 There's a number of ways of multiplying two numbers, but one of the more visually appealing is known as multiplication by an area model. And this is based on the following idea. From geometry, we know that the area of a rectangle is equal to the product of its length and width. And this means two things. If we want to find the area of a rectangle, we need to multiply. But also, since most things in mathematics swing both ways, if we need to multiply, we can find the area of a rectangle. For example, suppose we want to find the product 8 times 27. The product 8 times 27 is the area of a rectangle with width 8 and length 27. So we'll draw a picture. Now there's no point in drawing the picture and then calculating 8 times 27, since we could have done that without the picture. The reason the area model is useful is we can break the rectangle into smaller rectangles of convenient sizes. And here a useful thing to remember is that how you speak influences how you think. The length of 27 suggests that we can break this side into parts of length 20 and 7. And if we do that, this gives us two rectangles. A rectangle with length 20 and width 8 with area 8 times 20, which is 160, and another rectangle with length 7 and width 8 with area 8 times 7 or 56. We'll add the areas of the individual rectangles to get the area of the whole rectangle. And remember, the area is the product. So 8 times 27 is 216. Or how about 27 times 45? Now the product 27 times 45 is the same as the area of a rectangle that has length 45 and width 27. So we'll draw a picture. Now it's important to understand there's no point in drawing a picture and then computing 45 times 27 because we could have done that without the picture. The advantage to having the picture is that we could do things with a picture that might not be obvious with the numbers themselves. And in this case we'll break the sides of our rectangle into convenient parts. What's convenient? Well here it's useful to remember how you speak influences how you think. And so this length is 45. Well that suggests that we can break the length of 45 into a part of length 40 and a part of length 5. Likewise, the width of 27 can be broken into a part of length 20 and a part of length 7. This gives us four smaller rectangles whose areas are easier to compute. So this rectangle with width 40 and length 20 has area 40 times 20. That's 800. The rectangle with width 5 and length 20 has area 5 times 20. That's 100. The rectangle with width 40 and length 7 has area 40 times 7, or 280. And this rectangle with width 5 and length 7 has area 5 times 7, 35. And the area of the whole rectangle will be the sum of the four areas. And the area is the product. 27 times 45 is 1215. Or how about 154 times 37. So again, this will be equal to the area of a rectangle with length 154 and with 37, so we'll draw a picture. Again, how you speak influences how you think. One side is 154, so we'll split one side into parts of length 150 and 4. The other side, 37, we'll split the other side into parts of length 30 and 7. And this gives us the areas of six smaller rectangles to compute. So the area of the rectangle with sides 130 is going to be 100 times 30. That's 3000. Now let's stop a minute and smell the roses. Or remember, knowledge is power. In this particular case, notice that this rectangle is both longer and wider than any of the other rectangles. So in some sense, most of the area of the whole rectangle is contained in the area of this one rectangle. And so this suggests that the area of this one rectangle is approximately equal to our product. And so this suggests the approximation. 154 times 37 is about equal to 3000. The next two largest pieces are going to be the rectangle with sides of 100 and 7, whose area is 100 times 7. That's 700. And the rectangle with sides 50 and 30, which has area 1500. And again, these are the three largest parts of the area, so we can approximate 154 times 37 as the sum of the areas of these three pieces, 3000 plus 700 plus 1500 or 5200. Of course, this isn't the exact area because we still have a couple of bits of area left to calculate. So our last few rectangles have area 50 by 7, 4 by 30, and 4 by 7. And the area is the product. For a little bit of variety, let's add our partial sums vertically. So we'll add 3000 plus 700, we'll add 1500 plus 350, and we'll add 120 plus 28. And then we'll add our areas together to get our final answer.