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Arithmetic properties: Associative property
- Intro Lesson: a4:11
- Intro Lesson: b2:16
- Intro Lesson: c8:05
- Intro Lesson: d3:51
- Intro Lesson: e7:58
- Intro Lesson: f1:17
- Lesson: 1a1:56
- Lesson: 1b2:52
- Lesson: 1c2:36
- Lesson: 2a2:51
- Lesson: 2b3:52
- Lesson: 2c6:27
- Lesson: 3a2:12
- Lesson: 3b3:48
- Lesson: 3c4:45
- Lesson: 44:17
- Lesson: 53:28
- Lesson: 64:02
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.
- 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 = 25x
- 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; 44,22,1010)
- Ex. 43+42+41
- Group as: (43+41)+42=44+42=1+42=142
- Ex. 92+52+97+53+41
- Group as: (92+97)+(52+53)+41=(99)+(55)+41=1+1+41=241
- 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 = 80e
- Ex. 0.9 × 0.5 × 0.6
- Group as: (0.5 × 0.6) × 0.9 = (0.30) × 0.9 = 0.270
- Ex. 25 × 139 × 504
- Group as: 2x13x505x9x4 = (2x50)x13(5x4)x9 = (100)x13(20)x9=1300180
- The general formulas (where a, b and c are variables that represent real numbers) for the associative property are:
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- IntroductionIntroduction 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 propertyd)Showing that (a × b) × c = a × (b × c)e)Multiplication shortcuts using the associative propertyf)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)41×(61×51)=(41× ______ )×51c)(x + 3 + 8) + __ = (3 + 9 + x) + 8 - 2.Changing the grouping to add lists of numbers
Decide how to group the addends as a shortcut for addition. Double check your answer by adding without groups.a)0.1 + 0.6 + 0.9 + 0.4 =b)102+103+101+104+109+107+106+108=c)32+95+31+94+76= - 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 × 5b)0.8 × 0.9 × 0.5c)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- 6 × 40
- 48 × 5
- 6 × (8 × 5)
- 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.