 Alright, let's take a look at another way. We can multiply two things, and this one's actually a little bit more advanced. It's called multiplication using an area model. And this is based on two ideas, which are independent of all the arithmetic that we've done thus far. And the first is this notion of geometric area. If I have a rectangle that's A units wide by B units high, the area of that rectangle is the product A times B. A little bit later on, we'll look at the origin of this formula, because, again, it's a formula, and you can take it for what it is, but if you actually want to use it to find actual areas, it's important to understand what you're doing here, and not just that the formula is A times B. But here it's useful because it makes the connection that we can find the area by a product. The product is the area, the area is the product. The other idea that the area model is based on is the notion that area is a conserved quantity. And what that means is that if I have some area, whatever it looks like, and if I were to break it into a whole bunch of different pieces, the areas of all the individual pieces put together would equal the area of the whole thing. I can take an area, I can break it apart, but what I have left, what I have is still the same total amount of area. It may just be in a whole bunch of different pieces. So, for example, let's take a look at this. I want to find the product 24 times 8 using an area model. And, again, the idea here is that the area of a rectangle that's 24 by 8 is going to be the product 24 times 8. Now, at this point it does absolutely no good to then multiply 24 by 8 to get the area because, well, the problem is not to find the area of this rectangle, but rather to use the area of this rectangle to figure out how to do that product. So, if you just draw the rectangle and then do the multiplication, you're wasting time in performing this operation very inefficiently. The idea here is that if I break this rectangle up into pieces, I might be able to calculate the areas of the individual pieces much more easily. And the most important thing to remember is that the only way you should ever do this split is how you feel like computing. How you do the split depends on what you feel like computing. So, today maybe I'll be a little bit lazy and say, you know, I really don't feel like multiplying very large numbers. So, maybe I'll split that into a 10, a 10, and a 4. So, here I still have the rectangle that's 10, 24 by 8 because I wasn't feeling too ambitious. I just split it into bits, 10, 10, and 4. And now I can calculate the areas of the individual regions. So, 8 by 10, that's 80. 8 by 10, 10 by 8, that's another 80. 4 by 8, that's 32. And area is conserved. The sum of the areas is the product. So, I'm going to add 80 plus 80 plus 32. And that's going to be the same as the product, 24 by 8. And that works out to be 192. Well, I don't have to split it that way. I can use any split that I want to. So, again, I'll draw my 24 by 8 rectangle. And maybe you remember how to multiply by 12. So, I can split this 24 into 12 and 12. So, again, the important thing is I still have this 24 by 8 rectangle. So, this is 24 by 8. And if I remember that 12 times 8 is, what is it, that's going to be 96, then I can find the areas. And, again, the sum is the area. 96 plus 96, 192. The nice thing about this is I can extend it as far as I want to. So, 125 times 38. Well, let's set down an area model. We'll set down a rectangle that's 125 by 38. And nothing in the rules limits the number or size of the pieces that I can use. I can make the pieces as small or as large as I want to. The only requirement is that whatever size I choose, I have to work with. So, let's see this 125 by 38. Well, I'll split off lengths that are easy to work with. So, I might do 125 and 5. So, there's my 125 length. Now, I could do 100 by 38, 20 by 38, 5 by 38. But nothing in the rules says I can't split up the other factor as well. So, let's split up that 38 into 30 and 8. So, here's my rectangle once again, 125 by 38. And the area of this rectangle is going to correspond to this product. And, again, the advantage here is that the areas I have to compute are very easy to calculate. This is 100 by 30, 20 by 30, 5 by 30, and so on. So, I'll compute those areas, 30, and I have all those areas of all those rectangles, and the sum is the product. So, I'm going to add those together, 125 times 38. I'm going to add this way. This is 3,800. This is 760. This is 190. I'm going to now add these three together and get my final answer, 4,750.