Mathematics · Year 10

Active learning ideas

Solving Quadratic Equations by Factorising

Students often struggle to see the link between abstract factorising and solving equations. Active tasks like card sorts and relays make this connection concrete by letting students manipulate equations and roots in real time. Physical movement and peer discussion strengthen their understanding of null factor law and sign rules more effectively than worksheets alone.

• National Curriculum Attainment Targets
• GCSE: Mathematics - Algebra

• 25–40 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Collaborative Problem-Solving
• 35 min · Small Groups

Card Sort: Equation to Roots

Prepare cards with unsolved quadratics, factorised forms, graphs, and root pairs. Small groups sort and match sets on tables, then create their own cards to swap with another group. End with a class share-out of tricky matches.

Explain why the null factor law is fundamental to solving quadratic equations by factorising.

Facilitation Tip

For the Card Sort, circulate and ask each group to justify one equation-root pair to you before moving on, ensuring accountability.

What to look for

Present students with three quadratic equations: one easily factorised (e.g., x² + 5x + 6 = 0), one requiring a common factor (e.g., 2x² + 6x = 0), and one that does not factorise easily over integers (e.g., x² + 2x - 5 = 0). Ask students to solve the first two by factorising and state why the third is more efficiently solved by another method.

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

Activity 02

• Collaborative Problem-Solving
• 25 min · Pairs

Error Hunt: Faulty Solutions

Provide worksheets with five factorised solutions containing common errors like sign flips or ignored null factor steps. Pairs identify mistakes, explain corrections, and rewrite correctly. Groups present one to the class.

Evaluate the efficiency of factorising compared to other methods for specific quadratic equations.

Facilitation Tip

During Error Hunt, provide red pens so students can mark corrections directly on the faulty solutions, making the feedback visual and immediate.

What to look for

Give students the factorised equation (x - 3)(x + 5) = 0. Ask them to: 1. State the roots of the equation. 2. Write the expanded quadratic equation. 3. Explain in one sentence how the null factor law was used.

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

Activity 03

• Collaborative Problem-Solving
• 40 min · Small Groups

Relay Factorise: Chain Challenge

Divide class into teams lined up. First student factorises a quadratic on the board, tags next for null factor law application, and so on until roots found. Fastest accurate team wins; repeat with varied equations.

Predict the roots of a quadratic equation given its factorised form.

Facilitation Tip

In Relay Factorise, enforce a five-minute switch between stations so groups stay on pace and do not rush through the algebra.

What to look for

Pose the question: 'When solving a quadratic equation, is factorising always the best first step?' Facilitate a class discussion where students share examples of when factorising is efficient and when it might be time-consuming or impossible, leading them to consider other methods like completing the square or the quadratic formula.

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

Activity 04

• Collaborative Problem-Solving
• 30 min · Pairs

Prediction Pairs: Roots First

Give factorised forms like (x - 2)(x + 5) = 0; pairs predict roots, expand to verify, then solve reverse from expanded form. Switch roles and compare efficiencies.

Explain why the null factor law is fundamental to solving quadratic equations by factorising.

What to look for

Present students with three quadratic equations: one easily factorised (e.g., x² + 5x + 6 = 0), one requiring a common factor (e.g., 2x² + 6x = 0), and one that does not factorise easily over integers (e.g., x² + 2x - 5 = 0). Ask students to solve the first two by factorising and state why the third is more efficiently solved by another method.

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

A few notes on teaching this unit

Start with simple monic quadratics and have students list factor pairs aloud to build fluency with signs and sums. Avoid rushing to the general method—instead, let students discover the pattern through repeated exposure. Research shows that delayed introduction of the quadratic formula preserves conceptual understanding and reduces over-reliance on procedural shortcuts.

By the end, students will confidently rewrite quadratics as products of linear factors and use the null factor law to find roots. They will also recognize when factorising is suitable and when another method is better. Clear explanations during paired and group tasks will show their grasp of both process and reasoning.

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Watch Out for These Misconceptions

• During Card Sort: Equation to Roots, watch for students who pair factors without checking both the product and sum conditions.

  Have each group set the timer for two minutes to verify every pair: multiply to the constant term and add to the x coefficient. If a pair fails, they must swap cards until it matches.

• During Error Hunt: Faulty Solutions, watch for students who treat each linear factor as a new quadratic equation.

  Ask them to read their teammate’s solution aloud, stopping after each factor is set to zero. Prompt: ‘What does the null factor law say you do next?’

• During Relay Factorise: Chain Challenge, watch for students who ignore the signs of the constant term when listing factor pairs.

  Provide mini whiteboards with a sign reminder strip (+, -, +, -) taped to the edge so students check signs before recording pairs.

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Methods used in this brief

• Collaborative Problem-Solving