# Arithmetic properties: Associative property

### Arithmetic properties: Associative 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 associative property means that changing the grouping of numbers in an equation does NOT change the answer when you are performing ONLY addition or ONLY multiplication
• The numbers can be any real number (whole numbers, fractions, decimals, integers, etc.)
• To “associate” can mean to interact with a group of people/friends or to group together.
• No matter how you want to group (using brackets) the numbers in an addition or multiplication equation, it will not change the answer in the end.

• Ex. (1 + 2) + 3 = 1 + (2 + 3) will equal 6 either way
• Because (1 + 2) + 3 = (3) + 3 = 6
• As well, 1 + (2 + 3) = 1 + (5) = 6

• The associative property for addition can make shortcuts for adding whole numbers and decimals by making sums of 10 (i.e. 1 + 9, 2 + 8, 3 + 7, 4 + 6, and 5 + 5)
• Ex. 8 + 6 + 2 + 4 + 5 + $x$
• Group as: (8 + 2) + (6 + 4) + 5 + x = (10) + (10) + 5 + $x$ = 25$x$
• Ex. 0.9 + 0.7 + 0.3 + 0.1
• Group as: (0.9 + 0.1) + (0.7 + 0.3) = (1.0) + (1.0) = 2.0

• Shortcuts for adding fractions is also possible with the associative property by making wholes (i.e. same numerator and denominator; $\large \frac{4}{4}, \frac{2}{2},\frac{10}{10}$)
• Ex. $\large \frac{3}{4} + \frac{2}{4} + \frac{1}{4}$
• Group as: $\large (\frac{3}{4} + \frac{1}{4}) + \frac{2} {4} = \frac{4} {4} + \frac{2} {4} = 1 + \frac{2} {4} = 1 \frac{2}{4}$
• Ex. $\large \frac{2}{9} + \frac{2}{5} + \frac{7}{9} + \frac{3}{5} + \frac{1}{4}$
• Group as: $\large (\frac{2}{9} + \frac{7}{9}) + (\frac{2} {5} + \frac{3} {5}) + \frac{1} {4} = (\frac{9} {9}) + (\frac{5}{5}) + \frac{1}{4} = 1 + 1 + \frac{1}{4} = 2 \frac{1}{4}$

• For multiplication: the grouping of factors does not change the answer
• Ex. (2 × 3) × 4 = 2 × (3 × 4) will equal 24 either way
• Because (2 × 3) × 4 = (6) × 4 = 24
• As well, 2 × (3 × 4) = 2 × (12) = 24

• The associative property for multiplication can make shortcuts for multiplying any real numbers by making multiples of 10 (i.e. 10, 20, 30, 40…)
• Ex. 2 × 8 × 5 × $e$
• Group as: (2 × 5) × 8 × $e$ = (10) × 8 × $e$ = 80 × $e$ = 80$e$
• Ex. 0.9 × 0.5 × 0.6
• Group as: (0.5 × 0.6) × 0.9 = (0.30) × 0.9 = 0.270
• Ex. $\large \frac{5}{2}$ × $\frac{9}{13}$ × $\frac{4}{50}$
• Group as: $\large \frac{5 \, x \, 9 \, x \, 4}{2 \, x \, 13 \, x \, 50}$ = $\large \frac{(5 \, x \, 4) \, x \, 9}{(2 \, x \, 50) \, x \, 13}$ = $\large \frac{(20) \, x \, 9 }{(100) \, x \, 13 } = \frac{180}{1300}$

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

 Arithmetic Property Of Addition Of Multiplication Associative Property $(a + b) + c = a + (b + c)$ $(a × b) × c = a × (b × c)$
• Introduction
Introduction to the associative property of addition and multiplication:
a)
Showing that $(a + b) + c = a + (b + c)$

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

c)
Addition shortcuts using the associative property

d)
Showing that (a × b) × c = a × (b × c)

e)
Multiplication shortcuts using the associative property

f)
The general formulas for the associative property

• 1.
Associative Property Equations
Use the associative property (for addition and multiplication) to fill in the blanks.
a)
(7 + 2) + 8 = 7 + (2 + __)

b)
$\large \frac{1}{4} \, \times \, (\frac{1}{6} \, \times \, \frac{1}{5}) = (\frac{1}{4} \, \times \,$ ______ $\large ) \, \times \, \frac{1}{5}$

c)
($x$ + 3 + 8) + __ = (3 + 9 + x) + 8

• 2.
Changing the grouping to add lists of numbers
a)
0.1 + 0.6 + 0.9 + 0.4 =

b)
$\large \frac{2}{10} + \frac{3}{10} + \frac{1}{10} + \frac{4}{10} + \frac{9}{10} + \frac{7}{10} + \frac{6}{10} + \frac{8}{10} =$

c)
$\large \frac{2}{3} + \frac{5}{9} + \frac{1}{3} + \frac{4}{9} + \frac{6}{7} =$

• 3.
Changing the grouping to multiply lists of numbers
Decide how to group the factors as a shortcut for multiplication. Double check your answer by multiplying without groups.
a)
20 × 15 × 5

b)
0.8 × 0.9 × 0.5

c)
6 × 20 × 5 × 2

• 4.
Associative property of addition word problem
Ryan added these numbers together and his answer is correct. Show another way of adding numbers (with grouping) using the associative property!

• 5.
Associative property of multiplication word problem
Explain which choice is NOT an equal statement to: (6 × 8) × 5
1. 6 × 40
2. 48 × 5
3. 6 × (8 × 5)
4. 14 × 5

• 6.
Associative property and volume
The formula for the volume of a rectangular prism is given by:

Volume = length × width × height

Use the associative property of multiplication to show 2 ways to solve for this prism.