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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.
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.
How does this relate to 3rd grade math?
In order to use the associative property:
Use this quiz to check your grade 3 to 6 students’ understanding of properties of equality. 10+ questions with answers covering a range of 3rd, 5th and 6th grade properties of equality topics to identify areas of strength and support!
DOWNLOAD FREEUse this quiz to check your grade 3 to 6 students’ understanding of properties of equality. 10+ questions with answers covering a range of 3rd, 5th and 6th grade properties of equality topics to identify areas of strength and support!
DOWNLOAD FREEUse the associative property to solve 19 + 4 + 26.
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}Use the associative property to solve 7 \times 4 \times 5.
Check to see that the operation is addition or multiplication.
All the numbers are being multiplied, so the associative property can be used.
Change the grouping of the numbers and solve.
Use the associative property to solve 2 \times 4 \times 5 \times 2.
Check to see that the operation is addition or multiplication.
All the numbers are being multiplied, so the associative property can be used.
Change the grouping of the numbers and solve.
Use the associative property with friendly numbers to solve 22 + 49.
Check to see that the operation is addition or multiplication.
All the numbers are being added, so the associative property can be used.
Change the grouping of the numbers and solve.
Use the associative property with friendly numbers to solve 78 + 15.
Check to see that the operation is addition or multiplication.
All the numbers are being added, so the associative property can be used.
Change the grouping of the numbers and solve.
Use the associative property with friendly numbers to solve 5 \times 12.
Check to see that the operation is addition or multiplication.
All the numbers are being multiplied, so the associative property can be used.
Change the grouping of the numbers and solve.
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:
1. Which of the following equations shows an alternative way to solve 5 + 11 + 9 using the associative property?
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?
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?
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?
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?
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?
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}
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.
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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