# Arithmetic properties: Commutative property

### Arithmetic properties: Commutative property

#### Lessons

In this lesson, we will learn:

• What is the commutative property of addition?
• What is the commutative property of multiplication?
• How to write the general formulas/equations for the commutative properties
• Changing the order of a list of addends/factors does not change the answer
• How to solve word problems for the commutative property

Notes:

• The commutative property means that changing the order of numbers in an equation does NOT change the answer ONLY when you are performing addition or multiplication
• The numbers can be any real number (whole numbers, fractions, decimals, integers, etc.)
• To “commute” can mean to travel between home and school—when you switch places, the travel time does not change (going to school and going back home).
• If you switch the places of the numbers in an addition or multiplication equation, it will not change the answer in the end.

• Ex. 3 + 5 = 5 + 3 (equals 8 either way)
• Ex. 0.3 + 0.5 = 0.5 + 0.3 (equals 0.8 either way)
• Ex. $\frac{3}{10}$ + $\frac{5}{10}$ = $\frac{5}{10}$ + $\frac{3}{10}$ (equals $\frac{8}{10}$ either way)

• For multiplication: the order of factors does not change the answer
• Ex. 3 × 4 = 4 × 3 (equals 12 either way)
• Ex. 0.3 × 0.4 = 0.4 × 0.3 (equals 0.12 either way)
• Ex. $\frac{3}{10}$ + $\frac{4}{10}$ = $\frac{4}{10}$ + $\frac{3}{10}$ (equals $\frac{12}{100}$ either way)

• The general formulas (where a and b are variables that represent real numbers) for the commutative property are:

 Arithmetic Property Of Addition Of Multiplication Commutative Property $a + b = b + a$ $a \,x\,b = b\, x\, a$

• The commutative property does NOT work for subtraction nor division. In subtraction and division, the order of numbers DO matter and will change the answer
• Ex. 5 – 2 = 3, but if you switch the order, the answer will change: 2 – 5 = -3
• Ex. 10 ÷ 2 = 5, but if you switch the order, the answer will change: 2 ÷ 10 = 0.2
• Introduction
Introduction to the commutative property of addition and multiplication:
a)
Showing that a + b = b + a

b)
Why is it called the "commutative" property?

c)
Showing that a × b = b × a

d)
The general formulas for the commutative property

• 1.
Commutative property equations
Use the commutative property (for addition and multiplication) to fill in the blanks.
a)
3 + __ = 2 + 3

b)
$\frac{8}{10}$ + $\frac{5}{10}$ + __ = $\frac{5}{10}$ + $\frac{6}{10}$ + $\frac{8}{10}$

c)
0.8 × 0.5 = __ × 0.8

d)
4 × 9 × $a$ = 9 × __ × 4

• 2.
Changing the order to add/multiply lists of numbers
Do the operations in order, and then backwards order. Are the answers different?
a)
0.8 + 0.2 + 0.6 + 0.9 =

b)
15 + 23 + 37 + 44 =

c)
$\frac{2}{5}$ × $\frac{6}{10}$ × $\frac{8}{9}$ =

d)
5 × 3 × 4 × 2 × $g$ =

• 3.
Commutative property of addition word problem
James and Scarlet both need to buy a pair of skates and a helmet. Store A sells the skates for $83.67. Store B sells the helmet for$54.29. James goes to store A first, then store B; Scarlet goes to store B first, then store A. Did they pay different amounts? Explain.

• 4.
Commutative property of multiplication word problem
Felix has 24 pages of homework he must finish in 6 days. If he is going to do the same number of pages each day:
a)
How many pages must he complete each day?

b)
What other schedule(s) can he follow to finish his homework?

• 5.
Which wall will need more paint to cover its area?
• A wall that is 5 $\frac{1}{2}$m tall and 8m wide
• A wall that is 8m tall and 5 $\frac{1}{2}$ m wide