The Pythagorean Theorem in Circles
Lesson 5 of 6
Objective: Students will be able to solve problems involving properties of circles, the Pythagorean Theorem, special right triangles, and area.
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).
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.