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Properties of Addition and Multiplication
Students use the commutative, associative, identity, zero, and distributive properties to describe number patterns and rewrite expressions without changing their value.
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Beginner
15–20 minutes
Algebraic Expressions
Algebraic Expressions
Page 1 of 5
Inside Cover
Properties of Addition and Multiplication
At a Glance
Binder SectionAlgebraic Expressions
DifficultyBeginner
Estimated Time15–20 minutes
PrerequisitesNone
How can properties make calculations and algebraic expressions easier to understand?
Build Understanding
Properties are dependable rules about how numbers behave. The commutative property changes order: a + b = b + a and ab = ba. For example, 7 + 3 = 3 + 7 and 4 × 6 = 6 × 4. Subtraction and division are not commutative.
The associative property changes grouping when adding or multiplying: (a + b) + c = a + (b + c) and (ab)c = a(bc). For example, (2 + 8) + 5 = 2 + (8 + 5). It does not change the order of the numbers.
Identity properties leave a number unchanged. The additive identity is 0 because a + 0 = a. The multiplicative identity is 1 because a × 1 = a. The zero property of multiplication states that a × 0 = 0.
The distributive property connects multiplication with addition or subtraction: a(b + c) = ab + ac. For 6(10 + 2), multiply 6 by both terms: 60 + 12 = 72. In reverse, 18x + 12 can be factored as 6(3x + 2).
Worked example: Rewrite 4(x + 7). Multiply 4 by x and by 7, giving 4x + 28. A common error is multiplying only the first term. Every term inside the grouping symbols must be multiplied.
Visual Model
4Thousands
3Hundreds
2Tens
7Ones
4,327 = 4,000 + 300 + 20 + 7
Goals
Learning Targets
- Commutative means change the order.
- Associative means change the grouping.
- Adding 0 and multiplying by 1 leave a value unchanged.
- Any number multiplied by 0 equals 0.
- Distribute a factor to every term inside the grouping symbols.
- Subtraction and division are not commutative or associative.
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Interactive Vocabulary
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Reveal one step at a time
Worked Examples
Example 1
Mental computation
1
25+37+75
2
Reorder and regroup as (25+75)+37.
3
137.
Example 2
Distribute
1
6(x+4)
2
Multiply 6 by both terms.
3
6x+24.
Ask and explain
Common Questions
What is the difference between commutative and associative? Commutative changes order; associative changes grouping.
Can I use the commutative property with subtraction? No. Changing the order generally changes the answer.
Why is 1 the multiplicative identity? Multiplying any number by 1 keeps the original number.
What does distribute mean? Multiply the outside factor by every term inside the grouping symbols.
Can properties include variables? Yes. Properties apply to numerical and algebraic expressions.
Notice and correct
Common Misconceptions
Using commutative property with subtraction
Subtraction is not commutative; rewrite it as adding an opposite.
Distributing to only one term
Multiply the outside factor by every term inside the grouping symbols.
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Warm-Up
Guided
Independent
Challenge
1
Quick Start
Warm-Up
Name the property: 8+13=13+8.
2
We Do
Guided Practice
Name the property: 9 + 4 = 4 + 9. Answer: Commutative property of addition.
Name the property: (2 × 5) × 3 = 2 × (5 × 3). Answer: Associative property of multiplication.
Complete: 38 + ___ = 38. Answer: 0.
Complete: 17 × ___ = 17. Answer: 1.
Expand 5(x + 6). Answer: 5x + 30.
Expand 3(2y − 7). Answer: 6y − 21.
Factor 12x + 18 using the greatest common factor. Answer: 6(2x + 3).
3
You Do
Independent Practice
Name the property: (2×5)×7=2×(5×7).
Rewrite 9x+9y.
4
Stretch Your Thinking
Challenge & Real-World Practice
Create two different expressions that both equal 96. Use at least two properties in each rewrite and justify every step.
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IXL
Properties of Addition
Open resourceProperties of Multiplication
Open resourceDistributive Property
Open resourceKhan Academy
Properties of Addition
Open resourceProperties of Multiplication
Open resourceDistributive Property
Open resourceDeltaMath
DeltaMath student login—ask your teacher for an assigned properties activity
Open resource
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Before you begin, I can…
- I can distinguish a change in order from a change in grouping.
- I can distribute and factor correctly.
- I can name the property that justifies each rewrite.
1
Question 1
Which property shows 4+9=9+4?
2
Question 2
Which equals 5(x+3)?
3
Question 3
What is 17+0?
4
Question 4
Which property changes grouping?
5
Question 5
Factor 8x+16.
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