Factorisation by Taking Out Common Factors

Active learning helps students internalize the mechanics of factorisation by letting them manipulate concrete pieces of expressions. When they physically sort, move, and rewrite terms, the abstract process of identifying and extracting GCFs becomes visible and memorable. This kinesthetic and collaborative approach builds fluency faster than passive worksheets alone.

• Secondary 2
• Mathematics
• 4 activities
• 25 min–40 min

Learning Objectives

1. Identify the greatest common factor (GCF) of terms within algebraic expressions.
2. Calculate the GCF for expressions involving numerical coefficients, variables, and exponents.
3. Factor algebraic expressions by extracting the GCF, demonstrating the reversal of expansion.
4. Explain the relationship between expanding and factorizing algebraic expressions using examples.

Stations Rotation

⏱ 30 min·Pairs

Card Sort: Expanded to Factored Pairs

Prepare cards with expanded expressions on one set and factored forms on another. Pairs match them, such as 4x + 8 with 4(x + 2), then justify their GCF choice. Extend by having pairs create their own mismatched pairs for the class to sort.

Prepare & details

Explain the relationship between expansion and factorisation.

Facilitation Tip: During Card Sort, circulate and ask each pair to explain their pairing of expanded and factored forms before moving on.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

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

Stations Rotation

⏱ 25 min·Small Groups

Group Factor Relay

Divide class into teams. Each student factors one term from a multi-term expression on the board, passes to the next teammate for the next term, until the full GCF is extracted. Teams race while verifying each step aloud.

Prepare & details

Analyze how to identify the greatest common factor in an algebraic expression.

Facilitation Tip: For Group Factor Relay, set a strict 2-minute timer per station to keep energy high and prevent over-analysis.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

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

Stations Rotation

⏱ 40 min·Small Groups

Scavenger Hunt: GCF Stations

Set up stations with expressions on posters. Small groups visit each, factor on mini-whiteboards, and leave evidence of GCF identification. Rotate stations, then gallery walk to peer-review solutions.

Prepare & details

Justify why factorisation is a fundamental tool for simplifying expressions.

Facilitation Tip: At GCF Stations, provide only one answer key per group to encourage discussion and consensus-building.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

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

Stations Rotation

⏱ 35 min·Pairs

Peer Challenge Circuits

Students work individually on progressive expression sheets, then pair up to check and factor partner's work, discussing errors. Circulate to provide prompts before swapping partners.

Prepare & details

Explain the relationship between expansion and factorisation.

Facilitation Tip: During Peer Challenge Circuits, assign roles: factorer, checker, and presenter to ensure accountability.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

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

Teaching This Topic

Start with simple numerical pairs to establish the concept of common factors before introducing variables. Use color-coding on a whiteboard to highlight shared factors, which research shows improves GCF identification. Avoid rushing to abstract symbols; allow students to articulate their steps in full sentences. Emphasize that factorisation is a reversible operation, linking it back to expansion to reinforce the concept’s depth.

What to Expect

Students should confidently identify the GCF in any two-term expression and rewrite it in fully factorised form. They should explain their steps to peers and recognize when an expression cannot be simplified further. Successful learning is evident when students self-correct missteps during group work and justify their choices with clear reasoning.

Watch Out for These Misconceptions

Common Misconception

During Card Sort, watch for students who pair 6x + 9xy with 3(2x + 3xy) instead of 3x(2 + 3y).

What to Teach Instead

Prompt them to check if 3 can be factored out again from 2x + 3xy, guiding them to realize they need the greatest common factor, not just any common factor.

Common Misconception

During Group Factor Relay, watch for students who ignore the variable part entirely in expressions like x^2 + x^3.

What to Teach Instead

Ask them to write out the expanded form of x^2 + x^3 as xx + xxx and circle the common x terms, reinforcing that the GCF includes the lowest power of the variable.

Common Misconception

During Scavenger Hunt, watch for students who separate numbers and variables, factoring 2x + 4y as 2(x) + 4(y) instead of finding a shared factor.

What to Teach Instead

Have them list all common factors together, like 1, 2, x, or y, and guide them to see that only 2 is common to both terms, leading to 2(x + 2y).

Assessment Ideas

Quick Check

After Card Sort, present students with expressions like 4a + 8b and 15x^2 - 10x. Ask them to write down the GCF for each expression and then factorise the expression completely. Review their answers for accuracy in GCF identification and correct factorisation.

Exit Ticket

During Peer Challenge Circuits, give students the expression 9p^2q + 12pq^2. Ask them to: 1. List all common factors of the terms. 2. State the GCF. 3. Write the expression in factorised form. Collect these to gauge individual understanding of the process.

Discussion Prompt

After Scavenger Hunt, pose the question: 'How is factorising 10x + 15 like finding the common ingredients in a recipe, and how is it different?' Facilitate a class discussion where students compare the algebraic process to a real-world analogy, focusing on identifying shared components and separating them.

Extensions & Scaffolding

• Challenge: Ask students to create their own expression where the GCF includes both numbers and variables, then trade with a partner to factorise it.
• Scaffolding: Provide expressions with missing terms, such as 4x + ____, where students must determine the missing term that makes factorisation possible.
• Deeper: Introduce expressions with negative coefficients or fractions, like 0.5y - 1.5xy, to test their understanding of GCF extraction beyond whole numbers.

Key Vocabulary

Factor	A number or algebraic expression that divides another number or expression evenly. For example, 3 and x are factors of 6x.
Common Factor	A factor that two or more numbers or expressions share. For example, 3 is a common factor of 6 and 9.
Greatest Common Factor (GCF)	The largest factor that two or more numbers or expressions have in common. For example, the GCF of 12x and 18y is 6.
Factorisation	The process of expressing an algebraic expression as a product of its factors. This is the reverse of expansion.