Topics
1. Number Sense
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United States of America
Grade 6

1.12 Absolute values and number lines

Lesson
Practice

Introduction

Let's consider the following situation:

A scuba diver is diving at a depth of -50 feet. At the same time, a helicopter pilot is flying overhead at 30 feet above the surface. Which person is closer to sea level?

Although the scuba diver is at an altitude much lower than the helicopter, the helicopter pilot is closer to sea level.

When making this comparison, we are considering the absolute value of each measurement. The absolute value of a number is the distance from the number to zero on the number line.

Absolute values

Exploration

The applet below shows the absolute value, or distance from zero for different integers on the number line. Move the point left and right and consider the following questions:

  1. What do you notice about the absolute value of a positive number?

  2. What do you notice about the absolute value of a negative number?

  3. Can the absolute value of a number ever be a negative number? Why or why not?

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We can see that the absolute value of a positive number is the number itself. However, the absolute value of a negative number is its opposite. This is because the distance is always a positive number. This applies to all numbers on the number line.

The mathematical symbol for absolute value is |\,|. For example, we would read \left|-6\right| as "the absolute value of negative six."

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

Number line from negative 10 to 10 with jumps to the left of 0 by 3 units to negative 3 and to the right of 0 by 3 units to 3

The numbers -3 and 3 are both 3 units from 0, so they have the same absolute value.

The absolute value of a positive number is the number itself.

The absolute value of a negative number is its opposite.

For example, |3|=3 and |-3|=3.

Examples

Example 1

What is the value of \left|-155\right|?

Worked Solution
Create a strategy

The absolute value of a negative number is positive.

Apply the idea
\displaystyle \left|-155\right|\displaystyle =\displaystyle 155Evaluate

Example 2

Which of the following are smaller than \left|-20\right|?

A
-15
B
\ \left|-30\right|
C
\left|-5\right|
D
21
Worked Solution
Create a strategy

Evaluate the given expression and options and compare them.

Apply the idea

Original expression:

|-20|=20

Option B

|-30|=30

Option C

|-5|=5

The answers are Option A -15 and C |-5|.

Example 3

Evaluate each of these numbers, and order the results from smallest to largest:

\left| 19 \right|,\,\left| 0 \right|,\, \left| 41 \right|,\, \left| -31 \right|

Worked Solution
Create a strategy

Find the absoulute values of each given number and compare the results.

Apply the idea
\displaystyle \left|19\right|\displaystyle =\displaystyle 19Evaluate
\displaystyle \left|0\right|\displaystyle =\displaystyle 0Evaluate
\displaystyle \left|41\right|\displaystyle =\displaystyle 41Evaluate
\displaystyle \left|-31\right|\displaystyle =\displaystyle 31Evaluate

Compare the evaluated absolute values 0 < 19 < 31 < 41

Therefore, the order from smallest to largest is: \left| 0 \right|,\,\left| 19 \right|,\, \left| -31 \right|,\, \left| 41 \right|

Idea summary

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

The absolute value of a positive number is the number itself.

The absolute value of a negative number is its opposite.

Outcomes

6.NS.C.7

Understand ordering and absolute value of rational numbers.

6.NS.C.7.C

Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a realworld situation.

6.NS.C.7.D

Distinguish comparisons of absolute value from statements about order.

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