We looked at two diagrams as we finished up our unit on Circles. I knew that we didn’t have time for everyone to do both tasks. I wondered whether I could pull off giving students some choice in what they investigated.

Both tasks are from the Mathematics Assessment Project.

The first can be found as Circles in Triangles in the tasks section and Inscribing and Circumscribing Right Triangles in the formative assessment lessons section.

The second is Temple Geometry from the tasks section.

What mathematical question could we explore?

Some of you saw my tweet.

Yes. We are doing that “wonder” thing.

Most of the wonderings were about areas of various regions, and most of the questions were about the second diagram. I let each team decide what to explore. Only one team chose the first diagram.

I had copied a handout with questions about both diagrams, but I hesitated to give it to students, as it gave so much away. Instead, I gave the students a copy of the diagram only on which to work.

As has become our practice, students started working by themselves.

They practiced **look for and make use of structure**.

Some practiced **use appropriate tools strategically** to make sense of the problem.

Then they talked with their teams. Those working the second task almost immediately concluded that the radius of the smallest circle was half the radius of the largest circle. One student had used paper to measure the distance. Another “eyeballed” it correctly. Others constructed the diagram using dynamic geometry software and used the measuring tool to verify their conjecture. But no one was able to prove that the radius of the smallest circle was half the radius of the largest circle.

Time was ticking quickly. I still hadn’t even talked with the team who had chosen to work on the Circles in Triangles. What should I do to move student learning forward?

We moved into whole class mode. I selected a few students to share their thoughts about the length of the radius of the smallest circle. I made CS the Live Presenter so that he could show how his team translated the smallest circle so that its center lay on the point that partitioned the diameter of the medium circle into a 1:3 ratio.

So if the radius of the smallest circle is one-half the radius of the largest circle, then what is the area of the shaded region? Every team set to work, ensuring that all of members of their team could calculate the area of the shaded region, and they did so successfully. (The evidence is on my computer at school … we are at home today for a “snow” day.)

What would have happened if I had given the teams the handout from the beginning?

From Circles in Triangles:

From Temple Geometry:

Would they have learned more mathematics during the lesson? Would they have practiced **look for and make use of structure** or **use appropriate tools strategically **during the lesson? Would they have engaged in **productive struggle**?

What would have happened if I had given the teams the handout that provided the structure for them once they got to a certain point on their own … even though they wouldn’t have had time to complete the investigation together?

I am learning that what works in our classrooms has so much to do with the students we have. Productive struggle isn’t just for students: We can plan great ideas collaboratively, but even so, we must be attentive to meeting the students in our room where they are and moving those students’ learning forward. And so, thankfully, the journey continues …