In section 6.1, one of the things that you're going to start doing is looking at the two different interpretations of division that we have. One is how many groups and the other is how many units in one group. And you'll be asked to write problems of both types. You'll be asked to look at problems and identify which type they are.
And so I wanted to just kind of introduce you to the vocabulary. and get you started on some of the things that you'll need to be able to do that. So if we're talking about division of a problem that looks like A divided by B equals question mark, of course we want to label what A and B are and what the question mark represents so that we kind of know what's going on. And... Because we have two interpretations of division, there are two ways that we can label these division problems.
So A is always going to be the number of objects that are going to be put into equal groups. So that part isn't going to change. The B...
and the question mark will change depending on if you're talking about a how many groups problem or a how many units in one group problem. So for the first one that I want to talk about, I'm going to let b represent the number of units in one group. And we'll see what that means here momentarily. And then that means the question mark.
is going to be how many groups are there. And just to give you kind of some of the vocabulary that you're going to see in this chapter, A is called the dividend, B is called the divisor. And the answer when you do a divided by b is called the quotient. And that is going to be the same for both of the types of division that we're talking about. Those terms stay the same.
So that part won't change. Okay, so... There is a link between division and multiplication. And in fact, the way you rewrite your division problem as multiplication depends on whether you're talking about a how many groups problem or a how many units in one group problem. So if I want to take this division problem and rewrite it as multiplication, I'm going to have question mark.
which remember was how many groups times b which was the number of units in one group equals a so this is the number of groups this is the number of units or objects in one group and then a was our dividend or the total number of objects to be put into equal groups. So this should look familiar because this is exactly what we talked about in chapter 5 when we started doing multiplication or chapter 4. The first factor is the number of groups, the second factor is the number of units in each group. And this is the total number of units or objects.
So that part should look the same. So this is classified, this division with the labeling or the annotation in this way, is a how many groups division problem. And the reason this is a how many groups type.
is because that's what our unknown represents. If the unknown is how many groups, then that's the type of division that we have. This is one of the two types of division. Let's do the same thing over here. This A, our dividend, like I said, is always the number of objects to be put into equal groups.
And this time, B is the number of groups, and the question mark is how many units are in one group. And again, A is the dividend, B is the divisor. C or I'm sorry, the question mark is the quotient.
If I write this version of division as multiplication, if B is the number of groups, then B gets written first in our multiplication, because the first factor in multiplication is always the number of groups. The number of units in one group is second, so it's B times question mark equals A. So over here, we had question mark times b equals a. Now we have b times question mark equals a.
Because our unknown this time is how many units in one group, that's the type of problem that this is. It's a how many units in one group type of division problem. So the two interpretations of division are the how many groups. which was this kind, and the how many units in one group, which is this type. So last thing I want to do is just give you an example in context of each kind of problem.
So let's say Mrs. Gardner. bought 63 candies to make Halloween bags for her students. So that's the beginning context. Mrs. Gardner bought 63 candies to make Halloween bags for her students.
If we want to write a how many groups question, that means we need to know how many units she's putting in each group. So our bags, the Halloween bags that she's making, are going to be our groups. And we aren't going to know how many bags she's going to make. So let's suppose, so I'm going to branch off this way and do a how many groups problem.
Let's suppose... She puts seven candies in each bag. Then the question becomes, how many bags did she make? And like I said, the Halloween bags are the groups. And so we are looking for how many groups did we end up with if each Halloween bag is a group and each Halloween bag has seven candies in it.
So this particular problem is a how many groups problem. Suppose we want to write the other type of problem which is a how many units in one group. We can start off the same way.
Mrs. Gardner bought 63 candies to make Halloween bags for her students. The bags are still going to be our groups, but this time I want my unknown to be the number of units in one group, and I want my known divisor to be the number of Halloween bags. So let's say instead of the last part that we wrote over here, let's say she makes...
Seven Halloween bags. How many candies are in each bag? Or are in one bag? Either way. So this is a division problem for 63 divided by 7 equals question mark, where the question mark is how many units in one group.
This is a division problem for 63 divided by 7 equals question mark, where the question mark is how many groups. So this one was a how many groups problem, which means this one is a how many units in one group problem. And so like I said, one of the things that you will do in section 6.1 is read problems and identify which type of division problem they are based on the context. And so if you know how many groups you have, but you don't know how many items are in one group, then it's a how many units in one group problem. But if you know how many items you have in one group and don't know how many groups you have, then it's a how many groups problem.
I hope you enjoy section 6.1. I hope you're okay with this vocabulary and these examples. If you need help, please ask me and I would be happy to help you.