Creating volume expressions

0:00- [Instructor] What we wanna do in this video is figure out the volume of this rectangular prism here. So pause this video and see if you can do that on your own before we do this together. All right, now let's work through this together and we're actually gonna think about it in three different ways.

0:15And to help us do that in those three different ways, we'll visualize this volume in these three different ways. So the first way to visualize it is let's just think about it as if we think about it as cubic centimeters, five going up, four going across, or four wide, and then three deep.

0:33And so that's what all of these show. And in this first diagram, what we first can think about is how many cubes, how many orange cubes are in this first layer. Well, this first layer right over here is going to be four cubes in one dimension. Actually, let me scroll down a little bit.

0:51It's going to be four in this dimension right over here, and there's going to be three in this dimension. So if you think about how many cubes are there, that is four times three on just one base or one layer. And then how many layers are there? Well, we can see that there are one, two, three, four, five of those layers.

1:14So then you can multiply that times five. Or another way to think about it is, in that base layer there, if you did four times three, you're going to have 12 of those cubes, and then you're gonna have to multiply 12 times 5, which you might just know as being 60.

1:30Let me do that same color. As 60. Or you could do a repeated addition if you like. You could say, "Well, that's going to be equal to 12 plus 12 plus 12 plus 12 plus 12 plus 12, which also is equal to 60." And that's 60 of what? Well, each of these things is a cubic centimeter.

1:52So that is 60, 60 cubic centimeters. Now we could approach it a different way. We could think about one of these, I guess you could say these vertical layers first. So this vertical layer in this example, it is one, two, three, four, five high and then it, in each of those five rows, we can see it has one, two, three, four cubes.

2:19So how many cubes are just in this orange part here, this orange layer? Well, it's five times four, which is 20. And then how many of these layers are there? You can almost think of them as walls. Well, we could see that there's one, let me just see a different color.

2:35We could see that there are one, two, and three of them. So then you can multiply that times three. Well, five times four is 60, sorry, five times four. My brain is cutting to the chase. Five times four is 20, and then 20 times e is going to be 60. Once again 60 cubic centimeters.

2:59And you can do that with repeated addition. Five times four is 20, 20 plus, 20 plus 20 is 60. Now you can imagine where this last way of thinking about it could go. We could just think about how many cubes are in this right wall. We can see that it is one, two, three, four, five high.

3:19So it is five high and it is one, two, three deep. So in this last diagram, you have five times three, which is the same thing as 15 cubes. And then how many of these walls do you have? Well, you could see you have one, two, three, four of them. So times four.

3:40And so what is that going to get you? Probably not a surprise. Five times three is 15. 15 times four. You might know that that is going to be 60 cubic centimeters. Or you could say that that's the same thing as 15 plus 15 plus 15 plus 15. 15 plus 15 is 30. Plus 15 is 45 plus 15 is 60.

4:05It's 60 cubic centimeters. Why did I put cubic centimeters here? Well, that was what we were counting the entire time. Every one of these layers we're counting in terms of cubic centimeters. Each of these little cubes is, each of these little cubes right over here is a cubic centimeter.

4:23Now you might be saying, "What was the whole point of me doing this three times like this?" Well, so just show you that you can think of volume as multiplying the area of one side times the last or the area of one layer. Or you could think about the area of the base.

4:39So in this first example, if you think of the base as something like this, or if you think about the area of this base right over here, so three times four, which we did first or four times three, and then we multiplied that times the height. So base times height, that's what we did here.

4:55Or you could think of it the other way. You could think of it as, well, we could multiply. We could multiply, we could find the area of that side and then multiply it times the other dimension. So five times four would be the area of that backside. And then you could multiply it times the three once again to get to 60.

5:13Or last but not least, in this last example, you could say, "All right, let's find the area." I could say the area of this right side, right over here, which is gonna be five times three and then multiply it times that fourth side. But in all the situations, I'm just multiplying the sides.

5:29I have four times three times five, five times four times three, five times three times four. I use the parentheses to do some of that multiplication first. But when you're multiplying three numbers like this, it actually doesn't matter the order that you are multiplying them in.

5:43So you just have to think about the three sides here and multiply them in some order. And any way you do it, you're going to get 60 cubic centimeters.