 Okay, so let's take apart the standard algorithm for multiplication and see where that comes from and so the standard algorithm is based on three key ideas. First of all, the most fundamental concept is that multiplication is a repeated addition. Second, the properties of whole number multiplication, particularly the associative and commutative properties, and then also the base trade rate, which allows us to write our answer in a standardized form. So as I preliminary to the introduction of the standard algorithm, let's consider the problem 60 times 4. And so, literally read by the definition of multiplication, 60 times 4 is the sum of 64s. However, we also note that because multiplication is commutative, 60 times 4 is the same as 4 times 60, so 64s is going to be the same as 460s. And the reason that that interchange is useful is that I don't really know at this point how to add 64s together short of just adding them together, and that's a lot of things add together, but I can add 460s a lot more easily as follows. I'll use the place value chart and I'll set down the factor that I'll be adding repeatedly. So I'm adding 460s, so I want to start my place value chart by showing a 60. So here I have, here's my ones, here's my tens, 60 is 6 tens, 0 ones. Now I want four of each of these, so I'm going to take four 6s in the tens. And because the trade rate is 10 for 1, I really shouldn't write down this number, but rather I should write down what it's going to be once I simplify everything. So I will bundle sets of 10, that's a 20 and a 4, and then I'll trade that 20 at my trade rate of 10 for 1 for 2 in the next place over. So there's four 6s, it's going to be two in this column, hundreds, four tens. And I also want four of the zeros, because again I'm going to take four of everything, and four zeros is after a bunch of thought, zero, and all together arithmetic is bookkeeping. So what we want to do is we want to keep track of how many of each unit that we have. So here I have two hundreds, four tens, zero ones. So I'll just go ahead and write those down, and so that tells me that 60 times 4 is the same as 4 times 60, is two hundreds, four tens, and zero, 240 in standard x terms. Well, the reason that that example is important is let's consider the product 67 times 4. So again by definition 67 times 4 is the sum of 67 4s. And I could do the same thing I did the last time, convert it into the commutative product 4 times 67, and find the sum of 4, 67s, but I'm going to need to do the multiplication as it is, because there's a limited number of times I can do that commutativity usefully. So I want to view this as the sum of, well, let's see, that's 67 4s. Well, 67 4s is the same as 64s together with 7 4s. So let's go ahead and write that down. So again what I have is I'll set down 4 in my place value chart, and I want 64s, and I just figured out what 64s is. That's 240, so that's 240, and again I'll bundle. There's my set of 10, and trade at 10 for 1, I'll bundle, set of 10, and trade at 10 for 1, and there's my 64s. I also need 7 4s, and that's going to be 28. And again I should bundle and trade, and so there's my 64s, my 7 4s, and so again arithmetic is bookkeeping, so I want to determine how many of each unit I have. So here I have 200s, I have 6 10s, and I have 8 1s. And my product then is going to be 67 x 4, 200s, 6 10s, 8 1s, 268.