# Calculating Arcs and Areas of Sectors of Circles

## Mathematical goals

This lesson unit is intended to help you assess how well students are able to solve problems involving area and arc length of a sector of a circle using radians. It assumes familiarity with radians and should not be treated as an introduction to the topic. This lesson is intended to help you identify and assist students who have difficulties in:

- Computing perimeters, areas, and arc lengths of sectors using formulas.
- Finding the relationships between arc lengths, and areas of sectors after scaling.

## Introduction

- Before the lesson, students work individually on an assessment task that is designed to reveal their current understanding and difficulties. You then review their solutions and create questions for students to consider, in order to improve their work.
- During the lesson, small groups work on a collaborative task matching cards according to the arc length, area, or perimeter of the sectors.
- After considering in the same small groups the general results of changing the radius and/or sector angle of a sector, there is a whole-class discussion.
- In a follow-up lesson, students review their initial solutions and then use what they have learned to either revise the same introductory assessment task or complete a different task.

## Materials required

- Each student will need a copy of
*Sectors of Circles*and*Sectors of Circles (revisited),*the*Circles*sheet, a mini-whiteboard, pen, and eraser. - Each small group of students will need a glue stick, an enlarged copy of
*Arc Lengths and Areas of Sectors*, and the cut-up cards Dominos 1, Dominos 2, and*Changing the Angle**and**Radius of a Sector*. - There is a projector resource to support whole-class discussions. The
*Card Sets*may also be copied onto overhead transparencies to support these discussions.

## Time needed

20 minutes before the lesson, a 100-minute lesson (or two shorter lessons), and 20 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
- 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.A-REI: Solve equations and inequalities in one variable
- H.A-SSE: Interpret the structure of expressions
- H.G-C: Find arc lengths and areas of sectors of circles