# Adding and subtracting negative numbers

Here is everything you need to know about adding and subtracting negative numbers including adding and subtracting positive and negative numbers.

Students will first learn about adding and subtracting negative numbers as part of the number system in 7 th grade.

## What is adding and subtracting negative numbers?

Adding and subtracting negative numbers is operating with non-positive numbers. Negative integers are numbers that have a value less than zero, like - \; 5 and - \; 21.

To do this, you can use the number line. If you are adding, move to the right of the number line. If you are subtracting, move the opposite direction, to the left, of the number line.

For example,

To solve - \; 5+6 on a number line, start at - \; 5 and move 6 to the right.

- \; 5+6=1

To solve - \; 1-3, start at - \; 1 on the number line and move left 3.

- \; 1-3=- \; 4

Using the number line in this way can help you solve each type of problem, except when adding or subtracting a negative number. For this, a step needs to be added.

For example,

- \; 5-(- \; 2)

There are a few ways to think about how to solve this:

1. If you have 5 negatives and you take away 2 negatives, you are left with 3 negatives.
2. Think about the - symbol as the opposite \rightarrow so - \; ( - \; 2) means “the opposite of - \; 2 ” which is + \; 2.
3. Consider negatives to be cold and positives to be warm. That means - \; ( - \; 2) is like taking away cold. When you take away cold, it gets warmer. So 2 warmer than - \; 5.

All three ways show that - \; 5-( - \; 2)=- \; 3. From this you can establish the rule that subtracting a negative is the same as adding a positive.

- \; 5-( - \; 2)=- \; 5+2=- \; 3

The grid below can help to work out whether to add or subtract with a positive or negative number.

If you have two signs next to each other, change them to a single sign.

• If the signs are the same, add.
• If the signs are different, subtract.

## Common Core State Standards

How does this relate to 7 th grade math?

• Grade 7 – The Numbers System (7.NS.A.1d)
Apply properties of operations as strategies to add and subtract rational numbers.

## How to add and subtract negative numbers

In order to add and subtract negative numbers:

1. Change two adjacent signs to a single sign.
2. Circle the first number on the number line.
3. Use the number line to add or subtract.

## Adding and subtracting negative numbers examples

### Example 1: adding a positive number

Solve - \; 4+7.

1. Change two adjacent signs to a single sign.

If you have two signs next to each other, change them to a single sign.

• If the signs are the same, add.
• If the signs are different, subtract.

In this case you do not have two signs next to each other so no signs change.

2Circle the first number on the number line.

The first number in the question is - \; 4.

3Use the number line to add or subtract.

As you are calculating + \; 7, move 7 spaces right from - \; 4 on the number line:

- \; 4+7=3

### Example 2: adding a negative number

Solve - \; 2+- \; 3.

Change two adjacent signs to a single sign.

Circle the first number on the number line.

Use the number line to add or subtract.

### Example 3: subtracting a positive number

Solve - \; 5-2.

Change two adjacent signs to a single sign.

Circle the first number on the number line.

Use the number line to add or subtract.

### Example 4: subtracting a negative number

Solve: - \; 8-- \; 10.

Change two adjacent signs to a single sign.

Circle the first number on the number line.

Use the number line to add or subtract.

### Example 5: mixed operations

7-8- ( - \; 5)

Change two adjacent signs to a single sign.

Circle the first number on the number line.

Use the number line to add or subtract.

### Example 6: word problem

Alina had \$12 in her bank account. She purchased a t-shirt for \$ 20. By how much did she overdraw her account?

Change two adjacent signs to a single sign.

Circle the first number on the number line.

Use the number line to add or subtract.

### Teaching tips for adding and subtracting negative numbers

• Use visuals, such as number lines or counters, to illustrate the concept of adding and subtracting negative numbers.

• When teaching rules, such as when you add two negative numbers you get a positive, make sure to provide an explanation why it works and provide a visual if needed.

• When providing students with practice worksheets, make sure to begin with simple problems and progress to more challenging problems as students become confident with the concept.

• For students that are struggling with mastering the concept, consider the use of student math tutors. This is a strategy where students are in charge of the tutorials for other students. Sometimes allowing other students to use student language allows for deeper understanding.

### Easy mistakes to make

• Greater negative does mean a larger number
Students sometimes assume that the larger a negative number the greater it is. For example, students might incorrectly assume - \; 3 is greater than 2 because 3 is a larger number.

• Using the rules for two signs when the signs are not together
For example, thinking that - \; 5+7 would change to 5-7 (since there is a \; + and a \; - ). Changing the signs only applies when the signs are together.

So in the case of - \; 5+7, nothing would change and you would start at - \; 5 on the number line and move 7 spaces to the right. If the calculation was 5-+7 and the signs were together, you would then change it to 5-7.

### Practice adding and subtracting negative numbers questions

1. Solve: – \; 6 + 10

– \; 4

4

– \; 16

16

Start at – \; 6 on the number line and move to the right 10 numbers.

2. Solve: – \; 4 +- \; 8.

– \; 12

12

4

– \; 4

There is a positive (+) and negative sign (-) together so change them to a subtraction (-) only.

– \; 4+- \; 8=- \; 4-8

Start at – \; 4 on the number line and move to the left 8 numbers. As each jump is – \; 2, this is 4 jumps.

3. Solve: – \; 9-12.

– \; 3

3

21

– \; 21

Start at – \; 9 on the number line and move to the left 12 numbers.

4. Solve: – \; 15- ( – \; 6).

– \; 21

– \; 9

9

21

There are two negatives (-) together, so change the signs to an addition (+) sign only.

– \; 15- (- \; 6)=- \; 15+6

Start at – \; 15 on the number line and move to the right 6 numbers.

5. Solve: – \; 7+ ( – \; 8)-5+2.

– \; 2

– \; 8

– \; 18

– \; 12

There is a positive (+) and a negative (-) together, so change the signs to a subtraction (-) sign only.

– \; 7+(- \; 8)-5+2=- \; 7-8-5+2

Start at – \; 7 and move 8 places to the left, then move another 5 places to the left, then move 2 places to the right.

6. At 5 am in the morning, the temperature in Fargo, North Dakota was – \; 9^{\circ} \mathrm{F}. By 10 am the temperature had risen by 11^{\circ} \mathrm{F}. What was the new temperature?

20^{\circ} \mathrm{F}

– \; 20^{\circ} \mathrm{F}

– \; 2^{\circ} F

2^{\circ} \mathrm{F}

The equation you need to solve is – \; 9+11.

Start at – \; 9 on the number line and because it is an addition problem, move to the right 11 numbers.

At 10 am, it was 2^{\circ} \mathrm{F} in Fargo, North Dakota.

## Adding and subtracting negative numbers FAQs

The additive inverse when you add any number to its opposite and the sum will always be zero. For example, 4+( - \; 4)=0.

How is adding and subtracting negative numbers like adding and subtracting whole numbers?

Adding and subtracting negative numbers is similar to adding and subtracting whole numbers in a few ways.

For one, you can represent both on a number line to support the visualization of adding and subtracting. In both, addition and subtraction are inverse operations of one another.

Adding a negative value is equivalent to subtracting its absolute value, and subtracting a negative value is like adding its absolute value.

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