Learn on PengienVision, Mathematics, Grade 6Chapter 2: Integers and Rational Numbers

Lesson 2: Represent Rational Numbers on the Number Line

In this Grade 6 lesson from enVision Mathematics Chapter 2, students learn to represent rational numbers — including negative fractions and decimals — on both horizontal and vertical number lines by converting values like -4/3 and -1.5 into mixed numbers to find their precise positions. Students also practice comparing and ordering rational numbers using inequality symbols by plotting values such as -0.75, 2/3, and 1.75. Real-world contexts like ocean depths and temperatures help students interpret the meaning of negative rational numbers in everyday situations.

Section 1

Definition of Rational Numbers

Property

A rational number is a number that can be written in the form $\frac{p}{q}$ , where $p$ and $q$ are integers and $q \neq 0$ . All fractions, both positive and negative, are rational numbers.
Since any integer, terminating decimal, or repeating decimal can be written as a ratio of two integers, they are all rational numbers.

Examples

To write the integer $-25$ as a ratio of two integers, express it as a fraction with a denominator of 1: $\frac{-25}{1}$ .
The decimal $9.37$ can be written as a mixed number $9\frac{37}{100}$ , which converts to the improper fraction $\frac{937}{100}$ .
The mixed number $-4\frac{2}{3}$ is equivalent to the improper fraction $-\frac{14}{3}$ .

Explanation

Think of 'rational' as 'ratio-nal.' Any number that can be expressed as a simple fraction or ratio between two integers is a rational number. This includes whole numbers, integers, and decimals that either end or repeat predictably.

Section 2

Compare and Order Rational Numbers

Property

To compare or order rational numbers, plot them on a number line. For any two numbers $a$ and $b$ , if $a$ is to the left of $b$ , then $a < b$ . If $a$ is to the right of $b$ , then $a > b$ .

Examples

To compare $-\frac{3}{2}$ and $-0.8$ , we can plot them on a number line. Since $-\frac{3}{2} = -1.5$ , it is located to the left of $-0.8$ . Therefore, $-\frac{3}{2} < -0.8$ .
To order the numbers $1.25$ , $-2$ , and $\frac{1}{2}$ from least to greatest, we find their positions on the number line. $-2$ is furthest to the left, followed by $\frac{1}{2}$ (or $0.5$ ), and then $1.25$ . The correct order is $-2, \frac{1}{2}, 1.25$ .

Explanation

The number line provides a visual way to compare and order any set of rational numbers, including integers, fractions, and decimals. To make comparison easier, it can be helpful to convert all numbers to the same format, such as decimals. Once the numbers are plotted, their order from left to right corresponds to their order from least to greatest.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Definition of Rational Numbers

Property

Examples

To write the integer $-25$ as a ratio of two integers, express it as a fraction with a denominator of 1: $\frac{-25}{1}$ .
The decimal $9.37$ can be written as a mixed number $9\frac{37}{100}$ , which converts to the improper fraction $\frac{937}{100}$ .
The mixed number $-4\frac{2}{3}$ is equivalent to the improper fraction $-\frac{14}{3}$ .

Explanation

Section 2

Compare and Order Rational Numbers

Property

To compare or order rational numbers, plot them on a number line. For any two numbers $a$ and $b$ , if $a$ is to the left of $b$ , then $a < b$ . If $a$ is to the right of $b$ , then $a > b$ .

Examples

To compare $-\frac{3}{2}$ and $-0.8$ , we can plot them on a number line. Since $-\frac{3}{2} = -1.5$ , it is located to the left of $-0.8$ . Therefore, $-\frac{3}{2} < -0.8$ .
To order the numbers $1.25$ , $-2$ , and $\frac{1}{2}$ from least to greatest, we find their positions on the number line. $-2$ is furthest to the left, followed by $\frac{1}{2}$ (or $0.5$ ), and then $1.25$ . The correct order is $-2, \frac{1}{2}, 1.25$ .

Explanation

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Definition of Rational Numbers

Property

Examples

To write the integer $-25$ as a ratio of two integers, express it as a fraction with a denominator of 1: $\frac{-25}{1}$ .
The decimal $9.37$ can be written as a mixed number $9\frac{37}{100}$ , which converts to the improper fraction $\frac{937}{100}$ .
The mixed number $-4\frac{2}{3}$ is equivalent to the improper fraction $-\frac{14}{3}$ .

Explanation

Section 2

Compare and Order Rational Numbers

Property

To compare or order rational numbers, plot them on a number line. For any two numbers $a$ and $b$ , if $a$ is to the left of $b$ , then $a < b$ . If $a$ is to the right of $b$ , then $a > b$ .

Examples

To compare $-\frac{3}{2}$ and $-0.8$ , we can plot them on a number line. Since $-\frac{3}{2} = -1.5$ , it is located to the left of $-0.8$ . Therefore, $-\frac{3}{2} < -0.8$ .
To order the numbers $1.25$ , $-2$ , and $\frac{1}{2}$ from least to greatest, we find their positions on the number line. $-2$ is furthest to the left, followed by $\frac{1}{2}$ (or $0.5$ ), and then $1.25$ . The correct order is $-2, \frac{1}{2}, 1.25$ .