Making Sense of Division

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When solving division problems, students are often able to produce a quotient without realizing what that quotient represents. For example:

The bag has 695 pieces of candy corn, and Alyssa wants to put them into bags of 50. How many bags will she need?

Students may be able to mechanically produce the quotient 13. But do they fully understand the referential meaning of "13"? And let's not even get started about what the remainder represents!

Explicitly teaching students, with a model, that there are two ways to think about division can help them develop powerful mental structures for analyzing their results. I'm going to suggest Cuisenaire Rods as an excellent model for representing division.

Sharing (partitive) division is the situation your students are most comfortable with because it can be solved by dealing out. In sharing division, we know the total and we know the number of groups. The unknown is the number of items in each group. We can solve this by dealing out, or sharing, all of our items one at a time until there are no more to be distributed.

Sarah has 10 puppies and 2 doghouses. How many puppies can be grouped evenly in each doghouse?

The Cuisenaire Rods offer a nonlinguistic representation of the situation. We have a given total divided into 2 groups. We must find the number in each group.

There will be 5 puppies in each doghouse.

In the second division situation, the action to solve the problem is making groups. This is called measurement division, and lends itself well to the idea of repeated subtraction - repeatedly removing groups of a particular quantity until there are no more to be removed.
With a measurement division situation, we know the total, and we know the number in each group. The unknown is the number of groups.

Amiel has 10 puppies. He wants to place them 3 to a doghouse. How many doghouses will Amiel need?

We are presented with a given total, and the number per group.  We must find the number of groups.

We see that there will be 3 groups of 3 with one puppy left over. So Amiel will need to either give one puppy away are build a 4th doghouse!

Students should be challenged to solve both types of division problems, and to imprint the models into their mental toolbox. Eventually, students will develop strategies they will apply in both situations, and use their mental models to help them make sense of the results.