Commutativity Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Commutativity.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A property where swapping the order of two operands does not change the result: $a \ \star\ b = b\ \star\ a$ .

$3 + 5 = 5 + 3$ and $3 \times 5 = 5 \times 3$ . Swapping the order doesn't change the answer.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Commutativity says swapping the two operands of addition or multiplication leaves the result unchanged.

Common stuck point: The procedure for commutativity is the easy part; the trap is assuming subtraction or division commute. Asking "Can I swap these two operands of $+$ or $\times$ and still get the same result?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Can I swap these two operands of $+$ or $\times$ and still get the same result?

Worked Examples

Example 1

easy

Show that

6 + 9 = 9 + 6

. Does the order of adding matter?

Answer

Both equal 15; order does not matter

First step

Calculate

6 + 9

: count on from 9 to 15.

6 + 9 = 15

Full solution

2
Calculate $9 + 6$ : count on from 9 to 15. $9 + 6 = 15$ .
3
Both equal 15, so $6 + 9 = 9 + 6$ .
4
The order does NOT matter for addition.

The commutative property of addition states

a + b = b + a

. You get the same sum no matter which number you start with.

Example 2

medium

Verify that

4 \times 7 = 7 \times 4

using a rectangular array. Explain what changes and what stays the same.

Example 3

medium

Use commutativity to compute

7 + 23 + 3

quickly.

Example 4

medium

Show that

50 + 27 + 50 + 13

equals

140

by reordering.

Example 5

medium

Use commutativity to compute

25 \times 7 \times 4

Example 6

hard

Is the operation 'subtract then negate', defined as

a \star b = -(a - b)

, commutative?

Example 7

hard

Define

a \star b = \max(a, b)

. Is

\star

commutative?

Example 8

hard

Show that subtraction is anti-commutative:

a - b = -(b - a)

Example 9

challenge

Define

a \star b = a + b - ab

. Is

\star

commutative? Prove it.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

13 + 27 = 40

, what is

27 + 13

? Explain without calculating.

Example 2

medium

Does commutativity work for subtraction? Check with

10 - 3

and

3 - 10

Example 3

easy

Which property lets you rewrite

3+5

5+3

Example 4

easy

4\times 9

equal to

9\times 4

? Which property?

Example 5

easy

True or false:

7-2 = 2-7

Example 6

easy

Rewrite

6+11

using commutativity.

Example 7

easy

10\div 2 = 2\div 10

Example 8

easy

Fill the blank using commutativity:

8\times 5 = 5\times \underline{\ \ }

Example 9

easy

Which property makes

2+9+8

easy to compute as

2+8+9

Example 10

easy

Does swapping the order in

5+0

0+5

change the value?

Example 11

medium

Which of these uses commutativity (not associativity)? (a)

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

(b)

2+3=3+2

Example 12

medium

Use commutativity to compute

25\times 17\times 4

quickly.

Example 13

medium

Is the operation

a\star b = a^b

commutative? Test with

2

and

3

Example 14

medium

Find

x

so that

x+13 = 13+9

uses commutativity correctly.

Example 15

medium

A student writes

12-5 = 5-12

. Explain the error in one property term.

Example 16

medium

Rewrite

3\times 4 + 4\times 3

as a single product using commutativity.

Example 17

medium

Which property justifies

a+b+c = c+b+a

Example 18

medium

Is matrix-style 'subtract then negate'

a\star b=-(a-b)

commutative? Test

5,2

Example 19

challenge

Define

a\star b = a+b+ab

. Is

\star

commutative? Justify.

Example 20

challenge

Find all real

a,b

with

a-b=b-a

Example 21

challenge

For operation

a\star b=\frac{a+b}{1+ab}

, verify commutativity and compute

1\star 2

Example 22

medium

Does forming a fraction commute: is

\frac{2}{3}=\frac{3}{2}

Example 23

easy

25 + 18 = 43

, what is

18 + 25

Example 24

easy

6 \times 9 = 54

, what is

9 \times 6

Example 25

easy

True or false:

12 - 5 = 5 - 12

Example 26

easy

Use commutativity to rewrite

15 + 36

so the smaller addend comes first.

Example 27

easy

3 \times 5

array of dots has the same number of dots as a

5 \times 3

array. How many dots?

Example 28

medium

Use commutativity to compute

4 \times 17 \times 25

Example 29

medium

True or false: for all numbers

a, b

a + b = b + a

Example 30

medium

Find a pair

(a, b)

for which

a - b = b - a

Example 31

medium

Lucy says

7 \div 1 = 1 \div 7

because division is commutative. Is she correct?

Example 32

hard

Define

a \diamond b = a + 2b

. Is

\diamond

commutative?

Example 33

hard

Use commutativity to evaluate:

(-7) + 12 + 7 + (-12)

Example 34

hard

Use commutativity to make

5 \times 13 \times 2

easier, and compute the product.

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Related Concepts

Addition Multiplication Associativity Algebraic Representation

Background Knowledge

These ideas may be useful before you work through the harder examples.

additionmultiplication