Cognitively Guided Instruction, Lesson 9:

Grouping problems: Multiplication, Measurement Division, Partitive Division.

The structure of multiplication and division problems; the difference between multiplication, measurement and partitive division.

Multiplication and division problems are structured significantly differently from addition and subtraction problems. In addition problems you add apples to apples (and in subtraction, you subtract apples from apples), but in a multiplication problem, you multiply apples times bags:

Karen had 4 bags of apples.
Each bag had 3 apples.
She had 12 apples in all

When we have a multiplication problem that is structured like this, we give the different parts different names that go with the parts they play.  The number of groups is called the multiplier, the amount in each group is the multiplicand, and the total is called the product.  Later, we'll group the multiplier and multiplicand together and call them factors, but in a beginning introduction to multiplication and division, we use the terms multiplier and multiplicand to recognize that the numbers play different roles in how children visualize and act out the problem, and that it's not obvious that these numbers should be interchangeable (it is true but not at all obvious that 4 apples in each of 3 bags is the same amount as 3 apples in each of 4 bags).

A problem with this basic basic structure can be either a multiplication or a division problem depending on what the known and unknown values are.

If the multiplier and multiplicand are known and the product is unknown, then it is a multiplication problem:

I will write 4×3 to mean 4 groups of 3--so the first factor is the multiplier and the second factor is the multiplicand.  It isn't really fixed and unanimous which order we write multiplication in, but when you are first introducing multiplication to children, you should be consistent about which order you write multiplication sentences in.

If the product and the multiplicand are known and the multiplier is unknown, then this is called a measurement division problem. 

While we have a different name for the number of groups and size of groups when we write this as a missing number multiplication problem, we don't have different words for the number being divided by, so the known amount (whether a number of groups or an amount in each group) is called the divisor when we write a division problem, the total being divided is the dividend, and the result (whether a number of groups or amount in each group) is called the quotient.

Measurement division gets it's name because we know the size (the measure) of each group, and we can use that size to measure the total. 

When the known amounts are the total and the number of groups, and the amount in each group is unknown, then this is called a partitive division or partition division problem.

Partittive division gets its name from partition.  A partition divides something into smaller groups.  We find the amount in each group by partitioning the total into the given number of groups.

One thing that's a little tricky about remembering these names is that while the addition and subtraction problem types are named by the unknown (Result unknown, part unknown, difference unknown, etc.), division problem types are named by what is known about them. In a measurement division problem, the measure of each group (amount in each group) is known, and in a partition division, the number of groups (the partition--how many to split into ) is known.

Direct Modeling Solutions for multiplication, measurement division and partitive division.

There are several variations in how a child who is solving a multiplication or division problem by direct modeling may go about doing so.

Multiplication

For the problem:

Karen had 4 bags of apples.
Each bag had 3 apples.
How many apples did she have in all?

The basic strategy is to make 4 groups each of which have 3 apples in them. 

• Some children will just make groups of 3 counters (each of which represents an apple). All of the counters are counted to find the answer.
• Some children will first put out 4 counters to represent the bags, and then put 3 counters next to the "bag" counters to represent the apples in the bags. Then all of the "apple" counters are counted to find the answer.  (Even though it may not seem very clear to an observer, the children are generally quite good at keeping track of which counters represent bags and which represent apples.  You can see this strategy used by children in both the Children's Mathematics: Children's Strategies clip, and in the Children's Mathematics: Classroom Episodes (kindergarten) clip.)

Measurement division:

For the problem:

Karen had 12 apples.
Her apples were in bags, and each bag had 3 apples.
How many bags of apples did she have?

The basic strategy is to make groups of 3 counters with a total of 12 counters.

• Some children start by counting out 12 counters, and then move the counters into groups of 3.  The number of groups is counted to find the answer. (You can see this strategy in a clip on the Children's Mathematics: Children's Strategies CD.)
• Some children start by making groups of 3 counters, and then count the total in their groups and keep going until they have 12 total. The number of groups is counted to find the answer.

Partitive Division:

In the problem:

Karen had 12 apples.
Her apples were in 4 bags, and each bag had the same number of apples
How many apples were in each bag?

The basic strategy is to make 4 groups with a total of 12 counters.  This is more complicated than the measurement division strategy because you don't know how big to make each group.  As a result there is more variation in how children solve partitive problems by direct modeling.  All of these variations are discussed in Children's Mathematics, and are strategies different children will develop when given appropriate problems to sove:

Basic strategy #1: Count out the total (12) counters, and put/partition/deal them one at a time into 4 groups until you have used all of the counters.  Count the amount in 1 group to find the answer.

• variation 1: make the groups in 4 separate spaces on your work area.
• variation 2: use another counter to represent the groups (bags).  When you count the amount in a group to find the answer, count only the counters that represnet apples, and not the counter that represents the group.  (This variation can be seen in a clip on the Children's Mathematics: Children's Strategies CD).
  

Basic strategy #2: Count out 12 (total) counters.  Put equal numbers of counters into 4 groups, but not necessarily 1 at a time.  Adjust until all of the groups have an equal amount of counters.  Count the amount in a group to find the answer.  The following situations may arise:

• The child may initially put too few counters into each group, and may have to add more until all of the counters are used.
• The child may initially put too many counters into each group, and may have to take counters from a group with too many to even out with a group that has too few:

Basic Strategy #3: Deal out one counter at a time.  Count the total number of cubes you have put out as you go.  Stop when you reach 12 total counters.  Count the amount in a group to find the answer.

• Some children will make their groups just in a particular position  on the workspace
• Some children will use a counter to designate each group.

When and how to introduce multiplication and division.

Some multiplication and division problems are harder than others because of the context of the problem. One way of looking at relative difficulty is by looking at the table of multiplication and division problems (page 48 in the Children's Mathematics book).  As with the addition and subtraction problem types table, the type(s) in the upper left corner are the easiest, and problems get harder as you procede down or to the left on the table. 

Looking at the Common Core Standards, you'll see that there's a little bit of multiplication in second grade, and a lot of multiplication and division in third grade.  Everything in this lesson, and then some, is part of the third grade standards. Now, most of the types of problems in this lesson can be taught and learned by much younger children (K-1): when you watch the Children's Strategies clips on the Children's Mathematics CD, you'll see that they have children (I think first grade) solving their multiplication and division problems by direct modeling.  I noticed when I was watching those (especially the partitive divison example) that the problems were really set up in such a way as to suggest a direct modeling strategy for solving them.  So, this is the sort of thing I noticed.  You could write basically the same problem, tweaking it just a little, and make it more or less obvious to direct model:

Easier: Erin has 12 Pokemon cards. She can fit 4 cards on each page of her card collecting book. How many pages will she need to hold all of her Pokemon cards?

Harder: Erin has 12 Pokemon cards in her card collecting book. If there are 4 cards on each page, how many pages are filled with Pokemon cards?

I think the first one will be easier because you have this idea of acting out putting cards on the pages--if you are going to pretend you are Erin, then you are going to act out putting cards on pages.  In the second problem, if you are pretending to be Erin, then you already have cards on pages, and the action is done, not waiting to be done.

The previous problem was a measurement division problem (which are somewhat easier to solve anyway), so the distinction may be less important there.

For this partition division problem, the action is going to be even more helpful (just because it's sometimes hard to know where to start out solving a partition division problem):

Easier: Julia's mom made 12 cookies. She wants to put them on 3 plates, so each plate will have the same number of cookies. How many cookies should she put on each plate?

Harder: Julia has 12 cookies. The cookies are in 3 bags, and each bag has the same number of cookies. How many cookies are in each bag?

If you watch the partition division problem on the CD, you'll see that my first problem is really similar to the cupcake problem.  The person with the cookies (cupcakes, pencils, etc.) has some items that they want to put evenly into containers.  So, that action of putting things into containers, is an action you can act you.  In the second version, there's no action (the cookies are already in the bags), so it's not telling you what to do to act it out.

So, when doing multiplication or division problems with younger children (K-1, possibly 2), it's helpful to start with problems where there's an obvious action to act out, and, of course, with numbers that are small enough that they can easily solve problems with them.