 Alright, so let's take a look at how we would actually perform a division. And so, again, our definition of division, if I want to find a divided by b, by my definition, if I have a divided by b, what that's telling me is that a is c times b. And so that stems from our definition of division. And if we apply the definition of multiplication, that tells me that a is equal to the sum of cb's. And rearranging things a little bit by my definition of subtraction, that means that if I subtract bc times, then what I end up with is going to be zero, because there's going to be nothing left on the right-hand side. And so what that says is that I can perform a division, a divided by b, by answering the following questions. How many times do I have to subtract b, starting with a, until I get zero? How many times can I subtract b from a? And this is our basic division algorithm, and this is the basis for all methods of division. And so, this is everything you really need to know about how to subtract. Now, it turns out there is a problem here, which is that this approach to doing a division actually is at odds with how we interpret what a division is. And this leads to what's sometimes called cognitive dissonance. And the problem is that one of the things that makes division difficult is the most natural way to explain a divided by b is partitively. If I want to think about a division like 57 divided by 3, what I can do to explain what I'm talking about is I'm going to take 57, and I'm going to divide it into three parts. And so there's our partitive division. The problem is when we apply that repeated subtraction model for performing a division, we're actually solving it. We're actually finding it by interpreting this quotient quotatively. How many times can I subtract b from a? So again, 57 divided by 3. How many times can I subtract 3 from 57? And in particular, this concept requires a quotitive understanding of division. So again, let's take a look at that. So 36 divided by 12. Again, if you just read what this is saying, this sounds like I'm going to take 36, and I'm going to break it into 12 equal pieces. That is the partitive interpretation of this quotient. That's what the language sounds like we're doing. 36 divided by 12. And that's what the language tells us to do. But what we're actually doing is we are finding this quotient by subtraction. So I want to find out how many times I can subtract 12 from 36. And so I can actually do this. I'll show the subtractions using an arrow diagram here. So I'm starting with 36, and I'll subtract 12, and another 12, and another 12. And I'm at zero, so now what I've done is I've subtracted 12 one, two, three times. I'm able to subtract 12 three times, and so that tells me that the quotient is equal to three. Again, we can interpret this in terms of our multiplication. Working backwards, one, two, three, 12s together is 36. So 36 is three times 12, so 36 divided by 12 is equal to three. Well, let's do another division, 72 divided by three. And again, we view this. We read this as 72 divided into three equal pieces. But how we find it is through a repeated subtraction of threes. Now, that's actually going to take a while because this is a fairly large dividend. But as long as I keep track of how many threes I'm subtracting, I can not subtract single threes, but instead I can subtract sets of threes. And I could subtract large groups of threes and make my subtraction more efficient. So for example, one easy set of three, 10 threes, well that's the same as 30. So I might subtract, instead of subtracting a single three, I might subtract 10 threes all one go. And that means I'm going to start by subtracting 30, and I can subtract 30 again. And at this point I can't subtract 30, but I can subtract 3, and 3, and 3, and 3. And now I just have to keep track of how many threes I've subtracted all together. So let's go ahead and count them. So this 30 is 10 threes, so is this. So that's 10, 20, 21, 22, 23, 24. Altogether I've subtracted 3 24 times, and so that gives me my quotient. Now we might want to organize this repeated subtraction in a slightly different way, partly because it'll make it easier to keep track of everything. So what I might want to do is I might organize my subtraction vertically. And so here I have what I'm working with, what I'm subtracting, and in this column I'll just track the number of threes that I've subtracted. So I'm going to subtract 30, well that's 10 threes. And I can do that again, and I can't do that anymore, but I can now subtract 1, 3, another 3, another 3, and another 3. And so now same method, but we've just organized things a little bit differently. I've subtracted 10, 20, 21, 22, 23, 24 threes all together. So I'll add the number in that column, and there's the number of threes that I've subtracted all together, and I've ended up with a 0. So once again, my quotient, 72 divided by 3, is going to be 24, and there's nothing left over.