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.
 For addition: the grouping of addends does not change the answer
 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:




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

$(a × b) × c = a × (b × c)$
