 So we'll introduce another way that we can do multiplication, which we might call multiplication by factors. And this actually makes use of the associative property of multiplication, since I can do a multiplication A times quantity B times C, I can rearrange the factors A times B times C. And it may be that one of these products, A times B, or B times C, is going to be easier to do. And so this means that I can do multiplication by breaking a number apart into a couple of factors. So in general, it means that if I want to find M times N, if it's easy for me to break one of these two numbers into a bunch of factors, which are individually easier to work with, this multiplication by factors is going to be more efficient. For example, take 15 times 6. Now we might be able to do this multiplication by expanding 15 out as 10 plus 5, but then we can also note that 6 is 3 times 2, and it's easy to multiply by 3, or by 2, or vice versa. So I'll break that 6 apart into 3 times 2, and I can use the associative property and rearrange that. Rather than doing the 3 times 2 first, I'll do the 15 times 3 first. And that's going to be 45 times 2, and again, times 2 is easy. That's going to be 90. And as an alternative, I can display exactly the same multiplication using an arrow diagram. 15 times 3 gets us 45, times 2 gets us 90. Well, again, we can do the same problem, but this time let's do it in a slightly different way. Since again, 6 is 2 times 3, I can multiply first by 2, and then by 3. And that may make the problem easier. Let's take a look at that. So 6 splits into 2 and 3, and associative property says I can multiply by 2 first, and here, multiplying by 2, that's actually a pretty easy task to do, and I get 30 times 3. Again, easy multiplication to do equals 90. And by arrow diagram, 15 times 2 times 3, and there's my multiplication. Well, why not do the same thing? I have nothing in the rules, so I have to break the 6 apart. Maybe I can break the 15 apart, so I'll break apart the 15 into 3 times 5. Again, the associative property says I can do this multiplication starting with any pair, and here, again, times 5 is actually one of the easier multiplications to do, so I'll multiply 5 by 6 to get 30, and again, I'm back at 3 times 30, and this time, again, product is still 90. Now, the only requirement here is that I want to be able to find the factors easily, and again, the numbers that I'm working with are numbers that I have to want to work with. So I can use as many factors as I want to, and I can gain a lot in terms of efficiency and speed if I use a mixture of approaches. For example, consider the product 46 times 12. Now, if you really want to get a good sense of how this works, do this in your head. Don't write anything down, just do the multiplication mentally, and how we might do that if we were to do this mentally. 12, well, I might start out by thinking 12 is 3 times 4, and so I might begin. 36 times 12 is 46 times 3 times 4, so the associative property says go ahead, do the times 3 first, and again, do that multiplication in your head. Don't write anything down. That's 46 times 3, and the distributive property says we can do that as 3 times 40 plus 3 times 6. So that's 120 plus 18 is 138, still times 4, and while we could do 4 times 138, let's go ahead and break that 4 up. That's a 2 times 2, and so my 138 times 4 becomes 138 times 2 times 2, and the associative property says I can do that 138 times 2 first. So again, think distributive property, that's 138 times 2. That's 260 and 16, 260 plus 16, 276 times 2 again. And again, think distributive property, 276 times 2, that's 400, 140, and 12. So that's 400, 540 plus 12, 552 as my final answer. And again, we can think about this, drawing this out using an aero diagram, 46 times 3, that's 120 plus 18, 138 times 2, that's 260 and 16, 276 times 2 is 400, 140, and 12, 552. And as always, the more you know, the more efficiently you can apply the procedure, any procedure. So here's a problem, 36 times 24. And suppose you know the product tables for 12, maybe. So we could rewrite this problem, both 36 and 24 are multiples of 12, so I could rewrite this as 3 times 12 times 12 times 2. And multiplication is both associative and commutative, which means I can rewrite this in any order that I want to. So I'll write it this way, and I know my multiplication tables for 12, 12 times 12 is 144, and then I can, again, associativity allows me to rewrite this product any way I want to. So maybe I'll multiply by 2 first, and again 2 times 144, again thinking in terms of the distributive property, that's 144 times 2, that gives me 288, and then 3 times 288, and thinking in terms of the distributive property, that's 3 times 200, that's 600, 3 times 80, that's 240, 3 times 8 is 24, so there's my 600, 240, and 24, that's going to be 607, 800, 40, 860, 860 for as my product.