What Does Division Really Mean

Fraction division is a messy business.   Now that we have Common Core State Standards, students must be able to model fraction division in addition to performing the task algorithmically.   This is no simple task for many of them, because it forces them to grapple with the question of what it means to divide.   In order to do it successfully, they must really understand that division represents one of two things.   It can be dividing something into a specified group size to find the number of groups. (I have twenty four cookies and I want to make packages of 2 cookies, how many packages can I make?)  It can also be dividing something into a specified number of groups to find the group size.  (I have twenty four cookies and I want to serve 12 kids, how many cookies can they each have?)  They have to be able to read a problem and figure out which of those two types of problems it is and then form groups accordingly.     Making sense of these ideas and constructing a real understanding of fraction division is hard, even for a lot of adults.

As we were wrapping up our work with these ideas this week, I wanted to do a quick formative assessment to see where everyone was on these ideas.   I took two of the ACE  questions (this is the set of problems from which we draw homework assignments)  from the Connected Math textbook that I use.   Instead of using them as part of a homework assignment, I turned them into a Vote With Your Feet activity.

Vote With Your Feet is a Marzano high engagement strategy that incorporates movement into a lesson. In the activity, students are presented with a multiple choice item.   They move to different locations in the room based on their chosen response to the question (e.g., north wall for A, east wall for B, south wall for C, west wall for D).  The activity gives students a chance to get up and still stay focused on the task at hand.   It also is a really quick formative assessment, taking only a minute or two to see what each student thinks and where their misconceptions are (if the question is well-designed).

Here are the two questions that I posed.

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I had students vote on each of the two items without commenting on their choices.    Afterwards, we debriefed both questions by discussing what problems each model might represent.   Kids talked to partners and in table groups.   The class talked about it together.    As we talked, I asked students to give me two division problems for each model.   What is the problem if you are dividing by a specific group size?   What is the problem if you are dividing by a certain number of groups.

These are two of my favorite problems for a formative assessment on fraction division.   This year, I used them as a Vote With Your Feet activity.   Next year, I might take the four options and make them Quiz/Quiz/Trade cards.   I might take them and just use the model and present it as a “here is the answer, what is the question” formative assessment.   In the meantime, I will probably go ahead and make Quiz/Quiz/Trade cards with problems like this to use as a quick review of fraction division from time to time later in the year.

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