Mathematics · Grade 6

Active learning ideas

Identifying Equivalent Expressions

Active learning works for identifying equivalent expressions because students need to physically manipulate symbols and terms to see how operations transform expressions without changing their value. Concrete models and collaborative tasks help bridge the gap between abstract properties and students' intuitive understanding of equality.

• Ontario Curriculum Expectations
• 6.EE.A.3
• 6.EE.A.4

• 30–45 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Chalk Talk
• 35 min · Small Groups

Card Sort: Equivalent Expression Pairs

Print cards with expressions like 2(x + 4), 2x + 8, and 4 + 2x. Students in small groups sort into equivalent pairs, then justify using properties. Regroup to share one justification per group.

Explain how different looking expressions can be mathematically equivalent.

Facilitation Tip

For the Card Sort, circulate and listen for students explaining their matches using properties before confirming correctness, reinforcing verbal justifications.

What to look for

Provide students with two expressions, such as 4(x + 2) and 4x + 8. Ask them to write one sentence explaining why these expressions are equivalent and to show one step using a property of operations to prove their answer.

• Understand
• Analyze
• Evaluate
• Self-Awareness
• Self-Management

Activity 02

• Chalk Talk
• 40 min · Pairs

Algebra Tiles: Expand and Match

Provide algebra tiles for expressions. Pairs build models for given expressions, expand using distributive property, and match to equivalent written forms. Record properties used in journals.

Justify that two expressions are equivalent without testing every possible number.

Facilitation Tip

During the Algebra Tiles activity, explicitly connect the physical rearrangement of tiles to the written steps of expansion or factoring.

What to look for

Display a list of expressions on the board. Ask students to write down any pairs of expressions they believe are equivalent and to briefly state which property (distributive, commutative, or associative) could be used to show their equivalence.

• Understand
• Analyze
• Evaluate
• Self-Awareness
• Self-Management

Activity 03

• Chalk Talk
• 30 min · Small Groups

Expression Relay: Generate Equivalents

Divide class into teams. One student writes an expression, next generates an equivalent using a property, passes to teammate. First team to chain five correctly wins; debrief properties.

Construct an equivalent expression for a given algebraic expression.

Facilitation Tip

In the Expression Relay, circulate and ask guiding questions like 'Which property could you use to rewrite this next expression?' to keep students focused on the task's purpose.

What to look for

Pose the question: 'Can we always be sure two expressions are equivalent just by looking at them?' Guide students to discuss how properties of operations provide a reliable method for justification, rather than just testing a few numbers.

• Understand
• Analyze
• Evaluate
• Self-Awareness
• Self-Management

Activity 04

• Chalk Talk
• 45 min · Pairs

Properties Scavenger Hunt

Post expressions around room. Individuals or pairs find and rewrite three equivalents using different properties, photographing evidence. Whole class verifies with digital gallery walk.

Explain how different looking expressions can be mathematically equivalent.

Facilitation Tip

For the Properties Scavenger Hunt, assign each pair a different property to locate, ensuring all students engage with each one during sharing.

What to look for

Provide students with two expressions, such as 4(x + 2) and 4x + 8. Ask them to write one sentence explaining why these expressions are equivalent and to show one step using a property of operations to prove their answer.

• Understand
• Analyze
• Evaluate
• Self-Awareness
• Self-Management

A few notes on teaching this unit

Teach this topic by pairing concrete models with abstract reasoning, ensuring students see properties as tools rather than rules. Avoid rushing to symbolic representations; let students discover patterns through hands-on work and collaborative explanations. Research suggests that students who manipulate physical or visual models develop stronger justifications for equivalence than those who rely solely on symbolic manipulation or substitution.

Successful learning looks like students justifying equivalence using properties rather than testing values, pairing expressions confidently during card sorts, and explaining their reasoning with precise mathematical language. They should demonstrate flexibility in rewriting expressions in both directions and recognize patterns in equivalent forms.

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Watch Out for These Misconceptions

• During the Card Sort activity, watch for students who match expressions based solely on appearance without considering properties of operations.

  Prompt students to use their cards to test the property of their choice by rearranging terms or distributing, then ask them to explain how the property ensures the expressions are equivalent for all inputs.

• During the Algebra Tiles activity, watch for students who only expand expressions and do not practice factoring to create equivalent forms.

  After expanding with tiles, ask students to reverse the process and factor the expanded expression back into its original form, using the tiles as a guide.

• During the Expression Relay activity, watch for students who generate equivalents by testing numbers rather than using properties.

  Circulate and ask each pair to explain which property they used to rewrite their expression, and request they demonstrate the step with a written example on their paper.

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Methods used in this brief

• Chalk Talk