Properties of Operations: Commutative and Associative

Active learning works well for this topic because students need to see and feel how the commutative and associative properties change expressions without altering their value. Hands-on manipulation and movement reinforce abstract concepts, helping learners internalize the rules through repeated practice and peer discussion.

• Grade 6
• Mathematics
• 4 activities
• 20 min–45 min

Learning Objectives

1. Differentiate between the commutative and associative properties for addition and multiplication.
2. Apply the commutative and associative properties to simplify given algebraic expressions.
3. Construct an algebraic expression and demonstrate how applying commutative and associative properties simplifies it.
4. Analyze why subtraction and division do not follow the commutative and associative properties.

Peer Teaching

⏱ 25 min·Pairs

Manipulative Match: Commutative Pairs

Provide linking cubes or tiles labeled with numbers and variables. Pairs build expressions like 2x + 3 + x, then swap orders to verify equality. They record three simplified versions and share one non-commutative example with subtraction.

Prepare & details

Explain why some operations follow the commutative property while others do not.

Facilitation Tip: During Manipulative Match: Commutative Pairs, provide algebra tiles or counters so students physically rearrange terms to see that a + b equals b + a.

Setup: Presentation area at front, or multiple teaching stations

Materials: Topic assignment cards, Lesson planning template, Peer feedback form, Visual aid supplies

• Understand
• Apply
• Analyze
• Create
• Self-Management
• Relationship Skills

Peer Teaching

⏱ 30 min·Small Groups

Relay Race: Associative Grouping

Divide class into teams. Each student simplifies one part of a chain expression like ((5 + 2) + x) + 3 by regrouping associatively, passes to next teammate. First team to fully simplify wins; discuss results whole class.

Prepare & details

Differentiate between the commutative and associative properties.

Facilitation Tip: For Relay Race: Associative Grouping, set a timer and have teams physically move cubes or number cards to model regrouping before writing expressions.

Setup: Presentation area at front, or multiple teaching stations

Materials: Topic assignment cards, Lesson planning template, Peer feedback form, Visual aid supplies

• Understand
• Apply
• Analyze
• Create
• Self-Management
• Relationship Skills

Stations Rotation

⏱ 45 min·Small Groups

Stations Rotation: Property Sorts

Set up stations: commutative sort (match reordered pairs), associative regroup (rewrite with parentheses), mixed exceptions (identify subtraction/division). Groups rotate every 10 minutes, sorting cards and justifying choices on charts.

Prepare & details

Construct an example demonstrating how these properties can simplify an expression.

Facilitation Tip: In Station Rotation: Property Sorts, place examples and non-examples at each station so students categorize them by property, using color-coding or sticky notes for clarity.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Peer Teaching

⏱ 20 min·Pairs

Expression Builder: Partner Challenge

Partners draw variable cards and create expressions. One simplifies using properties; other checks with substitution. Switch roles, then combine to build a class anchor chart of examples.

Prepare & details

Explain why some operations follow the commutative property while others do not.

Setup: Presentation area at front, or multiple teaching stations

Materials: Topic assignment cards, Lesson planning template, Peer feedback form, Visual aid supplies

• Understand
• Apply
• Analyze
• Create
• Self-Management
• Relationship Skills

Teaching This Topic

Start with concrete examples using manipulatives to build intuition before moving to symbols. Avoid rushing to the abstract; let students discover the properties through guided exploration. Research suggests that pairing verbal explanations with physical actions strengthens retention, so encourage students to describe their moves as they work.

What to Expect

Successful learning looks like students confidently identifying when properties apply, using them to simplify expressions accurately, and explaining their reasoning clearly. They should also recognize when a property does not apply and justify their thinking with examples or counterexamples.

Watch Out for These Misconceptions

Common Misconception

During Manipulative Match: Commutative Pairs, watch for students who assume subtraction and division work the same way as addition and multiplication.

What to Teach Instead

Have pairs model 5 - 2 and 2 - 5 on a number line or with counters, then compare results. Ask them to explain why the outcomes differ and how this shows the commutative property does not apply.

Common Misconception

During Station Rotation: Property Sorts, watch for students who confuse commutative with associative properties.

What to Teach Instead

After sorting, ask students to present one example from each category and explain how the order or grouping changes in their own words. Peers can ask questions to clarify distinctions.

Common Misconception

During Relay Race: Associative Grouping, watch for students who apply the associative property to subtraction or division.

What to Teach Instead

Give teams two subtraction expressions to model with cubes, such as (7 - 3) - 2 and 7 - (3 - 2). Have them compare results and discuss why grouping matters differently in this operation.

Assessment Ideas

Quick Check

After Station Rotation: Property Sorts, provide students with several expressions and ask them to circle those simplified correctly using commutative or associative properties. Have them explain two examples in writing.

Exit Ticket

After Manipulative Match: Commutative Pairs, give each student a card with an operation (addition, subtraction, multiplication, division) and two variables. Ask them to write whether the operation is commutative and associative, with a brief justification.

Discussion Prompt

During Expression Builder: Partner Challenge, pose a problem like 5 + 2x + 7 + 3x. Ask partners to share how they would rearrange and simplify it using properties, then invite a few pairs to demonstrate their steps to the class.

Extensions & Scaffolding

• Challenge: Give students mixed operations (e.g., 5 + 2x - 3 + x) and ask them to simplify using properties where possible, explaining why they cannot apply both properties to subtraction.
• Scaffolding: Provide partially completed examples with blanks to fill in, such as (4 + ___) + 3 = 4 + (__ + 3), guiding students to see the regrouping.
• Deeper: Have students create their own word problems where commutative or associative properties simplify the solution, then trade with peers to solve.

Key Vocabulary

Commutative Property	A property stating that the order of operands does not change the outcome of an operation. For example, a + b = b + a and a × b = b × a.
Associative Property	A property stating that the grouping of operands does not change the outcome of an operation. For example, (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c).
Algebraic Expression	A mathematical phrase that can contain variables, numbers, and operation symbols. For example, 3x + 5.
Simplify	To rewrite an expression in a more concise form, often by combining like terms or performing indicated operations.