Learn on PengiReveal Math, AcceleratedUnit 5: Solve Problems Involving Operations with Integers and Rational Numbers

Lesson 5-1: Add Integers and Rational Numbers

In this Grade 7 lesson from Reveal Math, Accelerated (Unit 5, Lesson 5-1), students learn to add integers and rational numbers with the same sign and with different signs using two methods: number lines and absolute values. Real-world contexts like ocean depths and underground dens help students understand how to determine the sign of a sum based on the absolute values of the addends. By the end of the lesson, students can apply these rules to solve problems involving positive and negative rational numbers.

Section 1

Absolute value

Property

The absolute value of a number is its distance from zero on a number line.

|-3| = 3

Examples

The distance of $-8$ from zero is 8, so $|-8| = 8$ .
A submarine at $-200$ feet is the same distance from sea level as a helicopter at $+200$ feet, so $|-200| = |200| = 200$ .
Always simplify inside the bars first: $|5 - 12| = |-7| = 7$ .

Explanation

Absolute value is like asking, “How many jumps from zero are you?” It doesn’t care about direction (left or right), only distance! That's why the absolute value is always positive or zero—it represents pure, unfiltered distance. So, the absolute value of a number is its happy, positive twin, no matter how negative it started out.

Section 2

Adding Integers on a Number Line Using Arrows

Property

To add the integers $p$ and $q$ : begin at zero and draw the line segment (arrow) to $p$ .
Starting at the endpoint $p$ , draw the line segment representing $q$ . Where it ends is the sum $p + q$ .
An arrow pointing right is positive, and a negative arrow points left.
Each arrow is a quantity with both length (magnitude) and direction (sign).

Examples

To calculate $3 + 4$ , start at 0, move 3 units right, and then move 4 more units right. You land at 7. So, $3 + 4 = 7$ .
To find $-6 + 4$ , start at 0, move 6 units left to $-6$ , then move 4 units right. You land at $-2$ . So, $-6 + 4 = -2$ .
To compute $-3 + (-5)$ , start at 0, move 3 units left to $-3$ , then move 5 more units left. You land at $-8$ . So, $-3 + (-5) = -8$ .

Explanation

Think of adding on a number line as taking a journey. Positive numbers are steps to the right, and negative numbers are steps to the left. Your final position is the sum of the integers.

Section 3

Adding Integers

Property

To add integers with the same sign, add their absolute values and keep the common sign.

To add integers with different signs, subtract the smaller absolute value from the larger absolute value. The sum has the sign of the number with the larger absolute value.

Examples

To add $-15 + (-8)$ , the signs are the same. Add $15 + 8 = 23$ and keep the negative sign, so the sum is $-23$ .
To add $-10 + 25$ , the signs are different. Subtract $25 - 10 = 15$ . Since $25$ has the larger absolute value, the result is positive: $15$ .
To add $12 + (-30)$ , the signs are different. Subtract $30 - 12 = 18$ . Since $-30$ has the larger absolute value, the result is negative: $-18$ .

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Absolute value

Property

The absolute value of a number is its distance from zero on a number line.

|-3| = 3

Examples

The distance of $-8$ from zero is 8, so $|-8| = 8$ .
A submarine at $-200$ feet is the same distance from sea level as a helicopter at $+200$ feet, so $|-200| = |200| = 200$ .
Always simplify inside the bars first: $|5 - 12| = |-7| = 7$ .

Explanation

Section 2

Adding Integers on a Number Line Using Arrows

Property

Examples

To calculate $3 + 4$ , start at 0, move 3 units right, and then move 4 more units right. You land at 7. So, $3 + 4 = 7$ .
To find $-6 + 4$ , start at 0, move 6 units left to $-6$ , then move 4 units right. You land at $-2$ . So, $-6 + 4 = -2$ .
To compute $-3 + (-5)$ , start at 0, move 3 units left to $-3$ , then move 5 more units left. You land at $-8$ . So, $-3 + (-5) = -8$ .

Explanation

Think of adding on a number line as taking a journey. Positive numbers are steps to the right, and negative numbers are steps to the left. Your final position is the sum of the integers.

Section 3

Adding Integers

Property

To add integers with the same sign, add their absolute values and keep the common sign.

To add integers with different signs, subtract the smaller absolute value from the larger absolute value. The sum has the sign of the number with the larger absolute value.

Examples

To add $-15 + (-8)$ , the signs are the same. Add $15 + 8 = 23$ and keep the negative sign, so the sum is $-23$ .
To add $-10 + 25$ , the signs are different. Subtract $25 - 10 = 15$ . Since $25$ has the larger absolute value, the result is positive: $15$ .
To add $12 + (-30)$ , the signs are different. Subtract $30 - 12 = 18$ . Since $-30$ has the larger absolute value, the result is negative: $-18$ .

Overview

Lesson overview

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

Overview

Section 1

Absolute value

Property

The absolute value of a number is its distance from zero on a number line.

|-3| = 3

Examples

The distance of $-8$ from zero is 8, so $|-8| = 8$ .
A submarine at $-200$ feet is the same distance from sea level as a helicopter at $+200$ feet, so $|-200| = |200| = 200$ .
Always simplify inside the bars first: $|5 - 12| = |-7| = 7$ .

Explanation

Section 2

Adding Integers on a Number Line Using Arrows

Property

Examples

To calculate $3 + 4$ , start at 0, move 3 units right, and then move 4 more units right. You land at 7. So, $3 + 4 = 7$ .
To find $-6 + 4$ , start at 0, move 6 units left to $-6$ , then move 4 units right. You land at $-2$ . So, $-6 + 4 = -2$ .
To compute $-3 + (-5)$ , start at 0, move 3 units left to $-3$ , then move 5 more units left. You land at $-8$ . So, $-3 + (-5) = -8$ .

Explanation

Think of adding on a number line as taking a journey. Positive numbers are steps to the right, and negative numbers are steps to the left. Your final position is the sum of the integers.

Section 3

Adding Integers

Property

To add integers with the same sign, add their absolute values and keep the common sign.

To add integers with different signs, subtract the smaller absolute value from the larger absolute value. The sum has the sign of the number with the larger absolute value.

Examples

To add $-15 + (-8)$ , the signs are the same. Add $15 + 8 = 23$ and keep the negative sign, so the sum is $-23$ .
To add $-10 + 25$ , the signs are different. Subtract $25 - 10 = 15$ . Since $25$ has the larger absolute value, the result is positive: $15$ .
To add $12 + (-30)$ , the signs are different. Subtract $30 - 12 = 18$ . Since $-30$ has the larger absolute value, the result is negative: $-18$ .