Read more about the purpose of the MAP *Classroom Challenges*…

## Geometry Problems: 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 the following difficulties:

- 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 work collaboratively in small groups to produce an improved solution to the same problem.
- Working in the same small groups, students comment on and evaluate some solutions 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 the work they did on their individual solutions and write about what they learned.

### Materials required

- Each individual student will need a copy of the task sheet
*Circles and Triangles*, a ruler, calculator, pencil, mini-whiteboard, pen and eraser. - Each small group of students will need a copy of
*Sample Responses to Discuss,*and a large sheet of paper for making a poster. - There are also some slides to help with instructions and to support whole-class discussion.

### Time needed

15minutes before the lesson, a 1-hour lesson, and 10 minutes in a follow-up lesson (or for homework). Timings are approximate and will depend on the needs of the class.

### 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
- MP6: Attend to precision
- MP7: Look for and make use of structure

### Mathematical Content

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

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

### Resources

#### Lesson (complete)

- geometry_problems_complete.pdf ( 11760.9K PDF/Acrobat 27 Jun 2012)

#### Projector Resources

- geometry_problems_slides_beta.ppt ( 6264.5K MS PowerPoint 27 Jun 2012)

A draft Brief Guide for teachers and administrators (PDF) is now available, and is recommended for anybody using the MAP *Classroom Challenges* for the first time.

We have assigned lessons to grades based on the Common Core State Standards for Mathematical Content. During this transition period, you should use your judgement as to where they fit in your current course.

The Beta versions of the MAP Lesson Units may be distributed, unmodified, under the Creative Commons Attribution, Non-commercial, No Derivatives License 3.0. All other rights reserved. Please send any enquiries about commercial use or derived works to map.info@mathshell.org.

Can you help us by sending samples of students' work on the *Classroom Challenges?*