral. This is because the orthocenter is the intersection of the altitudes, which are also the medians and the angle bisectors in an equilateral triangle. The centroid is the intersection of the medians, which divide the triangle into six congruent triangles. Therefore, the orthocenter and the centroid coincide only when the triangle has three equal sides and angles.https://cdn.kastatic.org/ka-youtube-converted/pDZIvyeqX1o.mp4/pDZIvyeqX1o.mp4385yes https://puzzicle.org/math/geometry-home/v/review-of-triangle-propertieshttps://puzzicle.org/thumbs/326408404.jpgReview of triangle propertiesComparing perpendicular bisectors to angle bisectors to medians to altitudes.https://cdn.kastatic.org/ka-youtube-converted/KUhdMbx5ges.mp4/KUhdMbx5ges.mp4597yes https://puzzicle.org/math/geometry-home/v/euler-linehttps://puzzicle.org/thumbs/326395669.jpgEuler lineThe magic and mystery of the Euler Linehttps://cdn.kastatic.org/ka-youtube-converted/tUqyJgmGY7k.mp4/tUqyJgmGY7k.mp4372yes https://puzzicle.org/math/geometry-home/v/euler-s-line-proofhttps://puzzicle.org/thumbs/334522802.jpgEuler's line proofProving the somewhat mystical result that the circumcenter, centroid, and orthocenter all sit on the same line.https://cdn.kastatic.org/ka-youtube-converted/t_EgAi574sM.mp4/t_EgAi574sM.mp4590yes https://puzzicle.org/math/geometry-home/v/whether-a-special-quadrilateral-can-existhttps://puzzicle.org/thumbs/897531450.jpgWhether a special quadrilateral can existProving whether a special quadrilateral can exist or not.https://cdn.kastatic.org/ka-youtube-converted/6dIMIBO_2mc.mp4/6dIMIBO_2mc.mp4308yes https://puzzicle.org/math/geometry-home/v/rhombus-diagonalshttps://puzzicle.org/thumbs/13179268.jpgRhombus diagonalsTo show why the diagonals of a rhombus are perpendicular, we can rotate the rhombus to look like a diamond and draw one diagonal as a horizontal line. Then, we can use the fact that the top and bottom triangles are congruent and isosceles to drop altitudes from the vertices to the horizontal diagonal. These altitudes form the other diagonal, which is perpendicular and bisects the horizontal one. Similarly, the horizontal diagonal bisects the other one.https://cdn.kastatic.org/ka-youtube-converted/_QTFeOvPcbY.mp4/_QTFeOvPcbY.mp4278yes https://puzzicle.org/math/geometry-home/v/drawing-polygons-exercise-examplehttps://puzzicle.org/thumbs/x657ad7a6.jpgDrawing a trapezoid on the coordinate plane exampleWatch Sal solve an example problem where he draws an isosceles trapezoid on the coordinate plane.https://cdn.kastatic.org/ka-youtube-converted/yZw8z5BIwuQ.mp4/yZw8z5BIwuQ.mp4162yes https://puzzicle.org/math/geometry-home/v/measuring-area-with-partial-unit-squares-math-3rd-grade-khan-academyhttps://puzzicle.org/thumbs/x97dab21a.jpgMeasuring area with partial unit squaresLindsay finds the area of a shape by counting whole and partial unit squares.https://cdn.kastatic.org/ka-youtube-converted/YA7ZrKcbteA.mp4/YA7ZrKcbteA.mp4196yes https://puzzicle.org/math/geometry-home/v/perimeter-and-unit-conversionhttps://puzzicle.org/thumbs/153770150.jpgPerimeter and unit conversionSal converts metric unit to help solve a perimeter problem.https://cdn.kastatic.org/ka-youtube-converted/q9XCrpLPfGk.mp4/q9XCrpLPfGk.mp4373yes https://puzzicle.org/math/geometry-home/v/applying-the-metric-systemhttps://puzzicle.org/thumbs/32.jpgApplying the metric system to perimeterHow do you find width of a rectangle if given a length and a perimeter? First, make sure that your units match. Once the units match, use the formula for the perimeter of a rectangle (2 times the length plus 2 times the width) to solve for the width.https://cdn.kastatic.org/ka-youtube-converted/CDvPPsB3nEM.mp4/CDvPPsB3nEM.mp4343yes https://puzzicle.org/math/geometry-home/v/triangle-area-proofshttps://puzzicle.org/thumbs/283213902.jpgArea of triangle proofProof that shows that the area of any triangle is 1/2 b x h.https://cdn.kastatic.org/ka-youtube-converted/YOYQys52sPs.mp4/YOYQys52sPs.mp4623yes https://puzzicle.org/math/geometry-home/v/interesting-perimeter-and-area-problemshttps://puzzicle.org/thumbs/283213903.jpgChallenge problems: perimeter & areaThree example problems involving perimeter and area.https://cdn.kastatic.org/ka-youtube-converted/vWXMDIazHjA.mp4/vWXMDIazHjA.mp4493yes https://puzzicle.org/math/geometry-home/v/area-of-diagonal-generated-triangles-of-rectangle-are-equalhttps://puzzicle.org/thumbs/2600582.jpgArea of diagonal-generated trianglesWatch Sal prove that the areas of the pairs of triangles generated by the diagonals of a rectangle are equal.https://cdn.kastatic.org/ka-youtube-converted/YTRimTJ5nX4.mp4/YTRimTJ5nX4.mp4276yes https://puzzicle.org/math/geometry-home/v/conversion-between-metric-unitshttps://puzzicle.org/thumbs/30.jpgConversion between metric unitsWant to learn how to convert between different units of volume using prefixes like kilo, hecto, deka, deci, centi, and milli? In this video, you'll see how to use these prefixes to compare centiliters and dekaliters, and how to use dimensional analysis to convert one dekaliter to centiliters. You'll also get some tips on how to check your answers and avoid common mistakes. Watch now and master this useful skill!https://cdn.kastatic.org/ka-youtube-converted/SYkmadc2wOI.mp4/SYkmadc2wOI.mp4316yes https://puzzicle.org/math/geometry-home/v/density-example-blimphttps://puzzicle.org/thumbs/xa13c840d.jpgDensity word problem: blimpSal solves a word problem about the number of passengers a blimp can carry, using his knowledge of volume and density.https://cdn.kastatic.org/ka-youtube-converted/YbNVUhW-HJM.mp4/YbNVUhW-HJM.mp4365yes https://puzzicle.org/math/geometry-home/v/koch-snowflake-fractalhttps://puzzicle.org/thumbs/283213904.jpgKoch snowflake fractalA shape that has an infinite perimeter but finite areahttps://cdn.kastatic.org/ka-youtube-converted/azBNsPa1WC4.mp4/azBNsPa1WC4.mp4551yes https://puzzicle.org/math/geometry-home/v/area-of-koch-snowflake-part-1-advancedhttps://puzzicle.org/thumbs/294677477.jpgArea of Koch snowflake (1 of 2)Starting to figure out the area of a Koch Snowflake (which has an infinite perimeter). This is an advanced video.https://cdn.kastatic.org/ka-youtube-converted/64bH_27Ehoc.mp4/64bH_27Ehoc.mp4750yes https://puzzicle.org/math/geometry-home/v/area-of-koch-snowflake-part-2-advancedhttps://puzzicle.org/thumbs/294676175.jpgArea of Koch snowflake (2 of 2)Summing an infinite geometric series to finally find the finite area of a Koch Snowflake. This is an advanced video.https://cdn.kastatic.org/ka-youtube-converted/vBlR2xNAGmo.mp4/vBlR2xNAGmo.mp4392yes https://puzzicle.org/math/geometry-home/v/heron-s-formulahttps://puzzicle.org/thumbs/24971.jpgHeron's formulaUsing Heron's Formula to determine the area of a triangle while only knowing the lengths of the sideshttps://cdn.kastatic.org/ka-youtube-converted/-YI6UC4qVEY.mp4/-YI6UC4qVEY.mp4283yes https://puzzicle.org/math/geometry-home/v/part-1-of-proof-of-heron-s-formulahttps://puzzicle.org/thumbs/24973.jpgProof of Heron's formula (1 of 2)Sal proves Heron's Formula for finding the area of a triangle solely from its side lengths.https://cdn.kastatic.org/ka-youtube-converted/AreOMvPCxv8.mp4/AreOMvPCxv8.mp4659yes https://puzzicle.org/math/geometry-home/v/part-2-of-the-proof-of-heron-s-formulahttps://puzzicle.org/thumbs/24975.jpgProof of Heron's formula (2 of 2)Sal proves Heron's Formula for finding the area of a triangle solely from its side lengths.https://cdn.kastatic.org/ka-youtube-converted/nZu7IZLhJRI.mp4/nZu7IZLhJRI.mp4728yes https://puzzicle.org/math/geometry-home/v/rotating-about-arbitrary-pointhttps://puzzicle.org/thumbs/xd2c751b0.jpgRotating shapes: center ≠ (0,0)Sal is given a triangle on the coordinate plane and the definition of a rotation about an arbitrary point, and he manually draws the image of that rotation.https://cdn.kastatic.org/ka-youtube-converted/uuDQkhx6TAI.mp4/uuDQkhx6TAI.mp4632yes https://puzzicle.org/math/geometry-home/v/translations-of-polygonshttps://puzzicle.org/thumbs/x0bee44bd.jpgPerforming translations (old)An older video where Sal uses the interactive widget to find the image of a rectangle under a translation.https://cdn.kastatic.org/ka-youtube-converted/u9B-dV6r6rU.mp4/u9B-dV6r6rU.mp465yes https://puzzicle.org/math/geometry-home/v/rotating-segment-about-orgin-examplehttps://puzzicle.org/thumbs/x68aec32a.jpgPerforming rotations (old)An older video where Sal uses the interactive widget to find the image of a line segment under a rotation.https://cdn.kastatic.org/ka-youtube-converted/sQfsql8BPy0.mp4/sQfsql8BPy0.mp477yes https://puzzicle.org/math/geometry-home/v/reflection-and-mapping-points-examplehttps://puzzicle.org/thumbs/x0482ccff.jpgPerforming reflections: rectangle (old)An older video where Sal uses the interactive widget to find the image of a rectangle under a reflection.https://cdn.kastatic.org/ka-youtube-converted/Y70TTv5ZLkk.mp4/Y70TTv5ZLkk.mp4146yes https://puzzicle.org/math/geometry-home/v/reflecting-line-across-another-line-examplehttps://puzzicle.org/thumbs/x0ed71769.jpgPerforming reflections: line (old)An older video where Sal uses the interactive widget to find the image of a line under a reflection.https://cdn.kastatic.org/ka-youtube-converted/0VARkb_n-Bs.mp4/0VARkb_n-Bs.mp476yes https://puzzicle.org/math/geometry-home/v/determining-a-translation-for-a-shapehttps://puzzicle.org/thumbs/xdc65b657.jpgDetermining translations (old)An older video of Sal determining what translation was performed on a shape to result in another shape.https://cdn.kastatic.org/ka-youtube-converted/ZhROXmbUVKE.mp4/ZhROXmbUVKE.mp446yes https://puzzicle.org/math/geometry-home/v/rotation-of-polygons-examplehttps://puzzicle.org/thumbs/xd50b8c60.jpgRotation examples (old)An older video where Sal finds the images of shapes under various rotations, and where he determines the rotations that take one shape to another.https://cdn.kastatic.org/ka-youtube-converted/Hur3v1hrX3U.mp4/Hur3v1hrX3U.mp4147yes https://puzzicle.org/math/geometry-home/v/performing-a-rotation-to-match-figureshttps://puzzicle.org/thumbs/xd12e1d3c.jpgDetermining rotations (old)An older video where Sal determines the rotation that takes one shape to another.https://cdn.kastatic.org/ka-youtube-converted/QWX39H9kKvE.mp4/QWX39H9kKvE.mp4119yes https://puzzicle.org/math/geometry-home/v/dilating-one-line-onto-anotherhttps://puzzicle.org/thumbs/xa41fb2ec.jpgDilating linesSal shows how we can use dilations to map a line into another, parallel, line.https://cdn.kastatic.org/ka-youtube-converted/jq6WFvMikBQ.mp4/jq6WFvMikBQ.mp4239yes https://puzzicle.org/math/geometry-home/v/example-with-trig-functions-and-ratioshttps://puzzicle.org/thumbs/x835650f9.jpgTrig challenge problem: verify identitiesSal is given a diagram with multiple right triangles and is asked to verify identities relating to trig ratios in those triangles.https://cdn.kastatic.org/ka-youtube-converted/TugWqiUjOU4.mp4/TugWqiUjOU4.mp4307yes https://puzzicle.org/math/geometry-home/v/area-of-inscribed-equilateral-triangle-some-basic-trig-usedhttps://puzzicle.org/thumbs/24979.jpgArea of inscribed equilateral triangleA worked example of finding the area of an equilateral triangle inscribed within a circle who's area is known. This video uses Heron's formula and some trigonometry.https://cdn.kastatic.org/ka-youtube-converted/UmiZK6Hgm6c.mp4/UmiZK6Hgm6c.mp4699yes https://puzzicle.org/math/geometry-home/v/challenging-perimeter-problemhttps://puzzicle.org/thumbs/283213900.jpgChallenging perimeter problemPerimeter of rectangle covered by 9 non-overlapping squares. From 2000 American Invitational Math Examhttps://cdn.kastatic.org/ka-youtube-converted/1uWZNW5PF-s.mp4/1uWZNW5PF-s.mp4596yes https://puzzicle.org/math/geometry-home/v/ca-geometry-deductive-reasoninghttps://puzzicle.org/thumbs/24434.jpgCA Geometry: Deductive reasoning1-3, deductive reasoning and congruent angleshttps://cdn.kastatic.org/ka-youtube-converted/GluohfOedQE.mp4/GluohfOedQE.mp4617yes https://puzzicle.org/math/geometry-home/v/ca-geometry-proof-by-contradictionhttps://puzzicle.org/thumbs/24436.jpgCA Geometry: Proof by contradiction4-6, proof by contradictionhttps://cdn.kastatic.org/ka-youtube-converted/u6O0YHyarlI.mp4/u6O0YHyarlI.mp4716yes https://puzzicle.org/math/geometry-home/v/ca-geometry-more-proofshttps://puzzicle.org/thumbs/24438.jpgCA Geometry: More proofsHow do you evaluate answer choices about isosceles trapezoids? You start by drawing an isosceles trapezoid and comparing each answer choice against your drawing. The answer will then become obvious.https://cdn.kastatic.org/ka-youtube-converted/4PPMnI8-Zsc.mp4/4PPMnI8-Zsc.mp4732yes https://puzzicle.org/math/geometry-home/v/ca-geometry-similar-triangles-1https://puzzicle.org/thumbs/4802217.jpgCA Geometry: Similar triangles 110-14, similar triangleshttps://cdn.kastatic.org/ka-youtube-converted/bWTtHKSEcdI.mp4/bWTtHKSEcdI.mp4775yes https://puzzicle.org/math/geometry-home/v/ca-geometry-more-on-congruent-and-similar-triangleshttps://puzzicle.org/thumbs/24444.jpgCA Geometry: More on congruent and similar triangles17-20, more similar and congruent triangleshttps://cdn.kastatic.org/ka-youtube-converted/FVSgVMVZZ-4.mp4/FVSgVMVZZ-4.mp4671yes https://puzzicle.org/math/geometry-home/v/ca-geometry-triangles-and-parallelogramshttps://puzzicle.org/thumbs/24446.jpgCA Geometry: Triangles and parallelograms21-25, triangles and parallelogramshttps://cdn.kastatic.org/ka-youtube-converted/h0FFEBHBufo.mp4/h0FFEBHBufo.mp4674yes https://puzzicle.org/math/geometry-home/v/ca-geometry-area-pythagorean-theoremhttps://puzzicle.org/thumbs/24448.jpgCA Geometry: Area, pythagorean theorem26-30, area, circumference, pythagorean theoremhttps://cdn.kastatic.org/ka-youtube-converted/jRrRqMJbHKc.mp4/jRrRqMJbHKc.mp4619yes https://puzzicle.org/math/geometry-home/v/ca-geometry-area-circumference-volumehttps://puzzicle.org/thumbs/24450.jpgCA Geometry: Area, circumference, volume31-35, area, circumference and volumehttps://cdn.kastatic.org/ka-youtube-converted/BJSk1joCQsM.mp4/BJSk1joCQsM.mp4653yes https://puzzicle.org/math/geometry-home/v/ca-geometry-pythagorean-theorem-areahttps://puzzicle.org/thumbs/24452.jpgCA Geometry: Pythagorean theorem, area36-40, Pythagorean Theorem and Areahttps://cdn.kastatic.org/ka-youtube-converted/vaOXkt7uuac.mp4/vaOXkt7uuac.mp4644yes https://puzzicle.org/math/geometry-home/v/ca-geometry-exterior-angleshttps://puzzicle.org/thumbs/24454.jpgCA Geometry: Exterior angles41-45, polygons, exterior angleshttps://cdn.kastatic.org/ka-youtube-converted/Ncg1HB5uVLc.mp4/Ncg1HB5uVLc.mp4653yes https://puzzicle.org/math/geometry-home/v/ca-geometry-pythagorean-theorem-compass-constructionshttps://puzzicle.org/thumbs/24458.jpgCA Geometry: Pythagorean theorem, compass constructions51-55, Pythagorean Theorem, compass constructionshttps://cdn.kastatic.org/ka-youtube-converted/6EY0E3z-hsU.mp4/6EY0E3z-hsU.mp4755yes https://puzzicle.org/math/geometry-home/v/ca-geometry-compass-constructionhttps://puzzicle.org/thumbs/24460.jpgCA Geometry: Compass construction56-60, compass construction, coordinateshttps://cdn.kastatic.org/ka-youtube-converted/79lagkER9qY.mp4/79lagkER9qY.mp4769yes https://puzzicle.org/math/geometry-home/v/ca-geometry-basic-trigonometryhttps://puzzicle.org/thumbs/24462.jpgCA Geometry: Basic trigonometry61-65, basic trigonometryhttps://cdn.kastatic.org/ka-youtube-converted/MyvRxKM0xns.mp4/MyvRxKM0xns.mp4664yes https://puzzicle.org/math/geometry-home/v/ca-geometry-more-trighttps://puzzicle.org/thumbs/24464.jpgCA Geometry: More trig66-70, more trigonometryhttps://cdn.kastatic.org/ka-youtube-converted/8YBQQ0GeusM.mp4/8YBQQ0GeusM.mp4741yes https://puzzicle.org/math/geometry-home/v/ca-geometry-circle-area-chords-tangenthttps://puzzicle.org/thumbs/24466.jpgCA Geometry: Circle area chords tangent71-75, area, chords, tangents of circleshttps://cdn.kastatic.org/ka-youtube-converted/FJIZPvE3O1A.mp4/FJIZPvE3O1A.mp4606yes https://puzzicle.org/math/geometry-home/v/speed-translationhttps://puzzicle.org/thumbs/25779.jpgSpeed translationSal converts feet to inches to help solve a rate problem.https://cdn.kastatic.org/ka-youtube-converted/aTjNDKlz8G4.mp4/aTjNDKlz8G4.mp4597yes https://puzzicle.org/math/geometry-tx/v/how-sure-are-wehttps://puzzicle.org/thumbs/xc4de5f7be17bea94.jpgHow sure are we?In geometry, science, and in many areas of life, we have vocabulary to describe how sure we are. Conjecture: A statement that we believe is always true because we have seen many examples of it and no counterexamples. Postulate: A statement that we believe is always true because people generally agree it is true, but without proof. Theorem: A statement that we believe is always true because we prove it by a series of other definitions, postulates, and theorems.https://cdn.kastatic.org/ka-youtube-converted/a9TUKHWY3TA.mp4/a9TUKHWY3TA.mp4470yes https://puzzicle.org/math/geometry-tx/v/definitions-and-undefined-termshttps://puzzicle.org/thumbs/xb54ca5d9da4618bc.jpgDefinitions and undefined termsMuch of geometry depends on building our ideas from foundational pieces. We use definitions to be very clear about what we mean by a right angle, similar figures, or a line segments. If we go to the very base of even those definitions, we have to agree on some ideas, like what we mean by a point or a line, without being able to simply and clearly define them.https://cdn.kastatic.org/ka-youtube-converted/J_u_ROMJR8U.mp4/J_u_ROMJR8U.mp4252yes https://puzzicle.org/math/geometry-tx/v/dividing-line-segments-on-number-linehttps://puzzicle.org/thumbs/x1ac7ec290ea4af56.jpgDividing line segments on number lineGiven two points on a vertical number line with rational numbers, find the point that is a fractional distance between them.https://cdn.kastatic.org/ka-youtube-converted/aGAmPyfmLRA.mp4/aGAmPyfmLRA.mp4118yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/triangle-altitude-equationhttps://puzzicle.org/thumbs/xc1e9720bd4c91ffc.jpgTriangle altitude equationGiven a triangle on a coordinate plane and a base that is neither horizontal nor vertical, let's find the equation of the line that contains the altitude of the triangle relative to that base. We can use the fact that the slope of the altitude is perpendicular to the slope of the base. That means the two slopes are opposite reciprocals of each other.https://cdn.kastatic.org/ka-youtube-converted/Cb1XT3_hQwM.mp4/Cb1XT3_hQwM.mp4272yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/graphing-a-non-similar-pre-image-algebraicallyhttps://puzzicle.org/thumbs/x9995776eb6824399.jpgGraphing a non similar pre image algebraicallyGiven an image of a geometric figure and the algebraic rule that describes the transformations it is under, graph the pre-image. The composition of transformations the figure is under do not preserve congruence and similarity.https://cdn.kastatic.org/ka-youtube-converted/wkhQNXyW01U.mp4/wkhQNXyW01U.mp4219yes https://puzzicle.org/math/geometry-tx/v/surface-area-of-combination-of-solids-surface-area-and-volumehttps://puzzicle.org/thumbs/x3e873a83e0a930b6.jpgArea of combination of solidsLet's learn how to find the surface area of combination of solids by breaking them down into cuboids, cylinders, cones, and hemispheres with the help of an example.https://cdn.kastatic.org/ka-youtube-converted/QDQw66_TeaE.mp4/QDQw66_TeaE.mp4570yes https://puzzicle.org/math/geometry-tx/v/surface-area-of-combination-of-solids-problem-typeshttps://puzzicle.org/thumbs/x71bc517f7d43b4be.jpgProblem types: surface area of combination of solidsLet's look at a few common types of problems involving 'surface area of a combination of solids' and look at broad strategies on how to solve them and what mistakes to watch out for.https://cdn.kastatic.org/ka-youtube-converted/3CGBz05r1r4.mp4/3CGBz05r1r4.mp4358yes https://puzzicle.org/math/geometry-tx/v/volume-of-combination-of-solidshttps://puzzicle.org/thumbs/x80ad6049d7bd7c3e.jpgVolume of combination of solidsLet's learn how to find the volume of a complicated solid by breaking it down into cylinders, cones, hemispheres, and cuboids.https://cdn.kastatic.org/ka-youtube-converted/ANc85RET5Fk.mp4/ANc85RET5Fk.mp4531yes https://puzzicle.org/math/geometry-tx/v/volume-of-combination-solids-problem-typeshttps://puzzicle.org/thumbs/x4120b3651d8de779.jpgProblem types: volume of combination of solidsLet's look at a few common types of problems involving 'volume of a combination of solids' and look at broad strategies on how to solve them and the mistakes to watch out for.https://cdn.kastatic.org/ka-youtube-converted/feiSEqpPmAA.mp4/feiSEqpPmAA.mp4207yes https://puzzicle.org/math/geometry-tx/v/curved-surface-area-of-a-frustum-surface-areas-and-volumeshttps://puzzicle.org/thumbs/x9a90fe08f4a994e4.jpgCurved surface area of a frustumLet's learn how to find the curved surface area of a frustum of a cone through an example. Let's use the intuitive approach rather than directly plugging into the formula.https://cdn.kastatic.org/ka-youtube-converted/Ft-2DJ0UPN8.mp4/Ft-2DJ0UPN8.mp4454yes https://puzzicle.org/math/geometry-tx/v/volume-of-a-frustumhttps://puzzicle.org/thumbs/x3b57ab88f8c04520.jpgVolume of a frustumLet's learn how to find the volume of a frustum by looking at an example problem. Let's use an intuitive approach to find the volume rather than directly applying the formula.https://cdn.kastatic.org/ka-youtube-converted/Xn9pHi5u_Fw.mp4/Xn9pHi5u_Fw.mp4461yes https://puzzicle.org/math/precalculus/v/interpreting-general-multiplication-rulehttps://puzzicle.org/thumbs/xccee087782f2d0dc.jpgInterpreting general multiplication ruleWe can express the probability that two events both occur symbolically using the general multiplication rule, and we can interpret probability statements that are expressed symbolically.https://cdn.kastatic.org/ka-youtube-converted/yTJawPRzCzI.mp4/yTJawPRzCzI.mp4291yes https://puzzicle.org/math/statistics-probability/v/independent-events-1https://puzzicle.org/thumbs/27094.jpgIndependent & dependent probabilityThis time around we're not going to tell you whether we're working on a dependent or independent probability event problem. You tell us!https://cdn.kastatic.org/ka-youtube-converted/Za7G_eWKiF4.mp4/Za7G_eWKiF4.mp4159yes https://puzzicle.org/math/precalculus/v/factorial-and-counting-seat-arrangementshttps://puzzicle.org/thumbs/xc7e724ce.jpgFactorial and counting seat arrangementsLearn how to use permutations to solve problems involving ways to arrange things. Permutations involve using factorials to count all possible arrangements. This video also explores examples including arranging three people in three seats and five people in five seats.https://cdn.kastatic.org/ka-youtube-converted/eoxbgUIYhHo.mp4/eoxbgUIYhHo.mp4333yes https://puzzicle.org/math/geometry-tx/v/area-based-probabilityhttps://puzzicle.org/thumbs/x77208ef925158449.jpgFinding area-based probabilitiesLet's use area models to compare the theoretical probabilities of a dart landing in the shaded area of different dartboard designs. The probability of landing in the shaded region is (area of shaded region)/(total area).https://cdn.kastatic.org/ka-youtube-converted/5biGkYiu548.mp4/5biGkYiu548.mp4478yes https://puzzicle.org/math/get-ready-for-8th-grade/v/order-of-operationshttps://puzzicle.org/thumbs/2600695.jpgOrder of operations exampleThe order of operations tells us the order to solve steps in expressions with more than one operation. First, we solve any operations inside of parentheses or brackets. Second, we solve any exponents. Third, we solve all multiplication and division from left to right. Fourth, we solve all addition and subtraction from left to right.https://cdn.kastatic.org/ka-youtube-converted/GiSpzFKI5_w.mp4/GiSpzFKI5_w.mp4267yes https://puzzicle.org/math/precalculus/v/meaningfully-composing-functionshttps://puzzicle.org/thumbs/x467bc55a8fd4e34a.jpgMeaningfully composing functionsDecide which composed functions make sense by checking that the value that one function passes to the other is the right kind of input.https://cdn.kastatic.org/ka-youtube-converted/q9JGGVXdl9I.mp4/q9JGGVXdl9I.mp4221yes https://puzzicle.org/math/statistics-probability/v/monty-hall-problemhttps://puzzicle.org/thumbs/878209052.jpgThe Monty Hall problemHere we have a presentation and analysis of the famous thought experiment: the "Monty Hall" problem! This is fun.https://cdn.kastatic.org/ka-youtube-converted/Xp6V_lO1ZKA.mp4/Xp6V_lO1ZKA.mp4443yes https://puzzicle.org/math/statistics-probability/v/describing-subsets-of-sample-spaces-exercisehttps://puzzicle.org/thumbs/x9c78221b.jpgSubsets of sample spacesSal solves an example about subsets.https://cdn.kastatic.org/ka-youtube-converted/0uHhk7P9SNo.mp4/0uHhk7P9SNo.mp4344yes https://puzzicle.org/math/statistics-probability/v/relative-complement-or-difference-between-setshttps://puzzicle.org/thumbs/x4cc5bca0.jpgRelative complement or difference between setsSal shows an example finding the relative complement or difference of two sets A and B.https://cdn.kastatic.org/ka-youtube-converted/2B4EBvVvf9w.mp4/2B4EBvVvf9w.mp4375yes https://puzzicle.org/math/statistics-probability/v/universal-set-and-absolute-complementhttps://puzzicle.org/thumbs/x66808e41.jpgUniversal set and absolute complementSal moves onto more challenging set ideas and notation like the universal set and absolute complement.https://cdn.kastatic.org/ka-youtube-converted/GVZUpOm3XUg.mp4/GVZUpOm3XUg.mp4386yes https://puzzicle.org/math/statistics-probability/v/subset-strict-subset-and-supersethttps://puzzicle.org/thumbs/xf762eb2f.jpgSubset, strict subset, and supersetA subset of a set A is any set B such that every element of B is also an element of A. A strict subset is a subset that isn't equal to the original set (i.e. B must have at least one fewer element than A). A superset of A is any set C such that A is a subset of C.https://cdn.kastatic.org/ka-youtube-converted/1wsF9GpGd00.mp4/1wsF9GpGd00.mp4272yes https://puzzicle.org/math/statistics-probability/v/bringing-the-set-operations-togetherhttps://puzzicle.org/thumbs/x614a90bd.jpgBringing the set operations togetherSal summarizes the set operations that he has discussed in the previous videos.https://cdn.kastatic.org/ka-youtube-converted/OCNXS_m1HWU.mp4/OCNXS_m1HWU.mp4388yes https://puzzicle.org/math/grade-3-tx-teks/v/decomposing-whole-numbershttps://puzzicle.org/thumbs/x6c0238e88ea647fc.jpgDecomposing whole numbersSal looks at a number written in standard form and figures out how we can say the number by decomposing the parts.https://cdn.kastatic.org/ka-youtube-converted/JzdD_5SO9ns.mp4/JzdD_5SO9ns.mp4168yes https://puzzicle.org/math/grade-3-tx-teks/v/decomposing-whole-numbers-in-word-formhttps://puzzicle.org/thumbs/xc68a26c4dd26b66d.jpgDecomposing whole numbers in word formSal looks at a number presented in word form and uses a place value chart to put the number together to find the value.https://cdn.kastatic.org/ka-youtube-converted/jcRfhk2FAlA.mp4/jcRfhk2FAlA.mp4184yes https://puzzicle.org/math/grade-3-tx-teks/v/finding-consecutive-multiples-10-and-100https://puzzicle.org/thumbs/x8fc0056299fccf07.jpgFinding consecutive multiples (10 and 100)Sal uses a number line to find the nearest consecutive multiple of a given number.https://cdn.kastatic.org/ka-youtube-converted/vuq2HHx9lso.mp4/vuq2HHx9lso.mp494yes https://puzzicle.org/math/grade-3-tx-teks/v/finding-consecutive-multiples-1000-and-10000https://puzzicle.org/thumbs/xbc4e751b145dc4b8.jpgFinding consecutive multiples (1,000 and 10,000)Sal uses a number line to determine which of three points is closest to a given number. By using rounding, Sal finds the nearest consecutive multiple to the nearest 10,000.https://cdn.kastatic.org/ka-youtube-converted/pkNmgdcb7XY.mp4/pkNmgdcb7XY.mp481yes https://puzzicle.org/math/grade-3-tx-teks/v/creating-picture-and-bar-graphs-indian-accenthttps://puzzicle.org/thumbs/xe21eebba75272cb4.jpgCreating picture and bar graphsSal creates and interprets bar and picture graphs.https://cdn.kastatic.org/ka-youtube-converted/2QrLG7VuAG0.mp4/2QrLG7VuAG0.mp4279yes https://puzzicle.org/math/grade-3-tx-teks/v/creating-a-bar-graph-indian-accenthttps://puzzicle.org/thumbs/x115e9fa0847ee23b.jpgCreating a bar graphSal creates a bar chart using data from a survey.https://cdn.kastatic.org/ka-youtube-converted/DuaUm9dz5HQ.mp4/DuaUm9dz5HQ.mp467yes https://puzzicle.org/math/grade-3-tx-teks/v/what-is-labor-and-incomehttps://puzzicle.org/thumbs/x363d781c008685ea.jpgWhat is labor and income?Build on students’ understanding that money earned is called income by helping them explain how labor—which includes a person’s skills, education, and training—can affect how much income they earn. Students will explore different jobs, find out how much education or training each one usually requires, and compare the average salaries. They will look for patterns between how long someone goes to school and how much money they might make in different careers.https://cdn.kastatic.org/ka-youtube-converted/nHD2N5gHXj4.mp4/nHD2N5gHXj4.mp4337yes https://puzzicle.org/math/grade-3-tx-teks/v/what-is-credithttps://puzzicle.org/thumbs/x4b87a324e5d884bb.jpgWhat is credit?Learn all about credit! Credit is used when wants or needs exceed the ability to pay and that it is the borrower's responsibility to pay it back to the lender, usually with interest.https://cdn.kastatic.org/ka-youtube-converted/zdBrsvkhdeU.mp4/zdBrsvkhdeU.mp4289yes https://puzzicle.org/math/grade-3-tx-teks/v/what-is-a-savings-planhttps://puzzicle.org/thumbs/x870ae54d7c1a8721.jpgWhat is a savings plan?What is a savings plane? Understand the reasons to save and find out the benefits of saving (e.g., saving for college, surviving hard financial times, being prepared for unforeseen expenses, setting goals, etc.).https://cdn.kastatic.org/ka-youtube-converted/Q4T2ipCuAPo.mp4/Q4T2ipCuAPo.mp4243yes https://puzzicle.org/math/grade-3-tx-teks/v/what-are-costs-and-resourceshttps://puzzicle.org/thumbs/xc24c03ca7fd2572a.jpgWhat are costs and resources?Describe the relationship between the availability or scarcity of resources and how that impacts cost. Students learn to analyze data as they explore the relationship between supply and demand.https://cdn.kastatic.org/ka-youtube-converted/dMM2eiK-ejI.mp4/dMM2eiK-ejI.mp4334yes https://puzzicle.org/math/grade-3-tx-teks/v/classifying-three-dimensional-shapeshttps://puzzicle.org/thumbs/xbcfc1ea87c975ea0.jpgClassifying three dimensional shapesIdentify prisms, pyramids, cones, cylinders, and spheres.https://cdn.kastatic.org/ka-youtube-converted/4VgPJkh16_I.mp4/4VgPJkh16_I.mp4189yes https://puzzicle.org/math/grade-4-math-tx/v/representing-multi-step-problems-equationshttps://puzzicle.org/thumbs/x51b1362bf3c8c4b0.jpgRepresenting multi-step problems: equationsPractice using equations to represent multi-step word problems.https://cdn.kastatic.org/ka-youtube-converted/DGOGterRmp4.mp4/DGOGterRmp4.mp4160yes https://puzzicle.org/math/grade-4-math-tx/v/subtracting-decimals-farmers-markethttps://puzzicle.org/thumbs/x0c89b404d0ed1364.jpgSubtracting decimals: farmer's marketPractice solving problems by subtracting decimals.https://cdn.kastatic.org/ka-youtube-converted/DpEHVSdiRko.mp4/DpEHVSdiRko.mp4137yes https://puzzicle.org/math/grade-4-math-tx/v/input-output-tables-finding-the-outputhttps://puzzicle.org/thumbs/x95a19b145d90db2c.jpgInput & output tables: finding the outputPractice using pattern recognition to identify outputs in input and output tables.https://cdn.kastatic.org/ka-youtube-converted/9Hdo_OO19uM.mp4/9Hdo_OO19uM.mp455yes https://puzzicle.org/math/grade-4-math-tx/v/input-output-tables-identifying-the-rulehttps://puzzicle.org/thumbs/x43c0a18d96ee68e4.jpgInput & output tables: identifying the rulePractice using pattern recognition to identify rules in an input and output table.https://cdn.kastatic.org/ka-youtube-converted/Y9oW96dhzg0.mp4/Y9oW96dhzg0.mp4110yes https://puzzicle.org/math/grade-4-math-tx/v/calculating-profit-with-additions-and-subtractionhttps://puzzicle.org/thumbs/xaf77f2e7e2b273be.jpgCalculating profit with addition and subtractionExplore how to calculate profit using addition and subtraction in this clear, real-world 4th-grade math video with helpful examples.https://cdn.kastatic.org/ka-youtube-converted/kkzQfhQYr_E.mp4/kkzQfhQYr_E.mp4305yes https://puzzicle.org/math/grade-4-math-tx/v/calculating-profit-with-multiplicationhttps://puzzicle.org/thumbs/xdfd83434e77760c9.jpgCalculating profit with multiplicationLearn how to calculate profit using multiplication in this simple 4th grade video with real-world examples and easy math strategies.https://cdn.kastatic.org/ka-youtube-converted/iCA78yjz_HY.mp4/iCA78yjz_HY.mp4352yes https://puzzicle.org/math/grade-4-math-tx/v/finding-total-money-savedhttps://puzzicle.org/thumbs/x4caac52f7d7d8411.jpgFinding total money savedLearn how to find the total money saved by adding and multiplying.https://cdn.kastatic.org/ka-youtube-converted/3eaTDJKGRqw.mp4/3eaTDJKGRqw.mp493yes https://puzzicle.org/math/grade-4-math-tx/v/calculating-savingshttps://puzzicle.org/thumbs/x7c956fe75ee68c37.jpgCalculating savingsLet's calculate how long it takes to reach a savings goal.https://cdn.kastatic.org/ka-youtube-converted/jDRN7fMZ-P4.mp4/jDRN7fMZ-P4.mp457yes https://puzzicle.org/math/grade-5-math-tx/v/estimating-multi-digit-sumshttps://puzzicle.org/thumbs/xb11f0227129b67f1.jpgEstimating multi-digit sumsUse estimation to add multi-digit numbers.https://cdn.kastatic.org/ka-youtube-converted/NAXN_dlQ4JA.mp4/NAXN_dlQ4JA.mp4163yes https://puzzicle.org/math/grade-5-math-tx/v/estimating-to-subtract-multi-digit-numbershttps://puzzicle.org/thumbs/x6925339e71a2d542.jpgEstimating to subtract multi-digit numbersPractice using estimation as a strategy to subtract multi-digit numbers to find the difference.https://cdn.kastatic.org/ka-youtube-converted/GbosTJIyuSA.mp4/GbosTJIyuSA.mp4117yes https://puzzicle.org/math/grade-5-math-tx/v/dividing-2-digit-numbers-by-2-digit-numbershttps://puzzicle.org/thumbs/xef1b342be09bede8.jpgDividing 2-digit numbers by 2-digit numbersPractice dividing 2-digit numbers by 2-digit numbers.https://cdn.kastatic.org/ka-youtube-converted/sF1iIsW219s.mp4/sF1iIsW219s.mp4172yes https://puzzicle.org/math/grade-5-math-tx/v/dividing-3-digit-numbers-by-2-digit-numbershttps://puzzicle.org/thumbs/x848a8820d51fc4e4.jpgDividing 3-digit numbers by 2-digit numbersPractice dividing 3-digit whole numbers by 2-digit whole numbers using different strategies.https://cdn.kastatic.org/ka-youtube-converted/F6BNkk91XKc.mp4/F6BNkk91XKc.mp4203yes https://puzzicle.org/math/grade-5-math-tx/v/simplifying-numerical-expressionshttps://puzzicle.org/thumbs/xb4150d7796fa2444.jpgSimplifying numerical expressionsPractice using brackets and parentheses to simplify numerical expressions.https://cdn.kastatic.org/ka-youtube-converted/YrjnKwxdlr4.mp4/YrjnKwxdlr4.mp4182yes https://puzzicle.org/math/grade-5-math-tx/v/what-are-taxeshttps://puzzicle.org/thumbs/x39cde83790ec90c6.jpgWhat are taxes?Learn about the different types of taxes that impact you daily. Explore income tax, payroll tax, sales tax, and property tax. We can learn the definitions of each tax, while also finding out how they are used!https://cdn.kastatic.org/ka-youtube-converted/kdfk22Ck4nM.mp4/kdfk22Ck4nM.mp4319yes https://puzzicle.org/math/grade-5-math-tx/v/gross-and-net-incomehttps://puzzicle.org/thumbs/xe82cdd26dab9e6f2.jpgGross and net incomeLearn about gross and net income and how they impact the paycheck you would receive if working a job. We will discuss the high level overview of the differences between the two!https://cdn.kastatic.org/ka-youtube-converted/E9ZSeCh1GUE.mp4/E9ZSeCh1GUE.mp4125yes https://puzzicle.org/math/grade-5-math-tx/v/financial-record-keepinghttps://puzzicle.org/thumbs/xcd58f7ed01ea960a.jpgFinancial record keepingSal talks about the importance of financial record keeping and gives advice on how to stay organized - even at a young age!https://cdn.kastatic.org/ka-youtube-converted/xHntnwzXA4s.mp4/xHntnwzXA4s.mp4173yes https://puzzicle.org/math/grade-5-math-tx/v/balancing-a-budgethttps://puzzicle.org/thumbs/x9f74c9e396f15537.jpgBalancing a budgetLearn about how to create a budget! Practice seeing how you can increase your monthly income or decrease your expenses so you have a balanced budget.https://cdn.kastatic.org/ka-youtube-converted/zx98S3XR2KM.mp4/zx98S3XR2KM.mp4366yes https://puzzicle.org/math/grade-5-math-tx/v/rounding-decimals-to-the-hundredths-on-the-number-linehttps://puzzicle.org/thumbs/xe85ddba2e5d9c560.jpgRounding decimals to the hundredths on the number linePractice rounding decimals on a vertical number line. First, start by identifying where a point rounds on a number line given a place value. Then, use a number line with three given points to find which point rounds to a given decimal.https://cdn.kastatic.org/ka-youtube-converted/uu3RrUR-5iE.mp4/uu3RrUR-5iE.mp4196yes https://puzzicle.org/math/grade-5-math-tx/v/estimating-decimal-addition-thousandthshttps://puzzicle.org/thumbs/x08de2f914e5faaf0.jpgEstimating decimal addition (thousandths)Practice using estimation to add decimals to the thousandths. First, practice adding decimals greater than one whole and rounding to the nearest whole number. Then, practice adding two decimals less than one whole and rounding to the nearest tenth.https://cdn.kastatic.org/ka-youtube-converted/j1cY7Upjhbo.mp4/j1cY7Upjhbo.mp4239yes https://puzzicle.org/math/grade-5-math-tx/v/adding-decimals-with-thousandthshttps://puzzicle.org/thumbs/x168769d5e2e84754.jpgAdding decimals with thousandthsPractice using place value to line up decimals using the decimal point. Start by adding decimals in the farthest place value column to the right until all place value columns have been added.https://cdn.kastatic.org/ka-youtube-converted/_ayXtD83wBA.mp4/_ayXtD83wBA.mp4245yes https://puzzicle.org/math/grade-5-math-tx/v/adding-decimals-and-whole-numbers-thousandthshttps://puzzicle.org/thumbs/xc7c4e112bea312f4.jpgAdding decimals and whole numbers (thousandths)Practice adding decimals and whole numbers together using a variety of strategies. First, practice breaking apart the numbers into whole numbers and decimals. Then, line the decimals up by place to add.https://cdn.kastatic.org/ka-youtube-converted/V8LmXU8YiOM.mp4/V8LmXU8YiOM.mp4169yes https://puzzicle.org/math/grade-5-math-tx/v/estimating-decimal-subtractionhttps://puzzicle.org/thumbs/xa5c478a77fc873a2.jpgEstimating decimal subtraction (thousandths)Practice using estimation on a number line to help as a strategy for multiplying two decimals, up to the thousandths place.https://cdn.kastatic.org/ka-youtube-converted/pUsirzVGPrw.mp4/pUsirzVGPrw.mp4281yes https://puzzicle.org/math/grade-5-math-tx/v/coordinate-plane-relationshipshttps://puzzicle.org/thumbs/x7ef797e7831e1967.jpgCoordinate plane relationshipsPractice using your understanding of the x- and y-coordinates to determine the relationship between moving up, down, left or right on a coordinate plane.https://cdn.kastatic.org/ka-youtube-converted/oo-2ugRkgAk.mp4/oo-2ugRkgAk.mp4244yes https://puzzicle.org/math/grade-5-math-tx/v/additive-and-multiplicative-relationshipshttps://puzzicle.org/thumbs/x356555ea0d145787.jpgAdditive and multiplicative relationshipsPractice using additive and multiplicative relationships to determine patterns within tables.https://cdn.kastatic.org/ka-youtube-converted/Iti6JTcijE8.mp4/Iti6JTcijE8.mp4197yes https://puzzicle.org/math/grade-5-math-tx/v/constructing-scatterplotshttps://puzzicle.org/thumbs/x1440be8c35f36827.jpgConstructing scatterplotsPractice using data given in a table to construct a scatterplot on coordinate grid.https://cdn.kastatic.org/ka-youtube-converted/zgmMOXwDpFI.mp4/zgmMOXwDpFI.mp495yes https://puzzicle.org/math/grade-5-math-tx/v/understanding-scatterplotshttps://puzzicle.org/thumbs/x2e23a7b7a2adb73d.jpgUnderstanding scatterplotsPractice matching a table to a series of scatterplots based on the x- and y-values.https://cdn.kastatic.org/ka-youtube-converted/WaL02i4inE0.mp4/WaL02i4inE0.mp4121yes https://puzzicle.org/math/mappers/v/converting-decimals-to-percents-ex-1https://puzzicle.org/thumbs/915124662.jpgConverting decimals to percents: 0.601Percents can be written in decimal form. ​Per-cent means per-100. So, we divide the percent by 100 to get an equivalent decimal. Then, we remove the percent sign (%). For example, 65% can be converted to decimal form by solving 65÷100. So, 65%=0.65. Another way to think of dividing by 100 is moving the decimal two places to the left.https://cdn.kastatic.org/ka-youtube-converted/5z568fLBQyQ.mp4/5z568fLBQyQ.mp4156yes https://puzzicle.org/math/mappers/v/converting-percents-to-decimals-ex-2https://puzzicle.org/thumbs/x8959c8e0.jpgConverting percents to decimals: 113.9%Sal converts 113.9% to a decimal by demonstrating how to move the decimal pointhttps://cdn.kastatic.org/ka-youtube-converted/T4JKO0OGjpQ.mp4/T4JKO0OGjpQ.mp473yes https://puzzicle.org/math/pre-algebra/v/prime-factorizationhttps://puzzicle.org/thumbs/2600704.jpgPrime factorizationThis video explains the concept of prime numbers and how to find the prime factorization of a number using a factorization tree. It also shows how to write the prime factorization using exponential notation. A prime number is a number that is only divisible by itself and one.https://cdn.kastatic.org/ka-youtube-converted/ZKKDTfHcsG0.mp4/ZKKDTfHcsG0.mp4249yes https://puzzicle.org/math/pre-algebra/v/prime-factorization-exercisehttps://puzzicle.org/thumbs/xf27acc45.jpgPrime factorization exerciseLearn to find the prime factorization of any number by breaking it down into its prime factors. Understand the difference between prime and composite numbers. Learn from guided examples of finding the prime factorization of 36, 30, and 73.https://cdn.kastatic.org/ka-youtube-converted/XWq8bplP-_E.mp4/XWq8bplP-_E.mp4229yes https://puzzicle.org/math/pre-algebra/v/common-divisibility-exampleshttps://puzzicle.org/thumbs/56195550.jpgCommon divisibility examplesCommon Divisibility Exampleshttps://cdn.kastatic.org/ka-youtube-converted/zWcfVC-oCNw.mp4/zWcfVC-oCNw.mp4475yes https://puzzicle.org/math/grade-6-math-tx/v/classifying-rational-numbershttps://puzzicle.org/thumbs/x447359e7f4fa1273.jpgClassifying rational numbersClassify rational numbers including instances using a nested Venn diagram.https://cdn.kastatic.org/ka-youtube-converted/MqLUmHRy1L4.mp4/MqLUmHRy1L4.mp4176yes https://puzzicle.org/math/grade-6-math-tx/v/relating-fractions-and-divisionhttps://puzzicle.org/thumbs/x7a135b0e95cc1f76.jpgRelating fractions and divisionPractice relating fractions and division.https://cdn.kastatic.org/ka-youtube-converted/yC4yAhU8mNw.mp4/yC4yAhU8mNw.mp4223yes https://puzzicle.org/math/grade-6-math-tx/v/dividing-whole-numbers-by-decimals-tenthshttps://puzzicle.org/thumbs/x385e478d72179620.jpgDividing whole numbers by decimals (tenths)See worked examples of dividing whole numbers by decimals that are only to the tenths.https://cdn.kastatic.org/ka-youtube-converted/pVAmhlmDUMM.mp4/pVAmhlmDUMM.mp4312yes https://puzzicle.org/math/grade-6-math-tx/v/dividing-whole-numbers-by-decimals-hundredths-pencilshttps://puzzicle.org/thumbs/xdd7b9201948cb15d.jpgDividing whole numbers by decimals (hundredths) -pencilsSee a worked example about the sale of pencils to practice dividing whole numbers by decimals to the hundredths place.https://cdn.kastatic.org/ka-youtube-converted/DfnjfYxVdAc.mp4/DfnjfYxVdAc.mp4223yes https://puzzicle.org/math/grade-6-math-tx/v/dividing-whole-numbers-by-decimals-hundredths-ice-creamhttps://puzzicle.org/thumbs/xfcee8c2b51c6cc8a.jpgDividing whole numbers by decimals (hundredths) - ice creamWatch as Sal explains how to divide whole numbers by decimals to the hundredths using an ice cream store example.https://cdn.kastatic.org/ka-youtube-converted/S3MfUqTp0cg.mp4/S3MfUqTp0cg.mp4208yes https://puzzicle.org/math/grade-6-math-tx/v/adding-integers-numbers-on-the-number-line-6th-grade-khan-academyhttps://puzzicle.org/thumbs/x1df995eab6254590.jpgAdding integers on the number linePractice adding positive and negative integers using a number line.https://cdn.kastatic.org/ka-youtube-converted/rubKHy5GRvo.mp4/rubKHy5GRvo.mp4223yes https://puzzicle.org/math/grade-6-math-tx/v/multiplying-integers-word-problemhttps://puzzicle.org/thumbs/x61aca458e6ffc0d3.jpgMultiplying integers word problemIn this video, we'll explore both positive and negative rates of change through real-world applications.https://cdn.kastatic.org/ka-youtube-converted/4eAhLoYGSI0.mp4/4eAhLoYGSI0.mp491yes https://puzzicle.org/math/grade-6-math-tx/v/multiplying-integers-word-problems-temperaturehttps://puzzicle.org/thumbs/x3d93980180635067.jpgMultiplying integers word problems (temperature)Watch Sal practice multiplying integers, both negative and positive, using a temperature example.https://cdn.kastatic.org/ka-youtube-converted/p7IfPaQfqus.mp4/p7IfPaQfqus.mp468yes https://puzzicle.org/math/grade-6-math-tx/v/dividing-integers-word-problemshttps://puzzicle.org/thumbs/xe5ad46306c48945e.jpgDividing integers word problemsIn this video, we'll examine how quantities change over time to determine the rate of change. We will use division to calculate this rate in real-world contexts, demonstrating a straightforward approach to understanding these changes through practical examples.https://cdn.kastatic.org/ka-youtube-converted/siPU49HmRu0.mp4/siPU49HmRu0.mp4214yes https://puzzicle.org/math/grade-6-math-tx/v/problem-solving-with-integershttps://puzzicle.org/thumbs/x10a97b4197dae5a5.jpgProblem solving with integersIn this lesson, we'll use our understanding of integer operations to solve multi-step word problems.https://cdn.kastatic.org/ka-youtube-converted/STe5t3GPkC4.mp4/STe5t3GPkC4.mp4125yes https://puzzicle.org/math/grade-6-math-tx/v/rates-of-change-with-integershttps://puzzicle.org/thumbs/xa93d0f4a3df6cb33.jpgRates of change with integersSal practices solving a worked example involving rates of change with integers.https://cdn.kastatic.org/ka-youtube-converted/Zu7EupT2pdE.mp4/Zu7EupT2pdE.mp4390yes https://puzzicle.org/math/grade-6-math-tx/v/features-and-fees-of-checking-accountshttps://puzzicle.org/thumbs/x5310e6b5d5563cc2.jpgFeatures and fees of checking accountsLearn about the features of a checking account.https://cdn.kastatic.org/ka-youtube-converted/HBCsplRQ4Gg.mp4/HBCsplRQ4Gg.mp4388yes https://puzzicle.org/math/grade-6-math-tx/v/balancing-a-check-registerhttps://puzzicle.org/thumbs/xb01a50b5cfd4dbaf.jpgBalancing a check registerLearn how deposits, withdrawals, and transfers affect your account balance.https://cdn.kastatic.org/ka-youtube-converted/Tgr3trjhPs8.mp4/Tgr3trjhPs8.mp4179yes https://puzzicle.org/math/grade-6-math-tx/v/debit-and-credit-cardshttps://puzzicle.org/thumbs/x0e936d0c87cdeb52.jpgDebit and credit cardsLearn the key differences between debit and credit cards in this quick comparison. We'll cover how each works and what to consider when choosing between them.https://cdn.kastatic.org/ka-youtube-converted/_ZVEjFBclFQ.mp4/_ZVEjFBclFQ.mp4258yes https://puzzicle.org/math/grade-6-math-tx/v/credit-report-credit-historyhttps://puzzicle.org/thumbs/x260f1afe0632c2e6.jpgCredit report & credit historyExplore the essentials of credit reports and credit history. Learn how your credit history affects your financial future, what’s included in a credit report, and how to read it.https://cdn.kastatic.org/ka-youtube-converted/OB8bLkRscNc.mp4/OB8bLkRscNc.mp4268yes https://puzzicle.org/math/grade-6-math-tx/v/comparing-salarieshttps://puzzicle.org/thumbs/x3e63223b18de911d.jpgComparing salariesIn this video, we'll compare the salaries of different jobs. We'll look at how much people earn in certain careers and what affects their pay. This will help you understand which jobs pay more and why, giving you a better idea of what career might interest you.https://cdn.kastatic.org/ka-youtube-converted/EmbQ6s0v1fA.mp4/EmbQ6s0v1fA.mp4173yes https://puzzicle.org/math/mappers/v/equivalent-expressions-with-negative-numbershttps://puzzicle.org/thumbs/xedc4e33c.jpgEquivalent expressions with negative numbersEquivalent expressions with negative numbershttps://cdn.kastatic.org/ka-youtube-converted/fa__960kzVE.mp4/fa__960kzVE.mp4381yes https://puzzicle.org/math/statistics-probability/v/reading-pie-graphs-circle-graphshttps://puzzicle.org/thumbs/380854780.jpgReading pie graphs (circle graphs)Reading Pie Graphs (Circle Graphs)https://cdn.kastatic.org/ka-youtube-converted/4JqH55rLGKY.mp4/4JqH55rLGKY.mp4189yes https://puzzicle.org/math/grade-8-math-tx/v/relationships-between-sets-of-rational-numbershttps://puzzicle.org/thumbs/xc012ff7e26a8d685.jpgRelationships between sets of rational numbersA conceptual overview of visually representing relationships between sets of rational numbers (rational numbers, integers, whole numbers, natural numbers).https://cdn.kastatic.org/ka-youtube-converted/j68fKea4Q0M.mp4/j68fKea4Q0M.mp4229yes https://puzzicle.org/math/grade-7-math-tx/v/graphing-two-step-equationshttps://puzzicle.org/thumbs/xe1adb5ebc167138a.jpgGraphing two-step equationsWhen we have an equation in one variable, we can represent the solution on a number line. Since a two-step linear equation in one variable has a single solution, let's graph it as a single point.https://cdn.kastatic.org/ka-youtube-converted/1JxTtn3WFnQ.mp4/1JxTtn3WFnQ.mp4163yes https://puzzicle.org/math/grade-7-math-tx/v/family-budget-calculatorhttps://puzzicle.org/thumbs/x600dfff250bd6e4d.jpgFamily budget calculatorUse a family budget estimator to determine the minimum household budget and number of hours of work needed for a family to meet its basic needs.https://cdn.kastatic.org/ka-youtube-converted/Ady9CqoCNQM.mp4/Ady9CqoCNQM.mp4308yes https://puzzicle.org/math/grade-7-math-tx/v/finding-triangle-angles-with-algebrahttps://puzzicle.org/thumbs/xdeb1df5d76ec4241.jpgFinding triangle angles with algebraSolve for the interior angle measures of triangles given linear expressions for each angle measure.https://cdn.kastatic.org/ka-youtube-converted/wQik4r4_sV0.mp4/wQik4r4_sV0.mp4440yes https://puzzicle.org/math/grade-7-math-tx/v/identifying-equations-of-proportional-graphshttps://puzzicle.org/thumbs/x91bbde38b21cb769.jpgIdentifying equations of proportional graphsWe can write an equation for a linear graph that represents a proportional relationship in the form y = kx, where k is the constant of proportionality. Let's make a table of x- and y-coordinates from the graph to find the constant.https://cdn.kastatic.org/ka-youtube-converted/hUASd-tVWo4.mp4/hUASd-tVWo4.mp4174yes https://puzzicle.org/math/grade-7-math-tx/v/identifying-equations-of-linear-graphshttps://puzzicle.org/thumbs/x1ae2bd6c1011263c.jpgIdentifying equations of linear graphsVerify which equation represents a non-proportional linear graph by testing x-values in the equations. Every point on the graph of a line has an x- and y-coordinate. For an equation to represent the line, then the equation must be true when we substitute in each coordinate pair.https://cdn.kastatic.org/ka-youtube-converted/t9M3Wv0GDwQ.mp4/t9M3Wv0GDwQ.mp4187yes https://puzzicle.org/math/grade-7-math-tx/v/modeling-with-linear-tableshttps://puzzicle.org/thumbs/xf219cd0ccf69b4b7.jpgModeling with linear tablesGiven a context with an initial starting value and a constant rate of change, complete a table of values for the independent and dependent variables. Using an equation helps us to jump to particular values without needing to create a table for every possible x-value.https://cdn.kastatic.org/ka-youtube-converted/e-Cqegj3ya4.mp4/e-Cqegj3ya4.mp4348yes https://puzzicle.org/math/grade-7-math-tx/v/converting-between-measurement-systemshttps://puzzicle.org/thumbs/xb64cbcb776c444f9.jpgConverting between measurement systemsCreate and reason about a proportion to convert a mural measurement from US units to metric units.https://cdn.kastatic.org/ka-youtube-converted/TeOESA4r2TQ.mp4/TeOESA4r2TQ.mp4142yes https://puzzicle.org/math/grade-8-fl-best/v/orders-of-magnitude-exercise-example-2https://puzzicle.org/thumbs/xe58a764b.jpgOrders of magnitude exercise example 2Orders of magnitude exercise example 2 from 8th grade math (FL B.E.S.T.)https://cdn.kastatic.org/ka-youtube-converted/DaoJmvqU3FI.mp4/DaoJmvqU3FI.mp4348yes https://puzzicle.org/math/grade-8-math-tx/v/ordering-rational-numbers-in-contexthttps://puzzicle.org/thumbs/x0ebf379bf4d66f44.jpgOrdering rational numbers in contextPractice ordering rational numbers in a context.https://cdn.kastatic.org/ka-youtube-converted/T1mc4-_l5Ss.mp4/T1mc4-_l5Ss.mp4124yes https://puzzicle.org/math/grade-8-math-tx/v/ordering-rational-numbers-compared-to-an-averagehttps://puzzicle.org/thumbs/x01c541b2b45abaec.jpgOrdering rational numbers compared to an averageGiven a context with a set of rational numbers in a variety of forms, practice identifying which value represents some characteristic of the set.https://cdn.kastatic.org/ka-youtube-converted/jZd_FQdTRMY.mp4/jZd_FQdTRMY.mp4183yes https://puzzicle.org/math/grade-8-math-tx/v/simulating-samples-from-populations-example-1https://puzzicle.org/thumbs/x74a809d704f3ca4f.jpgSimulating samples from populations example 1Practice analyzing samples from populations to see if they are representative of their given population, and why those samples are or are not representative.https://cdn.kastatic.org/ka-youtube-converted/4-7u22MrnSs.mp4/4-7u22MrnSs.mp4141yes https://puzzicle.org/math/grade-8-math-tx/v/simulating-samples-from-populations-example-2https://puzzicle.org/thumbs/x373aecf3ff58141c.jpgSimulating samples from populations example 2Practice analyzing samples from populations to see if they are representative of their given population, and why those samples are or are not representative.https://cdn.kastatic.org/ka-youtube-converted/7Tio-lJ4kJ4.mp4/7Tio-lJ4kJ4.mp4124yes https://puzzicle.org/math/grade-8-math-tx/v/writing-equations-to-represent-geometric-problemshttps://puzzicle.org/thumbs/x028f7ed949c7052f.jpgWriting equations to represent geometric problemsPractice writing a one-variable equation with variables on both sides that can be used to represent a geometric problem.https://cdn.kastatic.org/ka-youtube-converted/3Xz8IlUgR8o.mp4/3Xz8IlUgR8o.mp4113yes https://puzzicle.org/math/grade-8-math-tx/v/writing-equations-to-represent-problemshttps://puzzicle.org/thumbs/x8968925936471856.jpgWriting equations to represent problemsExplore writing equations to represent real-world problems involving coral samples.https://cdn.kastatic.org/ka-youtube-converted/01I9vY0xOII.mp4/01I9vY0xOII.mp4118yes https://puzzicle.org/math/grade-8-math-tx/v/writing-inequalities-to-represent-problemshttps://puzzicle.org/thumbs/x7c79e16f628ef7fc.jpgWriting inequalities to represent problemsPractice writing a one-variable inequality with variables on both sides that can be used to represent a problem.https://cdn.kastatic.org/ka-youtube-converted/t4QhKsW_yCU.mp4/t4QhKsW_yCU.mp4163yes https://puzzicle.org/math/grade-8-math-tx/v/writing-inequalities-to-represent-real-world-problemshttps://puzzicle.org/thumbs/x89fcd04be79448d0.jpgWriting inequalities to represent real-world problemsPractice identifying which one-variable inequality with variables on both sides can be used to represent a problem.https://cdn.kastatic.org/ka-youtube-converted/J0tRjEi4pBw.mp4/J0tRjEi4pBw.mp4169yes https://puzzicle.org/math/grade-8-math-tx/v/using-similar-triangles-to-reason-about-slopehttps://puzzicle.org/thumbs/xfa0c7d9ccf985ed1.jpgUsing similar triangles to reason about slopeA conceptual overview video showing that the slope is the same between any two points on the same line.https://cdn.kastatic.org/ka-youtube-converted/JQgmzU23w20.mp4/JQgmzU23w20.mp4418yes https://puzzicle.org/math/grade-8-math-tx/v/similar-triangles-and-slope-proportion-of-segmentshttps://puzzicle.org/thumbs/x7dfa1cfdd60662b5.jpgSimilar triangles & slope: proportion of segmentsStudents use similar triangles to show that the slope is the same between any two points on a line in the coordinate plane. This worked example shows how to write the proportion with the letters for the sides.https://cdn.kastatic.org/ka-youtube-converted/u-r4MwyxdZ8.mp4/u-r4MwyxdZ8.mp4161yes https://puzzicle.org/math/grade-8-math-tx/v/similar-triangles-and-slope-proportion-using-coordinateshttps://puzzicle.org/thumbs/x93c70737d6593005.jpgSimilar triangles & slope: proportion using coordinatesStudents use similar triangles to show that the slope is the same between any two points on a line in the coordinate plane. This worked example shows how to write the proportion using the coordinates and the slope formula.https://cdn.kastatic.org/ka-youtube-converted/3TrT1vIR4to.mp4/3TrT1vIR4to.mp4174yes https://puzzicle.org/math/grade-8-math-tx/v/locating-y-intercepts-from-graphshttps://puzzicle.org/thumbs/xd5a902fd65cdbfb8.jpgLocating y-intercepts from graphsLocate the y-intercept from a graph of a line.https://cdn.kastatic.org/ka-youtube-converted/Hay980dASAk.mp4/Hay980dASAk.mp4135yes https://puzzicle.org/math/grade-8-math-tx/v/relating-unit-rate-to-slope-in-graphs-of-proportional-relationshipshttps://puzzicle.org/thumbs/xf68f8b44ef2fe784.jpgRelating unit rate to slope in graphs of proportional relationshipsCalculate a unit rate and choose which graph has a slope that equals the unit rate.https://cdn.kastatic.org/ka-youtube-converted/LCsMxvfeYd4.mp4/LCsMxvfeYd4.mp4159yes https://puzzicle.org/math/grade-8-math-tx/v/writing-y-mx-proportional-equations-worked-example-1https://puzzicle.org/thumbs/x906563924c1370cf.jpgWriting y = mx proportional equations worked example 1Practice writing equations in the form of y = mx to describe proportional relationships.https://cdn.kastatic.org/ka-youtube-converted/6h2co0jvTBM.mp4/6h2co0jvTBM.mp4150yes https://puzzicle.org/math/grade-8-math-tx/v/writing-y-mx-proportional-equations-worked-example-2https://puzzicle.org/thumbs/x4e7ce6d60d149653.jpgWriting y = mx proportional equations worked example 2Practice writing equations in the form of y = mx to describe proportional relationships.https://cdn.kastatic.org/ka-youtube-converted/XkMA_A7Q94Q.mp4/XkMA_A7Q94Q.mp4106yes https://puzzicle.org/math/grade-8-math-tx/v/locating-less-obvious-y-intercepts-on-graphshttps://puzzicle.org/thumbs/xd05fdd36126fe63d.jpgLocating less obvious y-intercepts on graphsGiven a graph of a linear relationship with two points labeled, locate the non-integer y-intercept.https://cdn.kastatic.org/ka-youtube-converted/j7igNR0hRkQ.mp4/j7igNR0hRkQ.mp4249yes https://puzzicle.org/math/grade-8-math-tx/v/identifying-proportional-non-proportional-functionshttps://puzzicle.org/thumbs/xe81cb8ec6d9fddbc.jpgIdentifying proportional & non-proportional functionsPractice identifying examples of proportional and non-proportional functions that arise from mathematical and real-world problems.https://cdn.kastatic.org/ka-youtube-converted/gnXCTo6lAnA.mp4/gnXCTo6lAnA.mp4205yes https://puzzicle.org/math/grade-8-math-tx/v/identifying-and-verifying-a-solution-to-a-systemhttps://puzzicle.org/thumbs/x992e8eefb1dce115.jpgIdentifying and verifying a solution to a systemGiven a graph of a system of two linear equations, identify and verify the values of x and y that simultaneously satisfy the two linear equations.https://cdn.kastatic.org/ka-youtube-converted/PQv5yEwXpeg.mp4/PQv5yEwXpeg.mp4112yes https://puzzicle.org/math/grade-8-math-tx/v/visually-verifying-a-solution-to-a-systemhttps://puzzicle.org/thumbs/x151f92318581943e.jpgVisually verifying a solution to a systemGiven a graph of a system of two linear equations, verify if a given value of x is a solution to the system.https://cdn.kastatic.org/ka-youtube-converted/Oehhq7kHOM4.mp4/Oehhq7kHOM4.mp475yes https://puzzicle.org/math/grade-8-math-tx/v/equivalent-ratios-in-similar-shapes2https://puzzicle.org/thumbs/xef175796cca0d417.jpgEquivalent ratios in similar shapesRelate side lengths within and between similar polygons and dilated shapes using proportion equations.https://cdn.kastatic.org/ka-youtube-converted/OQlMQ0e4AOs.mp4/OQlMQ0e4AOs.mp4220yes https://puzzicle.org/math/grade-8-math-tx/v/finding-equivalent-ratios-in-similar-triangleshttps://puzzicle.org/thumbs/xc83a0fd1174eca5c.jpgFinding equivalent ratios in similar trianglesPractice finding equivalent ratios of side lengths in similar shapes.https://cdn.kastatic.org/ka-youtube-converted/tgkfytxQjc0.mp4/tgkfytxQjc0.mp4182yes https://puzzicle.org/math/grade-8-math-tx/v/finding-equivalent-ratios-in-similar-quadrilateralshttps://puzzicle.org/thumbs/x4158161519d1bc16.jpgFinding equivalent ratios in similar quadrilateralsPractice finding equivalent ratios of side lengths in similar shapes.https://cdn.kastatic.org/ka-youtube-converted/IqjudxRCT40.mp4/IqjudxRCT40.mp4223yes https://puzzicle.org/math/grade-8-math-tx/v/lateral-total-surface-area-of-rectangular-prismshttps://puzzicle.org/thumbs/x6718d0fb086ea64a.jpgLateral & total surface area of rectangular prismsPractice finding the lateral surface area and total surface area of a rectangular prism using the formulas instead of nets.https://cdn.kastatic.org/ka-youtube-converted/EeaSMjBKRcA.mp4/EeaSMjBKRcA.mp4295yes https://puzzicle.org/math/grade-8-math-tx/v/lateral-total-surface-area-of-triangular-prismshttps://puzzicle.org/thumbs/xae3f73290ead4748.jpgLateral & total surface area of triangular prismsPractice finding the lateral surface area and total surface area of a triangular prism using the formulas instead of nets.https://cdn.kastatic.org/ka-youtube-converted/zFPCdP_zaa8.mp4/zFPCdP_zaa8.mp4404yes https://puzzicle.org/math/grade-8-math-tx/v/surface-area-of-cylinders-formulahttps://puzzicle.org/thumbs/xe3a94a65d9f316ae.jpgSurface area of cylinders formulaUse previous knowledge of surface area to make connections to the formulas for lateral and total surface area of cylinders.https://cdn.kastatic.org/ka-youtube-converted/GOLJkXcrZ2c.mp4/GOLJkXcrZ2c.mp4240yes https://puzzicle.org/math/grade-8-math-tx/v/lateral-total-surface-area-of-cylindershttps://puzzicle.org/thumbs/x01c88af0d33a7057.jpgLateral & total surface area of cylindersUse the formulas to find lateral and total surface area of a cylinder.https://cdn.kastatic.org/ka-youtube-converted/QJ69ms9xLU8.mp4/QJ69ms9xLU8.mp4246yes https://puzzicle.org/math/linear-algebra/v/vector-introduction-linear-algebrahttps://puzzicle.org/thumbs/xbb63d308.jpgVector intro for linear algebraA vector has both magnitude and direction. We use vectors to, for example, describe the velocity of moving objects. In this video, you'll learn how to write and draw vectors.https://cdn.kastatic.org/ka-youtube-converted/br7tS1t2SFE.mp4/br7tS1t2SFE.mp4349yes https://puzzicle.org/math/linear-algebra/v/real-coordinate-spaceshttps://puzzicle.org/thumbs/x184489c0.jpgReal coordinate spacesReal coordinate spaces from Linear algebrahttps://cdn.kastatic.org/ka-youtube-converted/lCsjJbZHhHU.mp4/lCsjJbZHhHU.mp4381yes https://puzzicle.org/math/linear-algebra/v/multiplying-vector-by-scalarhttps://puzzicle.org/thumbs/xf85df5ef.jpgMultiplying a vector by a scalarWatch Sal change the magnitude of a vector by multiplying it by a scalar.https://cdn.kastatic.org/ka-youtube-converted/ZN7YaSbY3-w.mp4/ZN7YaSbY3-w.mp4344yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-vector-exampleshttps://puzzicle.org/thumbs/25221.jpgVector examplesVisually understanding basic vector operationshttps://cdn.kastatic.org/ka-youtube-converted/r4bH66vYjss.mp4/r4bH66vYjss.mp41533yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-parametric-representations-of-lineshttps://puzzicle.org/thumbs/25223.jpgParametric representations of linesParametric Representations of Lines in R2 and R3https://cdn.kastatic.org/ka-youtube-converted/hWhs2cIj7Cw.mp4/hWhs2cIj7Cw.mp41486yes https://puzzicle.org/math/linear-algebra/v/linear-combinations-and-spanhttps://puzzicle.org/thumbs/25225.jpgLinear combinations and spanUnderstanding linear combinations and spans of vectorshttps://cdn.kastatic.org/ka-youtube-converted/Qm_OS-8COwU.mp4/Qm_OS-8COwU.mp41235yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-introduction-to-linear-independencehttps://puzzicle.org/thumbs/25227.jpgIntroduction to linear independenceIntroduction to linear dependence and independencehttps://cdn.kastatic.org/ka-youtube-converted/CrV1xCWdY-g.mp4/CrV1xCWdY-g.mp4946yes https://puzzicle.org/math/linear-algebra/v/more-on-linear-independencehttps://puzzicle.org/thumbs/25229.jpgMore on linear independenceMore examples determining linear dependence or independence.https://cdn.kastatic.org/ka-youtube-converted/Alhcv5d_XOs.mp4/Alhcv5d_XOs.mp41058yes https://puzzicle.org/math/linear-algebra/v/span-and-linear-independence-examplehttps://puzzicle.org/thumbs/25231.jpgSpan and linear independence exampleDetermining whether 3 vectors are linearly independent and/or span R3https://cdn.kastatic.org/ka-youtube-converted/9kW6zFK5E5c.mp4/9kW6zFK5E5c.mp41013yes https://puzzicle.org/math/linear-algebra/v/linear-subspaceshttps://puzzicle.org/thumbs/25233.jpgLinear subspacesIntroduction to linear subspaces of Rnhttps://cdn.kastatic.org/ka-youtube-converted/pMFv6liWK4M.mp4/pMFv6liWK4M.mp41409yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-basis-of-a-subspacehttps://puzzicle.org/thumbs/25235.jpgBasis of a subspaceUnderstanding the definition of a basis of a subspacehttps://cdn.kastatic.org/ka-youtube-converted/zntNi3-ybfQ.mp4/zntNi3-ybfQ.mp41140yes https://puzzicle.org/math/linear-algebra/v/vector-dot-product-and-vector-lengthhttps://puzzicle.org/thumbs/25237.jpgVector dot product and vector lengthDefinitions of the vector dot product and vector lengthhttps://cdn.kastatic.org/ka-youtube-converted/WNuIhXo39_k.mp4/WNuIhXo39_k.mp4550yes https://puzzicle.org/math/linear-algebra/v/proving-vector-dot-product-propertieshttps://puzzicle.org/thumbs/25239.jpgProving vector dot product propertiesProving the "associative", "distributive" and "commutative" properties for vector dot products.https://cdn.kastatic.org/ka-youtube-converted/rVQ3G9epCjw.mp4/rVQ3G9epCjw.mp4645yes https://puzzicle.org/math/linear-algebra/v/proof-of-the-cauchy-schwarz-inequalityhttps://puzzicle.org/thumbs/25241.jpgProof of the Cauchy-Schwarz inequalityProof of the Cauchy-Schwarz Inequalityhttps://cdn.kastatic.org/ka-youtube-converted/r2PogGDl8_U.mp4/r2PogGDl8_U.mp41015yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-vector-triangle-inequalityhttps://puzzicle.org/thumbs/25243.jpgVector triangle inequalityProving the triangle inequality for vectors in Rnhttps://cdn.kastatic.org/ka-youtube-converted/PsNidCBr5II.mp4/PsNidCBr5II.mp41133yes https://puzzicle.org/math/linear-algebra/v/defining-the-angle-between-vectorshttps://puzzicle.org/thumbs/25245.jpgDefining the angle between vectorsIntroducing the idea of an angle between two vectorshttps://cdn.kastatic.org/ka-youtube-converted/5AWob_z74Ks.mp4/5AWob_z74Ks.mp41511yes https://puzzicle.org/math/linear-algebra/v/defining-a-plane-in-r3-with-a-point-and-normal-vectorhttps://puzzicle.org/thumbs/25247.jpgDefining a plane in R3 with a point and normal vectorDetermining the equation for a plane in R3 using a point on the plane and a normal vectorhttps://cdn.kastatic.org/ka-youtube-converted/UJxgcVaNTqY.mp4/UJxgcVaNTqY.mp4833yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-cross-product-introductionhttps://puzzicle.org/thumbs/25249.jpgCross product introductionIntroduction to the cross producthttps://cdn.kastatic.org/ka-youtube-converted/pJzmiywagfY.mp4/pJzmiywagfY.mp4947yes https://puzzicle.org/math/linear-algebra/v/proof-relationship-between-cross-product-and-sin-of-anglehttps://puzzicle.org/thumbs/25251.jpgProof: Relationship between cross product and sin of angleProof: Relationship between the cross product and sin of angle between vectorshttps://cdn.kastatic.org/ka-youtube-converted/7MKA2QlKvHc.mp4/7MKA2QlKvHc.mp41089yes https://puzzicle.org/math/linear-algebra/v/dot-and-cross-product-comparison-intuitionhttps://puzzicle.org/thumbs/25253.jpgDot and cross product comparison/intuitionDot and Cross Product Comparison/Intuitionhttps://cdn.kastatic.org/ka-youtube-converted/tdwFdzVqito.mp4/tdwFdzVqito.mp41154yes https://puzzicle.org/math/linear-algebra/v/vector-triple-product-expansion-very-optionalhttps://puzzicle.org/thumbs/8187461.jpgVector triple product expansion (very optional)A shortcut for having to evaluate the cross product of three vectorshttps://cdn.kastatic.org/ka-youtube-converted/b7JTVLc_aMk.mp4/b7JTVLc_aMk.mp4865yes https://puzzicle.org/math/linear-algebra/v/normal-vector-from-plane-equationhttps://puzzicle.org/thumbs/8187465.jpgNormal vector from plane equationFiguring out a normal vector to a plane from its equationhttps://cdn.kastatic.org/ka-youtube-converted/gw-4wltP5tY.mp4/gw-4wltP5tY.mp4598yes https://puzzicle.org/math/linear-algebra/v/point-distance-to-planehttps://puzzicle.org/thumbs/8187466.jpgPoint distance to planeDistance between a point and a plane in three dimensionshttps://cdn.kastatic.org/ka-youtube-converted/7rIFO8hct9g.mp4/7rIFO8hct9g.mp4732yes https://puzzicle.org/math/linear-algebra/v/distance-between-planeshttps://puzzicle.org/thumbs/8187467.jpgDistance between planes2010 IIT JEE Paper 1 Problem 51 Distance Between Planeshttps://cdn.kastatic.org/ka-youtube-converted/v8plb6V8BQo.mp4/v8plb6V8BQo.mp4885yes https://puzzicle.org/math/linear-algebra/v/matrices-reduced-row-echelon-form-1https://puzzicle.org/thumbs/25255.jpgSolving a system of 3 equations and 4 variables using matrix row-echelon formSal solves a linear system with 3 equations and 4 variables by representing it with an augmented matrix and bringing the matrix to reduced row-echelon form.https://cdn.kastatic.org/ka-youtube-converted/L0CmbneYETs.mp4/L0CmbneYETs.mp41063yes https://puzzicle.org/math/linear-algebra/v/matrices-reduced-row-echelon-form-3https://puzzicle.org/thumbs/25259.jpgUsing matrix row-echelon form in order to show a linear system has no solutionsAnd another example of solving a system of linear equations by putting an augmented matrix into reduced row echelon formhttps://cdn.kastatic.org/ka-youtube-converted/JVDrlTdzxiI.mp4/JVDrlTdzxiI.mp4728yes https://puzzicle.org/math/linear-algebra/v/matrix-vector-productshttps://puzzicle.org/thumbs/25261.jpgMatrix vector productsDefining and understanding what it means to take the product of a matrix and a vectorhttps://cdn.kastatic.org/ka-youtube-converted/7Mo4S2wyMg4.mp4/7Mo4S2wyMg4.mp41270yes https://puzzicle.org/math/linear-algebra/v/introduction-to-the-null-space-of-a-matrixhttps://puzzicle.org/thumbs/25263.jpgIntroduction to the null space of a matrixShowing that the Null Space of a Matrix is a valid Subspacehttps://cdn.kastatic.org/ka-youtube-converted/jCwRV1QL_Xs.mp4/jCwRV1QL_Xs.mp4623yes https://puzzicle.org/math/linear-algebra/v/null-space-2-calculating-the-null-space-of-a-matrixhttps://puzzicle.org/thumbs/25265.jpgNull space 2: Calculating the null space of a matrixCalculating the null space of a matrixhttps://cdn.kastatic.org/ka-youtube-converted/qvyboGryeA8.mp4/qvyboGryeA8.mp4787yes https://puzzicle.org/math/linear-algebra/v/null-space-3-relation-to-linear-independencehttps://puzzicle.org/thumbs/25267.jpgNull space 3: Relation to linear independenceUnderstanding how the null space of a matrix relates to the linear independence of its column vectorshttps://cdn.kastatic.org/ka-youtube-converted/-fKh6SNEPr4.mp4/-fKh6SNEPr4.mp4572yes https://puzzicle.org/math/linear-algebra/v/column-space-of-a-matrixhttps://puzzicle.org/thumbs/25269.jpgColumn space of a matrixIntroduction to the column space of a matrixhttps://cdn.kastatic.org/ka-youtube-converted/st6D5OdFV9M.mp4/st6D5OdFV9M.mp4640yes https://puzzicle.org/math/linear-algebra/v/null-space-and-column-space-basishttps://puzzicle.org/thumbs/25271.jpgNull space and column space basisFiguring out the null space and a basis of a column space for a matrixhttps://cdn.kastatic.org/ka-youtube-converted/_uTAdf_AsfQ.mp4/_uTAdf_AsfQ.mp41513yes https://puzzicle.org/math/linear-algebra/v/visualizing-a-column-space-as-a-plane-in-r3https://puzzicle.org/thumbs/25273.jpgVisualizing a column space as a plane in R3Determining the planar equation for a column space in R3https://cdn.kastatic.org/ka-youtube-converted/EGNlXtjYABw.mp4/EGNlXtjYABw.mp41271yes https://puzzicle.org/math/linear-algebra/v/proof-any-subspace-basis-has-same-number-of-elementshttps://puzzicle.org/thumbs/25275.jpgProof: Any subspace basis has same number of elementsProof: Any subspace basis has same number of elementshttps://cdn.kastatic.org/ka-youtube-converted/Zn2K8UIT8r4.mp4/Zn2K8UIT8r4.mp41295yes https://puzzicle.org/math/linear-algebra/v/dimension-of-the-null-space-or-nullityhttps://puzzicle.org/thumbs/25277.jpgDimension of the null space or nullityDimension of the Null Space or Nullityhttps://cdn.kastatic.org/ka-youtube-converted/abYAUqs_n6I.mp4/abYAUqs_n6I.mp4839yes https://puzzicle.org/math/linear-algebra/v/dimension-of-the-column-space-or-rankhttps://puzzicle.org/thumbs/25279.jpgDimension of the column space or rankDimension of the Column Space or Rankhttps://cdn.kastatic.org/ka-youtube-converted/JUgrBkPteTg.mp4/JUgrBkPteTg.mp4768yes https://puzzicle.org/math/linear-algebra/v/showing-relation-between-basis-cols-and-pivot-colshttps://puzzicle.org/thumbs/25281.jpgShowing relation between basis cols and pivot colsShowing that linear independence of pivot columns implies linear independence of the corresponding columns in the original equationhttps://cdn.kastatic.org/ka-youtube-converted/BfVjTOjvI30.mp4/BfVjTOjvI30.mp4513yes https://puzzicle.org/math/linear-algebra/v/showing-that-the-candidate-basis-does-span-c-ahttps://puzzicle.org/thumbs/25283.jpgShowing that the candidate basis does span C(A)Showing that just the columns of A associated with the pivot columns of rref(A) do indeed span C(A).https://cdn.kastatic.org/ka-youtube-converted/CkQOCnLWPUA.mp4/CkQOCnLWPUA.mp4820yes https://puzzicle.org/math/linear-algebra/v/a-more-formal-understanding-of-functionshttps://puzzicle.org/thumbs/25285.jpgA more formal understanding of functionsA more formal understanding of functionshttps://cdn.kastatic.org/ka-youtube-converted/BQMyeQOLvpg.mp4/BQMyeQOLvpg.mp4961yes https://puzzicle.org/math/linear-algebra/v/vector-transformationshttps://puzzicle.org/thumbs/25287.jpgVector transformationsIntroduction to the notion of vector transformationshttps://cdn.kastatic.org/ka-youtube-converted/gAbadNuQEjI.mp4/gAbadNuQEjI.mp4859yes https://puzzicle.org/math/linear-algebra/v/linear-transformationshttps://puzzicle.org/thumbs/25289.jpgLinear transformationsIntroduction to linear transformationshttps://cdn.kastatic.org/ka-youtube-converted/4PCktDZJH8E.mp4/4PCktDZJH8E.mp4832yes https://puzzicle.org/math/linear-algebra/v/linear-transformations-as-matrix-vector-productshttps://puzzicle.org/thumbs/25293.jpgLinear transformations as matrix vector productsShowing how ANY linear transformation can be represented as a matrix vector producthttps://cdn.kastatic.org/ka-youtube-converted/PErhLkQcpZ8.mp4/PErhLkQcpZ8.mp41052yes https://puzzicle.org/math/linear-algebra/v/image-of-a-subset-under-a-transformationhttps://puzzicle.org/thumbs/25295.jpgImage of a subset under a transformationExploring what happens to a subset of the domain under a transformationhttps://cdn.kastatic.org/ka-youtube-converted/MIAmN5kgp3k.mp4/MIAmN5kgp3k.mp41091yes https://puzzicle.org/math/linear-algebra/v/im-t-image-of-a-transformationhttps://puzzicle.org/thumbs/25297.jpgim(T): Image of a transformationShowing that the image of a subspace under a transformation is also a subspace. Definition of the image of a Transformation.https://cdn.kastatic.org/ka-youtube-converted/hZ827mfh1Jo.mp4/hZ827mfh1Jo.mp4997yes https://puzzicle.org/math/linear-algebra/v/preimage-of-a-sethttps://puzzicle.org/thumbs/25299.jpgPreimage of a setDefinition of preimage of a sethttps://cdn.kastatic.org/ka-youtube-converted/zVsGtU8lIWs.mp4/zVsGtU8lIWs.mp4323yes https://puzzicle.org/math/linear-algebra/v/preimage-and-kernel-examplehttps://puzzicle.org/thumbs/25301.jpgPreimage and kernel exampleExample involving the preimage of a set under a transformation. Definition of kernel of a transformation.https://cdn.kastatic.org/ka-youtube-converted/6yrPU8rYOhs.mp4/6yrPU8rYOhs.mp4923yes https://puzzicle.org/math/linear-algebra/v/sums-and-scalar-multiples-of-linear-transformationshttps://puzzicle.org/thumbs/25303.jpgSums and scalar multiples of linear transformationsSums and Scalar Multiples of Linear Transformations. Definitions of matrix addition and scalar multiplication.https://cdn.kastatic.org/ka-youtube-converted/wHuY97vss18.mp4/wHuY97vss18.mp4909yes https://puzzicle.org/math/linear-algebra/v/more-on-matrix-addition-and-scalar-multiplicationhttps://puzzicle.org/thumbs/25305.jpgMore on matrix addition and scalar multiplicationMore on Matrix Addition and Scalar Multiplicationhttps://cdn.kastatic.org/ka-youtube-converted/Z19-ObbmHJE.mp4/Z19-ObbmHJE.mp4641yes https://puzzicle.org/math/linear-algebra/v/linear-transformation-examples-scaling-and-reflectionshttps://puzzicle.org/thumbs/25307.jpgLinear transformation examples: Scaling and reflectionsCreating scaling and reflection transformation matrices (which are diagonal)https://cdn.kastatic.org/ka-youtube-converted/qkfODKmZ-x4.mp4/qkfODKmZ-x4.mp4913yes https://puzzicle.org/math/linear-algebra/v/linear-transformation-examples-rotations-in-r2https://puzzicle.org/thumbs/25309.jpgLinear transformation examples: Rotations in R2Linear Transformation Examples: Rotations in R2https://cdn.kastatic.org/ka-youtube-converted/lPWfIq5DzqI.mp4/lPWfIq5DzqI.mp41072yes https://puzzicle.org/math/linear-algebra/v/rotation-in-r3-around-the-x-axishttps://puzzicle.org/thumbs/25311.jpgRotation in R3 around the x-axisConstruction a rotation transformation in R3https://cdn.kastatic.org/ka-youtube-converted/gkyuLPzfDV0.mp4/gkyuLPzfDV0.mp4738yes https://puzzicle.org/math/linear-algebra/v/unit-vectorshttps://puzzicle.org/thumbs/25313.jpgUnit vectorsWhat unit vectors are and how to construct themhttps://cdn.kastatic.org/ka-youtube-converted/lQn7fksaDq0.mp4/lQn7fksaDq0.mp4419yes https://puzzicle.org/math/linear-algebra/v/introduction-to-projectionshttps://puzzicle.org/thumbs/25315.jpgIntroduction to projectionsDetermining the projection of a vector on s linehttps://cdn.kastatic.org/ka-youtube-converted/27vT-NWuw0M.mp4/27vT-NWuw0M.mp4877yes https://puzzicle.org/math/linear-algebra/v/expressing-a-projection-on-to-a-line-as-a-matrix-vector-prodhttps://puzzicle.org/thumbs/25317.jpgExpressing a projection on to a line as a matrix vector prodExpressing a Projection on to a line as a Matrix Vector prodhttps://cdn.kastatic.org/ka-youtube-converted/JK-8XNIoAkI.mp4/JK-8XNIoAkI.mp41000yes https://puzzicle.org/math/linear-algebra/v/compositions-of-linear-transformations-1https://puzzicle.org/thumbs/25319.jpgCompositions of linear transformations 1Introduction to compositions of Linear Transformationshttps://cdn.kastatic.org/ka-youtube-converted/f_DTiXZpb8M.mp4/f_DTiXZpb8M.mp4741yes https://puzzicle.org/math/linear-algebra/v/compositions-of-linear-transformations-2https://puzzicle.org/thumbs/25321.jpgCompositions of linear transformations 2Providing the motivation for definition of matrix productshttps://cdn.kastatic.org/ka-youtube-converted/BuqcKpe5ZQs.mp4/BuqcKpe5ZQs.mp4990yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-matrix-product-exampleshttps://puzzicle.org/thumbs/25323.jpgMatrix product examplesExample of taking the product of two matriceshttps://cdn.kastatic.org/ka-youtube-converted/x1z0hOyjapU.mp4/x1z0hOyjapU.mp41094yes https://puzzicle.org/math/linear-algebra/v/matrix-product-associativityhttps://puzzicle.org/thumbs/25325.jpgMatrix product associativityShowing that matrix products are associativehttps://cdn.kastatic.org/ka-youtube-converted/Hhc96U_HvQE.mp4/Hhc96U_HvQE.mp4719yes https://puzzicle.org/math/linear-algebra/v/distributive-property-of-matrix-productshttps://puzzicle.org/thumbs/25327.jpgDistributive property of matrix productsShowing that matrix products exhibit the distributive propertyhttps://cdn.kastatic.org/ka-youtube-converted/oMWTMj78cwc.mp4/oMWTMj78cwc.mp4592yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-introduction-to-the-inverse-of-a-functionhttps://puzzicle.org/thumbs/25329.jpgIntroduction to the inverse of a functionIntroduction to the inverse of a functionhttps://cdn.kastatic.org/ka-youtube-converted/-eAzhBZgq28.mp4/-eAzhBZgq28.mp41133yes https://puzzicle.org/math/linear-algebra/v/proof-invertibility-implies-a-unique-solution-to-f-x-yhttps://puzzicle.org/thumbs/25331.jpgProof: Invertibility implies a unique solution to f(x)=yProof: Invertibility implies a unique solution to f(x)=y for all y in co-domain of f.https://cdn.kastatic.org/ka-youtube-converted/7GEUgRcnfVE.mp4/7GEUgRcnfVE.mp41361yes https://puzzicle.org/math/linear-algebra/v/surjective-onto-and-injective-one-to-one-functionshttps://puzzicle.org/thumbs/25333.jpgSurjective (onto) and injective (one-to-one) functionsIntroduction to surjective and injective functionshttps://cdn.kastatic.org/ka-youtube-converted/xKNX8BUWR0g.mp4/xKNX8BUWR0g.mp4571yes https://puzzicle.org/math/linear-algebra/v/relating-invertibility-to-being-onto-and-one-to-onehttps://puzzicle.org/thumbs/25335.jpgRelating invertibility to being onto and one-to-oneRelating invertibility to being onto (surjective) and one-to-one (injective)https://cdn.kastatic.org/ka-youtube-converted/QIU1daMN8fw.mp4/QIU1daMN8fw.mp4391yes https://puzzicle.org/math/linear-algebra/v/determining-whether-a-transformation-is-ontohttps://puzzicle.org/thumbs/25337.jpgDetermining whether a transformation is ontoDetermining whether a transformation is ontohttps://cdn.kastatic.org/ka-youtube-converted/eR8vEdJTvd0.mp4/eR8vEdJTvd0.mp41551yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-exploring-the-solution-set-of-ax-bhttps://puzzicle.org/thumbs/25339.jpgExploring the solution set of Ax = bExploring the solution set of Ax=b (non homogeneous equations)https://cdn.kastatic.org/ka-youtube-converted/1PsNIzUJPkc.mp4/1PsNIzUJPkc.mp4994yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-matrix-condition-for-one-to-one-transhttps://puzzicle.org/thumbs/25341.jpgMatrix condition for one-to-one transformationShowing that the rank of the of an mxn transformation matrix has to be n for the transformation to be one-to-one (injective)https://cdn.kastatic.org/ka-youtube-converted/M3FuL9qKTBs.mp4/M3FuL9qKTBs.mp41199yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-simplifying-conditions-for-invertibilityhttps://puzzicle.org/thumbs/25343.jpgSimplifying conditions for invertibilityShowing that a transformation is invertible if and only if rref(A) is equal to the identity matrixhttps://cdn.kastatic.org/ka-youtube-converted/Yz2OosyMTmY.mp4/Yz2OosyMTmY.mp4397yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-showing-that-inverses-are-linearhttps://puzzicle.org/thumbs/25345.jpgShowing that inverses are linearShowing that inverse transformations are also linearhttps://cdn.kastatic.org/ka-youtube-converted/mr9Tow8hpCg.mp4/mr9Tow8hpCg.mp41285yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-deriving-a-method-for-determining-inverseshttps://puzzicle.org/thumbs/25347.jpgDeriving a method for determining inversesDetermining a method for constructing inverse transformation matriceshttps://cdn.kastatic.org/ka-youtube-converted/6DpzCKJBsz0.mp4/6DpzCKJBsz0.mp41080yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-example-of-finding-matrix-inversehttps://puzzicle.org/thumbs/25349.jpgExample of finding matrix inverseExample of calculating the inverse of a matrixhttps://cdn.kastatic.org/ka-youtube-converted/r9aTLTN16V4.mp4/r9aTLTN16V4.mp4382yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-formula-for-2x2-inversehttps://puzzicle.org/thumbs/25351.jpgFormula for 2x2 inverseFiguring out the formula for a 2x2 matrix. Defining the determinant.https://cdn.kastatic.org/ka-youtube-converted/eEUK_ThrHuQ.mp4/eEUK_ThrHuQ.mp41100yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-3x3-determinanthttps://puzzicle.org/thumbs/25353.jpg3 x 3 determinantDeterminants: Finding the determinant of a 3x3 matrixhttps://cdn.kastatic.org/ka-youtube-converted/0c7dt2SQfLw.mp4/0c7dt2SQfLw.mp4601yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-nxn-determinanthttps://puzzicle.org/thumbs/25355.jpgn x n determinantDefining the determinant for nxn matrices. An example of a 4x4 determinant.https://cdn.kastatic.org/ka-youtube-converted/H9BWRYJNIv4.mp4/H9BWRYJNIv4.mp41120yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-determinants-along-other-rows-colshttps://puzzicle.org/thumbs/25357.jpgDeterminants along other rows/colsFinding the determinant by going along other rows or columnshttps://cdn.kastatic.org/ka-youtube-converted/nu87kfmwNfU.mp4/nu87kfmwNfU.mp4543yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-rule-of-sarrus-of-determinantshttps://puzzicle.org/thumbs/25359.jpgRule of Sarrus of determinantsA alternative "short cut" for calculating 3x3 determinants (Rule of Sarrus)https://cdn.kastatic.org/ka-youtube-converted/4xFIi0JF2AM.mp4/4xFIi0JF2AM.mp4438yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-determinant-when-row-multiplied-by-scalarhttps://puzzicle.org/thumbs/25361.jpgDeterminant when row multiplied by scalarThe determinant when a row is multiplied by a scalarhttps://cdn.kastatic.org/ka-youtube-converted/32rdijPB-rA.mp4/32rdijPB-rA.mp4800yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-correction-scalar-muliplication-of-rowhttps://puzzicle.org/thumbs/25363.jpg(correction) scalar multiplication of rowCorrection of last video showing that the determinant when one row is multiplied by a scalar is equal to the scalar times the determinanthttps://cdn.kastatic.org/ka-youtube-converted/WY6QG0mZ7uQ.mp4/WY6QG0mZ7uQ.mp4172yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-determinant-when-row-is-addedhttps://puzzicle.org/thumbs/25365.jpgDeterminant when row is addedThe determinant when one matrix has a row that is the sum of the rows of other matrices (and every other term is identical in the 3 matrices)https://cdn.kastatic.org/ka-youtube-converted/VrB3LaSD_uo.mp4/VrB3LaSD_uo.mp41015yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-duplicate-row-determinanthttps://puzzicle.org/thumbs/25367.jpgDuplicate row determinantDeterminant of a matrix with duplicate rowshttps://cdn.kastatic.org/ka-youtube-converted/gYv8sttBIqs.mp4/gYv8sttBIqs.mp4499yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-determinant-after-row-operationshttps://puzzicle.org/thumbs/25369.jpgDeterminant after row operationsWhat happens to the determinant when we perform a row operationhttps://cdn.kastatic.org/ka-youtube-converted/kpG7xySkivg.mp4/kpG7xySkivg.mp4625yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-upper-triangular-determinanthttps://puzzicle.org/thumbs/25371.jpgUpper triangular determinantThe determinant of an upper triangular matrixhttps://cdn.kastatic.org/ka-youtube-converted/VX7K8iqoiRc.mp4/VX7K8iqoiRc.mp4487yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-simpler-4x4-determinanthttps://puzzicle.org/thumbs/25373.jpgSimpler 4x4 determinantCalculating a 4x4 determinant by putting in in upper triangular form first.https://cdn.kastatic.org/ka-youtube-converted/QV0jsTiobU4.mp4/QV0jsTiobU4.mp4553yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-determinant-and-area-of-a-parallelogramhttps://puzzicle.org/thumbs/25375.jpgDeterminant and area of a parallelogramRealizing that the determinant of a 2x2 matrix is equal to the area of the parallelogram defined by the column vectors of the matrixhttps://cdn.kastatic.org/ka-youtube-converted/n-S63_goDFg.mp4/n-S63_goDFg.mp41297yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-determinant-as-scaling-factorhttps://puzzicle.org/thumbs/25377.jpgDeterminant as scaling factorViewing the determinant of the transformation matrix as a scaling factor of regionshttps://cdn.kastatic.org/ka-youtube-converted/g3MqSdO5Ubc.mp4/g3MqSdO5Ubc.mp41209yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-transpose-of-a-matrixhttps://puzzicle.org/thumbs/25379.jpgTranspose of a matrixTranspose of a matrixhttps://cdn.kastatic.org/ka-youtube-converted/2t0003_sxtU.mp4/2t0003_sxtU.mp4517yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-determinant-of-transposehttps://puzzicle.org/thumbs/25381.jpgDeterminant of transposeProof by induction that transposing a matrix does not change its determinanthttps://cdn.kastatic.org/ka-youtube-converted/x-GV2v0dRNE.mp4/x-GV2v0dRNE.mp4850yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-transpose-of-a-matrix-producthttps://puzzicle.org/thumbs/178920409.jpgTranspose of a matrix productTaking the transpose of the product of two matriceshttps://cdn.kastatic.org/ka-youtube-converted/UdHgWK13x5I.mp4/UdHgWK13x5I.mp4530yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-transposes-of-sums-and-inverseshttps://puzzicle.org/thumbs/25383.jpgTransposes of sums and inversesTransposes of sums and inverseshttps://cdn.kastatic.org/ka-youtube-converted/Br_8t58mdeI.mp4/Br_8t58mdeI.mp4521yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-transpose-of-a-vectorhttps://puzzicle.org/thumbs/25385.jpgTranspose of a vectorTranspose of a column vector. Matrix-matrix products using vectorshttps://cdn.kastatic.org/ka-youtube-converted/lSoaMCNKfAg.mp4/lSoaMCNKfAg.mp4726yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-rowspace-and-left-nullspacehttps://puzzicle.org/thumbs/25387.jpgRowspace and left nullspaceRowspace and Left Nullspacehttps://cdn.kastatic.org/ka-youtube-converted/qBfc57x_RSg.mp4/qBfc57x_RSg.mp41398yes https://puzzicle.org/math/linear-algebra/v/lin-alg-visualizations-of-left-nullspace-and-rowspacehttps://puzzicle.org/thumbs/25389.jpgVisualizations of left nullspace and rowspaceRelationship between left nullspace, rowspace, column space and nullspace.https://cdn.kastatic.org/ka-youtube-converted/GW8ZPjGlk24.mp4/GW8ZPjGlk24.mp41190yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-rank-a-rank-transpose-of-ahttps://puzzicle.org/thumbs/25393.jpgrank(a) = rank(transpose of a)Rank(A) = Rank(transpose of A)https://cdn.kastatic.org/ka-youtube-converted/tqqVOWm9YsU.mp4/tqqVOWm9YsU.mp4673yes https://puzzicle.org/math/linear-algebra/v/lin-alg-showing-that-a-transpose-x-a-is-invertiblehttps://puzzicle.org/thumbs/25407.jpgShowing that A-transpose x A is invertibleShowing that (transpose of A)(A) is invertible if A has linearly independent columnshttps://cdn.kastatic.org/ka-youtube-converted/ESSMQH6Y5OA.mp4/ESSMQH6Y5OA.mp4754yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-orthogonal-complementshttps://puzzicle.org/thumbs/25391.jpgOrthogonal complementsOrthogonal Complements as subspaces.https://cdn.kastatic.org/ka-youtube-converted/QOTjdgmNqlg.mp4/QOTjdgmNqlg.mp41328yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-dim-v-dim-orthogonal-complement-of-v-nhttps://puzzicle.org/thumbs/25395.jpgdim(v) + dim(orthogonal complement of v) = nShowing that if V is a subspace of Rn, then dim(V) + dim(V's orthogonal complement) = nhttps://cdn.kastatic.org/ka-youtube-converted/pGhBfkBqok0.mp4/pGhBfkBqok0.mp4567yes https://puzzicle.org/math/linear-algebra/v/lin-alg-representing-vectors-in-rn-using-subspace-membershttps://puzzicle.org/thumbs/25397.jpgRepresenting vectors in rn using subspace membersShowing that any member of Rn can be represented as a unique sum of a vector in subspace V and a vector in the orthogonal complement of V.https://cdn.kastatic.org/ka-youtube-converted/M6lFqkqvUgo.mp4/M6lFqkqvUgo.mp41620yes https://puzzicle.org/math/linear-algebra/v/lin-alg-orthogonal-complement-of-the-orthogonal-complementhttps://puzzicle.org/thumbs/25399.jpgOrthogonal complement of the orthogonal complementFinding that the orthogonal complement of the orthogonal complement of V is Vhttps://cdn.kastatic.org/ka-youtube-converted/zlI8mx8Hc8o.mp4/zlI8mx8Hc8o.mp4738yes https://puzzicle.org/math/linear-algebra/v/lin-alg-orthogonal-complement-of-the-nullspacehttps://puzzicle.org/thumbs/25401.jpgOrthogonal complement of the nullspaceThe orthogonal complement of the nullspace and left nullspace.https://cdn.kastatic.org/ka-youtube-converted/LE_aQAur7nI.mp4/LE_aQAur7nI.mp4207yes https://puzzicle.org/math/linear-algebra/v/lin-alg-unique-rowspace-solution-to-ax-bhttps://puzzicle.org/thumbs/25403.jpgUnique rowspace solution to Ax = bShowing that, for any b that is in the column space of A, there is a unique member of the row space that is the "smallest" solution to Ax=bhttps://cdn.kastatic.org/ka-youtube-converted/czIYvs9zj2g.mp4/czIYvs9zj2g.mp41152yes https://puzzicle.org/math/linear-algebra/v/linear-alg-rowspace-solution-to-ax-b-examplehttps://puzzicle.org/thumbs/25405.jpgRowspace solution to Ax = b exampleVisualizing the rowspace solution to Ax=bhttps://cdn.kastatic.org/ka-youtube-converted/qdf2CuMGdKs.mp4/qdf2CuMGdKs.mp41178yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-projections-onto-subspaceshttps://puzzicle.org/thumbs/25409.jpgProjections onto subspacesProjections onto subspaceshttps://cdn.kastatic.org/ka-youtube-converted/5B8XluiqdHM.mp4/5B8XluiqdHM.mp41046yes https://puzzicle.org/math/linear-algebra/v/linear-alg-visualizing-a-projection-onto-a-planehttps://puzzicle.org/thumbs/25411.jpgVisualizing a projection onto a planeVisualizing a projection onto a plane. Showing that the old and new definitions of projections aren't that different.https://cdn.kastatic.org/ka-youtube-converted/uWbZlJURkfA.mp4/uWbZlJURkfA.mp4568yes https://puzzicle.org/math/linear-algebra/v/lin-alg-a-projection-onto-a-subspace-is-a-linear-transformahttps://puzzicle.org/thumbs/25413.jpgA projection onto a subspace is a linear transformationShowing that a projection onto a subspace is a linear transformationhttps://cdn.kastatic.org/ka-youtube-converted/cTyNpXB92bQ.mp4/cTyNpXB92bQ.mp4975yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-subspace-projection-matrix-examplehttps://puzzicle.org/thumbs/25415.jpgSubspace projection matrix exampleExample of a transformation matrix for a projection onto a subspacehttps://cdn.kastatic.org/ka-youtube-converted/QTcSBB3uVP0.mp4/QTcSBB3uVP0.mp4784yes https://puzzicle.org/math/linear-algebra/v/lin-alg-another-example-of-a-projection-matrixhttps://puzzicle.org/thumbs/25417.jpgAnother example of a projection matrixFiguring out the transformation matrix for a projection onto a subspace by figuring out the matrix for the projection onto the subspace's orthogonal complement firsthttps://cdn.kastatic.org/ka-youtube-converted/XhCSHu6dHb8.mp4/XhCSHu6dHb8.mp41295yes https://puzzicle.org/math/linear-algebra/v/linear-alg-projection-is-closest-vector-in-subspacehttps://puzzicle.org/thumbs/25419.jpgProjection is closest vector in subspaceShowing that the projection of x onto a subspace is the closest vector in the subspace to xhttps://cdn.kastatic.org/ka-youtube-converted/b269qpILOpk.mp4/b269qpILOpk.mp4545yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-least-squares-approximationhttps://puzzicle.org/thumbs/25421.jpgLeast squares approximationThe least squares approximation for otherwise unsolvable equationshttps://cdn.kastatic.org/ka-youtube-converted/MC7l96tW8V8.mp4/MC7l96tW8V8.mp4932yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-least-squares-exampleshttps://puzzicle.org/thumbs/25423.jpgLeast squares examplesAn example using the least squares solution to an unsolvable systemhttps://cdn.kastatic.org/ka-youtube-converted/8mAZYv5wIcE.mp4/8mAZYv5wIcE.mp41130yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-another-least-squares-examplehttps://puzzicle.org/thumbs/25425.jpgAnother least squares exampleUsing least squares approximation to fit a line to pointshttps://cdn.kastatic.org/ka-youtube-converted/QkepM8Vv3kw.mp4/QkepM8Vv3kw.mp4805yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-coordinates-with-respect-to-a-basishttps://puzzicle.org/thumbs/25427.jpgCoordinates with respect to a basisUnderstanding alternate coordinate systemshttps://cdn.kastatic.org/ka-youtube-converted/uvgru6FcyxU.mp4/uvgru6FcyxU.mp4968yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-change-of-basis-matrixhttps://puzzicle.org/thumbs/25429.jpgChange of basis matrixUsing a change of basis matrix to get us from one coordinate system to another.https://cdn.kastatic.org/ka-youtube-converted/1j5WnqwMdCk.mp4/1j5WnqwMdCk.mp41075yes https://puzzicle.org/math/linear-algebra/v/lin-alg-invertible-change-of-basis-matrixhttps://puzzicle.org/thumbs/25431.jpgInvertible change of basis matrixUsing an invertible change of basis matrix to go between different coordinate systemshttps://cdn.kastatic.org/ka-youtube-converted/meibWcbGqt4.mp4/meibWcbGqt4.mp4814yes https://puzzicle.org/math/linear-algebra/v/lin-alg-transformation-matrix-with-respect-to-a-basishttps://puzzicle.org/thumbs/25433.jpgTransformation matrix with respect to a basisFinding the transformation matrix with respect to a non-standard basishttps://cdn.kastatic.org/ka-youtube-converted/PiuhTj0zCf4.mp4/PiuhTj0zCf4.mp41082yes https://puzzicle.org/math/linear-algebra/v/lin-alg-alternate-basis-tranformation-matrix-examplehttps://puzzicle.org/thumbs/25435.jpgAlternate basis transformation matrix exampleExample of finding the transformation matrix for an alternate basishttps://cdn.kastatic.org/ka-youtube-converted/lCRGNykWqFI.mp4/lCRGNykWqFI.mp4800yes https://puzzicle.org/math/linear-algebra/v/lin-alg-alternate-basis-tranformation-matrix-example-part-2https://puzzicle.org/thumbs/25437.jpgAlternate basis transformation matrix example part 2Showing that the transformation matrix with respect to basis B actually works. Brief point on why someone would want to operate in a different basis to begin with.https://cdn.kastatic.org/ka-youtube-converted/EyORbCPUAd4.mp4/EyORbCPUAd4.mp4756yes https://puzzicle.org/math/linear-algebra/v/lin-alg-changing-coordinate-systems-to-help-find-a-transformation-matrixhttps://puzzicle.org/thumbs/25439.jpgChanging coordinate systems to help find a transformation matrixChanging our coordinate system to find the transformation matrix with respect to standard coordinateshttps://cdn.kastatic.org/ka-youtube-converted/K_na_j1Msfg.mp4/K_na_j1Msfg.mp41739yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-introduction-to-orthonormal-baseshttps://puzzicle.org/thumbs/25441.jpgIntroduction to orthonormal basesLooking at sets and bases that are orthonormal -- or where all the vectors have length 1 and are orthogonal to each other.https://cdn.kastatic.org/ka-youtube-converted/7BFx8pt2aTQ.mp4/7BFx8pt2aTQ.mp4676yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-coordinates-with-respect-to-orthonormal-baseshttps://puzzicle.org/thumbs/25443.jpgCoordinates with respect to orthonormal basesSeeing that orthonormal bases make for good coordinate systemshttps://cdn.kastatic.org/ka-youtube-converted/SWbis2zWIvo.mp4/SWbis2zWIvo.mp4928yes https://puzzicle.org/math/linear-algebra/v/lin-alg-projections-onto-subspaces-with-orthonormal-baseshttps://puzzicle.org/thumbs/25445.jpgProjections onto subspaces with orthonormal basesProjections onto subspaces with orthonormal baseshttps://cdn.kastatic.org/ka-youtube-converted/we829K4NOq8.mp4/we829K4NOq8.mp4974yes https://puzzicle.org/math/linear-algebra/v/lin-alg-finding-projection-onto-subspace-with-orthonormal-basis-examplehttps://puzzicle.org/thumbs/25447.jpgFinding projection onto subspace with orthonormal basis exampleExample of finding the transformation matrix for the projection onto a subspace with an orthonormal basishttps://cdn.kastatic.org/ka-youtube-converted/SUoqA9srY1Q.mp4/SUoqA9srY1Q.mp4402yes https://puzzicle.org/math/linear-algebra/v/lin-alg-example-using-orthogonal-change-of-basis-matrix-to-find-transformation-mhttps://puzzicle.org/thumbs/25449.jpgExample using orthogonal change-of-basis matrix to find transformation matrixExample using orthogonal change-of-basis matrix to find transformation matrixhttps://cdn.kastatic.org/ka-youtube-converted/gfVY7cW7VLQ.mp4/gfVY7cW7VLQ.mp41624yes https://puzzicle.org/math/linear-algebra/v/lin-alg-orthogonal-matrices-preserve-angles-and-lengthshttps://puzzicle.org/thumbs/25451.jpgOrthogonal matrices preserve angles and lengthsShowing that orthogonal matrices preserve angles and lengthshttps://cdn.kastatic.org/ka-youtube-converted/yDwIfYjKEeo.mp4/yDwIfYjKEeo.mp4676yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-the-gram-schmidt-processhttps://puzzicle.org/thumbs/25453.jpgThe Gram-Schmidt processFinding an orthonormal basis for a subspace using the Gram-Schmidt Processhttps://cdn.kastatic.org/ka-youtube-converted/rHonltF77zI.mp4/rHonltF77zI.mp41164yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-gram-schmidt-process-examplehttps://puzzicle.org/thumbs/25455.jpgGram-Schmidt process exampleUsing Gram-Schmidt to find an orthonormal basis for a plane in R3https://cdn.kastatic.org/ka-youtube-converted/tu1GPtfsQ7M.mp4/tu1GPtfsQ7M.mp4794yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-gram-schmidt-example-with-3-basis-vectorshttps://puzzicle.org/thumbs/25457.jpgGram-Schmidt example with 3 basis vectorsGram-Schmidt example with 3 basis vectorshttps://cdn.kastatic.org/ka-youtube-converted/ZRRG386v6DI.mp4/ZRRG386v6DI.mp4837yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-introduction-to-eigenvalues-and-eigenvectorshttps://puzzicle.org/thumbs/25459.jpgIntroduction to eigenvalues and eigenvectorsWhat eigenvectors and eigenvalues are and why they are interestinghttps://cdn.kastatic.org/ka-youtube-converted/PhfbEr2btGQ.mp4/PhfbEr2btGQ.mp4463yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-proof-of-formula-for-determining-eigenvalueshttps://puzzicle.org/thumbs/25461.jpgProof of formula for determining eigenvaluesProof of formula for determining Eigenvalueshttps://cdn.kastatic.org/ka-youtube-converted/rfm0wQObxjk.mp4/rfm0wQObxjk.mp4559yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-example-solving-for-the-eigenvalues-of-a-2x2-matrixhttps://puzzicle.org/thumbs/25463.jpgExample solving for the eigenvalues of a 2x2 matrixExample solving for the eigenvalues of a 2x2 matrixhttps://cdn.kastatic.org/ka-youtube-converted/pZ6mMVEE89g.mp4/pZ6mMVEE89g.mp4339yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-finding-eigenvectors-and-eigenspaces-examplehttps://puzzicle.org/thumbs/25465.jpgFinding eigenvectors and eigenspaces exampleFinding the eigenvectors and eigenspaces of a 2x2 matrixhttps://cdn.kastatic.org/ka-youtube-converted/3-xfmbdzkqc.mp4/3-xfmbdzkqc.mp4874yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-eigenvalues-of-a-3x3-matrixhttps://puzzicle.org/thumbs/25467.jpgEigenvalues of a 3x3 matrixDetermining the eigenvalues of a 3x3 matrixhttps://cdn.kastatic.org/ka-youtube-converted/11dNghWC4HI.mp4/11dNghWC4HI.mp4848yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-eigenvectors-and-eigenspaces-for-a-3x3-matrixhttps://puzzicle.org/thumbs/25469.jpgEigenvectors and eigenspaces for a 3x3 matrixEigenvectors and eigenspaces for a 3x3 matrixhttps://cdn.kastatic.org/ka-youtube-converted/3Md5KCCQX-0.mp4/3Md5KCCQX-0.mp4934yes https://puzzicle.org/math/linear-algebra/v/linear-algebra-showing-that-an-eigenbasis-makes-for-good-coordinate-systemshttps://puzzicle.org/thumbs/25471.jpgShowing that an eigenbasis makes for good coordinate systemsShowing that an eigenbasis makes for good coordinate systemshttps://cdn.kastatic.org/ka-youtube-converted/C2PC9185gIw.mp4/C2PC9185gIw.mp4789yes https://puzzicle.org/math/mappers/v/common-fractions-halves-math-4th-grade-khan-academyhttps://puzzicle.org/thumbs/x3da30edc.jpgCommon fractions (halves)Lindsay discovers why 0.5 is equal to 1/2.https://cdn.kastatic.org/ka-youtube-converted/gWz08PB7b6E.mp4/gWz08PB7b6E.mp4240yes https://puzzicle.org/math/mappers/v/comparing-negatives-with-variableshttps://puzzicle.org/thumbs/x6e48d9c5.jpgNegative numbers, variables, number lineGiven a, b, and c shown on the number line, Sal determines if statements like -b < c are true.https://cdn.kastatic.org/ka-youtube-converted/kQkeayQke8A.mp4/kQkeayQke8A.mp4294yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/creating-diagrams-using-toolshttps://puzzicle.org/thumbs/x921f6a03d5a2e393.jpgCreating diagrams using toolsPractice using the foundational straightedge-and-compass moves that later constructions depend on.https://cdn.kastatic.org/ka-youtube-converted/CjUjy2XOANM.mp4/CjUjy2XOANM.mp4291yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/geometric-constructions-perpendicular-line-through-a-point-on-the-line-indian-achttps://puzzicle.org/thumbs/x2f11fab57d5f35a5.jpgGeometric constructions: perpendicular line through a point on the lineSal constructs a line perpendicular to a given line through a point on the line using compass and straightedge. Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now! (add course link here) Timestamps: Khan Academy India is a nonprofit organization with the mission of providing a free, world-class education for anyone, anywhere. We have videos and exercises that have been translated into multiple Indian languages, and 15 million people around the globe learn on Khan Academy every month. Support Us: https://india.khanacademy.org/donate Created byhttps://cdn.kastatic.org/ka-youtube-converted/-k7gKNInMyM.mp4/-k7gKNInMyM.mp467yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/using-geometric-methods-to-solve-design-problemshttps://puzzicle.org/thumbs/xf455b2d5adac667a.jpgUsing geometric methods to solve design problemsUse perpendicular bisectors to create diagrams that solve real-world allocation problems.https://cdn.kastatic.org/ka-youtube-converted/a9LwIWniinw.mp4/a9LwIWniinw.mp4115yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/drawing-a-reflectionhttps://puzzicle.org/thumbs/x6acb123cd5f6990a.jpgDrawing a reflectionExplore reflections on a grid by identifying a line of reflection and drawing reflected images of figures across a given line.https://cdn.kastatic.org/ka-youtube-converted/h83TDyZdBLY.mp4/h83TDyZdBLY.mp4145yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/determining-a-reflectionhttps://puzzicle.org/thumbs/x89e9b3359f17780b.jpgDetermining a reflectionExplore reflections on a grid by identifying a line of reflection and determining the reflected images of figures across a given line.https://cdn.kastatic.org/ka-youtube-converted/LNFTDsX_8jE.mp4/LNFTDsX_8jE.mp4143yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/defining-translationshttps://puzzicle.org/thumbs/x654975c96eea89d6.jpgDefining translationsExplore translations on a grid by specifying a directed line segment and drawing translated images of figures.https://cdn.kastatic.org/ka-youtube-converted/EcwQ1-P0DY4.mp4/EcwQ1-P0DY4.mp4112yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/defining-rotationshttps://puzzicle.org/thumbs/xc3cefac9ffb3111d.jpgDefining rotationsAnalyze whether one figure is a rotation of another on a grid.https://cdn.kastatic.org/ka-youtube-converted/sgvD0NoB-jM.mp4/sgvD0NoB-jM.mp4189yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/corresponding-parts-of-congruent-triangles-are-congruent-indian-accenthttps://puzzicle.org/thumbs/xf53c199a55087caf.jpgCorresponding parts of congruent triangles are congruentWhen two triangles are congruent, we can know that all of their corresponding sides and angles are congruent too!https://cdn.kastatic.org/ka-youtube-converted/FpMUWRr4QCY.mp4/FpMUWRr4QCY.mp4363yes https://puzzicle.org/math/math-1-illustrative-math-aligned/v/the-perpendicular-bisector-theoremhttps://puzzicle.org/thumbs/x009012eb37e1dbdc.jpgThe perpendicular bisector theoremProve the Perpendicular Bisector Theorem and its converse.https://cdn.kastatic.org/ka-youtube-converted/PatjYSGbfh8.mp4/PatjYSGbfh8.mp4465yes https://puzzicle.org/math/statistics-probability/v/dependent-probability-example-1https://puzzicle.org/thumbs/94879656.jpgDependent probability: coinsWe're thinking about how the probability of an event can be dependent on another event occuring in this example problem.https://cdn.kastatic.org/ka-youtube-converted/xPUm5SUVzTE.mp4/xPUm5SUVzTE.mp4541yes https://puzzicle.org/math/math3/v/types-statistical-studieshttps://puzzicle.org/thumbs/xdf317e8d.jpgTypes of statistical studiesTypes of statistical studies from Integrated math 3https://cdn.kastatic.org/ka-youtube-converted/z-Qi4w6Xkuc.mp4/z-Qi4w6Xkuc.mp4592yes https://puzzicle.org/math/statistics-probability/v/analyzing-statistical-studyhttps://puzzicle.org/thumbs/x177c39f1.jpgAppropriate statistical study exampleSal determines if a statistical study was a sample study, an experiment, or an observational study.https://cdn.kastatic.org/ka-youtube-converted/RQPC_rUvHas.mp4/RQPC_rUvHas.mp4319yes https://puzzicle.org/math/multivariable-calculus/v/multivariable-functionshttps://puzzicle.org/thumbs/x4aa5be13.jpgMultivariable functionsAn introduction to multivariable functions, and a welcome to the multivariable calculus content as a whole.https://cdn.kastatic.org/ka-youtube-converted/TrcCbdWwCBc.mp4/TrcCbdWwCBc.mp4362yes https://puzzicle.org/math/multivariable-calculus/v/representing-points-in-3dhttps://puzzicle.org/thumbs/x28725fc4.jpgRepresenting points in 3dLearn how to represent and think about points and vectors in three-dimensional space.https://cdn.kastatic.org/ka-youtube-converted/iBgOoaeLUcM.mp4/iBgOoaeLUcM.mp4399yes https://puzzicle.org/math/multivariable-calculus/v/introduction-to-3d-graphshttps://puzzicle.org/thumbs/x4fa0a163.jpgIntroduction to 3d graphsThree-dimensional graphs are a way to represent multi-variable functions with two inputs and one output. They're visualized by plotting input-output pairs in 3D space, resulting in a surface. Other methods of visualizing multi-variable functions include contour maps, parametric functions, and vector fields.https://cdn.kastatic.org/ka-youtube-converted/2DRmfxkH_VI.mp4/2DRmfxkH_VI.mp4426yes https://puzzicle.org/math/multivariable-calculus/v/interpreting-graphs-with-sliceshttps://puzzicle.org/thumbs/xb967e756.jpgInterpreting graphs with slices3d graphs can be a lot to take in, but it helps to imagine slicing them with planes parallel to the x-axis or y-axis and relate them with two-dimensional graphs.https://cdn.kastatic.org/ka-youtube-converted/E9bAwP7iK80.mp4/E9bAwP7iK80.mp4338yes https://puzzicle.org/math/multivariable-calculus/v/contour-plotshttps://puzzicle.org/thumbs/x3392e50a.jpgContour plotsAn alternative method to representing multivariable functions with a two-dimensional input and a one-dimensional output, contour maps involve drawing purely in the input space.https://cdn.kastatic.org/ka-youtube-converted/WsZj5Rb6do8.mp4/WsZj5Rb6do8.mp4336yes https://puzzicle.org/math/multivariable-calculus/v/parametric-curveshttps://puzzicle.org/thumbs/x3b1d7741.jpgParametric curvesWhen a function has a one-dimensional input, but a multidimensional output, you can think of it as drawing a curve in space.https://cdn.kastatic.org/ka-youtube-converted/bb4bSCjlFAw.mp4/bb4bSCjlFAw.mp4351yes https://puzzicle.org/math/multivariable-calculus/v/parametric-surfaceshttps://puzzicle.org/thumbs/x826fbf0d.jpgParametric surfacesFunctions that have a two-dimensional input and a three-dimensional output can be thought of as drawing a surface in three-dimensional space. This is actually pretty cool.https://cdn.kastatic.org/ka-youtube-converted/345SnWfahhY.mp4/345SnWfahhY.mp4381yes https://puzzicle.org/math/multivariable-calculus/v/vector-fields-introductionhttps://puzzicle.org/thumbs/x8f81fd3c.jpgVector fields, introductionVector fields let you visualize a function with a two-dimensional input and a two-dimensional output. You end up with, well, a field of vectors sitting at various points in two-dimensional space.https://cdn.kastatic.org/ka-youtube-converted/5FWAVmwMXWg.mp4/5FWAVmwMXWg.mp4305yes https://puzzicle.org/math/multivariable-calculus/v/fluid-flow-and-vector-fieldshttps://puzzicle.org/thumbs/xd3b831f4.jpgFluid flow and vector fieldsA neat way to interpret a vector field is to imagine that it represents some kind of fluid flow.https://cdn.kastatic.org/ka-youtube-converted/VJ2ZDLQk3IQ.mp4/VJ2ZDLQk3IQ.mp4215yes https://puzzicle.org/math/multivariable-calculus/v/3d-vector-fields-introductionhttps://puzzicle.org/thumbs/xd536672f.jpg3d vector fields, introductionVector fields can also be three-dimensional, though this can be a bit trickier to visualize.https://cdn.kastatic.org/ka-youtube-converted/Go3UvzbsSGo.mp4/Go3UvzbsSGo.mp4310yes https://puzzicle.org/math/multivariable-calculus/v/3d-vector-field-examplehttps://puzzicle.org/thumbs/xa91184f2.jpg3d vector field exampleSee an example of how you can start to understand how the formula for a three-dimensional vector field relates to the way it looks.https://cdn.kastatic.org/ka-youtube-converted/ZiMYRBtDB2M.mp4/ZiMYRBtDB2M.mp4244yes https://puzzicle.org/math/multivariable-calculus/v/transformations-part-1https://puzzicle.org/thumbs/xeec906c0.jpgTransformations, part 1One fun way to think about functions is to imagine that they literally move the points from the input space over to the output space. See what this looks like with some one-dimensional examples.https://cdn.kastatic.org/ka-youtube-converted/roRR6A3TozM.mp4/roRR6A3TozM.mp4316yes https://puzzicle.org/math/multivariable-calculus/v/transformations-part-2https://puzzicle.org/thumbs/x57b38402.jpgTransformations, part 2More transformations, but this time with a function that maps two dimensions to two dimensions.https://cdn.kastatic.org/ka-youtube-converted/Rkm0dLTYUuw.mp4/Rkm0dLTYUuw.mp4278yes https://puzzicle.org/math/multivariable-calculus/v/transformations-part-3https://puzzicle.org/thumbs/x75ec9be3.jpgTransformations, part 3Learn how you can think about a parametric surface as a certain kind of transformation.https://cdn.kastatic.org/ka-youtube-converted/U2SQXHMqclc.mp4/U2SQXHMqclc.mp4209yes https://puzzicle.org/math/multivariable-calculus/v/partial-derivatives-introductionhttps://puzzicle.org/thumbs/xa59f6489.jpgPartial derivatives, introductionPartial derivatives tell you how a multivariable function changes as you tweak just one of the variables in its input.https://cdn.kastatic.org/ka-youtube-converted/AXqhWeUEtQU.mp4/AXqhWeUEtQU.mp4656yes https://puzzicle.org/math/multivariable-calculus/v/partial-derivatives-and-graphshttps://puzzicle.org/thumbs/xa41cd13a.jpgGraphical understanding of partial derivativesOne of the best ways to think about partial derivatives is by slicing the graph of a multivariable function.https://cdn.kastatic.org/ka-youtube-converted/dfvnCHqzK54.mp4/dfvnCHqzK54.mp4415yes https://puzzicle.org/math/multivariable-calculus/v/formal-definition-of-partial-derivativeshttps://puzzicle.org/thumbs/xba09aabe.jpgFormal definition of partial derivativesPartial derivatives are formally defined using a limit, much like ordinary derivatives.https://cdn.kastatic.org/ka-youtube-converted/kdMep5GUOBw.mp4/kdMep5GUOBw.mp4478yes https://puzzicle.org/math/multivariable-calculus/v/symmetry-of-second-partial-derivativeshttps://puzzicle.org/thumbs/x5637b539.jpgSymmetry of second partial derivativesThere are many ways to take a "second partial derivative", but some of them secretly turn out to be the same thing.https://cdn.kastatic.org/ka-youtube-converted/J08-L2buigM.mp4/J08-L2buigM.mp4422yes https://puzzicle.org/math/multivariable-calculus/v/gradienthttps://puzzicle.org/thumbs/xe92f7935.jpgGradientThe gradient captures all the partial derivative information of a scalar-valued multivariable function.https://cdn.kastatic.org/ka-youtube-converted/tIpKfDc295M.mp4/tIpKfDc295M.mp4331yes https://puzzicle.org/math/multivariable-calculus/v/gradient-and-graphshttps://puzzicle.org/thumbs/xdada919e.jpgGradient and graphsLearn how the gradient can be thought of as pointing in the "direction of steepest ascent". This is a rather important interpretation for the gradient.https://cdn.kastatic.org/ka-youtube-converted/_-02ze7tf08.mp4/_-02ze7tf08.mp4371yes https://puzzicle.org/math/multivariable-calculus/v/gradient-and-contour-mapshttps://puzzicle.org/thumbs/x2478347d.jpgGradient and contour mapsGradient vectors always point perpendicular to contour lines.https://cdn.kastatic.org/ka-youtube-converted/ZTbTYEMvo10.mp4/ZTbTYEMvo10.mp4377yes https://puzzicle.org/math/multivariable-calculus/v/directional-derivativehttps://puzzicle.org/thumbs/xaebedeec.jpgDirectional derivativeDirectional derivatives tell you how a multivariable function changes as you move along some vector in its input space.https://cdn.kastatic.org/ka-youtube-converted/N_ZRcLheNv0.mp4/N_ZRcLheNv0.mp4434yes https://puzzicle.org/math/multivariable-calculus/v/directional-derivative-formal-definitionhttps://puzzicle.org/thumbs/x9dd381ba.jpgDirectional derivative, formal definitionLearn the limit definition of a directional derivative. This helps to clarify what it is really doing.https://cdn.kastatic.org/ka-youtube-converted/4RBkIJPG6Yo.mp4/4RBkIJPG6Yo.mp4399yes https://puzzicle.org/math/multivariable-calculus/v/directional-derivatives-and-slopehttps://puzzicle.org/thumbs/x0ce88a21.jpgDirectional derivatives and slopeThe directional derivative can be used to compute the slope of a slice of a graph, but you must be careful to use a unit vector.https://cdn.kastatic.org/ka-youtube-converted/4tdyIGIEtNU.mp4/4tdyIGIEtNU.mp4530yes https://puzzicle.org/math/multivariable-calculus/v/why-the-gradient-is-the-direction-of-steepest-ascenthttps://puzzicle.org/thumbs/x6a180b72.jpgWhy the gradient is the direction of steepest ascentThe way we compute the gradient seems unrelated to its interpretation as the direction of steepest ascent. Here you can see how the two relate.https://cdn.kastatic.org/ka-youtube-converted/TEB2z7ZlRAw.mp4/TEB2z7ZlRAw.mp4632yes https://puzzicle.org/math/multivariable-calculus/v/differential-of-a-vector-valued-functionhttps://puzzicle.org/thumbs/24320.jpgDifferential of a vector valued functionUnderstanding the differential of a vector valued functionhttps://cdn.kastatic.org/ka-youtube-converted/FYMn61HLw1k.mp4/FYMn61HLw1k.mp4304yes https://puzzicle.org/math/multivariable-calculus/v/vector-valued-function-derivative-examplehttps://puzzicle.org/thumbs/24322.jpgVector valued function derivative exampleConcrete example of the derivative of a vector valued function to better understand what it meanshttps://cdn.kastatic.org/ka-youtube-converted/vcwvzUVLPw0.mp4/vcwvzUVLPw0.mp4765yes https://puzzicle.org/math/multivariable-calculus/v/multivariable-chain-rulehttps://puzzicle.org/thumbs/x36d3e7c5.jpgMultivariable chain ruleThis is the simplest case of taking the derivative of a composition involving multivariable functions.https://cdn.kastatic.org/ka-youtube-converted/NO3AqAaAE6o.mp4/NO3AqAaAE6o.mp4573yes https://puzzicle.org/math/multivariable-calculus/v/multivariable-chain-rule-intuitionhttps://puzzicle.org/thumbs/x9c62350e.jpgMultivariable chain rule intuitionGet a feel for what the multivariable is really saying, and how thinking about various "nudges" in space makes it intuitive.https://cdn.kastatic.org/ka-youtube-converted/hFvBZf-Jx28.mp4/hFvBZf-Jx28.mp4467yes https://puzzicle.org/math/multivariable-calculus/v/vector-form-of-the-multivariable-chain-rulehttps://puzzicle.org/thumbs/x8e1e9957.jpgVector form of the multivariable chain ruleThe multivariable chain rule is more often expressed in terms of the gradient and a vector-valued derivative. This makes it look very analogous to the single-variable chain rule.https://cdn.kastatic.org/ka-youtube-converted/qZlBjnC3iro.mp4/qZlBjnC3iro.mp4325yes https://puzzicle.org/math/multivariable-calculus/v/multivariable-chain-rule-and-directional-derivativeshttps://puzzicle.org/thumbs/xddbd0aaf.jpgMultivariable chain rule and directional derivativesSee how the multivariable chain rule can be expressed in terms of the directional derivative.https://cdn.kastatic.org/ka-youtube-converted/m2mW2FQJgEE.mp4/m2mW2FQJgEE.mp4401yes https://puzzicle.org/math/multivariable-calculus/v/more-formal-treatment-of-multivariable-chain-rulehttps://puzzicle.org/thumbs/x49bfbf51.jpgMore formal treatment of multivariable chain ruleFor those of you who want to see how the multivariable chain rule looks in the context of the limit definitions of various forms of the derivative.https://cdn.kastatic.org/ka-youtube-converted/5mMLaK1ByZc.mp4/5mMLaK1ByZc.mp4715yes https://puzzicle.org/math/multivariable-calculus/v/curvature-intuitionhttps://puzzicle.org/thumbs/x97cc1c4f.jpgCurvature intuitionAn introduction to curvature, the radius of curvature, and how you can think about each one geometrically.https://cdn.kastatic.org/ka-youtube-converted/ugtUGhBSeE0.mp4/ugtUGhBSeE0.mp4252yes https://puzzicle.org/math/multivariable-calculus/v/curvature-formula-part-1https://puzzicle.org/thumbs/x221b8645.jpgCurvature formula, part 1Curvature is computed by first finding a unit tangent vector function, then finding its derivative with respect to arc length. Here we start thinking about what that means.https://cdn.kastatic.org/ka-youtube-converted/gspjhwSNMWs.mp4/gspjhwSNMWs.mp4366yes https://puzzicle.org/math/multivariable-calculus/v/curvature-formula-part-2https://puzzicle.org/thumbs/xbecb823e.jpgCurvature formula, part 2A continuation in explaining how curvature is computed, with the formula for a circle as a guiding example.https://cdn.kastatic.org/ka-youtube-converted/Q-BxnC-uWQo.mp4/Q-BxnC-uWQo.mp4426yes https://puzzicle.org/math/multivariable-calculus/v/curvature-formula-part-3https://puzzicle.org/thumbs/xdfb610a6.jpgCurvature formula, part 3Here, this concludes the explanation for how curvature is the derivative of a unit tangent vector with respect to length.https://cdn.kastatic.org/ka-youtube-converted/8V4_4M90RfA.mp4/8V4_4M90RfA.mp4517yes https://puzzicle.org/math/multivariable-calculus/v/curvature-formula-part-4https://puzzicle.org/thumbs/x55b36872.jpgCurvature formula, part 4After the last video made reference to an explicit curvature formula, here you can start to get an intuition for why that seemingly unrelated formula describes curvature.https://cdn.kastatic.org/ka-youtube-converted/WmdN_iPhxAI.mp4/WmdN_iPhxAI.mp4514yes https://puzzicle.org/math/multivariable-calculus/v/curvature-formula-part-5https://puzzicle.org/thumbs/xe8614242.jpgCurvature formula, part 5Here, we finish the intuition for how the curvature relates to the cross product between the first two derivatives of a parametric function.https://cdn.kastatic.org/ka-youtube-converted/8aHUrjgF6s0.mp4/8aHUrjgF6s0.mp4466yes https://puzzicle.org/math/multivariable-calculus/v/curvature-of-a-helix-part-1https://puzzicle.org/thumbs/x5c5962e2.jpgCurvature of a helix, part 1An example of computing curvature by finding the unit tangent vector function, then computing its derivative with respect to arc length.https://cdn.kastatic.org/ka-youtube-converted/AE8NVTyESd8.mp4/AE8NVTyESd8.mp4388yes https://puzzicle.org/math/multivariable-calculus/v/curvature-of-a-helix-part-2https://puzzicle.org/thumbs/x806cbe74.jpgCurvature of a helix, part 2This finishes up the helix-curvature example started in the last video.https://cdn.kastatic.org/ka-youtube-converted/kjNgXVZ1aZ4.mp4/kjNgXVZ1aZ4.mp4439yes https://puzzicle.org/math/multivariable-calculus/v/curvature-of-a-cycloidhttps://puzzicle.org/thumbs/x025231d3.jpgCurvature of a cycloidAn example of computing curvature with the explicit formula.https://cdn.kastatic.org/ka-youtube-converted/biK6_fWLDRc.mp4/biK6_fWLDRc.mp4283yes https://puzzicle.org/math/multivariable-calculus/v/computing-the-partial-derivative-of-a-vector-valued-functionhttps://puzzicle.org/thumbs/xfd75feb2.jpgComputing the partial derivative of a vector-valued functionWhen a function has a multidimensional input, and a multidimensional output, you can take its partial derivative by computing the partial derivative of each component in the output.https://cdn.kastatic.org/ka-youtube-converted/ln43wOC3iC4.mp4/ln43wOC3iC4.mp4138yes https://puzzicle.org/math/multivariable-calculus/v/partial-derivative-of-a-parametric-surface-part-1https://puzzicle.org/thumbs/xab9a6baf.jpgPartial derivative of a parametric surface, part 1When a vector-valued function represents a parametric surface, how do you interpret its partial derivative?https://cdn.kastatic.org/ka-youtube-converted/e7Nel0UoXR8.mp4/e7Nel0UoXR8.mp4569yes https://puzzicle.org/math/multivariable-calculus/v/partial-derivative-of-a-parametric-surface-part-2https://puzzicle.org/thumbs/xc71e0dbe.jpgPartial derivative of a parametric surface, part 2Taking the same example surface used in the last example, we now take a look at the partial derivative in the other direction.https://cdn.kastatic.org/ka-youtube-converted/lz8L-0SoMo0.mp4/lz8L-0SoMo0.mp4423yes https://puzzicle.org/math/multivariable-calculus/v/partial-derivatives-of-vector-fieldshttps://puzzicle.org/thumbs/x80d8b824.jpgPartial derivatives of vector fieldsHow do you intepret the partial derivatives of the function which defines a vector field?https://cdn.kastatic.org/ka-youtube-converted/ar9SeJpbPI8.mp4/ar9SeJpbPI8.mp4514yes https://puzzicle.org/math/multivariable-calculus/v/partial-derivatives-of-vector-fields-component-by-componenthttps://puzzicle.org/thumbs/xccb57798.jpgPartial derivatives of vector fields, component by componentHere we step through each partial derivative of each component in a vector field, and understand what each means geometrically.https://cdn.kastatic.org/ka-youtube-converted/VhtFSSq3lXo.mp4/VhtFSSq3lXo.mp4451yes https://puzzicle.org/math/multivariable-calculus/v/divergence-intuition-part-1https://puzzicle.org/thumbs/x044f3f1d.jpgDivergence intuition, part 1Vector fields can be thought of as representing fluid flow, and divergence is all about studying the change in fluid density during that flow.https://cdn.kastatic.org/ka-youtube-converted/c0MR-vWiUPU.mp4/c0MR-vWiUPU.mp4381yes https://puzzicle.org/math/multivariable-calculus/v/divergence-intuition-part-2https://puzzicle.org/thumbs/xbdd5f10c.jpgDivergence intuition, part 2In preparation for finding the formula for divergence, we start getting an intuition for what points of positive, negative and zero divergence should look like.https://cdn.kastatic.org/ka-youtube-converted/Yeie-aJT2eU.mp4/Yeie-aJT2eU.mp4328yes https://puzzicle.org/math/multivariable-calculus/v/divergence-formula-part-1https://puzzicle.org/thumbs/x32a14878.jpgDivergence formula, part 1How does the x-component of a vector field relate to the divergence?https://cdn.kastatic.org/ka-youtube-converted/uOX7SijjH9w.mp4/uOX7SijjH9w.mp4219yes https://puzzicle.org/math/multivariable-calculus/v/divergence-formula-part-2https://puzzicle.org/thumbs/x0f86b891.jpgDivergence formula, part 2Here we finish the line of reasoning which leads to the formula for divergence in two dimensions.https://cdn.kastatic.org/ka-youtube-converted/TKlpZ0UUJTQ.mp4/TKlpZ0UUJTQ.mp4364yes https://puzzicle.org/math/multivariable-calculus/v/divergence-examplehttps://puzzicle.org/thumbs/x1b43cae4.jpgDivergence exampleAn example of computing and interpreting the divergence of a two-dimensional vector field.https://cdn.kastatic.org/ka-youtube-converted/AlXVrAOls-8.mp4/AlXVrAOls-8.mp4251yes https://puzzicle.org/math/multivariable-calculus/v/divergence-notationhttps://puzzicle.org/thumbs/xb60334f1.jpgDivergence notationLearn how divergence is expressed using the same upsidedown triangle symbols that the gradient uses.https://cdn.kastatic.org/ka-youtube-converted/TyYlBXNETZE.mp4/TyYlBXNETZE.mp4289yes https://puzzicle.org/math/multivariable-calculus/v/2d-curl-intuitionhttps://puzzicle.org/thumbs/xc5160b5d.jpg2d curl intuitionA description of how vector fields relate to fluid rotation, laying the intuition for what the operation of curl represents.https://cdn.kastatic.org/ka-youtube-converted/eEwZeY51mT0.mp4/eEwZeY51mT0.mp4234yes https://puzzicle.org/math/multivariable-calculus/v/2d-curl-formulahttps://puzzicle.org/thumbs/x0deab6bd.jpg2d curl formulaHere we build up to the formula for computing the two-dimensional curl of a vector field, reasoning through what partial derivative information corresponds to fluid rotation.https://cdn.kastatic.org/ka-youtube-converted/qF9Kz37Ksq0.mp4/qF9Kz37Ksq0.mp4343yes https://puzzicle.org/math/multivariable-calculus/v/2d-curl-examplehttps://puzzicle.org/thumbs/x58b7877a.jpg2d curl exampleA worked example of computing and interpreting two-dimensional curl.https://cdn.kastatic.org/ka-youtube-converted/cAAEMza4GvQ.mp4/cAAEMza4GvQ.mp4339yes https://puzzicle.org/math/multivariable-calculus/v/2d-curl-nuancehttps://puzzicle.org/thumbs/x7dcabd7d.jpg2d curl nuanceThe meaning of positive curl in a fluid flow can sometimes look a bit different from the clear cut rotation-around-a-point examples discussed in previous videos.https://cdn.kastatic.org/ka-youtube-converted/szLkF7isbI0.mp4/szLkF7isbI0.mp4300yes https://puzzicle.org/math/multivariable-calculus/v/describing-rotation-in-3d-with-a-vectorhttps://puzzicle.org/thumbs/x62ccb697.jpgDescribing rotation in 3d with a vectorLearn how a three-dimensional vector can be used to describe three-dimensional rotation. This is important for understanding three-dimensional curl.https://cdn.kastatic.org/ka-youtube-converted/h5BmWo5_sc8.mp4/h5BmWo5_sc8.mp4366yes https://puzzicle.org/math/multivariable-calculus/v/3d-curl-intuition-part-1https://puzzicle.org/thumbs/x7f3a9ecf.jpg3d curl intuition, part 1Here we start transitioning from the understanding of two-dimensional curl into an understanding of three-dimensional curl.https://cdn.kastatic.org/ka-youtube-converted/erMopRd-MXg.mp4/erMopRd-MXg.mp4350yes https://puzzicle.org/math/multivariable-calculus/v/3d-curl-intuition-part-2https://puzzicle.org/thumbs/xbf11198c.jpg3d curl intuition, part 2Continuing the intuition for how three-dimensional curl represents rotation in three-dimensional fluid flow.https://cdn.kastatic.org/ka-youtube-converted/4yzV8m8X3Kk.mp4/4yzV8m8X3Kk.mp4497yes https://puzzicle.org/math/multivariable-calculus/v/3d-curl-formula-part-1https://puzzicle.org/thumbs/xca7d2a0c.jpg3d curl formula, part 1How to compute a three-dimensional curl, imagined as a cross product of sorts.https://cdn.kastatic.org/ka-youtube-converted/a_49iMi10kg.mp4/a_49iMi10kg.mp4461yes https://puzzicle.org/math/multivariable-calculus/v/3d-curl-formula-part-2https://puzzicle.org/thumbs/x29cb7d34.jpg3d curl formula, part 2This finishes the demonstration of how to compute three-dimensional curl using a certain determinant.https://cdn.kastatic.org/ka-youtube-converted/0k4tWeI3R0Y.mp4/0k4tWeI3R0Y.mp4372yes https://puzzicle.org/math/multivariable-calculus/v/3d-curl-computation-examplehttps://puzzicle.org/thumbs/x5d1b79ea.jpg3d curl computation exampleA worked example of a three-dimensional curl computation.https://cdn.kastatic.org/ka-youtube-converted/GRuEOLcdWjo.mp4/GRuEOLcdWjo.mp4335yes https://puzzicle.org/math/multivariable-calculus/v/laplacian-intuitionhttps://puzzicle.org/thumbs/x60573c87.jpgLaplacian intuitionA visual understanding for how the Laplace operator is an extension of the second derivative to multivariable functions.https://cdn.kastatic.org/ka-youtube-converted/EW08rD-GFh0.mp4/EW08rD-GFh0.mp4331yes https://puzzicle.org/math/multivariable-calculus/v/laplacian-computation-examplehttps://puzzicle.org/thumbs/x4c3be4fb.jpgLaplacian computation exampleA worked example of computing the laplacian of a two-variable function.https://cdn.kastatic.org/ka-youtube-converted/XbCvGRjjzgg.mp4/XbCvGRjjzgg.mp4321yes https://puzzicle.org/math/multivariable-calculus/v/explicit-laplacian-formulahttps://puzzicle.org/thumbs/x20f5346f.jpgExplicit Laplacian formulaThis is another way you might see the Laplace operator written.https://cdn.kastatic.org/ka-youtube-converted/6cMn1F6bNPU.mp4/6cMn1F6bNPU.mp4243yes https://puzzicle.org/math/multivariable-calculus/v/harmonic-functionshttps://puzzicle.org/thumbs/x0fad04bf.jpgHarmonic FunctionsIf the Laplacian of a function is zero everywhere, it is called Harmonic. Harmonic functions arise all the time in physics, capturing a certain notion of "stability", whenever one point in space is influenced by its neighbors.https://cdn.kastatic.org/ka-youtube-converted/JQSC0lCPG24.mp4/JQSC0lCPG24.mp4656yes https://puzzicle.org/math/multivariable-calculus/v/jacobian-prerequisite-knowledgehttps://puzzicle.org/thumbs/x89ac6576e2c8a11f.jpgJacobian prerequisite knowledgeBefore jumping into the Jacobian, it's important to make sure we all know how to think about matrices geometrically. This is targetted towards those who have seen linear algebra but may need a quick refresher.https://cdn.kastatic.org/ka-youtube-converted/VmfTXVG9S0U.mp4/VmfTXVG9S0U.mp4562yes https://puzzicle.org/math/multivariable-calculus/v/local-linearity-for-a-multivariable-functionhttps://puzzicle.org/thumbs/xc561d674a3d5a959.jpgLocal linearity for a multivariable functionA visual representation of local linearity for a function with a 2d input and a 2d output, in preparation for learning about the Jacobian matrix.https://cdn.kastatic.org/ka-youtube-converted/Vnga_psnCAo.mp4/Vnga_psnCAo.mp4323yes https://puzzicle.org/math/multivariable-calculus/v/the-jacobian-matrixhttps://puzzicle.org/thumbs/xb3c8591f3af73e71.jpgThe Jacobian matrixAn introduction to how the jacobian matrix represents what a multivariable function looks like locally, as a linear transformation.https://cdn.kastatic.org/ka-youtube-converted/bohL918kXQk.mp4/bohL918kXQk.mp4382yes https://puzzicle.org/math/multivariable-calculus/v/computing-a-jacobian-matrixhttps://puzzicle.org/thumbs/x15de8d5d7d1c48b6.jpgComputing a Jacobian matrixThis finishes the introduction of the Jacobian matrix, working out the computations for the example shown in the last video.https://cdn.kastatic.org/ka-youtube-converted/CGbBbH1e7Yw.mp4/CGbBbH1e7Yw.mp4233yes https://puzzicle.org/math/multivariable-calculus/v/the-jacobian-determinanthttps://puzzicle.org/thumbs/xe42b23090e815198.jpgThe Jacobian DeterminantHow to interpret the determinant of a Jacobian matrix, along with some examples.https://cdn.kastatic.org/ka-youtube-converted/p46QWyHQE6M.mp4/p46QWyHQE6M.mp4533yes https://puzzicle.org/math/multivariable-calculus/v/what-is-a-tangent-planehttps://puzzicle.org/thumbs/x3297f1fe.jpgWhat is a tangent planeThe "tangent plane" of the graph of a function is, well, a two-dimensional plane that is tangent to this graph. Here you can see what that looks like.https://cdn.kastatic.org/ka-youtube-converted/cHNT7_F8m1Y.mp4/cHNT7_F8m1Y.mp4200yes https://puzzicle.org/math/multivariable-calculus/v/controlling-a-plane-in-spacehttps://puzzicle.org/thumbs/x0b636b91.jpgControlling a plane in spaceHow can you describe a specified plane in space as the graph of a function?https://cdn.kastatic.org/ka-youtube-converted/H2hOwKszKRo.mp4/H2hOwKszKRo.mp4640yes https://puzzicle.org/math/multivariable-calculus/v/computing-a-tangent-planehttps://puzzicle.org/thumbs/x8763b3e1.jpgComputing a tangent planeHere you can see how to use the control over functions whose graphs are planes, as introduced in the last video, to find the tangent plane to a function graph.https://cdn.kastatic.org/ka-youtube-converted/QL6qb1h65hg.mp4/QL6qb1h65hg.mp4581yes https://puzzicle.org/math/multivariable-calculus/v/local-linearizationhttps://puzzicle.org/thumbs/xc8af53b7.jpgLocal linearizationA "local linearization" is the generalization of tangent plane functions; one that can apply to multivariable functions with any number of inputs.https://cdn.kastatic.org/ka-youtube-converted/o7_zS7Bx2VA.mp4/o7_zS7Bx2VA.mp4553yes https://puzzicle.org/math/multivariable-calculus/v/what-do-quadratic-approximations-look-likehttps://puzzicle.org/thumbs/xa0c2bd39.jpgWhat do quadratic approximations look likeAfter learning about local linearizations of multivariable functions, the next step is to understand how to approximate a function even more closely with a quadratic approximation.https://cdn.kastatic.org/ka-youtube-converted/80bJA_tSbo4.mp4/80bJA_tSbo4.mp4282yes https://puzzicle.org/math/multivariable-calculus/v/quadratic-approximation-formula-part-1https://puzzicle.org/thumbs/x48b93c30.jpgQuadratic approximation formula, part 1How to create a quadratic function that approximates an arbitrary two-variable function.https://cdn.kastatic.org/ka-youtube-converted/UV5yj5A3QIM.mp4/UV5yj5A3QIM.mp4429yes https://puzzicle.org/math/multivariable-calculus/v/quadratic-approximation-formula-part-2https://puzzicle.org/thumbs/xfd5eae14.jpgQuadratic approximation formula, part 2A continuation from the previous video, leading to the full formula for the quadratic approximation of a two-variable function.https://cdn.kastatic.org/ka-youtube-converted/szHMvVXxp-g.mp4/szHMvVXxp-g.mp4590yes https://puzzicle.org/math/multivariable-calculus/v/quadratic-approximation-examplehttps://puzzicle.org/thumbs/xf188596e.jpgQuadratic approximation exampleA worked example for finding the quadratic approximation of a two-variable function.https://cdn.kastatic.org/ka-youtube-converted/fW3snxnCPEY.mp4/fW3snxnCPEY.mp4729yes https://puzzicle.org/math/multivariable-calculus/v/the-hessian-matrixhttps://puzzicle.org/thumbs/x5346e9fe.jpgThe Hessian matrixThe Hessian matrix is a way of organizing all the second partial derivative information of a multivariable function.https://cdn.kastatic.org/ka-youtube-converted/LbBcuZukCAw.mp4/LbBcuZukCAw.mp4370yes https://puzzicle.org/math/multivariable-calculus/v/expressing-a-quadratic-form-with-a-matrixhttps://puzzicle.org/thumbs/xfa25e3b4.jpgExpressing a quadratic form with a matrixHow to write an expression like ax^2 + bxy + cy^2 using matrices and vectors.https://cdn.kastatic.org/ka-youtube-converted/0yEiCV-xEWQ.mp4/0yEiCV-xEWQ.mp4500yes https://puzzicle.org/math/multivariable-calculus/v/vector-form-of-multivariable-quadratic-approximationhttps://puzzicle.org/thumbs/xcad2e826.jpgVector form of multivariable quadratic approximationThis is the more general form of a quadratic approximation for a scalar-valued multivariable function. It is analogous to a quadratic Taylor polynomial in the single-variable world.https://cdn.kastatic.org/ka-youtube-converted/ClFrIg0PpnM.mp4/ClFrIg0PpnM.mp4515yes https://puzzicle.org/math/multivariable-calculus/v/multivariable-maxima-and-minimahttps://puzzicle.org/thumbs/xadaf4e40.jpgMultivariable maxima and minimaA description of maxima and minima of multivariable functions, what they look like, and a little bit about how to find them.https://cdn.kastatic.org/ka-youtube-converted/ux7EQ3ip2DU.mp4/ux7EQ3ip2DU.mp4484yes https://puzzicle.org/math/multivariable-calculus/v/saddle-pointshttps://puzzicle.org/thumbs/x5993ca15.jpgSaddle pointsJust because the tangent plane to a multivariable function is flat, it doesn't mean that point is a local minimum or a local maximum. There is a third possibility, new to multivariable calculus, called a "saddle point".https://cdn.kastatic.org/ka-youtube-converted/8aAU4r_pUUU.mp4/8aAU4r_pUUU.mp4350yes https://puzzicle.org/math/multivariable-calculus/v/warm-up-to-the-second-partial-derivative-testhttps://puzzicle.org/thumbs/x4affaf47.jpgWarm up to the second partial derivative testAn example of looking for local minima in a multivariable function by finding where tangent planes are flat, along with some of the intuitions that will underly the second partial derivative test.https://cdn.kastatic.org/ka-youtube-converted/nRJM4mY-Pq0.mp4/nRJM4mY-Pq0.mp4697yes https://puzzicle.org/math/multivariable-calculus/v/second-partial-derivative-testhttps://puzzicle.org/thumbs/x2cfadbec.jpgSecond partial derivative testHow to determine if the critical point of a two-variable function is a local minimum, a local maximum, or a saddle point.https://cdn.kastatic.org/ka-youtube-converted/m1FhUjMMv30.mp4/m1FhUjMMv30.mp4712yes https://puzzicle.org/math/multivariable-calculus/v/second-partial-derivative-test-intuitionhttps://puzzicle.org/thumbs/xd786588e.jpgSecond partial derivative test intuitionThe second partial derivative test is based on a formula which seems to come out of nowhere. Here, you can see a little more intuition for why it looks the way it does.https://cdn.kastatic.org/ka-youtube-converted/sJo7D74PAak.mp4/sJo7D74PAak.mp4643yes https://puzzicle.org/math/multivariable-calculus/v/second-partial-derivative-test-example-part-1https://puzzicle.org/thumbs/x3affa5ce.jpgSecond partial derivative test example, part 1A worked example of finding a classifying critical points of a two-variable function.https://cdn.kastatic.org/ka-youtube-converted/shWXeUn5BHk.mp4/shWXeUn5BHk.mp4371yes https://puzzicle.org/math/multivariable-calculus/v/second-partial-derivative-test-example-part-2https://puzzicle.org/thumbs/x88eeefe9.jpgSecond partial derivative test example, part 2Continuing the worked example from the previous video, now classifying each critical point.https://cdn.kastatic.org/ka-youtube-converted/TqslX-bUTD8.mp4/TqslX-bUTD8.mp4463yes https://puzzicle.org/math/multivariable-calculus/v/constrained-optimization-introductionhttps://puzzicle.org/thumbs/xf123b6a166983b0e.jpgConstrained optimization introductionThe Lagrange multiplier technique is how we take advantage of the observation made in the last video, that the solution to a constrained optimization problem occurs when the contour lines of the function being maximized are tangent to the constraint curve.https://cdn.kastatic.org/ka-youtube-converted/vwUV2IDLP8Q.mp4/vwUV2IDLP8Q.mp4389yes https://puzzicle.org/math/multivariable-calculus/v/lagrange-multipliers-using-tangency-to-solve-constrained-optimizationhttps://puzzicle.org/thumbs/x9645cf61a8256d54.jpgLagrange multipliers, using tangency to solve constrained optimizationThe Lagrange multiplier technique is how we take advantage of the observation made in the last video, that the solution to a constrained optimization problem occurs when the contour lines of the function being maximized are tangent to the constraint curve.https://cdn.kastatic.org/ka-youtube-converted/yuqB-d5MjZA.mp4/yuqB-d5MjZA.mp4523yes https://puzzicle.org/math/multivariable-calculus/v/finishing-the-intro-lagrange-multiplier-examplehttps://puzzicle.org/thumbs/xee1132860c2a5d35.jpgFinishing the intro lagrange multiplier exampleWorking out the algebra for the final solution the the example from the previous two videos.https://cdn.kastatic.org/ka-youtube-converted/aep6lwPqm6I.mp4/aep6lwPqm6I.mp4399yes https://puzzicle.org/math/multivariable-calculus/v/lagrange-multiplier-example-part-1https://puzzicle.org/thumbs/xbabfa1cb9a42466b.jpgLagrange multiplier example, part 1A Lagrange multipliers example of maximizing revenues subject to a budgetary constraint.https://cdn.kastatic.org/ka-youtube-converted/BSKtQcLQLWU.mp4/BSKtQcLQLWU.mp4470yes https://puzzicle.org/math/multivariable-calculus/v/lagrange-multiplier-example-part-2https://puzzicle.org/thumbs/x8b06c88b2bcfc0c9.jpgLagrange multiplier example, part 2Finishing off the example from the previous videohttps://cdn.kastatic.org/ka-youtube-converted/5gCx5YCmXbI.mp4/5gCx5YCmXbI.mp4411yes https://puzzicle.org/math/multivariable-calculus/v/the-lagrangianhttps://puzzicle.org/thumbs/x046926e43a1c2f91.jpgThe LagrangianHow a special function, called the "Lagrangian", can be used to package together all the steps needed to solve a constrained optimization problem.https://cdn.kastatic.org/ka-youtube-converted/hQ4UNu1P2kw.mp4/hQ4UNu1P2kw.mp4748yes https://puzzicle.org/math/multivariable-calculus/v/meaning-of-lagrange-multiplierhttps://puzzicle.org/thumbs/x3c5c1012d0728ba3.jpgMeaning of the Lagrange multiplierIn the previous videos on Lagrange multipliers, the Lagrange multiplier itself has just been some proportionality constant that we didn't care about. Here, you can see what its real meaning is.https://cdn.kastatic.org/ka-youtube-converted/m-G3K2GPmEQ.mp4/m-G3K2GPmEQ.mp4608yes https://puzzicle.org/math/multivariable-calculus/v/proof-for-the-meaning-of-lagrange-multipliershttps://puzzicle.org/thumbs/x39e4f4907a5382bc.jpgProof for the meaning of Lagrange multipliersHere, you can see a proof of the fact shown in the last video, that the Lagrange multiplier gives information about how altering a constraint can alter the solution to a constrained maximization problem. Note, this is somewhat technicalhttps://cdn.kastatic.org/ka-youtube-converted/b9B2FZ5cqbM.mp4/b9B2FZ5cqbM.mp41039yes https://puzzicle.org/math/multivariable-calculus/v/introduction-to-the-line-integralhttps://puzzicle.org/thumbs/24308.jpgIntroduction to the line integralIntroduction to the Line Integralhttps://cdn.kastatic.org/ka-youtube-converted/_60sKaoRmhU.mp4/_60sKaoRmhU.mp41128yes https://puzzicle.org/math/multivariable-calculus/v/line-integral-example-1https://puzzicle.org/thumbs/24310.jpgLine integral example 1Concrete example using a line integralhttps://cdn.kastatic.org/ka-youtube-converted/uXjQ8yc9Pdg.mp4/uXjQ8yc9Pdg.mp4782yes https://puzzicle.org/math/multivariable-calculus/v/line-integral-example-2-part-1https://puzzicle.org/thumbs/24312.jpgLine integral example 2 (part 1)Line integral over a closed path (part 1)https://cdn.kastatic.org/ka-youtube-converted/wyTjyQMVvc4.mp4/wyTjyQMVvc4.mp4773yes https://puzzicle.org/math/multivariable-calculus/v/line-integral-example-2-part-2https://puzzicle.org/thumbs/24314.jpgLine integral example 2 (part 2)Part 2 of an example of taking a line integral over a closed pathhttps://cdn.kastatic.org/ka-youtube-converted/Qqanbd3gLhw.mp4/Qqanbd3gLhw.mp4598yes https://puzzicle.org/math/multivariable-calculus/v/line-integrals-and-vector-fieldshttps://puzzicle.org/thumbs/24324.jpgLine integrals and vector fieldsUsing line integrals to find the work done on a particle moving through a vector fieldhttps://cdn.kastatic.org/ka-youtube-converted/t3cJYNdQLYg.mp4/t3cJYNdQLYg.mp41006yes https://puzzicle.org/math/multivariable-calculus/v/using-a-line-integral-to-find-the-work-done-by-a-vector-field-examplehttps://puzzicle.org/thumbs/24326.jpgUsing a line integral to find workUsing a line integral to find the work done by a vector field examplehttps://cdn.kastatic.org/ka-youtube-converted/AFF8FXxt5os.mp4/AFF8FXxt5os.mp4692yes https://puzzicle.org/math/multivariable-calculus/v/parametrization-of-a-reverse-pathhttps://puzzicle.org/thumbs/24328.jpgParametrization of a reverse pathUnderstanding how to parametrize a reverse path for the same curve.https://cdn.kastatic.org/ka-youtube-converted/eGRZKkmI_fo.mp4/eGRZKkmI_fo.mp4446yes https://puzzicle.org/math/multivariable-calculus/v/scalar-field-line-integral-independent-of-path-directionhttps://puzzicle.org/thumbs/24330.jpgScalar field line integral independent of path directionShowing that the line integral of a scalar field is independent of path directionhttps://cdn.kastatic.org/ka-youtube-converted/99pD1-6ZpuM.mp4/99pD1-6ZpuM.mp4982yes https://puzzicle.org/math/multivariable-calculus/v/vector-field-line-integrals-dependent-on-path-directionhttps://puzzicle.org/thumbs/24332.jpgVector field line integrals dependent on path directionShowing that, unlike line integrals of scalar fields, line integrals over vector fields are path direction dependenthttps://cdn.kastatic.org/ka-youtube-converted/fuSOY9r1R6w.mp4/fuSOY9r1R6w.mp4951yes https://puzzicle.org/math/multivariable-calculus/v/path-independence-for-line-integralshttps://puzzicle.org/thumbs/24334.jpgPath independence for line integralsShowing that if a vector field is the gradient of a scalar field, then its line integral is path independenthttps://cdn.kastatic.org/ka-youtube-converted/K_fgnCJOI8I.mp4/K_fgnCJOI8I.mp41034yes https://puzzicle.org/math/multivariable-calculus/v/closed-curve-line-integrals-of-conservative-vector-fieldshttps://puzzicle.org/thumbs/24336.jpgClosed curve line integrals of conservative vector fieldsShowing that the line integral along closed curves of conservative vector fields is zerohttps://cdn.kastatic.org/ka-youtube-converted/I2dbzp0zHuw.mp4/I2dbzp0zHuw.mp4505yes https://puzzicle.org/math/multivariable-calculus/v/example-of-closed-line-integral-of-conservative-fieldhttps://puzzicle.org/thumbs/24338.jpgExample of closed line integral of conservative fieldExample of taking a closed line integral of a conservative fieldhttps://cdn.kastatic.org/ka-youtube-converted/Q9t1LghwdGc.mp4/Q9t1LghwdGc.mp4662yes https://puzzicle.org/math/multivariable-calculus/v/second-example-of-line-integral-of-conservative-vector-fieldhttps://puzzicle.org/thumbs/24340.jpgSecond example of line integral of conservative vector fieldUsing path independence of a conservative vector field to solve a line integralhttps://cdn.kastatic.org/ka-youtube-converted/LpY8Qa3IP1w.mp4/LpY8Qa3IP1w.mp4619yes https://puzzicle.org/math/multivariable-calculus/v/double-integral-1https://puzzicle.org/thumbs/24288.jpgDouble integral 1Introduction to the double integralhttps://cdn.kastatic.org/ka-youtube-converted/85zGYB-34jQ.mp4/85zGYB-34jQ.mp4629yes https://puzzicle.org/math/multivariable-calculus/v/double-integrals-2https://puzzicle.org/thumbs/24290.jpgDouble integrals 2Figuring out the volume under z=xy^2https://cdn.kastatic.org/ka-youtube-converted/TdLD2Zh-nUQ.mp4/TdLD2Zh-nUQ.mp4590yes https://puzzicle.org/math/multivariable-calculus/v/double-integrals-3https://puzzicle.org/thumbs/24292.jpgDouble integrals 3Let's integrate dy first!https://cdn.kastatic.org/ka-youtube-converted/z8BM6cHifPA.mp4/z8BM6cHifPA.mp4483yes https://puzzicle.org/math/multivariable-calculus/v/double-integrals-4https://puzzicle.org/thumbs/24294.jpgDouble integrals 4Another way to conceptualize the double integral.https://cdn.kastatic.org/ka-youtube-converted/twT-WZChfZ8.mp4/twT-WZChfZ8.mp4564yes https://puzzicle.org/math/multivariable-calculus/v/double-integrals-5https://puzzicle.org/thumbs/24296.jpgDouble integrals 5Finding the volume when we have variable boundaries.https://cdn.kastatic.org/ka-youtube-converted/hrIPO8mQqtw.mp4/hrIPO8mQqtw.mp4591yes https://puzzicle.org/math/multivariable-calculus/v/double-integrals-6https://puzzicle.org/thumbs/24298.jpgDouble integrals 6Let's evaluate the double integrals with y=x^2 as one of the boundaries.https://cdn.kastatic.org/ka-youtube-converted/0pv0QtOi5l8.mp4/0pv0QtOi5l8.mp4598yes https://puzzicle.org/math/multivariable-calculus/v/polar-coordinates-1https://puzzicle.org/thumbs/xf16db78cc1c9b015.jpgPolar coordinatesIntroduction to polar coordinateshttps://cdn.kastatic.org/ka-youtube-converted/jexMSlSDubM.mp4/jexMSlSDubM.mp4620yes https://puzzicle.org/math/multivariable-calculus/v/triple-integrals-1https://puzzicle.org/thumbs/24300.jpgTriple integrals 1Introduction to the triple integralhttps://cdn.kastatic.org/ka-youtube-converted/vr0sTKbV7lI.mp4/vr0sTKbV7lI.mp4638yes https://puzzicle.org/math/multivariable-calculus/v/triple-integrals-2https://puzzicle.org/thumbs/24302.jpgTriple integrals 2Using a triple integral to find the mass of a volume of variable density.https://cdn.kastatic.org/ka-youtube-converted/vxQvL_WhBGU.mp4/vxQvL_WhBGU.mp4446yes https://puzzicle.org/math/multivariable-calculus/v/triple-integrals-3https://puzzicle.org/thumbs/24304.jpgTriple integrals 3Figuring out the boundaries of integration.https://cdn.kastatic.org/ka-youtube-converted/ZN2PfqZ4ihM.mp4/ZN2PfqZ4ihM.mp4708yes https://puzzicle.org/math/multivariable-calculus/v/introduction-to-parametrizing-a-surface-with-two-parametershttps://puzzicle.org/thumbs/2505283.jpgParametrizing a surface, part 1Introduction to Parametrizing a Surface with Two Parametershttps://cdn.kastatic.org/ka-youtube-converted/owKAHXf1y1A.mp4/owKAHXf1y1A.mp41142yes https://puzzicle.org/math/multivariable-calculus/v/determining-a-position-vector-valued-function-for-a-parametrization-of-two-paramhttps://puzzicle.org/thumbs/2542218.jpgDetermining a position vector-valued function for a parametrization of two parametersDetermining a Position Vector-Valued Function for a Parametrization of Two Parametershttps://cdn.kastatic.org/ka-youtube-converted/bJ_09eoCmag.mp4/bJ_09eoCmag.mp4992yes https://puzzicle.org/math/multivariable-calculus/v/partial-derivatives-of-vector-valued-functionshttps://puzzicle.org/thumbs/2565195.jpgPartial derivatives of vector-valued functionsPartial Derivatives of Vector-Valued Functionshttps://cdn.kastatic.org/ka-youtube-converted/c7ByaI3T7Dc.mp4/c7ByaI3T7Dc.mp4607yes https://puzzicle.org/math/multivariable-calculus/v/introduction-to-the-surface-integralhttps://puzzicle.org/thumbs/24356.jpgIntroduction to the surface integralIntroduction to the surface integralhttps://cdn.kastatic.org/ka-youtube-converted/9k97m8oWnaY.mp4/9k97m8oWnaY.mp41334yes https://puzzicle.org/math/multivariable-calculus/v/example-of-calculating-a-surface-integral-part-1https://puzzicle.org/thumbs/24358.jpgExample of calculating a surface integral part 1Example of calculating a surface integral part 1https://cdn.kastatic.org/ka-youtube-converted/7sQCcGlK2bY.mp4/7sQCcGlK2bY.mp4645yes https://puzzicle.org/math/multivariable-calculus/v/example-of-calculating-a-surface-integral-part-2https://puzzicle.org/thumbs/24360.jpgExample of calculating a surface integral part 2Example of calculating a surface integral part 2https://cdn.kastatic.org/ka-youtube-converted/qQAhhithHa8.mp4/qQAhhithHa8.mp4629yes https://puzzicle.org/math/multivariable-calculus/v/example-of-calculating-a-surface-integral-part-3https://puzzicle.org/thumbs/24362.jpgExample of calculating a surface integral part 3Example of calculating a surface integral part 3https://cdn.kastatic.org/ka-youtube-converted/s2_NTiISZl4.mp4/s2_NTiISZl4.mp4594yes https://puzzicle.org/math/multivariable-calculus/v/surface-integral-example-part-1-parameterizing-the-unit-spherehttps://puzzicle.org/thumbs/809362246.jpgSurface integral example, part 1Visualizing a suitable parameterizationhttps://cdn.kastatic.org/ka-youtube-converted/E_Hwhp74Rhc.mp4/E_Hwhp74Rhc.mp4665yes https://puzzicle.org/math/multivariable-calculus/v/surface-integral-example-part-2-calculating-the-surface-differentialhttps://puzzicle.org/thumbs/809263849.jpgSurface integral example part 2Taking the cross product to calculate the surface differential in terms of the parametershttps://cdn.kastatic.org/ka-youtube-converted/tyVCA_8MUV4.mp4/tyVCA_8MUV4.mp4771yes https://puzzicle.org/math/multivariable-calculus/v/surface-integral-example-part-3-the-home-stretchhttps://puzzicle.org/thumbs/809327989.jpgSurface integral example part 3: The home stretchUsing a few trigonometric identities to finally calculate the value of the surface integralhttps://cdn.kastatic.org/ka-youtube-converted/uzOXGgAgmPs.mp4/uzOXGgAgmPs.mp4759yes https://puzzicle.org/math/multivariable-calculus/v/surface-integral-ex2-part-1-parameterizing-the-surfacehttps://puzzicle.org/thumbs/815244209.jpgSurface integral ex2 part 1Parametrizing a surface that can be explictly made a function of x and y.https://cdn.kastatic.org/ka-youtube-converted/PpUWpxAb39w.mp4/PpUWpxAb39w.mp4310yes https://puzzicle.org/math/multivariable-calculus/v/surface-integral-ex2-part-2-evaluating-integralhttps://puzzicle.org/thumbs/815252175.jpgSurface integral ex2 part 2Evaluating the surface integralhttps://cdn.kastatic.org/ka-youtube-converted/ItW5CxLsLSo.mp4/ItW5CxLsLSo.mp4591yes https://puzzicle.org/math/multivariable-calculus/v/surface-integral-ex3-part-1-parameterizing-the-outside-surfacehttps://puzzicle.org/thumbs/815198581.jpgSurface integral ex3 part 1Breaking apart a larger surface into its componentshttps://cdn.kastatic.org/ka-youtube-converted/EOpzN8ZHqpU.mp4/EOpzN8ZHqpU.mp4540yes https://puzzicle.org/math/multivariable-calculus/v/surface-integral-ex3-part-2-evaluating-the-outside-surfacehttps://puzzicle.org/thumbs/815059993.jpgSurface integral ex3 part 2Evaluating the surface integral over the outside of the chopped cylinderhttps://cdn.kastatic.org/ka-youtube-converted/xGiILI906C4.mp4/xGiILI906C4.mp4550yes https://puzzicle.org/math/multivariable-calculus/v/surface-integral-ex3-part-3-top-surfacehttps://puzzicle.org/thumbs/815160761.jpgSurface integral ex3 part 3Parametrizing the top surfacehttps://cdn.kastatic.org/ka-youtube-converted/wiVWafc3JBM.mp4/wiVWafc3JBM.mp4761yes https://puzzicle.org/math/multivariable-calculus/v/surface-integral-ex3-part-4-home-stretchhttps://puzzicle.org/thumbs/815201348.jpgSurface integral ex3 part 4Evaluating the third surface integral and coming to the final answerhttps://cdn.kastatic.org/ka-youtube-converted/bhrmEy0wvD0.mp4/bhrmEy0wvD0.mp4242yes https://puzzicle.org/math/multivariable-calculus/v/conceputal-understanding-of-flux-in-three-dimensionshttps://puzzicle.org/thumbs/815159721.jpgConceptual understanding of fluxConceptual understanding of flux across a two-dimensional surfacehttps://cdn.kastatic.org/ka-youtube-converted/ivg3dLTarbs.mp4/ivg3dLTarbs.mp4503yes https://puzzicle.org/math/multivariable-calculus/v/constructing-a-unit-normal-vector-to-a-surfacehttps://puzzicle.org/thumbs/815232513.jpgConstructing a unit normal vectorDeriving a unit normal vector from the surface parametrizationhttps://cdn.kastatic.org/ka-youtube-converted/usH9VUi2-Xg.mp4/usH9VUi2-Xg.mp4383yes https://puzzicle.org/math/multivariable-calculus/v/vector-representation-of-a-surface-integralhttps://puzzicle.org/thumbs/815106014.jpgVector representation of a surface integralDifferent ways of representing a flux integralhttps://cdn.kastatic.org/ka-youtube-converted/NCwwfvPEVn8.mp4/NCwwfvPEVn8.mp4575yes https://puzzicle.org/math/multivariable-calculus/v/green-s-theorem-proof-part-1https://puzzicle.org/thumbs/24342.jpgGreen's theorem proof (part 1)Part 1 of the proof of Green's Theoremhttps://cdn.kastatic.org/ka-youtube-converted/l5zJvZKfMYE.mp4/l5zJvZKfMYE.mp4866yes https://puzzicle.org/math/multivariable-calculus/v/green-s-theorem-proof-part-2https://puzzicle.org/thumbs/24344.jpgGreen's theorem proof (part 2)Part 2 of the proof of Green's Theoremhttps://cdn.kastatic.org/ka-youtube-converted/qdFD-0OWBRo.mp4/qdFD-0OWBRo.mp41166yes https://puzzicle.org/math/multivariable-calculus/v/green-s-theorem-example-1https://puzzicle.org/thumbs/24346.jpgGreen's theorem example 1Using Green's Theorem to solve a line integral of a vector fieldhttps://cdn.kastatic.org/ka-youtube-converted/gGXnILbrhsM.mp4/gGXnILbrhsM.mp4631yes https://puzzicle.org/math/multivariable-calculus/v/green-s-theorem-example-2https://puzzicle.org/thumbs/24348.jpgGreen's theorem example 2Another example applying Green's Theoremhttps://cdn.kastatic.org/ka-youtube-converted/sSyPAAyL8nQ.mp4/sSyPAAyL8nQ.mp4427yes https://puzzicle.org/math/multivariable-calculus/v/constructing-a-unit-normal-vector-to-a-curvehttps://puzzicle.org/thumbs/809395096.jpgConstructing a unit normal vector to a curveFiguring out a unit normal vector at any point along a curve defined by a position vector functionhttps://cdn.kastatic.org/ka-youtube-converted/_9x2cqO7-Ig.mp4/_9x2cqO7-Ig.mp4608yes https://puzzicle.org/math/multivariable-calculus/v/2-d-divergence-theoremhttps://puzzicle.org/thumbs/809321651.jpg2D divergence theoremUsing Green's Theorem to establish a two dimensional version of the Divergence Theoremhttps://cdn.kastatic.org/ka-youtube-converted/eoJGBr-tJXo.mp4/eoJGBr-tJXo.mp4884yes https://puzzicle.org/math/multivariable-calculus/v/conceptual-clarification-for-2-d-divergence-theoremhttps://puzzicle.org/thumbs/809302581.jpgConceptual clarification for 2D divergence theoremUnderstanding the line integral as flux through a boundaryhttps://cdn.kastatic.org/ka-youtube-converted/qs0hRHfnzTQ.mp4/qs0hRHfnzTQ.mp4568yes https://puzzicle.org/math/multivariable-calculus/v/stokes-theorem-intuitionhttps://puzzicle.org/thumbs/873691665.jpgStokes' theorem intuitionConceptual understanding of why the curl of a vector field along a surface would relate to the line integral around the surface's boundaryhttps://cdn.kastatic.org/ka-youtube-converted/9iaYNaENVH4.mp4/9iaYNaENVH4.mp4732yes https://puzzicle.org/math/multivariable-calculus/v/green-s-and-stokes-theorem-relationshiphttps://puzzicle.org/thumbs/873865593.jpgGreen's and Stokes' theorem relationshipSeeing that Green's Theorem is just a special case of Stokes' Theoremhttps://cdn.kastatic.org/ka-youtube-converted/TPov_v9mLWc.mp4/TPov_v9mLWc.mp4415yes https://puzzicle.org/math/multivariable-calculus/v/orienting-boundary-with-surfacehttps://puzzicle.org/thumbs/873764943.jpgOrienting boundary with surfaceDetermining the proper orientation of the boundary given the orientation of the surfacehttps://cdn.kastatic.org/ka-youtube-converted/84NkvxfMPtM.mp4/84NkvxfMPtM.mp4209yes https://puzzicle.org/math/multivariable-calculus/v/orientation-and-stokeshttps://puzzicle.org/thumbs/874076304.jpgOrientation and stokesDetermining the proper orientation of a boundary given the orientation of the normal vectorhttps://cdn.kastatic.org/ka-youtube-converted/U6U4HhR3Fqc.mp4/U6U4HhR3Fqc.mp4266yes https://puzzicle.org/math/multivariable-calculus/v/conditions-for-stokes-theoremhttps://puzzicle.org/thumbs/878103964.jpgConditions for stokes theoremUnderstanding when you can use Stokes. Piecewise-smooth lines and surfaceshttps://cdn.kastatic.org/ka-youtube-converted/yDB-191sa2M.mp4/yDB-191sa2M.mp4284yes https://puzzicle.org/math/multivariable-calculus/v/stokes-example-part-1https://puzzicle.org/thumbs/878112005.jpgStokes example part 1Starting to apply Stokes theorem to solve a line integralhttps://cdn.kastatic.org/ka-youtube-converted/E2MRMdaPKcc.mp4/E2MRMdaPKcc.mp4190yes https://puzzicle.org/math/multivariable-calculus/v/part-2-parameterizing-the-surfacehttps://puzzicle.org/thumbs/878163443.jpgStokes example part 2Parameterizing the surfacehttps://cdn.kastatic.org/ka-youtube-converted/9bwy85M1Zxg.mp4/9bwy85M1Zxg.mp4242yes https://puzzicle.org/math/multivariable-calculus/v/stokes-example-part-3-surface-to-double-integralhttps://puzzicle.org/thumbs/878196180.jpgStokes example part 3Converting the surface integral to a double integralhttps://cdn.kastatic.org/ka-youtube-converted/VPz4bCQDGgk.mp4/VPz4bCQDGgk.mp4485yes https://puzzicle.org/math/multivariable-calculus/v/stokes-example-part-4-curl-and-final-answerhttps://puzzicle.org/thumbs/878133910.jpgStokes example part 4Finding the curl of the vector field and then evaluating the double integral in the parameter domainhttps://cdn.kastatic.org/ka-youtube-converted/clL2jLXtX4I.mp4/clL2jLXtX4I.mp4414yes https://puzzicle.org/math/multivariable-calculus/v/evaluating-line-integral-directly-part-1https://puzzicle.org/thumbs/1254589677.jpgEvaluating line integral directly - part 1Showing that we didn't need to use Stokes' Theorem to evaluate this line integralhttps://cdn.kastatic.org/ka-youtube-converted/A57_63yP1UU.mp4/A57_63yP1UU.mp4464yes https://puzzicle.org/math/multivariable-calculus/v/evaluating-line-integral-directly-part-2https://puzzicle.org/thumbs/1254590705.jpgEvaluating line integral directly - part 2Finishing up the line integral with a little trigonometric integrationhttps://cdn.kastatic.org/ka-youtube-converted/Jhu1M53AF0E.mp4/Jhu1M53AF0E.mp4383yes https://puzzicle.org/math/multivariable-calculus/v/3-d-divergence-theorem-intuitionhttps://puzzicle.org/thumbs/1254611464.jpg3D divergence theorem intuitionIntuition behind the Divergence Theorem in three dimensionshttps://cdn.kastatic.org/ka-youtube-converted/XyiQ2dwJHXE.mp4/XyiQ2dwJHXE.mp4554yes https://puzzicle.org/math/multivariable-calculus/v/divergence-theorem-example-1https://puzzicle.org/thumbs/1254640190.jpgDivergence theorem example 1Example of calculating the flux across a surface by using the Divergence Theoremhttps://cdn.kastatic.org/ka-youtube-converted/asyIsn59Lnc.mp4/asyIsn59Lnc.mp4645yes https://puzzicle.org/math/multivariable-calculus/v/why-we-got-zero-flux-in-divergence-theorem-example-1https://puzzicle.org/thumbs/1254539389.jpgExplanation of example 1Intuition as to why we got no net flux in the last worked examplehttps://cdn.kastatic.org/ka-youtube-converted/iFpyKTuC7jY.mp4/iFpyKTuC7jY.mp4214yes https://puzzicle.org/math/multivariable-calculus/v/stokes-theorem-proof-part-1https://puzzicle.org/thumbs/1254599500.jpgStokes' theorem proof part 1The beginning of a proof of Stokes' theorem for a special class of surfaces. Finding the curl of our vector field.https://cdn.kastatic.org/ka-youtube-converted/qIBqvr5_QqQ.mp4/qIBqvr5_QqQ.mp4314yes https://puzzicle.org/math/multivariable-calculus/v/stokes-theorem-proof-part-2https://puzzicle.org/thumbs/1254626652.jpgStokes' theorem proof part 2Figuring out a parameterization of our surface and representing dShttps://cdn.kastatic.org/ka-youtube-converted/apWQoMKVmW4.mp4/apWQoMKVmW4.mp4459yes https://puzzicle.org/math/multivariable-calculus/v/stokes-theorem-proof-part-3https://puzzicle.org/thumbs/1254632405.jpgStokes' theorem proof part 3Writing our surface integral as a double integral over the domain of our parametershttps://cdn.kastatic.org/ka-youtube-converted/4o4P7KrkzXg.mp4/4o4P7KrkzXg.mp4251yes https://puzzicle.org/math/multivariable-calculus/v/stokes-theorem-proof-part-4https://puzzicle.org/thumbs/1254257997.jpgStokes' theorem proof part 4Starting to work on the line integral about the surfacehttps://cdn.kastatic.org/ka-youtube-converted/1og25U-d3_8.mp4/1og25U-d3_8.mp4439yes https://puzzicle.org/math/multivariable-calculus/v/stokes-theorem-proof-part-5https://puzzicle.org/thumbs/1254588379.jpgStokes' theorem proof part 5Working on the integrals...https://cdn.kastatic.org/ka-youtube-converted/a29qZSRDBdE.mp4/a29qZSRDBdE.mp4328yes https://puzzicle.org/math/multivariable-calculus/v/stokes-theorem-proof-part-6https://puzzicle.org/thumbs/1254528847.jpgStokes' theorem proof part 6More manipulating the integrals...https://cdn.kastatic.org/ka-youtube-converted/02gVmoObrjM.mp4/02gVmoObrjM.mp4327yes https://puzzicle.org/math/multivariable-calculus/v/stokes-theorem-proof-part-7https://puzzicle.org/thumbs/1254530460.jpgStokes' theorem proof part 7Using Green's Theorem to complete the proofhttps://cdn.kastatic.org/ka-youtube-converted/YT6XwkcPcsw.mp4/YT6XwkcPcsw.mp4709yes https://puzzicle.org/math/multivariable-calculus/v/type-i-regions-in-three-dimensionshttps://puzzicle.org/thumbs/1254699123.jpgType I regions in three dimensionsDefinition of Type 1 regions. Visualizing what is and isn't a Type I regionhttps://cdn.kastatic.org/ka-youtube-converted/A11AfJ2-3rE.mp4/A11AfJ2-3rE.mp4519yes https://puzzicle.org/math/multivariable-calculus/v/type-ii-regions-in-three-dimensionshttps://puzzicle.org/thumbs/1254469905.jpgType II regions in three dimensionsDefinition and intuition for Type 2 Regionshttps://cdn.kastatic.org/ka-youtube-converted/pZ_2bATPi2I.mp4/pZ_2bATPi2I.mp4449yes https://puzzicle.org/math/multivariable-calculus/v/type-iii-regions-in-three-dimensionshttps://puzzicle.org/thumbs/1254671407.jpgType III regions in three dimensionsDefinition and intuition for type 3 regionshttps://cdn.kastatic.org/ka-youtube-converted/bxGnbVOtRYo.mp4/bxGnbVOtRYo.mp4393yes https://puzzicle.org/math/multivariable-calculus/v/divergence-theorem-proof-part-1https://puzzicle.org/thumbs/1254638405.jpgDivergence theorem proof (part 1)Setting up the proof for the divergence theoremhttps://cdn.kastatic.org/ka-youtube-converted/vCAFlt4pQSQ.mp4/vCAFlt4pQSQ.mp4509yes https://puzzicle.org/math/multivariable-calculus/v/divergence-theorem-proof-part-2https://puzzicle.org/thumbs/1254358889.jpgDivergence theorem proof (part 2)Breaking up the surface integralhttps://cdn.kastatic.org/ka-youtube-converted/OM3e-J3euXo.mp4/OM3e-J3euXo.mp4450yes https://puzzicle.org/math/multivariable-calculus/v/divergence-theorem-proof-part-3https://puzzicle.org/thumbs/1254688165.jpgDivergence theorem proof (part 3)Evaluating the surface integralhttps://cdn.kastatic.org/ka-youtube-converted/3_W6bGuTfzU.mp4/3_W6bGuTfzU.mp4381yes https://puzzicle.org/math/multivariable-calculus/v/divergence-theorem-proof-part-4https://puzzicle.org/thumbs/1254709098.jpgDivergence theorem proof (part 4)More evaluation of the surface integralhttps://cdn.kastatic.org/ka-youtube-converted/lUD7ti47IQc.mp4/lUD7ti47IQc.mp4489yes https://puzzicle.org/math/multivariable-calculus/v/divergence-theorem-proof-part-5https://puzzicle.org/thumbs/1254605394.jpgDivergence theorem proof (part 5)Home stretch. Proving the Type I parthttps://cdn.kastatic.org/ka-youtube-converted/DrRsXhln4S8.mp4/DrRsXhln4S8.mp4192yes https://puzzicle.org/math/precalculus/v/cosine-sine-and-tangent-of-6-and-3https://puzzicle.org/thumbs/xaa39178f36038ef7.jpgCosine, sine and tangent of π/6 and π/3With the unit circle and the Pythagorean theorem, we can find the exact sine, cosine, and tangent of the angles π/6 and π/3.https://cdn.kastatic.org/ka-youtube-converted/Tt_ATh5mCGw.mp4/Tt_ATh5mCGw.mp4528yes https://puzzicle.org/math/precalculus/v/cosine-equation-algebraic-solution-sethttps://puzzicle.org/thumbs/x6fa2fde28005f2a5.jpgCosine equation algebraic solution setSolve a cosine equation with an infinite number of solutions. Use trig identities to represent the whole solution set.https://cdn.kastatic.org/ka-youtube-converted/JoOfBpdaAiw.mp4/JoOfBpdaAiw.mp4373yes https://puzzicle.org/math/precalculus/v/cosine-equation-solution-set-in-an-intervalhttps://puzzicle.org/thumbs/x4f1bc4c6751cd4c4.jpgCosine equation solution set in an intervalGiven the algebraic solution set for a cosine equation, find which solutions fall within an interval.https://cdn.kastatic.org/ka-youtube-converted/0_4uTKBI99o.mp4/0_4uTKBI99o.mp4382yes https://puzzicle.org/math/precalculus/v/sine-equation-algebraic-solution-sethttps://puzzicle.org/thumbs/x096d3067db0caafb.jpgSine equation algebraic solution setSolve a sine equation with an infinite number of solutions. Use trig identities to represent the whole solution set.https://cdn.kastatic.org/ka-youtube-converted/ES15mSuqHM8.mp4/ES15mSuqHM8.mp4336yes https://puzzicle.org/math/precalculus/v/proof-of-the-tangent-angle-sum-and-difference-identitieshttps://puzzicle.org/thumbs/x2b010b5c159918fd.jpgProof of the tangent angle sum and difference identitiesUsing the sine and cosine of the sum or difference of two angles, we can prove: tan(x+y)=(tan(x)+tan(y))/(1-tan(x)tan(y))https://cdn.kastatic.org/ka-youtube-converted/nUlElr4LXz8.mp4/nUlElr4LXz8.mp4415yes https://puzzicle.org/math/precalculus/v/using-the-tangent-angle-addition-identityhttps://puzzicle.org/thumbs/xa5b3c26ba3b1a2e0.jpgUsing the tangent angle addition identityFind the tangent of 13pi/12 without a calculator using the tangent angle addition identity.https://cdn.kastatic.org/ka-youtube-converted/4nPMCRuts9o.mp4/4nPMCRuts9o.mp4389yes https://puzzicle.org/math/precalculus/v/complex-numbers-same-modulushttps://puzzicle.org/thumbs/x2b266422dad332be.jpgComplex numbers with the same modulus (absolute value)Sal shows how to determine which members in a set of complex numbers have the same modulus (or absolute value). He also shows how to visualize all of the complex numbers with a given modulus as a circle centered at the origin on the complex plane, since all points on such a circle are the same distance from the origin.https://cdn.kastatic.org/ka-youtube-converted/uB5QDraFefs.mp4/uB5QDraFefs.mp4172yes https://puzzicle.org/math/precalculus/v/multiply-complex-numbers-polarhttps://puzzicle.org/thumbs/xd88b64481289c668.jpgMultiplying complex numbers in polar formWe can multiply two complex numbers in polar form by multiplying their moduli and adding their arguments.https://cdn.kastatic.org/ka-youtube-converted/VkdXztTFsvM.mp4/VkdXztTFsvM.mp4147yes https://puzzicle.org/math/precalculus/v/divide-complex-polarhttps://puzzicle.org/thumbs/x34e958aff90c3f66.jpgDividing complex numbers in polar formWe can divide two complex numbers in polar form by dividing their moduli and subtracting their arguments.https://cdn.kastatic.org/ka-youtube-converted/lyWaNZ1ERMw.mp4/lyWaNZ1ERMw.mp4190yes https://puzzicle.org/math/precalculus/v/visualizing-powers-complex-numberhttps://puzzicle.org/thumbs/xd60fc5c65ef997ae.jpgTaking and visualizing powers of a complex numberGiven a complex number in rectangular form, we can convert it to polar form to show how to visualize powers of the complex number by scaling and rotating it by its own modulus and argument.https://cdn.kastatic.org/ka-youtube-converted/VZmnLQ3CMbE.mp4/VZmnLQ3CMbE.mp4381yes https://puzzicle.org/math/precalculus/v/parallelogram-rule-for-vector-additionhttps://puzzicle.org/thumbs/x64f19fb133aa602b.jpgParallelogram rule for vector additionThe parallelogram rule says that if we place two vectors so they have the same initial point, and then complete the vectors into a parallelogram, then the sum of the vectors is the directed diagonal that starts at the same point as the vectors.https://cdn.kastatic.org/ka-youtube-converted/Mep0foZMOCg.mp4/Mep0foZMOCg.mp4140yes https://puzzicle.org/math/precalculus/v/subtracting-vectors-with-parallelogram-rulehttps://puzzicle.org/thumbs/xa18760e499aacad7.jpgSubtracting vectors with parallelogram ruleThe parallelogram rule says that if we place two vectors so they have the same initial point, and then complete the vectors into a parallelogram, then the sum of the vectors is the directed diagonal that starts at the same point as the vectors. To subtract two vectors, we simply add the first vector and the opposite of the second vector, i.e., a+b=a+(-b).https://cdn.kastatic.org/ka-youtube-converted/4mRDJf6tShs.mp4/4mRDJf6tShs.mp4166yes https://puzzicle.org/math/precalculus/v/vector-word-problem-resultant-velocityhttps://puzzicle.org/thumbs/x689490466c9eb9af.jpgVector word problem: resultant velocityWhen an object, say, a boat, travels at a certain velocity, and the medium through which it travels, say, a river, has its own velocity, we can find the resultant velocity of the object by adding the two velocities. In this example, we find the resultant velocity vector of a boat.https://cdn.kastatic.org/ka-youtube-converted/Neon4dpvmUk.mp4/Neon4dpvmUk.mp4617yes https://puzzicle.org/math/precalculus/v/vector-word-problem-resultant-forcehttps://puzzicle.org/thumbs/x87912fdee3fc278a.jpgVector word problem: resultant forceWhen two different forces act on the same object, we can find the resultant force acting on the object by adding the two separate forces. In this example, we find a resultant force vector using geometry, specifically the laws of sines and the laws of cosines.https://cdn.kastatic.org/ka-youtube-converted/Ga1Di84GcgM.mp4/Ga1Di84GcgM.mp4432yes https://puzzicle.org/math/precalculus/v/matrices-as-transformations-of-the-planehttps://puzzicle.org/thumbs/x77c3c0ec61006c97.jpgMatrices as transformations of the planeWe can think of a 2X2 matrix as describing a special kind of transformation of the plane (called "linear transformation"). By telling us where the vectors [1,0] and [0,1] are mapped to, we can figure out where any other vector is mapped to.https://cdn.kastatic.org/ka-youtube-converted/PtjdhADnJi4.mp4/PtjdhADnJi4.mp4492yes https://puzzicle.org/math/precalculus/v/working-with-matrices-as-transformations-of-the-planehttps://puzzicle.org/thumbs/x5a4b500a33b2766d.jpgWorking with matrices as transformations of the planeOnce we know how 2X2 matrices define transformations of the plane, we can connect geometric transformations like rotations, reflections, and dilations to specific matrices.https://cdn.kastatic.org/ka-youtube-converted/vCkvVAvmJwY.mp4/vCkvVAvmJwY.mp4255yes https://puzzicle.org/math/precalculus/v/intro-to-determinantshttps://puzzicle.org/thumbs/xa2f9e84b3bcb95a7.jpgIntro to determinant notation and computationMatrix determinants are easy to define and hard to understand. So let's start with defining them and introducing related notation. In other videos we will learn what they mean and how to use them.https://cdn.kastatic.org/ka-youtube-converted/ZAkmBOCiNqU.mp4/ZAkmBOCiNqU.mp4206yes https://puzzicle.org/math/precalculus/v/interpreting-determinants-in-terms-of-areahttps://puzzicle.org/thumbs/xc5e499ea08b67bdc.jpgInterpreting determinants in terms of areaThe determinant of a 2X2 matrix tells us what the area of the image of a unit square would be under the matrix transformation. This, in turn, allows us to tell what the area of the image of any figure would be under the transformation.https://cdn.kastatic.org/ka-youtube-converted/ER_96Mx-KEE.mp4/ER_96Mx-KEE.mp4259yes https://puzzicle.org/math/precalculus/v/finding-area-of-figure-after-transformation-using-determinanthttps://puzzicle.org/thumbs/xa9ae3dc81e2633f3.jpgFinding area of figure after transformation using determinantIn this worked example, Sal finds the area of the image of a rectangle after a transformation defined by a given matrix. This is done by finding the area of the pre-image and multiplying by the matrix's determinant.https://cdn.kastatic.org/ka-youtube-converted/aVuCoUxt3vc.mp4/aVuCoUxt3vc.mp4127yes https://puzzicle.org/math/precalculus/v/proof-matrix-determinant-gives-area-of-image-of-unit-square-under-mappinghttps://puzzicle.org/thumbs/xe5735d39a2f26ed7.jpgProof: Matrix determinant gives area of image of unit square under mappingProof that the determinant of a 2X2 matrix gives the area of the image of the unit square under the mapping defined by the matrix.https://cdn.kastatic.org/ka-youtube-converted/_OiMiQGKvvc.mp4/_OiMiQGKvvc.mp4259yes https://puzzicle.org/math/precalculus/v/using-matrices-to-transform-a-4d-vectorhttps://puzzicle.org/thumbs/x218b7462a2078e66.jpgUsing matrices to transform a 4D vector4X4 matrices can define transformations of the 4D space. In this worked example, we see how to find the image of a given 4D vector under the transformation defined by a given matrix.https://cdn.kastatic.org/ka-youtube-converted/Foy8cAMqnAo.mp4/Foy8cAMqnAo.mp4379yes https://puzzicle.org/math/precalculus/v/composing-3x3-matriceshttps://puzzicle.org/thumbs/x9b5bb488dbd2fa0b.jpgComposing 3x3 matrices3X3 matrices can define transformations of the 3D space. In this worked example, we see how to find the matrix transformation that is the composition of two other matrices.https://cdn.kastatic.org/ka-youtube-converted/2O18RbWmrYA.mp4/2O18RbWmrYA.mp4218yes https://puzzicle.org/math/precalculus/v/inverse-matrix-introductionhttps://puzzicle.org/thumbs/x954f4bbe019f355a.jpgInverse matrix introductionThe inverse of a square matrix is another matrix (of the same dimensions), where the multiplication (or composition) of the two matrices results in the identity matrix. This is analogous to inverse functions (if we think of matrices as functions) or reciprocal numbers (if we think of matrices as special numbers). Fascinating!https://cdn.kastatic.org/ka-youtube-converted/sT9IgHjjhiY.mp4/sT9IgHjjhiY.mp4410yes https://puzzicle.org/math/precalculus/v/invertible-matrices-and-determinantshttps://puzzicle.org/thumbs/x056893a13619a4b9.jpgInvertible matrices and determinantsAn invertible matrix is a matrix that has an inverse. In this video, we investigate the relationship between a matrix's determinant, and whether that matrix is invertible.https://cdn.kastatic.org/ka-youtube-converted/GuFvSwriy1Q.mp4/GuFvSwriy1Q.mp4282yes https://puzzicle.org/math/precalculus/v/invertible-matrices-and-transformationshttps://puzzicle.org/thumbs/x678a9c56ef091bcf.jpgInvertible matrices and transformationsAn invertible matrix is a matrix that has an inverse. In this video, we compare the effect of two transformations, one defined by an invertible matrix and one by a non-invertible matrix. We see that the non-invertible matrix map the entire plane onto a single line.https://cdn.kastatic.org/ka-youtube-converted/S278zOWSsZM.mp4/S278zOWSsZM.mp4368yes https://puzzicle.org/math/precalculus/v/inverse-matrices-and-matrix-equationshttps://puzzicle.org/thumbs/x80af5561a2fc6840.jpgInverse matrices and matrix equationsIn other videos, we've seen how matrices can represent systems of equations, and we've also seen how matrices whose determinant is zero don't have an inverse. In this video, we connect this two understanding, and by doing that way we understand how we can solve systems of equations with matrices.https://cdn.kastatic.org/ka-youtube-converted/a2S0kuXO3gE.mp4/a2S0kuXO3gE.mp4450yes https://puzzicle.org/math/precalculus/v/probability-using-combinationshttps://puzzicle.org/thumbs/25851.jpgProbability using combinationsProbability of getting exactly 3 heads in 8 flips of a fair coin.https://cdn.kastatic.org/ka-youtube-converted/Xqfcy1rqMbI.mp4/Xqfcy1rqMbI.mp4550yes https://puzzicle.org/math/precalculus/v/probability-permutations-combinations-taste-testinghttps://puzzicle.org/thumbs/x4d467bb2723d85e4.jpgProbability with permutations & combinations example: taste testingWe can use combinations (when order does not matter) and permutations (when order does matter) to find probabilities.https://cdn.kastatic.org/ka-youtube-converted/C6gQZ7qKtdM.mp4/C6gQZ7qKtdM.mp4226yes https://puzzicle.org/math/precalculus/v/probability-combinations-choosing-groupshttps://puzzicle.org/thumbs/x14d325c1d92be14a.jpgProbability with combinations example: choosing groupsWe can use two combinations (when order does not matter) to find the probability of someone being included in the group that's chosen at random from a larger group.https://cdn.kastatic.org/ka-youtube-converted/gy8E0-wf4a0.mp4/gy8E0-wf4a0.mp4219yes https://puzzicle.org/math/precalculus/v/probability-combinations-example-choosing-cardshttps://puzzicle.org/thumbs/xafc50db74d0782e4.jpgProbability with combinations example: choosing cardsWe can use combinations (when order does not matter) to find the probability of drawing two aces and two kings in a draw of four cards.https://cdn.kastatic.org/ka-youtube-converted/Tp-SjgG11G8.mp4/Tp-SjgG11G8.mp4149yes https://puzzicle.org/math/precalculus/v/probability-distributions-empirical-datahttps://puzzicle.org/thumbs/xbbfc533d3121f47e.jpgProbability distributions from empirical dataWe can approximate a probability distribution by using empirical or observed data.https://cdn.kastatic.org/ka-youtube-converted/wztjEa7893c.mp4/wztjEa7893c.mp4172yes https://puzzicle.org/math/precalculus/v/probabilities-fair-decisions-examplehttps://puzzicle.org/thumbs/x5b6ea363f742d41d.jpgUsing probabilities to make fair decisions exampleWe can determine whether or not probabilities are being used to make a fair decision. In this example, we look at whether different outcomes have the same probability or not when we roll two dice to make a decision.https://cdn.kastatic.org/ka-youtube-converted/Z_5f3kys52o.mp4/Z_5f3kys52o.mp4214yes https://puzzicle.org/math/precalculus/v/expected-payoff-example-lotteryhttps://puzzicle.org/thumbs/x395fecff8361ddc2.jpgExpected payoff example: lottery ticketWe can find the expected payoff (or the expected net gain) of a certain lottery ticket by taking the weighted average the outcomes.https://cdn.kastatic.org/ka-youtube-converted/Ay1bVzqTKzg.mp4/Ay1bVzqTKzg.mp4244yes https://puzzicle.org/math/precalculus/v/expected-payoff-example-protection-planhttps://puzzicle.org/thumbs/x8a8f30a531ad189f.jpgExpected payoff example: protection planWe can find the expected payoff (or the expected net gain) of a protection plan offered by a store by taking the weighted average of the outcomes.https://cdn.kastatic.org/ka-youtube-converted/mKPeuVjPDo0.mp4/mKPeuVjPDo0.mp4158yes https://puzzicle.org/math/precalculus-tx/v/interpreting-a-transformed-polynomial-functions-graph-precalculus-khan-academyhttps://puzzicle.org/thumbs/x5cacfd43c8c54839.jpgInterpreting a transformed polynomial function's graphGiven a function and a transformed version of this function, interpret the new meaning of the key features in context.https://cdn.kastatic.org/ka-youtube-converted/Ftz3jDAEPfo.mp4/Ftz3jDAEPfo.mp4184yes https://puzzicle.org/math/precalculus-tx/v/solving-polynomial-functions-using-their-graphs-precalculus-khan-academyhttps://puzzicle.org/thumbs/x9e2110e526bfb84e.jpgSolving polynomial functions using their graphsFind the values of x when a function takes on certain y-values.https://cdn.kastatic.org/ka-youtube-converted/H-N4QpKJQoY.mp4/H-N4QpKJQoY.mp4195yes https://puzzicle.org/math/precalculus-tx/v/solving-polynomials-using-a-table-precalculus-khan-academyhttps://puzzicle.org/thumbs/x75e4a03990e669d6.jpgSolving polynomials using a tableFinding zeroes of a polynomial function from a table and estimating where a zero may live if it is not on the table.https://cdn.kastatic.org/ka-youtube-converted/R4l_ySDoZJ8.mp4/R4l_ySDoZJ8.mp4203yes https://puzzicle.org/math/precalculus-tx/v/solving-polynomial-inequalities-in-factored-formhttps://puzzicle.org/thumbs/x6f4a831b2f62952a.jpgSolving polynomial inequalities in factored formLearn how to solve a polynomial inequality when provided factored form.https://cdn.kastatic.org/ka-youtube-converted/4DUipIIwFfc.mp4/4DUipIIwFfc.mp4414yes https://puzzicle.org/math/precalculus-tx/v/solving-applications-of-polynomial-inequalities-precalculus-khan-academyhttps://puzzicle.org/thumbs/x7329f07b37bbce3c.jpgSolving applications of polynomial inequalitiesLearn how to apply knowledge of polynomial inequalities to real world contexts and apply solving skills to answer questions.https://cdn.kastatic.org/ka-youtube-converted/7AOvarRJy5w.mp4/7AOvarRJy5w.mp4342yes https://puzzicle.org/math/precalculus-tx/v/solving-factored-form-rational-inequalitieshttps://puzzicle.org/thumbs/x928b9c2467a894dc.jpgSolving factored form rational inequalities by testing intervalsIdentify when a rational inequality provided in factored form is true in terms of zero by testing intervals.https://cdn.kastatic.org/ka-youtube-converted/hER6MasSSww.mp4/hER6MasSSww.mp4482yes https://puzzicle.org/math/precalculus-tx/v/solving-rational-inequalities-with-rational-expressions-on-both-sideshttps://puzzicle.org/thumbs/x9515899b84d4fb45.jpgSolving rational inequalities with rational expressions on both sidesLearn how to solve rational inequalities where there is a rational expression on both sides of the inequality sign.https://cdn.kastatic.org/ka-youtube-converted/eQT08EvLW1k.mp4/eQT08EvLW1k.mp4479yes https://puzzicle.org/math/precalculus-tx/v/recursive-formula-for-geometric-sequencehttps://puzzicle.org/thumbs/x2000d28fbb4e08c2.jpgRecursive formula for geometric sequencesWrite the recursive formula for a geometric sequence by finding the common ratio and designated initial term.https://cdn.kastatic.org/ka-youtube-converted/GVLPBC-T2vU.mp4/GVLPBC-T2vU.mp4137yes https://puzzicle.org/math/precalculus-tx/v/equations-of-sequence-patternshttps://puzzicle.org/thumbs/26788.jpgSequences word problem: growth patternSal finds the equation that describes a growth pattern of shapes made of squares.https://cdn.kastatic.org/ka-youtube-converted/_3BnyEr5fG4.mp4/_3BnyEr5fG4.mp4516yes https://puzzicle.org/math/precalculus-tx/v/using-the-quotient-identities-to-prove-trig-identitieshttps://puzzicle.org/thumbs/x93f600e314e538af.jpgUsing the quotient identities to prove trig identitiesUsing the quotient identities to prove trig identities.https://cdn.kastatic.org/ka-youtube-converted/vcFiReQah40.mp4/vcFiReQah40.mp4122yes https://puzzicle.org/math/precalculus-tx/v/identifying-if-a-parabola-exists-from-given-key-featureshttps://puzzicle.org/thumbs/x577a85716c650fd9.jpgIdentifying if a parabola exists from given key featuresUse the distance formula to identify if a parabola can exist when provided a point, focus and directrix.https://cdn.kastatic.org/ka-youtube-converted/HyirU1YnzX0.mp4/HyirU1YnzX0.mp4279yes https://puzzicle.org/math/precalculus-tx/v/identifying-if-a-ellipse-or-hyperbola-exists-from-given-key-featureshttps://puzzicle.org/thumbs/x91e69c663864316b.jpgIdentifying if an ellipse or hyperbola exists from given key featuresIdentify if the provided points and foci can exist on an ellipse or hyperbola, or both.https://cdn.kastatic.org/ka-youtube-converted/NGKCekLhErg.mp4/NGKCekLhErg.mp4302yes https://puzzicle.org/math/precalculus-tx/v/identifying-the-foci-of-an-ellipse-using-the-radiihttps://puzzicle.org/thumbs/xed05297687a9eb01.jpgIdentifying the foci of an ellipse using the radiiUsing the graph of an ellipse to identify the foci by identifying the radii on the major and minor axes.https://cdn.kastatic.org/ka-youtube-converted/wGHqt0RwHxY.mp4/wGHqt0RwHxY.mp4150yes https://puzzicle.org/math/precalculus-tx/v/proving-the-formula-for-focal-length-of-an-ellipsehttps://puzzicle.org/thumbs/x34d88ec81133a03f.jpgProving the formula for focal length of an ellipseSal proves how to write the formula used for focal length of an ellipse.https://cdn.kastatic.org/ka-youtube-converted/ze-ij372f88.mp4/ze-ij372f88.mp4194yes https://puzzicle.org/math/statistics-probability/v/reading-pictographshttps://puzzicle.org/thumbs/380860451.jpgReading pictographsCommon sense tells you that a pictograph must be some kind of....picture. Yes! A pictograph is basically a way to represent data with pictures that relate to the data. Let's look at this word problem that asks to interpret a pictograph.https://cdn.kastatic.org/ka-youtube-converted/qrVvpYt3Vl0.mp4/qrVvpYt3Vl0.mp4141yes https://puzzicle.org/math/statistics-probability/v/reading-bar-graphshttps://puzzicle.org/thumbs/380854778.jpgReading bar graphsReading Bar Graphs.https://cdn.kastatic.org/ka-youtube-converted/kiQ6MUQZHSs.mp4/kiQ6MUQZHSs.mp4179yes https://puzzicle.org/math/statistics-probability/v/reading-bar-charts-3https://puzzicle.org/thumbs/xb3b8475c.jpgReading bar charts: putting it together with central tendencyLet's combine some learning! Here we looking at bar charts and using the information to determine the measures of central tendency.https://cdn.kastatic.org/ka-youtube-converted/DGZNaKnbQo0.mp4/DGZNaKnbQo0.mp4281yes https://puzzicle.org/math/statistics-probability/v/histogramshttps://puzzicle.org/thumbs/23768.jpgHistogramsHere's how we make a histogram: 1. Collect your data and decide on the number and size of bins (categories) you want to divide your data into. 2. Count the number of data points that fall within each bin. 3. Draw a graph with the bins as the x-axis and the frequency counts as the y-axis. 4. Draw vertical bars to represent the frequency count for each bin.https://cdn.kastatic.org/ka-youtube-converted/4eLJGG2Ad30.mp4/4eLJGG2Ad30.mp4368yes https://puzzicle.org/math/statistics-probability/v/u08-l1-t2-we2-reading-line-graphshttps://puzzicle.org/thumbs/380860452.jpgReading line graphsA line graph is way to visually represent data, especially data that changes over time. Let's take a look at an example.https://cdn.kastatic.org/ka-youtube-converted/36v2EXZRzUE.mp4/36v2EXZRzUE.mp4141yes https://puzzicle.org/math/statistics-probability/v/misleading-line-graphshttps://puzzicle.org/thumbs/380855564.jpgMisleading line graphsMisleading Line Graphshttps://cdn.kastatic.org/ka-youtube-converted/gs-OPF3KEGU.mp4/gs-OPF3KEGU.mp4293yes https://puzzicle.org/math/statistics-probability/v/statistics-alternate-variance-formulashttps://puzzicle.org/thumbs/26181.jpgStatistics: Alternate variance formulasSal explains a different variance formula and why it works! For a population, the variance is calculated as σ² = ( Σ (x-μ)² ) / N. Another equivalent formula is σ² = ( (Σ x²) / N ) - μ². If we need to calculate variance by hand, this alternate formula is easier to work with.https://cdn.kastatic.org/ka-youtube-converted/sRVGcYGjUk8.mp4/sRVGcYGjUk8.mp4737yes https://puzzicle.org/math/statistics-probability/v/range-and-mid-rangehttps://puzzicle.org/thumbs/380854781.jpgRange and mid-rangeThe range is the difference between the largest and smallest data points in a set of numerical data. The midrange is the average of the largest and smallest data points. Range is an easy to calculate measure of variability, while midrange is an easy to calculate measure of central tendency.https://cdn.kastatic.org/ka-youtube-converted/7DtWXEPB_AI.mp4/7DtWXEPB_AI.mp4123yes https://puzzicle.org/math/statistics-probability/v/ck12-org-exercise-standard-normal-distribution-and-the-empirical-rulehttps://puzzicle.org/thumbs/26204.jpgStandard normal distribution and the empirical rule (from ck12.org)Using the Empirical Rule with a standard normal distributionhttps://cdn.kastatic.org/ka-youtube-converted/2fzYE-Emar0.mp4/2fzYE-Emar0.mp4496yes https://puzzicle.org/math/statistics-probability/v/ck12-org-more-empirical-rule-and-z-score-practicehttps://puzzicle.org/thumbs/26206.jpgMore empirical rule and z-score practice (from ck12.org)More Empirical Rule and Z-score practicehttps://cdn.kastatic.org/ka-youtube-converted/itQEwESWDKg.mp4/itQEwESWDKg.mp4357yes https://puzzicle.org/math/statistics-probability/v/introduction-to-the-normal-distributionhttps://puzzicle.org/thumbs/26196.jpgDeep definition of the normal distributionWe take an extremely deep dive into the normal distribution to explore the parent function that generates normal distributions, and how to modify parameters in the function to produce a normal distribution with any given mean and standard deviation. We also look at relative frequency as area under the normal distribution.https://cdn.kastatic.org/ka-youtube-converted/hgtMWR3TFnY.mp4/hgtMWR3TFnY.mp41584yes https://puzzicle.org/math/statistics-probability/v/normal-distribution-excel-exercisehttps://puzzicle.org/thumbs/26194.jpgNormal distribution excel exercise(Long-26 minutes) Presentation on spreadsheet to show that the normal distribution approximates the binomial distribution for a large number of trials.https://cdn.kastatic.org/ka-youtube-converted/yTGEMoaWDCQ.mp4/yTGEMoaWDCQ.mp41564yes https://puzzicle.org/math/statistics-probability/v/standard-deviation-of-residuals-or-root-mean-square-error-rmsdhttps://puzzicle.org/thumbs/xa5ebe5224c12ce0f.jpgStandard deviation of residuals or Root-mean-square error (RMSD)Calculating the standard deviation of residuals (or root-mean-square error (RMSD) or root-mean-square deviation (RMSD)) to measure disagreement between a linear regression model and a set of data.https://cdn.kastatic.org/ka-youtube-converted/zMFdb__sUpw.mp4/zMFdb__sUpw.mp4403yes https://puzzicle.org/math/statistics-probability/v/squared-error-of-regression-linehttps://puzzicle.org/thumbs/8187292.jpgSquared error of regression lineIntroduction to the idea that one can find a line that minimizes the squared distances to the pointshttps://cdn.kastatic.org/ka-youtube-converted/6OvhLPS7rj4.mp4/6OvhLPS7rj4.mp4407yes https://puzzicle.org/math/statistics-probability/v/proof-part-1-minimizing-squared-error-to-regression-linehttps://puzzicle.org/thumbs/8187293.jpgProof (part 1) minimizing squared error to regression lineProof (Part 1) Minimizing Squared Error to Regression Linehttps://cdn.kastatic.org/ka-youtube-converted/mIx2Oj5y9Q8.mp4/mIx2Oj5y9Q8.mp4635yes https://puzzicle.org/math/statistics-probability/v/proof-part-2-minimizing-squared-error-to-linehttps://puzzicle.org/thumbs/8187297.jpgProof (part 2) minimizing squared error to regression lineProof Part 2 Minimizing Squared Error to Linehttps://cdn.kastatic.org/ka-youtube-converted/f6OnoxctvUk.mp4/f6OnoxctvUk.mp4594yes https://puzzicle.org/math/statistics-probability/v/proof-part-3-minimizing-squared-error-to-regression-linehttps://puzzicle.org/thumbs/8187294.jpgProof (part 3) minimizing squared error to regression lineProof (Part 3) Minimizing Squared Error to Regression Linehttps://cdn.kastatic.org/ka-youtube-converted/u1HhUB3NP8g.mp4/u1HhUB3NP8g.mp4654yes https://puzzicle.org/math/statistics-probability/v/proof-part-4-minimizing-squared-error-to-regression-linehttps://puzzicle.org/thumbs/8187295.jpgProof (part 4) minimizing squared error to regression lineProof (Part 4) Minimizing Squared Error to Regression Linehttps://cdn.kastatic.org/ka-youtube-converted/8RSTQl0bQuw.mp4/8RSTQl0bQuw.mp4258yes https://puzzicle.org/math/statistics-probability/v/regression-line-examplehttps://puzzicle.org/thumbs/8187296.jpgRegression line exampleRegression Line Examplehttps://cdn.kastatic.org/ka-youtube-converted/GAmzwIkGFgE.mp4/GAmzwIkGFgE.mp4567yes https://puzzicle.org/math/statistics-probability/v/second-regression-examplehttps://puzzicle.org/thumbs/8187299.jpgSecond regression exampleSecond Regression Examplehttps://cdn.kastatic.org/ka-youtube-converted/ww_yT9ckPWw.mp4/ww_yT9ckPWw.mp4555yes https://puzzicle.org/math/statistics-probability/v/calculating-r-squaredhttps://puzzicle.org/thumbs/8187300.jpgCalculating R-squaredCalculating R-Squared to see how well a regression line fits datahttps://cdn.kastatic.org/ka-youtube-converted/Fc5t_5r_7IU.mp4/Fc5t_5r_7IU.mp4585yes https://puzzicle.org/math/statistics-probability/v/covariance-and-the-regression-linehttps://puzzicle.org/thumbs/8187301.jpgCovariance and the regression lineCovariance, Variance and the Slope of the Regression Linehttps://cdn.kastatic.org/ka-youtube-converted/ualmyZiPs9w.mp4/ualmyZiPs9w.mp4908yes https://puzzicle.org/math/statistics-probability/v/probability-decisionshttps://puzzicle.org/thumbs/x9ecaa35a.jpgPicking fairlySal determines if a few different methods of a magician choosing a volunteer are fair.https://cdn.kastatic.org/ka-youtube-converted/vst006REeTQ.mp4/vst006REeTQ.mp4419yes https://puzzicle.org/math/statistics-probability/v/frequency-probability-and-unfair-coinshttps://puzzicle.org/thumbs/191769947.jpgProbability without equally likely eventsUp until now, we've looked at probabilities surrounding only equally likely events. What about probabilities when we don't have equally likely events? Say, we have unfair coins?https://cdn.kastatic.org/ka-youtube-converted/RI874OSJp1U.mp4/RI874OSJp1U.mp4538yes https://puzzicle.org/math/statistics-probability/v/independent-events-3https://puzzicle.org/thumbs/27096.jpgDie rolling probability with independent eventsWe hope you're not a gambler, but if you had to bet on whether you can roll even numbers three times in a row, you might want to figure this probability first.https://cdn.kastatic.org/ka-youtube-converted/2MYA8Ba2PvM.mp4/2MYA8Ba2PvM.mp4148yes https://puzzicle.org/math/statistics-probability/v/dependent-probability-example-2https://puzzicle.org/thumbs/94879657.jpgDependent probability exampleIt's important to practice these probability problems as they get more complex eventually. Take a stab on this one...with our help, of course.https://cdn.kastatic.org/ka-youtube-converted/7BkcNLOf56w.mp4/7BkcNLOf56w.mp4394yes https://puzzicle.org/math/statistics-probability/v/analyzing-dependent-probabilityhttps://puzzicle.org/thumbs/x850f9277.jpgDependent probabilitySal finds dependent probabilities like P(A | B) using an example rolling dice.https://cdn.kastatic.org/ka-youtube-converted/8iKsY8C5VW8.mp4/8iKsY8C5VW8.mp4570yes https://puzzicle.org/math/statistics-probability/v/probability-and-combinations-part-2https://puzzicle.org/thumbs/25853.jpgProbability & combinations (2 of 2)Making at least 3 out of 5 free throws.https://cdn.kastatic.org/ka-youtube-converted/W7DmsJKLoxc.mp4/W7DmsJKLoxc.mp4636yes https://puzzicle.org/math/statistics-probability/v/generalizing-with-binomial-coefficients-bit-advancedhttps://puzzicle.org/thumbs/193210639.jpgGeneralizing with binomial coefficients (bit advanced)Conceptual understanding of where the formula for binomial coefficients come fromhttps://cdn.kastatic.org/ka-youtube-converted/AOsWph2FNLw.mp4/AOsWph2FNLw.mp4700yes https://puzzicle.org/math/statistics-probability/v/conditional-probability-and-combinationshttps://puzzicle.org/thumbs/25855.jpgConditional probability and combinationsProbability that I picked a fair coin given that I flipped 4 heads out of 6 tosses.https://cdn.kastatic.org/ka-youtube-converted/xw6utjoyMi4.mp4/xw6utjoyMi4.mp41012yes https://puzzicle.org/math/statistics-probability/v/birthday-probability-problemhttps://puzzicle.org/thumbs/25857.jpgBirthday probability problemThe probability that at least 2 people in a room of 30 share the same birthday.https://cdn.kastatic.org/ka-youtube-converted/9G0w61pZPig.mp4/9G0w61pZPig.mp4796yes https://puzzicle.org/math/statistics-probability/v/probability-density-functionshttps://puzzicle.org/thumbs/25861.jpgProbability density functionsProbability density functions for continuous random variables.https://cdn.kastatic.org/ka-youtube-converted/Fvi9A_tEmXQ.mp4/Fvi9A_tEmXQ.mp4602yes https://puzzicle.org/math/statistics-probability/v/binomial-distributionhttps://puzzicle.org/thumbs/xaf13e4df.jpgBinomial distributionSal introduces the binomial distribution with an example.https://cdn.kastatic.org/ka-youtube-converted/WWv0RUxDfbs.mp4/WWv0RUxDfbs.mp4712yes https://puzzicle.org/math/statistics-probability/v/visualizing-a-binomial-distributionhttps://puzzicle.org/thumbs/x0f59b8e3.jpgVisualizing a binomial distributionSal walks through graphing a binomial distribution and connects it back to how to calculate binomial probabilities.https://cdn.kastatic.org/ka-youtube-converted/NF0lrkqXIkQ.mp4/NF0lrkqXIkQ.mp4567yes https://puzzicle.org/math/statistics-probability/v/mean-and-variance-of-bernoulli-distribution-examplehttps://puzzicle.org/thumbs/6022798.jpgMean and variance of Bernoulli distribution exampleSal calculates the mean and variance of a Bernoulli distribution (in this example the responses are either favorable or unfavorable).https://cdn.kastatic.org/ka-youtube-converted/O8vB1eInP_8.mp4/O8vB1eInP_8.mp4500yes https://puzzicle.org/math/statistics-probability/v/bernoulli-distribution-mean-and-variance-formulashttps://puzzicle.org/thumbs/6022799.jpgBernoulli distribution mean and variance formulasSal continues on from the previous video to derive the mean and variance formulas for the Bernoulli distribution.https://cdn.kastatic.org/ka-youtube-converted/ry81_iSHt6E.mp4/ry81_iSHt6E.mp4419yes https://puzzicle.org/math/statistics-probability/v/empirical-data-expected-valuehttps://puzzicle.org/thumbs/x63d8573a.jpgGetting data from expected valueExpected value refers to the average outcome you would expect from repeating an experiment over and over. It is calculated by multiplying each possible outcome by its probability of occurring, and summing those products together. If we know the expected value, we can go backwards and solve for frequency.https://cdn.kastatic.org/ka-youtube-converted/gHpl5UcgRkI.mp4/gHpl5UcgRkI.mp4592yes https://puzzicle.org/math/statistics-probability/v/law-of-large-numbershttps://puzzicle.org/thumbs/25879.jpgLaw of large numbersSal introduces the magic behind the law of large numbers.https://cdn.kastatic.org/ka-youtube-converted/VpuN8vCQ--M.mp4/VpuN8vCQ--M.mp4539yes https://puzzicle.org/math/statistics-probability/v/poisson-process-1https://puzzicle.org/thumbs/25875.jpgPoisson process 1Introduction to Poisson Processes and the Poisson Distribution.https://cdn.kastatic.org/ka-youtube-converted/3z-M6sbGIZ0.mp4/3z-M6sbGIZ0.mp4661yes https://puzzicle.org/math/statistics-probability/v/poisson-process-2https://puzzicle.org/thumbs/25877.jpgPoisson process 2More of the derivation of the Poisson Distribution.https://cdn.kastatic.org/ka-youtube-converted/Jkr4FSrNEVY.mp4/Jkr4FSrNEVY.mp4761yes https://puzzicle.org/math/statistics-probability/v/confidence-interval-examplehttps://puzzicle.org/thumbs/6022802.jpgConfidence interval exampleSal calculates a 99% confidence interval for the proportion of teachers who felt computers are an essential tool.https://cdn.kastatic.org/ka-youtube-converted/SeQeYVJZ2gE.mp4/SeQeYVJZ2gE.mp41116yes https://puzzicle.org/math/statistics-probability/v/margin-of-error-1https://puzzicle.org/thumbs/6022800.jpgMargin of error 1Finding the 95% confidence interval for the proportion of a population voting for a candidate.https://cdn.kastatic.org/ka-youtube-converted/OwPSuHXmiPw.mp4/OwPSuHXmiPw.mp4902yes https://puzzicle.org/math/statistics-probability/v/margin-of-error-2https://puzzicle.org/thumbs/6022801.jpgMargin of error 2Finding the 95% confidence interval for the proportion of a population voting for a candidate.https://cdn.kastatic.org/ka-youtube-converted/YclQE2XtaLw.mp4/YclQE2XtaLw.mp4605yes https://puzzicle.org/math/statistics-probability/v/sample-size-for-a-given-margin-of-error-for-a-meanhttps://puzzicle.org/thumbs/xeddcb5225895f2fd.jpgSample size for a given margin of error for a meanCalculate the approximate sample size required to obtain a desired margin of error in a confidence interval for a mean.https://cdn.kastatic.org/ka-youtube-converted/FSVDJoHZ3rM.mp4/FSVDJoHZ3rM.mp4462yes https://puzzicle.org/math/statistics-probability/v/t-statistic-confidence-intervalhttps://puzzicle.org/thumbs/6022809.jpgT-statistic confidence intervalSal computes a confidence interval for the emission from an engine with a new design.https://cdn.kastatic.org/ka-youtube-converted/hV4pdjHCKuA.mp4/hV4pdjHCKuA.mp4707yes https://puzzicle.org/math/statistics-probability/v/small-sample-size-confidence-intervalshttps://puzzicle.org/thumbs/6022803.jpgSmall sample size confidence intervalsConstructing small sample size confidence intervals using t-distributionshttps://cdn.kastatic.org/ka-youtube-converted/K4KDLWENXm0.mp4/K4KDLWENXm0.mp4671yes https://puzzicle.org/math/statistics-probability/v/simple-hypothesis-testinghttps://puzzicle.org/thumbs/xb0f85f82.jpgSimple hypothesis testingSal walks through an example about who should do the dishes that gets at the idea behind hypothesis testing.https://cdn.kastatic.org/ka-youtube-converted/5D1gV37bKXY.mp4/5D1gV37bKXY.mp4385yes https://puzzicle.org/math/statistics-probability/v/type-1-errorshttps://puzzicle.org/thumbs/6022807.jpgType 1 errorsA Type 1 error occurs when the null hypothesis is true, but we reject it because of an usual sample result.https://cdn.kastatic.org/ka-youtube-converted/EowIec7Y8HM.mp4/EowIec7Y8HM.mp4204yes https://puzzicle.org/math/statistics-probability/v/hypothesis-testing-and-p-valueshttps://puzzicle.org/thumbs/6022804.jpgHypothesis testing and p-valuesSal walks through an example about a neurologist testing the effect of a drug to discuss hypothesis testing and p-values.https://cdn.kastatic.org/ka-youtube-converted/-FtlH4svqx4.mp4/-FtlH4svqx4.mp4686yes https://puzzicle.org/math/statistics-probability/v/one-tailed-and-two-tailed-testshttps://puzzicle.org/thumbs/6022805.jpgOne-tailed and two-tailed testsSal continues his discussion of the effect of a drug to one-tailed and two-tailed hypothesis tests.https://cdn.kastatic.org/ka-youtube-converted/mvye6X_0upA.mp4/mvye6X_0upA.mp4394yes https://puzzicle.org/math/statistics-probability/v/z-statistics-vs-t-statisticshttps://puzzicle.org/thumbs/6022806.jpgZ-statistics vs. T-statisticsSal breaks down the difference between Z-statistics and T-statistics.https://cdn.kastatic.org/ka-youtube-converted/5ABpqVSx33I.mp4/5ABpqVSx33I.mp4399yes https://puzzicle.org/math/statistics-probability/v/small-sample-hypothesis-testhttps://puzzicle.org/thumbs/6022808.jpgSmall sample hypothesis testSal walks through an example of a hypothesis test where he determines if there is sufficient evidence to conclude that a new type of engine meets emission requirements.https://cdn.kastatic.org/ka-youtube-converted/D2sMsmL0ScQ.mp4/D2sMsmL0ScQ.mp4544yes https://puzzicle.org/math/statistics-probability/v/large-sample-proportion-hypothesis-testinghttps://puzzicle.org/thumbs/6022810.jpgLarge sample proportion hypothesis testingSal uses a large sample to test if more than 30% of US households have internet access.https://cdn.kastatic.org/ka-youtube-converted/1JT9oODsClE.mp4/1JT9oODsClE.mp4870yes https://puzzicle.org/math/statistics-probability/v/comparing-population-proportions-1https://puzzicle.org/thumbs/6022816.jpgComparing population proportions 1Sal uses an election example to compare population proportions.https://cdn.kastatic.org/ka-youtube-converted/a1Ye5RcWOqg.mp4/a1Ye5RcWOqg.mp4647yes https://puzzicle.org/math/statistics-probability/v/comparing-population-proportions-2https://puzzicle.org/thumbs/6022817.jpgComparing population proportions 2Sal continues the election example for population proportions.https://cdn.kastatic.org/ka-youtube-converted/MNbat1lrJW4.mp4/MNbat1lrJW4.mp4600yes https://puzzicle.org/math/statistics-probability/v/hypothesis-test-comparing-population-proportionshttps://puzzicle.org/thumbs/6022840.jpgHypothesis test comparing population proportionsOnce again, Sal continues the discussion of election results to run a hypothesis test comparing population proportions.https://cdn.kastatic.org/ka-youtube-converted/dvSa_tx04hw.mp4/dvSa_tx04hw.mp4973yes https://puzzicle.org/math/statistics-probability/v/statistical-significance-on-bus-speedshttps://puzzicle.org/thumbs/x5e1b1135.jpgStatistical significance on bus speedsSal determines if the results of an experiment about bus speeds are statistically significant.https://cdn.kastatic.org/ka-youtube-converted/W3C07uH-b9o.mp4/W3C07uH-b9o.mp4696yes https://puzzicle.org/math/statistics-probability/v/difference-of-sample-means-distributionhttps://puzzicle.org/thumbs/6022812.jpgDifference of sample means distributionSal walks through the difference of sample means distribution.https://cdn.kastatic.org/ka-youtube-converted/TcIDXqmt74A.mp4/TcIDXqmt74A.mp4738yes https://puzzicle.org/math/statistics-probability/v/confidence-interval-of-difference-of-meanshttps://puzzicle.org/thumbs/6022813.jpgConfidence interval of difference of meansSal uses a confidence interval to help figure out if a low-fat diet helps obese people lose weight.https://cdn.kastatic.org/ka-youtube-converted/hxZ6uooEJOk.mp4/hxZ6uooEJOk.mp4949yes https://puzzicle.org/math/statistics-probability/v/clarification-of-confidence-interval-of-difference-of-meanshttps://puzzicle.org/thumbs/6022814.jpgClarification of confidence interval of difference of meansClarification of Confidence Interval of Difference of Meanshttps://cdn.kastatic.org/ka-youtube-converted/yQsCMnz9wO8.mp4/yQsCMnz9wO8.mp4161yes https://puzzicle.org/math/statistics-probability/v/hypothesis-test-for-difference-of-meanshttps://puzzicle.org/thumbs/6022815.jpgHypothesis test for difference of meansHypothesis Test for Difference of Meanshttps://cdn.kastatic.org/ka-youtube-converted/N984XGLjQfs.mp4/N984XGLjQfs.mp4606yes https://puzzicle.org/math/statistics-probability/v/chi-square-distribution-introductionhttps://puzzicle.org/thumbs/8187302.jpgChi-square distribution introductionChi-Square Distribution Introductionhttps://cdn.kastatic.org/ka-youtube-converted/dXB3cUGnaxQ.mp4/dXB3cUGnaxQ.mp4623yes https://puzzicle.org/math/statistics-probability/v/pearson-s-chi-square-test-goodness-of-fithttps://puzzicle.org/thumbs/8187303.jpgPearson's chi square test (goodness of fit)Sal uses the chi square test to the hypothesis that the owner's distribution is correct.https://cdn.kastatic.org/ka-youtube-converted/2QeDRsxSF9M.mp4/2QeDRsxSF9M.mp4708yes https://puzzicle.org/math/statistics-probability/v/frequency-table-independent-eventshttps://puzzicle.org/thumbs/x4a98d13d.jpgFilling out frequency table for independent eventsGive the row and column totals, Sal fills in the cells of a frequency table so that the events are independent.https://cdn.kastatic.org/ka-youtube-converted/TQJQeTpH_lw.mp4/TQJQeTpH_lw.mp4296yes https://puzzicle.org/math/statistics-probability/v/contingency-table-chi-square-testhttps://puzzicle.org/thumbs/8187304.jpgContingency table chi-square testSal uses the contingency table chi-square test to see if a couple of different herbs prevent people from getting sick.https://cdn.kastatic.org/ka-youtube-converted/hpWdDmgsIRE.mp4/hpWdDmgsIRE.mp41057yes https://puzzicle.org/math/statistics-probability/v/anova-1-calculating-sst-total-sum-of-squareshttps://puzzicle.org/thumbs/8187305.jpgANOVA 1: Calculating SST (total sum of squares)Analysis of Variance 1 - Calculating SST (Total Sum of Squares)https://cdn.kastatic.org/ka-youtube-converted/EFdlFoHI_0I.mp4/EFdlFoHI_0I.mp4459yes https://puzzicle.org/math/statistics-probability/v/anova-2-calculating-ssw-and-ssb-total-sum-of-squares-within-and-between-avihttps://puzzicle.org/thumbs/8187306.jpgANOVA 2: Calculating SSW and SSB (total sum of squares within and between)Analysis of Variance 2 - Calculating SSW and SSB (Total Sum of Squares Within and Between)https://cdn.kastatic.org/ka-youtube-converted/j9ZPMlVHJVs.mp4/j9ZPMlVHJVs.mp4800yes https://puzzicle.org/math/statistics-probability/v/anova-3-hypothesis-test-with-f-statistichttps://puzzicle.org/thumbs/8187307.jpgANOVA 3: Hypothesis test with F-statisticAnalysis of Variance 3 -Hypothesis Test with F-Statistichttps://cdn.kastatic.org/ka-youtube-converted/Xg8_iSkJpAE.mp4/Xg8_iSkJpAE.mp4614yes https://puzzicle.org/math/statistics-tx/v/comparing-methods-for-determining-best-fithttps://puzzicle.org/thumbs/x6ae5f31302da057e.jpgComparing methods for determining best fitIn this video, Sal compares and contrasts different methods used to determine a line of best fit, including least squares, median-median, and least absolute value regression.https://cdn.kastatic.org/ka-youtube-converted/QkRX8vSmSJc.mp4/QkRX8vSmSJc.mp4127yes https://puzzicle.org/math/statistics-tx/v/determining-if-slope-or-y-intercept-is-reasonablehttps://puzzicle.org/thumbs/xc524b331fecdd48e.jpgDetermining if slope or y-intercept is reasonableIn this video, Sal shows how to determine if the slope or y-intercept of a linear model is reasonable in context.https://cdn.kastatic.org/ka-youtube-converted/tZxAWJTPveo.mp4/tZxAWJTPveo.mp4120yes https://puzzicle.org/math/statistics-tx/v/identifying-sources-of-variabilityhttps://puzzicle.org/thumbs/x0c2d1b8eb3b824db.jpgIdentifying sources of variabilitySal examines two pairs of scenarios, and identifies which scenario exhibits a specific source of variability and why.https://cdn.kastatic.org/ka-youtube-converted/cjiBPL0HJTc.mp4/cjiBPL0HJTc.mp4246yes https://puzzicle.org/math/statistics-tx/v/independent-events-example-test-taking-indian-accenthttps://puzzicle.org/thumbs/x72ce2672bff96e0c.jpgIndependent events example: test takingHave you ever taken a test and discovered you have no choice but to guess on a couple of problems? In this example problem, we are considering the probability of two independent events occurring.https://cdn.kastatic.org/ka-youtube-converted/DtQzDcIQYsw.mp4/DtQzDcIQYsw.mp4236yes https://puzzicle.org/math/statistics-tx/v/constructing-distributions-from-simulationshttps://puzzicle.org/thumbs/xcc38c0762f4dc943.jpgConstructing distributions from simulationsIn this video, Sal examines a simulation context to identify random variables and their possible values, organize simulation data into frequency tables, and choose an appropriate data representation.https://cdn.kastatic.org/ka-youtube-converted/1LnzkYvI6Fs.mp4/1LnzkYvI6Fs.mp4322yes https://puzzicle.org/math/statistics-tx/v/identifying-mathematical-and-statistical-modelshttps://puzzicle.org/thumbs/x0d7a5487ad0dd846.jpgIdentifying mathematical and statistical modelsIn this video, Sal explains how to distinguish between mathematical models (exact, deterministic relationships) and statistical models (relationships based on data, with variation and uncertainty).https://cdn.kastatic.org/ka-youtube-converted/OBBbgvA5euc.mp4/OBBbgvA5euc.mp4198yes https://puzzicle.org/math/statistics-tx/v/comparing-simulated-and-theoretical-sampling-distributions-proportionshttps://puzzicle.org/thumbs/xce4ef334fd8c22ce.jpgComparing simulated and theoretical sampling distributions - proportionsIn this video, Sal uses information about a population and the results of a simulated sampling distribution for a proportion to determine whether the simulation is consistent with the theoretical sampling distribution for that population.https://cdn.kastatic.org/ka-youtube-converted/3xo-3TogBcE.mp4/3xo-3TogBcE.mp4262yes https://puzzicle.org/math/statistics-tx/v/comparing-simulated-and-theoretical-sampling-distributions-meanshttps://puzzicle.org/thumbs/xfff8a1f4961f0ecc.jpgComparing simulated and theoretical sampling distributions - meansIn this video, Sal uses information about a population and the results of a simulated sampling distribution for a mean to determine whether the simulation is consistent with the theoretical sampling distribution for that population.https://cdn.kastatic.org/ka-youtube-converted/1nV77gNOPW4.mp4/1nV77gNOPW4.mp4209yes https://puzzicle.org/science/teacher-resources-tx-teks-science/v/how-to-use-teks-aligned-unit-guides-on-khan-academyhttps://puzzicle.org/thumbs/x90f5d0a59e0c4bd2.jpgHow to use TEKS-aligned unit guides on Khan AcademyJoin Jennifer, the Texas Professional Learning Specialist with Khan Academy as she introduces our TEKS-aligned Unit Guides to help you navigate the Texas Essential Knowledge and Skills (TEKS) standards more effectively. These comprehensive Unit Guides are designed to streamline your planning process. They outline key concepts and skills for each unit, helping you align your instruction with TEKS standards. With detailed overviews, objectives, standards, teaching tips, and best practices, these guides empower you to make a meaningful impact on your students’ learning experience.https://cdn.kastatic.org/ka-youtube-converted/IVjfIQbVGhk.mp4/IVjfIQbVGhk.mp4143yes https://puzzicle.org/math/trigonometry/v/pythagorean-trig-identity-from-soh-cah-toahttps://puzzicle.org/thumbs/x10c52471.jpgIntro to the Pythagorean trig identitySal introduces and proves the identity (sinθ)^2+(cosθ)^2=1, which arises from the Pythagorean theorem!https://cdn.kastatic.org/ka-youtube-converted/HnDvUaVjQ1I.mp4/HnDvUaVjQ1I.mp4256yes https://puzzicle.org/math/trigonometry/v/tau-versus-pihttps://puzzicle.org/thumbs/145890679.jpgTau versus piWhy Tau might be a better number to look at than Pihttps://cdn.kastatic.org/ka-youtube-converted/1jDDfkKKgmc.mp4/1jDDfkKKgmc.mp4982yes https://puzzicle.org/math/trigonometry/v/pi-is-still-wronghttps://puzzicle.org/thumbs/538086220.jpgPi is (still) wrongPlease excuse the simple math and spelling errors. You shouldn't believe anything I say without double-checking even at the best of times. Go here: http://tauday.com/ and here: http://www.math.utah.edu/~palais/pi.html Me: http://vihart.comhttps://cdn.kastatic.org/ka-youtube-converted/jG7vhMMXagQ.mp4/jG7vhMMXagQ.mp4317yes https://puzzicle.org/math/trigonometry/v/2003-aime-ii-problem-11-avihttps://puzzicle.org/thumbs/13179281.jpgTrig challenge problem: area of a triangleSal solves a very complicated geometrical trig problem that appeared as problem 11 in the 2003 AIME II exam.https://cdn.kastatic.org/ka-youtube-converted/smtrrefmC40.mp4/smtrrefmC40.mp4995yes https://puzzicle.org/math/trigonometry/v/2003-aime-ii-problem-14https://puzzicle.org/thumbs/13179282.jpgTrig challenge problem: area of a hexagonSal solves a very complicated geometrical trig problem that appeared as problem 14 in the 2003 AIME II exam.https://cdn.kastatic.org/ka-youtube-converted/Ec-BKdC8vOo.mp4/Ec-BKdC8vOo.mp41098yes https://puzzicle.org/math/trigonometry/v/cosine-angle-addition-2https://puzzicle.org/thumbs/x6130cff9.jpgTrig challenge problem: cosine of angle-sumSal is given cos(θ) and cos(φ) and he finds cos(θ+φ). To do that, he must first find sin(θ) and sin(φ) using the Pythagorean identity.https://cdn.kastatic.org/ka-youtube-converted/XGpEHj43kcc.mp4/XGpEHj43kcc.mp4434yes https://puzzicle.org/math/trigonometry/v/iit-jee-trigonometry-problem-1https://puzzicle.org/thumbs/8187432.jpgTrig challenge problem: arithmetic progressionSal solves a very complicated algebraic trig problem that appeared as problem 29 in the 2010 IIT JEE Paper I exam.https://cdn.kastatic.org/ka-youtube-converted/X7GT9JKoAbo.mp4/X7GT9JKoAbo.mp4913yes https://puzzicle.org/math/trigonometry/v/iit-jee-trigonometric-maximumhttps://puzzicle.org/thumbs/8187460.jpgTrig challenge problem: maximum valueSal solves a very complicated algebraic trig problem that appeared as problem 48 in the 2010 IIT JEE Paper I exam.https://cdn.kastatic.org/ka-youtube-converted/tzR9jUCSniQ.mp4/tzR9jUCSniQ.mp4594yes https://puzzicle.org/math/trigonometry/v/iit-jee-trigonometric-constraintshttps://puzzicle.org/thumbs/8187459.jpgTrig challenge problem: multiple constraintsSal solves a very complicated algebraic trig problem that appeared as problem 47 in the 2010 IIT JEE Paper I exam.https://cdn.kastatic.org/ka-youtube-converted/EjtjdJZ2x8w.mp4/EjtjdJZ2x8w.mp4769yes https://puzzicle.org/math/trigonometry/v/trigonometric-system-examplehttps://puzzicle.org/thumbs/8187470.jpgTrig challenge problem: system of equationsSal solves a very complicated algebraic trig problem that appeared as problem 55 in the 2010 IIT JEE Paper I exam.https://cdn.kastatic.org/ka-youtube-converted/Y2ed-g8Lpdc.mp4/Y2ed-g8Lpdc.mp4662yes https://puzzicle.org/science/ap-biology/v/hydrogen-bonding-in-waterhttps://puzzicle.org/thumbs/xc28468ba.jpgHydrogen bonding in waterWater is made up of two hydrogens and one oxygen atom, arranged in a tetrahedral shape. Oxygen is highly electronegative, which creates a partial negative charge on one end of the molecule, and a partial positive charge on the other. So, water molecules are able to form hydrogen bonds with one another, giving water many of its unique properties.https://cdn.kastatic.org/ka-youtube-converted/6G1evL7ELwE.mp4/6G1evL7ELwE.mp4407yes https://puzzicle.org/science/ap-biology/v/capillary-action-and-why-we-see-a-meniscushttps://puzzicle.org/thumbs/x451fa8cc.jpgCapillary action and why we see a meniscusA meniscus is a curved liquid surface that results from the interplay of adhesion (the liquid's attraction to its container) and cohesion (the liquid's attraction to itself). Adhesion also drives capillary action, which draws a liquid up a narrow tube.https://cdn.kastatic.org/ka-youtube-converted/eQXGpturk3A.mp4/eQXGpturk3A.mp4506yes https://puzzicle.org/science/ap-biology/v/surface-tensionhttps://puzzicle.org/thumbs/x097b2cfb.jpgSurface tensionWater molecules are held together by hydrogen bonds, which give water its unique properties. At the surface of water, molecules are more densely packed because they are not being pulled from above, resulting in stronger intermolecular forces. This creates surface tension, which allows for phenomena such as water droplets maintaining a round shape and insects walking on water.https://cdn.kastatic.org/ka-youtube-converted/_RTF0DAHBBM.mp4/_RTF0DAHBBM.mp4271yes https://puzzicle.org/science/ap-biology/v/water-as-a-solventhttps://puzzicle.org/thumbs/x139c1eb0.jpgWater as a solventWater—due to its polarity—acts as a fantastic solvent, especially for polar substances and ions. This is because the positive and negative ends of water molecules are attracted to ions, allowing them to dissolve. This property is crucial for chemical interactions within cells. However, non-polar substances like hydrocarbons do not dissolve well in water and are termed hydrophobic.https://cdn.kastatic.org/ka-youtube-converted/lCvBp73ZJ-A.mp4/lCvBp73ZJ-A.mp4552yes https://puzzicle.org/science/ap-biology/v/carbon-as-a-building-block-of-lifehttps://puzzicle.org/thumbs/x6f46b5d3.jpgCarbon as a building block of lifeCarbon is essential to life because its four valence electrons allow it to form four covalent bonds and create many different molecular structures. Carbon forms hydrocarbons such as methane as well as important biological molecules including DNA, proteins, glucose, and ATP. Its flexible bonding makes carbon a key building block of living organisms.https://cdn.kastatic.org/ka-youtube-converted/JgYlogdtJDo.mp4/JgYlogdtJDo.mp4542yes https://puzzicle.org/science/ap-biology/v/functional-groupshttps://puzzicle.org/thumbs/x7d3ab1b9.jpgFunctional groupsVarious functional groups can be attached to hydrocarbons, affecting chemical properties. Hydroxyl groups are polar, turn molecules into alcohols, and allow molecules to dissolve in water. Carboxyl groups can donate hydrogen ions, so they make molecules acidic. Amino groups can pick up hydrogen ions from the surroundings, so they make molecules basic. Other functional groups include sulfhydryl, carbonyl, methyl, and phosphate groups.https://cdn.kastatic.org/ka-youtube-converted/iuaYuCreEPk.mp4/iuaYuCreEPk.mp4611yes https://puzzicle.org/science/ap-biology/v/elemental-building-blocks-of-biological-moleculeshttps://puzzicle.org/thumbs/xbf070c5199e6197f.jpgElemental building blocks of biological moleculesBiological molecules share many of the same elemental building blocks, including carbon, hydrogen, oxygen, nitrogen, and phosphorus. These elements combine to form amino acids, ATP, fats, and DNA. Despite the complexity of living systems, their molecules use similar components, including sugars, phosphates, and nitrogenous bases, that ultimately come from the environment.https://cdn.kastatic.org/ka-youtube-converted/vnsrBHkYTq4.mp4/vnsrBHkYTq4.mp4271yes https://puzzicle.org/science/ap-biology/v/dehydration-synthesis-or-a-condensation-reactionhttps://puzzicle.org/thumbs/xc324ac4d.jpgDehydration synthesis or a condensation reactionThe monosaccharide glucose can be used as a building block for more complex sugars and carbohydrates. Two glucose molecules can be linked together through a dehydration synthesis reaction to form a disaccharide called maltose. This process can be repeated to form polysaccharides, such as starch and glycogen.https://cdn.kastatic.org/ka-youtube-converted/FEAXI5XeJ4M.mp4/FEAXI5XeJ4M.mp4603yes https://puzzicle.org/science/ap-biology/v/hydrolysishttps://puzzicle.org/thumbs/x5ff00476.jpgHydrolysisPolysaccharides (such as starch, chitin, glycogen, and cellulose) can be broken down into monosaccharides. This occurs through the process of hydrolysis, which uses water to break the bonds between monosaccharides.https://cdn.kastatic.org/ka-youtube-converted/SOQyiM6V3RQ.mp4/SOQyiM6V3RQ.mp4509yes https://puzzicle.org/science/ap-biology/v/introduction-to-carbohydrateshttps://puzzicle.org/thumbs/x05e30a4e3b15fddb.jpgIntroduction to carbohydratesA carbohydrate is a type of molecule that contains carbon, hydrogen, and oxygen. Carbohydrates can be simple sugars (monosaccharides) like glucose, or they can be made up of multiple sugar units (polysaccharides) like glycogen. They are important in biology as a source of energy and as structural components in plants.https://cdn.kastatic.org/ka-youtube-converted/6o4WL5jlpn0.mp4/6o4WL5jlpn0.mp4268yes https://puzzicle.org/science/ap-biology/v/molecular-structure-of-triglycerides-fatshttps://puzzicle.org/thumbs/x989f9684.jpgMolecular structure of triglycerides (fats)Triglycerides (or fats) are formed from the combination of glycerol and three fatty acid molecules. Triglycerides are formed through dehydration reactions. Another word for triglyceride is triacylglycerol. Fats can be solid (such as coconut oil) or liquid (such as vegetable oil) at room temperature.https://cdn.kastatic.org/ka-youtube-converted/OpyTJbzA7Fk.mp4/OpyTJbzA7Fk.mp4664yes https://puzzicle.org/science/ap-biology/v/saturated-fats-unsaturated-fats-and-trans-fatshttps://puzzicle.org/thumbs/xc3236192.jpgSaturated fats, unsaturated fats, and trans fatsA triglyceride is a type of lipid (or fat) molecule. It is made up of three fatty acid chains attached to a glycerol backbone. Triglycerides can be classified as saturated (having no double bonds) or unsaturated (having double bonds). Trans fats are a type of unsaturated fat that is produced artificially and can be unhealthy to consume.https://cdn.kastatic.org/ka-youtube-converted/O9lL2KStW9s.mp4/O9lL2KStW9s.mp4537yes https://puzzicle.org/science/ap-biology/v/dna-deoxyribonucleic-acidhttps://puzzicle.org/thumbs/x983f0c01.jpgDNADNA (or deoxyribonucleic acid) is the molecular basis for inheritance, discovered in the mid-1800s and understood in 1953 with the double helix structure. Composed of a sugar-phosphate backbone and base pairs (adenine-thymine, guanine-cytosine), DNA stores genetic information, replicates, and expresses traits. The human genome contains 6 billion base pairs across 46 chromosomes, making it a compact and efficient information storage system.https://cdn.kastatic.org/ka-youtube-converted/AmOO4j0E408.mp4/AmOO4j0E408.mp4782yes https://puzzicle.org/science/ap-biology/v/molecular-structure-of-dnahttps://puzzicle.org/thumbs/x5f5151df.jpgMolecular structure of DNADNA (deoxyribonucleic acid) consists of nucleotides forming a double helix structure. Nucleotides contain a phosphate group, deoxyribose sugar, and a nitrogenous base. The bases adenine, thymine, cytosine, and guanine pair up through hydrogen bonds, creating the rungs of the DNA ladder.https://cdn.kastatic.org/ka-youtube-converted/L677-Fl0joY.mp4/L677-Fl0joY.mp4781yes https://puzzicle.org/science/ap-biology/v/antiparallel-structure-of-dna-strandshttps://puzzicle.org/thumbs/xb0f3c430.jpgAntiparallel structure of DNA strandsDNA structure consists of two strands, each with a backbone of alternating phosphate and sugar groups. These strands are linked by complementary nitrogenous bases, forming a ladder-like structure. The strands are antiparallel, meaning they run in opposite directions, which is crucial for DNA replication.https://cdn.kastatic.org/ka-youtube-converted/0CQ5ls3Uc2Q.mp4/0CQ5ls3Uc2Q.mp4597yes https://puzzicle.org/science/ap-biology/v/molecular-structure-of-rnahttps://puzzicle.org/thumbs/x44533ed0.jpgMolecular structure of RNADNA and RNA have different sugar backbones: deoxyribose for DNA and ribose for RNA. Notably, RNA has uracil instead of thymine as a nitrogenous base. RNA plays various roles in cells, including messenger RNA (mRNA), transfer RNA (tRNA), ribosomal RNA (rRNA), and microRNA, and it is a vital part of processes such as transcription, translation, and cellular functions.https://cdn.kastatic.org/ka-youtube-converted/jUUJSOM1ihU.mp4/jUUJSOM1ihU.mp4690yes https://puzzicle.org/science/ap-biology/v/introduction-to-amino-acidshttps://puzzicle.org/thumbs/xe81203a5.jpgIntroduction to amino acidsAmino acids are the building blocks of proteins, molecules that perform many essential functions in organisms. Each amino acid contains an amino group, a carboxyl group, and a variable R group. These R groups affect properties such as polarity and help determine protein shape. Amino acids connect through peptide bonds to form polypeptide chains.https://cdn.kastatic.org/ka-youtube-converted/Pk4d9lY48GI.mp4/Pk4d9lY48GI.mp4594yes https://puzzicle.org/science/ap-biology/v/overview-of-protein-structurehttps://puzzicle.org/thumbs/x3759e606.jpgOverview of protein structureProtein structure determines how proteins function in living systems. Primary structure describes the amino acid sequence, while secondary structure includes alpha helices and beta-pleated sheets formed through backbone interactions. Side-chain interactions create tertiary structure, and proteins with multiple polypeptide chains have quaternary structure. Together, these levels produce specific three-dimensional shapes and functions.https://cdn.kastatic.org/ka-youtube-converted/MODnIkQvyz0.mp4/MODnIkQvyz0.mp4598yes https://puzzicle.org/science/ap-biology/v/introduction-to-the-cellhttps://puzzicle.org/thumbs/x42878b6111bb5bb2.jpgIntroduction to the cellCells are the most basic unit of life. All cells have a membrane that separates them from the outside world. Although cells are small, they are not simple. Cells contain different components, such as the cytoplasm, ribosomes, and genetic information in the form of DNA.https://cdn.kastatic.org/ka-youtube-converted/5KfHxF6Vhps.mp4/5KfHxF6Vhps.mp4352yes https://puzzicle.org/science/ap-biology/v/organelles-in-eukaryotic-cellshttps://puzzicle.org/thumbs/x9fee9f7cbfade35f.jpgOrganelles in eukaryotic cellsEukaryotic cells have membrane-bound organelles. The nucleus stores DNA. The endoplasmic reticulum and Golgi body are involved in protein maturation and transport. Mitochondria generate usable energy for the cell. Chloroplasts carry out photosynthesis. Vacuoles are storage compartments that sequester waste and help maintain water balance. Lysosomes contain enzymes that help break down waste.https://cdn.kastatic.org/ka-youtube-converted/bWPQvxElpLY.mp4/bWPQvxElpLY.mp4578yes https://puzzicle.org/science/ap-biology/v/endoplasmic-reticulum-and-golgi-bodieshttps://puzzicle.org/thumbs/x3b506ff8.jpgEndoplasmic reticulum and Golgi bodiesDuring protein synthesis, proteins meant for use within the cell are translated by free ribosomes. Proteins meant to be embedded in the cell membrane or used outside the cell are translated by ribosomes attached to the rough endoplasmic reticulum. These proteins are then transported to the Golgi body for further maturation and sorting before being released.https://cdn.kastatic.org/ka-youtube-converted/6UqtgH_Zy1Y.mp4/6UqtgH_Zy1Y.mp4700yes https://puzzicle.org/science/ap-biology/v/endomembrane-systemhttps://puzzicle.org/thumbs/xa41115fb.jpgEndomembrane systemThe endomembrane system in eukaryotic cells includes cell membrane, nuclear envelope, endoplasmic reticulum, Golgi apparatus, vesicles, vacuoles, and lysosomes. These parts (made of phospholipid bilayers) work together for protein and lipid synthesis, modification, transport, and recycling within the cell.https://cdn.kastatic.org/ka-youtube-converted/vC-cEWJxDRY.mp4/vC-cEWJxDRY.mp4380yes https://puzzicle.org/science/ap-biology/v/scale-of-cellshttps://puzzicle.org/thumbs/x0086be61.jpgScale of cellsEven though molecules, proteins, viruses, and cells are all tiny, there are significant size differences between them. The diameter of a water molecule is roughly 0.28 nanometers. The diameter of the protein hemoglobin is roughly 5 nanometers. The diameter of the HIV virus is roughly 120 nanometers. A red blood cell is 6-8 micrometers.https://cdn.kastatic.org/ka-youtube-converted/xKJ3txXIuQk.mp4/xKJ3txXIuQk.mp4440yes https://puzzicle.org/science/ap-biology/v/cell-sizehttps://puzzicle.org/thumbs/xffe5513f.jpgCell sizeCells range from tiny bacteria to much larger human egg cells, but their size has limits. Cells must be large enough to contain DNA and cellular machinery. As cells grow, volume increases faster than surface area, making nutrient and waste exchange less efficient. Cell shapes and adaptations can increase surface area and support efficient transport.https://cdn.kastatic.org/ka-youtube-converted/6xx5v3PKZZM.mp4/6xx5v3PKZZM.mp4536yes https://puzzicle.org/science/ap-biology/v/introduction-to-cilia-flagella-and-pseudopodiahttps://puzzicle.org/thumbs/xea5cff3f26fe7652.jpgIntroduction to cilia, flagella, and pseudopodiaAmoebas use pseudopods ("false feet") for movement and capturing prey, demonstrating the complexity of cellular structures. Unicellular organisms like Oxytricha trifallax use cilia (hair-like structures) for movement and have unique mating processes. Chlamydomonas (a green algae) uses a flagellum (tail-like structure) for movement. These examples highlight the intricate machinery of cells.https://cdn.kastatic.org/ka-youtube-converted/E0D1H74PMmI.mp4/E0D1H74PMmI.mp4265yes https://puzzicle.org/science/ap-biology/v/surface-area-to-volume-ratio-of-cellshttps://puzzicle.org/thumbs/xc2739ff2ca2f7492.jpgSurface area to volume ratio of cellsIn order to stay alive, cells exchange resources, waste, and energy with their environments. This exchange happens across the cell membrane. As the volume of a cell increases, the ratio of surface area to volume decreases, making these exchanges less efficient. Some cells have folds in their membranes, which increase their surface area to volume ratio.https://cdn.kastatic.org/ka-youtube-converted/uu9eHX6Tu8Q.mp4/uu9eHX6Tu8Q.mp4312yes https://puzzicle.org/science/ap-biology/v/fluid-mosaic-model-of-cell-membraneshttps://puzzicle.org/thumbs/x0c4fbd1d.jpgFluid mosaic model of cell membranesThe fluid mosaic model explains the structure of cell membranes. These membranes are composed of phospholipids, forming a bilayer with hydrophilic heads facing outwards and hydrophobic tails facing inwards. Proteins, glycolipids, glycoproteins, and cholesterol are embedded within this bilayer, creating a diverse "mosaic". The "fluid" aspect refers to the flexibility and movement within the membrane, maintaining separation between the cell's interior and exterior.https://cdn.kastatic.org/ka-youtube-converted/cP8iQu57dQo.mp4/cP8iQu57dQo.mp4527yes https://puzzicle.org/science/ap-biology/v/cell-membrane-proteinshttps://puzzicle.org/thumbs/xe53e7da2.jpgCell membrane proteinsMembrane proteins help cells transport substances, communicate, and maintain homeostasis. Integral and peripheral proteins differ in how they attach to the phospholipid bilayer. Channel and carrier proteins move ions and molecules across the membrane, sometimes using ATP, while glycoproteins combine sugars and proteins to support cell signaling and recognition.https://cdn.kastatic.org/ka-youtube-converted/0emD1AmfdjY.mp4/0emD1AmfdjY.mp4491yes https://puzzicle.org/science/ap-biology/v/cell-membrane-introductionhttps://puzzicle.org/thumbs/x3af34ed1.jpgCell membrane introductionThe cell membrane is made up of phospholipids, which have a hydrophilic phosphate head, a glycerol backbone, and two hydrophobic fatty acid tails. The phospholipids form a bilayer with the hydrophilic heads facing outward. This bilayer is semi-permeable meaning that it allows some substances to pass through while blocking others.https://cdn.kastatic.org/ka-youtube-converted/QpcACa39YtA.mp4/QpcACa39YtA.mp4532yes https://puzzicle.org/science/ap-biology/v/cell-membrane-overview-fluid-mosaic-modelhttps://puzzicle.org/thumbs/xb7ad5d15.jpgCell membrane overview and the fluid mosaic modelThe cell membrane protects cells and controls interactions with their surroundings. Its phospholipid bilayer contains cholesterol, which helps maintain fluidity, and proteins that support communication and transport. Carbohydrates help cells recognize one another. Together, these moving components form the fluid mosaic model, reflecting the membrane's flexible, dynamic structure.https://cdn.kastatic.org/ka-youtube-converted/LXaPt9i9hqk.mp4/LXaPt9i9hqk.mp4532yes https://puzzicle.org/science/ap-biology/v/plant-cell-wallshttps://puzzicle.org/thumbs/x927ae556.jpgPlant cell wallsThe cell wall surrounds the plant cell, providing both structure and protection. Plant cell walls are made up mostly of cellulose, and also include hemicellulose and pectin. Plant cells connect directly to one another via tunnels in their cell walls called plasmodesmata.https://cdn.kastatic.org/ka-youtube-converted/zdvKhaQxvag.mp4/zdvKhaQxvag.mp4463yes https://puzzicle.org/science/ap-biology/v/introduction-to-passive-and-active-transporthttps://puzzicle.org/thumbs/xceb2019f52466dba.jpgIntroduction to passive and active transportActive transport requires energy to move substances against a concentration or electrical gradient, like a canoeist paddling upstream. Passive transport, on the other hand, does not require energy as substances move along their gradient, similar to a canoeist drifting downstream. The sodium-potassium pump is a key example of active transport.https://cdn.kastatic.org/ka-youtube-converted/TsoAA7zOyj8.mp4/TsoAA7zOyj8.mp4486yes https://puzzicle.org/science/ap-biology/v/passive-transport-and-selective-permeabilityhttps://puzzicle.org/thumbs/x6525e15d.jpgPassive transport and selective permeabilityPassive transport describes the movement of substances across a cell membrane down their concentration gradients, which does not require energy. Cell membranes are selectively permeable, so only certain substances can passively diffuse directly across the membrane. These substances include small, nonpolar molecules such as carbon dioxide and molecular oxygen.https://cdn.kastatic.org/ka-youtube-converted/jQN07Hvq6WI.mp4/jQN07Hvq6WI.mp4420yes https://puzzicle.org/science/ap-biology/v/endocytosis-phagocytosis-and-pinocytosishttps://puzzicle.org/thumbs/xf31b726a.jpgEndocytosis, phagocytosis, and pinocytosisPhagocytosis is the process by which a cell engulfs and internalizes a large particle, such as a bacterium, by extending its membrane around it. Pinocytosis is similar, but instead the cell engulfs droplets of extracellular fluid, taking in any dissolved substances within. Both phagocytosis and pinocytosis are forms of endocytosis.https://cdn.kastatic.org/ka-youtube-converted/QspmZf_yWyU.mp4/QspmZf_yWyU.mp4278yes https://puzzicle.org/science/ap-biology/v/exocytosishttps://puzzicle.org/thumbs/xb8a748cc.jpgExocytosisExocytosis is a form of bulk transport during which large numbers of molecules are transported out of the cell. In exocytosis, a vesicle (a small, membrane-bound compartment) containing the molecules to be released fuses with the cell membrane, and the contents of the vesicle are expelled. Exocytosis is important for the transport of neurotransmitters.https://cdn.kastatic.org/ka-youtube-converted/VOzV4d0HKis.mp4/VOzV4d0HKis.mp4257yes https://puzzicle.org/science/ap-biology/v/facilitated-diffusionhttps://puzzicle.org/thumbs/x32334dc8.jpgFacilitated diffusionFacilitated diffusion is a type of passive transport that uses specialized proteins, such as channel proteins and carrier proteins, to help molecules move across a cell membrane. In this process, molecules can move down their concentration gradient without requiring any energy input from the cell.https://cdn.kastatic.org/ka-youtube-converted/8HlVy__J8XA.mp4/8HlVy__J8XA.mp4395yes https://puzzicle.org/science/ap-biology/v/diffusion-and-osmosishttps://puzzicle.org/thumbs/23984.jpgDiffusion and osmosisDiffusion refers to the movement of molecules from an area of high concentration to an area of lower concentration. Osmosis is a type of diffusion specifically for water molecules moving across a semi-permeable membrane. A concentration gradient is the difference in concentration of a substance between two areas, which drives diffusion or osmosis.https://cdn.kastatic.org/ka-youtube-converted/aubZU0iWtgI.mp4/aubZU0iWtgI.mp41139yes https://puzzicle.org/science/ap-biology/v/uniporters-symporters-and-antiportershttps://puzzicle.org/thumbs/x94a43831.jpgUniporters, symporters, and antiportersPassive transport is the movement of substances across a cell membrane without the use of energy. Examples include diffusion and facilitated diffusion. Active transport describes the use of energy to move molecules across a cell membrane, usually against their concentration gradients. Membrane proteins involved in active transport include symporters, antiporters, and the sodium-potassium pump.https://cdn.kastatic.org/ka-youtube-converted/-aGYytZ7K7M.mp4/-aGYytZ7K7M.mp4433yes https://puzzicle.org/science/ap-biology/v/osmosishttps://puzzicle.org/thumbs/xc57249c4.jpgOsmosisOsmosis is a process where water molecules move from an area of low solute concentration to high solute concentration through a semipermeable membrane. This movement can be influenced by factors like solute size and charge. Understanding osmosis helps us grasp how cells maintain balance and control their internal environment.https://cdn.kastatic.org/ka-youtube-converted/rCNlG_j_gSM.mp4/rCNlG_j_gSM.mp4484yes https://puzzicle.org/science/ap-biology/v/hypotonic-isotonic-and-hypertonic-solutions-tonicityhttps://puzzicle.org/thumbs/xf9dc3289.jpgHypotonic, isotonic, and hypertonic solutions (tonicity)Cells react differently in hypotonic, isotonic, and hypertonic solutions. In a hypotonic solution, water rushes into the cell causing it to expand or even burst. In an isotonic solution, there is no net flow of water, keeping the cell stable. In a hypertonic solution, water leaves the cell, causing it to shrivel. These reactions are due to the semipermeable nature of cell membranes and the concentration of solutes.https://cdn.kastatic.org/ka-youtube-converted/afWnU10ZNfg.mp4/afWnU10ZNfg.mp4390yes https://puzzicle.org/science/ap-biology/v/water-potential-examplehttps://puzzicle.org/thumbs/xc83d63fab651905c.jpgWater potential exampleWater potential helps explain how water moves into and out of plant cells through osmosis. By finding the sucrose concentration where a potato cube has no change in mass, its water potential can be calculated using solute potential. Concentration, temperature, and the ionization constant determine the final water potential in bars.https://cdn.kastatic.org/ka-youtube-converted/BsnJDe4Tl-4.mp4/BsnJDe4Tl-4.mp4405yes https://puzzicle.org/science/ap-biology/v/electrochemical-gradient-and-secondary-active-transporthttps://puzzicle.org/thumbs/xcf4785aa.jpgElectrochemical gradients and secondary active transportElectrochemical gradients combine differences in ion concentration and electric charge across a cell membrane. The sodium-potassium pump creates a sodium gradient that stores potential energy. A sodium-glucose symporter uses this energy to move glucose against its concentration gradient, an example of secondary active transport that helps cells bring in needed molecules.https://cdn.kastatic.org/ka-youtube-converted/wbWL2wfvsM8.mp4/wbWL2wfvsM8.mp4317yes https://puzzicle.org/science/ap-biology/v/sodium-potassium-pump-videohttps://puzzicle.org/thumbs/x9c826aa6.jpgSodium potassium pumpThe sodium-potassium pump is a trans-membrane protein that helps establish the resting membrane potential of cells, particularly neurons. In a process that requires ATP, the pump moves three sodium ions out of the cell for every two potassium ions it brings in. This contributes to the electric potential difference between the inside and outside of the cell.https://cdn.kastatic.org/ka-youtube-converted/vh166DKxYiM.mp4/vh166DKxYiM.mp4438yes https://puzzicle.org/science/ap-biology/v/prokaryotic-and-eukaryotic-cellshttps://puzzicle.org/thumbs/xed3c4701.jpgProkaryotic and eukaryotic cellsEukaryotic cells contain membrane-bound organelles (such as the nucleus and mitochondria), while prokaryotic cells do not. DNA in eukaryotic cells is found inside the nucleus, while DNA in prokaryotic cells is located in the cytoplasm. Eukaryotic cells are generally larger and more complex than prokaryotic cells. Eukaryotic organisms include animals, plants, fungi, and paramecia. Prokaryotic organisms include bacteria and archaea.https://cdn.kastatic.org/ka-youtube-converted/xTnNv7YplSo.mp4/xTnNv7YplSo.mp4410yes https://puzzicle.org/science/ap-biology/v/endosymbiosis-theoryhttps://puzzicle.org/thumbs/x5b44a315947b0f34.jpgEndosymbiosis theoryCertain characteristics of mitochondria and chloroplasts support the endosymbiotic theory, which posits that eukaryotic cells evolved from simpler prokaryotic cells that merged together. Lynn Margulis was a key figure in developing this theory. Margulis's work was at first met with skepticism, but now the endosymbiotic theory is widely accepted in the scientific community.https://cdn.kastatic.org/ka-youtube-converted/eCWwKrJyKxA.mp4/eCWwKrJyKxA.mp4250yes https://puzzicle.org/science/ap-biology/v/enzymeshttps://puzzicle.org/thumbs/x32f1ee82.jpgEnzymesEnzymes are biological catalysts that speed up reactions. The active site is where substrates bind to the enzyme. Induced fit occurs when the enzyme changes shape to better accommodate substrates, facilitating the reaction. Enzymes can be used multiple times and are affected by factors such as temperature and pH.https://cdn.kastatic.org/ka-youtube-converted/A90HKrgUOg0.mp4/A90HKrgUOg0.mp4493yes https://puzzicle.org/science/ap-biology/v/enzyme-reaction-velocity-and-phhttps://puzzicle.org/thumbs/xf648b619f7355779.jpgEnzyme reaction velocity and pHEnzymes are proteins that catalyze reactions in biological systems. The pH of an enzyme's surroundings affects the bonds that maintain the enzyme's structure, which can also affect the enzyme's activity. Different enzymes have optimal activity at different pH levels, with some enzymes working best in acidic environments and others in neutral or basic environments.https://cdn.kastatic.org/ka-youtube-converted/LtGCHzKAf8w.mp4/LtGCHzKAf8w.mp4403yes https://puzzicle.org/science/ap-biology/v/competitive-inhibitionhttps://puzzicle.org/thumbs/x7fa5fd48.jpgCompetitive inhibitionEnzymes catalyze reactions by binding substrates to their active sites. Inhibition can occur competitively (when a molecule competes for the active site) or non-competitively (when a molecule binds somewhere else on the enzyme, altering the enzyme's shape). Allosteric regulation describes any case where an enzyme's function at one site is affected by the binding of a molecule at another site.https://cdn.kastatic.org/ka-youtube-converted/O23i38B2EmA.mp4/O23i38B2EmA.mp4390yes https://puzzicle.org/science/ap-biology/v/noncompetitive-inhibitionhttps://puzzicle.org/thumbs/xf7618e41.jpgNoncompetitive inhibitionNoncompetitive inhibition occurs when an inhibitor reduces an enzyme's activity without preventing the substrate from binding. Both the substrate and inhibitor can bind to the enzyme at the same time, often at different sites. However, when the inhibitor is present, the enzyme cannot catalyze the reaction, distinguishing this process from competitive inhibition.https://cdn.kastatic.org/ka-youtube-converted/4xOsAuj0QZg.mp4/4xOsAuj0QZg.mp4295yes https://puzzicle.org/science/ap-biology/v/first-law-of-thermodynamics-introductionhttps://puzzicle.org/thumbs/x15363e45.jpgFirst law of thermodynamics introductionThe first law of thermodynamics states that energy cannot be created or destroyed, only converted from one form to another. For example, kinetic energy may be converted into thermal energy, or potential energy may be converted into kinetic energy. Energy is never "lost"; instead, it is transferred or converted in some way.https://cdn.kastatic.org/ka-youtube-converted/vny1qUaToHw.mp4/vny1qUaToHw.mp4558yes https://puzzicle.org/science/ap-biology/v/second-law-of-thermodynamics-and-entropyhttps://puzzicle.org/thumbs/x618d2187.jpgSecond law of thermodynamics and entropyEntropy is a measure of disorder or randomness in a system. It represents the number of possible states or configurations that a system can take on. According to the second law of thermodynamics, entropy tends to increase over time, meaning that systems naturally progress towards a more disordered or random state.https://cdn.kastatic.org/ka-youtube-converted/axG9HuqViDY.mp4/axG9HuqViDY.mp4512yes https://puzzicle.org/science/ap-biology/v/reaction-coupling-to-create-glucose-6-phosphatehttps://puzzicle.org/thumbs/xf7bec56a.jpgReaction coupling to create glucose-6-phosphateReaction coupling is a process in which two reactions are linked together, with one providing the energy needed for the other to occur. In coupled reactions, an energetically favorable reaction (exergonic) releases energy, which is then used to drive an energetically unfavorable reaction (endergonic). This allows the overall coupled reaction to be exergonic, meaning it can happen spontaneously.https://cdn.kastatic.org/ka-youtube-converted/L8ifD4I1o38.mp4/L8ifD4I1o38.mp4419yes https://puzzicle.org/science/ap-biology/v/introduction-to-metabolism-anabolism-and-catabolismhttps://puzzicle.org/thumbs/x6f1c22d2.jpgIntroduction to metabolism: Anabolism and catabolismMetabolism refers to the set of chemical reactions that occur within living organisms to maintain life. Anabolism is the process of building up larger, more complex molecules from smaller ones. This process requires energy. Catabolism is the process of breaking down larger molecules into smaller ones. This process releases energy that can be used by the organism.https://cdn.kastatic.org/ka-youtube-converted/KwNe9x0eChs.mp4/KwNe9x0eChs.mp4607yes https://puzzicle.org/science/ap-biology/v/conceptual-overview-of-light-dependent-reactionshttps://puzzicle.org/thumbs/x2ef70f81.jpgConceptual overview of light-dependent reactionsDuring the light-dependent reactions of photosynthesis, light energy excites electrons, which then move through a series of molecules in the thylakoid membrane of chloroplasts. As the electrons move to lower energy states, they help pump hydrogen ions into the thylakoid lumen, creating a concentration gradient. This gradient powers ATP synthase to produce ATP.https://cdn.kastatic.org/ka-youtube-converted/vEsAtC9d_MQ.mp4/vEsAtC9d_MQ.mp4545yes https://puzzicle.org/science/ap-biology/v/photosynthesis-evolutionhttps://puzzicle.org/thumbs/x19dd2b55504b41ad.jpgPhotosynthesis evolutionAncient cyanobacteria produced oxygen as a byproduct of photosynthesis, changing the composition of the atmosphere over time in the Great Oxygenation Event. These cyanobacteria also led to the evolution of chloroplasts when they started living inside other cells, a process called endosymbiosis. The cyanobacteria provided the larger cell with sugars from photosynthesis, and the larger cell gave protection and nutrients.https://cdn.kastatic.org/ka-youtube-converted/RnHCOBDSSfA.mp4/RnHCOBDSSfA.mp4488yes https://puzzicle.org/science/ap-biology/v/mitochondria-videohttps://puzzicle.org/thumbs/x7d4cfad5.jpgMitochondriaMitochondria are organelles that contain their own DNA, and have both inner and outer membranes. Mitochondria are often referred to as the "powerhouses of the cell" because they are responsible for producing most of the cell's energy in the form of ATP. According to the endosymbiont theory, mitochondria began as separate bacteria that were absorbed into another cell for their energy-producing capabilities.https://cdn.kastatic.org/ka-youtube-converted/i1dAnpSFbyI.mp4/i1dAnpSFbyI.mp4861yes https://puzzicle.org/science/ap-biology/v/introduction-to-cellular-respirationhttps://puzzicle.org/thumbs/23954.jpgCellular respiration introductionCellular respiration is the process by which cells derive energy from glucose. The chemical reaction for cellular respiration involves glucose and oxygen as inputs, and produces carbon dioxide, water, and energy (ATP) as outputs. There are three stages to cellular respiration: glycolysis, the Krebs cycle, and the electron transport chain.https://cdn.kastatic.org/ka-youtube-converted/2f7YwCtHcgk.mp4/2f7YwCtHcgk.mp4859yes https://puzzicle.org/science/ap-biology/v/overview-of-cellular-respirationhttps://puzzicle.org/thumbs/x19cf920e.jpgOverview of cellular respirationCellular respiration is how cells get energy from glucose. The process has three main parts: Glycolysis happens in the cytosol and breaks glucose into two pyruvate, producing 2 ATPs and 2 NADHs. The Krebs cycle occurs in the mitochondrial matrix, where pyruvate is turned into acetyl-CoA, which then goes through a series of reactions, producing ATP, NADH, and FADH2. Finally, oxidative phosphorylation uses NADH and FADH2 to create a proton gradient, which helps make more ATP.https://cdn.kastatic.org/ka-youtube-converted/9zoS5WGsmpc.mp4/9zoS5WGsmpc.mp4791yes https://puzzicle.org/science/ap-biology/v/oxidative-phosphorylation-and-the-electon-transport-chainhttps://puzzicle.org/thumbs/x5769d931.jpgOxidative phosphorylation and the electron transport chainOxidative phosphorylation produces ATP (energy) in our cells. NADH, a molecule produced during cellular respiration, gets oxidized and releases electrons. These electrons pass through a series of acceptors in the electron transport chain, releasing energy. This energy pumps hydrogen protons across a membrane, creating a gradient. Finally, the enzyme ATP synthase uses this gradient to produce ATP.https://cdn.kastatic.org/ka-youtube-converted/J30zpvbmw7s.mp4/J30zpvbmw7s.mp4937yes https://puzzicle.org/science/ap-biology/v/atp-synthasehttps://puzzicle.org/thumbs/xcf231b1f0cb6bdde.jpgATP synthaseATP synthase is a membrane-bound enzyme that uses the flow of protons (H+) across a membrane to drive the synthesis of ATP from ADP and phosphate. It is found in mitochondria and chloroplasts, and plays a crucial role in cellular energy production and photosynthesis.https://cdn.kastatic.org/ka-youtube-converted/rg8gpzFLQ-E.mp4/rg8gpzFLQ-E.mp4267yes https://puzzicle.org/science/ap-biology/v/cellular-communicationhttps://puzzicle.org/thumbs/x0f31c982.jpgCellular communicationCell communication is crucial for complex organisms. Cells can communicate directly through contact or indirectly via chemical signaling. This communication can occur over short distances, such as between neurons, or long distances, like hormonal signaling in the endocrine system. Understanding these interactions is key to grasping biological complexity.https://cdn.kastatic.org/ka-youtube-converted/5MoIg5lWLXA.mp4/5MoIg5lWLXA.mp4397yes https://puzzicle.org/science/ap-biology/v/overview-of-cell-signalinghttps://puzzicle.org/thumbs/xc8b3fbfd.jpgOverview of cell signalingCell communication is a crucial biological process where cells interact directly or signal over distances. Direct contact involves gap junctions or complementary surface proteins. Paracrine communication happens over short distances, while endocrine signaling involves hormones traveling through the bloodstream for long-distance communication. Autocrine processes allow a cell to signal itself.https://cdn.kastatic.org/ka-youtube-converted/FQFBygnIONU.mp4/FQFBygnIONU.mp4529yes https://puzzicle.org/science/ap-biology/v/intro-to-the-endocrine-systemhttps://puzzicle.org/thumbs/221376883.jpgIntroduction to the endocrine systemEndocrinology focuses on hormones, the body's chemical messengers. These hormones originate in one part of the body and journey to another to stimulate activity. They communicate through various methods, from direct contact to long-distance signaling. Crucial glands such as the pituitary, thyroid, and pancreas play significant roles in this process, regulating aspects like metabolism and growth.https://cdn.kastatic.org/ka-youtube-converted/f_Z1zsR9lFM.mp4/f_Z1zsR9lFM.mp4577yes https://puzzicle.org/science/ap-biology/v/membrane-receptorshttps://puzzicle.org/thumbs/x6589e16e.jpgMembrane receptorsMembrane receptors, integral proteins in the cell membrane, enable cells to communicate with the outside environment. They bind with specific signaling molecules or ligands, triggering changes inside the cell. This process, known as signal transduction, influences cell behavior. The specificity of ligand-receptor binding is crucial in body functions and pharmaceutical drug design.https://cdn.kastatic.org/ka-youtube-converted/8Otn9DVkm-g.mp4/8Otn9DVkm-g.mp4479yes https://puzzicle.org/science/ap-biology/v/g-protein-coupled-receptorshttps://puzzicle.org/thumbs/x7e31c319.jpgG-protein coupled receptorsG-protein coupled receptors (GPCRs) are a large, diverse group of membrane receptors in eukaryotes that respond to a wide range of external signals. When a ligand binds to a GPCR, it triggers a complex chain of events involving G-proteins and second messengers, influencing various cell functions. This process is key to many bodily functions, from sensation to growth to hormone response.https://cdn.kastatic.org/ka-youtube-converted/ZBSo_GFN3qI.mp4/ZBSo_GFN3qI.mp4768yes https://puzzicle.org/science/ap-biology/v/cellular-mechanism-hormone-actionhttps://puzzicle.org/thumbs/xd4489fab.jpgCellular mechanism of hormone actionHormones communicate with target cells via two primary methods: secondary messengers and primary messengers. Secondary messengers, such as cAMP, are released inside the cell after a hormone binds to a receptor, triggering a chain reaction. Primary messengers, such as steroid and thyroid hormones, can cross the cell membrane and directly affect transcription or translation. These processes enable efficient and amplified cellular responses.https://cdn.kastatic.org/ka-youtube-converted/TNGSzt2v4xY.mp4/TNGSzt2v4xY.mp4503yes https://puzzicle.org/science/ap-biology/v/example-of-signal-transduction-pathwayhttps://puzzicle.org/thumbs/x91f39d63.jpgExample of a signal transduction pathwaySignal transduction is a complex process where cells communicate through chemical signals. This involves a ligand binding to a receptor, initiating a cascade of events that can alter cell behavior. The Map kinase pathway, crucial in cell differentiation and replication, is a key player in this process. Mutations in this pathway can lead to diseases like cancer.https://cdn.kastatic.org/ka-youtube-converted/i7z7f-wnews.mp4/i7z7f-wnews.mp4395yes https://puzzicle.org/science/ap-biology/v/cell-signaling-in-yeast-reproductionhttps://puzzicle.org/thumbs/x512026aa.jpgCell signaling in yeast reproductionYeast can reproduce sexually through a signaling pathway known as the mating factor pathway. In this process, two haploid yeast cells combine to form a diploid cell. Yeast cells secrete a signal molecule called mating factor that attracts them to their mates. Once the mating factor of one yeast binds to the receptor on another yeast, an outgrowth called a "shmoo" forms, which allows the yeast cells to fuse together.https://cdn.kastatic.org/ka-youtube-converted/DJfNr8WoXW4.mp4/DJfNr8WoXW4.mp4482yes https://puzzicle.org/science/ap-biology/v/activation-and-inhibition-of-signal-transduction-pathwayshttps://puzzicle.org/thumbs/xa2d9f762996b1eb6.jpgActivation and inhibition of signal transduction pathwaysThe cholera toxin triggers a signal transduction pathway. This process starts when molecules outside the cell interact with cell receptors, setting off a chain of events. G-proteins, acting like molecular switches, play a key role here. Mutations or disruptions can affect the signaling pathway, leading to changes in how the cell responds.https://cdn.kastatic.org/ka-youtube-converted/zYX5a65LrXs.mp4/zYX5a65LrXs.mp4356yes https://puzzicle.org/science/ap-biology/v/homeostasishttps://puzzicle.org/thumbs/1262562541.jpgHomeostasisHomeostasis, or maintaining a steady body temperature, is achieved through feedback mechanisms. Exposure to extreme temperatures triggers physiological responses like shivering or sweating, ensuring a constant core temperature. Homeostasis also extends to regulating blood pressure and sugar levels.https://cdn.kastatic.org/ka-youtube-converted/rSBbnHLR_cg.mp4/rSBbnHLR_cg.mp4220yes https://puzzicle.org/science/ap-biology/v/hormone-concentration-metabolism-negative-feedbackhttps://puzzicle.org/thumbs/xcdd0a401.jpgHormone concentration metabolism and negative feedbackHormone concentration in our bodies is tightly regulated through metabolism, excretion, and feedback loops. The liver, kidneys, and sweat glands help remove excess hormones. Negative feedback loops, where hormone action suppresses further hormone release, play a crucial role in maintaining hormonal balance and homeostasis.https://cdn.kastatic.org/ka-youtube-converted/RycF0ub2Al0.mp4/RycF0ub2Al0.mp4287yes https://puzzicle.org/science/ap-biology/v/interphasehttps://puzzicle.org/thumbs/x78f186b2.jpgInterphaseA cell spends most of its life in interphase, which has three phases: G1, S, and G2. In the G1 phase, the cell grows and takes in nutrients. In the S phase, the cell's DNA is replicated. Each replicated chromosome consists of two sister chromatids connected at the centromere. The G2 phase is another growth phase, after which the cell is ready for mitosis.https://cdn.kastatic.org/ka-youtube-converted/VXLSTd_dlKg.mp4/VXLSTd_dlKg.mp4508yes https://puzzicle.org/science/ap-biology/v/mitosishttps://puzzicle.org/thumbs/x385762f9.jpgMitosisMitosis, a key part of the cell cycle, involves a series of stages (prophase, metaphase, anaphase, and telophase) that facilitate cell division and genetic information transmission. Centrosomes and microtubules play pivotal roles in orchestrating this complex process, ensuring the successful replication of cells.https://cdn.kastatic.org/ka-youtube-converted/TKGcfbyFXsw.mp4/TKGcfbyFXsw.mp4732yes https://puzzicle.org/science/ap-biology/v/cell-cycle-controlhttps://puzzicle.org/thumbs/x4662267b.jpgCell cycle controlThe cell cycle is tightly regulated by checkpoints between the G1 and S phase, and between the G2 and mitosis. Key proteins, cyclin-dependent kinases and cyclins, control this process. Cyclins are produced at specific times, activating the kinases and allowing progression through the cell cycle. This regulation prevents unchecked cell division, a hallmark of cancer.https://cdn.kastatic.org/ka-youtube-converted/542CMooowNY.mp4/542CMooowNY.mp4249yes https://puzzicle.org/science/ap-biology/v/loss-of-cell-cycle-control-in-cancerhttps://puzzicle.org/thumbs/x49a016da.jpgLoss of cell cycle control in cancerProteins like p53 and RB play a crucial role in regulating the cell cycle. These proteins, known as tumor suppressor genes, prevent uncontrolled cell division, potentially leading to cancer. The case of p21 is intriguing, as its absence can surprisingly trigger limb regeneration in mice.https://cdn.kastatic.org/ka-youtube-converted/UgZIULiNRoo.mp4/UgZIULiNRoo.mp4225yes https://puzzicle.org/science/ap-biology/v/fertilization-haploid-diploid-gamete-zygote-homologoushttps://puzzicle.org/thumbs/x49fa677e.jpgFertilization terminologySperm and egg cells, known as gametes, fuse during fertilization to create a zygote. Gametes have half the chromosomes (haploid) of a typical body cell, while zygotes have the full set (diploid). Homologous chromosomes from each parent determine traits, including sex. Understanding haploid and diploid numbers is essential in studying cell division and genetics.https://cdn.kastatic.org/ka-youtube-converted/dNp7vErqlaA.mp4/dNp7vErqlaA.mp4627yes https://puzzicle.org/science/ap-biology/v/phases-of-meiosis-ihttps://puzzicle.org/thumbs/x103b28b6.jpgPhases of meiosis IIn meiosis I, cells go through four phases: prophase I, metaphase I, anaphase I, and telophase I. During prophase I, chromosomes pair up and exchange genetic material, creating more variation. In metaphase I, chromosomes line up in the middle of the cell. Anaphase I separates homologous pairs, while telophase I forms two new cells with a haploid number of chromosomes.https://cdn.kastatic.org/ka-youtube-converted/XGWL9jfPHJ8.mp4/XGWL9jfPHJ8.mp4585yes https://puzzicle.org/science/ap-biology/v/phases-of-meiosis-iihttps://puzzicle.org/thumbs/xcc05c922.jpgPhases of meiosis IIMeiosis II is a process that helps cells divide and create gametes, which are needed for sexual reproduction. It starts with prophase II, where the nuclear envelope dissolves and chromosomes condense. Then, in metaphase II, chromosomes line up along the cell's middle. During anaphase II, sister chromatids separate and move to opposite ends of the cell. Finally, in telophase II, the nuclear envelope reforms, and the cell splits into four new cells, each with a haploid number of chromosomes.https://cdn.kastatic.org/ka-youtube-converted/mMCcBsSAlF4.mp4/mMCcBsSAlF4.mp4658yes https://puzzicle.org/science/ap-biology/v/comparing-mitosis-and-meiosishttps://puzzicle.org/thumbs/x9d72f94c.jpgComparing mitosis and meiosisMitosis and meiosis are two different types of cell division. Mitosis occurs in somatic cells and results in two identical daughter cells with a diploid (2n) number of chromosomes. This process is essential for growth and repair in the body. On the other hand, meiosis occurs in germ cells and produces four non-identical gametes with a haploid (n) number of chromosomes. These gametes, such as sperm and egg cells, are crucial for sexual reproduction and genetic diversity.https://cdn.kastatic.org/ka-youtube-converted/IQJ4DBkCnco.mp4/IQJ4DBkCnco.mp4351yes https://puzzicle.org/science/ap-biology/v/chromosomal-crossover-in-meiosis-ihttps://puzzicle.org/thumbs/x1cc0f1a3.jpgChromosomal crossover in meiosis IMeiosis is a process where germ cells divide to produce gametes, such as sperm and egg cells. In prophase I of meiosis, chromosomes condense and homologous recombination takes place, leading to genetic variation through chromosomal crossover. This forms a tetrad, which is made up of four chromatids (two sister chromatids per chromosome).https://cdn.kastatic.org/ka-youtube-converted/04gQ0bQu6xk.mp4/04gQ0bQu6xk.mp4736yes https://puzzicle.org/science/ap-biology/v/variation-in-a-specieshttps://puzzicle.org/thumbs/23926.jpgVariation in a speciesNatural selection relies on genetic variation within a species. Mutations alone aren't sufficient for creating diversity; sexual reproduction plays a crucial role as well. By combining genetic material from two parents, offspring inherit a unique mix of traits, resulting in vast potential for variation. This process allows for more efficient adaptation and survival, making sexual reproduction a widespread biological strategy.https://cdn.kastatic.org/ka-youtube-converted/DuArVnT1i-E.mp4/DuArVnT1i-E.mp41191yes https://puzzicle.org/science/ap-biology/v/introduction-to-heredityhttps://puzzicle.org/thumbs/23940.jpgIntroduction to heredityGenetics is a fascinating world, focusing on the inheritance of traits from parents to offspring. Gregor Mendel, the father of classical genetics, made significant contributions to our understanding of this process. Dominant and recessive traits play a crucial role in determining an individual's characteristics. The difference between genotype and phenotype is also a key concept, illustrated here through the example of eye color.https://cdn.kastatic.org/ka-youtube-converted/eEUvRrhmcxM.mp4/eEUvRrhmcxM.mp41047yes https://puzzicle.org/science/ap-biology/v/alleles-and-geneshttps://puzzicle.org/thumbs/x5155c0f0.jpgAlleles and genesGenes and alleles are key players in heredity. Genes are sections of DNA that code for specific proteins or functional RNA. Alleles, on the other hand, are different versions of the same gene. They can lead to variations in traits like eye color. Understanding these differences is crucial to studying inheritance patterns.https://cdn.kastatic.org/ka-youtube-converted/P0nMnPPdW_k.mp4/P0nMnPPdW_k.mp4498yes https://puzzicle.org/science/ap-biology/v/punnett-square-funhttps://puzzicle.org/thumbs/23942.jpgWorked example: Punnett squaresPunnett squares help predict offspring traits by showing possible gene combinations from parents. In addition, Punnett squares can illustrate trends among dominant and recessive traits, incomplete dominance, codominance, and dihybrid crosses. Punnett squares are useful for understanding genetics and inheritance patterns.https://cdn.kastatic.org/ka-youtube-converted/D5ymMYcLtv0.mp4/D5ymMYcLtv0.mp41516yes https://puzzicle.org/science/ap-biology/v/boveri-sutton-chromosome-theoryhttps://puzzicle.org/thumbs/xefeb7dbf.jpgBoveri-Sutton chromosome theoryGregor Mendel's groundbreaking 1866 work on heritable traits laid the foundation for modern genetics. His laws of segregation, independent assortment, and dominance provided a framework for understanding how traits pass from parents to offspring. However, it wasn't until the early 1900s that the Boveri-Sutton Chromosome Theory linked Mendel's laws to the behavior of chromosomes during meiosis, solidifying the molecular basis of inheritance.https://cdn.kastatic.org/ka-youtube-converted/rV6O3rGulaY.mp4/rV6O3rGulaY.mp4659yes https://puzzicle.org/science/ap-biology/v/pedigreeshttps://puzzicle.org/thumbs/xec86dfc45c44c9c2.jpgPedigreesPedigrees are a useful method to model the inheritance patterns of traits within families. By using symbols and analyzing phenotypes, we can infer genotypes and understand the role of autosomal dominant traits. The trait of freckles serves as a practical example here and shows the complex world of genetic inheritance patterns.https://cdn.kastatic.org/ka-youtube-converted/11s5Biyi9q4.mp4/11s5Biyi9q4.mp4368yes https://puzzicle.org/science/ap-biology/v/thomas-hunt-morgan-and-fruit-flieshttps://puzzicle.org/thumbs/xc4865c52.jpgThomas Hunt Morgan and fruit fliesThe groundbreaking work of Thomas Hunt Morgan in the early 1900s provided substantial evidence for the chromosome theory of inheritance. Using fruit flies as a model organism, Morgan discovered a mutant white-eyed male fly and traced its inheritance pattern, revealing a connection between the X sex chromosome and the gene for eye color. This pioneering research earned Morgan a Nobel Prize and cemented the role of chromosomes in genetic inheritance.https://cdn.kastatic.org/ka-youtube-converted/huDDaj0PjLU.mp4/huDDaj0PjLU.mp4640yes https://puzzicle.org/science/ap-biology/v/pedigree-for-determining-probability-of-exhibiting-sex-linked-recessive-traithttps://puzzicle.org/thumbs/xe9b4a7a106de559a.jpgPedigree for determining probability of exhibiting sex-linked recessive traitPedigree charts can be used to determine the probability of offspring exhibiting an X-linked recessive trait, such as color blindness. By scrutinizing genotypes and utilizing a Punnett square, the probability of color blindness in offspring can be calculated.https://cdn.kastatic.org/ka-youtube-converted/DzP7XO0Qg_Y.mp4/DzP7XO0Qg_Y.mp4319yes https://puzzicle.org/science/ap-biology/v/extranuclear-inheritance-1https://puzzicle.org/thumbs/x3ce4eef4.jpgExtranuclear inheritanceCertain organelles, like mitochondria and chloroplasts, have their own DNA and can replicate independently outside of the nucleus. This is called extranuclear inheritance and both mitochondria and chloroplasts show maternal inheritance. Mitochondria and chloroplasts are also involved in making energy (ATP) and food (glucose) for cells.https://cdn.kastatic.org/ka-youtube-converted/yyM0yIOJ2yE.mp4/yyM0yIOJ2yE.mp4434yes https://puzzicle.org/science/ap-biology/v/phenotype-plasticityhttps://puzzicle.org/thumbs/x7300199cf7c3dc96.jpgPhenotype plasticityIdentical twin astronauts, Mark Kelly and Scott Kelly, help illustrate differences between genotype (genetic info) and phenotype (observable traits). Despite their identical genotypes, environmental factors like the time Scott spent in space led to differences in the phenotypes of the twins, showcasing the concept of phenotype plasticity.https://cdn.kastatic.org/ka-youtube-converted/Mh3QXbVU6Eo.mp4/Mh3QXbVU6Eo.mp4197yes https://puzzicle.org/science/ap-biology/v/gene-environment-interactionhttps://puzzicle.org/thumbs/x777976b6.jpgGene environment interactionGenes and environment interact to influence behavior, and the term "nature through nurture" captures this relationship. The genetic condition phenylketonuria (PKU) is an example of how a genetic condition can be managed through environmental factors, like a specialized diet, to prevent negative outcomes.https://cdn.kastatic.org/ka-youtube-converted/sMyZO9YDlk8.mp4/sMyZO9YDlk8.mp4269yes https://puzzicle.org/science/ap-biology/v/introduction-to-nucleic-acids-and-nucleotideshttps://puzzicle.org/thumbs/x6c61c97a15e29d9c.jpgIntroduction to nucleic acids and nucleotidesNucleic acids, crucial macromolecules for life, were first discovered in cell nuclei and exhibit acidic properties. DNA and RNA, composed of nucleotide building blocks, store hereditary information. These polymers have a backbone of alternating ribose and phosphate groups, with nitrogenous bases forming ladder rungs. The sequence of nucleotides encodes genetic information, making nucleic acids fundamental to life.https://cdn.kastatic.org/ka-youtube-converted/hI4v7v8AdfI.mp4/hI4v7v8AdfI.mp4355yes https://puzzicle.org/science/ap-biology/v/leading-and-lagging-strands-in-dna-replicationhttps://puzzicle.org/thumbs/x5f21e0dd.jpgLeading and lagging strands in DNA replicationDNA replication is a precise process where DNA unwinds and splits into two strands. Each strand then serves as a template for a new DNA molecule. The leading strand is built continuously, while the lagging strand is built in fragments, called Okazaki fragments.https://cdn.kastatic.org/ka-youtube-converted/eM7arWJJ3zk.mp4/eM7arWJJ3zk.mp4619yes https://puzzicle.org/science/ap-biology/v/speed-and-precision-of-dna-replicationhttps://puzzicle.org/thumbs/x5e58fede.jpgSpeed and precision of DNA replicationDNA replication is a complex, fast, and precise process involving enzymes like topoisomerase, helicase, and polymerase. In E. coli, polymerase can replicate up to 700+ base pairs per second. Initially, one mistake occurs for every 10 million nucleotides, but proofreading reduces errors to one in a billion. This amazing cellular machinery keeps our DNA accurate and functional.https://cdn.kastatic.org/ka-youtube-converted/PSl6IXUCNuw.mp4/PSl6IXUCNuw.mp4235yes https://puzzicle.org/science/ap-biology/v/semi-conservative-replicationhttps://puzzicle.org/thumbs/xcdf631f4.jpgSemi-conservative replicationDNA replication has three possible methods: conservative, dispersive, and semi-conservative. Here, we focus on the Meselson-Stahl experiment, which proved DNA replication is semi-conservative. In this process, each new DNA pair consists of one old strand and one new strand, ensuring accurate genetic information transfer.https://cdn.kastatic.org/ka-youtube-converted/jaJ3NlqK5dA.mp4/jaJ3NlqK5dA.mp4132yes https://puzzicle.org/science/ap-biology/v/transcription-and-mrna-processinghttps://puzzicle.org/thumbs/x9ad4c3f8.jpgTranscription and mRNA processingTranscription involves rewriting genetic information from DNA to mRNA, with RNA polymerase playing a crucial role. In eukaryotic cells, DNA to mRNA transcription occurs within the nucleus, producing pre-mRNA. This pre-mRNA undergoes processing, including the addition of a 5' cap, a poly-A tail, and splicing out introns, resulting in mature mRNA, which then leaves the nucleus for protein translation.https://cdn.kastatic.org/ka-youtube-converted/JQIwwJqF5D0.mp4/JQIwwJqF5D0.mp4625yes https://puzzicle.org/science/ap-biology/v/post-translational-regulationhttps://puzzicle.org/thumbs/x11cb2084.jpgPost-transcriptional regulationPost-transcriptional regulation occurs in eukaryotes, stabilizing mRNA for translation. DNA transcribes into RNA, with exons coding for proteins and introns removed by spliceosomes. The mRNA acquires a 5' cap and 3' poly-A tail for protection and translation promotion. RNA editing, a rare process, alters RNA sequences through insertion, deletion, or base substitution, impacting viral enzymes and functions.https://cdn.kastatic.org/ka-youtube-converted/nz5rbvt64kE.mp4/nz5rbvt64kE.mp4368yes https://puzzicle.org/science/ap-biology/v/translation-mrna-to-proteinhttps://puzzicle.org/thumbs/xb510c64f.jpgTranslation (mRNA to protein)Translation is the process whereby mRNA is converted into proteins by ribosomes. Translation occurs in ribosomes, which are cellular structures made of proteins and ribosomal RNA (rRNA). Here, mRNA is converted into amino acid sequences, forming polypeptides. Transfer RNA (tRNA) carries amino acids and matches them with mRNA codons, allowing ribosomes to synthesize proteins.https://cdn.kastatic.org/ka-youtube-converted/ocAAkB32Hqs.mp4/ocAAkB32Hqs.mp4850yes https://puzzicle.org/science/ap-biology/v/rna-transcription-and-translationhttps://puzzicle.org/thumbs/x4626c3c0.jpgDNA replication and RNA transcription and translationDNA serves as the molecular basis of heredity through replication, expression, and translation processes. Replication creates identical DNA strands, while transcription converts DNA into messenger RNA (mRNA). Translation then decodes mRNA into amino acids, forming proteins essential for life functions.https://cdn.kastatic.org/ka-youtube-converted/6gUY5NoX1Lk.mp4/6gUY5NoX1Lk.mp4924yes https://puzzicle.org/science/ap-biology/v/differences-in-translation-between-prokaryotes-and-eukaryoteshttps://puzzicle.org/thumbs/xfee29e2b.jpgDifferences in translation between prokaryotes and eukaryotesThe process of mRNA translation differs between prokaryotes and eukaryotes. Prokaryotic mRNA has a Shine-Dalgarno sequence for ribosome binding, while eukaryotic mRNA has a 5' cap and poly-A tail for protection and ribosome binding. The first amino acid also differs: formylmethionine in prokaryotes and methionine in eukaryotes.https://cdn.kastatic.org/ka-youtube-converted/WNZf4ip_R9s.mp4/WNZf4ip_R9s.mp4396yes https://puzzicle.org/science/ap-biology/v/retroviruseshttps://puzzicle.org/thumbs/xa3a3b196.jpgRetrovirusesRetroviruses, like HIV, have a unique replication process combining lytic and lysogenic elements. They enter cells through direct fusion and use reverse transcriptase to create complementary DNA (cDNA) from their RNA. Integrase integrates viral DNA into the host genome, forming a provirus. Viral mRNA is produced, translated into proteins, and assembled into new viruses that bud off with an envelope. Protease matures the viruses before they infect other cells.https://cdn.kastatic.org/ka-youtube-converted/KhdX5eX4dMM.mp4/KhdX5eX4dMM.mp4448yes https://puzzicle.org/science/ap-biology/v/dna-and-chromatin-regulationhttps://puzzicle.org/thumbs/x4f8acd81.jpgDNA and chromatin regulationGene regulation controls when and how much a gene is expressed. Chromatin regulation and DNA methylation are two ways to regulate genes. Chromatin regulation involves histone modifications, while DNA methylation adds methyl groups. Both are examples of epigenetic regulation and are crucial for proper cell function.https://cdn.kastatic.org/ka-youtube-converted/OqRt723t33o.mp4/OqRt723t33o.mp4321yes https://puzzicle.org/science/ap-biology/v/regulation-of-transcriptionhttps://puzzicle.org/thumbs/x9fd7ee93.jpgRegulation of transcriptionGene regulation controls cell functions by determining which genes are transcribed. This process involves transcription factors, activators, enhancers, repressors, and silencers. Prokaryotes rely on gene regulation for environmental adaptation, while eukaryotes have more complex interactions and a nuclear envelope for added control. Understanding gene expression is crucial for studying cellular processes and responses.https://cdn.kastatic.org/ka-youtube-converted/ypH-hDKpCY0.mp4/ypH-hDKpCY0.mp4407yes https://puzzicle.org/science/ap-biology/v/operons-and-gene-regulation-in-bacteriahttps://puzzicle.org/thumbs/x565e1bd2.jpgOperons and gene regulation in bacteriaDNA regulation controls gene expression in cells, allowing different cell types to perform unique functions despite having the same DNA. Prokaryotes, like bacteria, use operons, which consist of a promoter and multiple genes. Repressors, corepressors, activators, and inducers can influence transcription by binding to regulatory DNA sequences, enabling cells to adapt to their environment.https://cdn.kastatic.org/ka-youtube-converted/N7_K0yl0Egk.mp4/N7_K0yl0Egk.mp4610yes https://puzzicle.org/science/ap-biology/v/lac-operonhttps://puzzicle.org/thumbs/xe8bc037d.jpgLac operonThe lac operon in E. coli regulates lactose metabolism. It consists of three genes (lacZ, lacY, and lacA) and is controlled by a promoter, operator, and CAP site. Lactose presence triggers allolactose production, which inhibits the lac repressor, allowing transcription. Glucose levels impact transcription through cyclic AMP and the catabolite activator protein.https://cdn.kastatic.org/ka-youtube-converted/LuOaEe89_HE.mp4/LuOaEe89_HE.mp4665yes https://puzzicle.org/science/ap-biology/v/trp-operonhttps://puzzicle.org/thumbs/x3bbb0a1a.jpgTrp operonThe Trp operon plays a crucial role in tryptophan production, an essential amino acid. In E. Coli, the operon consists of genes and regulatory DNA sequences, including the promoter and operator. Tryptophan acts as a co-repressor, binding to the Trp repressor and inhibiting transcription when abundant. Additionally, feedback inhibition occurs, where tryptophan directly inhibits enzyme activity. This regulation ensures efficient resource use and tryptophan biosynthesis.https://cdn.kastatic.org/ka-youtube-converted/EvLy_1_Y3tk.mp4/EvLy_1_Y3tk.mp4396yes https://puzzicle.org/science/ap-biology/v/cellular-specialization-differentiationhttps://puzzicle.org/thumbs/xb0aae61c.jpgCellular specialization (differentiation)Stem cells differentiate into specialized cells through gene expression, influenced by internal and external cues. Asymmetric segregation of cellular determinants and inductive signaling guide this process. Transcription factors activate genes, while induction involves cell-to-cell communication. These mechanisms shape cell specialization, forming diverse body parts during development.https://cdn.kastatic.org/ka-youtube-converted/YtvL-LQlPrU.mp4/YtvL-LQlPrU.mp4511yes https://puzzicle.org/science/ap-biology/v/non-coding-rna-ncrnahttps://puzzicle.org/thumbs/xcd2b2b61.jpgNon-coding RNA (ncRNA)Non-coding RNAs (ncRNAs) are functional RNA molecules that do not translate into proteins. They play vital roles in cellular processes, including transcription and translation. Key examples include micro RNAs (miRNAs), ribosomal RNAs, transfer RNAs, small nucleolar RNAs (snow RNAs), and small nuclear RNAs (snRNAs). These ncRNAs help regulate gene expression, protein synthesis, and maintain cellular structures.https://cdn.kastatic.org/ka-youtube-converted/9YrS35wxE7o.mp4/9YrS35wxE7o.mp4280yes https://puzzicle.org/science/ap-biology/v/an-introduction-to-genetic-mutationshttps://puzzicle.org/thumbs/xa20533ca.jpgAn introduction to genetic mutationsGenetic mutations originate at the DNA level, affecting protein synthesis. Mutations can be inherited from parents or occur spontaneously due to factors like replication errors or environmental influences. Sickle-cell disease exemplifies how a single amino acid change can impact protein function, leading to health complications.https://cdn.kastatic.org/ka-youtube-converted/g02RnGXCXrQ.mp4/g02RnGXCXrQ.mp4324yes https://puzzicle.org/science/ap-biology/v/mutagens-and-carcinogenshttps://puzzicle.org/thumbs/x149a9fec.jpgMutagens and carcinogensMutagens and carcinogens can cause DNA damage, affecting protein production. Mutagens, either endogenous or exogenous, increase genetic mutation probability. Endogenous mutagens, like reactive oxygen species (ROS), are produced by the body. Exogenous mutagens include intercalators and base analogs. Carcinogens lead to cancer, and while some are mutagenic, not all are. Examples include tobacco, asbestos, and UV radiation.https://cdn.kastatic.org/ka-youtube-converted/4mrp5uI9anE.mp4/4mrp5uI9anE.mp4337yes https://puzzicle.org/science/ap-biology/v/the-effects-of-mutationshttps://puzzicle.org/thumbs/x735334d0.jpgThe effects of mutationsMutations in DNA can lead to abnormal protein production, affecting organisms in various ways. These genetic changes can be advantageous, deleterious, or neutral, depending on factors like the environment. Examples include antibiotic-resistant bacteria, cystic fibrosis, and sickle-cell disease, which demonstrate the complex nature of mutations and their impact on living beings.https://cdn.kastatic.org/ka-youtube-converted/JxC2phkdY8Y.mp4/JxC2phkdY8Y.mp4251yes https://puzzicle.org/science/ap-biology/v/impact-of-mutations-on-translation-into-amino-acidshttps://puzzicle.org/thumbs/x8d81330713137fa1.jpgImpact of mutations on translation into amino acidsDNA transcription and RNA translation are key processes in converting genetic information into proteins. Amino acid translation tables help determine the resulting amino acids, while point mutations and frameshift mutations can impact protein synthesis. Mutations play a crucial role in providing variation and driving evolution in populations.https://cdn.kastatic.org/ka-youtube-converted/1Dk0_-fyyDI.mp4/1Dk0_-fyyDI.mp4532yes https://puzzicle.org/science/ap-biology/v/mutation-as-a-source-of-variationhttps://puzzicle.org/thumbs/x4775ddb6119931e5.jpgMutation as a source of variationNatural selection relies on variation in populations, driving evolution. Variation arises from sexual reproduction, shuffling gene combinations, and mutations, creating new alleles or genes. Phenotypes, observable traits, stem from genotypes, genetic information. Mutations can be neutral, advantageous, or detrimental, influencing the survival and reproduction of an organism.https://cdn.kastatic.org/ka-youtube-converted/LZCE-4q5f9c.mp4/LZCE-4q5f9c.mp4366yes https://puzzicle.org/science/ap-biology/v/introduction-to-genetic-engineeringhttps://puzzicle.org/thumbs/xb42c5cd2ce121bdc.jpgIntroduction to genetic engineeringGenetic engineering, although a recent concept, has roots in ancient human practices like selective breeding of animals and plants. Today, recombinant DNA technology enables precise gene manipulation, creating genetically modified organisms (GMOs) with enhanced traits. While offering benefits like robust crops and new medicines, it raises bioethical concerns, especially regarding human genome modification.https://cdn.kastatic.org/ka-youtube-converted/JNgONZ8Dq9I.mp4/JNgONZ8Dq9I.mp4391yes https://puzzicle.org/science/ap-biology/v/dna-cloning-and-recombinant-dnahttps://puzzicle.org/thumbs/xfe0194fc.jpgDNA cloning and recombinant DNADNA cloning involves creating identical copies of a specific gene using restriction enzymes to cut the desired gene, pasting it into a plasmid, and inserting it into bacteria like E. coli. The bacteria replicate the gene and can produce proteins, such as insulin. Antibiotic resistance genes in plasmids help select bacteria containing the desired gene.https://cdn.kastatic.org/ka-youtube-converted/5ffl-0OYVQU.mp4/5ffl-0OYVQU.mp4668yes https://puzzicle.org/science/ap-biology/v/the-polymerase-chain-reaction-pcrhttps://puzzicle.org/thumbs/x72f7bdcd.jpgPolymerase chain reaction (PCR)PCR (Polymerase Chain Reaction) is a vital technique in molecular biology, enabling researchers to amplify specific DNA fragments exponentially. Essential for cloning, forensics, and medical diagnostics, PCR involves three main steps: denaturation, annealing, and extension. By using heat-resistant Taq polymerase and repeating cycles, scientists can generate billions of DNA copies within hours, significantly enhancing their ability to study and manipulate genetic material.https://cdn.kastatic.org/ka-youtube-converted/nHi-3jP6Mvc.mp4/nHi-3jP6Mvc.mp4594yes https://puzzicle.org/science/ap-biology/v/gel-electrophoresis-dnahttps://puzzicle.org/thumbs/x5921c437.jpgGel electrophoresisGel electrophoresis is a technique used to separate DNA, RNA, or protein fragments by size. It involves a gel, electric charge, and migration of molecules. DNA samples are placed in wells within an agarose gel, and an electric field is applied. Smaller molecules move faster through the gel, allowing for separation and estimation of fragment lengths using a DNA ladder as a reference. Ethidium bromide or other dyes are used to visualize the separated fragments under UV light.https://cdn.kastatic.org/ka-youtube-converted/_EYsykjsfiY.mp4/_EYsykjsfiY.mp4660yes https://puzzicle.org/science/ap-biology/v/dna-sequencinghttps://puzzicle.org/thumbs/x1baa728e.jpgDNA sequencingDNA sequencing involves three main steps: 1) using PCR to amplify DNA fragments, 2) introducing dideoxynucleotides that halt DNA strand elongation, and 3) employing a computer to analyze the fluorescent labels on the DNA fragments to determine the sequence.https://cdn.kastatic.org/ka-youtube-converted/Jnk_4Maf5Fk.mp4/Jnk_4Maf5Fk.mp4281yes https://puzzicle.org/science/ap-biology/v/applications-of-dna-technologieshttps://puzzicle.org/thumbs/x37c99fd7.jpgApplications of DNA technologiesDNA technology revolutionizes medicine, forensics, and agriculture. Recombinant DNA creates insulin and growth hormones, improves vaccine safety, and helps solve crimes through genome analysis. Genetically modified crops enhance resistance, delay ripening, and support farmers, feeding more people and boosting the economy.https://cdn.kastatic.org/ka-youtube-converted/gqMqYYcJgz4.mp4/gqMqYYcJgz4.mp4301yes https://puzzicle.org/science/ap-biology/v/introduction-to-evolution-and-natural-selectionhttps://puzzicle.org/thumbs/23916.jpgIntroduction to evolution and natural selectionNatural selection is a mechanism of evolution and explains how species adapt to their environment over time through variations in traits. Examples include the peppered moth adapting to industrial pollution, yearly flu virus changes, and antibiotic-resistant bacteria. Understanding natural selection is essential for studying living systems.https://cdn.kastatic.org/ka-youtube-converted/GcjgWov7mTM.mp4/GcjgWov7mTM.mp41059yes https://puzzicle.org/science/ap-biology/v/natural-selection-and-the-owl-butterflyhttps://puzzicle.org/thumbs/23922.jpgNatural selection and the owl butterflyIntricate patterns, such as the owl butterfly's eye-like wing markings, evolve through natural selection. Over thousands of generations, small variations in wing patterns offer slight survival advantages, leading to the dominance of beneficial traits. This gradual process allows complex designs to emerge from random genetic mutations.https://cdn.kastatic.org/ka-youtube-converted/dR_BFmDMRaI.mp4/dR_BFmDMRaI.mp4809yes https://puzzicle.org/science/ap-biology/v/biodiversity-and-natural-selection-twohttps://puzzicle.org/thumbs/x29f3228a.jpgBiodiversity and natural selectionThe patterns and processes of evolution play a crucial role in biodiversity. Natural selection, adaptation, and inheritance serve as key mechanisms driving diversity. Examining relationships among organisms, genealogical trees, and environmental factors provides insight into the complex interplay shaping Earth's rich biodiversity.https://cdn.kastatic.org/ka-youtube-converted/sNgsCT0E7Es.mp4/sNgsCT0E7Es.mp4543yes https://puzzicle.org/science/ap-biology/v/molecular-variationhttps://puzzicle.org/thumbs/xe10e4777e8df7ebc.jpgMolecular variationMolecular variation in cells allows organisms to respond to a wider range of environmental stimuli, and allows populations to adapt to a wider range of environmental conditions. Examples of molecular variation include the different types of hemoglobin proteins, and the different forms of chlorophyll produced in cells.https://cdn.kastatic.org/ka-youtube-converted/RYz3d1meQQc.mp4/RYz3d1meQQc.mp4373yes https://puzzicle.org/science/ap-biology/v/artificial-selection-and-domesticationhttps://puzzicle.org/thumbs/xcadbfc6f4f94806e.jpgArtificial selection and domesticationArtificial selection and domestication are processes where humans influence the evolution of species. Dogs, farm animals, and crops have all been shaped by these methods. Humans select which individuals reproduce based on desired traits, accelerating changes that would take much longer in natural selection. This has led to the diverse breeds of dogs and varieties of crops we see today.https://cdn.kastatic.org/ka-youtube-converted/hFjGQXcwYIk.mp4/hFjGQXcwYIk.mp4251yes https://puzzicle.org/science/ap-biology/v/genetic-drift-bottleneck-effect-and-founder-effecthttps://puzzicle.org/thumbs/x8c483d3a.jpgGenetic drift, bottleneck effect, and founder effectEvolution has multiple mechanisms, including genetic drift, which involves random changes in trait frequency. In particular, genetic drift is more likely in small populations. Examples include the bottleneck effect, where a disaster reduces population size, and the founder effect, where a small group starts a new population; both result in less genetic variation.https://cdn.kastatic.org/ka-youtube-converted/IRTn0iNkAHI.mp4/IRTn0iNkAHI.mp4647yes https://puzzicle.org/science/ap-biology/v/dna-spells-evolutionhttps://puzzicle.org/thumbs/xe9279bc7.jpgDNA spells evolutionGenetic mutations lead to variations in traits. These variations can provide survival advantages, resulting in new species over generations. For example, brown bears evolved into polar bears due to beneficial mutations.https://cdn.kastatic.org/ka-youtube-converted/uI_Q1kjmfxo.mp4/uI_Q1kjmfxo.mp4134yes https://puzzicle.org/science/ap-biology/v/allele-frequencyhttps://puzzicle.org/thumbs/xf2057c13.jpgAllele frequencyAllele frequency describes how often an allele (a variant of a gene) appears in a population. In this video, eye color is used as an example, with brown (B) eyes being dominant and blue (b) eyes being recessive. Allele frequency (or genotype frequency) can also differ from phenotype frequency.https://cdn.kastatic.org/ka-youtube-converted/Bc9bhLk_AhI.mp4/Bc9bhLk_AhI.mp4447yes https://puzzicle.org/science/ap-biology/v/hardy-weinberghttps://puzzicle.org/thumbs/x5628ee19.jpgHardy-Weinberg equationThe Hardy-Weinberg equilibrium is a principle that helps to predict allele frequencies in a population. It assumes no selection, no mutation, no geneflow, random mating, and large populations for stable allele frequencies. The Hardy-Weinberg equation calculates probabilities of homozygous dominant, heterozygous, and homozygous recessive genotypes.https://cdn.kastatic.org/ka-youtube-converted/oc9fJCAIRJs.mp4/oc9fJCAIRJs.mp4494yes https://puzzicle.org/science/ap-biology/v/applying-hardy-weinberghttps://puzzicle.org/thumbs/x15272d4a.jpgApplying the Hardy-Weinberg equationThe Hardy-Weinberg equation can help to estimate allele frequencies in a population. Dominant (p) and recessive (q) allele frequencies and genotype frequencies can be calculated using the equation. In this video, eye color is used as an example to determine allele frequencies and genotype distribution.https://cdn.kastatic.org/ka-youtube-converted/D5NVlAaT-OA.mp4/D5NVlAaT-OA.mp4330yes https://puzzicle.org/science/ap-biology/v/discussions-of-conditions-for-hardy-weinberghttps://puzzicle.org/thumbs/x88d56ee0.jpgDiscussions of conditions for Hardy-WeinbergThe Hardy-Weinberg equation assumes stable allele frequencies in a population. Key conditions include no mutation, random mating, no gene flow, infinite population size, and no selection. Diploid organisms with sexual reproduction are also assumed. Although real-world populations may not strictly adhere to these conditions, the equation remains a useful approximation.https://cdn.kastatic.org/ka-youtube-converted/Q618DALi2eo.mp4/Q618DALi2eo.mp4414yes https://puzzicle.org/science/ap-biology/v/evidence-for-evolutionhttps://puzzicle.org/thumbs/x6bd5a4dc.jpgEvidence for evolutionThree types of evidence support evolution and natural selection: structural, microbiological, and direct observation. Structural evidence encompasses homologous structures and the fossil record. Microbiological evidence involves DNA similarities and shared processes across species. Direct observation showcases evolution in action, such as antibiotic resistance in bacteria.https://cdn.kastatic.org/ka-youtube-converted/Q-aGAX27SIo.mp4/Q-aGAX27SIo.mp4779yes https://puzzicle.org/science/ap-biology/v/fossils-rocking-the-earthhttps://puzzicle.org/thumbs/x1cf8e423.jpgFossils: Rocking the EarthFossils reveal connections between ancient and modern organisms, providing strong evidence for evolution. The fossil record shows gradual evolutionary change, physical proof of extinction, species splitting, and transitional forms. By studying fossils, we can trace the development of features in living organisms and understand how life on Earth has evolved over time.https://cdn.kastatic.org/ka-youtube-converted/rSWwmN5ozQ0.mp4/rSWwmN5ozQ0.mp4134yes https://puzzicle.org/science/ap-biology/v/biogeography-where-life-liveshttps://puzzicle.org/thumbs/x22546072.jpgBiogeography: Where life livesBiogeography explores the distribution of plants, animals, and organisms across Earth. New islands form habitats for unique species, like Galapagos finches and Hawaiian honeycreepers. Continental drift explains related species on different continents, with Pangaea's breakup separating organisms. Phylogenetic trees help trace species' movements, revealing Earth's dynamic biodiversity.https://cdn.kastatic.org/ka-youtube-converted/2UQC5ts6hUs.mp4/2UQC5ts6hUs.mp4110yes https://puzzicle.org/science/ap-biology/v/molecular-evidence-for-evolutionary-relationships-exampleshttps://puzzicle.org/thumbs/xb4f57f93bafc053f.jpgMolecular evidence for evolutionary relationships: ExamplesThis investigation focuses on determining the closest plant species to an unknown species by analyzing enzyme presence, amino acid sequence differences, and gel electrophoresis banding patterns. Plant species with the most similarities in these factors are considered closely related. The method also applies to finch species, using molecular evidence to establish evolutionary relationships.https://cdn.kastatic.org/ka-youtube-converted/mHH0sSm2cK8.mp4/mHH0sSm2cK8.mp4365yes https://puzzicle.org/science/ap-biology/v/carbon-14-dating-1https://puzzicle.org/thumbs/8187416.jpgCarbon-14 dating: Part 1Carbon-14 forms in the atmosphere through cosmic rays interacting with nitrogen-14, creating a small fraction of carbon-14 in living organisms. When an organism dies, its carbon-14 decays back into nitrogen-14 at a known rate, called a half-life (5,730 years). By measuring the remaining carbon-14 in fossils, scientists can estimate their age, helping us understand the history of life on Earth.https://cdn.kastatic.org/ka-youtube-converted/8wYvKeSK1IY.mp4/8wYvKeSK1IY.mp4603yes https://puzzicle.org/science/ap-biology/v/cellular-evidence-of-common-ancestryhttps://puzzicle.org/thumbs/xc5a0515c015fc65d.jpgCellular evidence of common ancestryAll living organisms share a common ancestor, evidenced by cellular similarities like DNA, RNA, and glycolysis. Evolutionary trees depict these relationships, with branches representing different species. Eukaryotes, a distinct group, possess membrane-bound organelles, linear chromosomes, and introns. DNA and RNA sequencing refines our understanding of these connections, enabling more precise evolutionary tree construction.https://cdn.kastatic.org/ka-youtube-converted/BwwElHbjTGo.mp4/BwwElHbjTGo.mp4229yes https://puzzicle.org/science/ap-biology/v/taxonomy-and-the-tree-of-lifehttps://puzzicle.org/thumbs/351477593.jpgTaxonomy and the tree of lifeCarl Linnaeus, the father of modern taxonomy, developed a system for classifying living organisms into categories like species, genus, order, class, and kingdom. This tree of life helps us understand relationships between organisms. Taxonomy is both an art and a science, with DNA analysis now playing a significant role. Humans are also classified within this system, highlighting our connections to other life forms.https://cdn.kastatic.org/ka-youtube-converted/oHvLlS_Sc54.mp4/oHvLlS_Sc54.mp4761yes https://puzzicle.org/science/ap-biology/v/discovering-the-tree-of-lifehttps://puzzicle.org/thumbs/xd6f10a27.jpgDiscovering the tree of lifeThe study of evolutionary lineages involves analyzing biodiversity over time. Phylogenetic systematics are a method to study the tree of life. By examining unique features in organisms, scientists can trace shared evolutionary histories and relationships. Modern techniques, such as DNA analysis, enhance our understanding of these connections, allowing us to better protect Earth's threatened biodiversity.https://cdn.kastatic.org/ka-youtube-converted/AjvLQJ6PIiU.mp4/AjvLQJ6PIiU.mp4470yes https://puzzicle.org/science/ap-biology/v/understanding-and-building-phylogenetic-trees-or-cladogramshttps://puzzicle.org/thumbs/xc14e1283.jpgUnderstanding and building phylogenetic treesConstructing a phylogenetic tree involves hypothesizing evolutionary relationships among species based on observable traits and genetic evidence. By practicing parsimony, we aim for the simplest explanation. This graphical representation helps determine relatedness and common ancestors, allowing us to understand the evolutionary history of different species.https://cdn.kastatic.org/ka-youtube-converted/6_XMKmFQ_w8.mp4/6_XMKmFQ_w8.mp4657yes https://puzzicle.org/science/ap-biology/v/specieshttps://puzzicle.org/thumbs/351477592.jpgSpeciesTaxonomy comes to life through the exploration of species, defined as groups of animals that can interbreed to produce fertile offspring. Examples of species include lions, tigers, and horses. Hybrids are the offspring of different species, which are usually infertile. Examples of hybrids include mules and ligers.https://cdn.kastatic.org/ka-youtube-converted/Tmt4zrDK3dA.mp4/Tmt4zrDK3dA.mp4529yes https://puzzicle.org/science/ap-biology/v/allopatric-and-sympatric-speciationhttps://puzzicle.org/thumbs/x1a6232a5.jpgSpeciationSpeciation occurs when populations become different enough that they no longer interbreed and produce fertile offspring. Allopatric speciation results from geographic separation, while sympatric speciation can occur without a physical barrier. Natural selection, genetic drift, behavior, and changes in chromosome number can contribute to the formation of new species.https://cdn.kastatic.org/ka-youtube-converted/gnku5AksWvU.mp4/gnku5AksWvU.mp4647yes https://puzzicle.org/science/ap-biology/v/reproductive-isolationhttps://puzzicle.org/thumbs/x2c49b090.jpgReproductive isolationSpecies are groups of organisms that can reproduce together; reproductive isolation keeps species separate. Pre-zygotic isolation prevents mating or fertilization (through temporal, habitat, behavioral, or mechanical isolation), while post-zygotic isolation affects zygotes or hybrids (via zygote mortality, hybrid inviability, or hybrid sterility). Although sexually-reproducing species are easier to classify, the species problem makes classifying asexually-reproducing organisms difficult.https://cdn.kastatic.org/ka-youtube-converted/vmgKC2IJjOI.mp4/vmgKC2IJjOI.mp4299yes https://puzzicle.org/science/ap-biology/v/new-localities-lead-to-new-biodiversityhttps://puzzicle.org/thumbs/xee15c10e.jpgNew localities lead to new biodiversityDispersal is the process by which species move and expand their ranges. It can lead to speciation, or new species forming. Islands represent excellent case studies; organisms on islands often experience isolation and restricted gene flow, resulting in unique speciation patterns (called island biogeography). Factors like distance, time, area, habitat, location, and serendipity influence the rate of colonization and speciation on islands.https://cdn.kastatic.org/ka-youtube-converted/ACxH7ZVUOLk.mp4/ACxH7ZVUOLk.mp4630yes https://puzzicle.org/science/ap-biology/v/environmental-change-and-adaptation-in-galapagos-fincheshttps://puzzicle.org/thumbs/x9dcd3b63fb2e570b.jpgEnvironmental change and adaptation in Galápagos finchesAdaptation by natural selection is one important process by which species change over time. After an environmental change, traits that support successful survival and reproduction in the new environment become more common; those that do not become less common. Thus, the distribution of traits in a population changes.https://cdn.kastatic.org/ka-youtube-converted/0G2uDKYBX-c.mp4/0G2uDKYBX-c.mp4141yes https://puzzicle.org/science/ap-biology/v/population-diversity-and-resiliencehttps://puzzicle.org/thumbs/xac52938959819e0e.jpgPopulation diversity and resilienceBiodiversity plays a critical role in the survival of populations, with genetic diversity providing resilience against environmental changes. The higher the genetic diversity in a population, the more resilient the population is to environmental change. This resilience ensures the survival and reproduction of populations, underscoring the importance of biodiversity and ecosystem conservation.https://cdn.kastatic.org/ka-youtube-converted/3NdTUkjwehs.mp4/3NdTUkjwehs.mp4269yes https://puzzicle.org/science/ap-biology/v/earth-formationhttps://puzzicle.org/thumbs/13179297.jpgEarth formationEarth ​was formed approximately 4.6 billion years ago, likely as the result of a supernova (star explosion). The debris from this explosion began to collapse in on itself due to gravity, forming the sun. Gravity continued to draw the remaining particles together, clumping them into larger bodies, ultimately forming Earth and the other planets in our solar system.https://cdn.kastatic.org/ka-youtube-converted/VbNXh0GaLYo.mp4/VbNXh0GaLYo.mp4608yes https://puzzicle.org/science/ap-biology/v/beginnings-of-lifehttps://puzzicle.org/thumbs/13179298.jpgBeginnings of lifeThe Archean Eon marks an early chapter in Earth's history when simple prokaryotic life existed in the oceans. Early organisms were likely anaerobic, but some bacteria evolved photosynthesis and began producing oxygen. This major change gradually transformed Earth's environment, setting the stage for oxygen-breathing organisms to evolve much later.https://cdn.kastatic.org/ka-youtube-converted/nYFuxTXDj90.mp4/nYFuxTXDj90.mp4626yes https://puzzicle.org/science/ap-biology/v/origins-of-lifehttps://puzzicle.org/thumbs/x24f81a83.jpgOrigins of lifeThe origin of life remains an open scientific question. Evidence suggests that simple molecules on early Earth could form organic molecules such as amino acids and nucleotides without life. Hypotheses including the RNA world and metabolism first explore how increasingly complex molecules and chemical processes may have eventually developed into the earliest life forms.https://cdn.kastatic.org/ka-youtube-converted/rMuj2MGeIAc.mp4/rMuj2MGeIAc.mp4632yes https://puzzicle.org/science/ap-biology/v/originoflifehttps://puzzicle.org/thumbs/x2d33c6b3.jpgThe RNA origin of lifeThe RNA world hypothesis proposes that RNA may have played a key role in the origin of life. Unlike DNA or proteins alone, RNA can store genetic information and perform cellular functions. Scientists hypothesize that early self-replicating RNA molecules evolved over time, eventually contributing to the development of proteins, DNA, and more complex life.https://cdn.kastatic.org/ka-youtube-converted/VYQQD0KNOis.mp4/VYQQD0KNOis.mp4190yes https://puzzicle.org/science/ap-biology/v/animal-communicationhttps://puzzicle.org/thumbs/x6ae75632.jpgAnimal communicationAnimal communication encompasses a diverse range of methods, including nonverbal and visual cues, to convey information within and among species, and even with humans. Key functions of animal communication include mating rituals, territorial proclamations, food discovery, and alarm signals. It is crucial to avoid anthropomorphizing animals by attributing human emotions and thoughts to their behaviors, as their primary goal is survival and adaptation within their environment.https://cdn.kastatic.org/ka-youtube-converted/1kbZ1JAoi5o.mp4/1kbZ1JAoi5o.mp4262yes https://puzzicle.org/science/ap-biology/v/animal-behavior-foraginghttps://puzzicle.org/thumbs/xda926539.jpgAnimal behavior: ForagingForaging—a crucial animal behavior—involves searching for food and balancing energy costs. Two primary strategies exist: solitary foraging (where animals hunt alone) and group foraging (where animals cooperate). While genetics play a significant role, animals also learn foraging techniques by observing and mimicking others in their group.https://cdn.kastatic.org/ka-youtube-converted/OrCoshLAOiQ.mp4/OrCoshLAOiQ.mp4134yes https://puzzicle.org/science/ap-biology/v/organism-life-history-and-fecundityhttps://puzzicle.org/thumbs/x0312bee7.jpgLife history strategies and fecunditySpecies exhibit diverse life histories, lifespans, and reproduction rates, as seen in elephants, rabbits, and salmon. Fecundity measures reproduction rate, while semelparity and iteroparity describe single or multiple reproductive events. Natural selection influences these traits based on factors like energy availability, predation, and disease, leading species to make trade-offs between growth, reproduction, and survival.https://cdn.kastatic.org/ka-youtube-converted/mAdUjImDx58.mp4/mAdUjImDx58.mp4707yes https://puzzicle.org/science/ap-biology/v/k-selected-and-r-selected-population-growth-strategieshttps://puzzicle.org/thumbs/x2d6cda30fa6cbafb.jpgr-selected and K-selected population growth strategiesr and K are two extremes for a range of population growth strategies. K-selected species, like elephants, are typically larger, live longer, and produce fewer offspring. Conversely, r-selected species, such as frogs, are generally smaller, have shorter lifespans, and produce a larger number of offspring. These strategies are shaped by various factors, including environmental conditions and the level of competition for resources.https://cdn.kastatic.org/ka-youtube-converted/ey4FTyYjiPw.mp4/ey4FTyYjiPw.mp4400yes https://puzzicle.org/science/ap-biology/v/flow-of-energy-and-matter-through-ecosystemshttps://puzzicle.org/thumbs/xe5f09d743abf7323.jpgFlow of energy and matter through ecosystemsEnergy flows and matter recycles in ecosystems, with the Sun as the primary energy source. Plants, as primary producers, convert sunlight into energy-storing biomolecules. Consumers, like animals, obtain energy by eating plants or other animals. Decomposers break down dead organisms, recycling matter and nutrients. Energy is conserved but often released as heat, while matter constantly cycles through the ecosystem.https://cdn.kastatic.org/ka-youtube-converted/TitrRpMUt0I.mp4/TitrRpMUt0I.mp4626yes https://puzzicle.org/science/ap-biology/v/impact-of-changes-to-trophic-pyramidshttps://puzzicle.org/thumbs/xadb4c9cef7f5b2bb.jpgImpact of changes to trophic pyramidsThe trophic pyramid illustrates energy transfer within ecosystems, highlighting primary producers, consumers, and decomposers. Energy from the sun is stored as biomass, with only 10% transferred between trophic levels. Ecosystem changes, such as reduced sunlight or pesticide use, can impact the entire pyramid, emphasizing the interconnectedness of ecosystems and the importance of energy cycling.https://cdn.kastatic.org/ka-youtube-converted/sV-KQ91Fnck.mp4/sV-KQ91Fnck.mp4432yes https://puzzicle.org/science/ap-biology/v/the-water-cyclehttps://puzzicle.org/thumbs/x46df8fd3c0a2c721.jpgThe water cycleThe water cycle—powered by the sun—involves water moving in various forms between sources and sinks. It starts with evaporation from bodies of water, condensation into clouds, and precipitation back to the earth. This cycle also includes transpiration from plants and sublimation from ice to vapor. The majority of Earth's water is saltwater, with only a small percentage being freshwater in lakes, rivers, and groundwater.https://cdn.kastatic.org/ka-youtube-converted/jFjI6y46QRk.mp4/jFjI6y46QRk.mp4494yes https://puzzicle.org/science/ap-biology/v/carbon-cyclehttps://puzzicle.org/thumbs/x33bc098853d5090c.jpgThe carbon cycleCarbon is crucial for life, forming molecules like glucose, ATP, amino acids, and DNA. The carbon cycle involves plants taking in carbon dioxide, fixing carbon, and creating organic molecules. Animals eat plants, metabolize the molecules, and release carbon dioxide. This cycle keeps carbon circulating in the biosphere.https://cdn.kastatic.org/ka-youtube-converted/_dYkByQ9Kmg.mp4/_dYkByQ9Kmg.mp4517yes https://puzzicle.org/science/ap-biology/v/phosphorous-cyclehttps://puzzicle.org/thumbs/xc07f92006e35dfec.jpgThe phosphorus cycleThe phosphorus cycle, essential for life, involves the movement of phosphorus atoms from rocks and soil into plants and animals. Unlike carbon or nitrogen, phosphorus doesn't exist in the atmosphere. It's often a limiting factor for plant growth, making it a key ingredient in fertilizers. Human activities like mining and farming can disrupt this cycle, leading to environmental consequences.https://cdn.kastatic.org/ka-youtube-converted/tm2LG5ScT1g.mp4/tm2LG5ScT1g.mp4331yes https://puzzicle.org/science/ap-biology/v/exponential-and-logistic-growth-in-populationshttps://puzzicle.org/thumbs/x32a950e6.jpgExponential and logistic growth in populationsRabbit populations grow exponentially when not limited by resources, space, or predators. Exponential growth has time in the exponent, causing a rapid increase in population size. In real-world situations, logistic growth is more accurate due to environmental constraints. Logistic growth models population growth with a natural carrying capacity, creating an S-shaped curve.https://cdn.kastatic.org/ka-youtube-converted/KyAKEisg2PQ.mp4/KyAKEisg2PQ.mp4452yes https://puzzicle.org/science/ap-biology/v/population-growth-rate-based-on-birth-and-death-rateshttps://puzzicle.org/thumbs/xc6c411beaccb5407.jpgPopulation growth rate based on birth and death ratesCalculating population growth rate is simpler than it seems. By comparing birth and death rates, we can intuitively find the growth rate. Here, we show the formula to calculate population growth rate, using calculus notation. Although this formula may look intimidating, it represents a straightforward concept, making it easy to grasp and apply in biology.https://cdn.kastatic.org/ka-youtube-converted/R2yPeLJsPOo.mp4/R2yPeLJsPOo.mp4194yes https://puzzicle.org/science/ap-biology/v/per-capita-population-growth-and-exponential-growthhttps://puzzicle.org/thumbs/x1bf413fba7f31c36.jpgPer capita population growth and exponential growthPopulation growth rate represents the net increase in population based on birth and death rates. Per capita population growth rate measures the average growth rate per individual. Exponential growth equations describe unconstrained population growth, while environmental carrying capacity sets real-world limits.https://cdn.kastatic.org/ka-youtube-converted/B_qsI4Q1nNk.mp4/B_qsI4Q1nNk.mp4525yes https://puzzicle.org/science/ap-biology/v/density-dependent-and-density-independent-population-regulationhttps://puzzicle.org/thumbs/x19eaca00.jpgPopulation regulationThere are two types of population regulation: density-dependent and density-independent. Density-dependent factors, like competition for resources, predation, disease, and waste accumulation, limit population growth based on population density. Density-independent factors, such as natural disasters, affect populations regardless of density. Understanding these factors helps us consider human population growth and potential carrying capacity.https://cdn.kastatic.org/ka-youtube-converted/6IS_M6CX7FE.mp4/6IS_M6CX7FE.mp4633yes https://puzzicle.org/science/ap-biology/v/logistic-growth-versus-exponential-growthhttps://puzzicle.org/thumbs/xab6648aeea47abe1.jpgLogistic growth versus exponential growthExponential growth presumes infinite resources, resulting in unrestrained population expansion. Conversely, logistic growth considers resource limitations and a carrying capacity (K): the maximum sustainable population. The logistic growth model modifies the exponential growth equation by including a factor that decelerates population growth as it nears the carrying capacity.https://cdn.kastatic.org/ka-youtube-converted/jfIPQ5_71Mc.mp4/jfIPQ5_71Mc.mp4602yes https://puzzicle.org/science/ap-biology/v/interactions-between-populationshttps://puzzicle.org/thumbs/x4b4b2de6.jpgInteractions between populationsInterspecific interactions are the relationships and interactions between different species living together in a shared habitat, such as competition, predation, and symbiosis. Competition involves species vying for resources, while predation entails one species consuming another. Symbiosis involves long-term relationships, including parasitism (one benefits, the other is harmed), mutualism (both benefit), and commensalism (one benefits, the other remains unaffected).https://cdn.kastatic.org/ka-youtube-converted/q2zdiLn3gSE.mp4/q2zdiLn3gSE.mp4492yes https://puzzicle.org/science/ap-biology/v/predator-prey-cyclehttps://puzzicle.org/thumbs/x3d85d1d2.jpgPredator-prey cyclesThe predator-prey cycle demonstrates how populations of predators and prey interact and influence each other's growth. As predator numbers increase, prey populations decline, leading to a decrease in predators. This allows prey populations to recover, restarting the cycle. The snowshoe hare and Canadian lynx exemplify this ecological relationship through observed population fluctuations over time.https://cdn.kastatic.org/ka-youtube-converted/NYq2078_xqc.mp4/NYq2078_xqc.mp4309yes https://puzzicle.org/science/ap-biology/v/exploring-ecosystems-tropical-rainforest-diversity-california-academy-of-sciencehttps://puzzicle.org/thumbs/xee183dd2.jpgTropical rainforest diversityTropical rainforests are among the most diverse, energy-rich ecosystems on the planet and host a vast array of species within distinct vertical layers. Researchers study these ecosystems' structure and diversity, focusing on species richness and evenness. Diverse ecological communities are more stable and resilient, helping rainforests adapt to disturbances like deforestation and climate change. Unraveling the mysteries of tropical rainforests can aid in their conservation.https://cdn.kastatic.org/ka-youtube-converted/LHPuo0rwM1w.mp4/LHPuo0rwM1w.mp4244yes https://puzzicle.org/science/ap-biology/v/simpsons-index-of-diversityhttps://puzzicle.org/thumbs/x88701f3862b56814.jpgSimpson's index of diversityLearn how to calculate Simpson's diversity index, a valuable tool for quantifying species diversity in communities. By comparing two communities with equal populations but different species distributions, the index reveals that a more evenly distributed community exhibits higher diversity. This mathematical method offers one way to evaluate biodiversity.https://cdn.kastatic.org/ka-youtube-converted/bfxhh7l-LQw.mp4/bfxhh7l-LQw.mp4343yes https://puzzicle.org/science/ap-biology/v/biodiversity-ecosystems-and-ecological-networkshttps://puzzicle.org/thumbs/x4baff49e.jpgEcosystems and ecological networksBiodiversity, specifically species richness, plays a crucial role in ecosystem stability and resilience. The emerging field of biodiversity-ecosystem function (BEF) studies these interactions. Higher biodiversity spreads interactions among more species, reducing the chance of ecological collapse. Ecosystems with more species and interactions are less likely to suffer negative impacts from disturbances, as crucial components are less likely to be removed.https://cdn.kastatic.org/ka-youtube-converted/-bE-Pydad7U.mp4/-bE-Pydad7U.mp4536yes https://puzzicle.org/science/ap-biology/v/introduced-species-and-biodiversityhttps://puzzicle.org/thumbs/x96adeb74.jpgIntroduced species and biodiversityIntroduced species, also known as exotic species, can disrupt ecosystems and harm native species. While some introduced species benefit humans and native species, invasive species outcompete natives, causing ecological damage. Invasive species often lack natural predators and can introduce diseases, parasites, and pathogens. It is important to recognize and mitigate the impact of invasive species on ecosystems to preserve biodiversity and maintain healthy ecological functions.https://cdn.kastatic.org/ka-youtube-converted/J-ftiWffNTc.mp4/J-ftiWffNTc.mp4596yes https://puzzicle.org/science/ap-biology/v/human-activities-that-threaten-biodiversityhttps://puzzicle.org/thumbs/xd1385475.jpgHuman activities that threaten biodiversityBiodiversity is threatened by human population growth and resource utilization, leading to decreased species richness. Local factors contributing to biodiversity loss include land-use changes, pollution, resource exploitation, and introduction of exotic species. Balancing human needs with ecosystem health requires doing more with less and preserving species richness for a sustainable future.https://cdn.kastatic.org/ka-youtube-converted/2RC3Hsk90t8.mp4/2RC3Hsk90t8.mp4771yes https://puzzicle.org/science/ap-biology/v/how-does-climate-change-affect-biodiversityhttps://puzzicle.org/thumbs/x3f29eda1.jpgHow does climate change affect biodiversity?Global warming, caused by increasing greenhouse gas emissions, threatens biodiversity. As temperatures rise, species struggle to adapt, leading to extinctions and ecosystem disruptions. Melting ice contributes to sea level rise, endangering low-lying habitats. Additionally, increased carbon dioxide levels affect plant growth and biodiversity, potentially exacerbating climate change through feedback loops. Urgent action is needed to address these challenges and protect our planet's diverse ecosystems.https://cdn.kastatic.org/ka-youtube-converted/XFmovUAWQUQ.mp4/XFmovUAWQUQ.mp4727yes https://puzzicle.org/science/ap-biology/v/biodiversity-and-extinction-then-and-nowhttps://puzzicle.org/thumbs/x778c92a4.jpgBiodiversity and extinction, then and nowExtinctions may occur because of direct, natural phenomena or the inability of species to adapt quickly enough to changing environmental conditions or ecological interactions. Five major mass extinctions have happened on Earth. A sixth mass extinction is currently underway and is driven by human activity, which underscores the need to learn from the past to safeguard our future.https://cdn.kastatic.org/ka-youtube-converted/v8PPsI0LqOY.mp4/v8PPsI0LqOY.mp4742yes https://puzzicle.org/science/ap-biology/v/how-did-dinosaurs-go-extincthttps://puzzicle.org/thumbs/x7e9ef16a.jpgHow did all dinosaurs except birds go extinct?The non-avian dinosaur extinction, excluding birds, remains a captivating mystery for paleontologists. The leading theory suggests an asteroid impact 66 million years ago caused tsunamis, blocked sunlight, halted photosynthesis, and induced freezing temperatures, acid rain, and global warming. Concurrently, immense lava flows in southwest India may have produced similar effects. While the asteroid impact is considered the final straw, both events likely contributed to the extinction.https://cdn.kastatic.org/ka-youtube-converted/mCKeogDioWE.mp4/mCKeogDioWE.mp4115yes https://puzzicle.org/science/ap-biology/v/dinosaur-declinehttps://puzzicle.org/thumbs/xc6e48ef4.jpgWere dinosaurs undergoing long-term decline before mass extinction?Were dinosaurs already undergoing a long-term decline before an asteroid hit at the end of the Cretaceous about 65.5 million years ago? A study led by museum scientists gives a multifaceted answer. Results suggest that (in general) large-bodied, bulk-feeding herbivores were declining during the last 12 million years of the Cretaceous Period. However, carnivorous dinosaurs and mid-sized herbivores were not.https://cdn.kastatic.org/ka-youtube-converted/cgGnvH0Ozvk.mp4/cgGnvH0Ozvk.mp4164yes https://puzzicle.org/science/ap-biology/v/introduction-to-simulation-based-exerciseshttps://puzzicle.org/thumbs/xa74da60b1d66487b.jpgIntroduction to simulation-based exercisesKhan Academy’s simulation-based exercises let students explore real-world phenomena through interactive experiments, data analysis, and evidence-based reasoning. Aligned to the Next Generation Science Standards or AP course CEDs, these engaging tools bring key concepts to life—helping students think and investigate like professional scientists.https://cdn.kastatic.org/ka-youtube-converted/JO68D_EieXA.mp4/JO68D_EieXA.mp4148yes https://puzzicle.org/science/ap-chemistry-beta/v/mass-spectrometryhttps://puzzicle.org/thumbs/x8a15ff9cea78f965.jpgMass spectrometryIn the analytical technique of mass spectrometry, atoms or molecules are ionized using a high-energy electron beam and then separated based on their mass-to-charge ratios (m/z). The results are presented as a mass spectrum, which shows the relative abundances of the ions on the y-axis and their m/z ratios on the x-axis. This data can be used to calculate the exact masses of the atoms or molecules in the sample.https://cdn.kastatic.org/ka-youtube-converted/myolF-h1kKI.mp4/myolF-h1kKI.mp4258yes https://puzzicle.org/science/ap-chemistry-beta/v/worked-example-identifying-element-from-mass-spectrumhttps://puzzicle.org/thumbs/xdeff1bb75f94717d.jpgWorked example: Identifying an element from its mass spectrumUsing mass spectrometry data, we can estimate an element's average atomic mass and determine its identity.https://cdn.kastatic.org/ka-youtube-converted/vU7MDe4ph5A.mp4/vU7MDe4ph5A.mp4201yes https://puzzicle.org/science/ap-chemistry-beta/v/worked-example-calculating-mass-of-substance-in-mixturehttps://puzzicle.org/thumbs/x66735a2a3d420f6d.jpgWorked example: Calculating the mass of a substance in a mixtureElemental analysis can be used to determine the amounts of substances in a mixture. For example, if elemental analysis tells us that a potassium supplement contains 22% K by mass, and we know that the K is present as KCl, we can calculate the grams of KCl in the supplement.https://cdn.kastatic.org/ka-youtube-converted/Zy3Y3ZbJ2xs.mp4/Zy3Y3ZbJ2xs.mp4321yes https://puzzicle.org/science/ap-chemistry-beta/v/worked-example-analyzing-purity-of-a-mixturehttps://puzzicle.org/thumbs/xfe065fa1d26a5da0.jpgWorked example: Analyzing the purity of a mixtureElemental analysis can be used to analyze the purity of a sample. For example, a pure sample of NaCl should contain 61% chlorine by mass. If elemental analysis tells us that the sample actually contains 73% chlorine by mass, this suggests that our sample has been contaminated by a compound containing a higher mass percent of chlorine.https://cdn.kastatic.org/ka-youtube-converted/K5HPsKdgxsI.mp4/K5HPsKdgxsI.mp4401yes https://puzzicle.org/science/hs-chemistry/v/orbitalshttps://puzzicle.org/thumbs/425003.jpgShells, subshells, and orbitalsThe electrons in an atom are arranged in shells that surround the nucleus, with each successive shell being farther from the nucleus. Electron shells consist of one or more subshells, and subshells consist of one or more atomic orbitals. Electrons in the same subshell have the same energy, while electrons in different shells or subshells have different energies.https://cdn.kastatic.org/ka-youtube-converted/26PF9Ctch9Y.mp4/26PF9Ctch9Y.mp4581yes https://puzzicle.org/science/hs-chemistry/v/introduction-to-electron-configurationshttps://puzzicle.org/thumbs/x946fe5f16f5f9997.jpgIntroduction to electron configurationsElectron configurations describe where electrons are located around the nucleus of an atom. For example, the electron configuration of lithium, 1s²2s¹, tells us that lithium has two electrons in the 1s subshell and one electron in the 2s subshell.https://cdn.kastatic.org/ka-youtube-converted/LyFzELsUgBU.mp4/LyFzELsUgBU.mp4308yes https://puzzicle.org/science/hs-chemistry/v/the-aufbau-principlehttps://puzzicle.org/thumbs/xed4472ca473f626c.jpgThe Aufbau principleThe Aufbau principle states that electrons fill lower-energy atomic orbitals before filling higher-energy ones (Aufbau is German for "building-up"). By following this rule, we can predict the electron configurations for atoms or ions. The Aufbau principle is most useful for the first 20 elements: from Sc on, the Aufbau principle does not accurately predict the order of electron filling in atoms.https://cdn.kastatic.org/ka-youtube-converted/Zersl0Ji3P0.mp4/Zersl0Ji3P0.mp4473yes https://puzzicle.org/science/ap-chemistry-beta/v/valence-electronshttps://puzzicle.org/thumbs/xa911bccf76ea72ea.jpgValence electronsValence electrons are the electrons in the outermost shell, or energy level, of an atom. For example, oxygen has six valence electrons, two in the 2s subshell and four in the 2p subshell. We can write the configuration of oxygen's valence electrons as 2s²2p⁴.https://cdn.kastatic.org/ka-youtube-converted/cZTdBsAjPPk.mp4/cZTdBsAjPPk.mp4311yes https://puzzicle.org/science/hs-chemistry/v/electron-configurations-of-ionshttps://puzzicle.org/thumbs/xfbba214cd1e413f4.jpgElectron configurations of ionsTo find the electron configuration for an ion, first identify the configuration for the neutral atom. Then, add or remove electrons depending on the ion's charge. For example, to find the configuration for the lithium ion (Li⁺), start with neutral lithium (1s²2s¹). Then, since the lithium ion has one less electron, remove an electron from the 2s subshell to get 1s².https://cdn.kastatic.org/ka-youtube-converted/oCajIGPK-WM.mp4/oCajIGPK-WM.mp4236yes https://puzzicle.org/science/ap-chemistry-beta/v/electron-configurations-in-the-3d-orbitalshttps://puzzicle.org/thumbs/x4d6604e9.jpgElectron configurations of the 3d transition metalsThe Aufbau principle predicts that the 4s orbital is always filled before the 3d orbitals, but this is actually not true for most elements! From Sc on, the 3d orbitals are actually lower in energy than the 4s orbital, which means that electrons enter the 3d orbitals first. In this video, we’ll discuss this in more depth and walk through all of the electron configurations for the 3d transition metals.https://cdn.kastatic.org/ka-youtube-converted/ddNsfv8iaps.mp4/ddNsfv8iaps.mp4753yes https://puzzicle.org/science/ap-chemistry-beta/v/periodic-trends-and-coulombs-lawhttps://puzzicle.org/thumbs/x3fd61bc1b1bf6d93.jpgPeriodic trends and Coulomb's lawPeriodic trends (such as electronegativity, electron affinity, atomic and ionic radii, and ionization energy) can be understood in terms of Coulomb's law, which is Fₑ = (q₁q₂)/r². For example, consider first ionization energy: Coulomb's law tells us that the greater the nuclear charge (q₁) and the shorter the distance between the nucleus and the outermost electron (r), the stronger the attraction between the nucleus and the electron. As a result, the electron will require more energy to remove.https://cdn.kastatic.org/ka-youtube-converted/gi2Y-kbrjCw.mp4/gi2Y-kbrjCw.mp4643yes https://puzzicle.org/science/ap-chemistry-beta/v/group-trend-for-ionization-energyhttps://puzzicle.org/thumbs/xe2b4547d.jpgIonization energy: group trendAn element's first ionization energy is the energy required to remove the outermost, or least bound, electron from a neutral atom of the element. On the periodic table, first ionization energy generally decreases as you move down a group. This is because the outermost electron is, on average, farther from the nucleus, meaning it is held less tightly and requires less energy to remove.https://cdn.kastatic.org/ka-youtube-converted/vQLcxlvK7Ik.mp4/vQLcxlvK7Ik.mp4619yes https://puzzicle.org/science/ap-chemistry-beta/v/period-trend-for-ionization-energyhttps://puzzicle.org/thumbs/xdb3555d5.jpgIonization energy: period trendAn element's first ionization energy is the energy required to remove the outermost, or least bound, electron from a neutral atom of the element. On the periodic table, first ionization energy generally increases as you move left to right across a period. This is due to increasing nuclear charge, which results in the outermost electron being more strongly bound to the nucleus.https://cdn.kastatic.org/ka-youtube-converted/649ZlWMp0LE.mp4/649ZlWMp0LE.mp4603yes https://puzzicle.org/science/ap-chemistry-beta/v/first-and-second-ionization-energyhttps://puzzicle.org/thumbs/x4562bba6.jpgFirst and second ionization energyAn element's second ionization energy is the energy required to remove the outermost, or least bound, electron from a 1+ ion of the element. Because positive charge binds electrons more strongly, the second ionization energy of an element is always higher than the first.https://cdn.kastatic.org/ka-youtube-converted/uVWquWFjnCw.mp4/uVWquWFjnCw.mp4455yes https://puzzicle.org/science/ap-chemistry-beta/v/worked-example-identifying-element-from-successive-ionization-energieshttps://puzzicle.org/thumbs/x8881d65fba31ef25.jpgWorked example: Identifying an element from successive ionization energiesWhen electrons are removed in succession from an element, the transition from removing valence electrons to removing core electrons results in a large jump in ionization energy. By looking for this large jump in energy, we can determine how many valence electrons an element has, which in turn can help us identify the element.https://cdn.kastatic.org/ka-youtube-converted/PfL-gzDdOBo.mp4/PfL-gzDdOBo.mp4151yes https://puzzicle.org/science/ap-chemistry-beta/v/electron-affinityhttps://puzzicle.org/thumbs/xb9779c70.jpgElectron affinity: period trendElectron affinity is the energy change that results from adding an electron to a gaseous atom. For example, when a fluorine atom in the gaseous state gains an electron to form F⁻(g), the associated energy change is -328 kJ/mol. Because this value is negative (energy is released), we say that the electron affinity of fluorine is favorable.https://cdn.kastatic.org/ka-youtube-converted/6WIBa32o63Q.mp4/6WIBa32o63Q.mp4786yes https://puzzicle.org/science/hs-chemistry/v/electronegativity-trendshttps://puzzicle.org/thumbs/x8dafd3b7.jpgElectronegativityElectronegativity is a measure of an atom's ability to attract shared electrons to itself. On the periodic table, electronegativity generally increases as you move from left to right across a period and decreases as you move down a group. As a result, the most electronegative elements are found on the top right of the periodic table, while the least electronegative elements are found on the bottom left.https://cdn.kastatic.org/ka-youtube-converted/Rr7LhdSKMxY.mp4/Rr7LhdSKMxY.mp4594yes https://puzzicle.org/science/ap-chemistry-beta/v/introduction-to-photoelectron-spectroscopyhttps://puzzicle.org/thumbs/x56731103acf29061.jpgIntroduction to photoelectron spectroscopyIn the analytical technique of photoelectron spectroscopy (PES), a sample is ionized using high-energy radiation, and the kinetic energies of the ejected electrons (called photoelectrons) are measured. From this, we can determine the binding or ionization energy of each electron in the atom or molecule. The results are presented as a PES spectrum, which shows the photoelectron count on the y-axis and binding energy on the x-axis.https://cdn.kastatic.org/ka-youtube-converted/UaTWrOTh4qU.mp4/UaTWrOTh4qU.mp4504yes https://puzzicle.org/science/hs-chemistry/v/predicting-bond-type-electronegativityhttps://puzzicle.org/thumbs/x8e340c87a1c11fc7.jpgPredicting bond type (electronegativity)One way to predict the type of bond that forms between two elements is to compare the electronegativities of the elements. In general, large differences in electronegativity result in ionic bonds, while smaller differences result in covalent bonds.https://cdn.kastatic.org/ka-youtube-converted/AHUHpVQXRsw.mp4/AHUHpVQXRsw.mp4283yes https://puzzicle.org/science/ap-chemistry-beta/v/bond-length-and-bond-energyhttps://puzzicle.org/thumbs/x4c2d3d6896637625.jpgBond length and bond energyA diatomic molecule can be represented using a potential energy curve, which graphs potential energy versus the distance between the two atoms (called the internuclear distance). From this graph, we can determine the equilibrium bond length (the internuclear distance at the potential energy minimum) and the bond energy (the energy required to separate the two atoms).https://cdn.kastatic.org/ka-youtube-converted/EMCWvehKpd4.mp4/EMCWvehKpd4.mp4402yes https://puzzicle.org/science/ap-chemistry-beta/v/worked-example-interpreting-potential-energy-curves-of-diatomic-moleculeshttps://puzzicle.org/thumbs/xdf2d44737d97e6c5.jpgWorked example: Interpreting potential energy curves of diatomic moleculesThe length and energy of a bond are influenced by both the bond order and the size of the atoms in the bond. In general, the higher the bond order and the smaller the atoms, the shorter and stronger the bond. As shown in this video, we can use these relationships to match diatomic molecules to their potential energy curves.https://cdn.kastatic.org/ka-youtube-converted/-mZAsdJnnMM.mp4/-mZAsdJnnMM.mp4321yes https://puzzicle.org/science/ap-chemistry-beta/v/lattice-energyhttps://puzzicle.org/thumbs/xec4d836b71d6b63a.jpgLattice energyThe energy required to separate the ions in a crystal lattice into individual gaseous ions is known as lattice energy. Lattice energy depends on the strength of interactions between cations and anions in the lattice, which we can estimate using Coulomb's law: Fₑ = (q₁q₂)/r². According to this equation, stronger interactions occur between ions with larger charges and smaller radii.https://cdn.kastatic.org/ka-youtube-converted/paXRIINiYlQ.mp4/paXRIINiYlQ.mp4490yes https://puzzicle.org/science/ap-chemistry-beta/v/drawing-lewis-diagramshttps://puzzicle.org/thumbs/x91256a3f5999fc0d.jpgDrawing Lewis diagramsA Lewis diagram shows how the valence electrons are distributed around the atoms in a molecule. Shared pairs of electrons are drawn as lines between atoms, while lone pairs of electrons are drawn as dots next to atoms. When constructing a Lewis diagram, keep in mind the octet rule, which refers to the tendency of atoms to gain, lose, or share electrons until they are surrounded by eight valence electrons (an octet).https://cdn.kastatic.org/ka-youtube-converted/9BZFphoY-vo.mp4/9BZFphoY-vo.mp4428yes https://puzzicle.org/science/ap-chemistry-beta/v/worked-example-lewis-diagram-of-formaldehydehttps://puzzicle.org/thumbs/x5b2a84d197de4686.jpgWorked example: Lewis diagram of formaldehyde (CH₂O)The Lewis diagram of a molecule can be constructed using the following stepwise procedure: (1) Find the number of valence electrons in the molecule. (2) Draw single bonds between bonded atoms. (3) Distribute the remaining electrons throughout the molecule, keeping in mind the duet and octet rules. In this video, we'll use these steps to construct the Lewis diagram of formaldehyde (CH₂O).https://cdn.kastatic.org/ka-youtube-converted/s0VM7WKnPeA.mp4/s0VM7WKnPeA.mp4368yes https://puzzicle.org/science/ap-chemistry-beta/v/worked-example-lewis-diagram-of-cyanide-ionhttps://puzzicle.org/thumbs/xc6e8f2162d590bfc.jpgWorked example: Lewis diagram of the cyanide ion (CN⁻)We can draw Lewis structures for polyatomic ions (ions containing multiple atoms) using the same stepwise procedure as for neutral molecules. In this video, we'll see how to construct the Lewis diagram of the cyanide ion (CN⁻).https://cdn.kastatic.org/ka-youtube-converted/Y8p4cthWB8I.mp4/Y8p4cthWB8I.mp4268yes https://puzzicle.org/science/ap-chemistry-beta/v/exceptions-to-the-octet-rulehttps://puzzicle.org/thumbs/x8266112775363add.jpgExceptions to the octet ruleWhile most atoms obey the duet and octet rules, there are some exceptions. For example, elements such as boron or beryllium often form compounds in which the central atom is surrounded by fewer than eight electrons (e.g., BF₃ or BeH₂). In contrast, many elements in the third-row and beyond have been observed to exceed the octet rule, forming compounds in which the central atom is surrounded by more than eight electrons (e.g. ICl₄⁻ or PF₅).https://cdn.kastatic.org/ka-youtube-converted/Tkb1OemB2KI.mp4/Tkb1OemB2KI.mp4430yes https://puzzicle.org/science/ap-chemistry-beta/v/worked-example-lewis-diagram-of-xenon-difluoridehttps://puzzicle.org/thumbs/x702bc808ee12c27c.jpgWorked example: Lewis diagram of xenon difluoride (XeF₂)In some molecules, the central atom exceeds the octet rule (is surrounded by more than eight electrons). See an example of a molecule that violates the octet rule (XeF₂) and learn how to draw its Lewis diagram in this video.https://cdn.kastatic.org/ka-youtube-converted/kfTm72uWt9o.mp4/kfTm72uWt9o.mp4191yes https://puzzicle.org/science/ap-chemistry-beta/v/introduction-to-resonancehttps://puzzicle.org/thumbs/xf0c549357879a0f5.jpgResonanceResonance arises when more than one valid Lewis structure can be drawn for a molecule or ion. The overall electronic structure of the molecule or ion is given by the weighted average of these resonance structures and is referred to as the resonance hybrid.https://cdn.kastatic.org/ka-youtube-converted/XRM9o1Sq5hg.mp4/XRM9o1Sq5hg.mp4444yes https://puzzicle.org/science/ap-chemistry-beta/v/formal-chargehttps://puzzicle.org/thumbs/x6c97c569abdf4021.jpgFormal chargeThe formal charge of an atom in a molecule is the charge that would reside on the atom if all of the bonding electrons were shared equally. We can calculate an atom's formal charge using the equation FC = VE - [LPE - ½(BE)], where VE = the number of valence electrons on the free atom, LPE = the number of lone pair electrons on the atom in the molecule, and BE = the number of bonding (shared) electrons around the atom in the molecule.https://cdn.kastatic.org/ka-youtube-converted/Kv5y3t9jsi4.mp4/Kv5y3t9jsi4.mp4423yes https://puzzicle.org/science/ap-chemistry-beta/v/worked-example-using-formal-charges-to-evaluate-nonequivalent-resonance-structurhttps://puzzicle.org/thumbs/xb31c81c773ae3ea6.jpgWorked example: Using formal charges to evaluate nonequivalent resonance structuresWhen a molecule has nonequivalent resonance structures, one structure may contribute more to the resonance hybrid than another. In terms of formal charge, a structure generally contributes more when (1) the formal charges on the atoms are minimized and (2) any negative formal charges are on more electronegative atoms and any positive charges are on more electropositive atoms. In this video, we use these guidelines to evaluate the nonequivalent resonance structures of SCN⁻.https://cdn.kastatic.org/ka-youtube-converted/N8qKGrChGC4.mp4/N8qKGrChGC4.mp4482yes https://puzzicle.org/science/ap-chemistry-beta/v/vsepr-for-2-electron-cloudshttps://puzzicle.org/thumbs/x072a19ee.jpgVSEPR for 2 electron cloudsThe valence shell electron-pair repulsion (VSEPR) model is used to predict the shapes of molecules and polyatomic ions. VSEPR is based on the idea that the “groups” or “clouds” of electrons surrounding an atom will adopt an arrangement that minimizes the repulsions between them. In this video, we look at examples of molecules in which there are two groups of electrons around the central atom.https://cdn.kastatic.org/ka-youtube-converted/97POZGcfoY8.mp4/97POZGcfoY8.mp4484yes https://puzzicle.org/science/ap-chemistry-beta/v/vsepr-for-3-electron-cloudshttps://puzzicle.org/thumbs/xdf5e267d.jpgVSEPR for 3 electron cloudsIn this video, we apply VSEPR theory to molecules and ions with three groups or “clouds” of electrons around the central atom. To minimize repulsions, three electron clouds will always adopt a trigonal planar electron geometry. If none of the clouds is a lone pair, the molecular geometry will also be trigonal planar. If one of the clouds is a lone pair, the molecular geometry will be bent.https://cdn.kastatic.org/ka-youtube-converted/3RDytvJYehY.mp4/3RDytvJYehY.mp4753yes https://puzzicle.org/science/ap-chemistry-beta/v/vsepr-for-4-electron-cloudshttps://puzzicle.org/thumbs/x13e2fe6f.jpgVSEPR for 4 electron cloudsIn this video, we apply VSEPR theory to molecules and ions with four groups or “clouds” of electrons around the central atom. To minimize repulsions, four electron clouds will always adopt a tetrahedral electron geometry. Depending on how many of the clouds are lone pairs, the molecular geometry will be tetrahedral (no lone pairs), trigonal pyramidal (one lone pair), or bent (two lone pairs).https://cdn.kastatic.org/ka-youtube-converted/ka8Yt4bTODs.mp4/ka8Yt4bTODs.mp4605yes https://puzzicle.org/science/ap-chemistry-beta/v/vsepr-for-5-electron-cloudshttps://puzzicle.org/thumbs/x6968389f.jpgVSEPR for 5 electron clouds (part 1)In this video, we apply VSEPR theory to molecules and ions with five groups or “clouds” of electrons around the central atom. To minimize repulsions, five electron clouds will always adopt a trigonal bipyramidal electron geometry. Depending on how many of the clouds are lone pairs, the molecular geometry will be trigonal bipyramidal (no lone pairs), seesaw (one lone pair), T-shaped (two lone pairs), or linear (three lone pairs). This video focuses on the first two molecular geometries, trigonal bipyramidal and seesaw.https://cdn.kastatic.org/ka-youtube-converted/0na0xtIHkXA.mp4/0na0xtIHkXA.mp4898yes https://puzzicle.org/science/ap-chemistry-beta/v/vsepr-for-5-electron-clouds-continuedhttps://puzzicle.org/thumbs/x14b19da7.jpgVSEPR for 5 electron clouds (part 2)In this video, we continue exploring VSEPR theory for molecules and ions with five regions or “clouds” of electrons around the central atom, focusing on examples where the molecular geometry is T-shaped (two of the clouds are lone pairs) or linear (three of the clouds are lone pairs).https://cdn.kastatic.org/ka-youtube-converted/ufzlgOyWBHE.mp4/ufzlgOyWBHE.mp4648yes https://puzzicle.org/science/ap-chemistry-beta/v/vsepr-for-6-electron-cloudshttps://puzzicle.org/thumbs/x0795dd14.jpgVSEPR for 6 electron cloudsIn this video, we apply VSEPR theory to molecules and ions with six groups or “clouds” of electrons around the central atom. To minimize repulsions, six electron clouds will always adopt a octahedral electron geometry. Depending on how many of the clouds are lone pairs, the molecular geometry will be octahedral (no lone pairs), square pyramidal (one lone pair), or square planer (two lone pairs).https://cdn.kastatic.org/ka-youtube-converted/WEoml1Fs26I.mp4/WEoml1Fs26I.mp4749yes https://puzzicle.org/science/ap-chemistry-beta/v/dipole-momenthttps://puzzicle.org/thumbs/x03aa879e.jpgMolecular polarityLike bonds, molecules can also be polar. In a polar molecule, electron density is unevenly distributed throughout the molecule, resulting in regions of partial negative charge and regions of partial positive charge. Molecular polarity depends on both individual bond polarities and molecular geometry, the latter of which we can predict using VSEPR theory.https://cdn.kastatic.org/ka-youtube-converted/q3g3jsmCOEQ.mp4/q3g3jsmCOEQ.mp4562yes https://puzzicle.org/science/ap-chemistry-beta/v/sp3-hybrid-orbital-jay-finalhttps://puzzicle.org/thumbs/x67a13ebc.jpgsp³ hybridizationIn sp³ hybridization, one s orbital and three p orbitals hybridize to form four sp³ orbitals, each consisting of 25% s character and 75% p character. This type of hybridization is required whenever an atom is surrounded by four groups of electrons.https://cdn.kastatic.org/ka-youtube-converted/JHgTNNX01r4.mp4/JHgTNNX01r4.mp4643yes https://puzzicle.org/science/ap-chemistry-beta/v/steric-number-sp3-hybridizationhttps://puzzicle.org/thumbs/x0213e20b.jpgSteric numberOne way to determine the hybridization of an atom is to calculate its steric number, which is equal to the number of sigma bonds surrounding the atom plus the number of lone pairs on the atoms. In this video, we focus on atoms with a steric number of 4, which corresponds to sp³ hybridization.https://cdn.kastatic.org/ka-youtube-converted/BM-My1AheLw.mp4/BM-My1AheLw.mp4587yes https://puzzicle.org/science/ap-chemistry-beta/v/sp2-hybridization-jay-finalhttps://puzzicle.org/thumbs/x3f29bc33.jpgsp² hybridizationIn sp² hybridization, one s orbital and two p orbitals hybridize to form three sp² orbitals, each consisting of 33% s character and 67% p character. This type of hybridization is required whenever an atom is surrounded by three groups of electrons.https://cdn.kastatic.org/ka-youtube-converted/ROzkyTgscGg.mp4/ROzkyTgscGg.mp4762yes https://puzzicle.org/science/ap-chemistry-beta/v/sp-hybridization-jay-finalhttps://puzzicle.org/thumbs/x3fb4948a.jpgsp hybridizationIn sp hybridization, one s orbital and one p orbital hybridize to form two sp orbitals, each consisting of 50% s character and 50% p character. This type of hybridization is required whenever an atom is surrounded by two groups of electrons.https://cdn.kastatic.org/ka-youtube-converted/BRHfy7envyQ.mp4/BRHfy7envyQ.mp4660yes https://puzzicle.org/science/ap-chemistry-beta/v/organic-hybridization-practicehttps://puzzicle.org/thumbs/xafb6fb15.jpgWorked examples: Finding the hybridization of atoms in organic moleculesWe can find the hybridization of an atom in a molecule by either looking at the types of bonds surrounding the atom or by calculating its steric number. In this video, we use both of these methods to determine the hybridizations of atoms in various organic molecules.https://cdn.kastatic.org/ka-youtube-converted/GuaozMpFS3w.mp4/GuaozMpFS3w.mp4621yes https://puzzicle.org/science/ap-chemistry-beta/v/london-dispersion-forceshttps://puzzicle.org/thumbs/x5c6a3803891c1d03.jpgLondon dispersion forcesLondon dispersion forces result from the coulombic interactions between instantaneous dipoles. Dispersion forces are present between all molecules (and atoms) and are typically greater for heavier, more polarizable molecules and molecules with larger surface areas.https://cdn.kastatic.org/ka-youtube-converted/5OT5l-NZS24.mp4/5OT5l-NZS24.mp4578yes https://puzzicle.org/science/hs-chemistry/v/dipole-dipole-forceshttps://puzzicle.org/thumbs/x68bf4a5bf43a24d0.jpgDipole–dipole forcesDipole-dipole forces occur between molecules with permanent dipoles (i.e., polar molecules). For molecules of similar size and mass, the strength of these forces increases with increasing polarity. Polar molecules can also induce dipoles in nonpolar molecules, resulting in dipole–induced dipole forces.https://cdn.kastatic.org/ka-youtube-converted/hg7jN32q9A0.mp4/hg7jN32q9A0.mp4406yes https://puzzicle.org/science/hs-chemistry/v/hydrogen-bondinghttps://puzzicle.org/thumbs/x2e43aa15619f9940.jpgHydrogen bondingHydrogen bonding is a special type of dipole-dipole interaction that occurs between the lone pair of a highly electronegative atom (typically N, O, or F) and the hydrogen atom in a N–H, O–H, or F–H bond. Hydrogen bonds can form between different molecules (intermolecular hydrogen bonding) or between different parts of the same molecule (intramolecular hydrogen bonding).https://cdn.kastatic.org/ka-youtube-converted/ltxqQbiI6-o.mp4/ltxqQbiI6-o.mp4399yes https://puzzicle.org/science/ap-chemistry-beta/v/ion-dipole-forceshttps://puzzicle.org/thumbs/x69b4fbdc3ec5950c.jpgIon–dipole forcesIon–dipole forces occur between ions and polar molecules. The relative strength of these forces can be understood in terms of Coulomb's law, which tells us that the electrostatic attraction between ion and dipole is directly related to the magnitudes of the ion charge and the dipole and inversely related to the distance between them.https://cdn.kastatic.org/ka-youtube-converted/7HCAGSkK1Do.mp4/7HCAGSkK1Do.mp4280yes https://puzzicle.org/science/hs-chemistry/v/intermolecular-forces-and-vapor-pressurehttps://puzzicle.org/thumbs/x400a8568b0d2774a.jpgIntermolecular forces and vapor pressureA liquid’s vapor pressure is directly related to the intermolecular forces present between its molecules. The stronger these forces, the lower the rate of evaporation and the lower the vapor pressure.https://cdn.kastatic.org/ka-youtube-converted/eubN8DwUh48.mp4/eubN8DwUh48.mp4364yes https://puzzicle.org/science/ap-chemistry-beta/v/ideal-gas-equation-pv-nrthttps://puzzicle.org/thumbs/408019.jpgThe ideal gas law (PV = nRT)The ideal gas law (PV = nRT) relates the macroscopic properties of ideal gases. An ideal gas is a gas in which the particles (a) do not attract or repel one another and (b) take up no space (have no volume). No gas is truly ideal, but the ideal gas law does provide a good approximation of real gas behavior under many conditions.https://cdn.kastatic.org/ka-youtube-converted/qObcdZj8YTM.mp4/qObcdZj8YTM.mp4379yes https://puzzicle.org/science/ap-chemistry-beta/v/worked-example-using-the-ideal-gas-law-to-calculate-number-of-moleshttps://puzzicle.org/thumbs/x2db232c8e659bd0c.jpgWorked example: Using the ideal gas law to calculate number of molesThe ideal gas law relates four macroscopic properties of ideal gases (pressure, volume, number of moles, and temperature). If we know the values of three of these properties, we can use the ideal gas law to solve for the fourth. In this video, we'll use the ideal gas law to solve for the number of moles (and ultimately molecules) in a sample of gas.https://cdn.kastatic.org/ka-youtube-converted/KnWEhXmJwdU.mp4/KnWEhXmJwdU.mp4437yes https://puzzicle.org/science/hs-chemistry/v/worked-example-using-the-ideal-gas-law-to-calculate-a-change-in-volumehttps://puzzicle.org/thumbs/x4b331a3152ed2cc0.jpgWorked example: Using the ideal gas law to calculate a change in volumeThe ideal gas law can be used to describe a change in state for an ideal gas. In this video, we'll apply the ideal gas law to the initial and final states of a gas to see how changes in temperature and pressure affect the volume of the gas.https://cdn.kastatic.org/ka-youtube-converted/bQ5vHcFlT78.mp4/bQ5vHcFlT78.mp4392yes https://puzzicle.org/science/hs-chemistry/v/introduction-to-partial-pressurehttps://puzzicle.org/thumbs/xed32e0e7.jpgGas mixtures and partial pressuresFor a mixture of ideal gases, the total pressure exerted by the mixture equals the sum of the pressures that each gas would exert on its own. This observation, known as Dalton's law of partial pressures, can be written as follows: P (total) = P ₁ + P ₂ + P ₃ + ... where P ₁, P ₂, and P ₃ are the partial pressures of the different gases in the mixture, and P (total) is the total pressure of the mixture.https://cdn.kastatic.org/ka-youtube-converted/A2xi-jEqpmg.mp4/A2xi-jEqpmg.mp4383yes https://puzzicle.org/science/ap-chemistry-beta/v/partial-pressurehttps://puzzicle.org/thumbs/428007.jpgWorked example: Calculating partial pressuresIn a mixture of ideal gases, each gas behaves independently of the other gases. As a result, we can use the ideal gas law to calculate the partial pressure of each gas in the mixture. Once we know the partial pressures of all of the gases, we can sum them using Dalton's law to find the total pressure of the mixture.https://cdn.kastatic.org/ka-youtube-converted/6jaMJHiza4s.mp4/6jaMJHiza4s.mp4483yes https://puzzicle.org/science/hs-chemistry/v/the-kinetic-molecular-theory-of-gaseshttps://puzzicle.org/thumbs/xe5baf4192ba83421.jpgThe kinetic molecular theory of gasesThe kinetic molecular theory (KMT) describes the behavior of ideal gases at the particle level. The five main postulates of the KMT are as follows: (1) the particles in a gas are in constant, random motion, (2) the combined volume of the particles is negligible, (3) the particles exert no forces on one another, (4) any collisions between the particles are completely elastic, and (5) the average kinetic energy of the particles is proportional to the temperature in kelvins.https://cdn.kastatic.org/ka-youtube-converted/HkSXiHz9vUc.mp4/HkSXiHz9vUc.mp4384yes https://puzzicle.org/science/hs-chemistry/v/kinetic-molecular-theory-and-the-gas-lawshttps://puzzicle.org/thumbs/xf8c4d2a22bf83fdd.jpgKinetic molecular theory and the gas lawsThe kinetic molecular theory (KMT) can be used to explain the macroscopic behavior of ideal gases. In this video, we'll see how the KMT accounts for the properties of gases as described by the various gas laws (Boyle's law, Gay-Lussac's law, Charles's law, Avogadro's law, and Dalton's law of partial pressures).https://cdn.kastatic.org/ka-youtube-converted/F6lhYxV6gEY.mp4/F6lhYxV6gEY.mp4491yes https://puzzicle.org/science/ap-chemistry-beta/v/real-gas-vs-ideal-gashttps://puzzicle.org/thumbs/xbd1a207f43dbb8bf.jpgIntroduction to real gasesReal gases do not always follow the assumptions of the kinetic molecular theory. While the particles of an ideal gas are assumed to occupy no volume and experience no interparticle attractions, the particles of a real gas do have finite volumes and do attract one another. As a result, real gases are often observed to deviate from ideal behavior.https://cdn.kastatic.org/ka-youtube-converted/yDZcVKy52LM.mp4/yDZcVKy52LM.mp4201yes https://puzzicle.org/science/ap-chemistry-beta/v/real-gases-deviations-from-ideal-behaviorhttps://puzzicle.org/thumbs/x5bc9e8e5be862592.jpgReal gases: Deviations from ideal behaviorIn this video, we examine the conditions under which real gases are most likely to deviate from ideal behavior: low temperatures and high pressures (small volumes). At low temperatures, attractions between gas particles cause the particles to collide less often with the container walls, resulting in a pressure lower than the ideal gas value. At high pressures (small volumes), finite particle volumes lower the actual volume available to the gas particles, resulting in a pressure higher than the ideal gas value.https://cdn.kastatic.org/ka-youtube-converted/UABFOI1sb7A.mp4/UABFOI1sb7A.mp4258yes https://puzzicle.org/science/ap-chemistry-beta/v/types-of-mixtureshttps://puzzicle.org/thumbs/x98e653184f76d83b.jpgTypes of mixturesA mixture is composed of one or more pure substances in varying composition. There are two types of mixtures: heterogeneous and homogeneous. Heterogeneous mixtures have visually distinguishable components, while homogeneous mixtures appear uniform throughout. The most common type of homogenous mixture is a solution, which can be a solid, liquid, or gas.https://cdn.kastatic.org/ka-youtube-converted/HgD6a4TVxLY.mp4/HgD6a4TVxLY.mp4375yes https://puzzicle.org/science/ap-chemistry-beta/v/dilutionhttps://puzzicle.org/thumbs/x47c67af7b90ba658.jpgDilutionA common method of making a solution of a given concentration involves taking a more concentration solution and adding water until the desired concentration is reached. This process is known as dilution. We can relate the concentrations and volumes before and after a dilution using the following equation: M₁V₁ = M₂V₂ where M₁ and V₁ represent the molarity and volume of the initial concentrated solution and M₂ and V₂ represent the molarity and volume of the final diluted solution.https://cdn.kastatic.org/ka-youtube-converted/T3H7YZ6-fto.mp4/T3H7YZ6-fto.mp4353yes https://puzzicle.org/science/ap-chemistry-beta/v/representing-solutions-using-particulate-modelshttps://puzzicle.org/thumbs/x056db2fcf75670c9.jpgRepresenting solutions using particulate modelsA solution is a homogeneous mixture composed of two or more pure substances. In this video, we'll learn how to represent the relative concentrations of the substances in a solution as well as the interactions between the substances using a particulate model.https://cdn.kastatic.org/ka-youtube-converted/NFcSRjLgDTk.mp4/NFcSRjLgDTk.mp4466yes https://puzzicle.org/science/ap-chemistry-beta/v/distillation-curveshttps://puzzicle.org/thumbs/xd66334de72193c5e.jpgDistillation curvesThe progress of a distillation can be represented using a distillation curve, which plots temperature versus volume of distillate collected. In this video, we'll learn how to interpret the various regions of a distillation curve for the fractional distillation of a 1:1 mixture of methyl and propyl acetate.https://cdn.kastatic.org/ka-youtube-converted/Aaz99lyLBf8.mp4/Aaz99lyLBf8.mp4384yes https://puzzicle.org/science/ap-chemistry-beta/v/thin-layer-chromatography-tlchttps://puzzicle.org/thumbs/xaf737a53fee16ba5.jpgThin-layer chromatography (TLC)Thin-layer chromatography (TLC) is a technique used to separate mixtures of compounds based on differences in polarity. In TLC, a glass plate coated with a stationary phase (typically silica gel) is spotted with the mixture to be separated. The plate is then placed in a mobile phase (solvent), which travels up the plate by capillary action. The rate at which each compound in the mixture moves up the plate depends on its relative attractions to the stationary and mobile phases.https://cdn.kastatic.org/ka-youtube-converted/GOWUw0-SEDE.mp4/GOWUw0-SEDE.mp4368yes https://puzzicle.org/science/ap-chemistry-beta/v/column-chromatography-introhttps://puzzicle.org/thumbs/x2bfff36aef54f50e.jpgColumn chromatographyColumn chromatography applies the same principles as TLC on a larger scale. In column chromatography, a glass column is filled with a stationary phase (typically silica gel), and the mixture of compounds to be separated is placed on top. Then, a mobile phase (solvent) is passed through the column from top to bottom. As with TLC, the rate at which each compound in the mixture moves down the column depends on its relative attractions to the stationary and mobile phases.https://cdn.kastatic.org/ka-youtube-converted/UHF1HXdkrZQ.mp4/UHF1HXdkrZQ.mp4240yes https://puzzicle.org/science/hs-chemistry/v/solubility-and-intermolecular-forceshttps://puzzicle.org/thumbs/410034.jpgSolubility and intermolecular forcesSubstances with similar polarities tend to be soluble in one another ("like dissolves like"). Nonpolar substances are generally more soluble in nonpolar solvents, while polar and ionic substances are generally more soluble in polar solvents.https://cdn.kastatic.org/ka-youtube-converted/ccDKr4TIWfk.mp4/ccDKr4TIWfk.mp4263yes https://puzzicle.org/science/ap-chemistry-beta/v/introduction-to-spectroscopyhttps://puzzicle.org/thumbs/x911639f72a122a76.jpgIntroduction to spectroscopySpectroscopy is the study of the interaction of light and matter. Many types of spectroscopy rely on the ability of atoms and molecules to absorb or emit electromagnetic (EM) radiation. The absorption or emission of different forms of EM radiation is related to different types of transitions. Microwave radiation is associated with molecular rotational transitions, infrared radiation is associated with molecular vibrational transitions, and UV/visible radiation is associated with electronic transitions.https://cdn.kastatic.org/ka-youtube-converted/LYseY0y2tgg.mp4/LYseY0y2tgg.mp4294yes https://puzzicle.org/science/ap-chemistry-beta/v/electronic-transitions-and-energyhttps://puzzicle.org/thumbs/x69960b6824c7681d.jpgElectronic transitions and energyElectronic transitions occur in atoms and molecules due to the absorption or emission of electromagnetic radiation (typically UV or visible). The energy change associated with a transition is related to the frequency of the electromagnetic wave by Planck's equation, E = h𝜈. In turn, the frequency of the wave is related to its wavelength and the speed of light by the equation c = 𝜆𝜈.https://cdn.kastatic.org/ka-youtube-converted/ukYSvaRyE3c.mp4/ukYSvaRyE3c.mp4472yes https://puzzicle.org/science/ap-chemistry-beta/v/worked-example-calculating-the-maximum-wavelength-capable-of-ionizationhttps://puzzicle.org/thumbs/x774545dc9cee7a62.jpgWorked example: Calculating the maximum wavelength capable of ionizationIf a photon has enough energy, it can completely remove an electron from an atom or molecule. In this video, we'll use the light equations (E = h𝜈 and c = 𝜆𝜈) to calculate the longest photon wavelength capable of removing an electron from a single atom of silver.https://cdn.kastatic.org/ka-youtube-converted/BR7Svm1s8fs.mp4/BR7Svm1s8fs.mp4380yes https://puzzicle.org/science/ap-chemistry-beta/v/spectrophotometry-introductionhttps://puzzicle.org/thumbs/6022723.jpgSpectrophotometry and the Beer–Lambert LawSpectrophotometry is a technique that uses light absorption to measure the concentration of an analyte in solution. The amount of light absorbed by a solution is related to the analyte concentration by the Beer–Lambert law, which is expressed as follows: A = εbc, where ε is the molar absorptivity of the analyte, b is the path length (the distance the light travels through the solution), and c is the concentration of the analyte.https://cdn.kastatic.org/ka-youtube-converted/fJRJLUYZe9c.mp4/fJRJLUYZe9c.mp4623yes https://puzzicle.org/science/ap-chemistry-beta/v/spectrophotometry-examplehttps://puzzicle.org/thumbs/6022724.jpgWorked example: Calculating concentration using the Beer–Lambert lawThe Beer–Lambert law relates the absorption of light by a solution to the properties of the solution according to the following equation: A = εbc, where ε is the molar absorptivity of the absorbing species, b is the path length, and c is the concentration of the absorbing species. In this video, we'll use the Beer–Lambert law to calculate the concentration of KMnO₄ in an unknown solution.https://cdn.kastatic.org/ka-youtube-converted/pbbHg_9zFbI.mp4/pbbHg_9zFbI.mp4228yes https://puzzicle.org/science/ap-chemistry-beta/v/visualizing-chemical-equations-using-particulate-modelshttps://puzzicle.org/thumbs/xfcbe0400bafdb1b1.jpgVisualizing chemical equations using particulate modelsA balanced chemical equation can be visualized using a particulate diagram, in which each of the atoms involved in the reaction is represented using a circle or a sphere. To be consistent with the law of conservation of mass, the diagram should depict the same numbers and types of atoms on each side of the reaction arrow.https://cdn.kastatic.org/ka-youtube-converted/3UoxWEZuwp8.mp4/3UoxWEZuwp8.mp4242yes https://puzzicle.org/science/ap-chemistry-beta/v/drawing-particulate-models-of-reaction-mixtureshttps://puzzicle.org/thumbs/x3a91038e40cef60e.jpgDrawing particulate models of reaction mixturesA given chemical reaction can be represented using a particulate diagram, in which the reaction mixture is depicted both before the reaction occurs and after the reaction has proceeded completely as possible. To be consistent with the law of conservation of mass, the "before reaction" and "after reaction" visualizations should each contain the same numbers and types of atoms.https://cdn.kastatic.org/ka-youtube-converted/kCkyk4YNUyY.mp4/kCkyk4YNUyY.mp4189yes https://puzzicle.org/science/ap-chemistry-beta/v/worked-example-calculating-the-amount-of-product-formed-from-a-limiting-reactanthttps://puzzicle.org/thumbs/x9e62d02e4448deea.jpgWorked example: Calculating the amount of product formed from a limiting reactantIn a chemical reaction, the reactant that is consumed first and limits how much product can be formed is called the limiting reactant (or limiting reagent). In this video, we'll determine the limiting reactant for a given reaction and use this information to calculate the theoretical yield of product.https://cdn.kastatic.org/ka-youtube-converted/cVqEfuJBVd8.mp4/cVqEfuJBVd8.mp4382yes https://puzzicle.org/science/ap-chemistry-beta/v/worked-example-relating-reaction-stoichiometry-and-the-ideal-gas-lawhttps://puzzicle.org/thumbs/x1ef7e362142729c4.jpgWorked example: Relating reaction stoichiometry and the ideal gas lawBy combining stoichiometric calculations with the ideal gas law, we can calculate amounts of reactants and products for chemical reactions involving gases.https://cdn.kastatic.org/ka-youtube-converted/FRNTJmqMko8.mp4/FRNTJmqMko8.mp4378yes https://puzzicle.org/science/ap-chemistry-beta/v/balance-and-redox-reactions1https://puzzicle.org/thumbs/x7e2a500f.jpgWorked example: Balancing a redox equation in acidic solutionWhen balancing equations for redox reactions occurring in acidic solution, it is often necessary to add H⁺ ions or the H⁺/H₂O pair to fully balance the equation. In this video, we'll walk through this process for the reaction between dichromate (Cr₂O₇²⁻) and chloride (Cl⁻) ions in acidic solution.https://cdn.kastatic.org/ka-youtube-converted/4tLrbMstfm8.mp4/4tLrbMstfm8.mp4724yes https://puzzicle.org/science/ap-chemistry-beta/v/balancing-redox-reactions1https://puzzicle.org/thumbs/x1cd3b06b.jpgWorked example: Balancing a redox equation in basic solutionWhen balancing equations for redox reactions occurring in basic solution, it is often necessary to add OH⁻ ions or the OH⁻/H₂O pair to fully balance the equation. In this video, we'll walk through this process for the reaction between ClO⁻ and Cr(OH)₄⁻ in basic solution.https://cdn.kastatic.org/ka-youtube-converted/Vjs8uK2ovQE.mp4/Vjs8uK2ovQE.mp4842yes https://puzzicle.org/science/ap-chemistry-beta/v/conjugate-acid-base-pairs-acids-and-bases-chemistry-khan-academyhttps://puzzicle.org/thumbs/x6dc05b65.jpgConjugate acid–base pairsIn the Brønsted–Lowry definition of acids and bases, a conjugate acid–base pair consists of two substances that differ only by the presence of a proton (H⁺). A conjugate acid is formed when a proton is added to a base, and a conjugate base is formed when a proton is removed from an acid.https://cdn.kastatic.org/ka-youtube-converted/4441EyWBPt8.mp4/4441EyWBPt8.mp4506yes https://puzzicle.org/science/ap-chemistry-beta/v/rate-law-and-reaction-orderhttps://puzzicle.org/thumbs/xba36c390.jpgRate law and reaction orderA rate law shows how the rate of a chemical reaction depends on reactant concentration. For a reaction such as aA → products, the rate law generally has the form rate = k[A]ⁿ, where k is a proportionality constant called the rate constant and n is the order of the reaction with respect to A. The value of n is not related to the reaction stoichiometry and must be determined by experiment.https://cdn.kastatic.org/ka-youtube-converted/6Ng8ayarWHw.mp4/6Ng8ayarWHw.mp4639yes https://puzzicle.org/science/ap-chemistry-beta/v/experimental-determination-of-rate-lawshttps://puzzicle.org/thumbs/xa3c0f3cf.jpgWorked example: Determining a rate law using initial rates dataThe rate law for a chemical reaction can be determined using the method of initial rates, which involves measuring the initial reaction rate at several different initial reactant concentrations. In this video, we'll use initial rates data to determine the rate law, overall order, and rate constant for the reaction between nitrogen dioxide and hydrogen gas.https://cdn.kastatic.org/ka-youtube-converted/Ad0aaYixFJg.mp4/Ad0aaYixFJg.mp4748yes https://puzzicle.org/science/ap-chemistry-beta/v/finding-units-of-rate-constant-khttps://puzzicle.org/thumbs/x6edd0754.jpgUnits of the rate constantThe units of the rate constant, k, depend on the overall reaction order. The units of k for a zero-order reaction are M/s, the units of k for a first-order reaction are 1/s, and the units of k for a second-order reaction are 1/(M·s).https://cdn.kastatic.org/ka-youtube-converted/fLL8xxjhDPI.mp4/fLL8xxjhDPI.mp4309yes https://puzzicle.org/science/ap-chemistry-beta/v/first-order-reactionshttps://puzzicle.org/thumbs/xe9b629531cb38ffa.jpgFirst-order reactionsThe integrated rate law for the first-order reaction A → products is ln[A]_t = -kt + ln[A]_0. Because this equation has the form y = mx + b, a plot of the natural log of [A] as a function of time yields a straight line. The rate constant for the reaction can be determined from the slope of the line, which is equal to -k.https://cdn.kastatic.org/ka-youtube-converted/PVqWOehVTOg.mp4/PVqWOehVTOg.mp4464yes https://puzzicle.org/science/ap-chemistry-beta/v/half-life-of-a-first-order-reactionhttps://puzzicle.org/thumbs/xfce7e24d.jpgHalf-life of a first-order reactionThe half-life of a reaction is the time required for a reactant to reach one-half its initial concentration or pressure. For a first-order reaction, the half-life is independent of concentration and constant over time.https://cdn.kastatic.org/ka-youtube-converted/e4qci9b2d3U.mp4/e4qci9b2d3U.mp4496yes https://puzzicle.org/science/ap-chemistry-beta/v/first-order-reaction-examplehttps://puzzicle.org/thumbs/xf66a9476.jpgWorked example: Using the first-order integrated rate law and half-life equationsIn this video, we'll use the first-order integrated rate law to calculate the concentration of a reactant after a given amount of time. We'll also calculate the amount of time it takes for the concentration to decrease to a certain value. Finally, we'll use the first-order half-life equation to calculate the half-life of the reaction.https://cdn.kastatic.org/ka-youtube-converted/fCdT2z0JcU8.mp4/fCdT2z0JcU8.mp4620yes https://puzzicle.org/science/ap-chemistry-beta/v/second-order-reactionshttps://puzzicle.org/thumbs/x43cc063e2c16cee8.jpgSecond-order reactionsThe integrated rate law for the second-order reaction A → products is 1/[A]_t = kt + 1/[A]_0. Because this equation has the form y = mx + b, a plot of the inverse of [A] as a function of time yields a straight line. The rate constant for the reaction can be determined from the slope of the line, which is equal to k.https://cdn.kastatic.org/ka-youtube-converted/YgWmHssKVvQ.mp4/YgWmHssKVvQ.mp4463yes https://puzzicle.org/science/ap-chemistry-beta/v/zero-order-reactionshttps://puzzicle.org/thumbs/x8426e58b1baf8027.jpgZero-order reactionsThe integrated rate law for the zero-order reaction A → products is [A]_t = -kt + [A]_0. Because this equation has the form y = mx + b, a plot of the concentration of A as a function of time yields a straight line. The rate constant for the reaction can be determined from the slope of the line, which is equal to -k.https://cdn.kastatic.org/ka-youtube-converted/aXf3e8a1teo.mp4/aXf3e8a1teo.mp4282yes https://puzzicle.org/science/ap-chemistry-beta/v/kinetics-of-radioactive-decayhttps://puzzicle.org/thumbs/x221fa86002de41cb.jpgKinetics of radioactive decayThe nuclei of radioactive elements decay according to first-order kinetics. As a result, the half-life equation and integrated rate law for radioactive decay processes can be derived from the rate laws for first-order reactions. The resulting equations can be used to find the rate constant k for a decay process and determine the amount of radioactive isotope remaining after a certain time period.https://cdn.kastatic.org/ka-youtube-converted/Xnu--929bo4.mp4/Xnu--929bo4.mp4473yes https://puzzicle.org/science/ap-chemistry-beta/v/elementary-rate-lawshttps://puzzicle.org/thumbs/xde3ff69d.jpgElementary reactionsAn elementary reaction is a reaction that occurs in a single step. The rate law for an elementary reaction can be derived from the coefficients of the reactants in the balanced equation. For example, the rate law for the elementary reaction 2A + B → products is rate = k[A]²[B].https://cdn.kastatic.org/ka-youtube-converted/vIZ0LnLo3jE.mp4/vIZ0LnLo3jE.mp4335yes https://puzzicle.org/science/ap-chemistry-beta/v/mechanisms-and-the-rate-determining-stephttps://puzzicle.org/thumbs/x3f370583.jpgReaction mechanism and rate lawA reaction mechanism is the sequence of elementary steps by which a chemical reaction occurs. Many reaction mechanisms contain one step that is much slower than the others; this step is known as the rate-determining step. If the rate-determining step is the first step in a mechanism, the rate law for the overall reaction can be derived directly from the stoichiometry of the step's balanced equation.https://cdn.kastatic.org/ka-youtube-converted/ShzW1LoQgoc.mp4/ShzW1LoQgoc.mp4522yes https://puzzicle.org/science/ap-chemistry-beta/v/the-pre-equilibrium-approximationhttps://puzzicle.org/thumbs/x5ce9cf605ad7a313.jpgThe pre-equilibrium approximationThe pre-equilibrium approximation is used to find the rate law for a reaction with a fast and reversible initial step. In this method, we first write the rate law based on the slow (rate-determining) step. Then, to eliminate any intermediates from the rate law, we use the fast initial step to solve for the intermediate concentration(s) in terms of reactant and/or product concentrations.https://cdn.kastatic.org/ka-youtube-converted/4czm7qIbUjA.mp4/4czm7qIbUjA.mp4443yes https://puzzicle.org/science/ap-chemistry-beta/v/arrhenius-equationhttps://puzzicle.org/thumbs/x81933fd6.jpgThe Arrhenius equationThe Arrhenius equation is k = Ae^(-Ea/RT), where A is the frequency or pre-exponential factor and e^(-Ea/RT) represents the fraction of collisions that have enough energy to overcome the activation barrier (i.e., have energy greater than or equal to the activation energy Ea) at temperature T. This equation can be used to understand how the rate of a chemical reaction depends on temperature.https://cdn.kastatic.org/ka-youtube-converted/zjhepBLtqug.mp4/zjhepBLtqug.mp4564yes https://puzzicle.org/science/ap-chemistry-beta/v/multistep-reaction-energy-profileshttps://puzzicle.org/thumbs/x9a07606446767ae4.jpgMultistep reaction energy profilesMany chemical reactions have mechanisms that consist of multiple elementary steps. The energy profile for a multistep reaction can be used to compare the activation energies of the different steps and identify the rate-determining step. The energy profile can also be used to determine the overall change in energy for the reaction.https://cdn.kastatic.org/ka-youtube-converted/l4VCiJulLKw.mp4/l4VCiJulLKw.mp4467yes https://puzzicle.org/science/ap-chemistry-beta/v/types-of-catalystshttps://puzzicle.org/thumbs/xa5de288495055b1e.jpgTypes of catalystsCatalysts can be categorized as homogeneous, heterogeneous, or enzymatic. Homogeneous catalysts exist in the same phase as the reactants, whereas heterogeneous catalysts exist in a different phase than the reactants. Enzymes are biological catalysts that speed up biochemical reactions in living organisms, often by many orders of magnitude.https://cdn.kastatic.org/ka-youtube-converted/5Tw9Yp-YuVU.mp4/5Tw9Yp-YuVU.mp4299yes https://puzzicle.org/science/ap-chemistry-beta/v/heat-transfer-and-thermal-equilibriumhttps://puzzicle.org/thumbs/x402b475eea799166.jpgHeat transfer and thermal equilibriumThe particles in a warmer object have a greater average kinetic energy than the particles in a cooler object. When two objects of different temperatures come into contact with one another, the particles at the surface of each object collide, resulting in the transfer of energy as heat. This process continues until thermal equilibrium is reached, at which point the average kinetic energies of the particles—and therefore the temperatures of the objects—is the same.https://cdn.kastatic.org/ka-youtube-converted/7jAPitc0bkY.mp4/7jAPitc0bkY.mp4189yes https://puzzicle.org/science/ap-chemistry-beta/v/heat-capacityhttps://puzzicle.org/thumbs/x1216e754323fa94e.jpgHeat capacityHeat capacity is the amount of heat required to change the temperature of a given amount of matter by 1°C. The heat capacity of 1 gram of a substance is called its specific heat capacity (or specific heat), while the heat capacity of 1 mole of a substance is called its molar heat capacity. The amount of heat gained or lost by a sample (q) can be calculated using the equation q = mcΔT, where m is the mass of the sample, c is the specific heat, and ΔT is the temperature change.https://cdn.kastatic.org/ka-youtube-converted/QlusUNM1ehA.mp4/QlusUNM1ehA.mp4521yes https://puzzicle.org/science/hs-chemistry/v/constant-pressure-calorimetryhttps://puzzicle.org/thumbs/xbce4b4dee42adb49.jpgConstant-pressure calorimetryConstant-pressure calorimetry is used to measure the change in enthalpy, ΔH, for a physical or chemical process. In this technique, a process is carried out in solution in a coffee cup calorimeter, an inexpensive device composed of two Styrofoam cups. The amount of heat transferred in the process (q) can be calculated from the mass, specific heat, and temperature change of the solution. Because the calorimeter is at constant (atmospheric) pressure, q is equal to ΔH for the process.https://cdn.kastatic.org/ka-youtube-converted/SkMeY6RBJZY.mp4/SkMeY6RBJZY.mp4560yes https://puzzicle.org/science/ap-chemistry-beta/v/worked-example-measuring-the-energy-content-of-foods-using-soda-can-calorimetryhttps://puzzicle.org/thumbs/x6ae8224cece922c3.jpgWorked example: Measuring the energy content of foods using soda-can calorimetryCalorimetry is often used to evaluate the energy content of different foods. In this video, we'll see how data from a soda-can calorimetry experiment can be used to calculate the energy content of a marshmallow in Calories per gram.https://cdn.kastatic.org/ka-youtube-converted/1Zx71Il5jaM.mp4/1Zx71Il5jaM.mp4466yes https://puzzicle.org/science/ap-chemistry-beta/v/constant-volume-calorimetryhttps://puzzicle.org/thumbs/x9dfabe8a466adac7.jpgConstant-volume calorimetryConstant-volume calorimetry is used to measure the change in internal energy, ΔE, for a combustion reaction. In this technique, a sample is burned under constant volume in a device called a bomb calorimeter. The amount of heat released in the reaction can be calculated using the equation q = -CΔT, where C is the heat capacity of the calorimeter and ΔT is the temperature change. Because the combustion occurs at constant volume, q is equal to ΔE for the reaction.https://cdn.kastatic.org/ka-youtube-converted/1NueJQpqkuc.mp4/1NueJQpqkuc.mp4478yes https://puzzicle.org/science/ap-chemistry-beta/v/heating-curve-for-waterhttps://puzzicle.org/thumbs/xf98674bfa32150a0.jpgHeating curve for waterThe heating curve for water shows how the temperature of a given quantity of water changes as heat is added at a constant rate. During a phase change, the temperature of the water remains constant, resulting in a plateau on the graph. We can use the heating curve to calculate the amount of heat required to raise the temperature of the water sample by a certain amount, such as from -25°C (when the water is present as a solid) to 125°C (when the water is present as a gas).https://cdn.kastatic.org/ka-youtube-converted/MqAVc_XaIXQ.mp4/MqAVc_XaIXQ.mp4631yes https://puzzicle.org/science/ap-chemistry-beta/v/enthalpy-of-reactionhttps://puzzicle.org/thumbs/x4a20fdc9722c1a28.jpgEnthalpy of reactionThe enthalpy change that accompanies a chemical reaction is referred to as the enthalpy of reaction and is abbreviated ΔH_rxn. The value of ΔH_rxn depends on how the balanced equation for the reaction is written and is typically given in units of kJ/mol-rxn.https://cdn.kastatic.org/ka-youtube-converted/cEzN33gfgVs.mp4/cEzN33gfgVs.mp4257yes https://puzzicle.org/science/ap-chemistry-beta/v/worked-example-measuring-enthalpy-of-reaction-using-coffee-cup-calorimetryhttps://puzzicle.org/thumbs/x1474d84a9ab0d884.jpgWorked example: Measuring enthalpy of reaction using coffee-cup calorimetryConstant-pressure calorimetry is often used to determine the changes in enthalpy for aqueous reactions. In this video, we'll see how data from a coffee-cup calorimetry experiment can be used to calculate ΔH_rxn for the reaction between AgNO₃(aq) and NaCl(aq).https://cdn.kastatic.org/ka-youtube-converted/pDCcRU5OGmA.mp4/pDCcRU5OGmA.mp4456yes https://puzzicle.org/science/ap-chemistry-beta/v/hess-s-law-introhttps://puzzicle.org/thumbs/x7e9abacd182248a6.jpgHess's lawHess's law states that if a process can be expressed as the sum of two or more steps, the enthalpy change for the overall process is the sum of the ΔH values for each step. To use Hess's law, two principles must be understood: one, if an equation is reversed, the sign of the ΔH value is also reversed. Two, if an equation is multiplied by a coefficient, the ΔH value is multiplied by the same coefficient.https://cdn.kastatic.org/ka-youtube-converted/QU3zVec5I_M.mp4/QU3zVec5I_M.mp4344yes https://puzzicle.org/science/ap-chemistry-beta/v/hess-s-law-examplehttps://puzzicle.org/thumbs/6022728.jpgWorked example: Using Hess's law to calculate enthalpy of reactionHess's law can be used to calculate enthalpy changes that are difficult to measure directly. In this video, we'll use Hess's law to calculate the enthalpy change for the formation of methane, CH₄, from solid carbon and hydrogen gas, a reaction that occurs too slowly to be measured in the laboratory.https://cdn.kastatic.org/ka-youtube-converted/8bCL8TQZFKo.mp4/8bCL8TQZFKo.mp4728yes https://puzzicle.org/science/ap-chemistry-beta/v/enthalpy-of-formation-introhttps://puzzicle.org/thumbs/x9a645835a14c858d.jpgEnthalpy of formationThe standard enthalpy of formation of a substance is the enthalpy change that occurs when 1 mole of the substance is formed from its constituent elements in their standard states. A pure element in its standard state has a standard enthalpy of formation of zero. For any chemical reaction, the standard enthalpy change is the sum of the standard enthalpies of formation of the products minus the sum of the standard enthalpies of formation of the reactants.https://cdn.kastatic.org/ka-youtube-converted/TNwGNHqwHxc.mp4/TNwGNHqwHxc.mp4565yes https://puzzicle.org/science/ap-chemistry-beta/v/writing-equilibrium-constant-and-reaction-quotient-expressionshttps://puzzicle.org/thumbs/x5ba6c83b65f6f515.jpgWriting equilibrium constant and reaction quotient expressionsThe equilibrium constant, K, describes the relative amounts of reaction species at equilibrium. The expression for K is equal to the concentrations (or partial pressures) of the products raised to their stoichiometric coefficients divided by the concentrations (or partial pressures) of the reactants raised to their stoichiometric coefficients. The reaction quotient, Q, has the same form as K but describes the relative amounts of reaction species at any point in time.https://cdn.kastatic.org/ka-youtube-converted/lG8tCeNzEjY.mp4/lG8tCeNzEjY.mp4423yes https://puzzicle.org/science/ap-chemistry-beta/v/worked-examples-calculating-equilibrium-constants-from-equilibrium-concentrationhttps://puzzicle.org/thumbs/x8e80cf5c2d1b7d81.jpgWorked examples: Calculating equilibrium constantsIn this video, we'll calculate equilibrium constants using measurements of concentration and partial pressures at equilibrium. First, we'll find Kc for an equilibrium system using equilibrium concentrations. Then, we'll find Kp for a different system using equilibrium partial pressures.https://cdn.kastatic.org/ka-youtube-converted/5HZbCNg9mIw.mp4/5HZbCNg9mIw.mp4492yes https://puzzicle.org/science/ap-chemistry-beta/v/magnitude-of-the-equilibrium-constanthttps://puzzicle.org/thumbs/x5b0c897613efccfb.jpgMagnitude of the equilibrium constantThe magnitude of the equilibrium constant provides information about the relative amounts of reactants and products at equilibrium. A large K value (greater than 1) indicates that there are more products than reactants at equilibrium, while a small K value (less than 1) indicates that there are more reactants than products at equilibrium.https://cdn.kastatic.org/ka-youtube-converted/CoRawW0_Kns.mp4/CoRawW0_Kns.mp4364yes https://puzzicle.org/science/ap-chemistry-beta/v/properties-of-the-equilibrium-constanthttps://puzzicle.org/thumbs/xae35ca847266af38.jpgProperties of the equilibrium constantThe equilibrium constant for a reaction depends on how the balanced equation is written. If the equation is reversed, K is inverted. If the equation is multiplied by a factor n, K is raised to the nth power. If multiple equations are added together, K for the overall equation is the product of the K values for the equations that were summed.https://cdn.kastatic.org/ka-youtube-converted/U39OqCEMjto.mp4/U39OqCEMjto.mp4396yes https://puzzicle.org/science/ap-chemistry-beta/v/worked-example-calculating-an-equilibrium-constant-from-initial-and-equilibrium-https://puzzicle.org/thumbs/xda43fd0e7a843470.jpgWorked example: Calculating an equilibrium constant from initial and equilibrium pressuresGiven an initial partial pressure and the total pressure at equilibrium, we can use Dalton's law to determine the equilibrium partial pressures of the gases in a reaction mixture. Once we know the equilibrium partial pressures, we can calculate the equilibrium constant for the reaction.https://cdn.kastatic.org/ka-youtube-converted/MKt3X0SGrtw.mp4/MKt3X0SGrtw.mp4291yes https://puzzicle.org/science/ap-chemistry-beta/v/worked-example-calculating-equilibrium-concentrations-from-initial-concentrationhttps://puzzicle.org/thumbs/x89e5acafaf8b5cef.jpgWorked example: Calculating equilibrium concentrations from initial concentrationsIn this video, we'll learn how to use initial concentrations along with the equilibrium constant to calculate the concentrations of reaction species at equilibrium.https://cdn.kastatic.org/ka-youtube-converted/l81swfmUaek.mp4/l81swfmUaek.mp4329yes https://puzzicle.org/science/ap-chemistry-beta/v/using-the-reaction-quotient-aphttps://puzzicle.org/thumbs/xfe3339a8f31de97a.jpgUsing the reaction quotientBy comparing the reaction quotient to the equilibrium constant, we can predict the direction a reaction will proceed to reach equilibrium. If Q < K, the reaction will proceed towards the products. If Q > K, the reaction will proceed towards the reactants. If Q = K, the reaction is already at equilibrium and will not change.https://cdn.kastatic.org/ka-youtube-converted/JDUNQMj9Yk0.mp4/JDUNQMj9Yk0.mp4470yes https://puzzicle.org/science/ap-chemistry-beta/v/worked-example-using-the-reaction-quotient-to-find-equilibrium-partial-pressureshttps://puzzicle.org/thumbs/x6f1ad9382238d63e.jpgWorked example: Using the reaction quotient to find equilibrium partial pressuresIn some equilibrium problems, we first need to use the reaction quotient to predict the direction a reaction will proceed to reach equilibrium. Once we know this, we can build an ICE table, which we can then use to calculate the concentrations or partial pressures of the reaction species at equilibrium.https://cdn.kastatic.org/ka-youtube-converted/U7U3e0r_8y8.mp4/U7U3e0r_8y8.mp4391yes https://puzzicle.org/science/ap-chemistry-beta/v/comparing-q-vs-k-examplehttps://puzzicle.org/thumbs/xb4136c2e.jpgWorked example: Using the reaction quotient to predict a pressure changeWe can predict how the total pressure of a reaction mixture will change as the system moves toward equilibrium by first calculating the reaction quotient, Qₚ, and then comparing Qₚ to Kₚ.https://cdn.kastatic.org/ka-youtube-converted/pnW1PNJlmWg.mp4/pnW1PNJlmWg.mp4353yes https://puzzicle.org/science/ap-chemistry-beta/v/le-chateliers-principle-changing-volumehttps://puzzicle.org/thumbs/x0f32a0d2f88ed955.jpgLe Chȃtelier’s principle: Changing volumeLe Chȃtelier’s principle can be used to predict the effect that a stress like changing volume has on a system at equilibrium. If the volume of the container is increased (at constant T), the system will shift in the direction that increases the number of moles of gas in the container. If the volume is decreased, the reverse effect will occur. If the system has the same number of moles of gas on both sides of the chemical equation, a change in volume will have no impact on the equilibrium position.https://cdn.kastatic.org/ka-youtube-converted/FmqyKCQo7Tk.mp4/FmqyKCQo7Tk.mp4669yes https://puzzicle.org/science/ap-chemistry-beta/v/solubility-product-constant-from-the-solubilityhttps://puzzicle.org/thumbs/x17fb6c6c.jpgIntroduction to solubility equilibriaThe solubility product constant, Kₛₚ, is an equilibrium constant that reflects the extent to which an ionic compound dissolves in water. For compounds that dissolve to produce the same number of ions, we can directly compare their Kₛₚ values to determine their relative solubilities. If we know the solubility of a salt, we can use this information to calculate the Kₛₚ value for the compound.https://cdn.kastatic.org/ka-youtube-converted/N9a2r01ToZk.mp4/N9a2r01ToZk.mp4497yes https://puzzicle.org/science/ap-chemistry-beta/v/solubility-from-the-solubility-product-constanthttps://puzzicle.org/thumbs/x9dbdb3a5.jpgWorked example: Calculating solubility from KₛₚA compound's molar solubility in water can be calculated from its Kₛₚ value at 25°C. To do so, first prepare an ICE (Initial, Change, and Equilibrium) table showing the equilibrium concentrations of the ions in terms of x, the molar solubility of the compound. Then, plug these expressions into the solubility-product expression for the compound and solve for the value of x.https://cdn.kastatic.org/ka-youtube-converted/Ywvw32DaxUk.mp4/Ywvw32DaxUk.mp4292yes https://puzzicle.org/science/ap-chemistry-beta/v/worked-example-predicting-whether-a-precipitate-forms-by-comparing-q-and-khttps://puzzicle.org/thumbs/xca4bf79a23423862.jpgWorked example: Predicting whether a precipitate forms by comparing Q and KₛₚWe can use the reaction quotient to predict whether a precipitate will form when two solutions containing dissolved ionic compounds are mixed. If Q < Kₛₚ, the newly mixed solution is undersaturated and no precipitate will form. If Q > Kₛₚ, the solution is oversaturated and a precipitate will form until Q = Kₛₚ.https://cdn.kastatic.org/ka-youtube-converted/vyENVkttVtw.mp4/vyENVkttVtw.mp4496yes https://puzzicle.org/science/ap-chemistry-beta/v/solubility-and-the-common-ion-effecthttps://puzzicle.org/thumbs/x33617989.jpgThe common-ion effectThe solubility of an ionic compound is decreased by the presence of a common ion (an ion that is also present in the compound); this is known as the common-ion effect.https://cdn.kastatic.org/ka-youtube-converted/fI3U9T7LigY.mp4/fI3U9T7LigY.mp4608yes https://puzzicle.org/science/ap-chemistry-beta/v/solubility-and-the-ph-of-the-solutionhttps://puzzicle.org/thumbs/x7212c366.jpgpH and solubilityFor ionic compounds containing basic anions, solubility increases as the pH of the solution is decreased. For ionic compounds containing anions of negligible basicity (such as the conjugate bases of strong acids), solubility is unaffected by changes in pH.https://cdn.kastatic.org/ka-youtube-converted/Z2bDnvDGQHQ.mp4/Z2bDnvDGQHQ.mp4314yes https://puzzicle.org/science/ap-chemistry-beta/v/autoionization-of-water-aphttps://puzzicle.org/thumbs/xf0385fb1aabd26dd.jpgAutoionization of waterIn the autoionization of water, a proton is transferred from one water molecule to another to produce a hydronium ion (H₃O⁺) and a hydroxide ion (OH⁻). The equilibrium expression for this reaction is Kw = [H₃O⁺][OH⁻], where Kw is the autoionization constant for water. At 25°C, the value of Kw is 1.0 x 10⁻¹⁴. In pure water, the concentrations of H₃O⁺ and OH⁻ are equal, and the water is considered to be neutral.https://cdn.kastatic.org/ka-youtube-converted/vt6WjF1O_08.mp4/vt6WjF1O_08.mp4541yes https://puzzicle.org/science/ap-chemistry-beta/v/the-ph-scalehttps://puzzicle.org/thumbs/xd2d23bef2d510733.jpgThe pH scaleThe pH scale is a convenient way to represent the acidity or basicity of a solution. We can calculate the pH of a solution by taking the negative logarithm of the hydronium ion concentration, or pH = -log[H₃O⁺]. At 25°C, a solution with pH < 7 is acidic, a solution with pH > 7 is basic, and a solution with pH = 7 is neutral.https://cdn.kastatic.org/ka-youtube-converted/75j1b1l6PWU.mp4/75j1b1l6PWU.mp4529yes https://puzzicle.org/science/ap-chemistry-beta/v/worked-examples-calculating-hydronium-ion-concentration-and-phhttps://puzzicle.org/thumbs/x305408b3c866c70f.jpgWorked examples: Calculating [H₃O⁺] and pHIn this video, we'll solve for [H₃O⁺] and pH in two different worked examples. First, we'll walk through the possible approaches for calculating [H₃O⁺] from pOH. Then, we'll find the pH of pure water at 50°C from the value of the autoionization constant at 50°C.https://cdn.kastatic.org/ka-youtube-converted/qK_YVKAat1w.mp4/qK_YVKAat1w.mp4425yes https://puzzicle.org/science/ap-chemistry-beta/v/strong-acid-solutionshttps://puzzicle.org/thumbs/x83c7f6c949baf649.jpgStrong acid solutionsStrong acids (such as HCl, HBr, HI, HNO₃, HClO₄, and H₂SO₄) ionize completely in water to produce hydronium ions. The concentration of H₃O⁺ in a strong acid solution is therefore equal to the initial concentration of the acid. For example, a solution of 0.1 M HNO₃ contains 0.1 M H₃O⁺ and has a pH of 1.0.https://cdn.kastatic.org/ka-youtube-converted/pZ43GbTechI.mp4/pZ43GbTechI.mp4378yes https://puzzicle.org/science/ap-chemistry-beta/v/worked-example-finding-the-percent-ionization-of-a-weak-acidhttps://puzzicle.org/thumbs/x47501ff0ec6cd797.jpgWorked example: Finding the percent ionization of a weak acidThe percent ionization of a weak acid, HA, is defined as the ratio of the equilibrium H₃O⁺ concentration to the initial HA concentration, multiplied by 100%. In this video, we'll use this relationship to find the percent ionization of acetic acid in a 0.20 M solution.https://cdn.kastatic.org/ka-youtube-converted/_jEe0Tkyn3w.mp4/_jEe0Tkyn3w.mp4410yes https://puzzicle.org/science/ap-chemistry-beta/v/weak-base-equilibriahttps://puzzicle.org/thumbs/xca521b040bbb24b1.jpgWeak base equilibriaWeak bases partially ionize in water to produce hydroxide ions. Because the ionization is not complete, the concentration of OH⁻ in a weak base solution is typically much less than the initial base concentration. The relative strength of a weak base is described by the base ionization constant, Kb, which is the equilibrium constant for the reaction of the weak base with water.https://cdn.kastatic.org/ka-youtube-converted/cvcgH3ohcNg.mp4/cvcgH3ohcNg.mp4570yes https://puzzicle.org/science/ap-chemistry-beta/v/acid-base-properties-of-saltshttps://puzzicle.org/thumbs/xcb0e59e6.jpgAcid–base properties of saltsWe can determine whether a salt solution will be acidic, basic, or neutral by considering the reactivity of both the cation and the anion with water. If neither species reacts with water, the solution will be neutral. If only the cation reacts with water, the solution will be acidic. If only the anion reacts with water, the solution will be basic. If both species react with water, the pH of the solution will depend on the relative strengths of the cation and the anion.https://cdn.kastatic.org/ka-youtube-converted/EDvtwQLJzEg.mp4/EDvtwQLJzEg.mp4813yes https://puzzicle.org/science/ap-chemistry-beta/v/factors-affecting-acid-strengthhttps://puzzicle.org/thumbs/x1f18a037178ed995.jpgFactors affecting acid strengthThe relative strength of an acid can be predicted based on its chemical structure. In general, an acid is stronger when the H–A bond is more polar. Acidity is also greater when the H–A bond is weaker and when the conjugate base, A⁻, is more stable.https://cdn.kastatic.org/ka-youtube-converted/v_94NsuBwDM.mp4/v_94NsuBwDM.mp4519yes https://puzzicle.org/science/ap-chemistry-beta/v/weak-acid-strong-base-reactionshttps://puzzicle.org/thumbs/xce5c79320ba60574.jpgWeak acid–strong base reactionsWhen a weak acid and a strong base are mixed, they react according to the following net-ionic equation: HA(aq) + OH⁻(aq) → A⁻(aq) + H₂O(l). If the acid and base are equimolar, the pH of the resulting solution can be determined by considering the equilibrium reaction of A⁻ with water. If the acid is in excess, the pH can be determined from the concentrations of HA and A⁻ after the reaction. If the base is in excess, the pH can be determined from the concentration of excess OH⁻.https://cdn.kastatic.org/ka-youtube-converted/VdAl4QK4-0Q.mp4/VdAl4QK4-0Q.mp4482yes https://puzzicle.org/science/ap-chemistry-beta/v/worked-example-calculating-the-ph-after-a-weak-acid-strong-base-reaction-excess-https://puzzicle.org/thumbs/xa623e9dd9076959e.jpgWorked example: Calculating the pH after a weak acid–strong base reaction (excess acid)When a strong base reacts with excess weak acid, the resulting solution contains both HA and A⁻. We can use reaction stoichiometry to determine the concentration of each species and then solve a common-ion equilibrium problem to find [H₃O⁺] and thus the pH.https://cdn.kastatic.org/ka-youtube-converted/9rRlLPx6w7M.mp4/9rRlLPx6w7M.mp4544yes https://puzzicle.org/science/ap-chemistry-beta/v/weak-base-strong-acid-reactionshttps://puzzicle.org/thumbs/x82de9a0fed7a5099.jpgWeak base–strong acid reactionsWhen a weak base and a strong acid are mixed, they react according to the following net-ionic equation: B(aq) + H₃O⁺(aq) → HB⁺(aq) + H₂O(l). If the acid and base are equimolar, the pH of the resulting solution can be determined by considering the equilibrium reaction of HB⁺ with water. If the base is in excess, the pH can be determined from the concentrations of B and HB⁺ after the reaction. If the acid is in excess, the pH can be determined from the concentration of excess H₃O⁺.https://cdn.kastatic.org/ka-youtube-converted/78Sg3RdEPVk.mp4/78Sg3RdEPVk.mp4388yes https://puzzicle.org/science/ap-chemistry-beta/v/weak-acid-weak-base-reactionshttps://puzzicle.org/thumbs/xb06a9cd3343723e0.jpgWeak acid–weak base reactionsWhen a weak acid and a weak base are mixed, they come to an equilibrium state represented by the following equation: HA(aq) + B(aq) ⇌ A⁻(aq) + HB⁺(aq). The equilibrium position lies on the side of the weaker acid and the weaker base, and we can determine whether the resulting solution is acidic, basic, or neutral by comparing the Ka and Kb values of these species.https://cdn.kastatic.org/ka-youtube-converted/G5FVxAzlmfM.mp4/G5FVxAzlmfM.mp4431yes https://puzzicle.org/science/ap-chemistry-beta/v/properties-of-buffershttps://puzzicle.org/thumbs/xfcfdc1868e2d9dcf.jpgProperties of buffersBuffer solutions contain high concentrations of both a weak acid and its conjugate base (or a weak base and its conjugate acid). Because these components can neutralize added H⁺ or OH⁻, buffers are highly resistant to changes in pH.https://cdn.kastatic.org/ka-youtube-converted/tZH6zvQ8Wl8.mp4/tZH6zvQ8Wl8.mp4419yes https://puzzicle.org/science/ap-chemistry-beta/v/hendersonhasselbalch-equationhttps://puzzicle.org/thumbs/x6949bb9400952fe6.jpgHenderson–Hasselbalch equationOne way to determine the pH of a buffer is by using the Henderson–Hasselbalch equation, which is pH = pKₐ + log([A⁻]/[HA]). In this equation, [HA] and [A⁻] refer to the equilibrium concentrations of the conjugate acid–base pair used to create the buffer solution. When [HA] = [A⁻], the solution pH is equal to the pKₐ of the acid.https://cdn.kastatic.org/ka-youtube-converted/7QgtdYiWH50.mp4/7QgtdYiWH50.mp4454yes https://puzzicle.org/science/ap-chemistry-beta/v/methods-for-preparing-buffershttps://puzzicle.org/thumbs/x3241eaf69b19a5ec.jpgMethods for preparing buffersIn this video, we'll explore two common methods for preparing buffer solutions. In the first approach, a certain amount of a weak acid (or weak base) is neutralized with a strong base (or strong acid), forming a conjugate acid–base pair in solution. In the second approach, a weak acid (or weak base) is combined with a salt containing its conjugate base (or conjugate acid).https://cdn.kastatic.org/ka-youtube-converted/OCPQaKhVC_0.mp4/OCPQaKhVC_0.mp4626yes https://puzzicle.org/science/ap-chemistry-beta/v/buffer-rangehttps://puzzicle.org/thumbs/x89606686d924813f.jpgBuffer rangeA buffer has an effective pH range of one pH unit on either side of the pKₐ value for the weak acid. If the pH of a buffer goes out of this range, the buffer will no longer be effective at resisting large changes in pH.https://cdn.kastatic.org/ka-youtube-converted/yia1GeEEVaM.mp4/yia1GeEEVaM.mp4317yes https://puzzicle.org/science/ap-chemistry-beta/v/buffer-capacity-aphttps://puzzicle.org/thumbs/x51def401cab09cf8.jpgBuffer capacityThe capacity of a buffer to neutralize added acid or base depends on the concentrations of HA and A⁻ in solution. For a given ratio of [HA] to [A⁻], the greater the concentrations, the higher the overall buffer capacity. When [HA] is greater than [A⁻], the capacity is higher for added base than acid. When [A⁻] is greater than [HA], the capacity is higher for added acid than base.https://cdn.kastatic.org/ka-youtube-converted/XcSZ5jeG5mA.mp4/XcSZ5jeG5mA.mp4618yes https://puzzicle.org/science/ap-chemistry-beta/v/strong-acid-strong-base-titrationshttps://puzzicle.org/thumbs/x733c3b3326d79057.jpgStrong acid–strong base titrationsThe results of an acid–base titration can be summarized using a titration curve, which plots pH vs. volume of titrant added. For the titration of a strong acid with a strong base, the curve begins acidic and then turns basic after the equivalence point, which occurs at pH = 7. For the titration of a strong base with a strong acid, the curve is similar except that it begins basic and turns acidic.https://cdn.kastatic.org/ka-youtube-converted/qkh0SMozSBQ.mp4/qkh0SMozSBQ.mp4540yes https://puzzicle.org/science/ap-chemistry-beta/v/weak-acid-strong-base-titrationshttps://puzzicle.org/thumbs/xc6bd0d5bfe42207b.jpgWeak acid–strong base titrationsFor the titration of a weak acid with a strong base, the pH curve is initially acidic and has a basic equivalence point (pH > 7). The section of curve between the initial point and the equivalence point is known as the buffer region. At the half-equivalence point, the concentrations of the buffer components are equal, resulting in pH = pKₐ.https://cdn.kastatic.org/ka-youtube-converted/sZW1qoQl3ng.mp4/sZW1qoQl3ng.mp4729yes https://puzzicle.org/science/ap-chemistry-beta/v/weak-base-strong-acid-titrationshttps://puzzicle.org/thumbs/x0a6a7576a293032e.jpgWeak base–strong acid titrationsFor the titration of a weak base with a strong acid, the pH curve is initially basic and has an acidic equivalence point (pH < 7). The section of curve between the initial point and the equivalence point is known as the buffer region. At the half-equivalence point, the concentrations of the buffer components are equal, resulting in pH = pKₐ (where pKₐ refers to the conjugate acid of the weak base).https://cdn.kastatic.org/ka-youtube-converted/IZ7NEYnuovI.mp4/IZ7NEYnuovI.mp4634yes https://puzzicle.org/science/ap-chemistry-beta/v/titrations-of-polyprotic-acidshttps://puzzicle.org/thumbs/x86414b1cfa2e2bed.jpgTitrations of polyprotic acidsTitrating a polyprotic acid with a strong base produces a pH curve with as many equivalence points as there are acidic protons on the acid. The pKₐ values for these protons can be estimated from the corresponding half-equivalence points on the curve, where pH = pKₐ.https://cdn.kastatic.org/ka-youtube-converted/AXKqh3za1Fg.mp4/AXKqh3za1Fg.mp4524yes https://puzzicle.org/science/ap-chemistry-beta/v/acid-base-indicatorshttps://puzzicle.org/thumbs/x7a817f82efe5cb51.jpgAcid–base indicatorsAcid–base indicators are compounds that change color when they become protonated or deprotonated. Because this color change occurs over a specific pH range, indicators can be used to approximate the equivalence point of an acid–base titration.https://cdn.kastatic.org/ka-youtube-converted/ZDgYWaeuwbw.mp4/ZDgYWaeuwbw.mp4425yes https://puzzicle.org/science/ap-chemistry-beta/v/introduction-to-entropy-aphttps://puzzicle.org/thumbs/xd817fa423dd049c3.jpgIntroduction to entropyAccording to the Boltzmann equation, entropy is a measure of the number of microstates available to a system. The number of available microstates increases when matter becomes more dispersed, such as when a liquid changes into a gas or when a gas is expanded at constant temperature. It also increases when energy becomes more dispersed, which occurs with an increase in temperature.https://cdn.kastatic.org/ka-youtube-converted/0Z2bIv4DXts.mp4/0Z2bIv4DXts.mp4714yes https://puzzicle.org/science/ap-chemistry-beta/v/absolute-entropy-and-entropy-changehttps://puzzicle.org/thumbs/x168191bf8a63545c.jpgAbsolute entropy and entropy changeThe standard molar entropy of a substance is the absolute entropy of 1 mole of the substance in the standard state. For any chemical reaction, the standard entropy change is the sum of the standard molar entropies of the products minus the sum of the standard molar entropies of the reactants.https://cdn.kastatic.org/ka-youtube-converted/ObridY7EHyk.mp4/ObridY7EHyk.mp4688yes https://puzzicle.org/science/ap-chemistry-beta/v/introduction-to-gibbs-free-energyhttps://puzzicle.org/thumbs/xb2f2d5ad49bb5347.jpgIntroduction to Gibbs free energyThe standard Gibbs free energy change, ΔG°, indicates the thermodynamic favorability of a physical or chemical process. When ΔG° < 0, the process is thermodynamically favored. For a given process, the value of ΔG° can be calculated directly from the values of ΔH° and ΔS° using the following equation: ΔG° = ΔH° - TΔS°.https://cdn.kastatic.org/ka-youtube-converted/_ikwCBn8wLE.mp4/_ikwCBn8wLE.mp4339yes https://puzzicle.org/science/ap-chemistry-beta/v/thermodynamic-favorability-and-temperaturehttps://puzzicle.org/thumbs/x335e7a51d00aef50.jpgThermodynamic favorability and temperatureThe temperature conditions under which a process is thermodynamically favored (ΔG° < 0) can be predicted from the signs of ΔH° and ΔS°. When ΔH° < 0 and ΔS° > 0, the process is favored at all temperatures. When ΔH° > 0 and ΔS° < 0, the process is favored at no temperature. When ΔH° < 0 and ΔS° < 0, the process is favored only at low temperatures. When ΔH° > 0 and ΔS° > 0, the process is favored only at high temperatures.https://cdn.kastatic.org/ka-youtube-converted/eejDYxFKGNs.mp4/eejDYxFKGNs.mp4834yes https://puzzicle.org/science/ap-chemistry-beta/v/worked-example-determining-the-effect-of-temperature-on-thermodynamic-favorabilihttps://puzzicle.org/thumbs/x58c4615bad858602.jpgWorked example: Determining the effect of temperature on thermodynamic favorabilityWhen ΔH° and ΔS° for a reaction have the same sign, the thermodynamic favorability of the reaction depends on temperature. In this video, we'll determine the thermodynamic favorability of a reaction with ΔH° < 0 and ΔS° < 0 at two different temperatures. We'll also calculate the temperature at which the reaction changes from being thermodynamically favored to thermodynamically unfavored.https://cdn.kastatic.org/ka-youtube-converted/_pRo2jlA-9I.mp4/_pRo2jlA-9I.mp4407yes https://puzzicle.org/science/ap-chemistry-beta/v/free-energy-of-formationhttps://puzzicle.org/thumbs/xeb4447c28cc6fa97.jpgFree energy of formationThe standard free energy of formation of a substance is the free energy change that occurs when 1 mole of the substance is formed from its constituent elements in their standard states. A pure element in its standard state has a standard free energy of formation of zero. For any chemical reaction, the standard free energy change is the sum of the standard free energies of formation of the products minus the sum of the standard free energies of formation of the reactants.https://cdn.kastatic.org/ka-youtube-converted/9igPBrHMnCc.mp4/9igPBrHMnCc.mp4326yes https://puzzicle.org/science/ap-chemistry-beta/v/free-energy-of-dissolutionhttps://puzzicle.org/thumbs/x0c4baca3ce7eafe5.jpgFree energy of dissolutionThe dissolution of a solid can be thought of as occurring in three hypothetical steps: (1) the breaking apart of the solid, (2) the spreading out of the solvent, and (3) the interaction of the dissolved solid with the solvent. As shown in this video, we can predict the signs of the enthalpy and entropy changes for each of these steps and relate this information to the solubility of the solid.https://cdn.kastatic.org/ka-youtube-converted/UJ_etGNuIOg.mp4/UJ_etGNuIOg.mp4856yes https://puzzicle.org/science/ap-chemistry-beta/v/thermodynamics-vs-kinetics-aphttps://puzzicle.org/thumbs/x3e03784ec6ce4835.jpgThermodynamics vs. kineticsThermodynamics tells us what can occur during a process, while kinetics tell us what actually occurs. Some processes, such as the conversion of diamond to graphite, are thermodynamically favored but kinetically unfavored. In these cases, the processes do not occur to any measurable extent.https://cdn.kastatic.org/ka-youtube-converted/ur7GBAQgiB0.mp4/ur7GBAQgiB0.mp4270yes https://puzzicle.org/science/ap-chemistry-beta/v/nonstandard-free-energy-changeshttps://puzzicle.org/thumbs/x7878a267e73e7757.jpgNonstandard free energy changesThe standard change in free energy, ΔG°, for a reaction applies only when the reactants and products are in their standard states. In all other circumstances, we must consider the nonstandard free energy change, ΔG, which can be calculated from ΔG° using the equation ΔG = ΔG° + RTlnQ. In this equation, R is the appropriate gas constant (8.314 J/mol·K), T is the temperature in kelvin, and Q is the reaction quotient.https://cdn.kastatic.org/ka-youtube-converted/P6dcloxLv2Q.mp4/P6dcloxLv2Q.mp4615yes https://puzzicle.org/science/ap-chemistry-beta/v/free-energy-and-equilibriumhttps://puzzicle.org/thumbs/x8ef554a85bfa3dd6.jpgFree energy and equilibriumThe standard change in free energy, ΔG°, for a reaction is related to its equilibrium constant, K, by the equation ΔG° = -RTlnK. When ΔG° < 0, K > 1, and the reaction is product-favored at equilibrium. When ΔG° > 0, K < 1, and the reaction is reactant-favored at equilibrium. When ΔG° = 0, K = 1, and the reaction is at equilibrium under standard state conditions.https://cdn.kastatic.org/ka-youtube-converted/4ZcTEiJru84.mp4/4ZcTEiJru84.mp4521yes https://puzzicle.org/science/ap-chemistry-beta/v/coupled-reactionshttps://puzzicle.org/thumbs/x318f183f0f88480b.jpgCoupled reactionsA thermodynamically unfavored reaction can be driven by coupling it to a favored reaction through one or more shared intermediates. The sum of the two reactions yields an overall reaction that has a negative ΔG° value.https://cdn.kastatic.org/ka-youtube-converted/NnSJsYa1IK0.mp4/NnSJsYa1IK0.mp4434yes https://puzzicle.org/science/ap-chemistry-beta/v/galvanic-voltaic-cellshttps://puzzicle.org/thumbs/x7343d24cfad6e4f0.jpgGalvanic (voltaic) cellsGalvanic (or voltaic) cells use a thermodynamically favored redox reaction to generate an electric current. Each half-reaction takes place in a separate compartment, or half-cell, containing an electrode. The electrode where oxidation occurs is the anode, and the electrode where reduction occurs is the cathode. Electrons flow through a wire connecting the two half-cells. To complete the circuit, a salt bridge is used, which allows ions to move between the half-cell solutions.https://cdn.kastatic.org/ka-youtube-converted/inXgVjz1BDQ.mp4/inXgVjz1BDQ.mp4552yes https://puzzicle.org/science/ap-chemistry-beta/v/standard-cell-potentialhttps://puzzicle.org/thumbs/xc9b387da1e92d475.jpgStandard cell potentialStandard cell potential is the potential of a cell when all reactants and products are in their standard states. The standard potential for any cell can be calculated by subtracting the standard reduction potential of the half-reaction occurring at the anode from the standard reduction potential of the half-reaction occurring at the cathode.https://cdn.kastatic.org/ka-youtube-converted/nBZ4PDBhSrY.mp4/nBZ4PDBhSrY.mp4419yes https://puzzicle.org/science/ap-chemistry-beta/v/worked-example-calculating-e-using-standard-reduction-potentialshttps://puzzicle.org/thumbs/x416b48408b0d8fea.jpgWorked example: Calculating E° using standard reduction potentialsThe standard potential, E°, for a redox reaction is the difference between the standard reduction potentials of the reduction and oxidation half-reactions. In this video, we'll use this relationship to calculate the E° for the redox reaction between Ag⁺(aq) and Cr(s).https://cdn.kastatic.org/ka-youtube-converted/PNolxY6rOKk.mp4/PNolxY6rOKk.mp4321yes https://puzzicle.org/science/ap-chemistry-beta/v/standard-potential-free-energy-and-the-equilibrium-constanthttps://puzzicle.org/thumbs/x31ef062bfe9caec9.jpgStandard potential, free energy, and the equilibrium constantThe standard potential, E°, for a redox reaction is related to the standard change in free energy, ΔG°, for the reaction by the equation ΔG° = -nFE°. When E° > 0, ΔG° < 0, and the reaction is thermodynamically favored. When E° < 0, ΔG° > 0, and the reaction is thermodynamically unfavored. It is also possible to relate E° to the equilibrium constant, K, for the reaction through the relationship between ΔG° and K (ΔG° = -RTlnK).https://cdn.kastatic.org/ka-youtube-converted/b2YI3AIs6l0.mp4/b2YI3AIs6l0.mp4548yes https://puzzicle.org/science/ap-chemistry-beta/v/the-nernst-equationhttps://puzzicle.org/thumbs/x560b890229e27653.jpgThe Nernst equationThe Nernst equation relates the instantaneous potential, E, to the standard potential, E°, and the reaction quotient, Q: E = E° - (0.0592 V/n)logQ at 298 K. When all reactants and products are in their standard states, Q = 1 and E = E°. When the reaction is at equilibrium, Q = K and E = 0.https://cdn.kastatic.org/ka-youtube-converted/PemB4mY4Jfk.mp4/PemB4mY4Jfk.mp4687yes https://puzzicle.org/science/ap-chemistry-beta/v/quantitative-electrolysis-aphttps://puzzicle.org/thumbs/x5d3b9e1304966117.jpgQuantitative electrolysisGiven the amount of electrical charge that passes through an electrolytic cell, we can calculate the quantity of substances consumed or produced during electrolysis (or vice versa). The total charge is related to the magnitude of the current and the time it runs by the equation I = Q/t, where I is the current in amperes, Q is the charge in coulombs, and t is the time in seconds.https://cdn.kastatic.org/ka-youtube-converted/v5qQusbk-j0.mp4/v5qQusbk-j0.mp4447yes https://puzzicle.org/science/biology/v/ecosystems-and-biomeshttps://puzzicle.org/thumbs/xd2ee49ae2ddb5919.jpgEcosystems and biomesEcosystems, made up of living and non-living things, can be as small as a tide pool or as big as a rainforest. Ecosystems can be classified as terrestrial or aquatic, with aquatic further divided into marine and freshwater. Land ecosystems are often categorized into biomes, such as tropical forests or deserts, based on climate and vegetation.https://cdn.kastatic.org/ka-youtube-converted/A495e31cDdE.mp4/A495e31cDdE.mp4453yes https://puzzicle.org/science/ap-college-environmental-science/v/primary-productivity-in-ecosystemshttps://puzzicle.org/thumbs/x97f65e2f807e38f8.jpgPrimary productivity in ecosystemsPrimary productivity is the rate at which solar energy (sunlight) is converted into organic compounds via photosynthesis over a unit of time. Net primary productivity is the rate of energy storage by photosynthetic organisms in a given area, after subtracting the energy lost to respiration.https://cdn.kastatic.org/ka-youtube-converted/8zO0algdgNM.mp4/8zO0algdgNM.mp4603yes https://puzzicle.org/science/ap-college-environmental-science/v/energy-flow-in-a-marine-ecosystemhttps://puzzicle.org/thumbs/x4f11e1ae0cd045f3.jpgEnergy flow in a marine ecosystemAll ecosystems depend on a continuous inflow of high-quality energy in order to maintain their structure and function of transferring matter between the environment and organisms via biogeochemical cycles. In terrestrial and near-surface marine communities, energy flows from the sun to producers in the lowest trophic levels and then upward to higher trophic levels. The 10% rule approximates that in the transfer of energy from one trophic level to the next, only about 10% of the energy is passed on.https://cdn.kastatic.org/ka-youtube-converted/fL0W_boPHLs.mp4/fL0W_boPHLs.mp4398yes https://puzzicle.org/science/ap-college-environmental-science/v/humans-and-ecosystems-how-do-vultures-provide-ecosystem-serviceshttps://puzzicle.org/thumbs/x01da1ff235f2c7fb.jpgHumans and ecosystems: how do vultures provide ecosystem services?The Lammergeier, a bone-eating vulture, exemplifies the importance of biodiversity and ecosystem services. These services, categorized into provisioning, regulating, supporting, and cultural, sustain the environment and human life. However, human disruptions, like harmful chemicals, can endanger species like vultures, leading to ecological and economic consequences.https://cdn.kastatic.org/ka-youtube-converted/bT_2MyIRsas.mp4/bT_2MyIRsas.mp4299yes https://puzzicle.org/science/biology/v/ecological-successionhttps://puzzicle.org/thumbs/x3d2d49b0.jpgEcological successionEcological succession, a dynamic process involving gradual changes in biological communities over time, encompasses two main types: primary succession, where life gradually colonizes a new habitat, and secondary succession, triggered by a disaster that disrupts an existing community, instigating shifts in species composition.https://cdn.kastatic.org/ka-youtube-converted/d7xbyNSxxrI.mp4/d7xbyNSxxrI.mp4280yes https://puzzicle.org/science/ap-college-environmental-science/v/human-population-dynamicshttps://puzzicle.org/thumbs/x83826b6c4ab54bbc.jpgHuman population dynamicsWorld population growth has surged over the last few centuries due to advancements in healthcare and agriculture. However, as societies develop and family planning is implemented, growth rates are declining. The Earth's carrying capacity for humans remains an open question, influenced by our ever-evolving technology and resource efficiency.https://cdn.kastatic.org/ka-youtube-converted/jKfojKK-8Zk.mp4/jKfojKK-8Zk.mp4389yes https://puzzicle.org/science/ap-college-environmental-science/v/rule-of-70-to-approximate-population-doubling-timehttps://puzzicle.org/thumbs/x8f8759d9d0ec55dc.jpgRule of 70 to approximate population doubling timeDoubling time is the period it takes for a population growing at a certain rate to double its size. One way to approximate doubling time is known as the rule of 70. This rule is used in various fields, including finance and environmental science. However, the rule of 70 is an approximation, not a precise calculation.https://cdn.kastatic.org/ka-youtube-converted/KBXec1ctCjg.mp4/KBXec1ctCjg.mp4201yes https://puzzicle.org/science/ap-college-environmental-science/v/demographic-transition-modelhttps://puzzicle.org/thumbs/x74ea28d2639e4347.jpgDemographic transition modelThe Demographic Transition Model illustrates how a country's birth and death rates change as it transitions from pre-industrial to post-industrial. High birth and death rates characterize pre-industrial societies, while industrialization leads to lower death rates and eventually lower birth rates. In post-industrial societies, birth rates may even fall below death rates, leading to population decline.https://cdn.kastatic.org/ka-youtube-converted/kEOgW5Rg0o8.mp4/kEOgW5Rg0o8.mp4439yes https://puzzicle.org/science/cosmology-and-astronomy/v/plate-tectonics-geological-features-of-divergent-plate-boundarieshttps://puzzicle.org/thumbs/13179439.jpgPlate tectonics: Geological features of divergent plate boundariesDivergent plate boundaries, where Earth's plates move apart, create unique geological features. These boundaries result in seafloor spreading, rift valleys, volcanoes, and earthquakes. The process begins with hot magma rising, causing the lithosphere to crack and form rift valleys. Over time, these valleys can fill with water, forming seas. As the land continues to spread, mid-oceanic ridges form, creating an underwater mountain chain.https://cdn.kastatic.org/ka-youtube-converted/FK1s1-OJ5BE.mp4/FK1s1-OJ5BE.mp4718yes https://puzzicle.org/science/cosmology-and-astronomy/v/plate-tectonics-geological-features-of-convergent-plate-boundarieshttps://puzzicle.org/thumbs/13179440.jpgPlate tectonics: Geological features of convergent plate boundariesPlate tectonics involves the movement and interaction of Earth's crustal plates, leading to geological changes. Divergent boundaries create new crust, while convergent boundaries, where plates collide, can form mountains, island arcs, earthquakes, and volcanoes. The subduction of one plate under another can result in deep trenches and underwater volcanoes.https://cdn.kastatic.org/ka-youtube-converted/Y0eWnOZpSpQ.mp4/Y0eWnOZpSpQ.mp4364yes https://puzzicle.org/science/ap-college-environmental-science/v/soil-texture-trianglehttps://puzzicle.org/thumbs/x7b074b37bab23ad4.jpgSoil texture triangleSoil texture, determined by the proportions of sand, silt, and clay, plays a crucial role in plant growth. The soil texture triangle helps categorize soil into 12 types based on these proportions. Understanding soil texture can guide optimal plant selection for different soil types.https://cdn.kastatic.org/ka-youtube-converted/Bn9ul9lxIdg.mp4/Bn9ul9lxIdg.mp4384yes https://puzzicle.org/science/ap-college-environmental-science/v/global-wind-patternshttps://puzzicle.org/thumbs/x45a66b603f628d93.jpgGlobal wind patternsGlobal wind patterns, crucial for life on Earth, are primarily driven by the Sun's intense radiation at the equator, resulting in atmospheric circulation. This process, involving rising hot air and sinking cool air, creates convection cells and prevailing winds that distribute heat and moisture unevenly across the globe, shaping diverse biomes and determining species survival.https://cdn.kastatic.org/ka-youtube-converted/xp50_ixPOhY.mp4/xp50_ixPOhY.mp4383yes https://puzzicle.org/science/ap-college-environmental-science/v/seasons-mshttps://puzzicle.org/thumbs/x37fbccdfe032e546.jpgSeasonsEarth's seasons are not dictated by its distance from the Sun, but rather by its axial tilt. The angle of the Sun's rays determines the intensity of solar radiation, with the equator receiving the highest intensity. This tilt also causes variations in daylight hours, contributing to the change of seasons.https://cdn.kastatic.org/ka-youtube-converted/0r9Lc7oYq5M.mp4/0r9Lc7oYq5M.mp4306yes https://puzzicle.org/science/ap-college-environmental-science/v/el-nino-and-la-ninahttps://puzzicle.org/thumbs/x37164ff4a860b19b.jpgEl Niño and La NiñaEl Niño and La Niña are climate patterns in the Pacific Ocean that can cause global changes in rainfall, wind, and ocean circulation. El Niño characterized by warmer ocean temperatures, can lead to heavy rainfall or severe drought in different regions. La Niña, on the other hand, is its cooler counterpart. These phenomena, influenced by geological and geographic factors, can drastically affect weather patterns worldwide.https://cdn.kastatic.org/ka-youtube-converted/U2-ACg2kbTA.mp4/U2-ACg2kbTA.mp4416yes https://puzzicle.org/science/ap-college-environmental-science/v/impacts-of-agricultural-practiceshttps://puzzicle.org/thumbs/x3903040a49a97dcc.jpgImpacts of agricultural practicesAgricultural practices like tilling, slash-and-burn farming, and the use of fertilizers can cause significant environmental damage. Tilling can lead to soil erosion, while slash-and-burn farming contributes to deforestation and climate change. Overuse of fertilizers can result in water pollution and harmful algal blooms. Sustainable alternatives exist, offering hope for more environmentally friendly agriculture.https://cdn.kastatic.org/ka-youtube-converted/dbEtcjNxGVQ.mp4/dbEtcjNxGVQ.mp4459yes https://puzzicle.org/science/ap-college-environmental-science/v/impacts-of-urbanizationhttps://puzzicle.org/thumbs/x6404e80a65961231.jpgImpacts of urbanizationUrbanization, the growth of cities, offers many advantages but also poses environmental challenges. High population densities can strain resources and increase pollution. However, sustainable urban planning, such as incorporating green spaces and permeable surfaces, can mitigate these effects. Thoughtful city design can reduce runoff, prevent saltwater intrusion, and decrease air pollution, making urban centers more sustainable.https://cdn.kastatic.org/ka-youtube-converted/2C7F_2OpRT8.mp4/2C7F_2OpRT8.mp4379yes https://puzzicle.org/science/ap-college-environmental-science/v/introduction-to-sustainabilityhttps://puzzicle.org/thumbs/x7d3f2679e9bcc89c.jpgIntroduction to sustainabilitySustainability refers to humans living on Earth and their use of resources without depletion of the resources for future generations. Environmental indicators that can guide humans to sustainability include biological diversity, food production, average global surface temperatures and CO2 concentrations, human population, and resource depletion. Sustainable yield is the amount of a renewable resource that can be taken without reducing the available supply.https://cdn.kastatic.org/ka-youtube-converted/k6jUaaQNY28.mp4/k6jUaaQNY28.mp4381yes https://puzzicle.org/science/ap-college-environmental-science/v/integrated-pest-managementhttps://puzzicle.org/thumbs/xdca3e64a3265acb1.jpgIntegrated pest managementIntegrated pest management (IPM) is a combination of methods used to effectively control pest species while minimizing the disruption to the environment. These methods include biological, physical, and limited chemical methods such as biocontrol, intercropping, crop rotation, and natural predators of the pests. Integrated pest management (IPM) minimizes disruptions to the environment and threats to human health but can be complex and expensive.https://cdn.kastatic.org/ka-youtube-converted/z2AMnP4nn2E.mp4/z2AMnP4nn2E.mp4403yes https://puzzicle.org/science/ap-college-environmental-science/v/nuclear-power-generationhttps://puzzicle.org/thumbs/x7ffc27b764b1bc3d.jpgNuclear power generationNuclear power plants generate electricity by using the heat from nuclear fission to create steam, which spins turbines. This process involves splitting atoms of Uranium-235, a heavy and unstable isotope, in a controlled chain reaction. Nuclear power is considered clean energy because it does not produce air pollutants. The resulting nuclear waste, however, poses significant disposal challenges due to its long-lasting radioactivity.https://cdn.kastatic.org/ka-youtube-converted/E7UIonbL4FU.mp4/E7UIonbL4FU.mp4439yes https://puzzicle.org/science/ap-college-environmental-science/v/solar-energyhttps://puzzicle.org/thumbs/x2add0be89e793438.jpgSolar energyPassive solar energy systems absorb heat directly from the sun without the use of mechanical and electric equipment, and energy cannot be collected or stored. Active solar energy systems use solar energy to heat a liquid through mechanical and electric equipment to collect and store the energy captured from the sun. Photovoltaic solar cells capture light energy from the sun and transform it directly into electrical energy. Their use is limited by the availability of sunlight.https://cdn.kastatic.org/ka-youtube-converted/OVODBZUJAYE.mp4/OVODBZUJAYE.mp4408yes https://puzzicle.org/science/ap-college-environmental-science/v/energy-conservation-methodshttps://puzzicle.org/thumbs/x9589d40917e5ed3a.jpgEnergy conservation methodsSome of the methods for conserving energy around the home include adjusting the thermostat to reduce the use of heat and air conditioning, conserving water, use of energy efficient appliances, and conservation landscaping. Methods for conserving energy on a large scale include improving fuel economy for vehicles, using BEVs and hybrid vehicles, using public transportation, and implementing green building design features.https://cdn.kastatic.org/ka-youtube-converted/GSc5zo4WjJs.mp4/GSc5zo4WjJs.mp4357yes https://puzzicle.org/science/ap-college-environmental-science/v/indoor-air-pollutantshttps://puzzicle.org/thumbs/xda3f60efaf345b1c.jpgIndoor air pollutantsIndoor air pollutants can come from natural sources, human-made sources, and combustion. Common natural pollutants include radon, mold, and dust. Common human-made pollutants include Volatile Organic Compounds (VOCs) and formaldehyde. Common combustion pollutants include carbon monoxide, nitrogen oxides, sulfur dioxide, particulates, and tobacco smoke.https://cdn.kastatic.org/ka-youtube-converted/nqRPRwO79kg.mp4/nqRPRwO79kg.mp4415yes https://puzzicle.org/science/ap-college-environmental-science/v/solid-waste-disposalhttps://puzzicle.org/thumbs/xfb680d074f409e05.jpgSolid waste disposalSolid waste is most often disposed of in open dumps and sanitary landfills. A sanitary municipal landfill consists of a bottom liner (plastic or clay), a storm water collection system, a leachate collection system, a cap, and a methane collection system. However, landfills can contaminate groundwater, cause leachate pollution, and release harmful gases. Solid waste can also be disposed of through incineration, where waste is burned at high temperatures.https://cdn.kastatic.org/ka-youtube-converted/LOzMugtKmN4.mp4/LOzMugtKmN4.mp4476yes https://puzzicle.org/science/ap-college-environmental-science/v/pathogens-and-the-environmenthttps://puzzicle.org/thumbs/x7087ff0baca6a5a6.jpgPathogens and the environmentPathogens, such as the pathogen that causes malaria, adapt to take advantage of new opportunities to infect and spread through human populations. As equatorial-type climate zones spread north and south into what are currently subtropical and temperate climate zones, pathogens, infectious diseases, and any associated vectors are spreading into these areas where the disease has not previously been know to occur.https://cdn.kastatic.org/ka-youtube-converted/wyd0ozxbWkk.mp4/wyd0ozxbWkk.mp4277yes https://puzzicle.org/science/ap-college-environmental-science/v/stratospheric-ozone-depletionhttps://puzzicle.org/thumbs/x997edc67965c5b93.jpgStratospheric ozone depletionOzone, a molecule composed of three oxygen atoms, plays a dual role in our atmosphere. In the stratosphere, it forms a protective layer that absorbs harmful UV light, reducing the risk of skin cancer. However, human activities like aerosol sprays and car exhaust can deplete this layer. Conversely, in the troposphere where we live, ozone is harmful, damaging living tissue and respiratory systems.https://cdn.kastatic.org/ka-youtube-converted/5CGtESB40Rc.mp4/5CGtESB40Rc.mp4309yes https://puzzicle.org/science/ap-college-environmental-science/v/ocean-acidificationhttps://puzzicle.org/thumbs/x988f7138251565ec.jpgOcean acidificationOcean acidification, a result of increased atmospheric carbon dioxide, is making our oceans more acidic, disrupting the formation of calcium carbonate in sea life and disturbing their homeostasis. This phenomenon is linked to emissions from human activities, such as the combustion of fossil fuels, driving vehicles, and deforestation.https://cdn.kastatic.org/ka-youtube-converted/8JUEga3MYsM.mp4/8JUEga3MYsM.mp4325yes https://puzzicle.org/science/ap-college-environmental-science/v/climate-emergency-feedback-loops-albedohttps://puzzicle.org/thumbs/x44f701903381efad.jpgClimate emergency: feedback loops - albedoClimate change is setting off dangerous feedback loops at the Poles. The melting of Arctic ice and snow decreases Earth' ability to reflect the Sun's rays, leading to further heating and melting of ice and snow. In Antarctica, the warmer climate is melting ice sheets, leading to raised sea levels, which melts ice sheets further in an amplifying loop.https://cdn.kastatic.org/ka-youtube-converted/HNqTxBHgC0Y.mp4/HNqTxBHgC0Y.mp4636yes https://puzzicle.org/science/ap-college-environmental-science/v/climate-emergency-feedback-loops-permafrosthttps://puzzicle.org/thumbs/x3e5c07edd70ee925.jpgClimate emergency: feedback loops - permafrostGlobal warming is causing the thawing of permafrost, the icy expanse of frozen ground covering one-quarter of the Northern Hemisphere. As it melts, previously frozen carbon stored in plant and animal remains is released into the atmosphere as heat-trapping greenhouse gases, warming the climate further, and melting more permafrost in a dangerous feedback loop.https://cdn.kastatic.org/ka-youtube-converted/e44IYZ-gQnE.mp4/e44IYZ-gQnE.mp4655yes https://puzzicle.org/science/ap-college-physics-1/v/introduction-to-ap-physics-1https://puzzicle.org/thumbs/xde8838aa7c5bd894.jpgIntroduction to AP Physics 1AP Physics 1 is an algebra-based, introductory college-level mechanics course. In this course you'll explore the following topics: Kinematics; Forces and translational dynamics; Work, energy, and power; Linear momentum; Torque and rotational dynamics; Energy and momentum of rotating systems; Oscillations; Fluids.https://cdn.kastatic.org/ka-youtube-converted/21p-yiQ9xy4.mp4/21p-yiQ9xy4.mp4188yes https://puzzicle.org/science/ap-college-physics-1/v/position-velocity-speedhttps://puzzicle.org/thumbs/xf119ffb760f77fa2.jpgPosition, velocity, and speedAn object's position specifies how far the object is from a reference point, and in what direction. The rate that an object's position changes is called velocity. Average velocity is calculated as change in position divided by change in time. Position and velocity are vector quantities. Average speed is calculated as distance travelled divided by time. Speed and distance are scalar quantities with no direction.https://cdn.kastatic.org/ka-youtube-converted/vOipqtdR23s.mp4/vOipqtdR23s.mp4624yes https://puzzicle.org/science/ap-college-physics-1/v/intro-to-relative-velocity-class-11-india-physics-khan-academyhttps://puzzicle.org/thumbs/x587cc05eb5ce0e1c.jpgRelative velocityVelocity depends on the point of view we take. Let's discuss what relative velocity is and how to visualize it.https://cdn.kastatic.org/ka-youtube-converted/EmOis0kQuxM.mp4/EmOis0kQuxM.mp4403yes https://puzzicle.org/science/highschool-physics/v/interpreting-motion-datahttps://puzzicle.org/thumbs/xf6edb2e65b2f7003.jpgInterpreting motion dataThe same motion data can be presented in various ways, including dot (oil drop) diagrams, position-time tables, and position vs. time graphs. The slope of a position vs. time graph represents the object's velocity: A straight line (constant slope) means the object moved with constant velocity, and a curved line (changing slope) means the object accelerated.https://cdn.kastatic.org/ka-youtube-converted/-1xbgz26ZNs.mp4/-1xbgz26ZNs.mp4606yes https://puzzicle.org/science/ap-college-physics-1/v/instantaneous-speed-and-velocityhttps://puzzicle.org/thumbs/xc7e30831.jpgInstantaneous speed and velocityInstantaneous speed is a measurement of how fast an object is moving at that particular moment. Instantaneous velocity is a vector quantity that includes both the speed and the direction in which the object is moving. Learn how to find an object’s instantaneous speed or velocity in three ways - by using calculus, by looking at the slope of a given point on a graph of an object’s rate vs. time, or by using kinematic formulas if the object’s acceleration is constant.https://cdn.kastatic.org/ka-youtube-converted/pfTTHx9kCHk.mp4/pfTTHx9kCHk.mp4278yes https://puzzicle.org/science/ap-college-physics-1/v/position-vs-time-graphshttps://puzzicle.org/thumbs/x92717942.jpgPosition vs. time graphsLearn how to interpret a position vs. time graph. Use a position vs. time graph to determine displacement, distance, average velocity, average speed, instantaneous velocity, and instantaneous speed.https://cdn.kastatic.org/ka-youtube-converted/GtoamALPOP0.mp4/GtoamALPOP0.mp4920yes https://puzzicle.org/science/ap-college-physics-1/v/intro-to-accelerationhttps://puzzicle.org/thumbs/x9430b28c45d8ba63.jpgAccelerationAcceleration is the rate of change of velocity (a = Δv/Δt). The unit for acceleration is m/s/s, or m/s^2. Acceleration is a vector with magnitude and direction. If velocity and acceleration point the same direction, the object is speeding up. If velocity and acceleration point opposite directions, the object is slowing down. Changing direction is also a form of acceleration.https://cdn.kastatic.org/ka-youtube-converted/JSPwCtIPfQw.mp4/JSPwCtIPfQw.mp4620yes https://puzzicle.org/science/ap-physics-c-mechanics/v/developing-kinematic-equations-from-datahttps://puzzicle.org/thumbs/xc28c1da86812be33.jpgDeveloping kinematic equations from dataExperimental data for an object moving with constant acceleration can be used to develop the kinematic equations. See how this can be accomplished using a process called linearization.https://cdn.kastatic.org/ka-youtube-converted/dr39gKnviAc.mp4/dr39gKnviAc.mp4785yes https://puzzicle.org/science/ap-physics-c-mechanics/v/proportional-reasoning-with-motionhttps://puzzicle.org/thumbs/x89e97e7d9a53e82f.jpgProportional reasoning with motionDifferent motion scenarios can be compared using proportional reasoning. Even if exact values aren't known for each value, the factor of change in one variable can be used to predict the factor of change in another.https://cdn.kastatic.org/ka-youtube-converted/pJWRXgqn8PA.mp4/pJWRXgqn8PA.mp4766yes https://puzzicle.org/science/ap-physics-c-mechanics/v/projectile-motionhttps://puzzicle.org/thumbs/x25aad12b6fe977c1.jpgProjectile motionMotion in multiple dimensions can be analyzed using one-dimensional kinematic relationships if the motion is separated into components. Projectile motion is a special case of two-dimensional motion that has zero acceleration in the horizontal dimension, and constant acceleration of magnitude g in the vertical dimension.https://cdn.kastatic.org/ka-youtube-converted/txJP95lBv98.mp4/txJP95lBv98.mp4687yes https://puzzicle.org/science/ap-physics-c-mechanics/v/forces-and-free-body-diagramshttps://puzzicle.org/thumbs/xb990f782a3eafd50.jpgForces and free-body diagramsA force is a vector quantity that describes a push or pull. A variety of contact forces (e.g., normal force, friction) and non-contact forces (e.g., gravity) may act on an object simultaneously. A free-body diagram is a model that shows each of the external forces exerted on an object. Forces are represented as vector arrows originating from the representation of the center of mass, such as a dot. A free-body diagram is useful for determining the net force on an object, which is the vector sum of all the individual forces acting on the object.https://cdn.kastatic.org/ka-youtube-converted/i447NWNpScM.mp4/i447NWNpScM.mp4734yes https://puzzicle.org/science/highschool-physics/v/newtons-second-law-calculationshttps://puzzicle.org/thumbs/x380fd629c47b0e53.jpgNewton's second law calculationsNewton's second law can be used to calculate the acceleration that an object will experience, if the net force acting on the object and its mass are known. The law can also be used to determine the net force required to produce a certain acceleration for a known mass. Remember that the force term in Newton's second law refers to the net force on the object.https://cdn.kastatic.org/ka-youtube-converted/O5G4uHe-qc4.mp4/O5G4uHe-qc4.mp4595yes https://puzzicle.org/science/ap-physics-c-mechanics/v/forces-at-an-anglehttps://puzzicle.org/thumbs/x2bcd7bcb1f04bcea.jpgForces at an angleForces may act on an object in more than one dimension. Such scenarios may require one or more the forces to be resolved into components. Learn how to analyze scenarios involving forces acting at different angles by first creating a free-body diagram, and then applying trigonometry and Newton’s second law.https://cdn.kastatic.org/ka-youtube-converted/W6rb-0AghZE.mp4/W6rb-0AghZE.mp4820yes https://puzzicle.org/science/ap-physics-c-mechanics/v/two-masses-hanging-from-a-pulleyhttps://puzzicle.org/thumbs/x483e06a2.jpgAnalyzing systemsLearn how to analyze the motion of one or more masses moving together as a system, using the example of two masses hanging from a pulley.https://cdn.kastatic.org/ka-youtube-converted/BrLbtCnrPaA.mp4/BrLbtCnrPaA.mp4530yes https://puzzicle.org/science/ap-physics-c-mechanics/v/gravitational-forces-and-fieldshttps://puzzicle.org/thumbs/x41b2aee0d318cf7a.jpgGravitational forces and fieldsNewton’s law of universal gravitation describes the gravitational forces two objects exert on each other. The gravitational force is always attractive and is exerted along the line connecting the two objects’ centers of mass. The gravitational field created by an object at a point in space is the gravitational force exerted on another test object at that location, divided by the mass of the test object. If the gravitational force is the only force exerted on an object, the observed acceleration of the object (in m/s^2) is numerically equal to the magnitude of the gravitational field strength (in N/kg) at that location.https://cdn.kastatic.org/ka-youtube-converted/CJV7RZNCN28.mp4/CJV7RZNCN28.mp4735yes https://puzzicle.org/science/ap-physics-c-mechanics/v/weight-apparent-weight-and-weightlessnesshttps://puzzicle.org/thumbs/xfe1787b7f502fea9.jpgWeight, apparent weight, and weightlessnessThe gravitational force exerted by an astronomical body on a relatively small nearby object is called weight. The magnitude of the apparent weight of an object is the magnitude of the normal force exerted on the object. If the system is accelerating vertically, the apparent weight of the object is not equal to the magnitude of the gravitational force exerted on the object. A system appears weightless when there are no forces exerted on the system, or when the force of gravity is the only force exerted on the system.https://cdn.kastatic.org/ka-youtube-converted/tVsYgAM8MV4.mp4/tVsYgAM8MV4.mp4750yes https://puzzicle.org/science/ap-physics-c-mechanics/v/inertial-mass-vs-gravitational-masshttps://puzzicle.org/thumbs/x4b66995f72c3636e.jpgInertial and gravitational massThe inertial and gravitational mass of an object have the same value, but they describe two different things about an object. Learn how to distinguish between inertial and gravitational mass and measure each independently of each other.https://cdn.kastatic.org/ka-youtube-converted/Ws3yB3QsKY4.mp4/Ws3yB3QsKY4.mp4492yes https://puzzicle.org/science/ap-college-physics-1/v/yo-yo-in-vertical-circle-examplehttps://puzzicle.org/thumbs/xd634e085.jpgMass swinging in vertical circle examplePractice finding the tension in a string that is swinging a mass in a vertical circle.https://cdn.kastatic.org/ka-youtube-converted/2lcaBPLLoLo.mp4/2lcaBPLLoLo.mp4631yes https://puzzicle.org/science/ap-college-physics-1/v/bowling-ball-in-vertical-loophttps://puzzicle.org/thumbs/xfa6de716.jpgBall in vertical loop examplePractice finding the normal force on a ball moving through a vertical, circular loop.https://cdn.kastatic.org/ka-youtube-converted/Xpgsg-fY4DY.mp4/Xpgsg-fY4DY.mp4529yes https://puzzicle.org/science/ap-college-physics-1/v/mass-swinging-in-a-horizontal-circlehttps://puzzicle.org/thumbs/x916bbdd0.jpgConical pendulum examplePractice solving for the tension and speed of a mass swinging at an angle in a horizontal circle.https://cdn.kastatic.org/ka-youtube-converted/0cDmgCcX_nU.mp4/0cDmgCcX_nU.mp4759yes https://puzzicle.org/science/ap-college-physics-1/v/work-energy-principle-examplehttps://puzzicle.org/thumbs/x21e9f77d9b2d17da.jpgWork-energy theorem exampleThe work-energy theorem tells us the amount of work done on an object or system will equal the change in kinetic energy for that object or system. Follow along in this worked example and understand about how to calculate the net work on an object and solve for the change in velocity.https://cdn.kastatic.org/ka-youtube-converted/BYkxKrbETb4.mp4/BYkxKrbETb4.mp4465yes https://puzzicle.org/science/ap-college-physics-1/v/work-as-area-under-curvehttps://puzzicle.org/thumbs/x4ec43ac2.jpgForce vs. displacement graphsExplore how the area under a force vs. displacement graph equals the work done by the force.https://cdn.kastatic.org/ka-youtube-converted/z01om-Qk2xM.mp4/z01om-Qk2xM.mp4751yes https://puzzicle.org/science/ap-college-physics-1/v/potential-energyhttps://puzzicle.org/thumbs/x2d81835ffb8a2805.jpgPotential energyPotential energy exists in a system of interacting objects. The amount of potential energy depends on the objects' relative positions. If two objects attract gravitationally or electrically, potential energy increases as they get farther apart. If two objects repel electrically, potential energy increases as they get closer. Elastic and chemical potential energy are fundamentally the result of electric potential energy at the particle level.https://cdn.kastatic.org/ka-youtube-converted/aUrms3VFn0I.mp4/aUrms3VFn0I.mp4840yes https://puzzicle.org/science/ap-physics-c-mechanics/v/conservation-of-energy-part-1https://puzzicle.org/thumbs/x90c1c34496cc87d8.jpgConservation of energy (part 1)Mechanical energy is the sum of a system’s kinetic and potential energies. If the work done on a selected system is zero and there are no nonconservative interactions within the system, the total mechanical energy of the system is constant. If the work done on a selected system is nonzero, energy is transferred between the system and the environment.https://cdn.kastatic.org/ka-youtube-converted/GdDrYgXzBew.mp4/GdDrYgXzBew.mp41043yes https://puzzicle.org/science/ap-physics-c-mechanics/v/conservation-of-energy-part-2https://puzzicle.org/thumbs/x99364af703a675e9.jpgConservation of energy (part 2)Energy is conserved in all interactions. Any change to a type of energy within a system must be balanced by an equivalent change of other types of energies within the system, or by a transfer of energy between the system and its surroundings.https://cdn.kastatic.org/ka-youtube-converted/JrVd-czHl_g.mp4/JrVd-czHl_g.mp4640yes https://puzzicle.org/science/ap-physics-c-mechanics/v/powerhttps://puzzicle.org/thumbs/x3e147b11.jpgPowerIn physics, power is defined as the rate at which work is done. It measures how quickly energy is being transferred or converted. Learn how to calculate both average and instantaneous power.https://cdn.kastatic.org/ka-youtube-converted/RpbxIG5HTf4.mp4/RpbxIG5HTf4.mp4346yes https://puzzicle.org/science/ap-college-physics-1/v/momentumhttps://puzzicle.org/thumbs/x6a3d52879f7f19e4.jpgMomentumMomentum is the product of an object's mass and its velocity. It is a vector quantity. Newton's second law can be rearranged to reveal that the net force on an object is equal to the rate of change of the object's momentum. The larger the net force experienced by an object, the more rapidly the object's momentum will change.https://cdn.kastatic.org/ka-youtube-converted/UoIIwzHug9M.mp4/UoIIwzHug9M.mp4601yes https://puzzicle.org/science/ap-college-physics-1/v/impulsehttps://puzzicle.org/thumbs/xd92e4729de316580.jpgImpulseImpulse is a change in an object's momentum. The impulse experienced by an object is equal to the net force acting on the object multiplied by the length of time the force acts (Δp = ΣFΔt). The same impulse can be caused by a large force acting for a short time, or a small force acting for a long time.https://cdn.kastatic.org/ka-youtube-converted/JlxJYSFPpQo.mp4/JlxJYSFPpQo.mp4764yes https://puzzicle.org/science/ap-college-physics-1/v/impulse-and-momentum-dodgeball-examplehttps://puzzicle.org/thumbs/x2680c555.jpgImpulse and momentum examplePractice solving for the impulse and force applied during a dodgeball collision using the impulse-momentum theorem.https://cdn.kastatic.org/ka-youtube-converted/uMYAc04D0ak.mp4/uMYAc04D0ak.mp4633yes https://puzzicle.org/science/ap-college-physics-1/v/force-vs-time-graphshttps://puzzicle.org/thumbs/x33970627.jpgForce vs. time graphsLearn how to use a force vs. time graph to find the change in momentum. Solve an example problem to find the final velocity of a spaceship.https://cdn.kastatic.org/ka-youtube-converted/8bHPj3ll0vs.mp4/8bHPj3ll0vs.mp4494yes https://puzzicle.org/science/ap-physics-c-mechanics/v/conservation-of-linear-momentumhttps://puzzicle.org/thumbs/x759ac7a51a8176c0.jpgConservation of linear momentumLinear momentum is conserved in all interactions. If the net external force on the selected system is zero, the total momentum of the system is constant. If the net external force on the selected system is nonzero, momentum is transferred between the system and the environment. Conservation of momentum can be used to determine the velocity of objects in a system immediately before or after a collision or explosion.https://cdn.kastatic.org/ka-youtube-converted/zQLxbN8dxJg.mp4/zQLxbN8dxJg.mp4909yes https://puzzicle.org/science/ap-college-physics-1/v/bouncing-fruit-collision-examplehttps://puzzicle.org/thumbs/xfe43c81b.jpgBouncing fruit collision exampleWhen an object with known velocity collides with an object at rest, and the velocity of one of the objects after the collision is known, conservation of momentum can be used to determine the velocity of the other object after the collision. It is important to consider the directions of velocity and momentum when interpreting signs in the calculation.https://cdn.kastatic.org/ka-youtube-converted/1i8VMBvMQ_E.mp4/1i8VMBvMQ_E.mp4553yes https://puzzicle.org/science/ap-physics-c-mechanics/v/torque-basicshttps://puzzicle.org/thumbs/x478911d9db4dedc9.jpgIntro to torqueA torque is the rotational equivalent of a force. When a torque is applied to an object it causes the object to rotate. Learn how forces create torques and how to calculate the net torque on a rotating object.https://cdn.kastatic.org/ka-youtube-converted/TQQXpFhACSU.mp4/TQQXpFhACSU.mp4580yes https://puzzicle.org/science/ap-physics-c-mechanics/v/finding-torque-for-angled-forceshttps://puzzicle.org/thumbs/x3956371b.jpgFinding torque for angled forcesDavid explains how to determine the torque exerted by a non-perpendicular force.https://cdn.kastatic.org/ka-youtube-converted/ZQnGh-t25tI.mp4/ZQnGh-t25tI.mp4714yes https://puzzicle.org/science/ap-physics-c-mechanics/v/rotational-version-of-newtons-second-lawhttps://puzzicle.org/thumbs/x88f76235.jpgRotational version of Newton's second lawDavid explains moment of inertia and the rotational version of Newton's second law and shows how to solve an example problem.https://cdn.kastatic.org/ka-youtube-converted/xusMSv2z77s.mp4/xusMSv2z77s.mp4836yes https://puzzicle.org/science/ap-physics-c-mechanics/v/more-on-moment-of-inertiahttps://puzzicle.org/thumbs/xfcb88944.jpgRotational inertiaLearn about rotational inertia, also called "moment of inertia," for commonly shaped objects.https://cdn.kastatic.org/ka-youtube-converted/lwO0V5FitAo.mp4/lwO0V5FitAo.mp4899yes https://puzzicle.org/science/ap-physics-c-mechanics/v/rotational-kinetic-energy-of-rigid-systemshttps://puzzicle.org/thumbs/x0ff40b625f3c3b0f.jpgRotational kinetic energy of rigid systemsThe rotational kinetic energy of a rigid system is related to the rotational inertia and angular velocity of the rigid system. The total kinetic energy of a rigid system is the sum of its rotational kinetic energy due to its rotation about its center of mass, and its translational kinetic energy due to the linear motion of its center of mass. Rotational kinetic energy is a scalar quantity.https://cdn.kastatic.org/ka-youtube-converted/pY1rwEsdCRo.mp4/pY1rwEsdCRo.mp4758yes https://puzzicle.org/science/ap-physics-c-mechanics/v/rolling-without-slipping-problemshttps://puzzicle.org/thumbs/xb0e01a66.jpgRolling without slippingLearn how to solve problems where an object rolls without slipping.https://cdn.kastatic.org/ka-youtube-converted/5eX5WnPDnvs.mp4/5eX5WnPDnvs.mp4900yes https://puzzicle.org/science/ap-college-physics-1/v/angular-momentum-of-rigid-systemshttps://puzzicle.org/thumbs/x5bbe90fc15a802c7.jpgAngular momentum of rigid systemsThe magnitude of the angular momentum of a rigid system about a specific axis is related to the rotational inertia and angular velocity of the rigid system. Angular impulse is defined as the product of the torque exerted on an object or rigid system and the time interval during which the torque is exerted. Angular impulse has the same direction as the torque imparting it. A rotational form of the impulse-momentum theorem relates the angular impulse delivered to a rigid system to the change in angular momentum of that rigid system.https://cdn.kastatic.org/ka-youtube-converted/noZr4jX8HIw.mp4/noZr4jX8HIw.mp4870yes https://puzzicle.org/science/ap-physics-c-mechanics/v/conservation-of-angular-momentum-aphttps://puzzicle.org/thumbs/x7fb54f2877e8c5fa.jpgConservation of angular momentumAngular momentum is conserved in all interactions. The angular impulse exerted by one rigid system on a second is equal and opposite to the angular impulse exerted by the second rigid system on the first. If the net external torque exerted on a selected system is nonzero, angular momentum is transferred between the system and the environment. The angular speed of a nonrigid system may change without the angular momentum of the system changing if the system changes shape by moving mass closer to or farther from the rotational axis.https://cdn.kastatic.org/ka-youtube-converted/-7trrETEE7k.mp4/-7trrETEE7k.mp4859yes https://puzzicle.org/science/ap-physics-c-mechanics/v/angular-momentum-of-satelliteshttps://puzzicle.org/thumbs/x921b86774920501e.jpgAngular momentum of satellitesPoint objects have linear momentum. The selection of the axis about which an object is considered to rotate influences the determination of the angular momentum of that object. The measured angular momentum of an object traveling in a straight line depends on the distance between the reference point and the object, the mass of the object, the speed of the object, and the angle between the radial distance and the velocity of the object. Angular momentum is constant for a satellite in a circular or elliptical orbit.https://cdn.kastatic.org/ka-youtube-converted/inus3WE5eTI.mp4/inus3WE5eTI.mp4931yes https://puzzicle.org/science/ap-physics-c-mechanics/v/energy-of-satellite-systemshttps://puzzicle.org/thumbs/xec735d999ce9cc24.jpgEnergy of satellite systemsIn circular and elliptical orbits, the satellite-central-object system’s total mechanical energy is constant. In elliptical orbits, the system’s gravitational potential energy and the satellite’s kinetic energy both change while the total remains constant. The total energy of a system consisting of a satellite orbiting a central object in a circular path can be written in terms of the gravitational potential energy of that system or the kinetic energy of the satellite. The escape velocity of a satellite is the satellite’s velocity such that the mechanical energy of the satellite-central-object system is equal to zero.https://cdn.kastatic.org/ka-youtube-converted/A-YeMNeq55E.mp4/A-YeMNeq55E.mp4945yes https://puzzicle.org/science/ap-physics-c-mechanics/v/shm-of-spring-mass-oscillatorshttps://puzzicle.org/thumbs/x71aeb67e4e299c68.jpgSHM of spring-mass oscillatorsSimple harmonic motion (SHM) is a special case of periodic motion which results when the magnitude of the restoring force exerted on an object is proportional to that object’s displacement from its equilibrium position. SHM is described by sinusoidal position, velocity, and acceleration functions. A spring-mass oscillator is one type of system that undergoes SHM. The period of a spring-mass oscillator is related to its mass and spring constant.https://cdn.kastatic.org/ka-youtube-converted/eRZSzGKDfwU.mp4/eRZSzGKDfwU.mp4913yes https://puzzicle.org/science/ap-physics-c-mechanics/v/energy-of-spring-mass-oscillatorshttps://puzzicle.org/thumbs/xb40960b7284455b1.jpgEnergy of spring-mass oscillatorsThe total energy of a system exhibiting simple harmonic motion (SHM) is the sum of the system’s kinetic and potential energies. The kinetic energy of a horizontal spring-mass system exhibiting SHM is at a maximum when the system’s potential energy is at a minimum, and vice versa. The minimum kinetic energy of a system exhibiting SHM is zero. The mechanical energy of an ideal spring-mass system is proportional to the amplitude of oscillation squared.https://cdn.kastatic.org/ka-youtube-converted/8dx1iJTZeGM.mp4/8dx1iJTZeGM.mp4895yes https://puzzicle.org/science/ap-physics-c-mechanics/v/shm-simple-pendulumshttps://puzzicle.org/thumbs/xe3e68eae0670555a.jpgSHM of simple pendulumsFor small amplitudes of motion, the small-angle approximation can be applied to the restoring torque exerted by gravity on a simple pendulum. Therefore, the motion of a pendulum with a small angular displacement can be modeled as simple harmonic motion because the restoring torque is proportional to the angular displacement. The period of a simple pendulum is related to its length and the magnitude of the gravitational field.https://cdn.kastatic.org/ka-youtube-converted/Dv5R543h3XM.mp4/Dv5R543h3XM.mp4734yes https://puzzicle.org/science/ap-college-physics-1/v/fluids-and-densityhttps://puzzicle.org/thumbs/xff493141dae7a7cc.jpgFluids and densityA fluid is a substance that has no fixed shape. Fluids include liquids and gases. Fluids can be characterized by their density, which is defined as the ratio of mass to volume. The ideal fluid model assumes a fluid is incompressible (same density everywhere) and has no viscosity (no resistance to flow).https://cdn.kastatic.org/ka-youtube-converted/AJxYCosjRH0.mp4/AJxYCosjRH0.mp4791yes https://puzzicle.org/science/ap-college-physics-1/v/fluid-pressurehttps://puzzicle.org/thumbs/x6a8251710da8a707.jpgFluid pressurePressure is the magnitude of the perpendicular force component exerted per unit area over a given surface area. Pressure is a scalar quantity. The pressure exerted by a fluid is the result of the interactions between the fluid’s particles and the surface with which those particles interact. The absolute pressure of a fluid at a given point is equal to the sum of a reference pressure, such as the atmospheric pressure, and the gauge pressure at that depth in the fluid. Pressure increases with depth in an incompressible fluid.https://cdn.kastatic.org/ka-youtube-converted/p_WLHzF1SdI.mp4/p_WLHzF1SdI.mp4948yes https://puzzicle.org/science/ap-college-physics-1/v/buoyant-forcehttps://puzzicle.org/thumbs/xec75b8131e0cf089.jpgBuoyant forceThe buoyant force is a net upward force exerted on an object by a fluid. The buoyant force results from the increase in fluid pressure with depth. The magnitude of the buoyant force exerted on an object by a fluid is equal to the weight of fluid displaced by the object (ρVg).https://cdn.kastatic.org/ka-youtube-converted/NYWG5eXRBFQ.mp4/NYWG5eXRBFQ.mp4761yes https://puzzicle.org/science/ap-college-physics-1/v/fluid-flowhttps://puzzicle.org/thumbs/xa0d04651dbce0c98.jpgFluid flowA difference in pressure between two locations causes a fluid to flow. Pressure differences can be created in various ways, such as removing air from one region (e.g., drinking through a straw) or maintaining a height of fluid (e.g., water towers). For an incompressible fluid, the rate at which matter enters a fluid-filled tube open at both ends must equal the rate at which fluid exits the tube. This conservation of mass is described by the continuity equation (A1v1 = A2v2).https://cdn.kastatic.org/ka-youtube-converted/ZBvQfi3P0l4.mp4/ZBvQfi3P0l4.mp4622yes https://puzzicle.org/science/ap-college-physics-1/v/ap-physics-1-review-of-1d-motionhttps://puzzicle.org/thumbs/x98abe201.jpgReview of 1D motionReview of 1D motionhttps://cdn.kastatic.org/ka-youtube-converted/NyPbcdE9v6o.mp4/NyPbcdE9v6o.mp4963yes https://puzzicle.org/science/ap-college-physics-1/v/ap-physics-1-review-of-2d-motion-and-vectorshttps://puzzicle.org/thumbs/xc28ac6fc.jpgReview of 2D motion and vectorsReview of 2D motion and vectorshttps://cdn.kastatic.org/ka-youtube-converted/YJEfSqlSxks.mp4/YJEfSqlSxks.mp4781yes https://puzzicle.org/science/ap-college-physics-1/v/ap-physics-1-review-of-forces-and-newtons-lawshttps://puzzicle.org/thumbs/x001423f1.jpgReview of forces and Newton's lawsReview of forces and Newton's lawshttps://cdn.kastatic.org/ka-youtube-converted/Bkl6Mn1Y23Q.mp4/Bkl6Mn1Y23Q.mp41036yes https://puzzicle.org/science/ap-college-physics-1/v/ap-physics-1-review-of-centripetal-forceshttps://puzzicle.org/thumbs/xe426ea78.jpgReview of circular motion and orbitsReview of circular motion and orbitshttps://cdn.kastatic.org/ka-youtube-converted/0r_JWUkXEkY.mp4/0r_JWUkXEkY.mp4954yes https://puzzicle.org/science/ap-college-physics-1/v/ap-physics-1-review-of-energy-and-workhttps://puzzicle.org/thumbs/xdfe60a80.jpgReview of energy and workReview of energy and workhttps://cdn.kastatic.org/ka-youtube-converted/65uG6v7YY2Q.mp4/65uG6v7YY2Q.mp4735yes https://puzzicle.org/science/ap-college-physics-1/v/ap-physics-1-review-of-momentum-and-impulsehttps://puzzicle.org/thumbs/xeab72a31.jpgReview of momentum and impulseReview of momentum and impulsehttps://cdn.kastatic.org/ka-youtube-converted/qMc6KOkmjTU.mp4/qMc6KOkmjTU.mp4801yes https://puzzicle.org/science/ap-college-physics-1/v/ap-physics-1-review-of-torque-and-angular-momentumhttps://puzzicle.org/thumbs/xf5c55bcd.jpgReview of rotationReview of rotationhttps://cdn.kastatic.org/ka-youtube-converted/uRA8vI2Gzvk.mp4/uRA8vI2Gzvk.mp41179yes https://puzzicle.org/science/ap-college-physics-1/v/ap-physics-1-review-of-waves-and-harmonic-motionhttps://puzzicle.org/thumbs/xc4addb94.jpgReview of oscillations and wavesReview of oscillations and waveshttps://cdn.kastatic.org/ka-youtube-converted/RA7x81NI8C4.mp4/RA7x81NI8C4.mp41195yes https://puzzicle.org/science/ap-physics-2/v/boltzmanns-constanthttps://puzzicle.org/thumbs/x9135ec72.jpgBoltzmann's constantExplore Ludwig Boltzmann's groundbreaking atomic theory and his constant's role in thermal physics. Understand the kinetic-molecular explanation of temperature, the ideal gas law, and the concept of entropy. Uncover the microscopic details of gas molecules and their impact on our understanding of heat energy.https://cdn.kastatic.org/ka-youtube-converted/rVD7HKafnnE.mp4/rVD7HKafnnE.mp4586yes https://puzzicle.org/science/hs-chemistry/v/thermal-energy-temperature-and-heathttps://puzzicle.org/thumbs/x842d6d2d2ab7398c.jpgThermal energy, temperature, and heatThermal energy refers to the kinetic energy of randomly moving particles in a substance. Particles can have translational, rotational, and/or vibrational kinetic energy, depending on the state of matter. Temperature is a measure of the average kinetic energy of the particles in a substance. Heat is thermal energy that transfers into or out of a system. Heat can transfer by conduction, convection, or radiation.https://cdn.kastatic.org/ka-youtube-converted/-7Gl-yKF6Y4.mp4/-7Gl-yKF6Y4.mp4692yes https://puzzicle.org/science/hs-chemistry/v/first-and-second-laws-of-thermodynamicshttps://puzzicle.org/thumbs/xfff5b5c700229413.jpgFirst and second laws of thermodynamicsThe first law of thermodynamics states that if a system gains/loses some amount of energy, the surroundings must lose/gain the same amount of energy. The second law of thermodynamics states that entropy does not spontaneously decrease. Entropy is a measure of how spread out energy is. As an example, if two systems at different temperatures are in contact, heat will flow from the warmer system to the cooler system until they reach thermal equilibrium (equal temperatures).https://cdn.kastatic.org/ka-youtube-converted/f1OokOgtcqg.mp4/f1OokOgtcqg.mp4740yes https://puzzicle.org/science/hs-chemistry/v/specific-heat-capacityhttps://puzzicle.org/thumbs/x42ab0ebb72ccdf09.jpgSpecific heat capacityIf the same amount of heat flows into (or out of) equal masses of different substances, their temperatures may not change by the same amount. Specific heat capacity (c) tells us how much heat energy is required to change one unit mass of a substance by one unit of temperature. Specific heat is related to heat (q), the mass of the sample (m), and the change in temperature (∆T) by the equation q=cm∆T.https://cdn.kastatic.org/ka-youtube-converted/GNelfJ6IAJw.mp4/GNelfJ6IAJw.mp4828yes https://puzzicle.org/science/ap-physics-2/v/first-law-of-thermodynamicshttps://puzzicle.org/thumbs/xcae8d5fc.jpgFirst law of thermodynamics equationEquation for the first law of thermodynamics.https://cdn.kastatic.org/ka-youtube-converted/u4-HYwo0Gsc.mp4/u4-HYwo0Gsc.mp4683yes https://puzzicle.org/science/ap-physics-2/v/first-law-of-thermodynamics-problem-solvinghttps://puzzicle.org/thumbs/x9552a749.jpgFirst law of thermodynamics problem solvingFirst law of thermodynamics problem solvinghttps://cdn.kastatic.org/ka-youtube-converted/6sP3kV-zgZk.mp4/6sP3kV-zgZk.mp4454yes https://puzzicle.org/science/ap-physics-2/v/pv-diagrams-part-1-work-and-isobaric-processeshttps://puzzicle.org/thumbs/xa78c7d17.jpgPV diagrams - part 1: Work and isobaric processesPV diagrams - part 1: Work and isobaric processeshttps://cdn.kastatic.org/ka-youtube-converted/Xcrco59p40o.mp4/Xcrco59p40o.mp4714yes https://puzzicle.org/science/ap-physics-2/v/pv-diagrams-part-2-isothermal-isometric-adiabatic-processeshttps://puzzicle.org/thumbs/x3fea420f.jpgPV diagrams - part 2: Isothermal, isometric, adiabatic processesThere are three types of thermodynamic processes you should know: isothermal, isometric, and adiabatic. An isothermal process is all about keeping the temperature constant, while an isometric process maintains a constant volume. The trickiest of the three is the adiabatic process, where no heat is exchanged and changes in volume can alter both pressure and temperature.https://cdn.kastatic.org/ka-youtube-converted/nkOtOMNS5bQ.mp4/nkOtOMNS5bQ.mp4780yes https://puzzicle.org/science/ap-physics-2/v/electric-charge-and-electric-forcehttps://puzzicle.org/thumbs/x0904666c9ca2fdbf.jpgElectric charge and electric forceElectric charge is a scalar quantity and is described as positive or negative. Coulomb’s law describes the electrostatic force between two charged objects as directly proportional to the magnitude of each of the charges and inversely proportional to the square of the distance between the objects. The direction of the electrostatic force depends on the signs of the charges of the interacting objects and is along the line of separation between the objects. Free space has a constant value of electric permittivity that appears in physical relationships.https://cdn.kastatic.org/ka-youtube-converted/lU8qZ0UyZUk.mp4/lU8qZ0UyZUk.mp4720yes https://puzzicle.org/science/ap-physics-2/v/conservation-of-electric-chargehttps://puzzicle.org/thumbs/xbfd65529fc7ca75a.jpgConservation of electric chargeThe net charge or charge distribution of a system can change in response to the presence of, or changes in, the net charge or charge distribution of other systems. Any change to a system’s net charge is due to a transfer of charge between the system and its surroundings. Grounding involves electrically connecting a charged object to a much larger and approximately neutral system (e.g., Earth).https://cdn.kastatic.org/ka-youtube-converted/1IXsgu44pMs.mp4/1IXsgu44pMs.mp4797yes https://puzzicle.org/science/ap-physics-2/v/electric-fields-part-1https://puzzicle.org/thumbs/xae2924f3faae10ab.jpgElectric fields (part 1)The electric field at a given point is the ratio of the electric force exerted on a test charge at the point to the charge of the test charge. An electric field points away from isolated positive charges and toward isolated negative charges. By superposition, the net electric field at a given location is the vector sum of individual electric fields created by nearby charged objects. The electric field is a vector quantity and can be represented in space using vector field maps.https://cdn.kastatic.org/ka-youtube-converted/D2Lh7Dlqn1o.mp4/D2Lh7Dlqn1o.mp4726yes https://puzzicle.org/science/ap-physics-2/v/electric-fields-part-2https://puzzicle.org/thumbs/x6535ea7a2300cda4.jpgElectric fields (part 2)While in electrostatic equilibrium, the excess charge of a conductor is distributed on the surface of the conductor, and the electric field within the conductor is zero. At the surface of a charged conductor, the electric field is perpendicular to the surface. While in electrostatic equilibrium, the excess charge of an insulator may be distributed throughout the interior of the insulator as well as at the surface, and the electric field within the insulator may have a nonzero value.https://cdn.kastatic.org/ka-youtube-converted/UypFZ2GNoKg.mp4/UypFZ2GNoKg.mp4593yes https://puzzicle.org/science/ap-physics-2/v/electric-potential-energy-aphttps://puzzicle.org/thumbs/xec63afba68616423.jpgElectric potential energyThe electric potential energy of a system of two point charges equals the amount of work required for an external force to bring the point charges to their current positions from infinitely far away. The total electric potential energy of a system can be determined by finding the sum of the electric potential energies of the individual interactions between each pair of charged objects in the system.https://cdn.kastatic.org/ka-youtube-converted/CNNLv7COpyY.mp4/CNNLv7COpyY.mp4671yes https://puzzicle.org/science/ap-physics-2/v/electric-potentialhttps://puzzicle.org/thumbs/xecdacbdec8ddd5c9.jpgElectric potentialElectric potential describes the electric potential energy per unit charge at a point in space. The electric potential due to multiple point charges can be determined by the principle of scalar superposition of the electric potential due to each of the point charges. The electric potential difference between two points is the change in electric potential energy per unit charge when a test charge is moved between the two points.https://cdn.kastatic.org/ka-youtube-converted/Fj_G5tN84Eg.mp4/Fj_G5tN84Eg.mp4740yes https://puzzicle.org/science/ap-physics-2/v/electric-potential-and-electric-fieldhttps://puzzicle.org/thumbs/x0744dcb081b12527.jpgElectric potential and electric fieldEquipotential lines represent lines of equal electric potential. These lines are also referred to as isolines of electric potential. Isolines are perpendicular to electric field vectors. An isoline map of electric potential can be constructed from an electric field vector map, and an electric field map may be constructed from an isoline map. An electric field vector points in the direction of decreasing potential. There is no component of an electric field along an isoline.https://cdn.kastatic.org/ka-youtube-converted/PrykTpvJDOM.mp4/PrykTpvJDOM.mp4878yes https://puzzicle.org/science/ap-physics-2/v/conservation-of-electric-energyhttps://puzzicle.org/thumbs/x5a0d89cd2eeacc9c.jpgConservation of electric energyWhen a charged particle moves between two locations with different electric potentials, the resulting change in the electric potential energy of the particle-field system can be found by multiplying the charge by the electric potential difference between the locations. Assuming no resistive forces, the movement of a charged particle between two points with different electric potentials results in a change in kinetic energy of the particle consistent with the conservation of energy. An electron volt (eV) is a unit of energy equal to the kinetic energy gained by an electron or proton when it accelerates through a potential difference of one volt under the electric force alone.https://cdn.kastatic.org/ka-youtube-converted/cPxnWRp4oPo.mp4/cPxnWRp4oPo.mp4825yes https://puzzicle.org/science/ap-physics-2/v/what-is-voltagehttps://puzzicle.org/thumbs/x294dfcab2adcb6fb.jpgVoltage (electric potential difference)The electric potential at a point represents the electric potential energy per unit charge that a particle would have at that location. The difference in electric potential between two points is called voltage. It's measured in volts (V), equivalent to joules per coulomb (J/C). When comparing two points, the point with the higher electric potential is labelled positive (+), and the point with the lower electric potential is negative (-). If two points are at the same electric potential, the voltage between them is zero.https://cdn.kastatic.org/ka-youtube-converted/NT6LYNq3jRE.mp4/NT6LYNq3jRE.mp4767yes https://puzzicle.org/science/ap-physics-2/v/electric-currenthttps://puzzicle.org/thumbs/x14a4196184e1f05d.jpgElectric currentElectric current is the net rate that charge flows through a cross-sectional area. Current is measured in amperes (A), equivalent to coulombs per second (C/s). Current flows when there is an electric potential difference (voltage) between two points, and there are charge carriers that are free to move between those points. Electrical conductors (metals) contain electrons that are free to move. Electrons are the drifting particles in a current-carrying wire, but conventional current is defined as the direction that positive charge would flow.https://cdn.kastatic.org/ka-youtube-converted/6K1Bl0cqecE.mp4/6K1Bl0cqecE.mp4809yes https://puzzicle.org/science/ap-physics-2/v/electric-circuits-part-1https://puzzicle.org/thumbs/x1f725db40c87ea1f.jpgIntro to electric circuitsElectric circuits are paths through which electric current can flow. Circuits are composed of specific building blocks known as circuit elements. Circuit elements include batteries and other voltage sources, wires, resistors, light bulbs and LEDs, and switches. Ammeters and voltmeters can be used to measure current and voltage in a circuit, respectively. An open circuit contains a broken path, preventing current flow. A short circuit contains a path with negligible resistance, resulting in very high current.https://cdn.kastatic.org/ka-youtube-converted/Nbb5Lw14wvo.mp4/Nbb5Lw14wvo.mp4718yes https://puzzicle.org/science/ap-physics-2/v/electric-circuits-part-2https://puzzicle.org/thumbs/xfca4d22d4f21f414.jpgSeries and parallel circuitsCircuit elements can be connected in series or in parallel. A series connection exists when elements are joined along a single chain. The same current flows through every element in the chain. A parallel connection exists when elements are connected across the same two points in a circuit, with no other components between them. The same voltage exists across each element in a parallel connection. Real-world circuits often utilize a combination of both series and parallel connections.https://cdn.kastatic.org/ka-youtube-converted/1ASc6piVKhQ.mp4/1ASc6piVKhQ.mp4852yes https://puzzicle.org/science/ap-physics-2/v/ohms-lawhttps://puzzicle.org/thumbs/x40f69842b7189e6e.jpgOhm's lawThrough experiment, we can uncover the relationship between the current through a circuit element and the voltage across it. Measurements show that the current is equal to the voltage divided by the element’s resistance (I=ΔV/R). This relationship is called Ohm’s law. Since the current is proportional to the voltage, a graph of current through an element versus the voltage across it is a straight line. The slope of the line is the element’s conductance, which is the reciprocal of resistance. Though many circuit elements obey Ohm’s law, some do not. Such components are called non-ohmic.https://cdn.kastatic.org/ka-youtube-converted/FuIn1L3qeqI.mp4/FuIn1L3qeqI.mp4738yes https://puzzicle.org/science/ap-physics-2/v/resistivity-and-conductivityhttps://puzzicle.org/thumbs/x2d278559.jpgResistivity and conductivityLet's look at the properties of materials that cause a resistor to resist the flow of current. Explore the principles of Ohm's law, resistance, and resistivity in electrical circuits. Understand how resistance is defined by voltage and current, and how it remains constant in Ohmic materials. Discover the factors that influence resistance, including the length, cross-sectional area, and material of a resistor.https://cdn.kastatic.org/ka-youtube-converted/4rsswT_Rv1M.mp4/4rsswT_Rv1M.mp4722yes https://puzzicle.org/science/ap-physics-2/v/ee-kirchhoffs-current-lawhttps://puzzicle.org/thumbs/x5d40a1b6.jpgKirchhoff's current lawKirchhoff's Current Law says the currents flowing into a node must add up to zero.https://cdn.kastatic.org/ka-youtube-converted/OYerdzZPSI0.mp4/OYerdzZPSI0.mp4363yes https://puzzicle.org/science/ap-physics-2/v/ee-kirchhoffs-voltage-lawhttps://puzzicle.org/thumbs/x2eab8586.jpgKirchhoff's voltage lawKirchhoff's Voltage Law says if you travel around any loop in a circuit, the voltages across the elements add up to zero.https://cdn.kastatic.org/ka-youtube-converted/Bt6V7D5av9A.mp4/Bt6V7D5av9A.mp4410yes https://puzzicle.org/science/ap-physics-2/v/electric-powerhttps://puzzicle.org/thumbs/x93fcbc34.jpgElectric powerIn this video David derives the formula to find the power used by a resistor.https://cdn.kastatic.org/ka-youtube-converted/3zt7DWUiaDE.mp4/3zt7DWUiaDE.mp4643yes https://puzzicle.org/science/ap-physics-2/v/ee-series-resistorshttps://puzzicle.org/thumbs/xa405e84f.jpgSeries resistorsResistors connected head-to-tail are in series. The equivalent overall resistance is the sum of the individual resistance values.https://cdn.kastatic.org/ka-youtube-converted/P_SELh0dPW0.mp4/P_SELh0dPW0.mp4717yes https://puzzicle.org/science/ap-physics-2/v/ee-parallel-resistors-3https://puzzicle.org/thumbs/xa85f5f14.jpgParallel resistorsPractice calculating equivalent resistance for resistors in parallel. In the special case where two equal resistors are in parallel, the equivalent resistance is half their value. Parallel resistance is always smaller than the smallest resistor, as it provides multiple current paths, effectively reducing the overall resistance in the circuit.https://cdn.kastatic.org/ka-youtube-converted/S0AJZjt0sEg.mp4/S0AJZjt0sEg.mp4561yes https://puzzicle.org/science/ap-physics-2/v/ee-simplifying-resistor-networkshttps://puzzicle.org/thumbs/x618abc88.jpgSimplifying resistor networksLearn strategies for combining resistors in series and parallel to simplify resistor networks.https://cdn.kastatic.org/ka-youtube-converted/tnYQ_tQmYrM.mp4/tnYQ_tQmYrM.mp4486yes https://puzzicle.org/science/ap-physics-2/v/solved-example-finding-current-voltage-in-a-circuithttps://puzzicle.org/thumbs/x29f42217e680a0a0.jpgSolved example: Finding current and voltage in a circuitLet's learn how to calculate current and voltage across each resistor in a circuit.https://cdn.kastatic.org/ka-youtube-converted/rkiBof2pvbE.mp4/rkiBof2pvbE.mp4805yes https://puzzicle.org/science/ap-physics-2/v/voltmeters-and-ammetershttps://puzzicle.org/thumbs/xa8591b23.jpgVoltmeters and ammetersLearn how to measure voltage and current in electrical circuits using voltmeters and ammeters. Discover why voltmeters are connected in parallel and ammeters in series. Understand the importance of resistance in these devices to ensure accurate readings and prevent damage.https://cdn.kastatic.org/ka-youtube-converted/yE3eQ6q39f4.mp4/yE3eQ6q39f4.mp4498yes https://puzzicle.org/science/ap-physics-2/v/ap-physics-1-review-of-charge-and-circuitshttps://puzzicle.org/thumbs/x83413413.jpgReview of electric charge and circuitsReview of electric charge and circuitshttps://cdn.kastatic.org/ka-youtube-converted/qdmYAXjhWiw.mp4/qdmYAXjhWiw.mp41152yes https://puzzicle.org/science/ap-physics-2/v/capacitors-and-capacitancehttps://puzzicle.org/thumbs/xc64de00d.jpgCapacitors and capacitanceCapacitors, essential components in electronics, store charge between two pieces of metal separated by an insulator. This video explains how capacitors work, the concept of capacitance, and how varying physical characteristics can alter a capacitor's ability to store chargeBy David Santo Pietro.https://cdn.kastatic.org/ka-youtube-converted/u-jigaMJT10.mp4/u-jigaMJT10.mp4343yes https://puzzicle.org/science/ap-physics-2/v/dielectrics-capacitorshttps://puzzicle.org/thumbs/xb0101987.jpgDielectrics in capacitorsCapacitors use non-conducting materials, or dielectrics, to store charge and increase capacitance. Dielectrics, when placed between charged capacitor plates, become polarized, reducing the voltage across the plates and increasing capacitance. The degree of capacitance increase depends on the dielectric constant of the material used. By David Santo Pietro.https://cdn.kastatic.org/ka-youtube-converted/rkntp3_cZl4.mp4/rkntp3_cZl4.mp4387yes https://puzzicle.org/science/ap-physics-2/v/energy-capacitorhttps://puzzicle.org/thumbs/x28f1d646.jpgEnergy of a capacitorCapacitors store energy as electrical potential. When charged, a capacitor's energy is 1/2 Q times V, not Q times V, because charges drop through less voltage over time. The energy can also be expressed as 1/2 times capacitance times voltage squared. Remember, the voltage refers to the voltage across the capacitor, not necessarily the battery voltage. By David Santo Pietro.https://cdn.kastatic.org/ka-youtube-converted/SIU_9SMd5q0.mp4/SIU_9SMd5q0.mp4383yes https://puzzicle.org/science/ap-physics-2/v/capacitors-serieshttps://puzzicle.org/thumbs/x276a1169.jpgCapacitors in seriesWhen capacitors are connected one after another, they are said to be in series. For capacitors in series, the total capacitance can be found by adding the reciprocals of the individual capacitances, and taking the reciprocal of the sum. Therefore, the total capacitance will be lower than the capacitance of any single capacitor in the circuit.https://cdn.kastatic.org/ka-youtube-converted/-MaD9Ycy3a4.mp4/-MaD9Ycy3a4.mp4575yes https://puzzicle.org/science/ap-physics-2/v/capacitors-parallelhttps://puzzicle.org/thumbs/x9e833752.jpgCapacitors in parallelA capacitator is a device that stores electrical energy in an electrical field. This video discusses the behavior of two capacitors connected in parallel. It compares two capacitators, and shows how to calculate the amount of charge each will receive. Finally, it discusses how to find the equivalent capacitance of the two capacitors combined.https://cdn.kastatic.org/ka-youtube-converted/P-1mWGeJjmg.mp4/P-1mWGeJjmg.mp4601yes https://puzzicle.org/science/ap-physics-2/v/magnetic-field-lines-their-propertieshttps://puzzicle.org/thumbs/x55e0ac08db74e462.jpgMagnetic field linesLet's explore what magnetic field lines are, why we draw them, and what their properties are. Here is the link to the simulation https://www.khanacademy.org/science/physics/discoveries/magnetic-fields/pi/tracing-a-magnetic-fieldhttps://cdn.kastatic.org/ka-youtube-converted/GgRCkduKCLE.mp4/GgRCkduKCLE.mp4656yes https://puzzicle.org/science/ap-physics-2/v/comparing-properties-of-magnetic-lorentz-electric-forceshttps://puzzicle.org/thumbs/x3b5be1f27ab788be.jpgComparing magnetic and electric forcesLet's compare and contrast properties of the magnetic force and the electric force acting on a charged particle.https://cdn.kastatic.org/ka-youtube-converted/9C8mRW43X0Q.mp4/9C8mRW43X0Q.mp4757yes https://puzzicle.org/science/ap-physics-2/v/oersteds-experiment-magnetic-field-due-to-currenthttps://puzzicle.org/thumbs/x741a474aea85f590.jpgMagnetic field due to currentElectric current produces a magnetic field around a wire, demonstrating that electricity and magnetism are fundamentally connected. The magnetic field becomes stronger with greater current and weaker with increasing distance from the wire. Around a straight current-carrying wire, magnetic field lines form concentric circles, and reversing the current reverses the field's direction.https://cdn.kastatic.org/ka-youtube-converted/V-irinjvhmc.mp4/V-irinjvhmc.mp4546yes https://puzzicle.org/science/ap-physics-2/v/right-hand-thumb-rule-solved-numericalhttps://puzzicle.org/thumbs/xe2ed3365612da2be.jpgRight hand thumb ruleLearn how to use the right-hand thumb rule to find the direction of the magnetic field around a current-carrying wire.https://cdn.kastatic.org/ka-youtube-converted/NLieiomI4sA.mp4/NLieiomI4sA.mp4652yes https://puzzicle.org/science/ap-physics-2/v/force-on-a-current-carrying-conductor-ni-a-magnetic-fieldhttps://puzzicle.org/thumbs/x1136bc2bae900274.jpgForce on a current-carrying conductor in a magnetic fieldCurrent carrying wires can push magnets. Can magnets push current carrying wires too? Let's find out.https://cdn.kastatic.org/ka-youtube-converted/uW8scRjaam8.mp4/uW8scRjaam8.mp4560yes https://puzzicle.org/science/ap-physics-2/v/electromagnetic-induction-faradays-experimentshttps://puzzicle.org/thumbs/xc34c4d81ecbdb697.jpgElectromagnetic inductionElectromagnetic induction is the production of an induced voltage or current by a changing magnetic field through a coil. A stationary magnetic field produces no current; only changes in magnetic field induce electricity. Relative motion between a magnet and coil or changing current in a nearby coil creates induction, forming the operating principle of electric generators.https://cdn.kastatic.org/ka-youtube-converted/yU--8Zk57-Y.mp4/yU--8Zk57-Y.mp4612yes https://puzzicle.org/science/ap-physics-2/v/faraday-s-law-n-thttps://puzzicle.org/thumbs/x9363bd5a00eb89d7.jpgFaraday's LawWe know that changing magnetic flux induces emf. In this video, let's figure out the magnitude of that induced emf.https://cdn.kastatic.org/ka-youtube-converted/MHRBjGCgb4U.mp4/MHRBjGCgb4U.mp4815yes https://puzzicle.org/science/ap-physics-2/v/lenzs-lawhttps://puzzicle.org/thumbs/xcc21a8f2.jpgLenz's lawWhen the magnetic flux through a conducting loop changes, current is induced in the loop. This current produces it's own magnetic field. Lenz's law tells us that the current flows in the direction that produces a magnetic field in the loop that opposes the change in magnetic flux.https://cdn.kastatic.org/ka-youtube-converted/xxZenoBs2Pg.mp4/xxZenoBs2Pg.mp4346yes https://puzzicle.org/science/ap-physics-2/v/emf-induced-in-rod-traveling-through-magnetic-fieldhttps://puzzicle.org/thumbs/xa173a477.jpgEmf induced in rod traveling through magnetic fieldAn emf induced by motion relative to a magnetic field is called a motional emf. This is represented by the equation emf = LvB, where L is length of the object moving at speed v relative to the strength of the magnetic field B.https://cdn.kastatic.org/ka-youtube-converted/rqq11Id_x5E.mp4/rqq11Id_x5E.mp4482yes https://puzzicle.org/science/ap-physics-2/v/paramagnetism-diamagnetismhttps://puzzicle.org/thumbs/xa8ebae998095d899.jpgParamagnetism & diamagnetismMost materials are either weakly attracted (paramagnets) or weakly repelled (diamagnets) by magnets. Diamagnetism is an inherent property of all materials, and it arises from Lenz's laws. Paramagnetism is seen in materials whose atoms have at least one unpaired electron.https://cdn.kastatic.org/ka-youtube-converted/wK7Jr1g4_ws.mp4/wK7Jr1g4_ws.mp4771yes https://puzzicle.org/science/ap-physics-2/v/ferromagnetism-curie-temperaturehttps://puzzicle.org/thumbs/x4bfc5beab8041c86.jpgFerromagnetism & curie temperatureFerromagnets are strongly attracted to magnets due to their magnetic domains. Magnetic domains are groups of atoms whose dipole moments are spontaneously aligned in the same direction. Above the Curie temperature, these magnetic domains break down and the material becomes a paramagnet.https://cdn.kastatic.org/ka-youtube-converted/8a2IZZdIeXw.mp4/8a2IZZdIeXw.mp4563yes https://puzzicle.org/science/ap-physics-2/v/specular-and-diffuse-reflectionhttps://puzzicle.org/thumbs/8187516.jpgSpecular and diffuse reflectionSpecular reflection, which occurs with smooth surfaces like mirrors, causes light rays to reflect at the same angle as they hit the surface. Diffuse reflection, which occurs with rougher surfaces, scatters light rays in different directions. The video also introduces double reflection, which occurs when light reflects off of one surface and then reflects off of another surface before reaching your eye.https://cdn.kastatic.org/ka-youtube-converted/sd0BOnN6aNY.mp4/sd0BOnN6aNY.mp4660yes https://puzzicle.org/science/ap-physics-2/v/virtual-imagehttps://puzzicle.org/thumbs/8187523.jpgVirtual imageLet's explore the concept of a virtual image.https://cdn.kastatic.org/ka-youtube-converted/nrOg85VPQgw.mp4/nrOg85VPQgw.mp4473yes https://puzzicle.org/science/ap-physics-2/v/concave-mirrorshttps://puzzicle.org/thumbs/x9c7bc095abef3bef.jpgConcave mirrorsIn this video, we will see how concave mirrors focus a parallel beam of light to a single point.https://cdn.kastatic.org/ka-youtube-converted/7p0_YK7hQ04.mp4/7p0_YK7hQ04.mp4756yes https://puzzicle.org/science/ap-physics-2/v/concave-mirror-applicationshttps://puzzicle.org/thumbs/xbd7e72589050a2d7.jpgConcave mirror applicationsIn this video, we will look at some applications of concave mirrors.https://cdn.kastatic.org/ka-youtube-converted/xhfKsl1WcO4.mp4/xhfKsl1WcO4.mp4517yes https://puzzicle.org/science/ap-physics-2/v/convex-mirror-applicationshttps://puzzicle.org/thumbs/xbef05574b73d9b97.jpgConvex mirror & applicationsWhen curved mirrors have a reflecting surface on the outside, they are called convex mirrors. (vex stands for bulging out). Let's learn what they do and their applications in our day to day life.https://cdn.kastatic.org/ka-youtube-converted/4sZZYkIwgqg.mp4/4sZZYkIwgqg.mp4774yes https://puzzicle.org/science/ap-physics-2/v/spherical-parabolic-mirrorshttps://puzzicle.org/thumbs/xe17bbcf06d5b73b7.jpgSpherical & parabolic mirrorsIn this video, we will explore two kinds of curved mirrors, parabolic and spherical mirrors.https://cdn.kastatic.org/ka-youtube-converted/djkBahaEHG4.mp4/djkBahaEHG4.mp4331yes https://puzzicle.org/science/ap-physics-2/v/spherical-mirrors-radius-of-curvature-focal-lengthhttps://puzzicle.org/thumbs/x26476658aa67c606.jpgSpherical mirrors, radius of curvature & focal lengthIn this video, we will introduce the terms radius of curvature, and see its connection with focal length. We will also introduce the term principal axis and see the need for it.https://cdn.kastatic.org/ka-youtube-converted/H1eRmNbTkm0.mp4/H1eRmNbTkm0.mp4536yes https://puzzicle.org/science/ap-physics-2/v/convex-concave-mirror-ray-diagrams-class-12-india-physics-khan-academyhttps://puzzicle.org/thumbs/x2ba94ea0892e0960.jpgConvex & concave mirror ray diagramsLet's explore the ray tracing technique to figure out the properties of images when things are kept in front of a concave or a convex mirror.https://cdn.kastatic.org/ka-youtube-converted/OHXOwz1NLh0.mp4/OHXOwz1NLh0.mp4803yes https://puzzicle.org/science/ap-physics-2/v/sign-conventions-for-mirror-lenseshttps://puzzicle.org/thumbs/xc1fbec526643e2ca.jpgSign convention for mirrors (& lenses)Let's explore the sign conventions (rules to assign +ve and -ve values) we use for mirrors and lenses. These rules are called Cartesian sign conventions.https://cdn.kastatic.org/ka-youtube-converted/Mc--Yz_YrsU.mp4/Mc--Yz_YrsU.mp4610yes https://puzzicle.org/science/ap-physics-2/v/mirror-formulahttps://puzzicle.org/thumbs/x4c269e893fe2032c.jpgMirror formulaLet's explore the mirror formula (1/f = 1/v+1/u) and see how to locate images without drawing any ray diagrams.https://cdn.kastatic.org/ka-youtube-converted/EPlr16b4XnQ.mp4/EPlr16b4XnQ.mp4533yes https://puzzicle.org/science/ap-physics-2/v/magnification-formula-for-mirrorshttps://puzzicle.org/thumbs/x2e50ac540380247c.jpgMagnification formula for mirrorsLet's explore how to calculate image height and it's nature (real or virtual) using magnification formula.https://cdn.kastatic.org/ka-youtube-converted/FfvuY8xsO1Y.mp4/FfvuY8xsO1Y.mp4613yes https://puzzicle.org/science/ap-physics-2/v/refraction-and-snell-s-lawhttps://puzzicle.org/thumbs/8187518.jpgRefraction and Snell's lawRefraction and Snell's Lawhttps://cdn.kastatic.org/ka-youtube-converted/y55tzg_jW9I.mp4/y55tzg_jW9I.mp4864yes https://puzzicle.org/science/ap-physics-2/v/refraction-through-glass-slabhttps://puzzicle.org/thumbs/x3ff4ed3d372136d3.jpgRefraction through glass slabLet's explore refraction of light through any parallel sided medium. Will also see why things appear to shift (lateral shift) when looked through such a medium.https://cdn.kastatic.org/ka-youtube-converted/YIYnSAb9Q9Y.mp4/YIYnSAb9Q9Y.mp4666yes https://puzzicle.org/science/ap-physics-2/v/snell-s-law-examples-1https://puzzicle.org/thumbs/8187520.jpgSnell's law exampleExample calculation using Snell's lawhttps://cdn.kastatic.org/ka-youtube-converted/HahjsBApxLE.mp4/HahjsBApxLE.mp4644yes https://puzzicle.org/science/ap-physics-2/v/total-internal-reflectionhttps://puzzicle.org/thumbs/8187522.jpgTotal internal reflectionCritical incident angle and total internal reflectionhttps://cdn.kastatic.org/ka-youtube-converted/WRuatAcd2WY.mp4/WRuatAcd2WY.mp4475yes https://puzzicle.org/science/ap-physics-2/v/convex-lenseshttps://puzzicle.org/thumbs/8187527.jpgConvex lensesExplore convex lenses, including how they refract and transmit light. Learn about a lens's focal point and the thin lens assumption.https://cdn.kastatic.org/ka-youtube-converted/K0sjZ5nqQ7g.mp4/K0sjZ5nqQ7g.mp4563yes https://puzzicle.org/science/ap-physics-2/v/convex-lens-exampleshttps://puzzicle.org/thumbs/8187528.jpgConvex lens examplesExplore how convex lenses form images, including different scenarios where an object is placed at various distances from the lens. Learn how light rays refract through a convex lens, forming real, inverted, or virtual images. This video emphasizes understanding the process rather than memorizing outcomes.https://cdn.kastatic.org/ka-youtube-converted/xxF_mdoZom0.mp4/xxF_mdoZom0.mp4576yes https://puzzicle.org/science/ap-physics-2/v/concave-lenseshttps://puzzicle.org/thumbs/8187529.jpgConcave lensesConcave lenses, with their inward curve, create divergent light rays. When a ray parallel to the principal axis hits the lens, it refracts outward as if coming from the focal point. Another ray, passing through the lens center, doesn't refract. These rays form a virtual, upright, and smaller image, appearing closer to the lens than the actual object.https://cdn.kastatic.org/ka-youtube-converted/Vh70PyitQzo.mp4/Vh70PyitQzo.mp4186yes https://puzzicle.org/science/ap-physics-2/v/lens-formulahttps://puzzicle.org/thumbs/x33600821156ee2df.jpgLens formulaLet's see how to use lens formula (1/v-1/u= 1/f) to locate images without having to draw ray diagrams.https://cdn.kastatic.org/ka-youtube-converted/NQ8OLvIcffg.mp4/NQ8OLvIcffg.mp4511yes https://puzzicle.org/science/ap-physics-2/v/solved-example-on-lens-formulahttps://puzzicle.org/thumbs/xe3e86662d3ba7542.jpgSolved example on lens formulaUsing the lens formula can be tricky because we need to substitute using signs. In this video, we will take one example and discuss how to use the lens formula.https://cdn.kastatic.org/ka-youtube-converted/HOaI6_F4ht8.mp4/HOaI6_F4ht8.mp4786yes https://puzzicle.org/science/ap-physics-2/v/magnification-formula-for-lenseshttps://puzzicle.org/thumbs/x56b2ca797ceaf438.jpgMagnification formula for lensesLet's explore the magnification formula (M= v/u) for lenses and see how to find the image height and its nature (whether it's real or virtual).https://cdn.kastatic.org/ka-youtube-converted/if-X8MCltso.mp4/if-X8MCltso.mp4596yes https://puzzicle.org/science/physics/v/introduction-to-waveshttps://puzzicle.org/thumbs/25743.jpgIntroduction to wavesTransverse and longitudinal waves are two types of mechanical waves, which involve the transfer of energy through a medium (e.g. water, air, a solid). Learn about transverse and longitudinal waves through the examples of a rope and a sound wave. Explore the difference between a single wave pulse and periodic waves.https://cdn.kastatic.org/ka-youtube-converted/c38H6UKt3_I.mp4/c38H6UKt3_I.mp4783yes https://puzzicle.org/science/physics/v/amplitude-period-frequency-and-wavelength-of-periodic-waveshttps://puzzicle.org/thumbs/25745.jpgProperties of periodic wavesLearn about different properties of waves, including amplitude, period, frequency, and wavelength. Explore how these properties are related to one another.https://cdn.kastatic.org/ka-youtube-converted/tJW_a6JeXD8.mp4/tJW_a6JeXD8.mp4866yes https://puzzicle.org/science/physics/v/wave-equationhttps://puzzicle.org/thumbs/xbb44b237.jpgThe equation of a waveIn this video David shows how to determine the equation of a wave, how that equation works, and what the equation represents.https://cdn.kastatic.org/ka-youtube-converted/9WZM68aVnGk.mp4/9WZM68aVnGk.mp4884yes https://puzzicle.org/science/physics/v/standing-waves-on-stringshttps://puzzicle.org/thumbs/x7e8ce62f.jpgStanding waves on stringsA standing wave occurs when two waves of the same frequency and amplitude are moving in opposite directions and interfere with each other. It has certain points (called nodes) where the amplitude is always zero, and other points (called antinodes) where the amplitude fluctuates with maximum intensity. Use the distance between two consecutive nodes or two consecutive antinodes to calculate the length of a standing wave.https://cdn.kastatic.org/ka-youtube-converted/gT0IqL1dyyk.mp4/gT0IqL1dyyk.mp4805yes https://puzzicle.org/science/physics/v/production-of-soundhttps://puzzicle.org/thumbs/xa4f83089.jpgProduction of soundSound is vibrating air. But how does the air start vibrating? Explore the intriguing science of sound waves and how they travel. Learn how speakers use oscillation to create sound, and discover why sound waves can transport energy without moving the medium itself. Uncover the mystery behind the difference in our voices in person and on recordings.https://cdn.kastatic.org/ka-youtube-converted/nGKffdaI4Pg.mp4/nGKffdaI4Pg.mp4226yes https://puzzicle.org/science/physics/v/sound-properties-amplitude-period-frequency-wavelengthhttps://puzzicle.org/thumbs/x3dd64aec.jpgSound properties: amplitude, period, frequency, wavelengthHow to find the amplitude, period, frequency, and wavelength for a sound wavehttps://cdn.kastatic.org/ka-youtube-converted/-_xZZt99MzY.mp4/-_xZZt99MzY.mp4316yes https://puzzicle.org/science/physics/v/speed-of-soundhttps://puzzicle.org/thumbs/x089b4e4b.jpgSpeed of soundHow fast does sound travel? Explore the nature of sound waves and how they travel. Discover why sound is a longitudinal wave and how its speed is influenced by the medium it travels through. Learn about the relationship between wavelength, frequency, and speed, and how changing the medium affects the speed of sound.https://cdn.kastatic.org/ka-youtube-converted/UgE2GIQwUCw.mp4/UgE2GIQwUCw.mp4225yes https://puzzicle.org/science/physics/v/relative-speed-of-sound-in-solids-liquids-and-gaseshttps://puzzicle.org/thumbs/x96f7dd45.jpgRelative speed of sound in solids, liquids, and gasesSounds travels at different speeds in different mediahttps://cdn.kastatic.org/ka-youtube-converted/yF4cvbAYjwI.mp4/yF4cvbAYjwI.mp4199yes https://puzzicle.org/science/physics/v/standing-waves-in-tubes-part-1https://puzzicle.org/thumbs/x1cd52325.jpgStanding waves in open tubesFind out how ​standing waves and resonance cause a flute makes such specific notes. Discover why blowing over a soda bottle creates a tone, how air molecules oscillate in an open-ended tube, and the mathematical representation of these phenomena.https://cdn.kastatic.org/ka-youtube-converted/BhQUW9s-R8M.mp4/BhQUW9s-R8M.mp4859yes https://puzzicle.org/science/physics/v/standing-waves-in-tubes-part-2https://puzzicle.org/thumbs/x255e2941.jpgStanding waves in closed tubesExplore the intriguing world of sound waves in open and closed tubes! This lesson breaks down how the length of a tube influences the wavelength and frequency of sound, revealing why you hear different notes as you sip your soda. Discover the unique patterns of harmonics in open-closed tubes, and how they differ from open-open tubes.https://cdn.kastatic.org/ka-youtube-converted/1S4DtuMY88I.mp4/1S4DtuMY88I.mp4597yes https://puzzicle.org/science/physics/v/beat-frequencyhttps://puzzicle.org/thumbs/xca74c9ea.jpgBeat frequencyBeat frequency occurs when two waves with different frequencies overlap, causing a cycle of alternating constructive and destructive interference between waves. Learn how this results in a fluctuation in sound loudness, and how the beat frequency can be calculated by finding the difference between the two original frequencies.https://cdn.kastatic.org/ka-youtube-converted/Ca91iOVGd9A.mp4/Ca91iOVGd9A.mp4708yes https://puzzicle.org/science/ap-physics-2/v/electromagnetic-waves-and-the-electromagnetic-spectrumhttps://puzzicle.org/thumbs/x93657b33.jpgElectromagnetic waves and the electromagnetic spectrumLearn how electromagnetic waves are formed from changing electric and magnetic fields. Understand the electromagnetic spectrum, including different regions from visible light to gamma rays and their uses.https://cdn.kastatic.org/ka-youtube-converted/7eutept5h0Q.mp4/7eutept5h0Q.mp4662yes https://puzzicle.org/science/ap-physics-2/v/polarization-of-light-linear-and-circularhttps://puzzicle.org/thumbs/x38701d13.jpgPolarization of lightLight, as electromagnetic waves, can oscillate in specific directions. This phenomenon is called polarization. Explore into the difference between polarized and non-polarized light, and see how polarization is important for everyday applications like sunglasses and 3D movies.https://cdn.kastatic.org/ka-youtube-converted/HH58VmUbOKM.mp4/HH58VmUbOKM.mp4870yes https://puzzicle.org/science/physics/v/constructive-and-destructive-interferencehttps://puzzicle.org/thumbs/x00cc1549.jpgConstructive and destructive interferenceConstructive interference happens when two waves overlap in such a way that they combine to create a larger wave. Destructive interference happens when two waves overlap in such a way that they cancel each other out. Wave interference also depends on the relative phase of the two waves, as this video shows through the examples of path length differences and pi shifts.https://cdn.kastatic.org/ka-youtube-converted/oTjTXS40pqs.mp4/oTjTXS40pqs.mp4824yes https://puzzicle.org/science/ap-physics-2/v/youngs-double-split-part-1https://puzzicle.org/thumbs/x7aa60791.jpgYoung's double slit introductionWe can see interference in action if we shine laser light through two slits onto a screen. Explore Young's Double Slit experiment, a cornerstone in understanding light as a wave. Discover how light waves spread out, overlap, and create patterns of constructive and destructive interference. Uncover the rules of wave interference in two dimensions, and how path length differences lead to these intriguing patterns.https://cdn.kastatic.org/ka-youtube-converted/Pk6s2OlKzKQ.mp4/Pk6s2OlKzKQ.mp4573yes https://puzzicle.org/science/ap-physics-2/v/youngs-double-slit-part-2https://puzzicle.org/thumbs/xd2a6554e.jpgYoung's double slit equationLet's derive a formula that relates all the variables in Young's double slit experiment as we explore Young's Double Slit Equation. Uncover how light's path length difference relates to the angle of incidence, and how this leads to constructive and destructive interference. Apply this knowledge to the world of wave patterns, trigonometry, and the wonders of diffraction!https://cdn.kastatic.org/ka-youtube-converted/1abpdO27KTo.mp4/1abpdO27KTo.mp4401yes https://puzzicle.org/science/ap-physics-2/v/youngs-double-slit-problem-solvinghttps://puzzicle.org/thumbs/xb810086a.jpgYoung's double slit problem solvingContinue to explore Young's double slit equation through a sample problem involving a double slit experiment. See how light waves of specific wavelengths interact when they pass through two slits, creating a pattern of bright spots on a screen. Get insights into the math behind calculating these distances and understand the impact of changing slit spacing.https://cdn.kastatic.org/ka-youtube-converted/5-2RZsU8e3U.mp4/5-2RZsU8e3U.mp4429yes https://puzzicle.org/science/ap-physics-2/v/diffraction-gratinghttps://puzzicle.org/thumbs/x917c7a7b.jpgDiffraction gratingWhat happens when there's way more than two holes? Uncover the power of diffraction gratings. Learn how adding more slits to a double slit experiment results in sharper, brighter interference patterns. Understand why this occurs and its significance in measuring light waves.https://cdn.kastatic.org/ka-youtube-converted/F6dZjuw1KUo.mp4/F6dZjuw1KUo.mp4861yes https://puzzicle.org/science/ap-physics-2/v/single-slit-interferencehttps://puzzicle.org/thumbs/xdc1d3db2.jpgSingle slit interferenceWhat happens when there's only one hole? Explore the concept of Single Slit Interference in light waves. Uncover how waves spread out at a hole due to Huygen's Principle, leading to diffraction. Understand how each point on a wavefront acts as a source of another wave, creating an interference pattern, and examine the math behind destructive points and the role of slit width in wave interference.https://cdn.kastatic.org/ka-youtube-converted/7CmbItRjM-Y.mp4/7CmbItRjM-Y.mp4900yes https://puzzicle.org/science/ap-physics-2/v/more-on-single-slit-interferencehttps://puzzicle.org/thumbs/x1a496461.jpgMore on single slit interferenceContinue to explore the phenomenon of single slit interference. See how there's actually constructive (rather than just destructive) interference at some points on the screen. single slit interference, a complex phenomenon in light physics. Learn about the mathematical reasoning behind destructive points and why it doesn't apply to constructive points, and relate this to diffraction gratings, phase differences, and the challenge of finding exact locations of constructive points.https://cdn.kastatic.org/ka-youtube-converted/T-kgoxhFSmU.mp4/T-kgoxhFSmU.mp4786yes https://puzzicle.org/science/ap-physics-2/v/thin-film-interference-part-1https://puzzicle.org/thumbs/x1408ebe1.jpgThin film interference (part 1)Thin film interference occurs when light waves reflecting off the top and bottom surfaces of a thin film interfere with one another. This type of interference is the reason that thin films, such as oil or soap bubbles, form colorful patterns.https://cdn.kastatic.org/ka-youtube-converted/oXowkdgJPO4.mp4/oXowkdgJPO4.mp4723yes https://puzzicle.org/science/ap-physics-2/v/thin-film-interference-part-2https://puzzicle.org/thumbs/x6ddace9a.jpgThin film interference (part 2)Let's work out a few details on how thin film interference works as we explore the concept of thin film interference as it relates to light waves. Understand how light's wavelength changes as it moves through different mediums, like air and oil. Learn how to calculate the wavelength in a new medium using either the speeds of light or the indices of refraction. Master the conditions for constructive and destructive interference.https://cdn.kastatic.org/ka-youtube-converted/yWkga94iBzU.mp4/yWkga94iBzU.mp4723yes https://puzzicle.org/science/ap-physics-2/v/photon-energyhttps://puzzicle.org/thumbs/x160e6164.jpgPhoton energyLearn what a photon is and how to determine the energy of a photon as you explore the dual nature of light as both a wave and a particle, known as wave-particle duality. Uncover the concept of quantum mechanics, where light deposits energy in discrete amounts called photons. Investigate Planck's constant and how it relates to the energy of a photon.https://cdn.kastatic.org/ka-youtube-converted/Zqt3btS1FwE.mp4/Zqt3btS1FwE.mp4558yes https://puzzicle.org/science/ap-physics-2/v/photon-momentumhttps://puzzicle.org/thumbs/x8d9d935e.jpgPhoton momentumLearn how to determine the momentum of a photon.https://cdn.kastatic.org/ka-youtube-converted/Qwtz-QSDyOM.mp4/Qwtz-QSDyOM.mp4175yes https://puzzicle.org/science/ap-physics-2/v/de-broglie-wavelengthhttps://puzzicle.org/thumbs/xe3fa6319.jpgde Broglie wavelengthIs it a particle or a wave? This is the question that physicists of the 1920s were asking about light. In 1924, Louis de Broglie took this question to another level as he explored how electrons - which are matter, and were thought to be simply particles - can behave like waves.https://cdn.kastatic.org/ka-youtube-converted/iTRxuBxttj8.mp4/iTRxuBxttj8.mp4680yes https://puzzicle.org/science/ap-physics-2/v/bohr-model-energy-levels-derivation-using-physicshttps://puzzicle.org/thumbs/xdb4337d3.jpgBohr model energy levels (derivation using physics)Using classical physics to calculate the energy of electrons in Bohr model. Solving for energy of ground state and more generally for level n.https://cdn.kastatic.org/ka-youtube-converted/7Zin8hG9Nhw.mp4/7Zin8hG9Nhw.mp4728yes https://puzzicle.org/science/hs-chemistry/v/atomic-spectrahttps://puzzicle.org/thumbs/xc8123f835ad11b76.jpgAtomic spectraElectrons only exist at specific, discrete energy levels in an atom. If an electron absorbs a photon with energy equal to the difference between two energy levels, the electron will transition to the higher energy level. The atom is then in an excited state. When an electron transitions to a lower energy level, it emits a photon with energy equal to the difference. Every element has a unique emission/absorption spectrum, making atomic spectral lines a valuable tool for identifying elements in distant astronomical objects.https://cdn.kastatic.org/ka-youtube-converted/N0lBsW3ZzN8.mp4/N0lBsW3ZzN8.mp4883yes https://puzzicle.org/science/ap-physics-2/v/the-photoelectric-and-photovoltaic-effectshttps://puzzicle.org/thumbs/xc0bd6659e907c854.jpgThe photoelectric and photovoltaic effectsWhen light at or above a threshold frequency shines on a metal surface, electrons are emitted from the surface. This phenomenon is called the photoelectric effect. The photoelectric effect is evidence that light is quantized—light exists as discrete packets of energy called photons. The greater the frequency of the light, the greater the energy of its photons. A closely related phenomenon is the photovoltaic effect, which is the underlying mechanism of solar cells.https://cdn.kastatic.org/ka-youtube-converted/HCCm0lFwqNw.mp4/HCCm0lFwqNw.mp4861yes https://puzzicle.org/science/ap-physics-2/v/einsteins-photoelectric-equation-work-functionhttps://puzzicle.org/thumbs/xf0aad8681f369332.jpgEinstein's photoelectric equation & work functionLet's explore what work function is, and why, in photoelectric effect, electrons come out with different kinetic energies. While exploring this, we will rediscover Einstein's photoelectric equation.https://cdn.kastatic.org/ka-youtube-converted/ARa6MWdmg2Y.mp4/ARa6MWdmg2Y.mp4589yes https://puzzicle.org/science/ap-physics-2/v/calculate-threshold-frequencyhttps://puzzicle.org/thumbs/x63421143a9eac300.jpgCalculate threshold frequencyWhen light of frequency 2.42 X 10^15 Hz is incident on a metal surface, the fastest photoelectrons are found to have a kinetic energy of 1.7eV. Find the threshold frequency of the metal. Let's explore how to use Einstein's photoelectric equation to solve such numerical on photoelectric effect.https://cdn.kastatic.org/ka-youtube-converted/IqmDELIaDKQ.mp4/IqmDELIaDKQ.mp4529yes https://puzzicle.org/science/ap-physics-2/v/which-metals-will-show-photoelectric-emissionhttps://puzzicle.org/thumbs/x9397f06593024a2f.jpgWhich metals will show photoelectric emissionLight of wavelength 430 nm is incident on each of the metals given in the table. We will explore how to predict which of the metals will show photoelectric emission.https://cdn.kastatic.org/ka-youtube-converted/rVLhBl6J4Qg.mp4/rVLhBl6J4Qg.mp4551yes https://puzzicle.org/science/ap-physics-2/v/blackbody-radiationhttps://puzzicle.org/thumbs/x2b3483cb8e21f684.jpgBlackbody radiationA blackbody is an idealized object that absorbs all electromagnetic radiation (light) that shines on it. Blackbodies do not perpetually warm up as they absorb light because the accelerating charged particles inside them radiate their own light. The spectrum radiated by a blackbody depends only on its temperature. The hotter a blackbody is, the more intense its radiation is, and the shorter its peak wavelength is. The relationship between temperature and spectrum is used to estimate a star's surface temperature based on its color.https://cdn.kastatic.org/ka-youtube-converted/DHG61XPuWyk.mp4/DHG61XPuWyk.mp4858yes https://puzzicle.org/science/hs-chemistry/v/intro-to-radioactive-decayhttps://puzzicle.org/thumbs/x5b2a641f29fd36b2.jpgIntro to radioactive decayDuring radioactive decay, an unstable nucleus (the "parent") spontaneously changes to become a different nucleus (the "daughter"), emitting radiation in the process. The nuclear radiation emitted during radioactive decay is high energy, ionizing radiation. Nuclear radiation is potentially dangerous to living things but also has many beneficial applications. Types of radioactive decay include alpha, beta, and gamma.https://cdn.kastatic.org/ka-youtube-converted/P_SD5Rt6XMk.mp4/P_SD5Rt6XMk.mp4482yes https://puzzicle.org/science/hs-chemistry/v/alpha-decayhttps://puzzicle.org/thumbs/x873157d4a30d924c.jpgAlpha decayDuring alpha decay, a large, unstable parent nucleus becomes a smaller daughter nucleus. It does this by emitting an alpha particle, a clump of two protons and two neutrons (a He-4 nucleus). The nucleus's atomic number decreases by two, and its mass number decreases by four. The alpha particle is high-energy ionizing radiation—it travels at high speed because it carries away the majority of the energy lost by the nucleus during the decay.https://cdn.kastatic.org/ka-youtube-converted/QCZQCi_uKpM.mp4/QCZQCi_uKpM.mp4666yes https://puzzicle.org/science/hs-chemistry/v/beta-decayhttps://puzzicle.org/thumbs/xe2c5e28638221f92.jpgBeta decayDuring beta minus decay, a neutron in an unstable nucleus transforms into a proton, releasing a high energy electron and antineutrino in the process. During beta plus decay, a proton in an unstable nucleus transforms into a neutron, releasing a high energy positron and neutrino in the process. Beta particles carry away energy lost by the nucleus and are ionizing radiation.https://cdn.kastatic.org/ka-youtube-converted/iGr6atucsBw.mp4/iGr6atucsBw.mp4708yes https://puzzicle.org/science/hs-chemistry/v/gamma-decayhttps://puzzicle.org/thumbs/xfee7add64448283a.jpgGamma decayDuring gamma decay, an excited nucleus releases energy in the form of a gamma photon. Gamma radiation is high-energy electromagnetic radiation (light) which has a shorter wavelength and higher frequency than visible light. Gamma rays are ionizing radiation. The number of protons and neutrons in the parent nucleus does not change, so the identity of the daughter nucleus is the same as the parent, but in a lower energy state.https://cdn.kastatic.org/ka-youtube-converted/ksLNmLQbNT4.mp4/ksLNmLQbNT4.mp4709yes https://puzzicle.org/science/hs-chemistry/v/half-life-radiometric-datinghttps://puzzicle.org/thumbs/xa4a3f55553bb9fba.jpgHalf-lifeHalf-life is the time required for half of a radioactive sample to decay. Half-life cannot be changed—nuclei cannot be forced to decay faster or slower. Additionally, we cannot predict when an individual nucleus will decay. However, when analyzing a large sample containing many nuclei, half-life allows us to predict how much of the sample will remain after a given amount of time. This is useful for radiometric dating.https://cdn.kastatic.org/ka-youtube-converted/gDpyBifg6-s.mp4/gDpyBifg6-s.mp4656yes https://puzzicle.org/science/ap-physics-2/v/half-life-of-radioactive-isotopeshttps://puzzicle.org/thumbs/xb514d30c.jpgHalf-life plotDefinition of half-life and graphing the decay of phosphorus-32. Calculating how much phosphorus-32 remains after 57.2 days.https://cdn.kastatic.org/ka-youtube-converted/dnYyMHSSb8M.mp4/dnYyMHSSb8M.mp4407yes https://puzzicle.org/science/hs-chemistry/v/nuclear-fusionhttps://puzzicle.org/thumbs/xbc30d5d0a48df1a1.jpgNuclear fusionDuring nuclear fusion, two or more nuclei combine to form a different nucleus. When light nuclei fuse to produce a nucleus lighter than iron, energy is released (exothermic). When heavy nuclei fuse to produce a nucleus heavier than iron, energy is absorbed (endothermic). The release or absorption of energy relates to the difference in total mass of the reactants and products by E=mc^2. Nuclear fusion powers the Sun and other stars.https://cdn.kastatic.org/ka-youtube-converted/b9qxAFLpz9U.mp4/b9qxAFLpz9U.mp4825yes https://puzzicle.org/science/hs-chemistry/v/nuclear-fissionhttps://puzzicle.org/thumbs/xfb5c32e6455a15bd.jpgNuclear fissionDuring a nuclear fission reaction, a fissile nucleus absorbs a neutron and splits into two smaller nuclei. One or more free neutrons are also released. If enough fissile nuclei are located close enough together, the neutrons released from one fission can induce further fissions, resulting in a chain reaction. Nuclear weapons utilize fast, uncontrolled chain reactions to produce an explosion. Nuclear reactors utilize slower, controlled chain reactions to generate electricity. The release of energy during fission relates to the difference in total mass of the reactants and products by E=mc^2.https://cdn.kastatic.org/ka-youtube-converted/5jDFqpoTgDo.mp4/5jDFqpoTgDo.mp4627yes https://puzzicle.org/science/ap-physics-c-mechanics/v/introduction-to-ap-physics-c-mechanicshttps://puzzicle.org/thumbs/x6359629ff75e7e8e.jpgIntroduction to AP Physics C: MechanicsAP Physics C: Mechanics is a calculus-based, introductory college-level physics course. In this course you'll explore the following topics: Kinematics; Forces and translational dynamics; Work, energy, and power; Linear momentum; Torque and rotational dynamics; Energy and momentum of rotating systems; Oscillations.https://cdn.kastatic.org/ka-youtube-converted/_pH22LAM7Ek.mp4/_pH22LAM7Ek.mp4189yes https://puzzicle.org/science/ap-physics-c-mechanics/v/kinematic-quantities-rates-of-changehttps://puzzicle.org/thumbs/x154ba44082ddfeaf.jpgKinematic quantities: Rates of changeAn object’s average velocity over some interval is the object’s change in position (displacement) divided by the time interval over which that displacement occurs. Taking the limit of this quotient as the time interval goes to zero gives the object’s instantaneous velocity at some time, which is the derivative of position with respect to time. An object’s average acceleration over some interval is the object’s change in velocity divided by the time interval over which that change in velocity occurs. Taking the limit of this quotient as the time interval goes to zero gives the object’s instantaneous acceleration at some time, which is the derivative of velocity with respect to time.https://cdn.kastatic.org/ka-youtube-converted/8bHhZ66tvfo.mp4/8bHhZ66tvfo.mp4924yes https://puzzicle.org/science/ap-physics-c-mechanics/v/kinematic-quantities-accumulations-of-changehttps://puzzicle.org/thumbs/x183e3a44832007d0.jpgKinematic quantities: Accumulations of changeAn object’s change in velocity during a time interval is equal to the integral of the object’s acceleration with respect to time over that interval. On a graph of acceleration as a function of time, the change in velocity is represented by the signed area between the curve and the time axis. An object’s change in position (displacement) during a time interval is equal to the integral of the object’s velocity with respect to time over that interval. On a graph of velocity as a function of time, the displacement is represented by the signed area between the curve and the time axis.https://cdn.kastatic.org/ka-youtube-converted/ClCSbLcKDyE.mp4/ClCSbLcKDyE.mp41030yes https://puzzicle.org/science/ap-physics-c-mechanics/v/motion-as-a-function-of-time-friction-examplehttps://puzzicle.org/thumbs/xae7468a1b94969f8.jpgMotion as a function of time: Friction exampleApplying Newton’s second law to a moving object experiencing kinetic friction results in a differential equation for velocity. The differential equation can be solved to find the object’s velocity as a function of time. The velocity function can then be used to find the object’s acceleration and position as functions of time through differentiation and integration, respectively. The force is constant while the object moves, so we find the velocity function is linear and the position function is quadratic.https://cdn.kastatic.org/ka-youtube-converted/T3UBiY4sHKA.mp4/T3UBiY4sHKA.mp4769yes https://puzzicle.org/science/ap-physics-c-mechanics/v/motion-as-a-function-of-time-resistive-force-example-part-1https://puzzicle.org/thumbs/x16b8196bff3106d4.jpgMotion as a function of time: Resistive force example (part 1)A resistive (drag) force is a velocity-dependent force which acts opposite the direction of an object’s velocity. Applying Newton’s second law to an object experiencing a resistive force results in a differential equation for velocity. Using the method of separation of variables, the velocity can be determined by integrating over the proper limits of integration. Terminal velocity is the maximum speed achieved by an object moving under the influence of a constant force and a resistive force that are exerted on the object in opposite directions. Terminal velocity is reached when the net force exerted on the object is zero.https://cdn.kastatic.org/ka-youtube-converted/d5BNFRylOZE.mp4/d5BNFRylOZE.mp4900yes https://puzzicle.org/science/ap-physics-c-mechanics/v/motion-as-a-function-of-time-resistive-force-example-part-2https://puzzicle.org/thumbs/x9b7b2e063d6f5dd3.jpgMotion as a function of time: Resistive force example (part 2)A resistive (drag) force is a velocity-dependent force which acts opposite the direction of an object’s velocity. Applying Newton’s second law to an object experiencing a resistive force results in a differential equation for velocity. The differential equation can be solved to find the object’s velocity as a function of time. The velocity function can then be used to find the object’s acceleration and position as functions of time through differentiation and integration, respectively. The functions are exponential and have asymptotes that are determined by the initial conditions and the forces exerted on the object.https://cdn.kastatic.org/ka-youtube-converted/F5Tg46DVWzo.mp4/F5Tg46DVWzo.mp4730yes https://puzzicle.org/science/ap-physics-c-mechanics/v/work-done-by-forces-part-1https://puzzicle.org/thumbs/xe7a8a84e31a79199.jpgWork done by forces (part 1)Work is the amount of energy transferred into or out of a system by a force exerted on that system over a distance. Work is a scalar quantity that may be positive, negative, or zero. Only the component of the force exerted on a system that is parallel to the displacement of the point of application of the force will change the system’s total energy. The work-energy theorem states that the change in an object’s kinetic energy is equal to the sum of the work (net work) being done by all forces exerted on the object.https://cdn.kastatic.org/ka-youtube-converted/Qf8hIBrsWZI.mp4/Qf8hIBrsWZI.mp4752yes https://puzzicle.org/science/ap-physics-c-mechanics/v/work-done-by-forces-part-2https://puzzicle.org/thumbs/x2376bbbf7994e559.jpgWork done by forces (part 2)If a force varies with position, the work done by that force cannot generally be calculated as a simple product. Instead, an integral is required to sum the infinitesimal work contributions along the object’s path. The work done by such a force is given by a line integral, evaluated along the path from the object’s initial to final position. For one-dimensional motion, work is equal to the area under the curve of the parallel component of force versus position.https://cdn.kastatic.org/ka-youtube-converted/gS2yei7d0_E.mp4/gS2yei7d0_E.mp41151yes https://puzzicle.org/science/ap-physics-c-mechanics/v/potential-energy-and-conservative-forces-part-1https://puzzicle.org/thumbs/x8dfc8a50b62104b0.jpgPotential energy and conservative forces (part 1)Potential energies are only associated with conservative forces. The change in potential energy between two configurations of a system is the negative of the work done by the conservative force during the change. Because the gravitational field near the surface of a planet is nearly constant, the change in gravitational potential energy of a system containing an object and a planet when the object is near the surface of the planet can be approximated by ΔU_g = mgΔy.https://cdn.kastatic.org/ka-youtube-converted/1eooA5q9GCs.mp4/1eooA5q9GCs.mp41026yes https://puzzicle.org/science/ap-physics-c-mechanics/v/potential-energy-and-conservative-forces-part-2https://puzzicle.org/thumbs/xfcf5dfe92985dc62.jpgPotential energy and conservative forces (part 2)The general form for the gravitational potential energy of a system consisting of two approximately spherical distributions of mass is given by the equation U_g = -GMm/r. The gravitational energy is negative because the zero reference configuration is chosen as the two masses being infinitely far away from each other (r = infinity). Therefore, the gravitational potential energy for any closer configuration is less than zero.https://cdn.kastatic.org/ka-youtube-converted/9o1WbcK7_-g.mp4/9o1WbcK7_-g.mp4853yes https://puzzicle.org/science/ap-physics-c-mechanics/v/potential-energy-and-conservative-forces-part-3https://puzzicle.org/thumbs/x018b777e5fd9c554.jpgPotential energy and conservative forces (part 3)The force exerted by an ideal spring is a conservative and obeys Hooke’s law. Applying the general equation for change in potential energy reveals that the elastic potential energy of a system containing an ideal spring is given by the following equation, where Δx is the distance the spring has been stretched or compressed from its equilibrium length: ΔU_s = ½k(Δx)^2. The elastic potential energy is zero when the spring is at its relaxed length.https://cdn.kastatic.org/ka-youtube-converted/Q6EgiGx7EUQ.mp4/Q6EgiGx7EUQ.mp4693yes https://puzzicle.org/science/ap-physics-c-mechanics/v/potential-energy-and-conservative-forces-part-4https://puzzicle.org/thumbs/xee5b6a72f7def80f.jpgPotential energy and conservative forces (part 4)A conservative force exerted on a system in a single dimension is equal to the negative derivative of the system’s potential energy with respect to position in that dimension. The force points in the direction of decreasing potential energy. The relationship is represented by the following equation: F_x = -dU(x)/dxhttps://cdn.kastatic.org/ka-youtube-converted/qKTBVL5jBIA.mp4/qKTBVL5jBIA.mp4760yes https://puzzicle.org/science/ap-physics-c-mechanics/v/potential-energy-and-conservative-forces-part-5https://puzzicle.org/thumbs/x0d74acb9c86b99fb.jpgPotential energy and conservative forces (part 5)Graphs of a system’s potential energy as a function of position can be useful in determining properties of that system. Stable equilibrium is a location at which a small displacement results in a force opposite the direction of the displacement, accelerating the object back toward the equilibrium position. Stable equilibrium positions exist where the potential energy as a function of position has a local minimum. Unstable equilibrium is a location at which a small displacement results in a force in the same direction as the displacement, accelerating the object away from the equilibrium position. Unstable equilibrium positions exist where the potential energy as a function of position has a local maximum.https://cdn.kastatic.org/ka-youtube-converted/gY8sqnmju-0.mp4/gY8sqnmju-0.mp4816yes https://puzzicle.org/science/ap-physics-c-mechanics/v/linear-momentum-and-impulsehttps://puzzicle.org/thumbs/x8c9df72bf62ec7b6.jpgLinear momentum and impulseAn object’s linear momentum is the product of its mass and velocity. Momentum is a vector quantity and has the same direction as the velocity. The rate of change of a system’s momentum is equal to the net external force exerted on that system. Impulse is defined as the integral of the net force exerted on an object or system over a time interval. The impulse-momentum theorem relates the impulse delivered to an object and the object’s change in momentum.https://cdn.kastatic.org/ka-youtube-converted/u1SOrLcHtVA.mp4/u1SOrLcHtVA.mp41002yes https://puzzicle.org/science/ap-physics-c-mechanics/v/time-dependent-forceshttps://puzzicle.org/thumbs/xc025371fd7bb309b.jpgTime-dependent forcesImpulse is defined as the integral of the net force exerted on an object or system over a time interval. The impulse-momentum theorem relates the impulse delivered to an object and the object’s change in momentum. If the force on an object is constant over an interval, the impulse delivered during that interval is simply the product of the constant force and the amount of time it acts. But if a force varies with time, the force as a function of time must be integrated with respect to time to find the impulse.https://cdn.kastatic.org/ka-youtube-converted/p8d72VuAjP8.mp4/p8d72VuAjP8.mp4544yes https://puzzicle.org/science/ap-physics-c-mechanics/v/angular-momentum-of-rigid-systems-with-calculushttps://puzzicle.org/thumbs/xb6947f193ef7fcb4.jpgAngular momentum of rigid systems (with calculus)The magnitude of the angular momentum of a rigid system about a specific axis is related to the rotational inertia and angular velocity of the rigid system. Angular impulse is defined as the product of the torque exerted on an object or rigid system and the time interval during which the torque is exerted. Angular impulse has the same direction as the torque imparting it. A rotational form of the impulse-momentum theorem relates the angular impulse delivered to a rigid system to the change in angular momentum of that rigid system.https://cdn.kastatic.org/ka-youtube-converted/UCREaEXFiIA.mp4/UCREaEXFiIA.mp4837yes https://puzzicle.org/science/ap-physics-c-mechanics/v/modeling-spring-mass-oscillatorshttps://puzzicle.org/thumbs/xa5d559731a61de1e.jpgModeling spring-mass oscillatorsSimple harmonic motion (SHM) can be modeled mathematically as a second-order differential equation derived using Newton’s second law. The solution to this differential equation is the object’s position as a function of time, which is sinusoidal. The position function can be differentiated with respect to time to find the object’s velocity and acceleration functions, which are also sinusoidal.https://cdn.kastatic.org/ka-youtube-converted/-inpCcFh9qk.mp4/-inpCcFh9qk.mp4812yes https://puzzicle.org/science/ap-physics-c-mechanics/v/modeling-pendulumshttps://puzzicle.org/thumbs/x0e23c64b3916ca29.jpgModeling pendulumsA physical pendulum is a rigid system that undergoes oscillation about a fixed axis. For small amplitudes of motion, the period of a physical pendulum is derived from the application of Newton’s second law in rotational form. A simple pendulum is a special case of physical pendulums in which the object can be modeled as a point mass hanging from the pivot point. A torsion pendulum involves a twisted wire or other material that behaves analogously to a spring. If the wire exerts a restoring torque proportional to its angle of twist, simple harmonic motion can result.https://cdn.kastatic.org/ka-youtube-converted/0NjBjpGQpUQ.mp4/0NjBjpGQpUQ.mp4746yes https://puzzicle.org/science/ap-physics-c-mechanics/v/resonance-and-damped-oscillationshttps://puzzicle.org/thumbs/x1f92e8f7f6a4ba9c.jpgResonanceThe natural frequency of a system is the frequency at which the system will oscillate when it is displaced from its equilibrium position. In the presence of a sinusoidal external force, a system may exhibit resonance. Resonance occurs when an external force is exerted at the natural frequency of the oscillating system. Resonance increases the amplitude of oscillating motion.https://cdn.kastatic.org/ka-youtube-converted/ubFGdZM1ziM.mp4/ubFGdZM1ziM.mp4897yes https://puzzicle.org/science/biology/v/overview-of-biologyhttps://puzzicle.org/thumbs/x7be07a3b.jpgBiology overviewBiology at Khan Academy explores life in all its forms, from humans to bacteria and how living things convert energy, grow, and reproduce. The course covers key biological molecules, the role of DNA, and the complexity of cells. It also touches on the intriguing world of viruses and how life continues to change and adapt through evolution and natural selection.https://cdn.kastatic.org/ka-youtube-converted/dQCsA2cCdvA.mp4/dQCsA2cCdvA.mp4274yes https://puzzicle.org/science/biology/v/lebron-asks-why-does-sweating-cool-you-downhttps://puzzicle.org/thumbs/1022193589.jpgLeBron Asks: Why does sweating cool you down?LeBron asks Sal why sweating helps cool the body down. Explore the molecular level of sweat and skin, understanding how temperature is related to molecular motion. Learn about evaporation and how high-energy water molecules escaping the skin's surface leads to cooling.https://cdn.kastatic.org/ka-youtube-converted/jgnNhL2jSXY.mp4/jgnNhL2jSXY.mp4341yes https://puzzicle.org/science/biology/v/evaporative-coolinghttps://puzzicle.org/thumbs/x57ce9014.jpgEvaporative coolingEvaporative cooling. Why sweating cools you down.https://cdn.kastatic.org/ka-youtube-converted/_eEONOJHnEs.mp4/_eEONOJHnEs.mp4314yes https://puzzicle.org/science/biology/v/heat-of-vaporization-of-water-and-ethanolhttps://puzzicle.org/thumbs/xac337cc2.jpgHeat of vaporization of water and ethanolHeat of vaporization of water and ethanol.https://cdn.kastatic.org/ka-youtube-converted/bVHuI_QpYIM.mp4/bVHuI_QpYIM.mp4436yes https://puzzicle.org/science/biology/v/specific-heat-of-waterhttps://puzzicle.org/thumbs/xd0570e1e.jpgSpecific heat of waterSpecific heat of water. A calorie as the specific heat of water. How water moderates temperature.https://cdn.kastatic.org/ka-youtube-converted/h-31O7CaF2o.mp4/h-31O7CaF2o.mp4650yes https://puzzicle.org/science/biology/v/liquid-water-denser-than-solid-water-icehttps://puzzicle.org/thumbs/x79241b59.jpgLiquid water denser than solid water (ice)Water less dense as ice. Why lakes don't freeze solid.https://cdn.kastatic.org/ka-youtube-converted/Y3ATc9he254.mp4/Y3ATc9he254.mp4343yes https://puzzicle.org/science/biology/v/autoionization-waterhttps://puzzicle.org/thumbs/x389f5189.jpgAutoionization of waterAutoionization of water into hydronium and hydroxide ions.https://cdn.kastatic.org/ka-youtube-converted/NUyYlRxMtcs.mp4/NUyYlRxMtcs.mp4654yes https://puzzicle.org/science/biology/v/introduction-to-definition-of-phhttps://puzzicle.org/thumbs/xff203ebc.jpgDefinition of pHIntroduction to pH and the pH scale. Examples of calculating pH of pure water, bleach, and orange juice.https://cdn.kastatic.org/ka-youtube-converted/J7-GewgqWUQ.mp4/J7-GewgqWUQ.mp4697yes https://puzzicle.org/science/biology/v/buffer-systemhttps://puzzicle.org/thumbs/x39758a10.jpgIntroduction to buffersIntroduction to buffer systems, which regulate pH in blood.https://cdn.kastatic.org/ka-youtube-converted/gjKmQ501sAg.mp4/gjKmQ501sAg.mp4380yes https://puzzicle.org/science/biology/v/molecular-structure-of-glucosehttps://puzzicle.org/thumbs/x66379952.jpgMolecular structure of glucoseMolecular structure of glucose. Monosaccharide structure. Linear and ring forms.https://cdn.kastatic.org/ka-youtube-converted/-Aj5BTnz-v0.mp4/-Aj5BTnz-v0.mp4559yes https://puzzicle.org/science/biology/v/molecular-structure-of-fructosehttps://puzzicle.org/thumbs/x6fc25ec1.jpgMolecular structure of fructoseFructose in comparison with glucose. Pyranose and furanose rings. Sucrose made from glucose and fructose.https://cdn.kastatic.org/ka-youtube-converted/662koqcIXSk.mp4/662koqcIXSk.mp4579yes https://puzzicle.org/science/biology/v/lipid-overviewhttps://puzzicle.org/thumbs/x3bc71445.jpgLipid overviewTypes of lipids including fats, waxes, steroids and phospholipids.https://cdn.kastatic.org/ka-youtube-converted/Ezp8F7XJHWE.mp4/Ezp8F7XJHWE.mp4691yes https://puzzicle.org/science/biology/v/tertiary-structure-of-proteinshttps://puzzicle.org/thumbs/x1cc9afc4.jpgTertiary structure of proteinsHow side chain interactions can impact the tertiary structure of proteins.https://cdn.kastatic.org/ka-youtube-converted/7udSVZ7bDb8.mp4/7udSVZ7bDb8.mp4448yes https://puzzicle.org/science/biology/v/introduction-to-energyhttps://puzzicle.org/thumbs/xad7ec6d2.jpgIntroduction to energyEnergy is defined as the ability to do work. Energy can be found in many things and can take different forms. For example, kinetic energy is the energy of motion, and potential energy is energy due to an object's position or structure. Energy is never lost, but it can be converted from one form to another.https://cdn.kastatic.org/ka-youtube-converted/qzvatDIDXwo.mp4/qzvatDIDXwo.mp4657yes https://puzzicle.org/science/biology/v/introduction-to-entropyhttps://puzzicle.org/thumbs/x84002b90.jpgIntroduction to entropyIntroduction to entropy, and how entropy relates to the number of possible states for a system.https://cdn.kastatic.org/ka-youtube-converted/Svvr5uF_FZ8.mp4/Svvr5uF_FZ8.mp4447yes https://puzzicle.org/science/biology/v/the-second-law-of-thermodynamicshttps://puzzicle.org/thumbs/x932d408d.jpgSecond Law of ThermodynamicsThe Second Law of Thermodynamics: there can be no spontaneous transfer of heat from cold to hot.https://cdn.kastatic.org/ka-youtube-converted/Vc8mmZTrNtI.mp4/Vc8mmZTrNtI.mp4622yes https://puzzicle.org/science/biology/v/why-heat-increases-entropyhttps://puzzicle.org/thumbs/x78208c62.jpgWhy heat increases entropyWhy heat increases entropy—even though some of it can do work!https://cdn.kastatic.org/ka-youtube-converted/cKAF1v5hJoE.mp4/cKAF1v5hJoE.mp4421yes https://puzzicle.org/science/biology/v/gibbs-free-energy-and-spontaneous-reactionshttps://puzzicle.org/thumbs/xca9e7640.jpgGibbs free energy and spontaneous reactionsGibbs free energy and spontaneous reactions.https://cdn.kastatic.org/ka-youtube-converted/CHHu-iTwHjg.mp4/CHHu-iTwHjg.mp4558yes https://puzzicle.org/science/biology/v/endergonic-exergonic-exothermic-and-endothermic-reactionshttps://puzzicle.org/thumbs/xb760620c.jpgEndergonic, exergonic, exothermic, and endothermicEndergonic, exergonic, exothermic, and endothermic reactions.https://cdn.kastatic.org/ka-youtube-converted/J2L-X2sUigs.mp4/J2L-X2sUigs.mp4712yes https://puzzicle.org/science/biology/v/adenosine-triphosphatehttps://puzzicle.org/thumbs/xb7f05d8d.jpgATP: Adenosine triphosphateATP, or Adenosine Triphosphate, is the energy currency in biological systems. It's made up of adenosine and three phosphate groups. Energy is stored when ATP is formed and released when it's broken down into ADP (Adenosine Diphosphate) and a phosphate group. This energy release powers various biological processes.https://cdn.kastatic.org/ka-youtube-converted/PK6HmIe2EAg.mp4/PK6HmIe2EAg.mp4379yes https://puzzicle.org/science/biology/v/atp-hydrolysis-mechanismhttps://puzzicle.org/thumbs/x20655c75.jpgATP hydrolysis mechanismATP hydrolysis mechanism. How energy is released when ATP is converted to ADP and phosphate.https://cdn.kastatic.org/ka-youtube-converted/-KE7jTXwNYs.mp4/-KE7jTXwNYs.mp4267yes https://puzzicle.org/science/biology/v/enzyme-cofactors-and-coenzymeshttps://puzzicle.org/thumbs/xf9e84d4d.jpgEnzyme cofactors and coenzymesThe cofactors and coenzymes (organic cofactors) that help enzymes catalyze reactions.https://cdn.kastatic.org/ka-youtube-converted/jYR0RlMft8Y.mp4/jYR0RlMft8Y.mp4599yes https://puzzicle.org/science/biology/v/cell-theoryhttps://puzzicle.org/thumbs/x7771d81a.jpgCell theoryHooke and Leeuwenhoek were two of the first scientists to use microscopes to study the microscopic world of cells. Hooke coined the term "cell" after observing the tiny compartments in cork, while Leeuwenhoek discovered a variety of living creatures in pond water, blood, and other samples. They contributed to cell theory, which states that 1) all living things are made of one or more cells, 2) cells are the basic unit of life, and 3) all cells come from other cells.https://cdn.kastatic.org/ka-youtube-converted/zk3vlhz1b6k.mp4/zk3vlhz1b6k.mp4483yes https://puzzicle.org/science/biology/v/cytoskeletonshttps://puzzicle.org/thumbs/xd96ae2b8.jpgThe cytoskeletonExploring the cell cytoskeleton, including microfilaments and microtubules (with a brief mention of intermediate filaments).https://cdn.kastatic.org/ka-youtube-converted/4BAGI6LbHeo.mp4/4BAGI6LbHeo.mp4413yes https://puzzicle.org/science/biology/v/extracellular-matrixhttps://puzzicle.org/thumbs/x5c4ff82c.jpgExtracellular matrixOverview of how cells interact with the extracellular matrix.https://cdn.kastatic.org/ka-youtube-converted/cMNx17H3dRU.mp4/cMNx17H3dRU.mp4414yes https://puzzicle.org/science/biology/v/overview-of-animal-and-plant-cellshttps://puzzicle.org/thumbs/x0e8b1264.jpgOverview of animal and plant cellsAnimal and plant cells share common elements like plasma membranes, cytoskeletons, and mitochondria. However, they differ in certain aspects. For example, plant cells have a cell wall and a central vacuole, while animal cells contain centrosomes. The understanding of these cellular structures continues to evolve through ongoing research.https://cdn.kastatic.org/ka-youtube-converted/24YMQ9GvLss.mp4/24YMQ9GvLss.mp4522yes https://puzzicle.org/science/biology/v/concentration-gradientshttps://puzzicle.org/thumbs/x094bd875.jpgConcentration gradientsParticles move from high to low concentration areas, a process called diffusion. Each particle type has its own unique concentration gradient. This movement isn't planned, but happens randomly. Over time, particles spread evenly, balancing the probability of movement in all directions. This is a key concept in biology and chemistry.https://cdn.kastatic.org/ka-youtube-converted/CNP-dCQ-Cmg.mp4/CNP-dCQ-Cmg.mp4184yes https://puzzicle.org/science/biology/v/oxidation-and-reduction-from-biological-viewhttps://puzzicle.org/thumbs/23956.jpgOxidation and reduction review from biological point-of-viewTaking a look at oxidation and reduction in a biological context. Discover how these processes, involving the loss and gain of electrons, are the same in both fields. Uncover why biologists often refer to oxidation as losing hydrogen atoms and reduction as gaining them.https://cdn.kastatic.org/ka-youtube-converted/orI2m6IarJg.mp4/orI2m6IarJg.mp4809yes https://puzzicle.org/science/biology/v/oxidation-and-reduction-in-cellular-respirationhttps://puzzicle.org/thumbs/23958.jpgOxidation and reduction in cellular respirationOxidation and reduction in cellular respiration. Reconciling the biology and chemistry definitions of oxidation and reduction.https://cdn.kastatic.org/ka-youtube-converted/_KyyVhlUDNU.mp4/_KyyVhlUDNU.mp41028yes https://puzzicle.org/science/biology/v/glycolysishttps://puzzicle.org/thumbs/23960.jpgOverview of glycolysisLet's explore the process of glycolysis, the first phase of cellular respiration. Learn how this process breaks down glucose into two 3-carbon compounds, using two ATPs in the investment phase and generating a net of two ATPs in the payoff phase.https://cdn.kastatic.org/ka-youtube-converted/FE2jfTXAJHg.mp4/FE2jfTXAJHg.mp4810yes https://puzzicle.org/science/biology/v/glycolysis-overviewhttps://puzzicle.org/thumbs/x66a00a38.jpgSteps of glycolysisIntroduction to glycolysis. Role of glycolysis in producing ATPs and NADHs and converting glucose to pyruvates.https://cdn.kastatic.org/ka-youtube-converted/ArmlWtDnuys.mp4/ArmlWtDnuys.mp4722yes https://puzzicle.org/science/biology/v/krebs-citric-acid-cyclehttps://puzzicle.org/thumbs/23962.jpgKrebs / citric acid cycleOverview of the Krebs or citric acid cycle, which is a series of reactions that takes in acetyl CoA and produces carbon dioxide, NADH, FADH2, and ATP or GTP.https://cdn.kastatic.org/ka-youtube-converted/juM2ROSLWfw.mp4/juM2ROSLWfw.mp41067yes https://puzzicle.org/science/biology/v/lactic-acid-fermentationhttps://puzzicle.org/thumbs/x5a5ece83.jpgLactic acid fermentationGlycolysis breaks down glucose into two pyruvate molecules, producing ATP and reducing NAD to NADH. If oxygen is unavailable, lactic acid fermentation occurs, recycling pyruvate and NADH to produce more NAD+ for glycolysis. This process is common in organisms like lactobacillus, which is used in making yogurt, kimchi, and sauerkraut.https://cdn.kastatic.org/ka-youtube-converted/lfeXuK8pbFw.mp4/lfeXuK8pbFw.mp4682yes https://puzzicle.org/science/biology/v/alcohol-or-ethanol-fermentationhttps://puzzicle.org/thumbs/xcc1b90e8.jpgAlcohol or ethanol fermentationGlycolysis breaks down glucose into two pyruvate molecules, producing ATP and reducing NAD+ to NADH. When oxygen isn't available, organisms can use lactic acid fermentation or alcohol fermentation to recycle NADH back to NAD+. In alcohol fermentation, pyruvate loses a carboxyl group, becoming acetaldehyde, which is then reduced to ethanol. This process, performed by yeast, is used in bread and alcohol production.https://cdn.kastatic.org/ka-youtube-converted/D6mRPgvAEOc.mp4/D6mRPgvAEOc.mp4383yes https://puzzicle.org/science/biology/v/light-dependent-reactions-actorshttps://puzzicle.org/thumbs/x614c1624.jpgLight dependent reactions actorsMore detailed description of the various molecular actors in the light dependent reactions.https://cdn.kastatic.org/ka-youtube-converted/KzpRivhr6HI.mp4/KzpRivhr6HI.mp4398yes https://puzzicle.org/science/biology/v/photosynthesis-light-reactions-1https://puzzicle.org/thumbs/23970.jpgPhotosynthesis: Overview of the light-dependent reactionsOverview of the light-dependent reactions, including the structure of the chloroplast, the photosystems, and how ATP is synthesized.https://cdn.kastatic.org/ka-youtube-converted/GR2GA7chA_c.mp4/GR2GA7chA_c.mp41216yes https://puzzicle.org/science/biology/v/photosynthesis-calvin-cyclehttps://puzzicle.org/thumbs/23974.jpgPhotosynthesis: Calvin cycleThe Calvin cycle, or the light-independent (dark) reactions of photosythesis.https://cdn.kastatic.org/ka-youtube-converted/slm6D2VEXYs.mp4/slm6D2VEXYs.mp4808yes https://puzzicle.org/science/biology/v/photorespirationhttps://puzzicle.org/thumbs/23976.jpgPhotorespirationThe Calvin cycle and a competing pathway called photorespiration (also initiated by Rubisco).https://cdn.kastatic.org/ka-youtube-converted/DnNqe8o0ehc.mp4/DnNqe8o0ehc.mp4665yes https://puzzicle.org/science/biology/v/c-4-photosynthesishttps://puzzicle.org/thumbs/23978.jpgC4 photosynthesisC4 photosynthesis: how some plants avoid photorespiration.https://cdn.kastatic.org/ka-youtube-converted/7ynX_F-SwNY.mp4/7ynX_F-SwNY.mp41018yes https://puzzicle.org/science/biology/v/cam-plantshttps://puzzicle.org/thumbs/23980.jpgCAM plantsHow CAM Plants are able to fix carbon at night so they don't have to keep their stomata open during the day.https://cdn.kastatic.org/ka-youtube-converted/xp6Zj24h8uA.mp4/xp6Zj24h8uA.mp4517yes https://puzzicle.org/science/biology/v/somatic-germ-cellshttps://puzzicle.org/thumbs/x754b170b.jpgZygote differentiating into somatic and germ cellsLearn how a zygote, the single cell produced by fertilization, divides by mitosis to produce all the tissues of the human body (including germ cells, which can undergo meiosis to make sperm and eggs).https://cdn.kastatic.org/ka-youtube-converted/PvoigrzODdE.mp4/PvoigrzODdE.mp4575yes https://puzzicle.org/science/biology/v/chromosomes-chromatids-chromatin-etchttps://puzzicle.org/thumbs/23928.jpgChromosomes, chromatids and chromatinDNA replication, transcription, and translation are key biological processes. Replication involves DNA duplicating itself. Transcription involves DNA creating mRNA, and translation converts mRNA into proteins.https://cdn.kastatic.org/ka-youtube-converted/s9HPNwXd9fk.mp4/s9HPNwXd9fk.mp41103yes https://puzzicle.org/science/biology/v/cancerhttps://puzzicle.org/thumbs/23938.jpgCancerCells replicate through a process called mitosis, making perfect copies of themselves. Sometimes, a cell's DNA can mutate during replication. Most mutations are harmless, but some can cause cells to grow uncontrollably, leading to cancer.https://cdn.kastatic.org/ka-youtube-converted/RZhL7LDPk8w.mp4/RZhL7LDPk8w.mp4756yes https://puzzicle.org/science/biology/v/apoptosishttps://puzzicle.org/thumbs/x7b82c8ab.jpgApoptosisApoptosis, or programmed cell death, is a natural and healthy process occurring in our bodies. It's like a cell's self-destruct button, activated when a cell is damaged beyond repair. This process prevents the spread of potential diseases like cancer. Apoptosis is also crucial during development, shaping our fingers from a block of tissue, for example. Unlike necrosis, where a cell explodes and can harm neighboring cells, apoptosis is a tidy process where cells are broken down and their components reused.https://cdn.kastatic.org/ka-youtube-converted/gNFDONSjB7Q.mp4/gNFDONSjB7Q.mp4648yes https://puzzicle.org/science/biology/v/establishing-dna-as-transformation-principlehttps://puzzicle.org/thumbs/x53fe520f.jpgDNA as the "transforming principle"How Griffith and later Avery, McCarty and MacLeod found strong evidence that DNA is the "transforming principle" that encodes genetic information.https://cdn.kastatic.org/ka-youtube-converted/9X12OVvOznU.mp4/9X12OVvOznU.mp4619yes https://puzzicle.org/science/biology/v/hershey-and-chase-conclusively-show-dna-genetic-materialhttps://puzzicle.org/thumbs/xc7b4e9cb.jpgHershey and Chase: DNA is the genetic materialHow Alfred Hershey and Martha Chase showed conclusively that DNA is the genetic material.https://cdn.kastatic.org/ka-youtube-converted/bNsSaxCsIIg.mp4/bNsSaxCsIIg.mp4529yes https://puzzicle.org/science/biology/v/the-discovery-of-the-double-helix-structure-of-dnahttps://puzzicle.org/thumbs/x658eb73b.jpgThe discovery of the double helix structure of DNAHow the double helix structure of DNA was discovered. Work of Watson, Crick, Franklin, Wilkins, and Chargaff.https://cdn.kastatic.org/ka-youtube-converted/hCUce36Ganc.mp4/hCUce36Ganc.mp4537yes https://puzzicle.org/science/biology/v/bacteriahttps://puzzicle.org/thumbs/23948.jpgBacteriaIntroduction to bacteria.https://cdn.kastatic.org/ka-youtube-converted/TDoGrbpJJ14.mp4/TDoGrbpJJ14.mp41106yes https://puzzicle.org/science/biology/v/viruseshttps://puzzicle.org/thumbs/23950.jpgVirusesIntroduction to viruses.https://cdn.kastatic.org/ka-youtube-converted/0h5Jd7sgQWY.mp4/0h5Jd7sgQWY.mp41397yes https://puzzicle.org/science/biology/v/ape-clarificationhttps://puzzicle.org/thumbs/42120192.jpgApe clarificationApes, tailless primates, are divided into two families: lesser apes like gibbons, and great apes, which include chimpanzees, gorillas, and humans. The absence of a tail is a key characteristic distinguishing apes. This clarification corrects a previous error where an ape was incorrectly depicted with a tail.https://cdn.kastatic.org/ka-youtube-converted/oFGkYA_diDA.mp4/oFGkYA_diDA.mp464yes https://puzzicle.org/science/biology/v/genetic-variation-gene-flow-and-new-specieshttps://puzzicle.org/thumbs/x92bc392c.jpgGenetic variation, gene flow, and new speciesNatural selection, DNA, and gene flow are key to understanding evolution. DNA, or Deoxyribonucleic acid, is the genetic code that determines traits in organisms. Variations in this code, called mutations, lead to unique traits. Gene flow, the mixing of genetic information, maintains consistency within a species. When gene flow is restricted, species can diverge, leading to new species. This process, called speciation, contributes to biodiversity.https://cdn.kastatic.org/ka-youtube-converted/11iYk0Yrx3g.mp4/11iYk0Yrx3g.mp4712yes https://puzzicle.org/science/biology/v/ozone-layer-and-eukaryotes-show-up-in-the-proterozoic-eonhttps://puzzicle.org/thumbs/13179299.jpgOzone layer and eukaryotes show up in the Proterozoic eonOzone layer and eukaryotes show up in the Proterozoic eon. Great oxygenation event (oxygen catastrophe).https://cdn.kastatic.org/ka-youtube-converted/E1P79uFLCMc.mp4/E1P79uFLCMc.mp4596yes https://puzzicle.org/science/biology/v/first-living-things-on-land-clarificationhttps://puzzicle.org/thumbs/13179435.jpgFirst living things on land clarificationFirst living things on land clarification (This video copyrighted under Create Commons Attribution and Share-Alike CC-BY-SA license)https://cdn.kastatic.org/ka-youtube-converted/3gUE_P9T-Wk.mp4/3gUE_P9T-Wk.mp4335yes https://puzzicle.org/science/biology/v/biodiversity-flourishes-in-phanerozoic-eonhttps://puzzicle.org/thumbs/13179300.jpgBiodiversity flourishes in Phanerozoic eonCambrian explosion and biodiversity in the Phanerozoic Eonhttps://cdn.kastatic.org/ka-youtube-converted/MS7x2hDEhrw.mp4/MS7x2hDEhrw.mp4628yes https://puzzicle.org/science/biology/v/human-evolution-overviewhttps://puzzicle.org/thumbs/99682014.jpgHuman evolution overviewThe extinction of dinosaurs 65 million years ago paved the way for a rodent-like creature to evolve into various species. This creature survived the mass extinction event and gradually evolved into primates, monkeys, apes, and us—Homo sapiens.https://cdn.kastatic.org/ka-youtube-converted/frE1rjhH77Y.mp4/frE1rjhH77Y.mp4746yes https://puzzicle.org/science/biology/v/chronometric-revolutionhttps://puzzicle.org/thumbs/334540860.jpgChronometric revolutionDiscussion of the relatively recent changes in our ability as a species to shine light on our deep pasthttps://cdn.kastatic.org/ka-youtube-converted/5mNTvtjDnP8.mp4/5mNTvtjDnP8.mp4593yes https://puzzicle.org/science/biology/v/carbon-14-dating-2https://puzzicle.org/thumbs/8187417.jpgCarbon 14 dating 2Carbon 14 Dating 2https://cdn.kastatic.org/ka-youtube-converted/4YUtnod-YuE.mp4/4YUtnod-YuE.mp4255yes https://puzzicle.org/science/biology/v/potassium-argon-k-ar-datinghttps://puzzicle.org/thumbs/328018950.jpgPotassium-argon (K-Ar) datingHow K-Ar dating can be used to date very old volcanic rock and the things that might be buried in betweenhttps://cdn.kastatic.org/ka-youtube-converted/NMZ5kJEviD0.mp4/NMZ5kJEviD0.mp4635yes https://puzzicle.org/science/biology/v/k-ar-dating-calculationhttps://puzzicle.org/thumbs/328024200.jpgK-Ar dating calculationWorking through a calculation for K-Ar dating (good to have some prior experience with e and logarithms)https://cdn.kastatic.org/ka-youtube-converted/iLFcSfzrlMk.mp4/iLFcSfzrlMk.mp4667yes https://puzzicle.org/science/biology/v/ecology-introductionhttps://puzzicle.org/thumbs/x598ab453.jpgEcology introductionEcology studies how life interacts with other life and their environment, focusing on both living (biotic) and non-living (abiotic) factors. It explores different scales, from individual organisms to populations, communities, ecosystems, and the biosphere. Each level offers unique insights into the complex, often balanced systems that form our world. Ecology also considers potential biospheres beyond Earth.https://cdn.kastatic.org/ka-youtube-converted/OfV3VNgjpvw.mp4/OfV3VNgjpvw.mp4589yes https://puzzicle.org/science/biology/v/thomas-malthus-and-population-growthhttps://puzzicle.org/thumbs/467947556.jpgThomas Malthus and population growthThomas Malthus's views on population. Malthusian limits.https://cdn.kastatic.org/ka-youtube-converted/r1ywppAJ1xs.mp4/r1ywppAJ1xs.mp4493yes https://puzzicle.org/science/biology/v/biogeochemical-cycleshttps://puzzicle.org/thumbs/x1f5392abf0304d17.jpgBiogeochemical cycles overviewBiogeochemical cycles are nature's way of recycling essential elements for life, like water, carbon, nitrogen, and phosphorus. These cycles circulate these molecules through ecosystems and the entire biosphere. From the water cycle to the carbon cycle, these processes ensure the constant reuse of matter, while energy flows through the ecosystem.https://cdn.kastatic.org/ka-youtube-converted/ccWUDlKC3dE.mp4/ccWUDlKC3dE.mp4475yes https://puzzicle.org/science/biology/v/distribution-howhttps://puzzicle.org/thumbs/x427d430d.jpgHow biodiversity is distributed globallyHow biodiversity is distributed globally. Video by California Academy of Sciences.https://cdn.kastatic.org/ka-youtube-converted/BTOHSRVqN20.mp4/BTOHSRVqN20.mp4438yes https://puzzicle.org/science/biology/v/distributionwhyhttps://puzzicle.org/thumbs/xb482835e.jpgWhy biodiversity is distributed unevenlyFactors that influence the global distribution of biodiversity. Video by California Academy of Sciences.https://cdn.kastatic.org/ka-youtube-converted/3OAfLjxQUg4.mp4/3OAfLjxQUg4.mp4649yes https://puzzicle.org/science/biology/v/ecosystem-serviceshttps://puzzicle.org/thumbs/x1be8fe42.jpgEcosystem servicesDescribing and quantifying the benefits that healthy ecosystems provide to humans. Video by California Academy of Sciences.https://cdn.kastatic.org/ka-youtube-converted/BCH1Gre3Mg0.mp4/BCH1Gre3Mg0.mp4543yes https://puzzicle.org/science/biology/v/biodiversity-hotspothttps://puzzicle.org/thumbs/xe98900f2.jpgWhat is a biodiversity hotspot?Biodiversity hotspots, unique areas on Earth with a high number of endemic species and significant habitat loss, are crucial for conservation efforts. These hotspots, representing less than three percent of the Earth's land surface, serve as a scientific tool to measure human impact on biodiversity and guide resource allocation for protection efforts.https://cdn.kastatic.org/ka-youtube-converted/RaQBaVeEbW8.mp4/RaQBaVeEbW8.mp4354yes https://puzzicle.org/science/biology/v/ocean-acidification-and-biodiversity-impactshttps://puzzicle.org/thumbs/xa74094f2.jpgDemystifying ocean acidification and biodiversity impactsWhy rising carbon dioxide levels in the atmosphere affect the acidity of the ocean, and what the consequences are for marine life. Video by California Academy of Sciences.https://cdn.kastatic.org/ka-youtube-converted/GL7qJYKzcsk.mp4/GL7qJYKzcsk.mp4733yes https://puzzicle.org/science/biology/v/conservation-and-the-race-to-save-biodiversityhttps://puzzicle.org/thumbs/x34bb666c.jpgConservation and the race to save biodiversityConservation, a study focused on preventing loss of Earth's biological diversity, uses two key approaches: protecting species and safeguarding their habitats. Laws, international treaties, and species survival plans help shield endangered species. Meanwhile, conserving places, like biodiversity hotspots and protected parks, ensures the survival of entire ecosystems. Emerging fields like conservation genetics offer new perspectives, emphasizing the importance of protecting genetic diversity within ecosystems.https://cdn.kastatic.org/ka-youtube-converted/-u8xi7KFHY4.mp4/-u8xi7KFHY4.mp4724yes https://puzzicle.org/science/biology/v/protecting-biodiversity-the-power-of-the-individualhttps://puzzicle.org/thumbs/xa0b8415e.jpgProtecting biodiversity: the power of the individualIndividuals hold significant power and responsibility in safeguarding biodiversity. This involves gaining knowledge about biodiversity, understanding our actions' effects on ecosystems, and making eco-friendly consumer choices. It's also crucial to reuse and recycle, engage in citizen science projects, and spread the passion for biodiversity.https://cdn.kastatic.org/ka-youtube-converted/PxAHbuyBvBY.mp4/PxAHbuyBvBY.mp4608yes https://puzzicle.org/science/biology/v/protecting-biodiversity-local-and-global-policieshttps://puzzicle.org/thumbs/x46fd888f.jpgProtecting biodiversity: local and global policiesPolicy making plays a crucial role in protecting biodiversity. Since organisms do not adhere to human-made borders, conservation efforts need to be local, regional, national, and international. While economic factors often overshadow conservation, long-term environmental effects are starting to impact short-term views. Scientific data is vital in shaping effective policies.https://cdn.kastatic.org/ka-youtube-converted/mXsFQONwIic.mp4/mXsFQONwIic.mp4640yes https://puzzicle.org/science/biology/v/genes-and-biodiversityhttps://puzzicle.org/thumbs/x79f3cdca.jpgGenes and biodiversityHow does the gene pool contribute to biodiversity? Learn how genetic diversity within populations contributes to species survival. Video by California Academy of Sciences.https://cdn.kastatic.org/ka-youtube-converted/XOxsjdB4-ZQ.mp4/XOxsjdB4-ZQ.mp4401yes https://puzzicle.org/science/biology/v/ecosystem-biodiversityhttps://puzzicle.org/thumbs/x11780233.jpgEcosystem biodiversityEcosystem diversity is a critical aspect of Earth's biodiversity. Ecosystems, defined by habitat and climate, are unique and diverse. Species within ecosystems are specialized and vary globally. Protecting ecosystem diversity is vital, focusing not just on the most biodiverse but on the interconnectedness of all ecosystems. Preserving these complex webs of interactions is essential for the stability of Earth's ecosystems.https://cdn.kastatic.org/ka-youtube-converted/ixf33dQDlXc.mp4/ixf33dQDlXc.mp4254yes https://puzzicle.org/science/biology/v/the-lungs-and-pulmonary-systemhttps://puzzicle.org/thumbs/23986.jpgThe lungs and pulmonary systemBreathing is a vital process that allows our bodies to take in necessary oxygen and expel waste carbon dioxide. Air enters through the nose or mouth, journeys down the throat, and reaches the lungs via tubes called bronchi. These bronchi split into smaller tubes, bronchioles, ending in tiny air sacs known as alveoli. Oxygen crosses the alveoli membrane into our bloodstream, while carbon dioxide moves from the blood into the alveoli, ready to be exhaled.https://cdn.kastatic.org/ka-youtube-converted/SPGRkexI_cs.mp4/SPGRkexI_cs.mp41233yes https://puzzicle.org/science/biology/v/red-blood-cellshttps://puzzicle.org/thumbs/23988.jpgRed blood cellsOxygen uptake by hemoglobin in red blood cells.https://cdn.kastatic.org/ka-youtube-converted/fLKOBQ6cZHA.mp4/fLKOBQ6cZHA.mp4990yes https://puzzicle.org/science/biology/v/circulatory-system-and-the-hearthttps://puzzicle.org/thumbs/23990.jpgCirculatory system and the heartHemoglobin, found in red blood cells, absorbs oxygen, increasing the diffusion gradient. This process encourages oxygen to cross the membrane. Hemoglobin is contained within red blood cells to prevent blood from becoming too thick. The blood flows from the heart to the lungs via the pulmonary artery, gets oxygenated, and returns to the heart via the pulmonary vein. The oxygenated blood is then pumped to the rest of the body.https://cdn.kastatic.org/ka-youtube-converted/QhiVnFvshZg.mp4/QhiVnFvshZg.mp4897yes https://puzzicle.org/science/biology/v/hemoglobinhttps://puzzicle.org/thumbs/23992.jpgHemoglobinHemoglobin and its role in the circulatory system.https://cdn.kastatic.org/ka-youtube-converted/LWtXthfG9_M.mp4/LWtXthfG9_M.mp4874yes https://puzzicle.org/science/biology/v/anatomy-of-a-neuronhttps://puzzicle.org/thumbs/23994.jpgAnatomy of a neuronNeurons (or nerve cells) are specialized cells that transmit and receive electrical signals in the body. Neurons are composed of three main parts: dendrites, a cell body, and an axon. Signals are received through the dendrites, travel to the cell body, and continue down the axon until they reach the synapse (the communication point between two neurons).https://cdn.kastatic.org/ka-youtube-converted/ob5U8zPbAX4.mp4/ob5U8zPbAX4.mp4373yes https://puzzicle.org/science/biology/v/electrotonic-action-potentialhttps://puzzicle.org/thumbs/xbfcf48e2.jpgElectrotonic and action potentialsTwo different types of changes in the membrane potential of a neuron.https://cdn.kastatic.org/ka-youtube-converted/kY8FEq0teOs.mp4/kY8FEq0teOs.mp4618yes https://puzzicle.org/science/biology/v/saltatory-conduction-neuronshttps://puzzicle.org/thumbs/x45263500.jpgSaltatory conduction in neuronsMyelin sheaths, nodes of Ranvier, and saltatory conduction in neurons.https://cdn.kastatic.org/ka-youtube-converted/ikFUv-gdNLQ.mp4/ikFUv-gdNLQ.mp4668yes https://puzzicle.org/science/biology/v/neuronal-synapses-chemicalhttps://puzzicle.org/thumbs/24002.jpgNeuronal synapses (chemical)How one neuron can stimulate (or inhibit) another neuron at a chemical synapse.https://cdn.kastatic.org/ka-youtube-converted/Tbq-KZaXiL4.mp4/Tbq-KZaXiL4.mp4953yes https://puzzicle.org/science/biology/v/overview-of-the-functions-of-the-cerebral-cortexhttps://puzzicle.org/thumbs/xda7ddf7a.jpgOverview of the functions of the cerebral cortexThe human brain, with its wrinkly cerebral cortex, is split into four lobes: frontal, parietal, occipital, and temporal. The frontal lobe controls body movements and executive functions, including thinking and problem-solving. The parietal lobe processes touch, pressure, temperature, and spatial orientation. The occipital lobe handles vision, while the temporal lobe manages auditory processing and language comprehension.https://cdn.kastatic.org/ka-youtube-converted/X-m0JDCw6TE.mp4/X-m0JDCw6TE.mp4344yes https://puzzicle.org/science/biology/v/the-kidney-and-nephronhttps://puzzicle.org/thumbs/24028.jpgThe kidney and nephronThe video provides a detailed overview of the kidney's smallest functional unit, the nephron. It explains how the nephron filters blood, excretes waste, and maintains water levels. The video also highlights the role of the renal cortex and renal medulla in this process. The loop of Henle's function in reabsorbing water and making the renal medulla salty is also discussed.https://cdn.kastatic.org/ka-youtube-converted/cc8sUv2SuaY.mp4/cc8sUv2SuaY.mp41118yes https://puzzicle.org/science/biology/v/secondary-active-transport-in-the-nephronhttps://puzzicle.org/thumbs/24030.jpgSecondary active transport in the nephronThis video provides an in-depth look at how the nephron, the functional unit of the kidney, actively pumps molecules against their concentration gradients. It details the process of secondary active transport, focusing on the roles of sodium-potassium pumps and cotransporters in reabsorbing essential molecules like glucose, salts, and calcium from the filtrate.https://cdn.kastatic.org/ka-youtube-converted/czY5nyvZ7cU.mp4/czY5nyvZ7cU.mp4853yes https://puzzicle.org/science/biology/v/myosin-and-actinhttps://puzzicle.org/thumbs/24004.jpgMyosin and actinLearn how proteins, specifically myosin and actin, use ATP to produce movement in muscles. Understand the role of ATP hydrolysis in this process. This is a key part of how muscles function, converting chemical energy into mechanical energy.https://cdn.kastatic.org/ka-youtube-converted/zopoN2i7ALQ.mp4/zopoN2i7ALQ.mp4578yes https://puzzicle.org/science/biology/v/tropomyosin-and-troponin-and-their-role-in-regulating-muscle-contractionhttps://puzzicle.org/thumbs/24006.jpgHow tropomyosin and troponin regulate muscle contractionLean how calcium ion concentration dictates whether a muscle contracts or not. Explore how myosin II, using ATP, interacts with actin filaments to enable muscle contraction. Learn about the regulatory roles of proteins tropomyosin and troponin, and the crucial part calcium ions play in muscle contraction and relaxation. This is a deep look into muscle cell anatomy and function.https://cdn.kastatic.org/ka-youtube-converted/LiOfeSsjrB8.mp4/LiOfeSsjrB8.mp4562yes https://puzzicle.org/science/biology/v/role-of-the-sarcoplasmic-reticulum-in-muscle-cellshttps://puzzicle.org/thumbs/24008.jpgRole of the sarcoplasmic reticulum in muscle cellsExplore the role of the sarcoplasmic reticulum in muscle cells, crucial for muscle contraction and relaxation. Learn how calcium ions are regulated, triggering muscle contraction when released into the cell's cytoplasm. Understand the interaction between troponin, tropomyosin, myosin, and actin, and how the nervous system controls these processes.https://cdn.kastatic.org/ka-youtube-converted/SauhB2fYQkM.mp4/SauhB2fYQkM.mp4882yes https://puzzicle.org/science/biology/v/anatomy-of-a-muscle-cellhttps://puzzicle.org/thumbs/24010.jpgAnatomy of a skeletal muscle fiberUnderstanding the structure of a muscle fiber.https://cdn.kastatic.org/ka-youtube-converted/uY2ZOsCnXIA.mp4/uY2ZOsCnXIA.mp4992yes https://puzzicle.org/science/biology/v/role-of-phagocytes-in-innate-or-nonspecific-immunityhttps://puzzicle.org/thumbs/24012.jpgRole of phagocytes in innate or nonspecific immunityThe immune system's primary role is to protect the body from harmful pathogens. It employs two lines of defense for non-specific immunity: the first line, which includes skin and mucus membranes, aims to keep pathogens out. The second line, featuring phagocytes, targets pathogens that have entered the body. This non-specific or innate immune system does not distinguish between types of pathogens, but effectively eliminates them.https://cdn.kastatic.org/ka-youtube-converted/O1N2rENXq_Y.mp4/O1N2rENXq_Y.mp4980yes https://puzzicle.org/science/biology/v/types-of-immune-responses-innate-and-adaptive-humoral-vs-cell-mediatedhttps://puzzicle.org/thumbs/24014.jpgTypes of immune responses: Innate and adaptive, humoral vs. cell-mediatedThe immune system is divided into nonspecific (innate) and specific (adaptive) responses. Nonspecific defenses, like skin and stomach acid, block pathogens indiscriminately. Specific defenses, such as B and T lymphocytes, adapt to specific pathogens after exposure. B lymphocytes are part of the humoral response, dealing with pathogens in body fluids.https://cdn.kastatic.org/ka-youtube-converted/rp7T4IItbtM.mp4/rp7T4IItbtM.mp4487yes https://puzzicle.org/science/biology/v/b-lymphocytes-b-cellshttps://puzzicle.org/thumbs/24016.jpgB lymphocytes (B cells)Learn about the role of B cells (B lymphocytes) in the humoral immune response. See how they are activated and produce antibodies that can recognize and bind to pathogens, leading to their destruction. The video also highlights the process of clonal selection and the importance of memory cells in immunity.https://cdn.kastatic.org/ka-youtube-converted/Z36dUduOk1Y.mp4/Z36dUduOk1Y.mp4853yes https://puzzicle.org/science/biology/v/professional-antigen-presenting-cells-apc-and-mhc-ii-complexeshttps://puzzicle.org/thumbs/24018.jpgProfessional antigen presenting cells (APC) and MHC II complexesProfessional antigen presenting cells (APCs) are immune cells that specialize in presenting an antigen to a T-cell. The main types of professional APCs are dendritic cells (DC), macrophages, and B cells. A professional APC takes up an antigen, processes it, and returns part of it to its surface, along with a class II major histocompatibility complex (MHC). The T-cell is activated when it interacts with the formed complex.https://cdn.kastatic.org/ka-youtube-converted/j_kSmmEpvQk.mp4/j_kSmmEpvQk.mp4693yes https://puzzicle.org/science/biology/v/helper-t-cellshttps://puzzicle.org/thumbs/24020.jpgHelper T cellsExplore the adaptive immune system's key players: B cells, T cells, and helper T cells. Learn how B cells generate antibodies to disable pathogens, while helper T cells raise the alarm. Discover the double-check system that prevents autoimmune diseases and the role of cytotoxic T cells in attacking infiltrated cells.https://cdn.kastatic.org/ka-youtube-converted/uwMYpTYsNZM.mp4/uwMYpTYsNZM.mp41233yes https://puzzicle.org/science/biology/v/cytotoxic-t-cell-mhc-i-complexeshttps://puzzicle.org/thumbs/xb1e2a45d.jpgCytotoxic T cells and MHC I complexesHow damaged cells present antigens from inside the cell on MHC I and are recognized by cytotoxic T cells.https://cdn.kastatic.org/ka-youtube-converted/YdBXHm3edL8.mp4/YdBXHm3edL8.mp4556yes https://puzzicle.org/science/biology/v/review-of-b-cells-cd4-t-cells-and-cd8-t-cellshttps://puzzicle.org/thumbs/24024.jpgReview of B cells, CD4+ T cells and CD8+ T cellsReview the roles of B cells, T cells, and plasma cells in the immune system. Understand how these cells respond to pathogens, differentiate into effector and memory cells, and produce antibodies. Learn the distinction between CD4 and CD8 T cells and their functions in fighting infections and diseases.https://cdn.kastatic.org/ka-youtube-converted/xaz5ftvZCyI.mp4/xaz5ftvZCyI.mp4667yes https://puzzicle.org/science/biology/v/inflammatory-responsehttps://puzzicle.org/thumbs/24026.jpgInflammatory responseOverview of the inflammatory response.https://cdn.kastatic.org/ka-youtube-converted/FXSuEIMrPQk.mp4/FXSuEIMrPQk.mp4875yes https://puzzicle.org/science/biology/v/phototropismhttps://puzzicle.org/thumbs/x5bd7eab9044a6e96.jpgPhototropismPhototropism and the role that auxin plays.https://cdn.kastatic.org/ka-youtube-converted/2A_LKrLlhxE.mp4/2A_LKrLlhxE.mp4415yes https://puzzicle.org/science/biology/v/photoperiodismhttps://puzzicle.org/thumbs/x56f1b2c995788144.jpgPhotoperiodismHow plants can regulate processes like flowering based on day length (photoperiod).https://cdn.kastatic.org/ka-youtube-converted/AQf3gKTCffs.mp4/AQf3gKTCffs.mp4476yes https://puzzicle.org/science/biology/v/2015-ap-biology-free-response-1-a-chttps://puzzicle.org/thumbs/xc288bd09.jpg1a-c, Responses to the environmentGenetic and environmental influences on mouse activity cycles.https://cdn.kastatic.org/ka-youtube-converted/Tbpj-dNGrhc.mp4/Tbpj-dNGrhc.mp4672yes https://puzzicle.org/science/biology/v/2015-ap-biology-free-response-1-d-ehttps://puzzicle.org/thumbs/x9754acf0.jpg1d-e, Responses to the environment & natural selectionLight-dark activity cycle of mutant mice and evolutionary implications.https://cdn.kastatic.org/ka-youtube-converted/LpryknYN2xg.mp4/LpryknYN2xg.mp4424yes https://puzzicle.org/science/biology/v/2015-ap-biology-free-response-2-a-bhttps://puzzicle.org/thumbs/xb2b6f0c6.jpg2a-b, Cellular respiration & common ancestryQuestions on cellular respiration and common ancestry.https://cdn.kastatic.org/ka-youtube-converted/UNT8uh9io2o.mp4/UNT8uh9io2o.mp4704yes https://puzzicle.org/science/biology/v/2015-ap-biology-free-response-2-c-dhttps://puzzicle.org/thumbs/x85d1a41b.jpg2c-d, Cellular respiration & cell compartmentalization and its originsQuestions on cellular respiration and the origins of cell compartmentalization.https://cdn.kastatic.org/ka-youtube-converted/dktGVfUWIv8.mp4/dktGVfUWIv8.mp4572yes https://puzzicle.org/science/biology/v/2015-ap-biology-free-response-3https://puzzicle.org/thumbs/x721ed261.jpg3a-b, PhylogenyThinking about evolutionary distance between species. Constructing phylogenetic trees (cladograms).https://cdn.kastatic.org/ka-youtube-converted/yjU49bK1gkA.mp4/yjU49bK1gkA.mp4511yes https://puzzicle.org/science/biology/v/2015-ap-biology-free-response-4https://puzzicle.org/thumbs/x32d512c5.jpg4a-b, Meiosis and genetic diversityCommonalities and differences between mitosis and meiosis.https://cdn.kastatic.org/ka-youtube-converted/nfqYCmt6RAc.mp4/nfqYCmt6RAc.mp4324yes https://puzzicle.org/science/biology/v/2015-ap-biology-free-response-5https://puzzicle.org/thumbs/x41f7e698.jpg5a-b, Responses to the environmentPhototropism investigation.https://cdn.kastatic.org/ka-youtube-converted/JVbdSwK5i1c.mp4/JVbdSwK5i1c.mp4414yes https://puzzicle.org/science/biology/v/2015-ap-biology-free-response-6https://puzzicle.org/thumbs/x1b47e25f.jpg6a-c, Population ecologyFixing a declining snake population.https://cdn.kastatic.org/ka-youtube-converted/NEm1AupTkto.mp4/NEm1AupTkto.mp4356yes https://puzzicle.org/science/chemistry/v/chem37-history-of-atomic-chemistryhttps://puzzicle.org/thumbs/x56806d67.jpgThe history of atomic chemistryHow did we get here? Well, in terms of Atomic Chemistry, Hank takes us on a tour of the folks that were part of the long chain of other folks who helped us get to these deeper understandings of the world. From Leucippus to Heisenberg to you - yes, YOU - the story of Atomic Chemistry is all wibbly-wobbly... and amazing. Writers: Edi Gonzalez Chief Editor: Blake de Pastino Consultant: Dr. Heiko Langner Director/Editor: Nicholas Jenkins Sound Designer: Michael Aranda Graphics: Thought Cafehttps://cdn.kastatic.org/ka-youtube-converted/agjG7kgixJs.mp4/agjG7kgixJs.mp4527yes https://puzzicle.org/science/chemistry/v/rutherfords-gold-foil-experimenthttps://puzzicle.org/thumbs/x72ec884f.jpgRutherford’s gold foil experimentLearn about Rutherford's discovery of the nucleus and the development of the nuclear model of the atom.https://cdn.kastatic.org/ka-youtube-converted/bVlwH1kfDeg.mp4/bVlwH1kfDeg.mp4684yes https://puzzicle.org/science/chemistry/v/heisenberg-uncertainty-principlehttps://puzzicle.org/thumbs/x5a9c24f5.jpgHeisenberg uncertainty principleThe Heisenberg uncertainty principle states that there is a limit to how precisely certain pairs of physical properties of a particle can be known simultaneously. Explore the Heisenberg uncertainty principle by calculating uncertainty in position given the uncertainty in momentum for Bohr model of hydrogen.https://cdn.kastatic.org/ka-youtube-converted/PZIoFD_Z73M.mp4/PZIoFD_Z73M.mp4618yes https://puzzicle.org/science/chemistry/v/quantum-numbershttps://puzzicle.org/thumbs/x7448f8b7.jpgQuantum numbersAn orbital is a region around an atom's nucleus where electrons are likely to be found. Different types of orbitals (s, p, d, f) have different shapes and can hold different numbers of electrons. Learn how quantum numbers are used to describe the orbitals, and compare Bohr model orbits with the quantum mechanical model of atom.https://cdn.kastatic.org/ka-youtube-converted/KrXE_SzRoqw.mp4/KrXE_SzRoqw.mp4720yes https://puzzicle.org/science/chemistry/v/quantum-numbers-for-the-first-four-shellshttps://puzzicle.org/thumbs/xfe7eabb6.jpgQuantum numbers for the first four shellsCalculates number of orbitals and number of electrons in different kinds of orbitals for n = 1 to 4. Explains that only two electrons are allowed per orbital, and gives shortcuts for calculating number of orbitals and total number of electrons for a given n.https://cdn.kastatic.org/ka-youtube-converted/qLU0X154wlE.mp4/qLU0X154wlE.mp4757yes https://puzzicle.org/science/chemistry/v/noble-gas-configuration-electronic-structure-of-atoms-chemistry-khan-academyhttps://puzzicle.org/thumbs/x1faeaecd.jpgNoble gas configurationHow to write electron configurations for atoms and monatomic ions using noble gas configurationhttps://cdn.kastatic.org/ka-youtube-converted/tlSJCG5DqzI.mp4/tlSJCG5DqzI.mp4407yes https://puzzicle.org/science/chemistry/v/electron-configurations-for-the-first-periodhttps://puzzicle.org/thumbs/xf39357d2.jpgElectron configurations for the first periodIntroduces aufbau principle, Pauli exclusion principle, and orbital notation. Writes out the quantum numbers for the electrons in H and He.https://cdn.kastatic.org/ka-youtube-converted/0MKsFD03Ukc.mp4/0MKsFD03Ukc.mp4382yes https://puzzicle.org/science/chemistry/v/electron-configurations-for-the-second-periodhttps://puzzicle.org/thumbs/x50988bfb.jpgElectron configurations for the second periodWrites out quantum numbers for all elements of the second period. Introduces Hund's rule, and connects blocks in periodic table with electron configuration.https://cdn.kastatic.org/ka-youtube-converted/UEPs-GA2s40.mp4/UEPs-GA2s40.mp4732yes https://puzzicle.org/science/chemistry/v/electron-configurations-for-the-third-and-fourth-periodshttps://puzzicle.org/thumbs/xf22e57ff.jpgElectron configurations for the third and fourth periodsHow to write the electron configurations for elements in the third and fourth periods of the periodic table. includes using noble gas notation.https://cdn.kastatic.org/ka-youtube-converted/3tyrkxruz88.mp4/3tyrkxruz88.mp4788yes https://puzzicle.org/science/chemistry/v/paramagnetism-and-diamagnetismhttps://puzzicle.org/thumbs/xb73055ce.jpgParamagnetism and diamagnetismParamagnetism and diamagnetism from Chemistry archivehttps://cdn.kastatic.org/ka-youtube-converted/-lIc5z3XjXQ.mp4/-lIc5z3XjXQ.mp4485yes https://puzzicle.org/science/chemistry/v/periodic-table-groupshttps://puzzicle.org/thumbs/x68205349.jpgGroups of the periodic tableThe s-, p-, and d-block elements of the periodic table are arranged into 18 numbered columns, or groups. The elements in each group have the same number of valence electrons. As a result, elements in the same group often display similar properties and reactivity.https://cdn.kastatic.org/ka-youtube-converted/jQGUkJc7_04.mp4/jQGUkJc7_04.mp4461yes https://puzzicle.org/science/chemistry/v/periodic-table-valence-electronshttps://puzzicle.org/thumbs/xf654ebd5.jpgCounting valence electrons for main group elementsHow to determine the number of valence electrons and draw Lewis structures for main group elements starting from the electron configuration.https://cdn.kastatic.org/ka-youtube-converted/akm5H2JsccI.mp4/akm5H2JsccI.mp4499yes https://puzzicle.org/science/chemistry/v/periodic-table-transition-metalshttps://puzzicle.org/thumbs/x545922ca.jpgThe periodic table - transition metalsThe definition of a transition metal, and how to write the electron configuration including examples for Fe and Zn.https://cdn.kastatic.org/ka-youtube-converted/kCM2mSb4qIU.mp4/kCM2mSb4qIU.mp4334yes https://puzzicle.org/science/chemistry/v/atomic-radius-trendhttps://puzzicle.org/thumbs/x2875bdf8.jpgAtomic radius trends on periodic tableHow atomic radius is defined, and trends across a period and down a group.https://cdn.kastatic.org/ka-youtube-converted/q--2WP8wXtk.mp4/q--2WP8wXtk.mp4580yes https://puzzicle.org/science/chemistry/v/mini-video-on-ion-sizehttps://puzzicle.org/thumbs/426005.jpgMini-video on ion sizeCorrecting a mistake and learning a bit about ion sizehttps://cdn.kastatic.org/ka-youtube-converted/HBi8xjMchZc.mp4/HBi8xjMchZc.mp4184yes https://puzzicle.org/science/chemistry/v/ionization-energy-trendshttps://puzzicle.org/thumbs/xaed272e3.jpgIonization energy trendsDefinition of ion and ionization energy, and trends in ionization energy across a period and down a group.https://cdn.kastatic.org/ka-youtube-converted/5CBs36jtZxY.mp4/5CBs36jtZxY.mp4602yes https://puzzicle.org/science/chemistry/v/electronegativity-and-chemical-bondshttps://puzzicle.org/thumbs/x57ecb61f.jpgElectronegativity and bondingElectronegativity differences in bonding using the Pauling scale. Classifying bonds as covalent, polar covalent, or ionic.https://cdn.kastatic.org/ka-youtube-converted/126N4hox9YA.mp4/126N4hox9YA.mp4699yes https://puzzicle.org/science/chemistry/v/ionic-bonds-and-coulombs-lawhttps://puzzicle.org/thumbs/x88a88398.jpgIonic bonds and Coulomb's lawIntroduction to how the strength of ionic bonds is related to Coulomb's law. Example of using Coulomb's law to explain differences in melting points of ionic compounds.https://cdn.kastatic.org/ka-youtube-converted/TwqMtO2FLYk.mp4/TwqMtO2FLYk.mp4452yes https://puzzicle.org/science/chemistry/v/drawing-dot-structureshttps://puzzicle.org/thumbs/x4b64d771.jpgDrawing dot structuresGuidelines for drawing Lewis dot structures.https://cdn.kastatic.org/ka-youtube-converted/p7Fsb21B2Xg.mp4/p7Fsb21B2Xg.mp4882yes https://puzzicle.org/science/chemistry/v/resonance-and-dot-structureshttps://puzzicle.org/thumbs/xc17639e8.jpgResonance and dot structuresDrawing resonance structures and resonance hybrid using example of nitrate anion.https://cdn.kastatic.org/ka-youtube-converted/bUCu7bPkZeI.mp4/bUCu7bPkZeI.mp4375yes https://puzzicle.org/science/chemistry/v/formal-charge-and-dot-structureshttps://puzzicle.org/thumbs/x999bc93d.jpgFormal charge and dot structuresDefinition of formal charge, and how minimization of formal charge can help choose the more stable dot structure.https://cdn.kastatic.org/ka-youtube-converted/dNPs-cr_6Bk.mp4/dNPs-cr_6Bk.mp4690yes https://puzzicle.org/science/chemistry/v/more-on-the-dot-structure-for-sulfur-dioxidehttps://puzzicle.org/thumbs/xdc597ee2.jpgMore on the dot structure for sulfur dioxideHow to analyze different ways to draw the dot structure for sulfur dioxidehttps://cdn.kastatic.org/ka-youtube-converted/Fr_La6MrAfM.mp4/Fr_La6MrAfM.mp4268yes https://puzzicle.org/science/chemistry/v/2015-ap-chemistry-free-response-2-d-ehttps://puzzicle.org/thumbs/x1dae2933.jpg2015 AP Chemistry free response 2d and eLewis electron-dot diagram and bond angles for ethanol. From 2015 AP Chemistry free response 2d and 2e.https://cdn.kastatic.org/ka-youtube-converted/ImTwySWxA4U.mp4/ImTwySWxA4U.mp4380yes https://puzzicle.org/science/hs-chemistry/v/balancing-chemical-equations-introductionhttps://puzzicle.org/thumbs/xc8ecefc8.jpgBalancing chemical equationsA balanced chemical equation shows the same number of each type of atom on both sides of the arrow.https://cdn.kastatic.org/ka-youtube-converted/TUuABq95BBM.mp4/TUuABq95BBM.mp4304yes https://puzzicle.org/science/hs-chemistry/v/balancing-more-complex-chemical-equationhttps://puzzicle.org/thumbs/x068874ad.jpgBalancing more complex chemical equationsA balanced chemical equation shows the same number of each type of atom on both sides of the arrow. In this example, we balance the combustion reaction of ethylene, C₂H₄, and provide tips on how to balance more complex chemical equations.https://cdn.kastatic.org/ka-youtube-converted/xqpYeiefZl8.mp4/xqpYeiefZl8.mp4277yes https://puzzicle.org/science/hs-chemistry/v/visually-understanding-balancing-chemical-equationshttps://puzzicle.org/thumbs/x58fe4869.jpgVisually understanding balancing chemical equationsA balanced chemical equation shows the same number of each type of atom on both sides of the arrow. We can relate a balanced chemical equation to the structural formulas of its reactants and products.https://cdn.kastatic.org/ka-youtube-converted/HbIdp5DURJA.mp4/HbIdp5DURJA.mp4277yes https://puzzicle.org/science/hs-chemistry/v/balancing-another-combustion-reactionhttps://puzzicle.org/thumbs/xf98cead0.jpgBalancing another combustion reactionA balanced chemical equation shows the same number of each type of atom on both sides of the arrow. In this example, we balance the combustion reaction of ethane, C₂H₆.https://cdn.kastatic.org/ka-youtube-converted/cmi8EnJQA84.mp4/cmi8EnJQA84.mp4298yes https://puzzicle.org/science/chemistry/v/balancing-chemical-equation-with-substitutionhttps://puzzicle.org/thumbs/xf8a51a8a.jpgBalancing chemical equation with substitutionUsing the substitution method to balance a reaction where a reactant and product have a polyatomic ion in common.https://cdn.kastatic.org/ka-youtube-converted/8KXWJCmshEE.mp4/8KXWJCmshEE.mp4252yes https://puzzicle.org/science/chemistry/v/chem09-precipitation-reactionshttps://puzzicle.org/thumbs/xb7a71439.jpgPrecipitation reactionsA lot of ionic compounds dissolve in water, dissociating into individual ions. But when two ions find each other that form an insoluble compound, they suddenly fall out of solution in what's called a precipitation reaction. In this episode of Crash Course Chemistry, we learn about precipitation, precipitates, anions, cations, and how to describe and discuss ionic reactions. Writer: Kim Krieger Chief Editor: Blake de Pastino Consultant: Dr. Heiko Langner Director/Editor: Nicholas Jenkins Sound Designer: Michael Aranda Graphics: Thought Cafehttps://cdn.kastatic.org/ka-youtube-converted/cdvV8O90HJM.mp4/cdvV8O90HJM.mp4634yes https://puzzicle.org/science/chemistry/v/vapor-pressure-examplehttps://puzzicle.org/thumbs/6022729.jpgWorked example: Vapor pressure and the ideal gas lawVapor pressure example using the ideal gas law.https://cdn.kastatic.org/ka-youtube-converted/-QpkmwIoMaY.mp4/-QpkmwIoMaY.mp4962yes https://puzzicle.org/science/chemistry/v/van-der-waals-equationhttps://puzzicle.org/thumbs/xc0a2763a9023469c.jpgThe van der Waals equationBy adding corrections for interparticle attractions and particle volumes to the ideal gas law, we can derive a new equation that more accurately describes real gas behavior. This equation, known as the van der Waals equation, can be used to calculate the properties of a gas under non-ideal conditions.https://cdn.kastatic.org/ka-youtube-converted/9_epJ4vUsNI.mp4/9_epJ4vUsNI.mp4317yes https://puzzicle.org/science/chemistry/v/specific-heat-and-latent-leat-of-fusion-and-vaporization-2https://puzzicle.org/thumbs/xf8a20837.jpgSpecific heat and latent heat of fusion and vaporizationDefining specific heat, heat of fusion, and heat of vaporization. How to calculate the amount of heat to change the temperature of water and the energy required to change for a phase change.https://cdn.kastatic.org/ka-youtube-converted/WU7TfO-iaK8.mp4/WU7TfO-iaK8.mp4897yes https://puzzicle.org/science/chemistry/v/specific-heat-heat-of-fusion-and-vaporizationhttps://puzzicle.org/thumbs/418012.jpgSpecific heat, heat of fusion and vaporization exampleSpecific heat and phase changes: Calculating how much heat is needed to convert 200 g of ice at -10 degrees C to 110 degree steam.https://cdn.kastatic.org/ka-youtube-converted/zz4KbvF_X-0.mp4/zz4KbvF_X-0.mp4889yes https://puzzicle.org/science/chemistry/v/chilling-water-problemhttps://puzzicle.org/thumbs/432003.jpgChilling water problemHow much ice at -10 degrees C is necessary to get 500 g of water down to 0 degrees C?https://cdn.kastatic.org/ka-youtube-converted/lsXcKgjg8Hs.mp4/lsXcKgjg8Hs.mp4683yes https://puzzicle.org/science/chemistry/v/change-of-state-examplehttps://puzzicle.org/thumbs/6022730.jpgChange of state exampleSpecific heat capacity and enthalpy of vaporization example: calculating how much energy it takes to vaporize 1.00 kg of ethanol starting from 20 degrees C.https://cdn.kastatic.org/ka-youtube-converted/tvO0358YUYM.mp4/tvO0358YUYM.mp4458yes https://puzzicle.org/science/chemistry/v/vapor-pressurehttps://puzzicle.org/thumbs/417006.jpgVapor pressureVapor pressure, volatility, and evaporation.https://cdn.kastatic.org/ka-youtube-converted/hA5jddDYcyg.mp4/hA5jddDYcyg.mp41084yes https://puzzicle.org/science/chemistry/v/phase-diagramshttps://puzzicle.org/thumbs/373049.jpgPhase diagramsUnderstanding and interpreting phase diagramshttps://cdn.kastatic.org/ka-youtube-converted/Qp87Z4m8R-w.mp4/Qp87Z4m8R-w.mp4756yes https://puzzicle.org/science/chemistry/v/boiling-points-of-organic-compoundshttps://puzzicle.org/thumbs/xe50d545d.jpgBoiling points of organic compoundsHow to analyze the different boiling points of organic compounds using intermolecular forceshttps://cdn.kastatic.org/ka-youtube-converted/pILGRZ0nT4o.mp4/pILGRZ0nT4o.mp4667yes https://puzzicle.org/science/chemistry/v/boiling-points-ap-chemistry-multiple-choice-samplehttps://puzzicle.org/thumbs/x92f9f310.jpgBoiling point comparison: AP Chemistry multiple choiceComparing the boiling point of nonane and 2,3,4-trifluoropentane. A sample multiple choice problem from the 2014 AP course description.https://cdn.kastatic.org/ka-youtube-converted/JRxFsmvsmHY.mp4/JRxFsmvsmHY.mp4379yes https://puzzicle.org/science/chemistry/v/solubility-of-organic-compounds-redohttps://puzzicle.org/thumbs/xd9a60692.jpgSolubility of organic compoundsOrganic compounds tend to dissolve well in solvents that have similar properties to themselves. This principle is often referred to as "like dissolves like," which means that polar molecules will generally dissolve well in polar solvents and non-polar molecules will generally dissolve in non-polar solvents.https://cdn.kastatic.org/ka-youtube-converted/_mWHWJiMKXU.mp4/_mWHWJiMKXU.mp4741yes https://puzzicle.org/science/chemistry/v/le-chateliers-principle-worked-example-chemical-equilibrium-chemistry-khan-acadehttps://puzzicle.org/thumbs/xee318355.jpgLe Chatelier's principle: Worked exampleA worked example using Le Chatelier's principle to predict how concentrations will shift for different perturbations. Example includes changing reaction vessel volume, changing amount of solid product, adding inert gas, and adding a catalyst.https://cdn.kastatic.org/ka-youtube-converted/5gujU2QcGcY.mp4/5gujU2QcGcY.mp4477yes https://puzzicle.org/science/chemistry/v/intro-to-reaction-quotient-qhttps://puzzicle.org/thumbs/x353b5962.jpgIntroduction to reaction quotient QcIntroduction to the reaction quotient Qc, and comparing the reaction quotient with the equilibrium constant to predict how concentrations will change.https://cdn.kastatic.org/ka-youtube-converted/2PM1yc_z4Bk.mp4/2PM1yc_z4Bk.mp4456yes https://puzzicle.org/science/chemistry/v/acid-strength-anion-size-and-bond-energyhttps://puzzicle.org/thumbs/x13d9aabd.jpgAcid strength, anion size, and bond energyHow anion size and bond dissociation energies affect acid strengthhttps://cdn.kastatic.org/ka-youtube-converted/Ce4BGV1DVVg.mp4/Ce4BGV1DVVg.mp4347yes https://puzzicle.org/science/chemistry/v/relationship-between-ka-and-kbhttps://puzzicle.org/thumbs/xe0b51228.jpgRelationship between Ka and KbThe relationship between Ka and Kb, and pKa and pKb. Examples of finding Ka of a weak acid given Kb of the conjugate base.https://cdn.kastatic.org/ka-youtube-converted/DGMs81-Rp1o.mp4/DGMs81-Rp1o.mp4680yes https://puzzicle.org/science/chemistry/v/ka-and-acid-strengthhttps://puzzicle.org/thumbs/x4c27d237.jpgKa and acid strengthHow to write an equilibrium expression for an acid-base reaction and how to evaluate the strength of an acid using Kahttps://cdn.kastatic.org/ka-youtube-converted/BeHOvYchtBg.mp4/BeHOvYchtBg.mp4558yes https://puzzicle.org/science/chemistry/v/weak-acid-equilibriumhttps://puzzicle.org/thumbs/xe962fbd5.jpgWeak acid equilibriumQuick overview of Ka and pKa. Example of calculating the pH of a weak acid solution.https://cdn.kastatic.org/ka-youtube-converted/icNOBmKjDLo.mp4/icNOBmKjDLo.mp4571yes https://puzzicle.org/science/chemistry/v/weak-base-equilibriumhttps://puzzicle.org/thumbs/x1676f258.jpgWeak base equilibriumQuick overview of Kb and pKb. Examples of calculating the pH of a weak base solution.https://cdn.kastatic.org/ka-youtube-converted/xGE3sEz6U9Y.mp4/xGE3sEz6U9Y.mp4682yes https://puzzicle.org/science/chemistry/v/ph-of-salt-solutionshttps://puzzicle.org/thumbs/x3029a604.jpgpH of salt solutionsThe pH of a salt solution is determined by the relative strength of its ​conjugated acid-base pair. Salts can be acidic, neutral, or basic. Salts that form from a strong acid and a weak base are acid salts, like ammonium chloride (NH4Cl). Salts that form from a weak acid and a strong base are basic salts, like sodium bicarbonate (NaHCO3).https://cdn.kastatic.org/ka-youtube-converted/XZWoMXVANww.mp4/XZWoMXVANww.mp4855yes https://puzzicle.org/science/chemistry/v/common-ion-effecthttps://puzzicle.org/thumbs/x177cf25c.jpgCommon ion effect and buffersThe common ion effect describes the effect on ​equilibrium that occurs when a common ion (an ion that is already contained in the solution) is added to a solution. The common ion effect generally decreases ​solubility of a solute. It also can have an effect on buffering solutions, as adding more conjugate ions may shift the pH of the solution.https://cdn.kastatic.org/ka-youtube-converted/223KLPnJCBI.mp4/223KLPnJCBI.mp4799yes https://puzzicle.org/science/chemistry/v/buffer-solution-calculationshttps://puzzicle.org/thumbs/x81deb5b6.jpgBuffer solution pH calculationsExample of calculating the pH of a buffer solution using the Henderson-Hasselbalch equation, including the pH of the buffer solution after adding some NaOH.https://cdn.kastatic.org/ka-youtube-converted/lsHq5aqz4uQ.mp4/lsHq5aqz4uQ.mp4699yes https://puzzicle.org/science/chemistry/v/ph-and-pka-relationship-for-buffershttps://puzzicle.org/thumbs/xe0d3b723.jpgpH and pKa relationship for buffersHow the Henderson-Hasselbalch equation can be used to look at the ratio of conjugate acid and base using relationship between buffer pH and pKa.https://cdn.kastatic.org/ka-youtube-converted/7ZLjzp7thNI.mp4/7ZLjzp7thNI.mp4470yes https://puzzicle.org/science/chemistry/v/buffer-capacityhttps://puzzicle.org/thumbs/x8be0d8f0.jpgBuffer capacityThe definition of buffer capacity, and an example showing why it depends on the absolute concentrations of the conjugate acid and base.https://cdn.kastatic.org/ka-youtube-converted/pnPWdDOR9XE.mp4/pnPWdDOR9XE.mp4644yes https://puzzicle.org/science/chemistry/v/titration-of-a-strong-acid-with-a-strong-basehttps://puzzicle.org/thumbs/x884a1838.jpgTitration of a strong acid with a strong baseIn a strong acid-strong base titration, the acid and base will react to form a neutral solution. At the equivalence point of the reaction, hydronium (H+) and hydroxide (OH-) ions will react to form water, leading to a pH of 7. This is true of all strong acid-strong base titrations. An example of this would be the titration of hydrochloric acid (strong acid) and sodium hydroxide (strong base) to form sodium chloride and water.https://cdn.kastatic.org/ka-youtube-converted/JoGQYSTlOKo.mp4/JoGQYSTlOKo.mp4612yes https://puzzicle.org/science/chemistry/v/titration-of-a-strong-acid-with-a-strong-base-continuedhttps://puzzicle.org/thumbs/x0f5bb35f.jpgTitration of a strong acid with a strong base (continued)Calculating the pH at and after the equivalence point for titration of strong acid, hydrochloric acid, with strong base, NaOH.https://cdn.kastatic.org/ka-youtube-converted/HO1dsNrzE58.mp4/HO1dsNrzE58.mp4617yes https://puzzicle.org/science/chemistry/v/titration-of-a-weak-acid-with-a-strong-basehttps://puzzicle.org/thumbs/xe3933619.jpgTitration of a weak acid with a strong baseCalculating the pH for titration of acetic acid with strong base NaOH before adding any base and at half-equivalence point.https://cdn.kastatic.org/ka-youtube-converted/x3CbfUr449Y.mp4/x3CbfUr449Y.mp4867yes https://puzzicle.org/science/chemistry/v/titration-of-a-weak-acid-with-a-strong-base-continuedhttps://puzzicle.org/thumbs/xcf6e8167.jpgTitration of a weak acid with a strong base (continued)Calculating the pH for titration of acetic acid with strong base NaOH at equivalence point and past the equivalence point.https://cdn.kastatic.org/ka-youtube-converted/WbDL7xN-Pn0.mp4/WbDL7xN-Pn0.mp4913yes https://puzzicle.org/science/chemistry/v/titration-of-a-weak-base-with-a-strong-acidhttps://puzzicle.org/thumbs/xa23021eb.jpgTitration of a weak base with a strong acidIn a weak base-strong acid titration, the acid and base will react to form an acidic solution. A conjugate acid will be produced during the titration, which then reacts with water to form hydronium ions. This results in a solution with a pH lower than 7. An example of this is the titration of hydrochloric acid (strong acid) into ammonia (weak base), which forms the conjugate acid ammonium and produces an acidic solution.https://cdn.kastatic.org/ka-youtube-converted/pmHlKhv7C0E.mp4/pmHlKhv7C0E.mp4898yes https://puzzicle.org/science/chemistry/v/titration-of-a-weak-base-with-a-strong-acid-continuedhttps://puzzicle.org/thumbs/xf9342364.jpgTitration of a weak base with a strong acid (continued)Calculating the pH for titration of weak base, ammonia, with strong acid, HCl, at the equivalence point and past the equivalence point.https://cdn.kastatic.org/ka-youtube-converted/kWucfgOkCIQ.mp4/kWucfgOkCIQ.mp4888yes https://puzzicle.org/science/chemistry/v/titration-curves-and-acid-base-indicatorshttps://puzzicle.org/thumbs/x394a1fa6.jpgTitration curves and acid-base indicatorsChoosing the best indicator for different titrations depending on the pH at the equivalence point.https://cdn.kastatic.org/ka-youtube-converted/1YlXtpF-19A.mp4/1YlXtpF-19A.mp4430yes https://puzzicle.org/science/chemistry/v/2015-ap-chemistry-free-response-4https://puzzicle.org/thumbs/x252359c2.jpg2015 AP Chemistry free response 4Calculating the molar solubility of calcium hydroxide in a 0.10 M calcium nitrate solution. From 2015 AP Chemistry free response 4.https://cdn.kastatic.org/ka-youtube-converted/JbLRFt_Z8nI.mp4/JbLRFt_Z8nI.mp4560yes https://puzzicle.org/science/chemistry/v/solubility-and-complex-ion-formationhttps://puzzicle.org/thumbs/xd322f5e7.jpgSolubility and complex ion formationCalculating the solubility of silver chloride in pure water and 3.0 M ammonia. Calculating new equilibrium constant to account for formation of complex ion.https://cdn.kastatic.org/ka-youtube-converted/r7gTH_5XfOI.mp4/r7gTH_5XfOI.mp4890yes https://puzzicle.org/science/chemistry/v/disproportionationhttps://puzzicle.org/thumbs/x528f68ef.jpgDisproportionationAssigning oxidation states for the decomposition of hydrogen peroxide, a disproportionation reaction.https://cdn.kastatic.org/ka-youtube-converted/PXlvKtpvUEk.mp4/PXlvKtpvUEk.mp4303yes https://puzzicle.org/science/chemistry/v/redox-reaction-from-dissolving-zinc-in-copper-sulfatehttps://puzzicle.org/thumbs/xfc2d5f2d.jpgRedox reaction from dissolving zinc in copper sulfateWhat happens when you add zinc to a solution of copper sulfate? Identifying the half reactions to see what got oxidized and reduced.https://cdn.kastatic.org/ka-youtube-converted/wVMe8ZmCZo0.mp4/wVMe8ZmCZo0.mp4301yes https://puzzicle.org/science/chemistry/v/galvanic-cell-voltaic-cellhttps://puzzicle.org/thumbs/xe7416dc9.jpgIntroduction to galvanic/voltaic cellsHow to use a redox reaction to construct a galvanic/voltaic cell to produce a flow of current.. Shows the flow of electrons and ions, and explains the role of the salt bridge.https://cdn.kastatic.org/ka-youtube-converted/9Xncz_mMc5g.mp4/9Xncz_mMc5g.mp4360yes https://puzzicle.org/science/chemistry/v/electrodes-and-voltage-of-galvanic-cellhttps://puzzicle.org/thumbs/x1773beb6.jpgElectrodes and voltage of Galvanic cellIdentifying the anode and cathode in a galvanic cell, and calculating the voltage using standard electrode potentials.https://cdn.kastatic.org/ka-youtube-converted/m0SFJt_0v28.mp4/m0SFJt_0v28.mp4273yes https://puzzicle.org/science/chemistry/v/shorthand-notation-for-galvanicvoltaic-cellshttps://puzzicle.org/thumbs/x13a28207.jpgShorthand notation for galvanic/voltaic cellsGet a grip on Galvanic cells in this electrochemistry tutorial! You'll see how redox reactions generate electric currents, understand the functions of anodes and cathodes, and learn the shorthand notation for these cells.https://cdn.kastatic.org/ka-youtube-converted/xHuLZHasldg.mp4/xHuLZHasldg.mp4437yes https://puzzicle.org/science/chemistry/v/lead-storage-batteryhttps://puzzicle.org/thumbs/x51f5d849.jpgLead storage batteryExplore the process of balancing redox reactions in acidic solutions using a lead storage battery as an example. Understand the assignment of oxidation states, the creation of oxidation and reduction half-reactions, and how to balance atoms and charge. Discover how this reaction powers car batteries and can be reversed for recharging.https://cdn.kastatic.org/ka-youtube-converted/PQ48N5jaG2w.mp4/PQ48N5jaG2w.mp4546yes https://puzzicle.org/science/chemistry/v/nickel-cadmium-batteryhttps://puzzicle.org/thumbs/xc191ac11.jpgNickel-cadmium batteryGet charged up about the nickel-cadmium battery! This tutorial breaks down the redox reaction that powers these rechargeable batteries. Learn how solid cadmium and nickel oxide hydroxide transform into cadmium hydroxide and nickel hydroxide, and how this process is easily reversed, making recharging a breeze.https://cdn.kastatic.org/ka-youtube-converted/tUvpwdqdLlI.mp4/tUvpwdqdLlI.mp4667yes https://puzzicle.org/science/chemistry/v/standard-reduction-potentialshttps://puzzicle.org/thumbs/x9f0707ce.jpgStandard reduction potentialsHow to use a table of standard reduction potentials to calculate standard cell potential. Identifying trends in oxidizing and reducing agent strength.https://cdn.kastatic.org/ka-youtube-converted/oAOq55MPJpM.mp4/oAOq55MPJpM.mp4550yes https://puzzicle.org/science/chemistry/v/voltage-as-an-intensive-propertyhttps://puzzicle.org/thumbs/xda15e076.jpgVoltage as an intensive propertyDemonstrating that voltage is an intensive property by calculating the voltage and change in Gibbs free energy for a half reaction.https://cdn.kastatic.org/ka-youtube-converted/xsSLvlonxs4.mp4/xsSLvlonxs4.mp4322yes https://puzzicle.org/science/chemistry/v/using-reduction-potentialshttps://puzzicle.org/thumbs/xa118949b.jpgUsing reduction potentialsLearn how to calculate the standard cell potential using standard reduction potentials. This video explains the process of oxidation and reduction, how to identify half-reactions, and the role of moles in these reactions.https://cdn.kastatic.org/ka-youtube-converted/79OmpES_9OE.mp4/79OmpES_9OE.mp4364yes https://puzzicle.org/science/chemistry/v/spontaneity-and-redox-reactionshttps://puzzicle.org/thumbs/x1c447583.jpgSpontaneity and redox reactionsLearn how lead can oxidize aluminum but not copper in standard conditions. Using reduction potential tables and the diagonal rule, we confirm predictions by calculating standard cell potentials. Understand the spontaneity of reactions and the activity series in chemistry.https://cdn.kastatic.org/ka-youtube-converted/sorB6XFJ8Jc.mp4/sorB6XFJ8Jc.mp4742yes https://puzzicle.org/science/chemistry/v/free-energy-and-cell-potentialhttps://puzzicle.org/thumbs/xf918ece2.jpgFree energy and cell potentialRelationship between Gibbs free energy and standard cell potential.https://cdn.kastatic.org/ka-youtube-converted/4tcLzVJdxZs.mp4/4tcLzVJdxZs.mp4494yes https://puzzicle.org/science/chemistry/v/standard-cell-potential-and-the-equilibrium-constanthttps://puzzicle.org/thumbs/x4f2f97b8.jpgStandard cell potential and the equilibrium constantFinding the relationship between standard cell potential and equilibrium constant K.https://cdn.kastatic.org/ka-youtube-converted/CdE1Sa18gIo.mp4/CdE1Sa18gIo.mp4322yes https://puzzicle.org/science/chemistry/v/calculating-the-equilibrium-constant-from-the-standard-cell-potentialhttps://puzzicle.org/thumbs/x0a77eceb.jpgCalculating the equilibrium constant from the standard cell potentialExample problem for calculating the equilibrium constant K using the standard cell potential.https://cdn.kastatic.org/ka-youtube-converted/kxNaVzQ07yc.mp4/kxNaVzQ07yc.mp4583yes https://puzzicle.org/science/chemistry/v/galvanic-cells-and-changes-in-free-energyhttps://puzzicle.org/thumbs/x0c9df053.jpgGalvanic cells and changes in free energyRelationship between Gibbs free energy, reaction quotient Q, and cell voltage.https://cdn.kastatic.org/ka-youtube-converted/TYFAJHx5LFA.mp4/TYFAJHx5LFA.mp4758yes https://puzzicle.org/science/chemistry/v/nernst-equationhttps://puzzicle.org/thumbs/xac7a2c03.jpgNernst equationDeriving a few different forms of the Nernst equation, the relationship between Gibbs free energy and reaction quotient Q.https://cdn.kastatic.org/ka-youtube-converted/QaOtsYouet0.mp4/QaOtsYouet0.mp4393yes https://puzzicle.org/science/chemistry/v/using-the-nernst-equationhttps://puzzicle.org/thumbs/x0442f790.jpgUsing the Nernst equationUsing the Nernst equation to calculate the cell potential when concentrations are not standard conditions.https://cdn.kastatic.org/ka-youtube-converted/WqsJNtBEKtg.mp4/WqsJNtBEKtg.mp4690yes https://puzzicle.org/science/chemistry/v/concentration-cellhttps://puzzicle.org/thumbs/x2c135e90.jpgConcentration cellConcentration cells are galvanic (or voltaic) cells made of two half-cells, each of which containing the same electrodes, but different concentrations. Concentration cells work to establish equilibrium by transferring electrons from the cell with the lower concentration to the cell with the higher concentration. The electrode potential difference between the two half-cells can be calculated using the Nernst equation.https://cdn.kastatic.org/ka-youtube-converted/mMzbr5rm4j8.mp4/mMzbr5rm4j8.mp4578yes https://puzzicle.org/science/chemistry/v/quantitative-electrolysishttps://puzzicle.org/thumbs/x37e22dab.jpgQuantitative electrolysisCalculating how much zinc deposits on the zinc electrode after 1.0 h when a current of 5.0 A is applied to the battery.https://cdn.kastatic.org/ka-youtube-converted/QEUsZeKYsgo.mp4/QEUsZeKYsgo.mp4420yes https://puzzicle.org/science/chemistry/v/electrolysis-of-molten-sodium-chloridehttps://puzzicle.org/thumbs/xdf4ac844.jpgElectrolysis of molten sodium chlorideExample quantitative electrolysis problem using molten sodium chloride.https://cdn.kastatic.org/ka-youtube-converted/Q9BtOLDr0vg.mp4/Q9BtOLDr0vg.mp4744yes https://puzzicle.org/science/chemistry/v/first-order-reaction-with-calculushttps://puzzicle.org/thumbs/x6e4aa8e2.jpgFirst-order reaction (with calculus)Deriving the integrated rate law for first-order reactions using calculus. How you can graph first-order rate data to see a linear relationship.https://cdn.kastatic.org/ka-youtube-converted/Bt0mz4mGddk.mp4/Bt0mz4mGddk.mp4434yes https://puzzicle.org/science/chemistry/v/plotting-data-for-a-first-order-reactionhttps://puzzicle.org/thumbs/x2fe55283.jpgPlotting data for a first-order reactionExample of graphing first-order rate data to see a linear relationship, and calculating rate constant k from the slope.https://cdn.kastatic.org/ka-youtube-converted/vN7O7zsRKb8.mp4/vN7O7zsRKb8.mp4561yes https://puzzicle.org/science/chemistry/v/second-order-reaction-with-calculushttps://puzzicle.org/thumbs/xde3bccae.jpgSecond-order reaction (with calculus)Deriving the integrated rate law for second order reactions using calculus. How you can graph second order rate data to see a linear relationship.https://cdn.kastatic.org/ka-youtube-converted/BRVPlDCMgMg.mp4/BRVPlDCMgMg.mp4434yes https://puzzicle.org/science/chemistry/v/half-life-of-a-second-order-reactionhttps://puzzicle.org/thumbs/x1d7ae02d.jpgHalf-life of a second-order reactionDeriving half-life equation of a second-order reaction starting from the integrated rate law.https://cdn.kastatic.org/ka-youtube-converted/XUmB0reCB-U.mp4/XUmB0reCB-U.mp4429yes https://puzzicle.org/science/chemistry/v/zero-order-reaction-with-calculushttps://puzzicle.org/thumbs/xa0344f0e.jpgZero-order reaction (with calculus)Deriving the integrated rate law for zeroth order reactions using calculus. How you can graph zero order rate data to see a linear relationship.https://cdn.kastatic.org/ka-youtube-converted/49LcF9Zf9TI.mp4/49LcF9Zf9TI.mp4593yes https://puzzicle.org/science/chemistry/v/forms-of-the-arrhenius-equationhttps://puzzicle.org/thumbs/x71322e6f.jpgForms of the Arrhenius equationHow to write different forms of the Arrhenius equation. Using the Arrhenius equation to look at how changing temperature and activation energy affects collisions.https://cdn.kastatic.org/ka-youtube-converted/J0ndBMMQW20.mp4/J0ndBMMQW20.mp4401yes https://puzzicle.org/science/chemistry/v/using-the-arrhenius-equationhttps://puzzicle.org/thumbs/x260a2638.jpgUsing the Arrhenius equationHow to use the Arrhenius equation to calculate the activation energy.https://cdn.kastatic.org/ka-youtube-converted/aIvYdSzLGoI.mp4/aIvYdSzLGoI.mp4666yes https://puzzicle.org/science/cosmology-and-astronomy/v/scale-of-the-largehttps://puzzicle.org/thumbs/89868304.jpgScale of the largeAttempting to comprehend the scale of the largehttps://cdn.kastatic.org/ka-youtube-converted/5FEjrStgcF8.mp4/5FEjrStgcF8.mp4662yes https://puzzicle.org/science/cosmology-and-astronomy/v/scale-of-the-smallhttps://puzzicle.org/thumbs/91063693.jpgScale of the smallFrom honey bees to cells, viruses, and atoms -- understanding the scale of the very smallhttps://cdn.kastatic.org/ka-youtube-converted/ERKx3Oa2omo.mp4/ERKx3Oa2omo.mp4794yes https://puzzicle.org/science/cosmology-and-astronomy/v/scale-of-earth-and-sunhttps://puzzicle.org/thumbs/8187268.jpgScale of earth and sunScale of Earth and Sunhttps://cdn.kastatic.org/ka-youtube-converted/GZx3U0dbASg.mp4/GZx3U0dbASg.mp4571yes https://puzzicle.org/science/cosmology-and-astronomy/v/scale-of-solar-systemhttps://puzzicle.org/thumbs/8187269.jpgScale of solar systemSal Khan describes the scale of our Solar System.https://cdn.kastatic.org/ka-youtube-converted/GP53b__h4ew.mp4/GP53b__h4ew.mp4751yes https://puzzicle.org/science/cosmology-and-astronomy/v/scale-of-distance-to-closest-starshttps://puzzicle.org/thumbs/8187270.jpgScale of distance to closest starsThe nearest stars to our solar system, are located in the Alpha Centauri system. The two main stars are Alpha Centauri A and Alpha Centauri B, which are approximately 4.3 light years away from Earth. The third star in the Alpha Centauri system, Proxima Centauri, is actually the closest star to our solar system at approximately 4.2 ​lightyears away from Earth.https://cdn.kastatic.org/ka-youtube-converted/jEeJkkMXt6c.mp4/jEeJkkMXt6c.mp4498yes https://puzzicle.org/science/cosmology-and-astronomy/v/scale-of-the-galaxyhttps://puzzicle.org/thumbs/8187271.jpgScale of the galaxyScale of the Galaxyhttps://cdn.kastatic.org/ka-youtube-converted/rcLnMe1ELPA.mp4/rcLnMe1ELPA.mp4775yes https://puzzicle.org/science/cosmology-and-astronomy/v/intergalactic-scalehttps://puzzicle.org/thumbs/8187272.jpgIntergalactic scaleIntergalactic Scalehttps://cdn.kastatic.org/ka-youtube-converted/JiE_kNk3ucI.mp4/JiE_kNk3ucI.mp4650yes https://puzzicle.org/science/cosmology-and-astronomy/v/hubble-image-of-galaxieshttps://puzzicle.org/thumbs/8187273.jpgHubble image of galaxiesHubble Image of Galaxieshttps://cdn.kastatic.org/ka-youtube-converted/Wl4re38deh0.mp4/Wl4re38deh0.mp4252yes https://puzzicle.org/science/cosmology-and-astronomy/v/cosmological-time-scale-1https://puzzicle.org/thumbs/8187354.jpgCosmological time scale 1Cosmological Time Scale 1https://cdn.kastatic.org/ka-youtube-converted/DRtLXagrMHw.mp4/DRtLXagrMHw.mp4427yes https://puzzicle.org/science/cosmology-and-astronomy/v/cosmological-time-scale-2https://puzzicle.org/thumbs/8187355.jpgCosmological time scale 2Cosmological Time Scale 2https://cdn.kastatic.org/ka-youtube-converted/LO7-3MpWijU.mp4/LO7-3MpWijU.mp4378yes https://puzzicle.org/science/cosmology-and-astronomy/v/introduction-to-lighthttps://puzzicle.org/thumbs/148049940.jpgIntroduction to lightLight and the electromagnetic radiation spectrum. Wave and particle-like behavior, and how to calculate the wavelength or frequency of a light wave.https://cdn.kastatic.org/ka-youtube-converted/rLNM8zI4Q_M.mp4/rLNM8zI4Q_M.mp4577yes https://puzzicle.org/science/cosmology-and-astronomy/v/four-fundamental-forceshttps://puzzicle.org/thumbs/8187356.jpgFour fundamental forcesGravity, Weak, Electromagnetic and Stronghttps://cdn.kastatic.org/ka-youtube-converted/FEF6PxWOvsk.mp4/FEF6PxWOvsk.mp4629yes https://puzzicle.org/science/cosmology-and-astronomy/v/light-and-the-luminiferous-etherhttps://puzzicle.org/thumbs/x761aba17.jpgLight and the luminiferous etherIf waves are disruptions traveling through a medium and light is a wave, what medium does it travel through? A luminiferous ether?https://cdn.kastatic.org/ka-youtube-converted/iAPYsOaq-VY.mp4/iAPYsOaq-VY.mp4478yes https://puzzicle.org/science/cosmology-and-astronomy/v/potential-ways-to-detect-an-ether-windhttps://puzzicle.org/thumbs/x3a23ded7.jpgPotential ways to detect an ether windThinking about how Earth's relative velocity to hypothetical luminiferous ether can be used to detect ether.https://cdn.kastatic.org/ka-youtube-converted/88hs5LcCoX4.mp4/88hs5LcCoX4.mp4444yes https://puzzicle.org/science/cosmology-and-astronomy/v/michelson-morley-experiment-introductionhttps://puzzicle.org/thumbs/x926ab958.jpgMichelson–Morley Experiment introductionSimple idea behind the Michelson‒Morley experiment to detect ether wind.https://cdn.kastatic.org/ka-youtube-converted/3G_Q6AggQF8.mp4/3G_Q6AggQF8.mp4492yes https://puzzicle.org/science/cosmology-and-astronomy/v/starting-to-set-up-a-newtonian-space-time-diagramhttps://puzzicle.org/thumbs/xe37b647e.jpgStarting to set up a Newtonian path–time diagramWe introduce a graph that's a lot like your good old position-vs-time graph—but with a twist!https://cdn.kastatic.org/ka-youtube-converted/PRx_R9iIWk4.mp4/PRx_R9iIWk4.mp4475yes https://puzzicle.org/science/cosmology-and-astronomy/v/visualizing-multiple-newtonian-space-time-diagramshttps://puzzicle.org/thumbs/xd3080b6e.jpgVisualizing multiple Newtonian path–time diagramsNow that we know what path–time diagrams are, let's see what happens when we bring along a friend!https://cdn.kastatic.org/ka-youtube-converted/F7BU1sXtul4.mp4/F7BU1sXtul4.mp4578yes https://puzzicle.org/science/cosmology-and-astronomy/v/galilean-transformation-and-contradictions-with-lighthttps://puzzicle.org/thumbs/x134b3407.jpgGalilean transformation and contradictions with lightHere we'll see how classical physics predicts scenarios that disagree with what we observe in nature. Something's got to give…https://cdn.kastatic.org/ka-youtube-converted/OIwp8m3S30c.mp4/OIwp8m3S30c.mp4484yes https://puzzicle.org/science/cosmology-and-astronomy/v/introduction-to-special-relativity-and-minkowski-spacetime-diagramshttps://puzzicle.org/thumbs/xb19d0f64.jpgIntroduction to special relativity and Minkowski spacetime diagramsIncluding multiple observers in the "most obvious" way led to some problems. Let's see how we can start to solve those problems by introducing (what we'll later call) Minkowski spacetime diagrams.https://cdn.kastatic.org/ka-youtube-converted/ii-KBKSODek.mp4/ii-KBKSODek.mp4824yes https://puzzicle.org/science/cosmology-and-astronomy/v/measuring-time-in-meters-in-minkowski-spacetimehttps://puzzicle.org/thumbs/x0c31cbce.jpgMeasuring time in meters in Minkowski spacetimeAs counterintuitive as it might sound, it will become extremely convenient to use the same units for space and time. Happily, we can do that just by multiplying the time axis by c, the speed of light. Here's how.https://cdn.kastatic.org/ka-youtube-converted/nEqexIckVCM.mp4/nEqexIckVCM.mp4298yes https://puzzicle.org/science/cosmology-and-astronomy/v/angle-of-x-axis-in-minkowski-spacetimehttps://puzzicle.org/thumbs/x0b9d5690.jpgAngle of x' axis in Minkowski spacetimeIt looks like there might be some relationship between the relative angles of the axes in the two references frames we've considered. Let's take a closer look.https://cdn.kastatic.org/ka-youtube-converted/qzNokxFaAAg.mp4/qzNokxFaAAg.mp4442yes https://puzzicle.org/science/cosmology-and-astronomy/v/introduction-to-the-lorentz-transformationhttps://puzzicle.org/thumbs/xd92fd692.jpgIntroduction to the Lorentz transformationSo we've got two coordinate systems from the perspectives of two observers. How can we convert spacetime coordinates between these? Enter the Lorentz transformation!https://cdn.kastatic.org/ka-youtube-converted/HIQ5hnm61LQ.mp4/HIQ5hnm61LQ.mp4501yes https://puzzicle.org/science/cosmology-and-astronomy/v/evaluating-a-lorentz-transformationhttps://puzzicle.org/thumbs/x817d8cc7.jpgEvaluating a Lorentz transformationWe'll consider an example of the Lorentz transformation with actual numbers, and analyze the results we get.https://cdn.kastatic.org/ka-youtube-converted/G-aTU0RXJww.mp4/G-aTU0RXJww.mp4521yes https://puzzicle.org/science/cosmology-and-astronomy/v/algebraically-manipulating-lorentz-transformationhttps://puzzicle.org/thumbs/x02c6977d.jpgAlgebraically manipulating Lorentz transformationThe forms in which we've introduced the Lorentz transformations are really nice in that they reveal the symmetry of the two axes of spacetime. But there are other representations: some are just more common in practical use, while others reveal other interesting similarities with classical mechanics!https://cdn.kastatic.org/ka-youtube-converted/Z9DuKef9jkw.mp4/Z9DuKef9jkw.mp4481yes https://puzzicle.org/science/cosmology-and-astronomy/v/lorentz-transformation-derivation-part-1https://puzzicle.org/thumbs/x8cac90b6.jpgLorentz transformation derivation part 1Using symmetry of frames of reference and the absolute velocity of the speed of light (regardless of frame of reference) to begin to solve for the Lorentz factor.https://cdn.kastatic.org/ka-youtube-converted/_qiFIMviIeM.mp4/_qiFIMviIeM.mp4655yes https://puzzicle.org/science/cosmology-and-astronomy/v/deriving-lorentz-transformation-part-2https://puzzicle.org/thumbs/xd60dd018.jpgDeriving Lorentz transformation part 2Continuing the algebra to solve for the Lorentz factor.https://cdn.kastatic.org/ka-youtube-converted/Lm_Yn_MZU6k.mp4/Lm_Yn_MZU6k.mp4368yes https://puzzicle.org/science/cosmology-and-astronomy/v/lorentz-transformation-derivation-part-3https://puzzicle.org/thumbs/x2a444ed5.jpgLorentz transformation derivation part 3Finishing our Lorentz transformation derivation for t'.https://cdn.kastatic.org/ka-youtube-converted/xb8nBY4MDC8.mp4/xb8nBY4MDC8.mp4682yes https://puzzicle.org/science/cosmology-and-astronomy/v/lorentz-transformation-for-change-in-coordinateshttps://puzzicle.org/thumbs/xbfc52e3d.jpgLorentz transformation for change in coordinatesLorentz transformation for change in coordinateshttps://cdn.kastatic.org/ka-youtube-converted/1F1tFouUGTU.mp4/1F1tFouUGTU.mp4306yes https://puzzicle.org/science/cosmology-and-astronomy/v/einstein-velocity-addition-formula-derivationhttps://puzzicle.org/thumbs/xd39ef880.jpgEinstein velocity addition formula derivationEinstein velocity addition formula derivationhttps://cdn.kastatic.org/ka-youtube-converted/F9BW57exRn8.mp4/F9BW57exRn8.mp4456yes https://puzzicle.org/science/cosmology-and-astronomy/v/applying-einstein-velocity-additionhttps://puzzicle.org/thumbs/xdb3d50bd.jpgApplying Einstein velocity additionApplying Einstein velocity additionhttps://cdn.kastatic.org/ka-youtube-converted/0XkyAPXq55k.mp4/0XkyAPXq55k.mp4382yes https://puzzicle.org/science/cosmology-and-astronomy/v/finding-an-in-between-frame-of-referencehttps://puzzicle.org/thumbs/x1e47fcb4.jpgFinding an in-between frame of referenceLet's use Einstein velocity addition to find a frame of reference where A and B are traveling in opposite directions at the same speed.https://cdn.kastatic.org/ka-youtube-converted/Biua3BgRf54.mp4/Biua3BgRf54.mp4492yes https://puzzicle.org/science/cosmology-and-astronomy/v/calculating-neutral-velocityhttps://puzzicle.org/thumbs/xcbc0b377.jpgCalculating neutral velocityLet's now do the math to calculate the equal speed at which both A and B could be traveling away from a "neutral" observer.https://cdn.kastatic.org/ka-youtube-converted/iTyBlnTK2ac.mp4/iTyBlnTK2ac.mp4405yes https://puzzicle.org/science/cosmology-and-astronomy/v/time-dilationhttps://puzzicle.org/thumbs/xfbd3b322.jpgTime dilationWe'll start thinking about time dilation in special relativity and how a Loedel diagram can help up appreciate symmetry between inertial frames.https://cdn.kastatic.org/ka-youtube-converted/2BVGig1LXLs.mp4/2BVGig1LXLs.mp4554yes https://puzzicle.org/science/cosmology-and-astronomy/v/big-bang-introductionhttps://puzzicle.org/thumbs/8187274.jpgBig bang introductionBig bang introduction from Cosmology and astronomyhttps://cdn.kastatic.org/ka-youtube-converted/eUF59jCFcyQ.mp4/eUF59jCFcyQ.mp4663yes https://puzzicle.org/science/cosmology-and-astronomy/v/radius-of-observable-universehttps://puzzicle.org/thumbs/8187275.jpgRadius of observable universeRadius of Observable Universehttps://cdn.kastatic.org/ka-youtube-converted/6nVysrZQnOQ.mp4/6nVysrZQnOQ.mp4999yes https://puzzicle.org/science/cosmology-and-astronomy/v/correction-radius-of-observable-universehttps://puzzicle.org/thumbs/8187276.jpgRadius of observable universe (correction)(Correction) Radius of Observable Universehttps://cdn.kastatic.org/ka-youtube-converted/b6VQv76BQDs.mp4/b6VQv76BQDs.mp4134yes https://puzzicle.org/science/cosmology-and-astronomy/v/red-shifthttps://puzzicle.org/thumbs/8187351.jpgRed shiftRed shift from Cosmology and astronomyhttps://cdn.kastatic.org/ka-youtube-converted/mx2M_ZKXM_c.mp4/mx2M_ZKXM_c.mp4604yes https://puzzicle.org/science/cosmology-and-astronomy/v/cosmic-background-radiationhttps://puzzicle.org/thumbs/8187352.jpgCosmic background radiationCosmic Background Radiationhttps://cdn.kastatic.org/ka-youtube-converted/sxbPwl_KRuA.mp4/sxbPwl_KRuA.mp4696yes https://puzzicle.org/science/cosmology-and-astronomy/v/cosmic-background-radiation-2https://puzzicle.org/thumbs/8187353.jpgCosmic background radiation 2Cosmic Background Radiation 2 - Redshift of the Cosmic Background Radiationhttps://cdn.kastatic.org/ka-youtube-converted/06z7Q8TWPyU.mp4/06z7Q8TWPyU.mp4615yes https://puzzicle.org/science/cosmology-and-astronomy/v/hubble-s-lawhttps://puzzicle.org/thumbs/8187365.jpgHubble's lawHubble's Lawhttps://cdn.kastatic.org/ka-youtube-converted/1V9wVmO0Tfg.mp4/1V9wVmO0Tfg.mp4672yes https://puzzicle.org/science/cosmology-and-astronomy/v/a-universe-smaller-than-the-observablehttps://puzzicle.org/thumbs/8187279.jpgA universe smaller than the observableA Universe Smaller than the Observablehttps://cdn.kastatic.org/ka-youtube-converted/0w9R_foNLrg.mp4/0w9R_foNLrg.mp4538yes https://puzzicle.org/science/cosmology-and-astronomy/v/birth-of-starshttps://puzzicle.org/thumbs/8187357.jpgBirth of starsStars are formed from massive clouds of hydrogen atoms in space. Under intense pressure and temperature, these atoms undergo fusion, combining to form helium and releasing energy. This energy provides an outward pressure, preventing the star from collapsing, thus creating a star.https://cdn.kastatic.org/ka-youtube-converted/i-NNWI8Ccas.mp4/i-NNWI8Ccas.mp4478yes https://puzzicle.org/science/cosmology-and-astronomy/v/accreting-mass-due-to-gravity-simulationhttps://puzzicle.org/thumbs/1272530969.jpgAccreting mass due to gravity simulationThe simulation Sal mentions in the video can be found directly following this video in the lesson.https://cdn.kastatic.org/ka-youtube-converted/hD4ySbQYYyA.mp4/hD4ySbQYYyA.mp4214yes https://puzzicle.org/science/cosmology-and-astronomy/v/becoming-a-red-gianthttps://puzzicle.org/thumbs/8187358.jpgBecoming a red giantBecoming a Red Gianthttps://cdn.kastatic.org/ka-youtube-converted/kJSOqlcFpJw.mp4/kJSOqlcFpJw.mp4690yes https://puzzicle.org/science/cosmology-and-astronomy/v/white-and-black-dwarfshttps://puzzicle.org/thumbs/8187359.jpgWhite and black dwarfsWhite and Black Dwarfshttps://cdn.kastatic.org/ka-youtube-converted/EdYyuUUY-nc.mp4/EdYyuUUY-nc.mp4774yes https://puzzicle.org/science/cosmology-and-astronomy/v/star-field-and-nebula-imageshttps://puzzicle.org/thumbs/8187360.jpgStar field and nebula imagesStar Field and Nebula Imageshttps://cdn.kastatic.org/ka-youtube-converted/w3IKEa_GOYs.mp4/w3IKEa_GOYs.mp4424yes https://puzzicle.org/science/cosmology-and-astronomy/v/lifecycle-of-massive-starshttps://puzzicle.org/thumbs/8187386.jpgLifecycle of massive starsMassive stars begin as large hydrogen clouds that condense under gravity. They burn hotter and faster than smaller stars, undergoing fusion to create heavier elements. This fusion process continues until iron is formed. Beyond iron, fusion requires energy, leading to a supernova.https://cdn.kastatic.org/ka-youtube-converted/UhIwMAhZpCo.mp4/UhIwMAhZpCo.mp4401yes https://puzzicle.org/science/cosmology-and-astronomy/v/supernova-supernovaehttps://puzzicle.org/thumbs/8187387.jpgSupernova (supernovae)A supernova is a powerful explosion of a massive star at the end of its life cycle. This event releases an enormous amount of energy, temporarily outshining an entire galaxy. It also disperses heavier elements formed during the star's life and the explosion itself into the universe.https://cdn.kastatic.org/ka-youtube-converted/qOwCpnQsDLM.mp4/qOwCpnQsDLM.mp4718yes https://puzzicle.org/science/cosmology-and-astronomy/v/supernova-clarificationhttps://puzzicle.org/thumbs/8187388.jpgSupernova clarificationSupernova clarificationhttps://cdn.kastatic.org/ka-youtube-converted/QMv4kxATfzs.mp4/QMv4kxATfzs.mp492yes https://puzzicle.org/science/cosmology-and-astronomy/v/black-holeshttps://puzzicle.org/thumbs/8187389.jpgBlack holesBlack Holeshttps://cdn.kastatic.org/ka-youtube-converted/JoL2pO3O0rg.mp4/JoL2pO3O0rg.mp4609yes https://puzzicle.org/science/cosmology-and-astronomy/v/supermassive-black-holeshttps://puzzicle.org/thumbs/8187393.jpgSupermassive black holesSupermassive Black Holeshttps://cdn.kastatic.org/ka-youtube-converted/DxkkAHnqlpY.mp4/DxkkAHnqlpY.mp4568yes https://puzzicle.org/science/cosmology-and-astronomy/v/quasarshttps://puzzicle.org/thumbs/8187394.jpgQuasarsQuasarshttps://cdn.kastatic.org/ka-youtube-converted/4LmIyMyAuN0.mp4/4LmIyMyAuN0.mp4555yes https://puzzicle.org/science/cosmology-and-astronomy/v/quasar-correctionhttps://puzzicle.org/thumbs/8187395.jpgQuasar correctionQuasar Correctionhttps://cdn.kastatic.org/ka-youtube-converted/PX_XSnVWlNc.mp4/PX_XSnVWlNc.mp4106yes https://puzzicle.org/science/cosmology-and-astronomy/v/galactic-collisionshttps://puzzicle.org/thumbs/8187396.jpgGalactic collisionsCollision of the Milky Way Galaxy with Andromeda (forming "Milkomeda")https://cdn.kastatic.org/ka-youtube-converted/QXYbGZ3T3_k.mp4/QXYbGZ3T3_k.mp4214yes https://puzzicle.org/science/cosmology-and-astronomy/v/parallax-in-observing-starshttps://puzzicle.org/thumbs/8187361.jpgParallax in observing starsParallax in Observing Starshttps://cdn.kastatic.org/ka-youtube-converted/ETzUpoqZIHY.mp4/ETzUpoqZIHY.mp4441yes https://puzzicle.org/science/cosmology-and-astronomy/v/stellar-parallaxhttps://puzzicle.org/thumbs/145846956.jpgStellar parallaxAnother version of the stellar parallax introductionhttps://cdn.kastatic.org/ka-youtube-converted/omvNINaRdxg.mp4/omvNINaRdxg.mp4566yes https://puzzicle.org/science/cosmology-and-astronomy/v/stellar-distance-using-parallaxhttps://puzzicle.org/thumbs/8187362.jpgStellar distance using parallaxStellar Distance Using Parallaxhttps://cdn.kastatic.org/ka-youtube-converted/lVadjWOjvV8.mp4/lVadjWOjvV8.mp4700yes https://puzzicle.org/science/cosmology-and-astronomy/v/stellar-parallax-clarificationhttps://puzzicle.org/thumbs/8187363.jpgStellar parallax clarificationStellar Parallax Clarificationhttps://cdn.kastatic.org/ka-youtube-converted/6FP-hLuAlr4.mp4/6FP-hLuAlr4.mp4142yes https://puzzicle.org/science/cosmology-and-astronomy/v/parsec-definitionhttps://puzzicle.org/thumbs/8187364.jpgParsec definitionParsec Definitionhttps://cdn.kastatic.org/ka-youtube-converted/6zV3JEjLoyE.mp4/6zV3JEjLoyE.mp4453yes https://puzzicle.org/science/cosmology-and-astronomy/v/cepheid-variables-1https://puzzicle.org/thumbs/8187390.jpgCepheid variables 1Cepheid Variables 1https://cdn.kastatic.org/ka-youtube-converted/BWs-ONRDDG4.mp4/BWs-ONRDDG4.mp4563yes https://puzzicle.org/science/cosmology-and-astronomy/v/why-cepheids-pulsatehttps://puzzicle.org/thumbs/8187391.jpgWhy cepheids pulsateWhy Cepheids Pulsatehttps://cdn.kastatic.org/ka-youtube-converted/X_3QAB3o4Vw.mp4/X_3QAB3o4Vw.mp4430yes https://puzzicle.org/science/cosmology-and-astronomy/v/why-gravity-gets-so-strong-near-dense-objectshttps://puzzicle.org/thumbs/8187392.jpgWhy gravity gets so strong near dense objectsWhy Gravity Gets So Strong Near Dense Objectshttps://cdn.kastatic.org/ka-youtube-converted/RpOHZc6cDIw.mp4/RpOHZc6cDIw.mp4493yes https://puzzicle.org/science/cosmology-and-astronomy/v/plate-tectonics-difference-between-crust-and-lithospherehttps://puzzicle.org/thumbs/13179436.jpgPlate tectonics: Difference between crust and lithospherePlate Tectonics Introduction and Difference between crust and lithospherehttps://cdn.kastatic.org/ka-youtube-converted/f2BWsPVN7c4.mp4/f2BWsPVN7c4.mp4479yes https://puzzicle.org/science/cosmology-and-astronomy/v/structure-of-the-earthhttps://puzzicle.org/thumbs/13179437.jpgStructure of the earthStructure of the Earth - crust, mantle, corehttps://cdn.kastatic.org/ka-youtube-converted/4AxZ-6MOznY.mp4/4AxZ-6MOznY.mp4573yes https://puzzicle.org/science/cosmology-and-astronomy/v/plate-tectonics-evidence-of-plate-movementhttps://puzzicle.org/thumbs/13179438.jpgPlate tectonics: Evidence of plate movementPlate Tectonics -- Evidence of plate movementhttps://cdn.kastatic.org/ka-youtube-converted/6EdsBabSZ4g.mp4/6EdsBabSZ4g.mp4798yes https://puzzicle.org/science/cosmology-and-astronomy/v/plates-moving-due-to-convection-in-mantlehttps://puzzicle.org/thumbs/13179441.jpgPlates moving due to convection in mantlePlates Moving Due to Convection in Mantlehttps://cdn.kastatic.org/ka-youtube-converted/f8GK2oEN-uI.mp4/f8GK2oEN-uI.mp4362yes https://puzzicle.org/science/cosmology-and-astronomy/v/hawaiian-islands-formationhttps://puzzicle.org/thumbs/13179442.jpgHawaiian islands formationThe Hawaiian Islands formed primarily as a result of volcanic activity. While most islands ​form near tectonic plate boundaries, the Hawaiian Islands are nearly 2000 miles away from the nearest plate margin. Therefore, scientists believe that the islands formed due to the presence of the Hawaiian "hot spot," a region deep in the Earth's mantle from which heat rises. This heat produces melted rock (magma), which then pushes through the crust and solidifies.https://cdn.kastatic.org/ka-youtube-converted/D1eibbfAEVk.mp4/D1eibbfAEVk.mp4481yes https://puzzicle.org/science/cosmology-and-astronomy/v/pangaeahttps://puzzicle.org/thumbs/13179459.jpgPangaeaPangaea - the idea of Pangaea and some of the evidence behind ithttps://cdn.kastatic.org/ka-youtube-converted/axB6uhEx628.mp4/axB6uhEx628.mp4358yes https://puzzicle.org/science/cosmology-and-astronomy/v/compositional-and-mechanical-layers-of-the-earthhttps://puzzicle.org/thumbs/13179443.jpgCompositional and mechanical layers of the earthThe crust is the outermost layer of Earth and is composed of solid rock. The lithosphere contains the crust and upper mantle. The mantle is the thickest layer and is made up of hot, mostly solid rock. The core is divided into two parts: the inner core and outer core. The inner core is solid and the outer core is liquid.https://cdn.kastatic.org/ka-youtube-converted/hHteUIS0OFY.mp4/hHteUIS0OFY.mp4710yes https://puzzicle.org/science/cosmology-and-astronomy/v/how-we-know-about-the-earth-s-corehttps://puzzicle.org/thumbs/13179458.jpgHow we know about the earth's coreS-wave shadow and P-wave detection patterns give us information about the corehttps://cdn.kastatic.org/ka-youtube-converted/KL0i1RSnpfI.mp4/KL0i1RSnpfI.mp4349yes https://puzzicle.org/science/cosmology-and-astronomy/v/seismic-waveshttps://puzzicle.org/thumbs/13179444.jpgSeismic wavesS-waves and P-waveshttps://cdn.kastatic.org/ka-youtube-converted/NhioAAdYDJM.mp4/NhioAAdYDJM.mp4518yes https://puzzicle.org/science/cosmology-and-astronomy/v/why-s-waves-only-travel-in-solidshttps://puzzicle.org/thumbs/13179455.jpgWhy S-waves only travel in solidsWhy S-Waves Only Travel in Solidshttps://cdn.kastatic.org/ka-youtube-converted/BTWLwoaNeBA.mp4/BTWLwoaNeBA.mp4255yes https://puzzicle.org/science/cosmology-and-astronomy/v/refraction-of-seismic-waveshttps://puzzicle.org/thumbs/13179456.jpgRefraction of seismic wavesRefraction of Seismic Waveshttps://cdn.kastatic.org/ka-youtube-converted/UNZ171fDja4.mp4/UNZ171fDja4.mp4562yes https://puzzicle.org/science/cosmology-and-astronomy/v/the-mohorovicic-seismic-discontinuityhttps://puzzicle.org/thumbs/13179457.jpgThe mohorovicic seismic discontinuityThe Mohorovicic Seismic Discontinuity- boundary between crust and mantle (or Moho)https://cdn.kastatic.org/ka-youtube-converted/yAQSucmHrAk.mp4/yAQSucmHrAk.mp4373yes https://puzzicle.org/science/cosmology-and-astronomy/v/seasons-aren-t-dictated-by-closeness-to-sunhttps://puzzicle.org/thumbs/150055884.jpgSeasons aren't dictated by closeness to sunWhy our closeness to the sun does not dictate the seasonshttps://cdn.kastatic.org/ka-youtube-converted/SJUd5du0T08.mp4/SJUd5du0T08.mp4279yes https://puzzicle.org/science/cosmology-and-astronomy/v/how-earth-s-tilt-causes-seasonshttps://puzzicle.org/thumbs/150055883.jpgHow Earth's tilt causes seasonsHow Earth's tilt causes seasonshttps://cdn.kastatic.org/ka-youtube-converted/05qDIjKevJo.mp4/05qDIjKevJo.mp41247yes https://puzzicle.org/science/cosmology-and-astronomy/v/are-southern-hemisphere-seasons-more-severehttps://puzzicle.org/thumbs/152085900.jpgAre southern hemisphere seasons more severe?Are Southern Hemisphere seasons more severe because of the eccentricity in Earth's orbit?https://cdn.kastatic.org/ka-youtube-converted/umvNQj-zmq4.mp4/umvNQj-zmq4.mp4437yes https://puzzicle.org/science/cosmology-and-astronomy/v/milankovitch-cycles-precession-and-obliquityhttps://puzzicle.org/thumbs/150055882.jpgMilankovitch cycles precession and obliquityHow changes in Earth's rotation can effect Earth's seasons and climatehttps://cdn.kastatic.org/ka-youtube-converted/ZD8THEz18gc.mp4/ZD8THEz18gc.mp4777yes https://puzzicle.org/science/cosmology-and-astronomy/v/precession-causing-perihelion-to-happen-laterhttps://puzzicle.org/thumbs/152085903.jpgPrecession causing perihelion to happen laterClarifying the effect of axial precession on the calendar and the date of perihelion and aphelionhttps://cdn.kastatic.org/ka-youtube-converted/2o-Sef6wllg.mp4/2o-Sef6wllg.mp4724yes https://puzzicle.org/science/cosmology-and-astronomy/v/what-causes-precession-and-other-orbital-changeshttps://puzzicle.org/thumbs/152085901.jpgWhat causes precession and other orbital changesWhat causes precession and other orbital changeshttps://cdn.kastatic.org/ka-youtube-converted/nL6LMX8-bPY.mp4/nL6LMX8-bPY.mp4126yes https://puzzicle.org/science/cosmology-and-astronomy/v/apsidal-precession-perihelion-precession-and-milankovitch-cycleshttps://puzzicle.org/thumbs/152085902.jpgApsidal precession (perihelion precession) and Milankovitch cyclesApsidal Precession (Perihelion Precession)https://cdn.kastatic.org/ka-youtube-converted/m6lMGoZTJnc.mp4/m6lMGoZTJnc.mp4427yes https://puzzicle.org/science/cosmology-and-astronomy/v/intro-to-moon-phaseshttps://puzzicle.org/thumbs/x71004a2e3a9c632b.jpgIntro to Moon phasesThe Moon and the sun look roughly the same size in the sky because although the sun’s diameter is ~400 times greater than the Moon’s, the sun is ~400 times farther away from the Earth as the Moon is! The Moon goes through phases because as it rotates around the Earth, different parts of the Moon are made visible to us from the sun’s light.https://cdn.kastatic.org/ka-youtube-converted/ZQcf9xLABa0.mp4/ZQcf9xLABa0.mp4334yes https://puzzicle.org/science/cosmology-and-astronomy/v/solar-and-lunar-eclipseshttps://puzzicle.org/thumbs/x65c91ab6fe86292e.jpgSolar and lunar eclipsesUnderstanding solar eclipses and lunar eclipses. Why don't we have a solar eclipse every new moon (every time the Moon is between the Earth and Sun)? Why don't we have a lunar eclipse every time the Earth is between the Sun and Moon.https://cdn.kastatic.org/ka-youtube-converted/qJqGg3Xe4iE.mp4/qJqGg3Xe4iE.mp4332yes https://puzzicle.org/science/cosmology-and-astronomy/v/understanding-calendar-notationhttps://puzzicle.org/thumbs/101104459.jpgUnderstanding calendar notationDifference between BC, BCE, AD and CE. A little bit about the birth of Jesus as well.https://cdn.kastatic.org/ka-youtube-converted/EKKe7DBZVhI.mp4/EKKe7DBZVhI.mp4581yes https://puzzicle.org/science/cosmology-and-astronomy/v/correction-calendar-notationhttps://puzzicle.org/thumbs/105033394.jpgCorrection calendar notationCorrecting the time difference calculation by taking into account that there is no year 0https://cdn.kastatic.org/ka-youtube-converted/N06Jy-gQog8.mp4/N06Jy-gQog8.mp4252yes https://puzzicle.org/science/cosmology-and-astronomy/v/energy-inputs-for-tilling-a-hectare-of-landhttps://puzzicle.org/thumbs/465986949.jpgEnergy inputs for tilling a hectare of landThinking about energy consumption for tilling landhttps://cdn.kastatic.org/ka-youtube-converted/5I9dH5im24U.mp4/5I9dH5im24U.mp4821yes https://puzzicle.org/science/cosmology-and-astronomy/v/random-predictions-for-2060https://puzzicle.org/thumbs/466052144.jpgRandom predictions for 2060Sal's random predictions for the year 2060https://cdn.kastatic.org/ka-youtube-converted/E4rbS0oxg30.mp4/E4rbS0oxg30.mp4667yes https://puzzicle.org/science/cosmology-and-astronomy/v/detectable-civilizations-in-our-galaxy-1https://puzzicle.org/thumbs/91824745.jpgDetectable civilizations in our galaxy 1A framework for thinking about how many detectable civilizations are out there.https://cdn.kastatic.org/ka-youtube-converted/YYEgq1bweN4.mp4/YYEgq1bweN4.mp4783yes https://puzzicle.org/science/cosmology-and-astronomy/v/detectable-civilizations-in-our-galaxy-2https://puzzicle.org/thumbs/92633418.jpgDetectable civilizations in our galaxy 2Why do we even care about the Drake Equation. Thinking about the fraction of a planet's life when a civilization might be detectable.https://cdn.kastatic.org/ka-youtube-converted/jN03g05TSWM.mp4/jN03g05TSWM.mp4533yes https://puzzicle.org/science/cosmology-and-astronomy/v/detectable-civilizations-in-our-galaxy-3https://puzzicle.org/thumbs/92633419.jpgDetectable civilizations in our galaxy 3Reconciling with the traditional Drake Equationhttps://cdn.kastatic.org/ka-youtube-converted/LRmwUsxNSL0.mp4/LRmwUsxNSL0.mp4438yes https://puzzicle.org/science/cosmology-and-astronomy/v/detectable-civilizations-in-our-galaxy-4https://puzzicle.org/thumbs/92633420.jpgDetectable civilizations in our galaxy 4Taking a shot at estimating the number of detectable civilizationshttps://cdn.kastatic.org/ka-youtube-converted/T5DGZIsfK-0.mp4/T5DGZIsfK-0.mp4375yes https://puzzicle.org/science/cosmology-and-astronomy/v/detectable-civilizations-in-our-galaxy-5https://puzzicle.org/thumbs/94896541.jpgDetectable civilizations in our galaxy 5More reasons why we haven't detected anything yet.https://cdn.kastatic.org/ka-youtube-converted/xL9ejqb53ms.mp4/xL9ejqb53ms.mp4309yes https://puzzicle.org/science/electrical-engineering/v/ee-currenthttps://puzzicle.org/thumbs/x91cda9c9.jpgElectric currentCurrent is the flow of charge. Copper and silver atoms have loose electrons, making them good conductors. Current is measured by counting charges passing through a boundary per second. It also shows how saltwater conducts current with positive and negative ions moving in opposite directions.https://cdn.kastatic.org/ka-youtube-converted/ZRLXDiiUv8Q.mp4/ZRLXDiiUv8Q.mp4558yes https://puzzicle.org/science/electrical-engineering/v/current-directionhttps://puzzicle.org/thumbs/x6c7f6193.jpgCurrent directionThe positive sign for current corresponds to the direction a positive charge would move. In metal wires, current is carried by negatively charged electrons, so the positive current arrow points in the opposite direction the electrons move. This has been the sign convention for 270 years, ever since Ben Franklin named electric charges with + and - signs. This convention came about 150 years before the discovery of the electron.https://cdn.kastatic.org/ka-youtube-converted/4frpZ4Q0q58.mp4/4frpZ4Q0q58.mp4328yes https://puzzicle.org/science/electrical-engineering/v/ee-voltage-intuitionhttps://puzzicle.org/thumbs/xd11f8925.jpgVoltageVoltage is a cornerstone concept in electricity. We create an intuitive mental image by comparing voltage to gravity.https://cdn.kastatic.org/ka-youtube-converted/k9SwNST1eW0.mp4/k9SwNST1eW0.mp4179yes https://puzzicle.org/science/electrical-engineering/v/ee-conventional-currenthttps://puzzicle.org/thumbs/x267eb7e6.jpgConventional current directionBy convention, we define positive direction of current to be in the direction a positive charge would move. Electrons (with their negative charge) move in the opposite direction of the positive current arrow.https://cdn.kastatic.org/ka-youtube-converted/17EhKw2tsu4.mp4/17EhKw2tsu4.mp4237yes https://puzzicle.org/science/electrical-engineering/v/ideal-circuit-elementshttps://puzzicle.org/thumbs/x0085d2b0.jpgIdeal circuit elementsExplore the world of circuit analysis with the three fundamental circuit elements: resistors, capacitors, and inductors. Understand their unique symbols, properties, and the key equations that connect voltage and current for each component. Master Ohm's law for resistors, the capacitor equation, and the inductor equation to build a strong foundation in circuit design and analysis. These ideal component equations will unlock the door to creating and understanding a wide variety of electrical circuits.https://cdn.kastatic.org/ka-youtube-converted/l-h72j2-X0o.mp4/l-h72j2-X0o.mp4391yes https://puzzicle.org/science/electrical-engineering/v/ee-ideal-sourceshttps://puzzicle.org/thumbs/x99d7332b.jpgIdeal sourcesThe ideal voltage source has a constant voltage. The ideal current source has a constant current. Both sources can be plotted on an IV curve. These sources are crucial for providing power and signals within electrical circuits.https://cdn.kastatic.org/ka-youtube-converted/CBlfcuUXzUY.mp4/CBlfcuUXzUY.mp4175yes https://puzzicle.org/science/electrical-engineering/v/circuit-terminologyhttps://puzzicle.org/thumbs/x05b9c9e6.jpgCircuit terminologyThe key terms for circuits are components (resistors, capacitors, inductors), elements, nodes (junctions), branches, meshes (loops), and loops. In our analysis of circuits, we will learn how to identify nodes, branches, and meshes.https://cdn.kastatic.org/ka-youtube-converted/-82UNytyrCQ.mp4/-82UNytyrCQ.mp4483yes https://puzzicle.org/science/electrical-engineering/v/ee-passive-sign-conventionhttps://puzzicle.org/thumbs/xb4f026f4.jpgSign convention for passive componentsThe sign convention for passive components defines what we mean by positive and negative voltage and current.https://cdn.kastatic.org/ka-youtube-converted/1Oopq5U0b08.mp4/1Oopq5U0b08.mp4269yes https://puzzicle.org/science/electrical-engineering/v/ee-parallel-resistors-1https://puzzicle.org/thumbs/xf40ba5f6.jpgParallel resistors (derivation)Discover the fascinating world of parallel resistors in electrical circuits! Learn how to identify parallel resistors, understand the concept of shared nodes and voltage, and apply Ohm's law to calculate equivalent resistance. Master the formula for parallel resistors and enhance your circuit analysis skills with real-life examples.https://cdn.kastatic.org/ka-youtube-converted/MHADYILlW1E.mp4/MHADYILlW1E.mp4668yes https://puzzicle.org/science/electrical-engineering/v/ee-parallel-resistors-2https://puzzicle.org/thumbs/xd00091cc.jpgParallel resistors (derivation continued)Learn how to calculate the equivalent resistance for multiple parallel resistors. This method simplifies complex resistor networks by replacing them with a single equivalent parallel resistor, streamlining circuit analysis and improving efficiency. Grasp the concept of current splitting among resistors and understand that all parallel resistors share the same voltage, making this technique applicable to various electrical engineering scenarios.https://cdn.kastatic.org/ka-youtube-converted/0IeWoAiHbAY.mp4/0IeWoAiHbAY.mp4255yes https://puzzicle.org/science/electrical-engineering/v/ee-voltage-dividerhttps://puzzicle.org/thumbs/xa22475c5.jpgVoltage dividerExplore the voltage divider, a fundamental circuit pattern in electrical engineering, consisting of two series resistors that control output voltage. Master the voltage divider expression, which calculates output voltage based on input voltage and resistor values. Dive into practical examples, learn when to apply this versatile circuit, and understand the importance of considering current flow for accurate voltage division in real-world applications.https://cdn.kastatic.org/ka-youtube-converted/t_hPrz7rs34.mp4/t_hPrz7rs34.mp4399yes https://puzzicle.org/science/electrical-engineering/v/analyzing-a-resistor-circuit-with-two-batterieshttps://puzzicle.org/thumbs/x159d4673.jpgAnalyzing a resistor circuit with two batteriesDiscover how to solve complex circuit problems in physics by simplifying schematics, combining parallel resistors, and applying Ohm's law. Master the art of analyzing circuits with multiple batteries and resistors to find the current flowing through each component. Boost your electrical engineering skills and conquer challenging circuit questions.https://cdn.kastatic.org/ka-youtube-converted/j-iR7puLj6M.mp4/j-iR7puLj6M.mp4584yes https://puzzicle.org/science/electrical-engineering/v/ee-labeling-voltageshttps://puzzicle.org/thumbs/xbbbe6a4d.jpgLabeling voltagesLabeling voltages on a schematic is not a matter of "right" and "wrong". It simply establishes how the voltage appears in the analysis equations.https://cdn.kastatic.org/ka-youtube-converted/ZjPs0V6tiR0.mp4/ZjPs0V6tiR0.mp4667yes https://puzzicle.org/science/electrical-engineering/v/ee-application-of-the-fundamental-laws-setuphttps://puzzicle.org/thumbs/x501e0a36.jpgApplication of the fundamental laws (setup)We can solve circuits by the direct application of the fundamental laws: Ohm's Law and Kirchhoff's Laws. (Part 1 of 2)https://cdn.kastatic.org/ka-youtube-converted/1VaKLjvbWZ0.mp4/1VaKLjvbWZ0.mp4482yes https://puzzicle.org/science/electrical-engineering/v/ee-application-of-the-fundamental-laws-solvehttps://puzzicle.org/thumbs/x889b22a3.jpgApplication of the fundamental laws (solve)We solve the equations created by direct application of the fundamental laws: Ohm's Law and Kirchhoff's Laws. (Part 2 of 2)https://cdn.kastatic.org/ka-youtube-converted/ShxyPUl7asE.mp4/ShxyPUl7asE.mp4520yes https://puzzicle.org/science/electrical-engineering/v/ee-node-voltage-method-steps-1-to-4https://puzzicle.org/thumbs/x86bd1a36.jpgNode voltage method (steps 1 to 4)The Node Voltage Method solves circuits with the minimum number of KCL equations. Steps 1 to 4 out of 5.https://cdn.kastatic.org/ka-youtube-converted/2lY757QaaKs.mp4/2lY757QaaKs.mp4597yes https://puzzicle.org/science/electrical-engineering/v/ee-node-voltage-method-step-5https://puzzicle.org/thumbs/x7c70cce8.jpgNode voltage method (step 5)Last step of the Node voltage method for analyzing circuits: Solve the system of simultaneous equationshttps://cdn.kastatic.org/ka-youtube-converted/LMC3jcNomNc.mp4/LMC3jcNomNc.mp4385yes https://puzzicle.org/science/electrical-engineering/v/ee-mesh-current-steps-1-to-3https://puzzicle.org/thumbs/x8d04abc2.jpgMesh current method (steps 1 to 3)We solve a circuit by writing Kirchhoff's Voltage Law in terms of "mesh currents." This video covers the first three steps out of four.https://cdn.kastatic.org/ka-youtube-converted/j8LHrm3_brk.mp4/j8LHrm3_brk.mp4556yes https://puzzicle.org/science/electrical-engineering/v/ee-mesh-current-method-step-4-solvehttps://puzzicle.org/thumbs/x7c56cf3c.jpgMesh current method (step 4)We complete the solution of a circuit using the mesh current method. Fourth step (solve) out of four.https://cdn.kastatic.org/ka-youtube-converted/7I3-HAW8rmM.mp4/7I3-HAW8rmM.mp4732yes https://puzzicle.org/science/electrical-engineering/v/ee-capacitor-i-v-equationshttps://puzzicle.org/thumbs/xb62c6631.jpgCapacitor i-v equationsThe capacitor current-voltage equation in derivative form and integral formhttps://cdn.kastatic.org/ka-youtube-converted/EqlsSqET1kk.mp4/EqlsSqET1kk.mp4521yes https://puzzicle.org/science/electrical-engineering/v/ee-capacitor-integrates-currenthttps://puzzicle.org/thumbs/xf4894782.jpgA capacitor integrates currentA current flowing into a capacitor causes charge to accumulate. The voltage rises according to q = Cv and we say the capacitor integrates current.https://cdn.kastatic.org/ka-youtube-converted/JxOMm24lSR4.mp4/JxOMm24lSR4.mp4561yes https://puzzicle.org/science/electrical-engineering/v/ee-inductor-equationshttps://puzzicle.org/thumbs/x192b1d27.jpgInductor equationsWe look at the inductor i-v equations in derivative and integral form.https://cdn.kastatic.org/ka-youtube-converted/w1NR5sjsChI.mp4/w1NR5sjsChI.mp4767yes https://puzzicle.org/science/electrical-engineering/v/ee-inductor-kickback-1https://puzzicle.org/thumbs/x3d6518e7.jpgInductor kickback (1 of 2)We notice how important it is to give inductor current a place to flowhttps://cdn.kastatic.org/ka-youtube-converted/YKKd5p4a5Lo.mp4/YKKd5p4a5Lo.mp4769yes https://puzzicle.org/science/electrical-engineering/v/ee-inductor-kickback-2https://puzzicle.org/thumbs/x5c5c4af3.jpgInductor kickback (2 of 2)We give the inductor's kickback current a path to flow by adding a diodehttps://cdn.kastatic.org/ka-youtube-converted/xOoMotHFOKk.mp4/xOoMotHFOKk.mp4446yes https://puzzicle.org/science/electrical-engineering/v/ee-rc-natural-response-intuitionhttps://puzzicle.org/thumbs/xed38ea01.jpgRC natural response - intuitionWe put some charge on a capacitor in an RC circuit and observe what happens to the voltage and current. This is called the natural response.https://cdn.kastatic.org/ka-youtube-converted/PZM6acXvsoQ.mp4/PZM6acXvsoQ.mp4446yes https://puzzicle.org/science/electrical-engineering/v/ee-rc-natural-response-derivationhttps://puzzicle.org/thumbs/x9e7f9050.jpgRC natural response - derivationWe derive the natural response of an RC circuit and discover it has an exponential form.https://cdn.kastatic.org/ka-youtube-converted/4qgLz0TF8-M.mp4/4qgLz0TF8-M.mp4659yes https://puzzicle.org/science/electrical-engineering/v/rc-natural-response-examplehttps://puzzicle.org/thumbs/x6879a53e.jpgRC natural response - exampleAn example of the exponential natural response of an RC circuit, with real component valueshttps://cdn.kastatic.org/ka-youtube-converted/2NRBn6ZLmDc.mp4/2NRBn6ZLmDc.mp4173yes https://puzzicle.org/science/electrical-engineering/v/ee-rc-step-response-intuitionhttps://puzzicle.org/thumbs/x3d9229e2.jpgRC step response - intuitionAn RC circuit responds to a step of voltage with a smooth transition between the starting and ending voltage. The RC step response is a fundamental behavior of all digital circuits.https://cdn.kastatic.org/ka-youtube-converted/zfqJRZyeGRE.mp4/zfqJRZyeGRE.mp4609yes https://puzzicle.org/science/electrical-engineering/v/ee-rc-step-response-1-of-3-setuphttps://puzzicle.org/thumbs/xd7e40b7e.jpgRC step response setup (1 of 3)The RC step response is an exponential curve. We set up the circuit and create the differential equation we need to solve. We introduce the technique of Natural response + Forced response.https://cdn.kastatic.org/ka-youtube-converted/k6KH4WmcgaU.mp4/k6KH4WmcgaU.mp4548yes https://puzzicle.org/science/electrical-engineering/v/ee-rc-step-response-2-of-3-solvehttps://puzzicle.org/thumbs/xa7268fa5.jpgRC step response solve (2 of 3)We find the RC step response using the method of Natural + Forced response = Total response.https://cdn.kastatic.org/ka-youtube-converted/A_1N561qdSw.mp4/A_1N561qdSw.mp4567yes https://puzzicle.org/science/electrical-engineering/v/ee-rc-step-response-3-of-3-examplehttps://puzzicle.org/thumbs/x7270efde.jpgRC step response example (3 of 3)An example solution of an RC step response, with real component values.https://cdn.kastatic.org/ka-youtube-converted/A4aoLMC1-5Y.mp4/A4aoLMC1-5Y.mp4220yes https://puzzicle.org/science/electrical-engineering/v/ee-sketching-exponentialshttps://puzzicle.org/thumbs/x74c74eda.jpgSketching exponentialsVoltages and currents often have an exponential shape. We look at some properties of exponential curves and learn how to rapidly sketch accurate waveforms.https://cdn.kastatic.org/ka-youtube-converted/4XHMqvVByFQ.mp4/4XHMqvVByFQ.mp4470yes https://puzzicle.org/science/electrical-engineering/v/ee-sketching-exponentials-exampleshttps://puzzicle.org/thumbs/xa7d45c96.jpgSketching exponentials - examplesSketch exponentials with different time constantshttps://cdn.kastatic.org/ka-youtube-converted/kxE7wipiMps.mp4/kxE7wipiMps.mp4441yes https://puzzicle.org/science/electrical-engineering/v/ee-lc-natural-response-intuition1https://puzzicle.org/thumbs/x6a27615d.jpgLC natural response intuition 1The inductor-capacitor (LC) circuit is the place where sinewaves are born. We talk about how this circuit works by tracking the movement of an initial charge we placed on the capacitor.https://cdn.kastatic.org/ka-youtube-converted/FuuR_SqwXm4.mp4/FuuR_SqwXm4.mp4313yes https://puzzicle.org/science/electrical-engineering/v/ee-lc-natural-response-intuition2https://puzzicle.org/thumbs/xadf4d02b.jpgLC natural response intuition 2We can predict the shape of voltage and current in an LC circuit by tracking the motion of charge as it flows back and forth.https://cdn.kastatic.org/ka-youtube-converted/DIxU2l204HU.mp4/DIxU2l204HU.mp4573yes https://puzzicle.org/science/electrical-engineering/v/ee-lc-natural-response-derivation1https://puzzicle.org/thumbs/x1a672ae3.jpgLC natural response derivation 1We begin the derivation of the natural response of the LC circuit, by modeling it with a 2nd-order differential equation.https://cdn.kastatic.org/ka-youtube-converted/vvf-qiN9SWA.mp4/vvf-qiN9SWA.mp4399yes https://puzzicle.org/science/electrical-engineering/v/ee-lc-natural-response-derivation2https://puzzicle.org/thumbs/x77ef60da.jpgLC natural response derivation 2Starting from the differential equation, we come up with a proposed exponential solution and plug it into the equation. This gives us a characteristic equation. A "natural frequency" emerges.https://cdn.kastatic.org/ka-youtube-converted/aSTvToHuG3I.mp4/aSTvToHuG3I.mp4570yes https://puzzicle.org/science/electrical-engineering/v/ee-lc-natural-response-derivation3https://puzzicle.org/thumbs/x57ae5231.jpgLC natural response derivation 3We use Euler's Formula to change our complex exponential solution into a solution expressed in terms of sines and cosines.https://cdn.kastatic.org/ka-youtube-converted/p-b6k9jyIkE.mp4/p-b6k9jyIkE.mp4431yes https://puzzicle.org/science/electrical-engineering/v/ee-lc-natural-response-derivation4https://puzzicle.org/thumbs/xccd5c0d0.jpgLC natural response derivation 4In this final step of the derivation, we find two initial conditions and use them to come up with a sinusoidal solution for the LC natural response.https://cdn.kastatic.org/ka-youtube-converted/51UHJnjUOSs.mp4/51UHJnjUOSs.mp4452yes https://puzzicle.org/science/electrical-engineering/v/ee-lc-natural-response-examplehttps://puzzicle.org/thumbs/xb469f4a4.jpgLC natural response exampleWe solve the voltage and current of an example LC circuit with given values for L and C, and an initial charge on the capacitorhttps://cdn.kastatic.org/ka-youtube-converted/vkIBypq8MTE.mp4/vkIBypq8MTE.mp4444yes https://puzzicle.org/science/electrical-engineering/v/ee-ac-analysis-intro1https://puzzicle.org/thumbs/xd05ad841.jpgAC analysis intro 1Solving circuits with differential equations is hard. If we limit ourselves to sinusoidal input signals, a whole new method of AC analysis emerges.https://cdn.kastatic.org/ka-youtube-converted/7UJi2GQCi00.mp4/7UJi2GQCi00.mp4403yes https://puzzicle.org/science/electrical-engineering/v/ee-ac-analysis-intro2https://puzzicle.org/thumbs/xbf82f3c6.jpgAC analysis intro 2Here's a preview of how AC analysis is going to work. To get ready we need to review some of the ideas from trig and complex numbers.https://cdn.kastatic.org/ka-youtube-converted/NuAgmNaX58M.mp4/NuAgmNaX58M.mp4538yes https://puzzicle.org/science/electrical-engineering/v/ee-ac-trig-reviewhttps://puzzicle.org/thumbs/xeca5168a.jpgTrigonometry reviewA quick review of some ideas from trigonometry that will help us with AC analysis. The definitions of sine, cosine, and tangent, circles, degrees, radians.https://cdn.kastatic.org/ka-youtube-converted/6hjr4n4MCPA.mp4/6hjr4n4MCPA.mp4450yes https://puzzicle.org/science/electrical-engineering/v/ee-sine-cosine-circleshttps://puzzicle.org/thumbs/x2c67c16e.jpgSine and cosine come from circlesSine and cosine can be generated by projecting the tip of a vector onto the y-axis and x-axis as the vector rotates about the origin.https://cdn.kastatic.org/ka-youtube-converted/LmxGmUOiP2A.mp4/LmxGmUOiP2A.mp4280yes https://puzzicle.org/science/electrical-engineering/v/ee-sine-of-timehttps://puzzicle.org/thumbs/xb88ed089.jpgSine of timeIf we introduce time as the argument to the sine function, we get a sine wave. Animated.https://cdn.kastatic.org/ka-youtube-converted/R3en76G8_W0.mp4/R3en76G8_W0.mp4638yes https://puzzicle.org/science/electrical-engineering/v/ee-sine-cosine-from-rotating-vectorhttps://puzzicle.org/thumbs/xdde03158.jpgSine and cosine from rotating vectorOne way to generate sine and cosine is to track the motion of a rotating vector. Animated.https://cdn.kastatic.org/ka-youtube-converted/a_zReGTxdlQ.mp4/a_zReGTxdlQ.mp4237yes https://puzzicle.org/science/electrical-engineering/v/ee-lead-laghttps://puzzicle.org/thumbs/x6d4df74e.jpgLead LagSine and cosine look similar, except they are out of phase. When we talk about sine and cosine as a function of time, the difference is called "lead" or "lag".https://cdn.kastatic.org/ka-youtube-converted/G0h9Sa_s18A.mp4/G0h9Sa_s18A.mp4446yes https://puzzicle.org/science/electrical-engineering/v/ee-complex-numbershttps://puzzicle.org/thumbs/x53c95fe4.jpgComplex numbersComplex numbers can be represented three ways on the complex plane: cartesian coordinates, radius and angle, and exponential form.https://cdn.kastatic.org/ka-youtube-converted/wbEEWmJFDVg.mp4/wbEEWmJFDVg.mp4516yes https://puzzicle.org/science/electrical-engineering/v/ee-multiplying-j-rotationhttps://puzzicle.org/thumbs/x99f14726.jpgMultiplying by j is rotationSuccessive powers of the imaginary unit j correspond to a rotation.https://cdn.kastatic.org/ka-youtube-converted/giO5ACdU2LA.mp4/giO5ACdU2LA.mp4373yes https://puzzicle.org/science/electrical-engineering/v/ee-complex-rotationhttps://puzzicle.org/thumbs/x7079f83a.jpgComplex rotationMultiplying a real or complex number by the imaginary unit j corresponds to a rotation by +90 degrees. This is the key feature of j that makes it such a useful number.https://cdn.kastatic.org/ka-youtube-converted/CIQmq1wOjnc.mp4/CIQmq1wOjnc.mp4380yes https://puzzicle.org/science/electrical-engineering/v/ee-eulers-formulahttps://puzzicle.org/thumbs/x9bea2f03.jpgEuler's formulaEuler's formula relates the complex exponential to the cosine and sine functions. This formula is the most important tool in AC analysis. It is why electrical engineers need to understand complex numbers.https://cdn.kastatic.org/ka-youtube-converted/LE2uwd9V5vw.mp4/LE2uwd9V5vw.mp4522yes https://puzzicle.org/science/electrical-engineering/v/ee-complex-exponential-magnitudehttps://puzzicle.org/thumbs/x7d155a8b.jpgComplex exponential magnitudeA closer look at the complex exponential term in Euler's Formula. We see that it represents a complex number, a distance of 1 from the origin of the complex plane.https://cdn.kastatic.org/ka-youtube-converted/XxP6auFa44M.mp4/XxP6auFa44M.mp4370yes https://puzzicle.org/science/electrical-engineering/v/ee-complex-exponentials-spinhttps://puzzicle.org/thumbs/xb095c462.jpgComplex exponentials spinWhen we put time in the exponent of a complex exponential, the complex number it represent rotates in a circle on the complex plane. You can think of it as a spinning number!https://cdn.kastatic.org/ka-youtube-converted/4-gVIRgZi2U.mp4/4-gVIRgZi2U.mp4400yes https://puzzicle.org/science/electrical-engineering/v/ee-eulers-sine-wavehttps://puzzicle.org/thumbs/x1ecb1090.jpgEuler's sine waveA sine wave emerges from Euler's Formula. Music, no narration. Animated with d3.js.https://cdn.kastatic.org/ka-youtube-converted/1tSrRYU6LKM.mp4/1tSrRYU6LKM.mp4203yes https://puzzicle.org/science/electrical-engineering/v/ee-euler-cosinehttps://puzzicle.org/thumbs/xf6747f03.jpgEuler's cosine waveA cosine wave emerges from Euler's Formula. Music, no narration. Animated with d3.js.https://cdn.kastatic.org/ka-youtube-converted/zGTzWgSDFR0.mp4/zGTzWgSDFR0.mp4184yes https://puzzicle.org/science/electrical-engineering/v/ee-negative-frequencyhttps://puzzicle.org/thumbs/xbb83765d.jpgNegative frequencyThe concept of negative frequency is perplexing, but it makes sense when you think about rotating complex exponentialshttps://cdn.kastatic.org/ka-youtube-converted/SvFfVtRT1UI.mp4/SvFfVtRT1UI.mp4401yes https://puzzicle.org/science/electrical-engineering/v/ee-ac-analysis-superpositionhttps://puzzicle.org/thumbs/xafc6252e.jpgAC analysis superpositionWe break a sinusoidal input voltage into two complex exponentials. Using superposition, we can recover the complex output signals and reassemble them into a real sinusoidal output voltage.https://cdn.kastatic.org/ka-youtube-converted/UjiLH7RE8_Y.mp4/UjiLH7RE8_Y.mp4439yes https://puzzicle.org/science/electrical-engineering/v/ee-impedancehttps://puzzicle.org/thumbs/x7ea223b6.jpgImpedanceThe ratio of a sinusoidal voltage to a sinusoidal current is called "impedance". This is a generalization of Ohm's Law for resistors. We derive the impedance of a resistor, inductor, and capacitor. The inductor and capacitor impedance includes a term for frequency, so the impedance of these components depends on frequency.https://cdn.kastatic.org/ka-youtube-converted/WmTlioVfS78.mp4/WmTlioVfS78.mp4701yes https://puzzicle.org/science/electrical-engineering/v/ee-impedance-vs-frequencyhttps://puzzicle.org/thumbs/x9dbef6a8.jpgImpedance vs frequencyThe impedance of capacitors and inductors in a circuit depend on the frequency of the electric signal. The impedance of an inductor is directly proportional to frequency, while the impedance of a capacitor is inversely proportional to frequency.https://cdn.kastatic.org/ka-youtube-converted/I21hginkKk0.mp4/I21hginkKk0.mp4536yes https://puzzicle.org/science/electrical-engineering/v/ee-eli-the-ice-manhttps://puzzicle.org/thumbs/xbb84e633.jpgELI the ICE manRemember the relationship between current and voltage in an inductor and capacitor with the help of ELI the ICE man.https://cdn.kastatic.org/ka-youtube-converted/2yqjMiFUMlA.mp4/2yqjMiFUMlA.mp4629yes https://puzzicle.org/science/electrical-engineering/v/ee-impedance-of-simple-networkshttps://puzzicle.org/thumbs/x022a2cf2.jpgImpedance of simple networksImpedance of 2 elements in series is a complex number. Impedance terminology: reactance, susceptance, admittance.https://cdn.kastatic.org/ka-youtube-converted/smLN8WKJjas.mp4/smLN8WKJjas.mp4703yes https://puzzicle.org/science/electrical-engineering/v/ee-kvl-frequency-domainhttps://puzzicle.org/thumbs/x04d23e29.jpgKVL in the frequency domainDemonstration that Kirchhoff's voltage law applies in the frequency domain. The voltage phase offsets around a loop sum to zero.https://cdn.kastatic.org/ka-youtube-converted/779k4-oFjuc.mp4/779k4-oFjuc.mp4771yes https://puzzicle.org/science/electrical-engineering/v/ee-opamp-introhttps://puzzicle.org/thumbs/xaf155f88.jpgWhat is an operational amplifier?The "operational amplifier" has two differential inputs and very high gain. Willy describes the symbol and properties of an op-amp. Op-amps are the backbone of analog circuit design.https://cdn.kastatic.org/ka-youtube-converted/lJDjWZqhpVc.mp4/lJDjWZqhpVc.mp4694yes https://puzzicle.org/science/electrical-engineering/v/ee-noninverting-opamphttps://puzzicle.org/thumbs/xb80be02c.jpgNon-inverting op-ampAn op-amp can be used to amplify a voltage. The gain is determined by the ratio of two resistors.https://cdn.kastatic.org/ka-youtube-converted/_Ut-nQ535iE.mp4/_Ut-nQ535iE.mp4752yes https://puzzicle.org/science/electrical-engineering/v/ee-feedbackhttps://puzzicle.org/thumbs/xb159b105.jpgFeedbackFeedback is the design technique where a part of the amplifier output "feeds back" to the input of the amplifier. The overall effect creates a very stable gain determined by resistor ratios.https://cdn.kastatic.org/ka-youtube-converted/ua4eMZTxpQ0.mp4/ua4eMZTxpQ0.mp4356yes https://puzzicle.org/science/electrical-engineering/v/ee-inverting-opamphttps://puzzicle.org/thumbs/xd6dd3ffd.jpgInverting op-ampWe analyze the inverting op-amp configuration, doing all the algebra from first principles.https://cdn.kastatic.org/ka-youtube-converted/U1KbM4ffiLg.mp4/U1KbM4ffiLg.mp4770yes https://puzzicle.org/science/electrical-engineering/v/ee-virtual-groundhttps://puzzicle.org/thumbs/xa9088670.jpgVirtual groundThe idea of a "virtual ground" makes some simplifying assumptions, giving us an elegant and simple way to look at op-amp circuits.https://cdn.kastatic.org/ka-youtube-converted/pxKLeIjzxAk.mp4/pxKLeIjzxAk.mp4358yes https://puzzicle.org/science/electrical-engineering/v/ee-virtual-ground-exampleshttps://puzzicle.org/thumbs/x83b79b90.jpgVirtual ground - examplesUsing the "virtual ground" idea, we reexamine the inverting op-amp and find a solution much quicker. We apply the virtual ground method to a unity-gain buffer (amplifier with gain = 1).https://cdn.kastatic.org/ka-youtube-converted/aD6l5bNnJgE.mp4/aD6l5bNnJgE.mp4387yes https://puzzicle.org/science/electrical-engineering/v/ee-summing-opamphttps://puzzicle.org/thumbs/x78eefffe.jpgSumming op-ampThis op-amp circuit outputs the sum of two input voltages. You can do analog arithmetic on signals.https://cdn.kastatic.org/ka-youtube-converted/jlwcIRKCqek.mp4/jlwcIRKCqek.mp4548yes https://puzzicle.org/science/electrical-engineering/v/ee-diodehttps://puzzicle.org/thumbs/x4ce88ac1.jpgDiodeIntroducing the diode as a circuit element. A diode conducts current in one direction but not the other.https://cdn.kastatic.org/ka-youtube-converted/0FbeUMPZFjU.mp4/0FbeUMPZFjU.mp4660yes https://puzzicle.org/science/electrical-engineering/v/ee-diode-graphical-solutionhttps://puzzicle.org/thumbs/x3c002e02.jpgDiode graphical solutionWe solve a diode circuit graphically by plotting a diode i-v curve and resistor to find the intersection. Graphical solution methods are a common way to work with non-linear components like diodes.https://cdn.kastatic.org/ka-youtube-converted/IMZ2D8Nz6sA.mp4/IMZ2D8Nz6sA.mp4487yes https://puzzicle.org/science/electrical-engineering/v/proof-advanced-field-from-infinite-plate-part-1https://puzzicle.org/thumbs/25687.jpgProof: Field from infinite plate (part 1)Advanced proof of the formula for the electric field generated by a uniformly charged, infinite platehttps://cdn.kastatic.org/ka-youtube-converted/prLfVucoxpw.mp4/prLfVucoxpw.mp4807yes https://puzzicle.org/science/electrical-engineering/v/proof-advanced-field-from-infinite-plate-part-2https://puzzicle.org/thumbs/25689.jpgProof: Field from infinite plate (part 2)We see that the infinite, uniformly charged plate generates a constant electric field (independent of the height above the plate)https://cdn.kastatic.org/ka-youtube-converted/TxwE4_dXo8s.mp4/TxwE4_dXo8s.mp4688yes https://puzzicle.org/science/electrical-engineering/v/electric-potential-energyhttps://puzzicle.org/thumbs/25691.jpgElectric potential energyExplore the concept of electrical potential energy and its similarities to gravitational potential energy. Understand how work is required to move an object within a gravitational or electric field, and how this work translates into potential energy. Discover how potential energy changes can impact kinetic energy and velocity.https://cdn.kastatic.org/ka-youtube-converted/elJUghWSVh4.mp4/elJUghWSVh4.mp4757yes https://puzzicle.org/science/electrical-engineering/v/ee-fourier-series-introhttps://puzzicle.org/thumbs/x05009cb1.jpgFourier Series introductionFourier series are a powerful tool that can help us break down complex signals into their constituent parts. By using some basic mathematics, we can deconstruct signals into simple sine waves, making them much easier to understand and analyze.https://cdn.kastatic.org/ka-youtube-converted/UKHBWzoOKsY.mp4/UKHBWzoOKsY.mp4313yes https://puzzicle.org/science/electrical-engineering/v/ee-integral-of-sinmt-and-cosmthttps://puzzicle.org/thumbs/xb8db33db.jpgIntegral of sin(mt) and cos(mt)Integrating sin(mt) and cos(mt) over a full period equals zero.https://cdn.kastatic.org/ka-youtube-converted/6AWZT8owvO8.mp4/6AWZT8owvO8.mp4385yes https://puzzicle.org/science/electrical-engineering/v/ee-integral-of-sine-times-cosinehttps://puzzicle.org/thumbs/xee75f2fb.jpgIntegral of sine times cosineDefinite integral of the product of sine and cosine. Integrating sin(mt) * cos(nt) over a full period equals zero for any integer m and n.https://cdn.kastatic.org/ka-youtube-converted/EDhYig0bEpA.mp4/EDhYig0bEpA.mp4267yes https://puzzicle.org/science/electrical-engineering/v/ee-integral-sine-times-sinehttps://puzzicle.org/thumbs/x90a87d13.jpgIntegral of product of sinesDefinite integral of product of sines. The integral of sin(mt) * sin(nt) = 0, except for the special case when m = n. When m = n, the integral evaluates to pi.https://cdn.kastatic.org/ka-youtube-converted/PLxHkjrVBgE.mp4/PLxHkjrVBgE.mp4495yes https://puzzicle.org/science/electrical-engineering/v/ee-integral-cosine-times-cosinehttps://puzzicle.org/thumbs/x7e684a10.jpgIntegral of product of cosinesDefinite integral of the product of cosines. The integral of cos(mt) * cos(nt) = 0, except for the special case when m = n. When m = n, the integral evaluates to pi.https://cdn.kastatic.org/ka-youtube-converted/ki4VK_dQlPs.mp4/ki4VK_dQlPs.mp4435yes https://puzzicle.org/science/electrical-engineering/v/ee-first-term-fourier-serieshttps://puzzicle.org/thumbs/x19e42e4e.jpgFirst term in a Fourier seriesThe first term in a Fourier series is the average value (DC value) of the function being approximated.https://cdn.kastatic.org/ka-youtube-converted/59RIABaGTcs.mp4/59RIABaGTcs.mp4409yes https://puzzicle.org/science/electrical-engineering/v/ee-fourier-coefficients-cosinehttps://puzzicle.org/thumbs/x24abab90.jpgFourier coefficients for cosine termsFourier coefficients for cosine termshttps://cdn.kastatic.org/ka-youtube-converted/kwqNDIacqt0.mp4/kwqNDIacqt0.mp4368yes https://puzzicle.org/science/electrical-engineering/v/ee-fourier-coefficients-sinehttps://puzzicle.org/thumbs/x91ce05b4.jpgFourier coefficients for sine termsFourier coefficients for sine termshttps://cdn.kastatic.org/ka-youtube-converted/khDD_9g5YTM.mp4/khDD_9g5YTM.mp4393yes https://puzzicle.org/science/electrical-engineering/v/ee-fourier-coefficients-for-square-wavehttps://puzzicle.org/thumbs/xc0806f77.jpgFinding Fourier coefficients for square waveFinding Fourier coefficients for a square wavehttps://cdn.kastatic.org/ka-youtube-converted/WTD8jg8fXoE.mp4/WTD8jg8fXoE.mp4640yes https://puzzicle.org/science/electrical-engineering/v/ee-visualize-fourier-series-square-wavehttps://puzzicle.org/thumbs/xe5597c6d.jpgVisualizing the Fourier expansion of a square waveVisualize the Fourier expansion of a square wave. A square wave can be approximated by adding odd harmonics of a sine wave.https://cdn.kastatic.org/ka-youtube-converted/aalANrz7bi8.mp4/aalANrz7bi8.mp4210yes