# Division Clusters

Question: My child is in the 5th grade and they are doing division clusters. He is having a difficult time in learning this particular form of math, and the way that I am showing him how to do this is confusing him even more. Please help me explain this type of math to him better than what I am doing (i.e. 135 ÷ 5 = ).
Parent

Answer: Cluster problems are groups of related problems that help students use what they know to solve harder problems. "Students choose problems from the cluster to help them solve the final problem. Sometimes, more than one strategy is suggested by the group of problems, depending on which problems are used. Students need not solve all the problems in the cluster, and they may add additional problems to the cluster that they find useful in solving the final problem." (Building on Numbers You Know, p. 82)

For example, consider the following cluster problem, which is designed to help students solve 12 x 25.

 2 x 25 = 4 x 25 = 10 x 25 = 12 x 25 =

The question is "What do you know that would help you solve 12 x 25? If you know that 4 x 25 = 100, you might think of 12 x 25 as (4 x 25) + (4 x 25) + (4 x 25) or 100 + 100 + 100. Or, you might start with 10 x 25. If you know that 10 x 25 = 250, then you can start with 250 and add two more 25's to get 300." (Mathematical Thinking at Grade 5, p. 62)

Cluster problems are often used to aid students as they begin working with a mathematical operation. As the students work with the clusters, they "learn how they can pull apart problems into more manageable components, then combine the answers to each component; they learn to use their knowledge of factor pairs, multiples, and the relationship between multiplication and division; and they become familiar with what happens when we multiply a number by 10, 100, or their multiples." (Building on Numbers You Know, p. 82) In other words, cluster problems provide students with ways to think about solving a problem that help them become more efficient at the operation.

The original question, however, is particularly concerned with division. This is a common question, as "many elementary students are more comfortable with multiplication than division, just as they are often more comfortable with addition than subtraction." (Mathematical Thinking at Grade 5, p. 59)

There are a variety of ways a fifth grader might solve the problem 135 divided by 5. The first step is thinking about what you know that will help you solve the problem.

Some students will think, "135 divided by 5 means how many 5's are in 135. I know there are 20 5's in 100. Then how many 5's in 35 is 7. So there are 27 5's in 135."

100÷5=20
35÷5=7
135÷5=27

Another student may solve the problem in a very similar way, but will think of the problem primarily as multiplication. They might say, "I think of 135 divided by 5 as 5 times what equals 135? I know 5 times 20 is 100. That leaves 35. 5 times 7 is 35. So the answer is 27."

5 X ? = 135
5 X 20 = 100
5 X 7 = 35
5 X 27 = 135 so 135÷5=27

Therefore, a cluster problem to help students solve 135 ÷ 5 might include both multiplication and division problems:

 5 x 20 = 5 x 7 = 5 x 20 = 5 x 7 = 135 ÷ 5 =

Note that strategies such as these rely on the fact that "multiplication and division are related operations: both involve two factors and the multiple created by multiplying those two factors. For example, here is a set of linked multiplication and division relationships:

 24 x 3 = 72 72 ÷ 24 = 3 3 x 24 = 72 72 ÷ 3 = 24
" (Mathematical Thinking at Grade 5, p. 63)

To help your child with division cluster problems, there are several questions you can ask, and ways you can support their thinking. Continuing to use 135 ÷ 5 as an example, you might ask, "What is the problem asking you to figure out?" (How many groups of 5 are in 135, or how many would be in a group if you made 5 groups?) Sometimes making up a story (or context) to go with the problem is also helpful. For example, there are 135 people that need to be split into 5 teams. How many people would be on each team? Or, there are 135 people. They need to be put into teams of 5. How many teams would there be?

Always encourage your child to look at the cluster of problems, and ask, "Which of these do I already know the answer to?" (Students should be able to solve the cluster problems mentally.) "If you know 100 ÷ 5 = 20, how does that help you solve 135 ÷ 5? What do you still have to figure out?" The goal is for children to think about the simpler problems, how they relate to each other, and how they relate to the final problem.

Megan Murray, TERC
With thanks to Keith Cochran, Katie Bloomfield and Lucy Wittenberg
February 2004

You can find additional information related to this question and how to support your child using Investigations in previous Ask the Author features:

This information was reprinted with permission of CESAME, Northeastern Univ., and the Educational Alliance, Brown University.