 Alright, let's take a look at division. And it turns out that division actually encompasses two different ideas, which are referred to as the partitive form and the quotative form. So let's go ahead and take a look at that. So if I have a division problem like a divided by b, there's two different ways that I can think about this problem. First, I can think about it from a partitive viewpoint. So what I'm going to do is I'm going to take a and divide it into b equal parts. And the question we're trying to answer, how big is each of these parts? The other possibility is I might look at a quotative division. And this comes from the idea of having a quota. And so what I'm going to do is I'm going to divide a into a bunch of parts, each with a quota, each with a size of b. And so the question I want to answer here is how many parts do I have? Now, we can use the tape diagram to model both forms of division. So let's take a look at that. So for example, I want to take a look at the division 24 divided by 4. First, I want to show it partitively. And I also want to show it quotitively using a tape diagram. So let's take a look at our partitive division first. So here I'm going to divide this into four equal parts. So there's where our partitive comes from. And so I'm going to take my tape representing 24 units, and I want to divide that into four equal pieces. So I'll draw the dividing lines looking something like that. And the question I want to answer is how big is each piece? And the answer to that is I don't know, but let's take a guess. Suppose I guess that, oh, I don't know, maybe each piece is size 5. So I'll make this piece size 5. And since all the other pieces are the same size, then all the other pieces are also size 5. And I count, multiply however I want to do this. The important observation I make here is this is not 24. These four pieces together only add up to 20. So five, not quite enough. So let's make it a little bit larger. And again, we might note that we have 20. So we're pretty close. So let's not make this a lot larger. Let's make it, for example, one unit larger. And so if I make it one unit larger, then each of these parts is 5 and 1. Each of these is 6. And I have 20, 21, 22, 23, 24, which is all that I have to divide. So that gives us our total of 24. And so now I've taken 24. I've divided it into four pieces of equal size. And each piece is of size 5 and 1, otherwise known as 6. So I can say that 24 divided into four equal pieces is going to give us 6 in each piece. And there's our partitive division. What about the quotitive division? Well, again, in this case, what we're going to do is we're going to take the 24, and we're going to make a bunch of pieces each with four in each piece. There's our quota, and each of our pieces is going to have four in each of them. So we'll go ahead and start with our figure representing 24 and just start putting down pieces of size 4. So I'll throw down a piece of size 4. I'll throw down another piece, another piece, another piece, another piece. And let's see, how much do I have here? 4, 8, 12, 16, 20. I could throw down one more piece of size 4. And there we have our 24. And the question is, how many pieces do I have? Well, I can count them 1, 2, 3, 4, 5, 6. So if I break 24 into pieces of size 4, I get 6 pieces each. And so I can say 24 divided by 4 is 6, and there's our quotitive division.