 So let's take a look at a couple of different methods of multiplication, and we'll start out with multiplication by the area model, because this is a fairly important and fairly powerful method of doing multiplication. And it's based on two principal ideas. First off, the area of a rectangle that's A units wide by B units high is the same as the product. So that means that if I can find the area of the rectangle, I can find the product, and vice versa. Now, that by itself isn't particularly useful except areas a conserved quantity. And what that means is that if I break an area, if I break a region into several pieces, the area of the region is just the sum of the areas of the individual pieces. And that's really useful because it means I can find whatever area I want to, however complicated the region is, I can break the region into a bunch of bits and find the areas of each individual bit. So let's take a look at a problem. We're going to use an area model for this. So our first step is we want to draw a rectangle with sites 24 and 8. So there's our rectangle. And if I can compute the area of this rectangle, then I also can find what the product is. Now, the important thing here is we don't just find the area of this rectangle by multiplying 24 times 8, because that defeats the purpose of having an area model in the first place. The key idea here is we want to split this up into a bunch of pieces and use areas we are comfortable calculating. So I want to break this rectangle into a couple of pieces, and the only requirement for those pieces is I should be able to calculate what the areas of those pieces are. So let's take a byte out of this rectangle here. Let's keep the height of that rectangle 8, but let's take a chunk of that with 10. And because the rectangle is 24 across, I know I have some extras left over. Let's take another chunk of size 10, and let's see that's 10 and 10. That leaves me four left over for this last bit. And since area is a conserved quantity, I can find the area of the whole rectangle by finding the area of each of the individual pieces. So this first piece here has area 80. This is 10 by 8, also area 80. 4 by 8, area 32. And the sum is the area, so the sum is the product, so I'm going to add those areas together. The product is the sum, and 80 and 80 and 32 is 192, and there's my product. Do we have to do the split this way? Not at all. We can find it using an entirely different split, and again, it depends on what we feel comfortable working with. So we can use any split of the factors that we want to. So again, we'll start off with our 24 by 8 rectangle, and suppose you know how to multiply by 12. You memorized your 12 times tables. Well, you can break off a chunk of 12, and there's my first bit, and again, the rectangle is 24 wide, so if that first bit is 12, it turns out the last bit is also going to be 12, and you can say, ah, I know how to multiply 12 times 8. 12 times 8 is 96. And so I have the two areas, and now I can just add the two areas together. The product is the area is the sum. So I'll add them together, and I should get 192 once again. Just as a quick review of how we might add, this is almost 100 plus almost 100. This is four short of 100 twice. So this sum here, 200, but I've gone too far, subtract 8. That gives me my 192. Well, you can also find products of multi-digit numbers. So here's an example, 125 multiplied by 38. So again, we'll set down our 125 by 38 rectangle, and again, it doesn't do us any good to just multiply these two numbers to find the area to find the product that's defeating the purpose of using the area model in the first place. I want to break up the factors into pieces that I am willing to work with. So what might those be? Well, let's start with this 125. So maybe I'll take a first piece that's 100 wide, and there's still a bunch left over, so maybe I'll take another piece that's 20 wide, and then that's 120, and the last bit is going to be a width of 5. And so now I have 1, 2, 3 areas, and they're a lot easier to work with. Well, nothing in the rule says you only can split the top. I could also split the side as well. This length of 38, maybe I'll split that further. That's going to be a length of 30. I like working with 30, and then leftover length is going to be 8. So here's my setup for the multiplication, 125 times 38, and it's worth noting that what's going to make this easy is that any rectangle whose area I have to find, I can find by multiplying two numbers, both of which have a single non-zero digit. Now, I didn't have to do it this way, but it's convenient because if I know how to multiply one digit numbers 20 by 8, then I can find the product fairly easily. So let's go ahead and find what those areas are. Useful idea for things to come later on. If we're interested in estimating the product, notice that this rectangle here is actually the largest. It has the greatest width and it has the greatest height. So if I wanted to estimate the product 125 times 38, I could look at the biggest piece, which is 100 by 30 or 3,000. So as a quick estimate of what the product is, it's someplace in the vicinity of 3,000. Well, if I want to find the actual product, I need to find the remaining areas. So this rectangle here, 20 by 30. This rectangle here, 5 by 30. This rectangle here, 100 wide by 8 high. This rectangle here, 20 wide by 8. This rectangle, 5 by 40. And those are the areas of the individual pieces, and the sum is the product. So how you add them together, it depends on how you want to add them together, but we might add them this way. I might go to this first set of rectangles here, 3,000 plus 800, 600 plus 160, 150 plus 40 gives me 190, and I'm going to add these three numbers together. And skipping over the details, I get 4,750 as my final sum and product.