Mathematics · Year 9

Active learning ideas

Factorising by Grouping and Special Products

Active learning works here because factorising by grouping and special products rely on pattern recognition and spatial reasoning. Students need to see how terms rearrange and match, not just follow steps. Hands-on sorting, building, and racing let them experience those moments of insight directly.

• ACARA Content Descriptions
• AC9M9A03

• 25–40 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Collaborative Problem-Solving
• 35 min · Small Groups

Card Sort: Grouping Matches

Prepare cards with unfactored four-term expressions, partially grouped steps, and final factors. In small groups, students sort and sequence them correctly, then justify choices on mini-whiteboards. Extend by having groups create one new set for another group.

Explain when factorising by grouping is an appropriate strategy.

Facilitation Tip

For Card Sort: Grouping Matches, circulate and ask each pair to explain why they paired two expressions, focusing on the shared binomial rather than isolated numbers.

What to look for

Present students with the expression 6x + 10y + 9xz + 15yz. Ask them to identify the common binomial factor after grouping and write the fully factorised form. Observe their grouping strategy and accuracy.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Activity 02

• Collaborative Problem-Solving
• 40 min · Pairs

Algebra Tiles: Special Products

Provide algebra tiles for students to build difference of squares and perfect square expansions in pairs. They photograph arrangements, factor back, and compare with peers. Discuss how tiles reveal the patterns visually.

Compare the process of factorising a difference of two squares with other methods.

Facilitation Tip

For Algebra Tiles: Special Products, model how to rotate tiles to match the binomial side before students build their own pairs. Pause to ask what the middle term represents.

What to look for

On a slip of paper, ask students to write one expression that is a difference of two squares and its factorised form. Then, ask them to write one expression that is a perfect square trinomial and its factorised form.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Activity 03

• Collaborative Problem-Solving
• 30 min · Whole Class

Relay Race: Factor Design

Divide class into teams. Each student runs to board, factors a given expression or designs one using grouping or special products, tags next teammate. Teams verify all steps collaboratively at end.

Design an expression that can be factorised using the grouping method.

Facilitation Tip

For Relay Race: Factor Design, write the next expression only after the previous group has shown their fully factorised form to prevent skipping steps.

What to look for

Pose the question: 'When might factorising by grouping be a less efficient strategy than other factorisation methods you know?' Facilitate a class discussion where students compare scenarios and justify their reasoning.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Activity 04

• Collaborative Problem-Solving
• 25 min · Pairs

Partner Proof: Recognition Hunt

Pairs hunt for special products in a list of quadratics, factor them, and prove by expanding. Switch roles, then share strongest examples with class via gallery walk.

Explain when factorising by grouping is an appropriate strategy.

Facilitation Tip

For Partner Proof: Recognition Hunt, set a timer and instruct partners to find three correct matches before they justify their choices to another pair.

What to look for

Present students with the expression 6x + 10y + 9xz + 15yz. Ask them to identify the common binomial factor after grouping and write the fully factorised form. Observe their grouping strategy and accuracy.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

A few notes on teaching this unit

Teach by letting students feel the difference between success and failure. Start with expressions that look similar but require different strategies, so they learn to pause and choose. Use algebra tiles to ground abstract signs in concrete shapes, especially for perfect squares where the sign of the middle term matters. Research shows that students who manipulate physical models before symbolic work make fewer sign errors later.

Successful learning shows when students can confidently group terms, identify common binomials, and apply special product rules without hesitation. They explain their steps aloud and check their work by expanding back. Groups debate until all agree on the correct factorisation.

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Watch Out for These Misconceptions

• During Card Sort: Grouping Matches, watch for students who match pairs solely by number rather than by shared binomial factors.

  Prompt them to verbalise what the common factor is inside each pair and to write it on the back of the card before gluing.

• During Algebra Tiles: Special Products, watch for students who believe (a - b)² = a² - b².

  Have them build (a - b)(a + b) and (a - b)² side by side, then count the number of a-tiles in the middle term to correct the sign.

• During Partner Proof: Recognition Hunt, watch for students who treat all perfect squares as having positive middle terms.

  Ask them to sort the cards into two piles: those with positive middle terms and those with negative ones, then justify the rule aloud.

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Methods used in this brief

• Collaborative Problem-Solving