 Alright, so once we know a few multiplication facts, once we know how to multiply a couple of fairly small numbers, we might try to find out how to multiply larger numbers. And one of the easy places to start is what's called multiplication by factors. And the basic idea is the following. We rely on the associative property of multiplication, and that is to say that if I have three numbers A, B, and C, the product A times B times C, I can multiply A and B together first, then multiply by C, or I can multiply B and C together, then multiply by A, and I get the same result either way. And the reason that this is useful is in general one of these multiplications, B times C or A times B, is going to be easier. And so I look for the easy one, and I might be able to multiply much more efficiently. And if I wanted to generalize that, if I wanted to find a product M times N, I might be able to break one or both of the factors into lots and lots of pieces, and individually those pieces might be easier to work with. So for example, let's take 6 times 15. Well, on the one hand, I know that 6 is 3 times 2. Again, we have to know some multiplication facts, but we can use very simple ones. Since 6 is 3 times 2, I can multiply by 3, and then multiply by 2. So if I want to do 6 times 15 times 6, that's 15 times 3 times 2, but I don't have to do it that way. I can regroup it as 15 times 3 times 2. And well, let's see, 3 15-minute quarters is 45 minutes. And well, times 2, that's easy. I can do that in my head. That's going to be 90. And one way I can organize this, I don't have to write all this out, because the relevant pieces of information really are these partial products, 45 and 90. So what I might do is I might show this using an arrow diagram. What I did was I took 15 times 3 to 45 times 2 to 90, all together times 3 times 2 was a product of 6. Same example, but I can do this in a different order, possibly an easier order, because multiplication is commutative. 2 times 3 is the same as 3 times 2, so 15 times 6, 15 times again, 6, but I don't have to do it in that order. I can do the 15 times 2 first. Okay, that's a lot easier. Times 2, that's pretty easy. That's 30. Times 3, well, again, that's an easy multiplication to do. It's going to be 90. And my corresponding arrow diagram looks very similar. Times 2 times 3, and all together multiplied by 6. Still the same example. Nothing in the rules says I have to break up the second factor. So I could break the 15 apart. So 15 times 6, well, again, I have to know some multiplication. 15 is 3 times 5. And again, I can group however I feel like it. 5 times 6, I like that one. 5 times 6, easy to do. That's 30. Times 30 is 90 once again. And we can use as many factors as we can find. And as usual, we can gain significant improvements in efficiency and speed if we switch between our approaches. Again, this is the basis behind the concept of adaptive expertise, the more ways of solving a problem that you know, and the more you understand about what makes a particular approach easy or difficult to use, the more efficiently you can solve any problem. If you only have a hammer, your only alternative is to bash things apart with it. Works all right if you're trying to crack a nut, not so much if you're trying to get into a door. So let's take 46 times 12. Well, one thing I might do, I know 12 is 3 times 4. So 46 times 12, 46 times 3 times 4, 46 times 3. At this point, I might think about the distributive property. That's 40 and 6. So that's 3 times 40 is 120. 3 times 6 is 18. So 3 times 46, 120 plus 18. That's 138. Still times 4, I haven't done that yet. Now I could multiply by 4, but I'm going to take the easy way out. 4 is 2 times 2. And so that means I can times 2 double and then double again. So 138 doubled. Let's see, that's 130 and 8. That's 260 and 16. That's 276. And doubled again. That's 276 double. That's 400, 140, 12. That's 552 as my final product. And again, as an arrow diagram, times 3 times 2 times 2. And notice that we only required two things here. We had to know how to multiply by 3 and by 2. I didn't have to know how to multiply by 12. All I had to do, all I did have to know was that 12 is 3 times 4. I had to know a multiplication fact, a couple other multiplication facts, and then the ability to multiply by fairly small numbers. Certainly, I didn't have to know how to multiply by a two digit number. Well, suppose I did know how to multiply by 12. So for example, let's take 36 times 24. If you do know how to multiply by 12, and some of us learned at some point, and we might actually be able to do this as, well, let's split this up. 36 is 3 times 12. 24 is 12 times 2. And the associative property combined with the communicative property says I can rearrange things and multiply however I feel like it. And well, if I know how to multiply by 12, I know what 12 times 12 is. And then I need to multiply by 2 and then by 3. So times 2 times 3. Here, I'm applying the distributive property 3 times 280 at 8. That's 3 times 200. 3 times 80. 3 times 8. And I'll add those together, 864. Now, just as a quick little note, this is a lot of writing. And you may say, well, that's not efficient to do a lot of writing, which is true. But it's only a lot of writing because we want to go through what our steps actually are. In practice, you could actually do this in your head. 36 times 24 is 3 times 2 times 12 times 12. That's 3 times 2 times 144, which is 288 times 3, which is 240. That's 840 plus 48 is 864. And again, you can actually do this product in your head because these are all things that are relatively easy to keep track of. Trying to do this using the standard algorithm in your head, a lot more challenging.