Two Pizzas and Five People

I’m thinking a lot about how my 6th graders responded to a pre-lesson task in “Interpreting Multiplication and Division” — a lesson from Mathematics Assessment Project (MAP) .

MAP lessons begin with a set structure:

Before the lesson, students work individually on a task designed to reveal their current levels of understanding. You review their scripts and write questions to help them improve their work.

I gave the students this pre-lesson task for homework, and 57 students completed the task.


I’m sharing students’ responses to question 2 (of 4) only because there’s already a lot here to process. I’m grouping the kids’ calculations and answers based on their diagrams.

Each pizza is cut into fifths.

About 44% (25/57) of the kids split the pizzas into fifths. I think I would have done the same, and my hand-drawn fifths would only be slightly less sloppy than theirs. The answer of 2 must mean 2 slices, and that makes sense when we see 10 slices total. The answer 10 might be a reflection of the completed example in the first row. “P divided by 5 x 2 or 5 divided by P x 2” suggests that division is commutative, and P here must mean pizza.

 Each pizza is cut into eighths.



Next to cutting a circle into fourths, cutting into eighths is pretty easy and straightforward.

 Each pizza is cut into tenths.



I’m a little bit surprised to see tenths because it’s tedious to sketch them in, but then ten is a nice round number. The answer 20, like before, might be a mimic of the completed example in the first row.

 Each pizza is cut into fourths.



I’m thinking the student sketched the diagram to illustrate that the pizzas get cut into some number of pieces — the fourths are out of convenience.  The larger number 5 divided by the smaller number 2 is not surprising.

 Each pizza is cut into sixths.



It’s easier to divide a circle by hand into even sections, even though the calculations do not show 6 or 12.

 Each pizza is cut into fifths, vertically.



Oy. I need to introduce these 3 students to rectangular pizzas. :)

 Five people? Here, five slices.



Mom and Dad are bigger people, so they should get the larger slices. This seems fair. We just need to examine the commutative property more closely.

 Circles drawn, but uncut.



I’m wondering about the calculation of 5 divided by 2.

 Only one pizza drawn, cut into fifths.



Twenty percent fits with the diagram, if each person is getting one of the five slices. The 100 in the calculation might be due to the student thinking about percentage.

 Only one pizza drawn, cut into tenths, but like this.



I wonder if the student has forgotten what the question is asking for because his/her focus has now shifted to the diagram.

 Each rectangular pizza is cut into fifths.



Three kids after my own heart.

Five portions set out, each with pizza sticks.


I wonder where the 10 comes from in his calculation.

Five plates set out, each plate with pizza slices.



Kids don’t always know what we mean by “draw a picture” or “sketch a diagram.” This student has already portioned out the slices.

What diagrams and calculations would you expect to see for question 3?


There’s important work ahead for us. The kids have been working on matching calculation, diagram, and problem cards. They’re thinking and talking to one another. I have a lot of questions to ask them, and hopefully they’ll come up with questions of their own as they try to make sense of it all. If I were just looking for the answer of 2/5 or 0.4, then only 12 of the 57 papers had this answer. But I saw more “correct” answers that may not necessarily match the key. We starve ourselves of kids’ thinking and reasoning if we only give multiple-choice tests or seek only for the answer.

That’s why Max Ray wants to remind us of why 2 > 4.

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