# Size of Group vs. Number of Groups: When Division Meets Context

My 4th grader recently came home with a division math assignment and declared, “I don’t get this math!”  (This is not usually his M.O.)  When I asked him what he was working on, he showed me this:

So I began asking questions to determine where his understanding stopped and his confusion began.  I first asked if he knew why his teacher had him decompose his thousands into hundreds.  He said he wasn’t sure.  I then asked if he thought four thousand was the same of forty hundreds.  After some thought, he used him homework to show how each thousand had been “broken up” into ten hundreds, totaling forty hundreds.  I nodded in agreement and pushed forward.  Perhaps this was his point of misconception.

My next question seemed too obvious to ask, but I did it anyway.  I asked him what division means.  I felt like this would be a question he would have an answer to, as we have worked quite diligently on interpreting multiplication of “groups of” over the past year and I thought he would make the connection.  When he looked at me with a blank stare, I realized that his misconception was at the foundation of the operation.  So I offered a context I knew he would be familiar with to support his conceptual understanding of division:  Money!

Here is the problem I posed:

You have \$4000.  You want to divide it equally among 10 of your friends.  How much will each friend get?

He began looking at the representation he had drawn with his teacher.  He could see the four thousands and could also identify the forty hundreds, but could not figure out how his picture gave him an answer to the question I had asked.

I, too, took another look at his drawing and realized that even though my story problem could be solved with the equation 4000 ÷ 10, just like the problem he had represented, I was asking him to make ten equal groups and his representation showed equal groups of ten. What a happy accident!  Not only were we going to get to the bottom of the meaning of division, but we were going to be able compare the two types of division with the same problem.

(If you haven’t yet seen this progression video linked above, it is worth your time!)

So I asked him, does this picture show ten equal groups to represent the ten friends?  He said no.  I asked if he could create a drawing that would help him find the answer using ten equal groups.  He thought about this for a while, then leaned on his understanding of “fair shares” to find an answer.  His thought process was very interesting.  One might call it the beginnings of partial quotients.

Wanting to push his thinking forward on these two definitions of division using the place value chart, I asked him to identify where his four hundreds would be in his previous representation.

After some serious thought, he made a vertical ring around the four dots in the left-most portion of his drawing.  I then asked how many of those four hundreds did he have in his drawing.  At this point he answered confidently, “Ten”, without needing his representation for support.  I then circled back to the questions I asked at the very beginning, which was why the four thousands needed to be decomposed.  He stated that if there were ten friends and only four thousands that there wouldn’t be enough thousands to share equally.  By making hundreds, he could share them fairly.  I nodded and asked one final question, “What does it mean to divide?”.  He said it means to share fairly.  With a smile on his face, I knew we had reached the point of understanding.

Walking away from this learning episode, I was reminded of two very important elements of teaching division:

1. There are two interpretations of division, so we must be sure to model them both in our lessons.
2. Context matters!  Give students problems worth solving and allow the context of the problem to drive their interpretation of division.

I am a math consultant with the Stanislaus County Office of Education and a preservice math content instructor at California State University, Stanislaus. It is my passion to work with teachers to improve math content understanding and instruction.