# Associative property

Here you will learn about the associative property, including what it is, and how to use it to solve problems.

Students will first learn about the associative property as part of operations and algebraic thinking in 3rd grade.

## What is the associative property?

The associative property says that when you add or multiply numbers, they can be grouped in different ways and the answer will still be the same.

It can also be referred to as the associative property of addition and the associative property of multiplication.

For example,

When adding 5 + 6 + 1, you can group the numbers in different ways:

\begin{aligned} \hspace{.5cm}& (5+6)+1 \hspace{.5cm} \\ & =11+1 \\ & =12 \end{aligned}

\begin{aligned} \hspace{.5cm} & 5+(6+1) \hspace{.5cm} \\ & =5+7 \\ & =12 \end{aligned}

Notice that even with different groupings, the sum of the numbers is the same.

This is also true when multiplying a set of numbers.

For example,

When multiplying 2 \times 5 \times 3, you can group the numbers in different ways:

\begin{aligned} \hspace{.5cm} & (2 \times 5) \times 3 \hspace{.5cm} \\ & =10 \times 3 \\ & =30 \end{aligned}

\begin{aligned} \hspace{.5cm} & 2 \times(5 \times 3)\hspace{.5cm} \\ & =2 \times 15 \\ & =30 \end{aligned}

Notice that even with different groupings, the product of the numbers is the same.

The associative property can be used to find friendly numbers when solving. Friendly numbers are numbers that are easy to add or multiply mentally – like multiples of 10.

For example,

\begin{aligned} & 44+59 \\\\ & =(43+1)+59 \hspace{0.4cm} \text{ **Break up 44 to be 43 + 1} \\\\ & =43+(1+59) \hspace{0.4cm} \text{ **Regroup the 1 with 59} \\\\ & =43+60 \end{aligned}

The associative property lets us regroup and create friendlier numbers. 43 + 60 is easier to solve mentally than 44 + 59.

For example,

\begin{aligned} & 7 \times 5 \times 6 \\\\ & =7 \times(5 \times 6) \hspace{0.4cm} \text{ **Regroup to multiply } 5 \times 6 \text{ first} \\\\ & =7 \times 30 \end{aligned}

The associative property lets us regroup and create friendlier numbers. 7 \times 30 is easier to solve mentally than (7 \times 5) \times 6=35 \times 6.

### What is the associative property? ## Common Core State Standards

How does this relate to 3rd grade math?

• Grade 3 – Operations and Algebraic Thinking (3.OA.B.5)
Apply properties of operations as strategies to multiply and divide.
Examples: If 6 \times 4 = 24 is known, then 4 \times 6 = 24 is also known. (Commutative property of multiplication.)
3 \times 5 \times 2 can be found by 3 \times 5 = 15, then 15 \times 2 = 30, or by 5 \times 2 = 10, then 3 \times 10 = 30. (Associative property of multiplication.)
Knowing that 8 \times 5 = 40 and 8 \times 2 = 16, one can find 8 \times 7 as 8 \times (5 + 2) = (8 \times 5) + (8 \times 2) = 40 + 16 = 56. (Distributive property.)

## How to use the associative property

In order to use the associative property:

1. Check to see that the operation is addition or multiplication.
2. Change the grouping of the numbers and solve.

## Associative property examples

### Example 1: simple associative property with addition

Use the associative property to solve 19 + 4 + 26.

1. Check to see that the operation is addition or multiplication.

All the numbers are being added, so the associative property can be used.

2Change the grouping of the numbers and solve.

\begin{aligned} & 19+4+26 \\\\ & =19+(4+26) \hspace{0.4cm} \text{ *Group and add these numbers first} \\\\ & =19+30 \\\\ & =49 \end{aligned}

### Example 2: simple associative property with multiplication

Use the associative property to solve 7 \times 4 \times 5.

Check to see that the operation is addition or multiplication.

Change the grouping of the numbers and solve.

### Example 3: associative property with multiplication

Use the associative property to solve 2 \times 4 \times 5 \times 2.

Check to see that the operation is addition or multiplication.

Change the grouping of the numbers and solve.

### Example 4: associative property – addition with friendly numbers

Use the associative property with friendly numbers to solve 22 + 49.

Check to see that the operation is addition or multiplication.

Change the grouping of the numbers and solve.

### Example 5: associative property – addition with friendly numbers

Use the associative property with friendly numbers to solve 78 + 15.

Check to see that the operation is addition or multiplication.

Change the grouping of the numbers and solve.

### Example 6: associative property – multiplication with friendly numbers

Use the associative property with friendly numbers to solve 5 \times 12.

Check to see that the operation is addition or multiplication.

Change the grouping of the numbers and solve.

### Teaching tips for the associative property

• Intentionally choose practice problems that lend themselves to being solved with the associative property, as it is not always necessary or useful in all solving situations.

• Instead of just giving students the associative property definition, draw attention to examples of the associative property as they come up in daily math activities. You may even keep an anchor chart of different examples. Over time, students will start using it and recognizing it on their own and then you can introduce them to the property and its official definition through their own examples.

• Include plenty of student discourse around this topic to ensure that students understand that regrouping the numbers when adding or multiplying does not change the sum or product. This could include students sharing their thinking or critiquing the thinking of others.

### Easy mistakes to make

• Using the associative property for subtraction or division
The associative property only works when grouping the numbers differently doesn’t change the answer. This does not work for subtraction or division.
For example,

\begin{aligned} & 10-5-2 \hspace{1cm} 10-(5-2)\\ & =5-2 \hspace{1.4cm} =10-3 \\ & =3 \hspace{1.95cm} =7 \\ \end{aligned}
Changing the grouping of the numbers, changes the answer.

• Thinking there is only one way to use the associative property to solve with friendly numbers
Often, there is more than one way to use the associative property when solving.
For example,

\begin{aligned} & 45+48 \hspace{2.8cm} 45+48 \\ & =(43+2)+48 \hspace{1.7cm} =45+(5+43) \\ & =43+(2+48) \hspace{0.6cm} \text{OR} \hspace{0.6cm} =(45+5)+43 \\ & =43+50 \hspace{2.5cm} =50+43 \\ & =93 \hspace{3.23cm} =93 \\ \end{aligned}

• Confusing the order of operations
Equations are always solved starting from the left-hand side and moving to the right-hand side. While students do not need to be introduced formally to the order of all operations, it is important that they read and understand equations in this way. Otherwise, the different groupings or use of parentheses may not mean anything to them.

This associative property topic guide is part of our series on properties of equality. You may find it helpful to start with the main properties of equality topic guide for a summary of what to expect or use the step-by-step guides below for further detail on individual topics. Other topic guides in this series include:

### Practice associative property questions

1. Which of the following equations shows an alternative way to solve 5 + 11 + 9 using the associative property?

5 + (11 + 9) = 5 + 20 11 + 5 + 9 = 16 + 9 (5 + 11) + 9 = 16 + 9 5 + 9 + 11 = 5 + 20 All the numbers are being added, so the associative property can be used.

Change the grouping of the numbers and solve.

\begin{aligned} & 5+11+9 \\\\ & =5+(11+9) \hspace{0.3cm} \text{ *Group and add these numbers first}\\\\ & =5+20 \\\\ & =25 \end{aligned}

2. Which of the following equations showsan alternative way to solve 5 \times 5 \times 8 using the associative property?

5 \times 5 \times 8=25 \times 8 (5 \times 5) \times 8=(5 \times 5) \times 8 5 \times(5 \times 8)=5 \times 40 5 \times 5 \times 8=8 \times 5 \times 5 All the numbers are being multiplied, so the associative property can be used.

Change the grouping of the numbers and solve.

\begin{aligned} & 5 \times 5 \times 8 \\\\ & =5 \times(5 \times 8) \hspace{0.3cm} \text{ *Group and multiply these numbers first}\\\\ & =5 \times 40 \\\\ & =200 \end{aligned}

3. Which equation shows the associative property?

4 \times 3 \times 10=10 \times 3 \times 4 3 \times 4=2 \times 6 (11 + 4) + 1 = (11 + 4) + 1 (6 + 7) + 13 = 6 + (7 + 13) When solving 6 + 7 + 13, the associative property says you can group the numbers differently and still get the same answer, so (6 + 7) + 13 = 6 + (7 + 13).

4. Which equation shows the associative property?

6 \times 4 \times 11=11 \times 4 \times 6 (7 \times 5) \times 10=7 \times(5 \times 10) (14 + 31) + 19 = (14 + 31) + 19 34 + 12 = 12 + 34 When solving 7 \times 5 \times 10 the associative property says you can group the numbers differently and still get the same answer, so (7 \times 5) \times 10=7 \times(5 \times 10).

5. Which shows how to use the associative property AND friendly numbers to solve 26 + 45?

\begin{aligned} & 26+45 \\ & =(22+4)+45 \\ & =22+(4+45) \end{aligned} \begin{aligned} & 26+45 \\ & =26+(4+41) \\ & =(26+4)+41 \end{aligned} \begin{aligned} & 26+45 \\ & =45+26 \\ & =65+6 \end{aligned} \begin{aligned} & 26+45 \\ & =20+6+45 \\ & =26+45 \end{aligned} Friendly numbers are numbers that are easy to add mentally – like multiples of 10.

Change the grouping of the numbers and use friendly numbers to solve.

\begin{aligned} & 26+45 \\\\ & =26+(4+41) \hspace{0.3cm} \text{ *Break apart 45} \\\\ & =(26+4)+41 \hspace{0.3cm} \text{ *Regroup and add these numbers first} \\\\ & =30+41 \hspace{1.1cm} \text{ *30 is a friendly number because it is a multiple of 10}\\\\ & =71 \end{aligned}

6. Which shows how to use the associative property AND friendly numbers to solve 8 \times 5?

\begin{aligned} & 8 \times 5 \\ & =(4 \times 2) \times 5 \\ & =4 \times(2 \times 5) \end{aligned} \begin{aligned} & 8 \times 5 \\ & =(4 \times 2) \times 5 \\ & =8 \times 5 \end{aligned} \begin{aligned} & 8 \times 5 \\ & =5 \times 8 \\ & =8 \times 5 \end{aligned} \begin{aligned} & 8 \times 5 \\ & =(3+5) \times(4+1) \\ & =(5+3) \times(1+4) \end{aligned} Friendly numbers are numbers that are easy to multiply mentally – like multiples of 10.

Change the grouping of the numbers and use friendly numbers to solve.

\begin{aligned} & 8 \times 5 \\\\ & =(4 \times 2) \times 5 \hspace{0.3cm} \text{ *Show two factors of 8 } \\\\ & =4 \times(2 \times 5) \hspace{0.3cm} \text{ *Regroup and multiply these numbers first} \\\\ & =4 \times 10 \hspace{0.9cm} \text{ *10 is a friendly number}\\\\ & =40 \end{aligned}

## Associative property FAQs

Does the associative property work with other groupings of numbers besides whole numbers?

Yes, the associative property can be used with fractions, decimals, negative numbers and rational numbers, as long as they are all being added or multiplied.

How are the associative and commutative properties different?

The associative property changes the grouping of the numbers but not their location within the equation. The commutative property changes the order, or the location, of numbers within an equation. They both affect the order of operations, which can make them easy to confuse.

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