I attended a “Toward Greater Focus & Coherence” Illustrative Mathematics Conference in Tucson last May. While I was there, someone gave me an idea for introducing sectors: give students a sector of a circle (or piece of pie, or piece of pizza … whatever you prefer) and ask them to tell you everything they can about the whole from which it came.

So I tried it this year.

We gave each student a copy of the sector and asked what they could tell us about the whole.

I loved watching students **use appropriate tools strategically** while they worked.

Some used their protractors.

Some used the sector to create the whole circle.

Some measured with a ruler.

Then we talked.

What did you find out about the whole?

There were six pieces of this size in the whole.

How do you know?

I measured the angle, and it was 60˚. There are six 60s in 360˚.

Students figured out the whole area of the circle using the length of the radius that they measured with their rulers. And then they divided by 6 to find the area of the sector.

Students figured out the whole circumference of the circle using the length of the radius that they measured with their rulers. And then they divided by 6 to find the length of the sector.

Did everyone get 2.5 inches for the radius?

One student, our new student from Iran, got 6 cm for the radius (yay!). The other 31 students had measured in inches.

We went on to briefly review the Geometry Nspired activity Arc Length and Sectors, but we were able to spend most of our time on practice problems, since the students had figured out the part to whole relationship themselves through their exploration with the part of the whole.

What can you say about the measure in degrees of the red arc compared to the measure in degrees of the blue arc?

What can you say about the length in units of the red arc compared to the length in units of the blue arc?

What can you say about the measure in degrees of the red arc compared to the measure in degrees of the blue arc?

What can you say about the length in units of the red arc compared to the length in units of the blue arc?

This is a good reminder that **all circles are similar**, not just concentric circles (G-SRT.A.1).

I don’t treat arc length and area of a sector as a new formula to learn, as many textbooks do. I have found that it is more intuitive for students to use proportional reasoning to make sense of arc length and area without memorizing a formula.

And so the journey continues, trying what I learn from others to provide my students the opportunity to make sense of mathematics …