Use a number line to explore the absolute value of an integer

**Absolute Value**

Essential Question:

How can a number line help you find two integers that are the same distance from zero?

Several planes were flying at the same height. The green line below represents their starting point. Which two planes moved the same distance but in opposite directions?

+8

-10

+10

-15

In the diagram below, +8 means Plane A climbed 8 feet and -10 means Plane B moved down 10 feet.

Use the diagram to compare the distance each plane moved from the green line.

Step One

Complete the chart to compare the distance

each plane moved from the green line.

Plane

Integer

Direction

Distance Moved (ft)

C

D

E

0

+10

-15

none

0

Step Two

Determine which two planes moved the same distance away from the dashed line.

So, plane

____

and plane

____

moved

_____

feet from the green line.

Work with a partner to complete the table.

The first one is done for you.

Integer

Distance Between

Integer & Zero

Opposite

Integer

Distance Between

Opposite Integer

and Zero

3

3

-3

3

4

7

-11

-13

19

-21

0

What can you conclude

about the distance from zero for

both an integer and its opposite?

The movement of Plane B in the Investigation

was represented by the number -10. What does zero represent in the investigation?

How can a number line help you find two integers that are the same distance from zero?

The distance between a number and zero

on the number line

is called

**absolute**

value

value

Each mark on the number line

indicates one yard. Draw a tree

three yards west of the house. Draw a

mailbox three yards east of the house.

The distance between the house and the tree is

_______

the

distance between the house and the mailbox.

The tree and the mailbox are in

_______

directions from the house.

Jesse leaves home and walks 4

blocks west to the grocery store to buy milk then returns home. He then walks

another 4 blocks east to the Post Office. Compare the distance and the direction of the grocery store and the Post Office from Jesse's house.

Positive numbers, such as 2, are graphed to the right (or above) zero on a number line. Negative numbers, such as -2, are graphed to the left (or below) zero on a number line.

Opposites are numbers that are the same distance from zero in opposite directions. Since 0 is not negative nor positive, 0 is its own opposite. The opposite of the opposite of a number is the number itself.

For example, the opposite of the opposite of 3, -(-3), is still 3.

0

1

2

-1

-2

-2 is 2 units

to the left of

zero.

2 is 2 units

to the right of

zero.

Find the opposite of -5.

Method 1: Use a number line.

Draw a number line and graph -5.

-5 is 5 units to the left of 0. The integer of 5 is 5 units to the right of 0.

So, 5 is the opposite of -5.

Find the opposite of -5.

Method 2: Use symbols.

The integer of -5 uses the negative symbol.

The opposite of a negative symbol is a positive symbol.

So, the opposite of -5 is +5, or 5.

Got it? Try these...

What is the opposite of 3?

What is the opposite of the opposite of -2?

The absolute value of a number is the distance between the number and zero on a number line.

0

1

2

3

-1

-2

-3

3 units

3 units

| 3 | = 3 The absolute value of 3 is 3.

| -3 | = 3 The absolute value of -3 is 3.

The integers -3 and 3 are each 3 units from 0, even though they are on opposite sides of 0. | -3 | is read "absolute value of negative three."

Since distance cannot be negative, the absolute value of a number is always positive or zero.

**Examples**

Evaluate | -7 |.

Evaluate | 5 | + | -6 |.

Evaluate | -7 | - | 3 |.

A seagull is flying 25 feet above sea level. Nevaeh is diving 15 feet below sea level. What is the distance between Nevaeh and the seagull?

The expression | 25 | describes the seagull's distance above sea level. The expression | -15 | describes Nevaeh's distance below sea level.

To find the distance, add the absolute values.

| 25 | + | -15 | =

25 + |-15| =

25 + 15 =

40, so the total distance between the seagull and Nevaeh is 40 feet.

**You Try**

What is the opposite of 0?

What is the opposite of the opposite of 6?

Evaluate each expression.

| -5 | =

| 20 | - | -17 | =

| -16 | + | -12 | =

A game show contestant lost 15 points. He

answered another question incorrectly and lost another 15 points. How many points has he lost?

A

B

C

D

E