# Absolute value

Here you will learn about absolute value, including what it is, how to find it, and its connection to real-world situations.

Students will first learn about absolute value as part of the number system in 6th grade.

## What is absolute value?

Absolute value is the distance a number is from 0.

To find the absolute value, place the number on a number line and measure the distance from 0.

For example,

What is the absolute value of -2?

-2 is 2 away from 0, so the absolute value is 2.

To write this mathematically, use the absolute value symbol, which is two vertical bars around a number or expression: |-2|=2.

This equation reads “the absolute value of -2 is 2 ”.

Now, what is |2|= ?

2 is also 2 away from 0, so the absolute value is 2. Both -2 and 2 have the same absolute value, because they are the same distance from 0.

### What is absolute value? ## Common Core State Standards

How does this relate to 6th grade math?

• Grade 6 – The Number System (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 a real-world situation. For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars.

## How to find the absolute value

In order to find the absolute value:

1. Find the value on a number line.
2. Measure the distance from zero.

In order to understand absolute value in a real-world situation:

1. Find the absolute value.
2. Explain the absolute value in terms of the real-world situation.

## Absolute value examples

### Example 1: absolute value of a positive number

Solve | 6.8 |= \, ?

1. Find the value on a number line.

2Measure the distance from zero.

6.8 is 6.8 away from 0, so |6.8|= 6.8.

Notice that even though the direction from 6.8 to 0 is down (or negative), the absolute distance is not – it only shows the distance, not the direction.

### Example 2: absolute value of a fraction

Solve \bigl|-\cfrac{7}{8}\bigr|= \, ?

-\cfrac{7}{8} \, is \, \cfrac{7}{8} \, away from 0, so \bigl|-\cfrac{7}{8}\bigr|=\cfrac{7}{8} \, .

### Example 3: absolute value of a negative number

Solve |-15|= \, ?

-15 is 15 away from 0, so |-15|= 15.

### Example 4: absolute value, real-world with a positive value

The number line shows the temperature in ^{\circ}C.

3^{\circ} is 3^{\circ} away from 0^{\circ}, so |3^{\circ}|= 3^{\circ}.

The actual temperature is 3^{\circ} above freezing, so the | 3^{\circ} | is the distance from the freezing/melting point, which is at 0^{\circ}C.

### Example 5: absolute value, real-world with a decimal

The number line shows the balance on a cafeteria account.

-\$3.50 is \$3.50 away from \$0, so |-\$3.50|= \$3.50. The actual balance is -\$3.50, so the |-\$3.50| is the distance from having no money (neither debt nor credit), which is at \$0.

### Example 6: absolute value, real-world with a mixed number

The number line shows the elevation in kilometers.

1 \, \cfrac{6}{10} \mathrm{~km} is 1 \, \cfrac{6}{10} \mathrm{~km} away from 0 \mathrm{~km}, so \bigl|1 \cfrac{6}{10} \mathrm{~km}\bigr|=1 \cfrac{6}{10} \mathrm{~km}.

The actual elevation is 1 \, \cfrac{6}{10} \mathrm{~km} above sea level, so the \bigl|1 \, \cfrac{6}{10} \mathrm{~km}\bigr| is the distance from sea level, which is 0 \mathrm{~km}.

### Teaching tips for absolute value

• Like all other foundational skills, it is important not to jump right to teaching students to just drop the negative sign. Given sufficient time working with a number line, students will catch on to this rule in a meaningful way. This is important for a few reasons;

One, students often confuse absolute value with additive opposites or just completely forget what it is. If they spend enough time using the number line and building meaning, this is less likely to happen. Also, they will be expected to solve and graph much more complicated absolute value equations in upper grades, where just the “dropping the negative” may end up being more confusing than helpful.

• Worksheets have their place when teaching absolute value, especially when they incorporate number lines, but also give students many opportunities to think about absolute value in real life. The most common connections are to temperature, account balances, and sea level, but students may also come up with other ways to think about and apply absolute value to the real world.

### Easy mistakes to make

• Using parentheses instead of absolute value bars
The symbol for absolute value, | \; |, is similar to parentheses and brackets, but it is not the same. In the beginning, students will need to be reminded of this until they have had enough exposure to remember on their own.

• Thinking absolute value can be negative
This either happens because it gets confused with something else, like additive opposites, or students do not have a conceptual understanding of what absolute value is. One way to help them understand that absolute value cannot be negative is the following demonstration;

Have two students stand side by side and then have them each take 5 steps away from each other (or any distance that is the same, but in opposite directions). Then ask, who traveled farther? The students should reason that both students traveled 5 steps – even though they went different directions. Just like with this example, in regards to absolute value, only the distance is being observed. You can also bring to their attention that a statement like “One student traveled - \, 5 steps” doesn’t make sense in this context, unless a + and - direction is defined.

### Practice absolute value questions

1. Complete the equation: |18|= \, ?

18 -18 \cfrac{1}{18} -\cfrac{1}{18} Show 18 on a number line and measure the distance from 0. 18 is 18 away from 0, so |18|= 18.

Notice that even though the direction from 18 to 0 is down (or negative), the absolute distance is not – it only shows the distance, not direction.

2. Complete the equation: |-0.2|= \, ?

2.0 -2.0 0.2 -0.2 Show -0.2 on a number line and measure the distance from 0. -0.2 is 0.2 away from 0, so |-0.2|= 0.2.

3. Complete the equation: \bigl|9 \cfrac{3}{7}\bigr|= \, ?

\cfrac{7}{66} -\cfrac{66}{7} -9 \, \cfrac{3}{7} 9 \, \cfrac{3}{7} Show 9 \, \cfrac{3}{7} \, on a number line and measure the distance from 0. 9 \, \cfrac{3}{7} \, is 9 \, \cfrac{3}{7} \, away from 0, so \bigl|9 \, \cfrac{3}{7}\bigr|=9 \, \cfrac{3}{7} \, .

4. Which statement is true about the temperature 21^{\circ} \mathrm{C} ? The absolute value is -21^{\circ} \mathrm{C}. 21^{\circ} \mathrm{C} is 21^{\circ} \mathrm{C} away from the freezing point. -21^{\circ} \mathrm{C} is 21^{\circ} \mathrm{C} away from the freezing point. The absolute value is 0^{\circ} \mathrm{C} Find the absolute value. 21^{\circ} \mathrm{C} is 21^{\circ} \mathrm{C} away from 0^{\circ} \mathrm{C}, so \left|21^{\circ} \mathrm{C}\right|=21^{\circ} \mathrm{C}.

The temperature is 21^{\circ} \mathrm{C} above freezing, so the \left|21^{\circ} \mathrm{C}\right| is the distance from the freezing point, which is 0^{\circ} \mathrm{C}.

5. Which statement is true for an account balance -\$0.75? The absolute value is \$0.00. The absolute value is -\$0.75. \$0.75 is -\$0.75 away from no debt. -\$0.75 is \$0.75 away from no debt. Find the absolute value. -\$0.75 is \$0.75 away from \$0.00, so \left|-\$0.75 \right|=\$0.75.

The account balance is a debt of \$0.75, so the \left|-\$0.75 \right| is the distance from \\$0.00, which is no debt.

6. Which statement is true for the elevation -45m? \left|-45 \, m \right| is the distance from sea level. 45 \, m is -45 \, m away from sea level. \left|0 \, m \right| is the distance from sea level. The absolute value is -45 \, m. Find the absolute value. -45 \, m is 45 \, m away from 0 \, m, so \left|-45 \, m \right|= 45 \, m.

-45 \, m is 45 \, m below sea level, so the \left|-45 \, m \right| is the distance from sea level, which is 0 \, m.

## Absolute value FAQs

How else is absolute value used in math?

This page is just an introduction to absolute value. In later standards, students will learn how to solve and graph absolute value equations and absolute value functions that involve any real number, including square roots. They will also learn how to solve and graph inequalities that include absolute value.

Can there be multiple absolute value signs in an absolute value expression?

Yes, an expression can have more than one, such as |4-3|+\left|5^2+2\right|. Just like an expression can have multiple operations or sets of parentheses, it can have multiple absolute value operations. To solve, follow the order of operations within the absolute value symbols first.

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