Farrand kicked this one off with a figure composed of 18 paper squares that had a perimeter of 20. “Can you come up with a figure using 18 squares that has a larger perimeter than this one?” he asked participants. “How about a smaller perimeter?”
Farrand then passed out baggies of paper squares that we cautiously manipulated on our laps. This session was begging for tables. With squares placed corner to corner, we found a maximum perimeter of 72. We agreed on a minimum perimeter of 18. One participant stacked the squares on top of each other for a perimeter of 4, but this only resulted in clarifying conditions for the shapes.
West, Farrand, and Stetson teamed up to ask a lot of questions. On the side, we were asked which questions perplexed us the most.
Through a series of these questions we were coaxed into thinking about what happens to the perimeter as we add squares. We quickly discovered that when squares are added, sometimes perimeter increases, but at other times it doesn’t. Here’s what we found:
The group’s discovery was that changing the number of contiguous squares in a row or column has no effect on the perimeter. The perimeter is equivalent to a figure with the row or column maximized. Putting a finger on why this is the case and using language to explain it pulled in some rich math practices.
Farrand then challenged us to use this understanding to calculate the perimeter of a figure without counting:
Throughout the session Stetson facilitated “Teacher Time-Out” moments, highlighting approaches that were used to get us thinking more deeply about the problem. “Did you hear the question Rick just asked? What sort of thinking would that require you to do? What mathematical practices are in play here?” Participants reflected on the importance of capitalizing on student curiosity and facilitating student sharing.
We then explored figures composed of equilateral triangles with the guiding question, “How does adding triangles affect perimeter?” One big surprise of the session came at this point: adding an equilateral triangle to a figure of five triangles can actually decrease the perimeter.
I found this problem to be refreshingly devoid of context. There was no talk of minimizing fence material for a dog’s pen, for example. It was a great opportunity to reason and justify conclusions. Discoveries were made and a few surprises propelled participants into further questioning. West, Farrand and Stetson reminded us that problem solving tasks are key sources for the use of mathematical practices in our classrooms.