# Arithmetic properties: Associative property

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##### Intros
###### Lessons
1. Introduction to the associative property of addition and multiplication:
2. Showing that $(a + b) + c = a + (b + c)$
3. Why is it called the "associative" property?
4. Addition shortcuts using the associative property
5. Showing that (a × b) × c = a × (b × c)
6. Multiplication shortcuts using the associative property
7. The general formulas for the associative property
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##### Examples
###### Lessons
1. Associative Property Equations
Use the associative property (for addition and multiplication) to fill in the blanks.
1. (7 + 2) + 8 = 7 + (2 + __)
2. $\large \frac{1}{4} \, \times \, (\frac{1}{6} \, \times \, \frac{1}{5}) = (\frac{1}{4} \, \times \,$ ______ $\large ) \, \times \, \frac{1}{5}$
3. ($x$ + 3 + 8) + __ = (3 + 9 + x) + 8
2. Changing the grouping to add lists of numbers
1. 0.1 + 0.6 + 0.9 + 0.4 =
2. $\large \frac{2}{10} + \frac{3}{10} + \frac{1}{10} + \frac{4}{10} + \frac{9}{10} + \frac{7}{10} + \frac{6}{10} + \frac{8}{10} =$
3. $\large \frac{2}{3} + \frac{5}{9} + \frac{1}{3} + \frac{4}{9} + \frac{6}{7} =$
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.
1. 20 × 15 × 5
2. 0.8 × 0.9 × 0.5
3. 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!

1. 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
1. 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.
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##### Practice
###### Topic Notes

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 $(a + b) + c = a + (b + c)$ $(a × b) × c = a × (b × c)$