Rates of change with integers

0:00- [Lecturer] We're told, "Samantha used her kiln to fire pottery pieces she crafted. When she initiated the cooling process, her kiln was at 1,500 degrees Celsius. After five hours, the new temperature of Samantha's kiln was 280 degrees Celsius. On average, how much did the temperature change each hour the kiln was cooling?"

0:21Pause this video and try to figure this yourself. Figure it out yourself before we do it together. All right, so the way I would approach this is first I would figure out what was the total change. So let me write this down, total change. Let's figure that out first.

0:36And then after we figure out the total change, then we could divide that by five to figure out the average per hour change. So what's the total change going to be? So when you think about the change in something, you wanna figure out, well, where did it start and where did it end, and subtract where it started from, from where it ended.

0:56So we ended at 280 degrees, and we started at 1,500 degrees. So the total change is going to be, 280 minus 1,500. Now some of you might say, hey look, 1,500's, a bigger number. Maybe you want to go from 1,500, and from that subtract 280, and you would get a positive result.

1:21But we have to be very careful. If they just said, how much did the temperature drop? Then this one would've been okay, this would've figured out how much the temperature dropped. But if we wanna know change, and especially if we are dropping, then the change is going to be negative.

1:36So we actually want a negative value. So we actually do want to figure out what this is. Of course, this is just the negative of that, so it's still useful to know what 1,500 minus 280 actually is, and then this will just be the negative of that. So let's just figure this out.

1:54We might be able to do this in our head, but I will do it with a little bit of regrouping. Zero minus zero, is zero. We have a zero here, so we need to regroup. Let's take one from the place 10 times that place, and so that makes this one 10. 10 minus 8 is 2. 4 minus 2 is 2.

2:15And then you bring down that one, you get 1220. This one right over here would've been 1,220. But this is the negative of that. We've swapped the signs. So this one is -1,220 degrees Celsius. This is the total change. So what is the average change going to be?

2:34Well, this happened over five hours. So what we are going to want to do is divide this by five. So if you take, -1220 and we divide it by 5, let me do this right over here. So if we just figure out, let's just figure out what 1220 is, divided by 5. And of course if you have a negative divided by a positive, you're going to get a negative answer.

3:02So we know this is going to be negative. Make the clearer negative sign there. But five doesn't go into one, it goes into 12 two times. 2 times 5 is 10. Subtract. 12 minus 10 is 2. Bring down that next 2, 5 goes into 22 four times. 4 times five is 20. Subtract, and then we get a 2 .

3:25And then bring down that zero. Five goes into 20 exactly four times. 4 times 5 is 20, and we have no remainder and we are done. So it's going to be 244. So this is going to be -244 right over here when we factor in the actual negative sign. So on average, the temperature changed -244 degrees celsius per hour.

3:53Let's do another example. So here we are told that, "A submarine was at a depth of 100 meters below sea level." That below is a clue. That tells us that the depth, if I were to write it, was at -100 meters. That's where the submarine started. "It ascended for five hours to avoid a school of fish."

4:16Oh, that's very nice of them. "After ascending, the new location of the submarine was 25 meters below sea level." So it's still below sea level. So it went from -100 meters to a -25 meters below sea level. "On average, how many meters did the submarine ascend each hour?"

4:39So let's think about this. What is the change in the depth? So we can take the finishing point, so -25, and from that, subtract the starting point, minus -100. Well, that's going to be the same thing as -25 plus 100. One way to think about it as well. Let me just rewrite it so we don't get confused.

5:03So this is the same thing as -25 plus 100, or 100 minus 25, and that's going to be 75 meters. So our change is a positive 75 meters. And whenever you're dealing with negative numbers, it's good to just reason it through. We went from a deeper depth to a less deep depth.

5:26So it makes sense that we went up by 75 meters. So it makes sense that this is a positive right over here. So over the, what does it say, five hours, we ascended by 75 meters. But that's not what they're asking us. They wanna know how many meters per hour on average did we ascend.

5:45Well ,then we'd have to divide this by five hours. Now some of you might be able to do this in your head. You might know that that's 15 meters per hour. But let's just do it little of mini long division, right over here. Seven goes, sorry. 5 goes into 7 one time.

6:031 times 5 is 5. We subtract. 7 minus 5 is 2. We bring down that 5. 5 goes into 25 exactly five times. 5 times 5 is 25. Just be very clear that we have no remainder there. So this is going to be equal to 15 meters per hour. And we feel good about the sign here, because we are ascending 15 meters per hour.

6:25We're getting less and less deep and we're done.