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.

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

Learning Objectives

1. Analyze given algebraic expressions to identify common factors and terms.
2. Compare two algebraic expressions to determine if they are equivalent using properties of operations.
3. Generate equivalent algebraic expressions for a given expression by applying the distributive, commutative, and associative properties.
4. Explain how the properties of operations ensure that two different-looking algebraic expressions represent the same quantity for any value of the variable.
5. Evaluate the equivalence of two algebraic expressions by constructing a simplified form for each.

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.

Prepare & details

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.

Setup: Large papers on tables or walls, space to circulate

Materials: Large paper with central prompt, Markers (one per student), Quiet music (optional)

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

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.

Prepare & details

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.

Setup: Large papers on tables or walls, space to circulate

Materials: Large paper with central prompt, Markers (one per student), Quiet music (optional)

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

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.

Prepare & details

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.

Setup: Large papers on tables or walls, space to circulate

Materials: Large paper with central prompt, Markers (one per student), Quiet music (optional)

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

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.

Prepare & details

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.

Setup: Large papers on tables or walls, space to circulate

Materials: Large paper with central prompt, Markers (one per student), Quiet music (optional)

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

Teaching This Topic

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.

What to Expect

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.

Watch Out for These Misconceptions

Common Misconception

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

What to Teach Instead

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.

Common Misconception

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

What to Teach Instead

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.

Common Misconception

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

What to Teach Instead

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.

Assessment Ideas

Exit Ticket

After the Card Sort activity, provide students with two expressions, such as 3(4 + x) and 12 + 3x. 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.

Quick Check

After the Properties Scavenger Hunt, 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.

Discussion Prompt

During the Algebra Tiles activity, pose the question: 'Can we always be sure two expressions are equivalent just by testing numbers?' Guide students to discuss how properties of operations provide a reliable method for justification, rather than relying on a few examples.

Extensions & Scaffolding

• Challenge: Ask students to create their own set of three equivalent expressions and write a step-by-step justification using properties.
• Scaffolding: Provide a partially completed expression tree or diagram for students to finish, highlighting the property used at each step.
• Deeper exploration: Introduce expressions with exponents, such as 2(x + 3)^2 and 2x^2 + 12x + 18, and ask students to verify equivalence using area models or expanded forms.

Key Vocabulary

Equivalent Expressions	Expressions that have the same value for all possible values of the variable(s). For example, 2x + 3 and 3 + 2x are equivalent.
Distributive Property	A property that states multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. For example, a(b + c) = ab + ac.
Commutative Property	A property that states the order of numbers in an operation does not change the result. For addition: a + b = b + a. For multiplication: a × b = b × a.
Associative Property	A property that states the way in which numbers are grouped in an operation does not change the result. For addition: (a + b) + c = a + (b + c). For multiplication: (a × b) × c = a × (b × c).
Algebraic Expression	A mathematical phrase that can contain numbers, variables, and operation symbols. For example, 5y + 2.