Factorisation by Grouping

Active learning helps students practice pattern recognition and strategic thinking, both essential for factorisation by grouping. When students sort, rearrange, and manipulate terms with their hands or collaboratively, they move beyond abstract rules to concrete understanding of how terms interact.

• Secondary 3
• Mathematics
• 4 activities
• 25 min–40 min

Learning Objectives

1. Identify pairs of terms within a four-term expression that share common factors.
2. Apply the distributive property in reverse to factorize four-term algebraic expressions by grouping.
3. Analyze the structure of algebraic expressions to determine the most efficient grouping strategy.
4. Justify the sequence of steps taken to factorize a four-term expression using grouping.
5. Create equivalent factorized forms of four-term expressions, verifying accuracy through expansion.

Collaborative Problem-Solving

⏱ 35 min·Pairs

Card Sort: Identifying Groupable Expressions

Create cards with 20 four-term expressions, half factorable by grouping and half not. In pairs, students sort into 'yes' and 'no' piles, justify choices, then factor the 'yes' ones. Follow with whole-class sharing of tricky cases.

Prepare & details

Analyze how grouping terms reveals common binomial factors.

Facilitation Tip: In Think-Pair-Share, provide sentence starters like 'I chose this grouping because...' to scaffold justification before students discuss in pairs.

Setup: Groups at tables with problem materials

Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric

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

Collaborative Problem-Solving

⏱ 40 min·Small Groups

Relay Race: Step-by-Step Factorisation

Divide into small groups and line up. Provide a four-term expression; the first student groups and factors one pair on paper, passes to the next for the second pair and final factor, until complete. Time teams and review expansions.

Prepare & details

Predict when factorisation by grouping might be the most effective strategy.

Setup: Groups at tables with problem materials

Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric

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

Collaborative Problem-Solving

⏱ 30 min·Pairs

Algebra Tiles Grouping Challenge

Use algebra tiles to represent four-term expressions on mats. Students in pairs physically group tiles into pairs, remove common factors, and photograph before writing algebraic steps. Extend by creating their own expressions.

Prepare & details

Justify the steps involved in factorising a four-term expression by grouping.

Setup: Groups at tables with problem materials

Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric

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

Think-Pair-Share

⏱ 25 min·Whole Class

Think-Pair-Share: Strategy Prediction

Pose expressions on board; students think individually when grouping works best, pair to predict and factor, then share justifications with class. Vote on most effective strategies.

Prepare & details

Analyze how grouping terms reveals common binomial factors.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Teaching This Topic

Start with expressions that clearly group, then introduce ones requiring rearrangement to build resilience. Avoid rushing to the final answer; instead, model confusion and problem-solving by trying pairs aloud. Research shows students learn factorisation more deeply when they experience failed attempts and correct them.

What to Expect

By the end of these activities, students should confidently pair terms, extract common binomial factors, and verify results by expansion. They should also demonstrate flexibility by rearranging terms and justifying their grouping choices during discussions.

Watch Out for These Misconceptions

Common Misconception

During Card Sort, watch for students who group terms based solely on numerical coefficients without considering binomial factors.

What to Teach Instead

Prompt students to verbalize the binomial factor they extract from each pair, returning to the card layout if they cannot name it.

Common Misconception

During Relay Race, watch for students who insist the original order of terms cannot change.

What to Teach Instead

Have students physically swap the order of terms on the relay sheet and re-attempt grouping, discussing why the new order works better.

Common Misconception

During Algebra Tiles Grouping Challenge, watch for students who stop factoring after the first extraction.

What to Teach Instead

Ask students to rebuild the tiles with their final answer to check for further common factors, using the visual model to confirm completeness.

Assessment Ideas

Quick Check

After Card Sort, present the expression 8mn + 12m + 6n + 9. Ask students to write down two possible groupings on mini-whiteboards and hold them up for immediate feedback.

Exit Ticket

After Relay Race, give students the expression 5pq + 10q + 3pr + 6r and ask them to factorise completely, writing one sentence about why their grouping strategy was effective.

Discussion Prompt

During Think-Pair-Share, pose the expression 4x^2 + 8x + 3x + 6. Ask pairs to discuss whether more than one grouping is possible, then share findings with the class to address flexibility in grouping.

Extensions & Scaffolding

• Challenge early finishers to create their own four-term expression that can be factorised by grouping in two different ways and explain why the results are equivalent.
• Scaffolding for struggling students: Provide partially completed groupings on a worksheet so they only need to fill in the missing steps or common factors.
• Deeper exploration: Ask students to design a poster explaining the steps of factorisation by grouping, including a section on when and why rearranging terms is necessary.

Key Vocabulary

Common Binomial Factor	A binomial expression that is a factor of two or more terms within a larger expression. In factorisation by grouping, it's the factor that emerges after grouping terms.
Grouping	The process of pairing terms within a four-term expression, typically two pairs, to identify and extract common factors from each pair.
Distributive Property (Reverse)	The principle applied in factorisation where a common factor is 'distributed' out of terms. For example, ax + ay = a(x + y).
Algebraic Expression	A mathematical phrase that can contain variables, constants, and operation signs. This topic focuses on expressions with four terms.