 Alright, so again one of the important things to keep in mind is that the value of basic arithmetic is not because someday you may have to learn how to multiply 27 by 15 without a calculator, but rather that sooner or later you're going to have to do some algebra. And understanding what's going on in basic arithmetic helps you with an understanding of what's going on in algebra because algebra is the generalization of arithmetic. And everything you do in basic arithmetic has an analog and algebra. And in this case let's take a look at that definition of multiplication. So our definition of A times B is that A times B is the sum of A sum ends of B. And this carries over as long as A is a whole number. We can use this definition even if we have no idea what B is. So if B is the unknown x, then 5x, 5 times x, well that is the sum of 1, 2, 3, 4, 5 x's. And so 5x is equal to that, 3xy, well that's the sum of 3xy's and 100x squared's, well that's the sum of x squared plus itself 100 times. I'm not going to write that out but I have a whole bunch of these x squared's added together. And if you understand that then a lot of those basic rules of algebra that you get started with are pretty obvious. For example 3x plus 2x, what can you do to simplify that? Well again to remember how to answer this question just remember that A B is the sum of A B's. So what's 3x? Well 3x is x plus x plus x, 2x is x plus x, and when I add these two together, 3x plus 2x, so I'm going to put these together with these, I'm going to get this mess here. And well I don't have to carry that around, I know the definition of multiplication. I have x added to itself 1, 2, 3, 4, 5 times and so that thing on the right hand side is 5x. Well here's a very common mistake in algebra, somebody writes down 3x plus 2y and they add them together and they get 5xy. So two things we want to do, first of all we want to identify that this is not actually the correct addition of these two terms but equally importantly, why is it incorrect? Of course you could just resort to the tactics of saying you're wrong, that's not how you do it, but that's not very helpful for a couple of reasons. One, it's fairly confrontational and there's no need for it. But really the goal of this is twofold. First of all, if you have to explain something you have to understand what you know about. You have to go over what you understand about the concept and in particular you also have to consider what it is you don't understand. The other one is in every field, it doesn't matter what you go into, if you're never going to do algebra in your life you still benefit from the ability to communicate and so this is as much a problem in the practice of communication as it is a problem in mathematics. Well let's think about this. In terms of the first aspect, we have to understand what we're talking about. Well we have to go back to this question of what do we really mean when we say 3x, 3y, and 5xy? What do we really mean by these ideas? And again, so 2x, well that's the sum of two x's. 3y is the sum of three y's. And if I add 2x and 3y together, well addition is a putting together of two different things and as I put these together I get this thing. And there's not a whole lot I can do with that. I can see that this is 2x, I can see that this is 3y, but I can't really combine these in any meaningful fashion. On the other hand, let's take a look at that 5xy. Well I know the definition of multiplication, so 5xy is the sum of 5xy's. So here's what 5xy looks like, here's what 2x plus 3y looks like, and the important thing to notice here is they're not the same. They are distinctly different. Now I might point out some other things here. There's definitely five things here, but the problem is it's not 5x's, it's not 5y's, it's not 5xy's. There's five of something, but they're different things, so I can't really write this down as five of anything. Here I have five things all the same, so I can write down 5xy. But here when I add 2x and 3y, while I do have five of something, I can't describe the something in any useful fashion. And so this sum is this, this sum is this, and they're very different.