Mathematics · Year 9

Active learning ideas

Factorising by Highest Common Factor

Active learning works for factorising by HCF because students must manipulate and compare expressions directly, making abstract concepts like ‘largest common factor’ tangible. Physically grouping terms and matching factorised forms to their expanded versions helps students see the connection between expansion and factorisation in real time.

• ACARA Content Descriptions
• AC9M9A03

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

Activity 01

• Peer Teaching
• 30 min · Pairs

Card Sort: Expression Matching

Prepare cards with expanded expressions on one set and factored forms on another. Pairs match them, discussing HCF choices. Extend by having pairs create mismatched sets for the class to fix.

Justify why finding the highest common factor is the first step in efficient factorisation.

Facilitation Tip

During Card Sort: Expression Matching, circulate to listen for students’ reasoning about why certain factors are ‘larger’ or more efficient than others.

What to look for

Present students with three algebraic expressions, such as 8a + 12b, 15x^2 - 10x, and 9y^3 + 6y^2. Ask them to write down the HCF for each expression and then factorise it completely.

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

Activity 02

• Peer Teaching
• 40 min · Small Groups

Group Relay: Factorise Chain

Divide class into teams. First student factorises an expression on board, next expands a given factored form, alternating until chain completes. Teams justify HCF steps aloud.

Analyze the relationship between expanding and factorising algebraic expressions.

Facilitation Tip

For Group Relay: Factorise Chain, set a visible timer to create urgency and encourage quick, accurate factorisation under pressure.

What to look for

Pose the question: 'Imagine you have two ways to factorise an expression, one starting with the HCF and one starting with a smaller common factor. Explain why starting with the HCF is always more efficient, using an example like 24x^2y + 36xy^2.'

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

Activity 03

• Peer Teaching
• 35 min · Small Groups

Visual Tiles: HCF Build

Provide algebra tiles for terms. Small groups pull out HCF tiles first, then group remainders. Photograph before-after for portfolios and peer review.

Construct an example where factorisation simplifies a complex problem.

Facilitation Tip

In Visual Tiles: HCF Build, provide coloured tiles for each variable power to help students quickly see which terms share common factors.

What to look for

Give each student an expression that has been factorised, for example, 5(2x + 3). Ask them to expand it and then factorise the resulting expression by its HCF, showing their steps.

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

Activity 04

• Peer Teaching
• 25 min · Individual

Individual Challenge: Simplify Puzzles

Students get worksheets with multi-step expressions needing HCF first. They check work by expanding back, then swap with partners for verification.

Justify why finding the highest common factor is the first step in efficient factorisation.

Facilitation Tip

During Individual Challenge: Simplify Puzzles, give worked examples of incorrect factorisations for students to diagnose and correct.

What to look for

Present students with three algebraic expressions, such as 8a + 12b, 15x^2 - 10x, and 9y^3 + 6y^2. Ask them to write down the HCF for each expression and then factorise it completely.

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

A few notes on teaching this unit

Teachers should model the process of scanning both coefficients and variables systematically, asking students to verbalise their steps. Avoid rushing to the answer—instead, pause to compare multiple approaches, such as factoring out a smaller common factor first versus the HCF. Research suggests that students learn best when they repeatedly test reversibility, so always pair factoring with immediate expansion to confirm correctness.

Successful learning shows when students can identify the HCF with confidence, factorise completely without leaving common factors, and explain why the HCF is the most efficient starting point. They should also justify their process and verify results by expanding the factored form back to the original expression.

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Watch Out for These Misconceptions

• During Card Sort: Expression Matching, watch for students who select any common factor rather than insisting on the highest.

  Ask students to compare their chosen factor with others in their group, prompting them to calculate the largest number and variable power that divides all terms before finalising their choice.

• During Visual Tiles: HCF Build, watch for students who ignore variables and only focus on the numbers in the coefficients.

  Have students physically group tiles by variable power first, then by numerical value, to reinforce that the HCF must include both.

• During Group Relay: Factorise Chain, watch for students who assume the factored form cannot be expanded back to the original expression.

  Instruct groups to assign a teammate to immediately expand their factored result, using a calculator to verify correctness before moving on to the next expression.

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Methods used in this brief

• Peer Teaching