Division with area models

- [Voiceover] Let's say that this rectangle, this green rectangle right over here, let's say it had an area of 268 square units, whatever those units are. You could imagine them being square centimeters, or if you imagine this being a big field that you're looking at from space, it could be square miles or something. So I'm just gonna write square units. And lets say you knew the dimensions of one side of the field. So let's say that you knew, you knew that this... let me just hit a color, let's say you knew that this side of the field right over here, the length of this side of the field is two units. Two units. And I haven't really drawn this to scale. If I wanted to draw it to scale, it would be like, it would be much shorter, it would be like that. But you would have trouble seeing the rectangle then. But let's just assume it's two units. So if you know the whole area is 268 square units, and one side is two units, what's the other side going to be? What is the other side going to be? What is, what is... let me pick another color. What is this side, what is this side of this rectangular, of this rectangle, this field, whatever this might be, what is the length of that side? Well if you multiply these two sides, you get the area, so if you start with the area, if you start with 268, 268, and you divide by the other side, divide by the other side, you're going to get the length of this side right over here. So if we wanted to figure out the length of that side, it would be 268 divided by two. And you've already seen, or we've already seen, multiple ways to figure out what 268 divided by two is, but the whole reason of me drawing this rectangle, or this aerial view of this field, or whatever you want to call it, is so that we visualize it using area. And so one way to do it is, is to break up this 268 square unit area into areas that are easier to imagine dividing by two. So here I have the same field, but I've just broken it up. So it's the same field, this dimension right over here is still two units, two units, but I've broken it up. This blue area is 200... let me do that in another color. So this blue area is 200 units, this yellow area is 60 units, and this magenta area is eight units. Now why is that useful? Well, now each of these, it's much easier to divide by two. All I did is I took the 268, and I said, "Well look, this is the same thing as "200 plus 60... "plus 60, plus eight." I just broke up the 268 into things that are easier to divide by two, and now I can take those things and divide by two. I can take each of them and divide by two. So what is, what is this, what is this dimension going to be, right over here? Actually let me do that in a different color. What is this dimension right over here going to be? Well two times that is going to be a hundred, so this is going to be 100. And how did we get that? Well we got that by 200 divided by two. 200 divided by two is 100. What's 60 divided by two? Well, 60 divided by two is going to be 30. So this part of the field is going to be 30 in that direction and two in this direction. And once again, not drawn to scale. And then finally, what's this section going to be? It's going to be, it's going to be, eight divided by two, which is four. Notice, hundred times two is 200, 30 times two is 60, four times two is eight. And so this whole length up here is going to be 100, plus 30, plus four, or 134. Now we've already seen other ways of coming up with this. You say, "Look, two hundreds divided by two is 100, "six tens divided by two is three tens, "eight ones divided by two is four ones." And that's exactly what we just did over here, but we visualized it using this kind of a rectangle, breaking it up into chunks that are, maybe easier to imagine dividing by two. We broke it up into two hundreds, two hundreds right over here. We broke it up into six tens, or 60, or 60 right over here. And we broke it up into eight ones, eight ones. We broke up the area, and then we took each of those areas and we divided it by two to find that part of the length. And when you add them all together, you get the entire, you get the entire length. Now, that's just one way to do it, where you take each of the place values and you break up your field or your rectangle like that, but you could do it other ways. You don't always have to break it up that much. For example, let's say, let's say that this... let's say that this area is 856 square units. Square units... Now let's say that this dimension right over here is eight, is eight units. Is eight units. So how could we break this up so it's easier to think about what the other dimension would be, what the length is going to be? And this length, once again, is going to be 856 divided by eight. 856 divided by eight is this length right over here. Well you could break it up in to eight hundreds, five tens, and six ones, but you might notice, "Well five tens, it's not so easy to divide that by eight." But we can divide 56 by eight. We know that eight times seven is 56. So what you could do is, you could break this rectangle into the 856 square units, you could break it up in to 800 square units, and then another 56, and another 56 square units. So once again, we broke it up in to eight hundreds, and then 56 ones. And then 50... 56... 856 ones. Same area, I just broke it up. And now if you say, "Look, this is eight." if you say, if you say, "This right over here is eight, "well what is going to be, what is going to be this length? "What is this length up here going to be?" Well it's going to be 800 divided by, it's going to be 800 divided by eight. So actually, let me write it below it. So 800 divided by eight, this length right over here is going to be, is going to be 100. How did I get that? I got eight hundreds divided by eight is 100. 800 divided by eight is 100. And then you have this other magenta part. What is 56 divided by eight? Well that's seven. 56 divided by eight, this length right over here, is going to be seven. So what's this entire length? What's this entire length? It's going to be a hundred plus seven, which is equal to 107. So once again, you could have said, "Look, eight goes into 800 a hundred times, "eight goes into 56 seven times, 107, "why did Sal draw these rectangles?" Well just so that you can help... it's sometimes helpful to visualize, and say, "Look, you're trying, this is an area problem. "If this is the total area, if 856 is the total area, "and eight is one of the dimensions, "well then 856 divided by eight "is going to be the other dimension." And one way, when you're breaking this up by place value, you're breaking up the numbers into pieces that are easier to divide by eight, you can think of it as you're just breaking up that area and you're just trying to figure out parts of this length. So that's a hundred right over there, and then this right over here is seven. So anyway, hopefully this broadens your visualization capabilities when you are dividing multiple, when you're doing division.