Factorisation of Quadratic Expressions (Cross Method)

Active learning turns the abstract steps of the cross method into concrete, visual processes. Students see how splitting terms and checking products builds reliable factorisation, which reduces frustration for complex quadratics. Movement and collaboration keep energy high while practice cements patterns that trial and error misses.

• Secondary 3
• Mathematics
• 4 activities
• 30 min–45 min

Learning Objectives

1. Demonstrate the application of the cross method to factorise quadratic expressions of the form ax^2 + bx + c.
2. Analyze the relationship between the product of two numbers and their sum in the context of factorising quadratics.
3. Compare the efficiency of the cross method versus trial and error for factorising complex quadratic expressions.
4. Evaluate whether a given quadratic expression can be factorised using integer coefficients.

Problem-Based Learning

⏱ 30 min·Pairs

Card Matching: Cross Method Pairs

Prepare cards with quadratics on one set and factored forms on another. Students work in pairs to match using the cross method, then swap and check by expanding. Discuss mismatches as a class.

Prepare & details

Explain the logic behind the cross method for factorising quadratics.

Facilitation Tip: During Card Matching, circulate and listen for students explaining their pairings aloud, as this verbalisation builds clarity and catches missteps early.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

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

Stations Rotation

⏱ 45 min·Small Groups

Stations Rotation: Factorisation Challenges

Set up stations with increasing difficulty: a=1, a>1, negative coefficients. Small groups solve one per station using cross method, record steps on mini-whiteboards, rotate every 10 minutes.

Prepare & details

Compare the efficiency of the cross method with trial and error for complex quadratics.

Facilitation Tip: For Station Rotation, set timers so groups rotate before losing momentum, and leave hint cards at each station for students to self-correct.

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

Problem-Based Learning

⏱ 35 min·Small Groups

Relay Race: Quadratic Factorisation

Divide class into teams. One student solves a quadratic at the board using cross method, tags next teammate. First team to finish all correctly wins; review errors together.

Prepare & details

Assess the conditions under which a quadratic expression can be factorised using integers.

Facilitation Tip: In the Relay Race, enforce the rule that each runner must complete the factorisation before passing the baton, ensuring full participation.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

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

Problem-Based Learning

⏱ 40 min·Pairs

Peer Tutoring: Custom Quadratics

Pairs create quadratics for each other using cross method guidelines, swap to factorise, then verify expansions. Circulate to prompt discussions on integer conditions.

Prepare & details

Explain the logic behind the cross method for factorising quadratics.

Facilitation Tip: When students tutor each other in Peer Tutoring, provide a checklist for feedback so tutors focus on signs and products, not just answers.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

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

Teaching This Topic

Start by modelling the cross method with two examples: one where a=1 and one where a>1, explicitly naming each step and why it matters. Avoid rushing to shortcuts; instead, insist on writing all working so patterns in signs and products become visible. Research shows that students who practise verifying their work through expansion develop stronger retention and fewer errors in later topics like solving quadratics.

What to Expect

By the end of these activities, students should confidently apply the cross method to any quadratic expression, explain the role of a×c and b, and verify their results by expanding. They should also recognise when expressions cannot be factorised over integers and justify their reasoning.

Watch Out for These Misconceptions

Common Misconception

During Card Matching: Cross Method Pairs, watch for students assuming the cross method only works when a=1.

What to Teach Instead

Ask pairs to sort the cards into two piles: those where a=1 and those where a>1, then factorise one from each pile on the board together.

Common Misconception

During Station Rotation: Factorisation Challenges, watch for students ignoring negative signs when identifying factors of a×c.

What to Teach Instead

At each station, include a poster showing sign rules and ask groups to annotate their work with these rules before starting the next expression.

Common Misconception

During Relay Race: Quadratic Factorisation, watch for students skipping the expansion step after factorising.

What to Teach Instead

Require each runner to write the expanded form on the back of their card before passing it on, so the next student can check for errors.

Assessment Ideas

Quick Check

After Card Matching, present three expressions on the board and ask students to hold up cards showing whether each can be factorised using the cross method or not, explaining their choice.

Discussion Prompt

During Station Rotation, ask each group to share one example where the cross method was clearly faster than trial and error, and one where trial and error might have been easier, fostering metacognitive awareness.

Exit Ticket

After the Relay Race, give each student an expression like 4x² - 12x + 9 and ask them to show the cross method steps on a half-sheet, then expand their factors to verify the answer before leaving.

Extensions & Scaffolding

• Challenge early finishers with a set of three quadratics where two share a common factor but look different, asking them to factorise and justify their choice.
• For students who struggle, provide partially completed cross diagrams with missing signs or products, asking them to fill in the gaps before attempting full factorisation.
• Deeper exploration: invite students to create their own quadratic expressions with specified roots and coefficients, then swap with peers to factorise and verify.

Key Vocabulary

Quadratic Expression	An algebraic expression of the form ax^2 + bx + c, where a, b, and c are constants and a is not equal to zero.
Factorisation	The process of breaking down a mathematical expression into a product of simpler expressions, called factors.
Cross Method	A visual technique used to factorise quadratic expressions by arranging terms in a cross shape to identify pairs of numbers that satisfy specific product and sum conditions.
Constant Term	The term in a polynomial that does not contain any variables; in ax^2 + bx + c, the constant term is c.
Coefficient	A numerical or constant quantity placed before and multiplying the variable in an algebraic expression; for example, 'a' and 'b' are coefficients in ax^2 + bx + c.