# Tackling Tanton’s Pinwheel Area Problem

While our middle school math summer success program has ended I wanted to share how the students grappled with James Tanton’s Pinwheel Area problem. I was only working with seven students that day because the rest of the class was on an EL field trip.

I gave each student a copy of the problem and 10 minutes of quiet think time before solving as a group.

After 7-8 minutes none of the students were able to get started so I asked, “What shapes do you see?” Their first response was the pinwheel and squares. Then I asked, “What other shapes to you see?” There as a very long pause. I sensed the students eyes were solely focused on the pinwheel and grid so I asked them to look at the entire image. Another long pause.

“Do you see any other shapes besides the pinwheel and squares?”

A few moments passed and a student responded, “I see triangles.”

I should have anticipated this task better. I was expecting to scaffold, but as we moved through the task, the students needed additional support.

“What triangles do you see?” My eyes naturally saw the larger triangles, but his eyes didn’t.  I asked the student to point to the triangles.

At this point I recognized the grid was a distractor–which is great in terms of adding rigor to the problem while providing another scaffolding opportunity.

Let’s draw this on the board.

Now we were able to discuss a strategy. It took some time, but we agreed that if we were able to subtract the non-shaded area from the 5×5 grid, we would find the area of the pinwheel.

Students enjoy working on the whiteboard and wanted to shade the triangles.

Students recalled how to find the area of a square or rectangle, but could not remember how to find the area of a triangle. I drew a rectangle, and labeled the length and width as 3×2. “What’s the area?”

“Six.”

I cut the rectangle in half horizontally and asked, “What’s the area of one section?”

“Three.”

I erased the horizontal line and replaced it by cutting the rectangle in half vertically. “What’s the area of this section?”

“Three.”

I erased the vertical line and cut the rectangle in half diagonally. “What’s the area of this section?”

“Three.”

So if we multiply the base and height of a triangle and cut it in half (divide by 2) we can find the area. We revisited the printout of the problem to find the measurements and began calculating.

Now we were able to put it all together.

To achieve productive struggle this group required more scaffolding than I originally anticipated. I need to reread Five Practices.

## 3 thoughts on “Tackling Tanton’s Pinwheel Area Problem”

1. It’s interesting that the grid was initially a distractor. I thought that would’ve been more of a help in determining the units and give a starting point. Did you by chance have any students attempt to estimate the area first? I’m wondering if any students looked at the shape and thought it was less/more than half of the entire area.

I enjoyed reading about how you integrated a mini-lesson on triangle area within the task. Some students that I see initially use the same formula to find the areas of rectangles and triangles. While reading your post I thought of how often I ask students if there answer is reasonable. Here the units might have been helpful to estimate the reasonableness of their answer. I appreciate the many different entry points for this problem as well as the multiple standards that are evident. About how long did the task take?

1. I should have asked students to estimate first, but I didn’t. It would have been an a good starting point and would have made for good discussion afterwards.

The task took nearly 50 minutes from start of finish. I gave a considerable amount of wait time at various points. At one point I considered pausing and revisiting the next day, but since half the class was on a field trip, revisiting it the next day would have been problematic.

Eliminating the grid reduced the amount of “clutter” and “busyness”. For this group I’m glad I did that.

1. The “pausing and revisiting” dilemma is something I consider too. It’s more of an issue of making sure everyone has been exposed to the task or giving students an extension opportunity (which doesn’t happen as much as I’d like). I also have to consider the scope-and-sequence and how many sessions students are missing because of other time commitments. Most of my students are also part of orchestra/band. I could easily turn the 50 minute time into two class sessions, but then I revert back to what standards are being explored (not necessarily just the posted objective).

Thanks for sharing, Mary. I’m hoping you and your team have another chance to use this task at some point next year.