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Finding area with fractional sides 1

CCSS Math: 5.NF.B.4

Video transcript

- [Voiceover] So we've got a rectangle here, it's five-ninths of a meter tall, and seven-eighths of a meter wide. What is its area? And I encourage you to pause the video to think about that. Well one way to think about it, is you can say our area, our area is just going to be the width times the height. We're just going to multiply these two dimensions. And so the width is seven-eighths of a meter. So it's going to be seven-eighths of a meter times the height, times the height which is five-ninths of a meter. Times five-ninths of a meter. And what's that going to get us? Well, that's just going to be equal to the meters times the meters give us square meters, so meters squared. We could write it like that. And then we're going have, and then we're going to have seven times, this in a new color, we're going to have seven times five in the numerator to get us 35, and then in the denominator, in the denominator we are going to have eight times nine to give us 72. And we'd be done. This is the area of this rectangle here. It's 35-72nds of a square meter. What I want to do now is think a little bit deeper about why that actually makes sense. Or just really another way of thinking about it. And to do that, what I'm going to do is I'm going to split this region into equal rectangles. So let's split it into equal rectangles. And we see that we have seven, if we go in the horizontal direction we have one, two, three, four, five, six, seven or you could say in each row we have seven of these rectangles. In each column you have one, two, three, four, five of these rectangles. So you can see we have five times one, two, three, four, five, six, seven. So we have five times seven of these rectangles. So, we have--so 35, we have 35 rectangles. I'll just write this, 35 rectangles. And what's the area of each of those rectangles? Well, if this is seven-eighths meters wide, and this is divided into seven equal sections in the horizontal direction, that means that each of these is exactly one-eighth of a meter wide. And by that same logic, each of these, if this whole thing is five-ninths, and the height of each of these is one-fifth because we have five rectangles per column, then the height of each of these is going to be one-ninth of a meter. So what's the area of just this character right over here? Well, it's going to be one-ninth of a meter times one-eighth of a meter. So this area, this area right over there is just going to be one-ninth of a meter times one-eighth of a meter which is equal to one times one is one, nine times eight is 72, and meters times meters is square meters. So the area of each of these 35 is one-72nd of a square meter. So, if I say 35, so the area of all of them combined is going to be 35 times the area of each of them. 35 times one-72nd of a square meter. And what's that going to be? Well, that's going to be exactly what we got up here. 35 times one-72nd of a square meter is going to be 35, 35-72nds of a square meter. And this 35 is the same one that we had in yellow. That's this one right over there. So once again, you can just multiply five-ninths times seven-eighths to get what we have got here. But hopefully when we thought about the area of each of these rectangles, it might make a little bit more intuitive sense where this number came from.