1.08 Absolute values and number lines

Lesson

Let's consider the following situation:

 A scuba diver is diving at a depth of $-50$−50 feet. At the same time, a helicopter pilot is flying overhead at $30$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.

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?

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 "$\text{| |}$| |". For example, we would read "$\left|-6\right|$|6|" as "the absolute value of negative six."

Absolute value

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

The numbers $-3$3 and $3$3 are both $3$3 units from $0$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, $\left|3\right|=3$|3|=3 and $\left|-3\right|=3$|3|=3

Worked example

Question 1

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

A) $-15$15    B) $\left|-30\right|$|30|     C) $\left|-5\right|$|5|     D) $21$21

Think: We need to evaluate each of these terms, then compare them to $\left|-20\right|$|20|.

Do: Let's start by evaluating all the absolute values:

$\left|-20\right|=20$|20|=20, $\left|-30\right|=30$|30|=30 and $\left|-5\right|=5$|5|=5

Which of the four possible answers are smaller than $20$20?

So $\left|-20\right|$|20| is greater than A) $-15$15 and C) $\left|-5\right|$|5|

Practice questions

Question 2

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

Question 3

Laura is in a hot air balloon $8.5$8.5m above sea level and Fred is exploring a cave $6.7$6.7m below sea level. Which of these statements are true? Select ALL correct responses that apply.

1. Fred has greater elevation than Laura.

A

Fred is further from sea level than Laura.

B

Laura has greater elevation than Fred.

C

Laura is further from sea level than Fred.

D

Fred has greater elevation than Laura.

A

Fred is further from sea level than Laura.

B

Laura has greater elevation than Fred.

C

Laura is further from sea level than Fred.

D

Question 4

Amelia has a bank account balance of $-\$16.60$$16.60. Marge has a bank account balance of -\53.30$$53.30.

1. Who has the biggest debt?

Amelia

A

Marge

B

Amelia

A

Marge

B
2. Who's account balance is furthest from a $\$00 balance?

Marge

A

Amelia

B

Marge

A

Amelia

B

Outcomes

6.NS.C.7

Understand ordering and absolute value of rational numbers.

6.NS.C.7c

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 real-world context.

6.NS.C.7d

Distinguish comparisons of absolute value from statements about order in mathematical problems and problems in real-world context.