Learn on PengiPengi Math (Grade 6)Chapter 1: Rational Numbers — Whole Numbers, Fractions, and Decimals

Lesson 4: Comparing and Ordering Rational Numbers

In this Grade 6 Pengi Math lesson from Chapter 1, students learn to compare and order rational numbers — including integers, fractions, and decimals — using number line position and inequality symbols. Students practice ordering mixed sets of rational numbers and translating comparisons into real-world contexts, while explaining their reasoning both verbally and visually.

Section 1

Order positive and negative numbers

Property

On a number line, numbers increase in value from left to right. We use inequality symbols to show ordering:

$a < b$ (read $a$ is less than $b$ ) when $a$ is to the left of $b$ .
$a > b$ (read $a$ is greater than $b$ ) when $a$ is to the right of $b$ .

Examples

To compare 15 and 8, we see 15 is to the right of 8 on the number line, so $15 > 8$ .

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.

Section 3

Integer Problem Solving and Reasoning

Property

Real-world problems can be solved by translating scenarios into integer inequalities.
To find the greatest or least integer value satisfying an inequality, use a number line to identify the integer closest to the boundary value that is within the solution set.

Examples

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Order positive and negative numbers

Property

On a number line, numbers increase in value from left to right. We use inequality symbols to show ordering:

$a < b$ (read $a$ is less than $b$ ) when $a$ is to the left of $b$ .
$a > b$ (read $a$ is greater than $b$ ) when $a$ is to the right of $b$ .

Examples

To compare 15 and 8, we see 15 is to the right of 8 on the number line, so $15 > 8$ .

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

Section 3

Integer Problem Solving and Reasoning

Property

Examples

Overview

Lesson overview

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

Overview

Section 1

Order positive and negative numbers

Property

On a number line, numbers increase in value from left to right. We use inequality symbols to show ordering:

$a < b$ (read $a$ is less than $b$ ) when $a$ is to the left of $b$ .
$a > b$ (read $a$ is greater than $b$ ) when $a$ is to the right of $b$ .

Examples

To compare 15 and 8, we see 15 is to the right of 8 on the number line, so $15 > 8$ .

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 $-2, \frac{1}{2}, 1.25$ .