Learn on PengiPengi Math (Grade 7)Chapter 1: The Integer System and Operations

Lesson 1: Integers, Opposites, and Absolute Value

In this Grade 7 Pengi Math lesson, students learn to define integers and locate them on the number line, including negative numbers and their opposites using the additive inverse property. The lesson covers absolute value as a measure of distance from zero and explains why a greater absolute value means a lesser value among negative numbers. Students also practice comparing and ordering negative integers, building foundational skills for Chapter 1: The Integer System and Operations.

Section 1

Defining Integers and Opposites

Property

Negative numbers are numbers less than 0.
The opposite of a number is the number that is the same distance from zero on the number line but on the opposite side of zero.
The notation $-a$ is read as “the opposite of $a$ .”
The whole numbers and their opposites are called the integers.
The integers are the numbers $\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$ .

Examples

The opposite of $15$ is $-15$ , as both are 15 units from zero.
The opposite of $-9$ is $9$ . This can be written as $-(-9) = 9$ .
If $y = -25$ , then $-y$ means the opposite of $-25$ , which is $-(-25) = 25$ .

Explanation

Integers expand our number system to include negative values, which are like mirror images of positive numbers across zero. The 'opposite' of a number is simply its reflection on the other side of the number line.

Section 2

Absolute Value

Property

The absolute value of a number is its distance from $0$ on the number line. The absolute value of a number $n$ is written as $|n|$ .

Property of Absolute Value
$|n| \geq 0$ for all numbers. Absolute values are always greater than or equal to zero!

Examples

The absolute value of $-18$ is its distance from 0, so $|-18| = 18$ .
To simplify $30 - |12 - 4(5)|$ , we calculate inside the bars first: $30 - |12 - 20| = 30 - |-8| = 30 - 8 = 22$ .
Compare $|-11|$ and $-|-11|$ . We get $11$ and $-11$ . So, $|-11| > -|-11|$ .

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Defining Integers and Opposites

Property

Examples

The opposite of $15$ is $-15$ , as both are 15 units from zero.
The opposite of $-9$ is $9$ . This can be written as $-(-9) = 9$ .
If $y = -25$ , then $-y$ means the opposite of $-25$ , which is $-(-25) = 25$ .

Explanation

Section 2

Absolute Value

Property

The absolute value of a number is its distance from $0$ on the number line. The absolute value of a number $n$ is written as $|n|$ .

Property of Absolute Value
$|n| \geq 0$ for all numbers. Absolute values are always greater than or equal to zero!

Examples

The absolute value of $-18$ is its distance from 0, so $|-18| = 18$ .
To simplify $30 - |12 - 4(5)|$ , we calculate inside the bars first: $30 - |12 - 20| = 30 - |-8| = 30 - 8 = 22$ .
Compare $|-11|$ and $-|-11|$ . We get $11$ and $-11$ . So, $|-11| > -|-11|$ .

Overview

Lesson overview

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

Overview

Section 1

Defining Integers and Opposites

Property

Examples

The opposite of $15$ is $-15$ , as both are 15 units from zero.
The opposite of $-9$ is $9$ . This can be written as $-(-9) = 9$ .
If $y = -25$ , then $-y$ means the opposite of $-25$ , which is $-(-25) = 25$ .

Explanation

Section 2

Absolute Value

Property

The absolute value of a number is its distance from $0$ on the number line. The absolute value of a number $n$ is written as $|n|$ .

Property of Absolute Value
$|n| \geq 0$ for all numbers. Absolute values are always greater than or equal to zero!

Examples

The absolute value of $-18$ is its distance from 0, so $|-18| = 18$ .
To simplify $30 - |12 - 4(5)|$ , we calculate inside the bars first: $30 - |12 - 20| = 30 - |-8| = 30 - 8 = 22$ .
Compare $|-11|$ and $-|-11|$ . We get $11$ and $-11$ . So, $|-11| > -|-11|$ .