Factorising into Single Brackets

Active learning works for factorising into single brackets because students must physically manipulate terms, compare coefficients and variables, and correct mistakes in real time. This hands-on approach builds fluency in identifying the highest common factor (HCF) while turning abstract rules into concrete patterns they can see and discuss.

• Year 8
• Mathematics
• 4 activities
• 25 min–40 min

Learning Objectives

1. Identify the highest common factor (HCF) in algebraic expressions containing integers and variables.
2. Factorise algebraic expressions into a single bracket by extracting the HCF.
3. Explain the relationship between expanding and factorising algebraic expressions.
4. Analyze common errors when factorising expressions involving negative coefficients or multiple variables.

Stations Rotation

⏱ 25 min·Pairs

Card Match: Expression Pairs

Prepare cards with unfactorised expressions on one set and factorised forms on another. Students work in pairs to match them, then factorise any mismatches. Pairs swap sets with neighbours to verify and discuss choices.

Prepare & details

Explain how factorising is the reverse process of expanding brackets.

Facilitation Tip: During Card Match, circulate and listen for students explaining their factor pairs aloud to uncover hidden misconceptions before they write anything down.

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·Small Groups

Error Hunt: Spot the Mistake

Distribute worksheets with 10 common factorisation errors, like incorrect HCF or sign flips. Small groups identify and correct three each, then teach their fixes to the class via mini-presentations.

Prepare & details

Construct factorised expressions by identifying the highest common factor.

Facilitation Tip: When running Error Hunt, ask groups to present one mistake they found and the corrected version, forcing them to articulate why the original was wrong.

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

⏱ 30 min·Small Groups

Relay Factorise: Team Chain

Divide class into teams. One student factorises an expression on the board, tags the next for expansion check, then refactorisation. First team to complete five rounds correctly wins; rotate roles.

Prepare & details

Analyze common errors when factorising expressions with negative terms or multiple variables.

Facilitation Tip: In Relay Factorise, stand at the back of the room to watch how students divide tasks and communicate, intervening only if the process breaks down.

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·Individual

Visual Tiles: Build and Factor

Provide algebra tiles or printed mats. Individuals build expressions by placing tiles, then regroup to factorise. Share photos of their models in a class gallery walk for peer feedback.

Prepare & details

Explain how factorising is the reverse process of expanding brackets.

Facilitation Tip: With Visual Tiles, model the first grouping yourself under a visualiser to show how terms split into equal parts before letting students try independently.

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 concrete examples students can expand and then reverse, using visual tiles or algebra tiles to connect expansion to factorisation. Avoid rushing to abstract rules before students see the connection between expanded and factorised forms. Research shows that students who work in small, structured groups make fewer sign and variable errors because they verbalise each step before writing it down.

What to Expect

By the end of these activities, students will confidently identify the HCF across coefficients and variables, rewrite expressions by extracting the factor, and explain their reasoning to peers. Success looks like quick recognition of patterns, accurate distribution of signs, and clear verbal justifications during group work.

Watch Out for These Misconceptions

Common Misconception

During Card Match, watch for students who pull out the largest coefficient without checking all terms, like taking 6 from 6x + 9y instead of 3.

What to Teach Instead

Have pairs list all factor pairs of each coefficient on a mini-whiteboard before matching, then compare their lists to agree on the HCF as a group.

Common Misconception

During Error Hunt, watch for sign errors, such as factorising -3x + 6 as -3(x - 2) instead of 3(-x + 2).

What to Teach Instead

Ask students to verbalise the sign as they distribute, then check their work by expanding the bracket to see if it matches the original expression.

Common Misconception

During Visual Tiles, watch for students who overlook variables in the HCF, like factorising 2xy + 4x as 2(y + 2) missing the x.

What to Teach Instead

Have students physically group tiles by colour and shape, then ask them to identify what is common to every tile before writing the factorised form.

Assessment Ideas

Quick Check

After Card Match, collect one pair of cards from each group and ask them to explain why their expressions form a correct pair, focusing on the HCF and distribution.

Discussion Prompt

During Relay Factorise, pause after each round and ask one student to share their team’s strategy for finding the HCF, then have the class agree or challenge their approach.

Peer Assessment

After Visual Tiles, have students swap their factorised expressions and use the tiles to check their partner’s work, marking any terms that were incorrectly grouped or missed.

Extensions & Scaffolding

• Challenge: Give students expressions with three or four terms, like 6xy + 9xz - 3x, and ask them to factorise completely.
• Scaffolding: Provide a template where students list factors of each coefficient and variable separately before attempting to write the bracket.
• Deeper exploration: Introduce a matching activity where students create their own expressions and factorised pairs for peers to solve, justifying their choices in writing.

Key Vocabulary

Factor	A number or algebraic term that divides another number or term exactly. For example, 3 and x are factors of 6x.
Highest Common Factor (HCF)	The largest factor that two or more numbers or algebraic terms share. For example, the HCF of 12x and 18y is 6.
Algebraic Expression	A mathematical phrase that contains numbers, variables, and operation signs. For example, 4x + 8 is an algebraic expression.
Factorise	To rewrite an algebraic expression as a product of its factors. This is the reverse of expanding brackets.
Coefficient	A numerical or constant quantity placed before and multiplying the variable in an algebraic expression. For example, 4 is the coefficient in 4x.