# Understanding Rational Numbers

## Overview

# Lesson Overview

Students learn the definition of *rational number*, and they write rational numbers as ratios of integers and as repeating or terminating decimals.

# Key Concepts

Students have been working with rational numbers throughout this unit, but the term *rational* *number* is formally defined in this lesson. A rational number is a number that can be written in the form $\frac{p}{q},$ where *p* and *q* are integers. All the integers, fractions, decimals, and percents students have worked with so far in their math classes are rational numbers. Following are some rational numbers written as ratios of integers:

$36=\frac{36}{1}$

$-1.2=\frac{-12}{10}$

$5\%=\frac{5}{100}$ $-\frac{1}{2}=\frac{-1}{2}$

Any rational number can also be written as a decimal that terminates or that repeats forever in a regular pattern. For example, $\frac{3}{5}$ = 0.6 and $\frac{7}{11}$ = 0.63636363… Repeating decimals are often written with a bar over the digits that repeat. For example, 0.63636363… can be written as $0.\overline{63}$.

There are numbers that are irrational. These numbers include π and the square root of any whole number that is not a perfect square, such as $\sqrt{2}$. The decimal form of an irrational number does not terminate, and the digits do not follow a repeating pattern. Students will study irrational numbers in Grade 8.

# Goals and Learning Objectives

- Understand the definition of
*rational number*. - Write rational numbers as ratios of integers.
- Write rational numbers as terminating or repeating decimals.

SWD: Students with disabilities may have difficulty working with decimals and fractions, especially moving between the two. If students demonstrate difficulty to the point of frustration, provide direct instruction on the basics for finding equivalent fractions and decimals.

ELL: Target and model key language and vocabulary. Specifically, focus on the term *rational*, as well as terms such as *terminate*. As you’re discussing the key points, write the words on the board or on large sheets of paper and explain/demonstrate what the words mean. Since these are important points that students will be using throughout the module, write them on large poster board so that students can use them as a reference. Have students record new terms, definitions, and examples in their Notebook.

# Fractions to Decimals

# Lesson Guide

Students should use a calculator to change the fractions to decimals and discuss the results with a partner before discussing the results as a class.

# Mathematics

Explain to students that the decimals in the second column repeat forever. The calculator shows as many digits as fit in the display and rounds the last digit. So, although the display shows $\frac{11}{12}$ as 0.9166666667, the 7 is the result of rounding. The pattern of 6s actually repeats forever. Tell students that one way to show that a decimal continues forever is to use an ellipsis (…). Write $\frac{11}{12}$ as 0.9166666… to demonstrate.

## Opening

# Fractions to Decimals

Use a calculator to change each of these fractions to a decimal. Write all the digits that appear in the calculator’s display.

- How are the decimals in the first column different from those in the second column?

# Rational Numbers

# Mathematics

Point out that the fractions students just worked are all rational numbers. The decimal forms of the fractions in the first column terminate, or end, while the decimal forms of the fractions in the second column repeat in a regular pattern forever.

## Opening

# Rational Numbers

**Read and Discuss**

These fractions are examples of rational numbers.

A *rational number* is a number that can be written in the form $\frac{p}{q}$, where *p* and *q* are integers and *q* is not 0. That is, a rational number is a number that can be written as a *ratio* of integers. All the integers, fractions, decimals, and percents you have worked with are rational numbers.

All rational numbers can be written as decimals that either terminate (end), like those in the first column, or repeat forever in a pattern, like those in the second column.

# Math Mission

# Lesson Guide

Discuss the Math Mission. Students will explore rational numbers and write them in various forms.

## Opening

Explore rational numbers and write them in various forms.

# Rational Numbers as Ratios

# Lesson Guide

Have students work individually or with a partner on the problems.

Select both correct and incorrect answers for all the problems to be presented during Ways of Thinking.

# Mathematical Practices

**Mathematical Practice 2: Reason abstractly and quantitatively.**

Students must work with rational numbers both in a general sense and with specific examples.

**Mathematical Practice 7: Look for and make use of structure.**

Students must discern and describe a pattern in the decimal forms in order to write numbers as ratios.

# Answers

Answers will vary. Possible answers:

- $\frac{7}{10}$ or $\frac{70}{100}$
- $\frac{-6}{1}$ or $\frac{6}{-1}$
- $\frac{15}{100}$ or $\frac{3}{20}$
- $\frac{-4}{11}$ or $\frac{4}{-11}$
- $\frac{-43}{10}$

## Work Time

# Rational Numbers as Ratios

Show that each number is a rational number by writing it as a ratio of integers.

- 0.7
- −6
- 15%
- $-\frac{4}{11}$
- −4.3

## Hint:

When you write a number as a ratio of integers, $\frac{p}{q}$, the value of *q* can be 1.

# Work With Rational Numbers

# Interventions

**Student has difficulty showing that $-\frac{p}{q}=\frac{-p}{q}=\frac{p}{-q}$.**

- Try to first show that $\frac{-p}{q}=-\frac{p}{q}$, and then show that $\frac{p}{-q}=-\frac{p}{q}$.
- How can you use the multiplication property of −1 to rewrite the numerator of $\frac{-p}{q}$?
- Now what can you do? Remember you are trying to get to $-\frac{p}{q}$.

# Answers

Answers will vary. Possible answer:

$-\frac{3}{4}=-(3\xf74)=-0.75$

$\frac{-3}{4}=-3\xf74=-0.75$$\frac{3}{-4}=3\xf7\left(-4\right)=-0.75$All three expressions equal −0.75, so they are all equal to each other.

Answers will vary. Possible answer:

$\frac{-p}{q}=\frac{-1\cdot p}{q}=-1\cdot \frac{p}{q}=-\frac{p}{q}$

$\frac{p}{-q}=\frac{p}{-1\cdot q}=\frac{1}{-1}\cdot \frac{p}{q}=-1\cdot \frac{p}{q}=-\frac{p}{q}$

Since both $\frac{-p}{q}$ and $\frac{p}{-q}$ equal $-\frac{p}{q}$, they are also equal to each other, so $\frac{-p}{q}=\frac{p}{-q}=-\frac{p}{q}$.

## Work Time

# Work With Rational Numbers

- Show that $-\left(\frac{3}{4}\right)=\frac{-3}{4}=\frac{3}{-4}$.
- Explain why for any integers
*p*and*q*, where*q*is not 0, $-\left(\frac{p}{q}\right)=\frac{-p}{q}=\frac{p}{-q}$

## Hint:

First show that $-\left(\frac{p}{q}\right)=\frac{-p}{q}$, and then show that $\frac{-p}{q}=\frac{p}{-q}$.

# Repeating Decimals

# Mathematics

[common error] Watch for students who incorrectly place the bar in a repeating decimal. The bar should only be over the digits that repeat and over the smallest possible set of digits. So, for example, $\frac{11}{12}=0.916666666\dots $ and is written as $0.91\overline{6}$, not as $0.\overline{916}$ or$0.91\overline{66}$.

# Interventions

**Student incorrectly uses bar notation to show a repeating decimal.**

- Did you draw the bar only over the digits that repeat?
- What is the smallest set of digits that repeats?
- Did you include too many digits under your bar?

ELL: During class discussions, make sure you provide wait time (5–10 seconds) and acknowledge student responses, both verbally and with gestures.

# Answers

- $0.\overline{3}$
- $0.\overline{6}$
- $0.91\overline{6}$
- $0.\overline{27}$

# Challenge Problem

## Answer

Let *x* = 0.7777777777…

Then:

10x | = | 7.7777777777… |
---|---|---|

−(x | = | 0.7777777777…) |

9x | = | 7 |

So, $x=\frac{7}{9}$

## Work Time

# Repeating Decimals

To indicate that a decimal repeats forever in a specific pattern, you write a bar over the repeating digits.

For example, $\frac{6}{11}=0.54545454545\dots $. You can write this decimal as $0.\overline{54}$. The use of a line to show the repeating digits in a decimal is called *bar notation*.

Write each of these fractions in decimal form using bar notation.

- $\frac{1}{3}$
- $\frac{2}{3}$
- $\frac{11}{12}$
- $\frac{3}{11}$

# Challenge Problem

It is easy to change a terminating decimal to a fraction; for example, $0.09=\frac{9}{100}$ and $3.2=\frac{32}{10}$.

Changing a repeating decimal to a fraction is trickier. The steps that follow describe a method for changing the repeating decimal $0.\overline{12}$ to a fraction.

- Write the decimal to show the repeating pattern: For $0.\overline{12}$, you write 0.1212121212…

- Let
*x*equal the repeating decimal:*x*= 0.1212121212…

- Multiply both sides of the preceding equation by whatever power of 10 (10, 100, 1000, and so on) moves “one set” of the repeating digits to the left of the decimal point. One set of repeating digits in 0.1212121212… is “12.” To move this set to the left of the decimal point, you need to multiply
*x*by 100: 100*x*= 12.1212121212… Subtract

*x*from the equation in the previous step:100 *x*= 12.1212121212… − *x*= 0.1212121212… 99 *x*= 12 - Solve $x=\frac{12}{99}=\frac{4}{33}$

** • Use this method to change $0.\overline{7}$ to a fraction.**

# Make Connections

# Lesson Guide

Have students present their solutions as their classmates listen and ask questions.

Questions to pose for discussion:

- How would you define rational number?
- What types of numbers are rational numbers?
- What do you know about the decimal form of a rational number?
- How can you show that a decimal goes on forever in a repeating pattern?

Students may wonder if there are numbers that are not rational. If so, tell them that there are and that such numbers are called irrational numbers. Irrational numbers have decimal forms that go on forever but do not have any sequence of digits that repeats forever.

An example of an irrational number that students may have heard of is π. Have students press the π key on their calculator to see the first several digits of π. Explain that the calculator could show any number of digits but there would never be any sequence of digits that repeats forever.

## Performance Task

# Ways of Thinking: Make Connections

Take notes about your classmates’ methods and explanations for working with rational numbers.

## Hint:

As your classmates present, ask questions such as:

- For the first group of problems, can you explain how you changed that number to a ratio of integers?
- When you used bar notation, how did you know which digits to write the bar over?
- How is a rational number different from an integer?
- Can you give some examples of numbers that are rational numbers but not integers?
- How is a repeating decimal different from a terminating decimal?

# Rational Numbers

# A Possible Summary

A rational number is a number that can be written as a ratio of integers. This means it can be written as $\frac{p}{q}$ where *p* and *q* are integers and *q* is not equal to 0. The integers, fractions, decimals, and percents we have worked with are all rational numbers. For example, the integer −3 is rational because it can be written as $-\frac{3}{1}\mathrm{.}$ The decimal 0.35 is rational because it can be written as $\frac{35}{100}\mathrm{.}$ And 12% is rational because it can be written as $\frac{12}{100}\mathrm{.}$ Decimal equivalents of rational numbers either terminate or go on forever in a repeating sequence. For example, $\frac{1}{4}$ = 0.25 and $\frac{1}{3}$ = 0.333333… We can use a bar to show the repeating digits in a repeating decimal. For example, $\frac{1}{3}=0.333333\dots =0.\overline{3}$

SWD: Refer to the Hint questions as a checklist of the information they need to include in their summaries.

## Formative Assessment

# Summary of the Math: Rational Numbers

Write a summary about rational numbers.

## Hint:

Check your summary.

- Do you define
*rational number*? - Do you explain and give examples of how to write integers, decimals, and percents as a ratio of integers?
- Do you define and give examples of the two decimal forms of rational numbers?
- Do you explain how to use bar notation?

# Inequalities with Rational Numbers

# Lesson Guide

This task allows you to assess students’ work and determine what difficulties they are having. The results of the Self Check will help you determine which students should work on the Gallery problems and which students would benefit from review before the assessment. Have students work on the Self Check individually.

# Assessment

Have students submit their work to you. Make notes about what their work reveals about their current levels of understanding and their different problem-solving approaches.

Do not score students’ work. Share with each student the most appropriate Interventions to guide their thought process. Also note students with a particular issue so that you can work with them in the Putting It Together lesson that follows.

# Interventions

**Student does not know the rules for determining the sign of a sum or product.**

- What did you learn in prior lessons that would be useful for this problem?
- Is the product of a positive number and a negative number positive or negative? What about the sign of a product of two negative numbers?
- Is the quotient of a positive number and a negative number positive or negative? What about the sign of a quotient of two negative numbers?

**Student has difficulty getting started.**

- Will the value of
*x*that makes the statement true be positive or negative? - Try guessing a number for
*x*and testing to see if it makes the statement true. If it doesn’t, how can you use the result to make another guess? - If the inequality were an equal sign, what value of
*x*would make the equation true? Now test values greater than this value and less than this value.

**Student works inefficiently.**

- What do you need to consider to make an “educated” first guess?
- How can you keep track of your guesses and the results? Can you make a table?
- Can your answer and work from one part help you find an answer to other parts?

# Answers

Answers will vary. The answer for each could be any number that satisfies the given inequality. Possible answers:

*x*= −10 or any other number*x*such that*x*< −3*x*= 10 or any other number*x*such that*x*> 3*x*= −$\frac{1}{10}$ or any other number*x*such that $-\frac{1}{4}<x<0$*x*= $\frac{1}{10}$ or any other number*x*such that $0<x<\frac{1}{4}$*x*= −10 or any other number*x*such that*x*< −6*x*= 10 or any other number*x*such that*x*> 6

## Formative Assessment

# Inequalities With Rational Numbers

Complete this Self Check by yourself.

Find a value of *x* that makes each inequality true.

- −4
*x*> 12 - −4
*x*< −12 - 0 < −4
*x*< 1 - −1 < −4
*x*< 0 *x*÷ 3 < −2*x*÷ 3 > 2

# Reflect On Your Work

# Lesson Guide

Have each student write a brief reflection before the end of the class. Review the reflections to find out what students still find confusing about rational numbers.

## Work Time

# Reflection

Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.

**Something that confuses me about rational numbers is …**