Mathematics · Year 7 · Algebraic Thinking · Autumn Term

Factorising into Single Brackets

Reversing the process of expanding by finding common factors to factorise expressions.

• National Curriculum Attainment Targets
• KS3: Mathematics - Algebra

About This Topic

Factorising into single brackets reverses the expansion process. Students identify the highest common factor among terms in expressions like 6x + 9 or 4a + 12b, then rewrite as 3(2x + 3) or 4(a + 3b). This builds fluency in recognising patterns and strengthens algebraic manipulation skills essential for Year 7 algebraic thinking.

In the UK National Curriculum for KS3 Mathematics, this topic sits within Algebra, linking directly to expanding brackets studied earlier. Students analyse the inverse relationship, differentiate expanded from factorised forms, and construct their own factorisable expressions. These steps foster relational understanding and prepare for quadratic factorising and equation solving in later units.

Active learning suits this topic well. Collaborative matching games or sorting tasks make abstract factoring concrete, while peer explanation during group challenges reinforces correct methods and exposes errors early. Hands-on practice with visual aids like algebra tiles helps students internalise the process, boosting confidence and retention.

Key Questions

1. Analyze the relationship between expanding and factorising expressions.
2. Differentiate between an expanded and a factorised expression.
3. Construct an expression that can be factorised into a single bracket.

Learning Objectives

• Identify the highest common factor (HCF) of terms within algebraic expressions.
• Factorise algebraic expressions into the form a(bx + c) or a(bx + cy).
• Compare factorised expressions with their expanded forms to verify correctness.
• Construct algebraic expressions that can be factorised into a single bracket.

Before You Start

Multiplying Terms and Expanding Single Brackets

Why: Students need to be fluent in multiplying a number or variable by each term inside a bracket before they can reverse the process.

Identifying Multiples and Factors

Why: Understanding how to find common factors of numbers is essential for finding the highest common factor of algebraic terms.

Key Vocabulary

Factorise	To rewrite an algebraic expression as a product of its factors, often by finding a common factor to place outside a bracket.
Highest Common Factor (HCF)	The largest number or algebraic term that divides exactly into two or more numbers or algebraic terms.
Expand	To multiply the terms inside a bracket by the factor outside the bracket, removing the brackets.
Term	A single number or variable, or numbers and variables multiplied together, such as 5x or 7.

Watch Out for These Misconceptions

Common Misconception

Always factor out the smallest coefficient.

What to Teach Instead

Students must identify the highest common factor across all terms, not just the smallest number. Group discussions during matching activities reveal this error when pairs compare factorisations, leading to consensus on HCF rules.

Common Misconception

Factorising applies only to numerical terms, ignoring variables.

What to Teach Instead

Every term shares both numerical and variable factors if possible, like x^2 + 3x factors as x(x + 3). Visual sorting tasks with algebra tiles help students see common variable factors, correcting this through manipulation and peer teaching.

Common Misconception

Factorised form is always simpler than expanded.

What to Teach Instead

Both forms are equivalent; factorising aids solving equations later. Collaborative error hunts expose over-simplification, as groups debate and test expansions to verify equality, building deeper understanding.

Active Learning Ideas

Card Match: Expanded to Factorised

Prepare cards with expanded expressions on one set and factorised forms on another. Pairs match them, such as 5x + 10 with 5(x + 2), then justify choices verbally. Extend by having pairs create new pairs for the class to match.

30 min·Pairs

Factor Race: Small Groups

Divide class into small groups with expression sheets. First group to factor all correctly wins a point; rotate roles as timer and checker. Discuss strategies after each round to highlight common factors.

35 min·Small Groups

Error Hunt: Whole Class

Project expressions with deliberate mistakes in factorisation. Class votes on errors via mini-whiteboards, then whole class corrects and explains using HCF rules. Follow with individual practice on similar items.

25 min·Whole Class

Build and Factor: Individual then Pairs

Students individually expand brackets to create expressions, then swap with a partner to factorise. Pairs verify each other's work and refine if needed, building a shared set of examples.

40 min·Individual

Real-World Connections

• Architects use factorisation principles when designing modular building components, ensuring that standard sizes can be combined efficiently to create complex structures, reducing waste and cost.
• Computer programmers utilize factorisation when optimising code, simplifying complex algorithms into smaller, reusable functions to make programs run faster and use less memory.

Assessment Ideas

Quick Check

Present students with a list of expressions (e.g., 8x + 12, 5y - 10, 3a + 7b). Ask them to circle the expressions that can be factorised into a single bracket and underline the HCF for each.

Exit Ticket

Give each student an expression like 9m + 15. Ask them to factorise it into a single bracket. Then, ask them to write one sentence explaining how they found the HCF.

Discussion Prompt

Pose the question: 'If you expand 4(3x + 2), you get 12x + 8. How does understanding the reverse process, factorising, help you check your expansion?' Facilitate a brief class discussion on the inverse relationship.

Frequently Asked Questions

What is the key skill in factorising single brackets?

The core skill is finding the highest common factor (HCF) of all terms, numerical and algebraic, then dividing each term by it to form the bracketed expression. Practice with varied examples like 9y + 6 or 2m + 4n ensures students check all terms systematically. This links directly to reversing expansion for algebraic fluency.

How does factorising connect to expanding brackets?

Factorising is the inverse: expanding distributes a factor into the bracket, while factorising extracts it. Students analyse this by expanding factorised forms to verify, such as checking 4(x + 2) = 4x + 8. This bidirectional practice cements the relationship and prepares for more complex algebra.

How can active learning help students master factorising?

Active approaches like card matching and group races make factoring interactive and social. Students physically pair expressions, explain reasoning to peers, and compete, which reveals misconceptions instantly. Visual tools such as algebra tiles allow manipulation of factors, turning abstract rules into tangible experiences that improve accuracy and speed.

What differentiation strategies work for this topic?

Provide scaffolded sheets: simpler numerical factors for some, variables for others, and multi-term challenges for advanced. Pair stronger students with those needing support during races. Extension tasks include constructing and factorising original expressions, ensuring all access the key questions at their level.