Math resources Number and quantity Add & subtract

# Adding and subtracting rational numbers

Here you will learn strategies on how to add and subtract rational numbers including using visual models, the number line, as well as using standard algorithms.

Students will first learn about integers in 6 th grade math as part of their work with the number system and expand that knowledge to operations with integers in the 7 th grade.

## What are adding and subtracting rational numbers?

Adding and subtracting rational numbers is when you add or subtract two or more rational numbers together.

To add or subtract two rational numbers with the same denominator, we can add or subtract the numerators and write the result over the common denominator.

To add or subtract two rational numbers with different denominators, we must find a common denominator to add or subtract the numerators.

You can also add and subtract rational numbers using visual models or a number line.

## Common Core State Standards

How does this apply to 7 th grade math?

• Grade 7: The Number System (7.NS.A.1)
Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

• Grade 7: The Number System (7.NS.A.1a)
Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.

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

## How to add and subtract rational numbers

In order to add and subtract rational numbers using counters:

1. Model the problem with counters and use zero pairs when necessary.
2. The answer is the leftover counters.

In order to add and subtract rational numbers using a number line:

1. To add, start at the first number and move to the second number; to subtract, start from the second number and move to the first number.

## Adding and subtracting rational numbers examples

### Example 1: adding negative numbers using counters

-2 \, \cfrac{1}{2}+7= \, ?

1. Model the problem with counters and use zero pairs when necessary.

Use what you know about integer zero pairs.

There is \cfrac{1}{2} of a positive counter and 4 whole positive counters left.

2The answer is the leftover counters.

-2 \, \cfrac{1}{2}+7=4 \cfrac{1}{2}

### Example 2: subtracting a negative number using counters

-4.5-(-6)= \, ?

Model the problem with counters and use zero pairs when necessary.

The answer is the leftover counters.

### Example 3: adding decimals with a number line

-11.1+(-0.3)= \, ?

To add, start at the first number and move to the second number; to subtract, start from the second number and move to the first number.

## How to add and subtract rational numbers with algorithms

In order to add and subtract rational numbers using algorithms:

1. Make sense of the calculation – relating to positive numbers when necessary.
2. Use an algorithm.
3. Decide if the final answer is positive or negative.

### Example 4: subtracting fractions (mixed numbers) with different denominators

Solve 3 \, \cfrac{4}{5}-5 \, \cfrac{1}{6} \, .

Make sense of the calculation – relating to positive numbers when necessary.

Use an algorithm.

Decide if the final answer is positive or negative.

### Example 5: adding negative decimals

Solve -125.5+(-34.56).

Make sense of the calculation – relating to positive numbers when necessary.

Use an algorithm.

Decide if the final answer is positive or negative.

### Example 6: adding a positive and negative decimal

Solve 125.5+(-34.56).

Make sense of the calculation – relating to positive numbers when necessary.

Use an algorithm.

Decide if the final answer is positive or negative.

### Teaching tips for adding and subtracting rational numbers

• Start with adding and subtracting rational expressions that involve only positive numbers. Students will use these skills to then operate with negative numbers. For students who are struggling to remember all algorithms, it may be necessary to provide them with step-by-step guides.

• When choosing worksheets, look for ones that include word problems. These types of problems give the rational numbers a real world context and allow students to practice figuring out which operation is appropriate.

### Easy mistakes to make

• Getting the positive and negative direction on the number line backwards
Right is positive and left is negative.

• Confusing the different signs and rules
The rules for adding and subtracting integers can be used to solve with rational numbers. However, just like with integers it is important to think about why the rule works when using it, to avoid using the wrong rule.
For example,

-5 + 3 = \, ?
-5 + (-3) = -8

-5 + 3 = \, ?
5 + 3 = -2

It doesn’t make sense that you would add positive 3 to a -5 and get a more negative number. You should get a less negative number, which is shown in the correct strategy.

### Practice adding and subtracting rational numbers questions

1) Use the model below to add   -7 \, \cfrac{3}{4}+6.

-13 \, \cfrac{3}{4}

1 \, \cfrac{3}{4}

-1 \, \cfrac{3}{4}

13 \, \cfrac{3}{4}

There are  6 zero pairs with one negative counter and   \cfrac{3}{4} of a negative counter leftover.

-7 \, \cfrac{3}{4}+6=-1 \, \cfrac{3}{4}

2) Use the number line to subtract -1.5-(- 0.9).

-0.6

0.6

2.4

-2.4

Solving -1.5-(-0.9) is the same as finding the distance from -0.9 to -1.5.

From -0.9 move left 0.6 and you get to -1.5. Moving left is in the negative direction.

So, -1.5-(-0.9) = -0.6

3) Use the number line to add   0.75 + (-1.25).

-2

2

-0.5

0.5

Start at 0.75 and move 1.25 places in the negative direction (left).

You land at -0.5.

0.75 + (-1.25) = -0.5

4) Subtract: -14 \, \cfrac{2}{3}-\left(-8 \, \cfrac{1}{2}\right)= \, ?

6 \, \cfrac{1}{6}

-22

22

-6 \, \cfrac{1}{6}

-14 \, \cfrac{2}{3}-\left(-8 \, \cfrac{1}{2}\right) is starting with -14 \, \cfrac{2}{3} taking away -8 \, \cfrac{1}{2}.

It can also be thought of as the distance from -8 \, \cfrac{1}{2} \, to -14 \, \cfrac{2}{3}.

Because of absolute value, this is the same as the distance from 8 \, \cfrac{1}{2} \, to 14 \, \cfrac{2}{3}.

So, you can solve for the difference with 14 \, \cfrac{2}{3}-8 \, \cfrac{1}{2}.

\begin{aligned} & 14 \, \cfrac{2}{3}-8 \frac{1}{2} \\\\ & =14 \, \cfrac{2 \times 2}{3 \times 2}-8 \frac{1 \times 3}{2 \times 3} \\\\ & =14 \, \cfrac{4}{6}-8 \frac{3}{6} \\\\ & =6 \, \cfrac{1}{6} \end{aligned}

Since the original equation started with more negatives, the result will still be negative – although there are less now after subtracting 8 \, \cfrac{1}{2} negatives. Also, moving from -8 \, \cfrac{1}{2} to -14 \, \cfrac{2}{3}  is to the left or in the negative direction.

-14 \, \cfrac{2}{3}-\left(-8 \, \cfrac{1}{2}\right)=-6 \, \cfrac{1}{6}

5) Add: -13.2 + (-4.8) = \, ?

-18

18

8.4

-8.4

-13.2 + (-4.8) combines two negative numbers, so the sum will be more negative.

You can solve for the sum of the negatives like you would positives since there are no zero pairs canceling values out:

13.2 + 4.8 = 18.

Since the original equation is adding negatives, the answer will also be negative.

-13.2 + (-4.8) = -18

6) On a January day in New York, the morning temperature was -3.5 degrees Fahrenheit. Later that day, the temperature increased by 14.3 degrees. What is the new temperature in degrees?

-10.8 degree

17.8 degrees

-17.8 degrees

10.8 degree

-3.5 increased by 14.3 degrees is -3.5 + 14.3.

Since you can add in any order, you can also solve for 14.3 + (-3.5).

14.3-3.5 = 10.8, so 14.3 + (-3.5) = 10.8 which means -3.5 + 14.3 = 10.8.

## Adding and subtracting rational numbers FAQs

How are positive and negative numbers related?

All positive numbers have an inverse that is negative and all negative numbers have an inverse that is positive. For example, -4.5 and 4.5 are inverses. When you add inverses together their sum is always 0 – think zero pairs. Thinking about inverses is one way to solve addition and subtraction of rational numbers.

How do you add fractions with uncommon denominators?

In order to add negative fractions and mixed fractions (mixed numbers), each fraction must have the same denominator. A common strategy is to find the least common multiple (LCM) of the denominators and use this to solve (sometimes called the least common denominator or LCD). Once the denominators are common, you can add the numerators to solve.

For what other topics is addition and subtraction of rational numbers useful?

Operations with rational numbers can be expanded to solve algebraic expressions and equations, including ones with polynomials and inequalities.

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