# Translating Between Repeating Decimals and Fractions

## Mathematical goals

This lesson unit is intended to help you assess how well students are able to:

- Translate between decimal and fraction notation, particularly when the decimals are repeating.
- Create and solve simple linear equations to find the fractional equivalent of a repeating decimal.
- Understand the effect of multiplying a decimal by a power of 10.

## Introduction

- Before the lesson, students attempt the
*Repeating Decimals*task individually. You review their work and write questions for students to answer in order to improve their solutions. - A whole-class discussion is used to introduce students to some of the patterns in repeating decimals and a method for converting such decimals to fractions.
- Students work collaboratively in small groups on a matching task that requires them to translate between fractions, decimals, and equations.
- In a final whole-class discussion, students compare and evaluate the methods.
- In a follow-up lesson, students review their initial solutions and then complete a different
*Repeating Decimals*task.

## Materials required

- Each student will need the task sheets
*Repeating Decimals, Repeating Decimals (revisited)*, and*Fraction Conversion Challenge,*a calculator, a mini-whiteboard, a pen, and an eraser. - Each small group of students will need Card Sets A:
*Decimals*, B:*Equations*, C:*Fractions*, a large sheet of paper for making a poster, and a glue stick.

You may want to copy the Card Sets onto transparencies for use on an overhead projector.

There is a projector resource to remind students of the instructions for the group activity.

## Time needed

15 minutes before the lesson, a 1-hour lesson and 20 minutes in a follow-up lesson (or for homework). Timings are only approximate. Exact timings will depend on the needs of your 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
- MP3: Construct viable arguments and critique the reasoning of others
- 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:*

- 8.EE: Analyze and solve linear equations and pairs of simultaneous linear equations.
- 8.NS: Know that there are numbers that are not rational, and approximate them by rational numbers.