# Inscribing and Circumscribing Right Triangles

## Mathematical goals

This lesson unit is intended to help you assess how well students are able to use geometric properties to solve problems. In particular, it will help you identify and help students who have difficulty:

- Decomposing complex shapes into simpler ones in order to solve a problem.
- Bringing together several geometric concepts to solve a problem.
- Finding the relationship between radii of inscribed and circumscribed circles of right triangles.

## Introduction

This lesson is designed to enable students to develop strategies for describing relationships between right triangles and the radii of their inscribed and circumscribed circles.

- Before the lesson students attempt the problem individually. You review their work and formulate questions for students to answer in order to improve their solutions.
- At the start of the lesson, students work alone answering your questions. They then work in groups in a collaborative discussion of the same task before evaluating and commenting on some sample solutions.
- In a whole-class discussion, students explain and compare the alternative solution strategies they have seen and used.
- Finally, in a follow-up lesson students review what they have learned.

## Materials required

- Each student will need a mini-whiteboard, pen, and eraser and a copy of
*Inscribing and Circumscribing Right Triangles,**Circle Theorems*, and the*How Did You Work?*questionnaire. - Each group of students will need an enlarged copy of
*Inscribing and Circumscribing Right Triangles*and the*Sample Responses to Discuss*. - Compasses, rules, pencils and protractors can be given to students if requested.

## Time needed

15 minutes before the lesson, a 100-minute lesson (or two 50-minute lessons), and 10 minutes in a follow-up lesson. Exact timings will depend on the needs of the class.

## Lesson Type

## Mathematical Practices

This lesson involves a range of *mathematical practices* from the standards, with emphasis on:

- MP1: Make sense of problems and persevere in solving them
- MP2: Reason abstractly and quantitatively
- MP3: Construct viable arguments and critique the reasoning of others
- MP4: Model with mathematics
- MP5: Use appropriate tools strategically
- MP6: Attend to precision
- MP7: Look for and make use of structure
- MP8: Look for and express regularity in repeated reasoning

## Mathematical Content Standards

This lesson asks students to select and apply mathematical content from across the grades, including the *content standards:*

- H.A-CED: Create equations that describe numbers or relationships
- H.G-C: Understand and apply theorems about circles
- H.G-SRT: Define trigonometric ratios and solve problems involving right triangles