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,

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.

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.

What is adding and subtracting negative numbers?

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.
4. Write your final answer.

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:

4Write your final answer.

Example 2: adding a negative number

Solve - \; 2+- \; 3.

Circle the first number on the number line.

The first number in the question is - \; 2.

Use the number line to add or subtract.

In this case, you are subtracting 3 , so move 3 spaces left from - \; 2 on the number line:

Example 3: subtracting a positive number

Solve - \; 5-2.

Circle the first number on the number line.

The first number in the question is - \; 5.

Use the number line to add or subtract.

In this case, you are calculating - \; 2 so move 2 spaces left from - \; 5 on the number line:

Example 4: subtracting a negative number

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

Circle the first number on the number line.

The first number in the question is - \; 8.

Use the number line to add or subtract.

In this case you are calculating + \; 10, so move 10 spaces right from - \; 8 on the number line:

Example 5: mixed operations

Answer the calculation below:

Circle the first number on the number line.

The first number in the question is 7.

Use the number line to add or subtract.

In this case, you are calculating - \; 8 , so move 8 spaces left from 7 on the number line:

Nex,t you are calculating + \; 5 , so move 5 spaces right from - \; 1 on the number line:

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?

Circle the first number on the number line.

The first number in the question is 12.

Use the number line to add or subtract.

In this case you are calculating - \; 20.

Instead of subtracting 1 each time, each jump is subtracting 2 so we can subtract 20 in 10 jumps.

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.

Related addition and subtraction lessons

• Adding and subtracting integers
• Adding and subtracting rational numbers
• Add and subtract within 100
• Standard algorithm addition
• Order of operations
• Absolute value

The next lessons are

• Multiplication and division
• Types of numbers
• Rounding numbers
• Adding and subtracting fractions
• Multiplying and dividing negative numbers

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