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* *Reflection: Connection to Prior Knowledge
The Pythagorean Theorem in Circles - Section 2: Pair Classwork and Reflection

This assignment features challenging problems that integrate the current content—the Pythagorean Theorem—with content students had explored in the past: area, perimeter, properties of isosceles triangles, properties of circles. I explicitly told students they would have to apply the Pythagorean Theorem in problems that united several geometric ideas, which helped them to anticipate some of the challenges they would face.

One change I made this year was to ask students to formally reflect after they finished working through these problems. I asked them to identify two or three problems they found difficult, to specifically explain why the problems were challenging, and to write about a strategy or tool they used to work through each problem. My goal in asking them to reflect on their work and their feelings about their work was to empower them to see that through their effective effort and persistence, they were able to make sense of difficult problems.

*Tricky Triangles!*

*Connection to Prior Knowledge: Tricky Triangles!*

# 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.

#### Warm-Up

*20 min*

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**).

#### Resources

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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.

*expand content*

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- UNIT 1: Creating Classroom Culture to Develop the Math Practices
- UNIT 2: Introducing Geometry
- UNIT 3: Transformations
- UNIT 4: Discovering and Proving Angle Relationships
- UNIT 5: Constructions
- UNIT 6: Midterm Exam Review
- UNIT 7: Discovering and Proving Triangle Properties
- UNIT 8: Discovering and Proving Polygon Properties
- UNIT 9: Discovering and Proving Circles Properties
- UNIT 10: Geometric Measurement and Dimension
- UNIT 11: The Pythagorean Theorem
- UNIT 12: Triangle Similarity and Trigonometric Ratios
- UNIT 13: Final Exam Review

- LESSON 1: The Pythagorean Theorem
- LESSON 2: Converse of the Pythagorean Theorem and Special Right Triangles Investigation
- LESSON 3: Pythagorean Triples and Special Right Triangles
- LESSON 4: Distance in Geometry
- LESSON 5: The Pythagorean Theorem in Circles
- LESSON 6: Pythagorean Theorem Unit Assessment