Multiplying 3-digit by 2-digit with area models

0:00- What we're gonna do in this video is try to figure out what 259 times 35 is. And there's a lot of ways to approach this, but one way to think about it is, imagine we had a rectangle. I'll make it a really wide rectangle. So that's the width of our rectangle, and then that's the height of our rectangle.

0:22And let me just draw the entire rectangle. And let's imagine that this width of this rectangle over here is 259 units, whatever those units might be, that's its width. And then this height over here is 35 units. Well then the area of this rectangle would be 259 times 35, whatever that product is.

0:49Now, why am I even talking about this? Why is this useful? Well, we can actually break up 259 into 200 plus 50 plus nine and think about those chunks of the area and do the same thing for 35. What do I mean by that? Well, let's first imagine 200. So if this whole thing is 259, then maybe 200.

1:09And I'm not going to get it exactly right, but maybe 200 is going to be about, about, let's see, am eyeballing it. It's about that much of it right over there. And then the 50, 50 would be about that much of it. And then the nine, I'll do this in a new color, might be that much of it.

1:36That much of it. So 259 is the same thing as 200 plus 50 plus nine. And I can divide this rectangle into these areas. So one way to think about it is that this area over here is going to be 200 times 35. This area is gonna be 50 times 35, and this area is nine times 35.

1:56Well, as I said, I could do the same thing with 35. Let's say that, let's say that this is 30. That is 30 right over there, and then the rest of it is five. So this right over here is five. Well, I can further, I can split up this bigger rectangle into even more rectangles.

2:20Now, why is this interesting? Well, this top left area is going to be what? It's going to be 30 times 200. And we might be able to do that in our heads. If I did three times two it would be six, 30 times two is 60, 30 times 20 is 630 times 200 is 6,000. Or some of you might say, okay, three times two is six.

2:40And then I have these three zeros, which is the equivalent to multiplying by ten three times or multiplying by a thousand. So it's 6,000. And now what is the area of this section right over here? Well, that's going to be 30 times 50. So 30 times 50 is going to be equal to three times five is 15.

3:01And then I have two more zeros. So it's 1500. And then this over here is going to be 30 times nine, which, three times nine is 27. So 30 times nine is 270. And now if I go down here, this a little area is going to be, I probably should have drawn it a little bit bigger, but it's going to be five times 200, which is equal to 1000.

3:25This one over here is gonna be five times 50, five times 50, which is equal to 250. And then last but not least five times nine is 45. Now why is this interesting? Because each of these little smaller areas, if I add them all up, I get the bigger area. So the bigger area is going to be 6,000 plus 1500 plus 270.

3:54And one way to think about it is each of these are partial products. They're partial products of just 30 times 200 or 30 times 50 or 30 times nine. But I'm gonna keep going. And then I have this a thousand here, which was the five times 200. Then I have the 250 here, and then I have the 45.

4:11If I add up all of these partial products or partial areas, I could say, then I'm gonna get the total product or the total area of this rectangle. So let's see, in the one's place, I just end up with a five, in the tens place, seven tens plus five, that's 12 plus four is 16 10s.

4:28So six tens and a hundred. One plus five is six, plus two is eight, plus two more is 10. So that gets regrouped that one, and we get one plus six is seven, plus one is eight, plus one more is nine. So we get to 9,065 in total.