Mathematics · Grade 6

Active learning ideas

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.

• Ontario Curriculum Expectations
• 6.EE.A.3

• 20–45 min
• Pairs → Whole Class
• 4 activities

Activity 01

• 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.

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.

What to look for

Present students with several expressions, some simplified using commutative/associative properties and others not. Ask them to circle the expressions that have been correctly simplified and briefly explain why for two examples.

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

Activity 02

• 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.

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.

What to look for

Give each student a card with a different operation (e.g., addition, subtraction, multiplication, division) and two variables. Ask them to write one sentence stating whether the operation is commutative and associative, providing a brief justification.

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

Activity 03

• 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.

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.

What to look for

Pose the question: 'How can using the commutative and associative properties save time when solving a problem like 5 + 2x + 7 + 3x?' Facilitate a brief class discussion where students share their strategies and demonstrate how they would rearrange and combine terms.

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

Activity 04

• 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.

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

What to look for

Present students with several expressions, some simplified using commutative/associative properties and others not. Ask them to circle the expressions that have been correctly simplified and briefly explain why for two examples.

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

A few notes on teaching this unit

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.

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.

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Watch Out for These Misconceptions

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

  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.

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

  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.

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

  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.

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Methods used in this brief

• Peer Teaching
• Stations Rotation