Multiplying 3 digit by 2 digit with partial products

0:00- [Instructor] We're going to do in this video is figure out what 295 times 39 is. And we're going to use a technique known as partial products, which sounds very fancy, but hopefully when we see it, it's just breaking down these numbers and multiplying all of the different parts of it and then adding up everything together.

0:15But we're also going to visualize it with what's known as an area model. Now, an area model just means thinking about an area of a rectangle where one side is 295 units and the other is 39 units, and then breaking up the different pieces. So what do I mean by that?

0:31So let's say that, let me do one side over here. Actually, let me make it... So let's say this is 200, that's 200 there. Actually, maybe make it like that. So let's say that that is 200. This is 90, I'm gonna try to draw it as close to scale as I can, but I'm probably not gonna get it perfect.

0:53So this is 90, if that's 200, then 90 would look something like that. And then five would be pretty short. Five would look something like this. And I'm gonna multiply that times 39. So 30 might look something like this and then nine would look something like this.

1:15Whoops, let me make it a little bit cleaner. So nine, maybe something like that. And I can make a grid. So let me make it very clear, this is 200 plus 90, 90, and that is five. And I'm going to multiply that times 39. So the rectangle, the area of the rectangle that I'm about to draw is the area that we're trying to figure out, the area of this rectangle right over, right over there.

1:47And I'm gonna set it up as a grid so I can do the different parts. This will be useful for the technique of partial products. So let me just do each of these parts here. So in partial products, what you would do is you would just write these numbers. Actually, maybe I'll write it...

2:07Maybe I'll write it like this, just in one color so I don't have to keep switching. 295 times 39. We get the place values under each other. Now in partial products, what you wanna do is make sure that you multiply each of the numbers by each of them, what do I mean?

2:23Well first, we might say, all right, what is nine times five? Well, nine times five we know is 45. Now, how is that... Let me do this in a different color since I already used that purple color. So nine times five, right over there, is 45. Now, how does that relate to the area of this whole rectangle?

2:45Well, nine times five would be nine times five. That would be this area right over there. So it's part of the whole rectangle, but it's not the whole thing. So this is a partial product. That's why we call it partial products. Well then, we can go on and we could say, well, what is nine times nine?

3:05Well, nine, or let me be careful. That's not a nine, that's a nine in the tens place. It's nine times 90. Well, nine times nine is 81. So nine times 90 is 810. Now, where is that in our area model? Well, here it's nine times 90 is 810. And then we can move on and we can try to figure out what is nine times 200.

3:33Well, nine times 200 is going to be 1,800. Where is that in this area model? nine times 200 is 1,800. Then, we can go on to this 30. What is 30 times five? 30 times five is 150. Where is that here? That's right over... It's hard to see, let me make it in a slightly different color.

4:00So 150, right in there, 150. Then, we can do... I'm really running outta colors here. Then I can do 30 times 90. So one way to think about it is three times nine is 27, 30 times nine is 270, 30 times 90 is 2,700. And then last, and where is that on this area model?

4:25Three, or I should say, oh, let me be clear, this is 30. This is 30 right now. I don't know how, why I wrote three here, 39. What is 30 times 90? That is 2,700, what I just wrote down. So that is this 2,700. And then last but not least, and that's why this area, one of the reasons this area model is useful, I can make sure I'm getting all of the combinations.

4:48I wanna figure out what 30 times 200 is. So let's think about it, 30 times 200. If I had three times two, that is six, 30 times two is 60, 30 times 20 is 600, 30 times 200 is 6,000. So there, this right over here is 6,000. And if I want the area of the whole thing, or if I want this entire product, I just have to add all of these things up, all of these partial products up.

5:21So let's do that, and I have it all nicely set up here. I only have five ones. Let me put a five there. See, four plus one plus five is 10 tens, which is equal to 100. So let me just put it as 100. Let me put it up here to show where I'm regrouping. One plus eight plus eight is 17, plus one is 18, plus seven is 25, so that's five hundreds and two thousands.

5:50Two plus one is three, plus two is five, plus six is 11 thousands. I guess I could write it like that. And then I'm done: 11,505. 11,505, which is this product, which would be the area of this rectangle, which would be if I added up all of these partial areas or partial products.