Cooper is having a great time getting to know his pen pal, Riley, in New Zealand. They are both the same age and love sports. Cooper wants to call Riley, but he's having a hard time figuring out when it would be OK to call him because of the time zone difference.

Cooper figures that it would be best if someone called *him* some time after he got home from school at 3pm and before he went to bed at 10pm. He knows that New Zealand is 18 hours ahead of Santa Fe, NM, where Cooper lives, but he doesn't know what time that makes it in New Zealand.

In this concept, you will learn how to add integers using a number line.

### Adding Integers Using a Number Line

An **integer **is any positive whole number or its opposite. Here, opposite means sign. So a positive integer has a negative opposite and vice versa.

A **positive number **is a number greater than zero. It can be written with or without a + symbol in front of it. A gain in something is written with a positive number.

A **negative number **is a number that is less than zero. It is always written with a - symbol in front of it. A loss is written with a negative number.

A **number line **is a line on which numbers are marked at intervals, used to illustrate simple numerical operations. Using a number line allows you to see where a number is in relationship to other numbers and from zero.

Number lines are useful tools for visualizing simple arithmetic operations, especially addition and subtraction. Start at the position indicated by the first number. If the second number is negative, move that many units to the left. If it's positive, move that many units to the right. The final mark is the answer.

Here is an example of using a number line in addition.

-5 + 7 = ____

First, mark the initial integer.

Next, move seven units in the positive direction.

The result is +2.

That is the answer of this addition.

Here is another example.

6 + -9 = _____

First, mark the initial integer, +6.

Next, move to left (the negative direction) 9 units.

The final mark is at -3.

The answer is -3.

### Examples

#### Example 1

Earlier, you were given a problem about Cooper and his time zone difficulty.

He wants to call his friend, Riley, in New Zealand some time between 3pm and 10pm, but he doesn't know what time that is in Santa Fe, where he lives. He knows Santa Fe is 18 hours later than New Zealand.

In order to solve this problem, Cooper can use number lines and addition.

First, Cooper can draw a special number line that repeats every twelve hours.

Next, to figure out the early limit, Cooper marks 3pm and then adds 18 hours.

3 pm + 18 hours = 9 am

So, 3pm in New Zealand is 9am in Santa Fe.

Then, he repeats this process to find the later limit.

10 pm + 18 hours = 4pm

So, if Cooper wants to call Riley some time between the time Riley comes home from school at 3pm and before he goes to bed at 10pm, Cooper will have to call him between 9am and 4pm. But Cooper, himself, doesn't get home until 3pm. So he concludes that the only time he can call Riley is as soon as he gets home from school at 3pm.

#### Example 2

Draw a number line from -6 to 9 and use it to find the sum.

\begin{align*}-5 + -1 + 7\end{align*}

First, draw the number line.

Next, start at -5 and move 1 to the left.

Then, add 7 to the right.

You are now at +1.

The answer is \begin{align*}1\end{align*}

#### Example 3

Draw a number line and use it to find the sum.

-5 + 9 = ____

First, draw the number line.

Next, start at -5 and move 9 to the right.

You are now at +4.

The answer is 4.

#### Example 4

Draw a number line and use it to find the sum.

-1 + -8 = ____

First, draw the number line.

Next, start at -1 and move 8 to the left.

You are now at -9.

The answer is -9.

#### Example 5

Draw a number line and use it to find the sum.

5 + -7 = ____

First, draw the number line.

Next, start at 5 and move 7 to the left.

You are now at -2.

The answer is -2.

### Review

Add the following integers that have the same sign by using a number line. You may figure out a pattern. If so, try finding the sums without a number line.

- 6 + 7 = _____
- -9 + -7 = _____
- -3 + -4 = _____
- 5 + 12 = _____
- -12 + -23 = _____
- 27 + 11 = _____
- -34 + -13 = _____
- 25 + 16 = _____
- -9 + -29 = _____
- -16 + -12 = _____

Add the following integers using a number line. Notice that they have different signs.

- -9 + 3 = _____
- -7 + 5 = _____
- 1 + -12 = _____
- 3 + -8 = _____
- -19 + 11 = _____
- 7 + -12 = _____
- 23 + -10 = _____
- -4 + 16 = _____
- 15 + -18 = _____
- -15 + 9 = _____

### Review (Answers)

To see the Review answers, open this PDF file and look for section 11.5.