The Pythagorean Theorem in Circles

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Students will be able to solve problems involving properties of circles, the Pythagorean Theorem, special right triangles, and area.

Big Idea

By working in pairs, students will work through and reflect on a variety of problems that require them to synthesize several geometry topics.


20 minutes

At this point in the year, I spiral in prior topics to keep them fresh in students' minds and to help them to discover new applications. In today's warm-up my students:

  • Find the circumference of a circle by using several properties of circles alongside the Pythagorean Theorem
  • Use the distance formula to prove that a triangle is isosceles
  • Solve a problem about the area of a figure and justify their reasoning in general terms.

Like most warm-ups, I give students time to work individually before sharing out their answer in groups. This approach gives me the opportunity to circulate the room to see how individual students are progressing. I am also on the lookout for the strategies that my students are using. 

We will debrief today's warm-up by having student volunteers come up to present their work under the document camera.  Problem #3 can be particularly interesting for whole-class discussion because students might have reasoned about the area of each figure in different ways. If so, I like to take advantage of this opportunity to work on constructing viable arguments and critiquing each other’s reasoning (MP3).  

Pair Classwork and Reflection

55 minutes

Next I plan to give my students a classwork assignment, which I adapted from Lesson 9.6 of Discovering Geometry. The assignment focuses on similar concepts as the warm-up.  I want my students to continue to practice integrating properties of circles into problem solving with the Pythagorean Theorem, special right triangles, and area calculation. 

I ask my students to work in pairs because it is the kind of practice that would be difficult to do individually. I want students to have a resource available to them if they are stuck. Moreover, when my students work in pairs, I find that there tends to be increased accountability with respect to making sense of and documenting individual work. I find that pairs works better than individual work and larger groups. I think that in a pair my students may take risks more safely than in group of four.

As students begin, I will ask them to call me over then they have completed their work. At this point I will give students the answer key. I'll invite them to correct their work using a different colored pen.  After checking their work, I will ask students to take out a piece of notebook paper and to reflect on one or two problems they believe helped them to learn the most and to explain why, which I collect at the end of the lesson.