Associative Law

Overview

In this lesson, we are going to learn about the associative law. Make sure you fully understand the commutative law before reviewing this lesson!

Here’s a quick review of the commutative law.

The commutative law tells us that we can switch numbers when adding and multiplying and still get the same answer. We will only be focusing on understanding how the associative law applies to addition.

5 +4 = 9 is the same as 4 + 5 = 9 (This is the commutative law. We switched the numbers 5 and 4, and we always get the same answer which is 9.)

Awesome! So what is the associative law? It’s very similar to the commutative law, and it allows us to “re-group” numbers to still get to the same answer.

Sometimes we have more than two groups to add together! For example, in the picture below, there are three groups of stars. How many stars are there in all?

We can add the three groups to find the sum!

We can add them like this:

4 + 6 + 2 = 12

When we add groups or numbers, the order does not matter.

We can use parentheses ( ) to add two group of numbers together.

(4 + 6) + 2 = 12

We will add like this:

4 + 6 = 10

Then we will add our sum to the rest of the equation.

10 + 2 =12

We can also solve it like this:

4 + (6 + 2) = 12

6 + 2 = 8

and   8 + 4 = 12

Notice that we moved the parenthesis from the first example and we still ended up with the same answer. This is the associative property. Numbers can be grouped together differently when adding, and still have the same answer.

Let’s take a look at another problem below.

We have three groups of stars. How many stars are there in all?

We can add the three groups to find the sum!

We can add them like this:

7 + 3 + 5 = 15

When we add groups or numbers, the order does not matter.

We can use parentheses ( ) to add two group of numbers together.

(7 + 3) + 5 = 15

We will add like this:

7 + 3 = 10

Then we will add our sum to the rest of the equation.

10 + 5 =15

We can also solve it like this (using associative property):

7 + (3 + 5) = 15

We will add like this:

3 + 5 = 8

and     8 + 7 = 15

Numbers can be grouped together to add. As long as you add carefully, you will get the correct answer. In this example, the sum is 15.

Let’s try some more examples! In the next few problems, find the missing addend or sum. Use what you have learned about the associative property to help you!

To solve, let’s look at the first problem:

5 + (2 + ___ ) = 10

We can figure out the missing addend! We know that

5 + 2 = 7

We also know that

 7 + ___ = 10

7 + 3 = 10

Now we will solve the right side of the equation.

 2 + (3 + ___ ) = 10

We can figure out the missing addend! We know that

2 + 3 = 5

We also know that

5 + ___ = 10

5 + 5 = 10

Now we can complete the math sentence:

5 + (2 + 3) = 10 = 2 + (3 + 5)

Both equations equal 10!
Here is another example. We can solve it in the same way!

To solve, let’s look at the first problem:

4 + (3 + ___ ) = 9

We can figure out the missing addend! We know that

4 + 3 = 7

We also know that

7 + ___ = 9

7 + 2 = 9

Now we will solve the right side of the equation.

3 + (2 + ___) = 9

We can figure out the missing addend! We know that

3 + 2 = 5

We also know that

5 + ___ = 9

5 + 4 = 9

Now we can complete the math sentence:

4 + (3 + 2) = 9 = 3 + (2 + 4)

Both equations equal 9!

Practice Question 1

Let’s practice! Look at the group below. Write the addition sentence and solve the problem below.

( ______ + ______ ) + ______ =

______ + ______ =

______ + ______ = _____

REVEAL THE ANSWER

Practice Question 2

 Look at the group below. Write the addition sentence and solve the problem below.

( ______ + ______ ) + ______ =

______ + ______ =

______ + ______ = _____

REVEAL THE ANSWER

Practice Question 3

Take a look at the math sentence below. Draw a picture to match it. Then solve.

1 + 3 + 9 = 

REVEAL THE ANSWER

Practice Question 4

Take a look at the math sentence below. Draw a picture to match it. Then solve.

4 + 4 + 7 =

REVEAL THE ANSWER

Practice Question 5

Take a look at the math sentence below. Draw a picture to match it. Then solve.

2 + 9 + 5 =

REVEAL THE ANSWER