Commutative & Associative Properties

As we learned in the past lesson, the Order of Operations is a key rule for working through mathematical expressions. Within these laws are the Commutative Properties and Associative Properties which can only be applied to addition and multiplication. The Commutative Property states that you can move the places of numbers in an expression and still get the same answer. The Associative Property states you can regroup numbers in an expression and you will still get the same answer. Using these methods teaches us a skill we can use to manipulate expressions and equations to simplify and solve them!

Question 1

Add this sequence of numbers without the aid of a calculator. The general convention is add from left to right. However, with the Commutative Property we can rearrange these numbers to make the long addition easier. State the Commutative Property, then regroup these numbers in a simpler way to add.
5 + 7 + 6 + 3 + 9 + 5 + 4 + 2 + 1 + 8

Firstly, the Commutative Property states that you can move the places of numbers in an expression and still get the same answer.

Now, rather than add in the order i.e. 5 + 8 = 13 + 3 = 16 + ..., we can simplify by looking at pairing together the numbers so that each pair gives us an easier number to add with. Often we can look for pairs that will add together to get numbers like 10 or 20.

By looking at the numbers and finding combinations, following the Commutative Property, we can get:

5 + 5 + 7 + 3 + 6 + 4 + 9 + 1 + 8 + 2

Now, each pair adds together to give 10,

10 + 10 + 10 + 10 + 10 >= 50

Question 2.1

Use the Commutative Property for Addition and Multiplication to rearrange and show that the arrangements are equal.

i) 2 + 7

ii) 3 x 9

2 + 7

Rearranging gives us 7 + 2,

2 + 7 = 9, 7 + 2 = 9, they are both equal to 9

3 x 9

Rearranging gives us 9 x 3,

3 x 9 = 27, 9 x 3 = 27, they are both equal to 27

Question 2.2

State the Associative Property. Use the Associative Property for Addition and Multiplication to rearrange and show that the arrangements are equal.

iii) (1 + 4 ) + 7

iv) (3 x 6) x 5

First, state the Associative Property,

The Associative Property states you can regroup numbers in an expression and you will still get the same answer.

For the Associative Property, regrouping refers to brackets.

(1 + 4) + 7 = 12

This is the same as,

1 + (4 +7) = 12

They both equal 12.

(3 x 6) x 5 = 90

This is the same as,

3 x (6 x 5) = 90

They are both equal to 90

Question 3

The Commutative Property applies to addition and multiplication only. Show an expression rearranged with the Commutative Property to show it does not work for subtraction.

Start with any expression involving a subtraction,

3 - 4 + 5 ( = 4 )

Now swap the numbers, but keep the +/- signs in the same spot,

5 - 3 + 4 ( = 6)

As we can see these two expressions are not the same and therefore not Commutative.

When dealing with subtracting numbers, it is often beneficial to think about adding a negative or “The addition of opposites”.

For example 5 - 3 is the same as 5 + (-3)

Question 4

Rearrange the expression below to only involve addition. Then use the Commutative and Associative Properties to organise the expression, then determine its value.

9 - (-4) + 6 - 2 + 3 - 8 + 7 - 1

Let's start by changing all the subtraction into addition of the opposites or adding a negative,

Firstly though, we have ... - (-4)..., subtraction of a negative is the same as adding a positive (you can think of the two lines negative symbols (-) combining to form a plus sign),

So -(-4) becomes + 4,

9 + 4 + 6 - 2 + 3 - 8 + 7 - 1

Now lets change all of our subtractions into a addition of a negative ...-... >... + (-...)

9 + 4 + 6 + (-2) + 3 + (-8) + 7 + (-1)

Use the Commutative and Associative Properties to first, rearrange, then second, regroup the numbers into positives on one side and negatives on the other,

Commutative gives us.

9 + 4 + 6 + 3 + 7 + (-2) + (-8) + (-1)

Associative gives us,

( 9 + 4 + 6 + 3 + 7 ) + ( (-2) + (-8) + (-1) )

Now, evaluate each group of brackets.

(29) + (-11) = 18

Before we move on to exercises involving variables that we don't know, let's quickly recall the rules for addition and subtraction of variables with the same variable but different coefficients.

2x + 3x =

A reminder that 2x is the same as x + x and 3x is the same as x + x + x

So, 2x + 3x can be written as,

(x + x) + (x + x + x) =

x + x + x + x + x =

5x

However, for these purposes, we want you to be able to recognise that 2x + 3x = 5x

-4x + (-2x)

It can help to forget about the “x” for the addition/subtraction, as long as you remember to add it back in!

-4 + (-2) = -6; However, start to become familiar with working with these sorts of addition and subtraction.

-4x + (-2x) = -6x

9x - 4x = 5x

Question 6

Use the Commutative and Associative Properties to rearrange and simplify the following expression.

(4x + 3) + (x + 7) =

Using Associative we can either rearrange or remove groups or brackets,

4x + 3 + x + 7 =

Using Commutative we can rearrange the position, to get the x variables on one side and the numbers on the other,

4x + x + 3 + 7 =

Using Associative, we can now regroup the expression into an “x” group and a number group,

(4x + x) + (3 + 7) =

Adding the values in the groups together gives us,

5x + 10

Question 7

Simplify the expressions below using what we have learned above.

x + 3 + (-3x) - 9

First, start by changing any subtraction into addition of a negative, ( (-3x) is already done for us)

x + 3 + (-3x) + (-9)

Then sort the x’s on one side and the numbers on the other,

x + (-3x) + 3 + (-9)

Then group them,

(x + (-3x)) + (3 + (-9))

Then simply the groups,

(-2x) + (-6) or -2x - 6

6y - 7 - 2y + 8

A different variable, y is being used, do not be thrown off, the exact same rules apply!

First, start by changing any subtraction into addition of a negative,

6y +(-7) + (-2y) + 8

Then sort the y’s on one side and the numbers on the other,

6y + (-2y) +(-7) + 8

Then group them,

( 6y + (-2y) ) + ( (-7) + 8 )

Then simply the groups,

4y + 1

-3x + 2 + 7x + 7

First, start by changing any subtraction into addition of a negative, in this one however there aren't any so onto the next step.

Sort the x’s on one side and the numbers on the other.

-3x + 7x + 2 + 7

Then group them,

(-3x + 7x) + (2 + 7)

Then simply the groups,

4x + 9

And Finally ...

Apply the Associative and Commutative Properties to the expression below to simplify so that we are left with only two terms in the expression.

-7x + 4 - x + 1 + 12x - 9

Writing a subtraction as an addition of a negative can help to avoid mistakes. So, changing all subtractions to addition of a negative gives us.

-7x + 4 + (-x) + 1 + 12x + (-9)

Now we use the Commutative Property to rearrange the order of the terms, grouping the x-terms on one side and the numerical terms on the other,

-7x + (-x) + 12x + 4 + 1 + (-9)

Now, we use the associative property to group these like terms and simplify them,

(-7x +(-x) + 12x) + ( 4 + 1 + (-9))

[in the brackets we have (-7x + (-x) + 12x ) = 4x and (4 + 1 + (-9)) = -4]

(4x) + (4)

So, our simplified answer is,

4x + 4