# Solving Problems with Circles and 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, the lesson will help you identify and help students who have difficulty:

- Solving problems by determining the lengths of the sides in right triangles.
- Finding the measurements of shapes by decomposing complex shapes into simpler ones.

The lesson unit will also help students to recognize that there may be different approaches to geometrical problems and to understand the relative strengths and weaknesses of those approaches.

## Introduction

- Before the lesson, students attempt the problem individually. You then review their work and create questions for students to answer in order to improve their solutions.
- During the lesson, students review their individual solutions before working collaboratively in small groups to produce an improved solution to the problem. They then comment on and evaluate some solutions to the same problem, produced by students in another class.
- In a whole-class discussion, students explain and compare the alternative solution strategies they have seen and used.
- Finally, students review and write about what they learned.

## Materials required

- Each individual student will need the
*Circles and Triangles*task, the*How Did You**Work?*questionnaire, a ruler, calculator, pencil, mini-whiteboard, pen, and eraser. - Each small group of students will need a copy of the
*Sample Responses to Discuss,*a large sheet of paper for making a poster, and a felt-tipped pen. - There is a projector resource to help with instructions and to support whole-class discussion.

## Time needed

15 minutes before the lesson, an 80-minute lesson (or two 45-minute lessons), and 15 minutes in a follow-up lesson. Timings are approximate and 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

## Mathematical Content Standards

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

- H.G-C: Understand and apply theorems about circles
- H.G-SRT: Prove theorems involving similarity
- H.G-SRT: Define trigonometric ratios and solve problems involving right triangles