Mathematics · Year 11

Active learning ideas

Solving Quadratic Equations by Factorising

Active learning builds fluency in algebraic manipulation, which is essential when solving systems involving linear and quadratic equations. Students need repeated, scaffolded practice to transfer skills from basic factorising to solving for coordinates, making hands-on and collaborative tasks ideal for this topic.

• National Curriculum Attainment Targets
• GCSE: Mathematics - Algebra

• 15–30 min
• Pairs → Whole Class
• 3 activities

Activity 01

• Think-Pair-Share
• 15 min · Pairs

Think-Pair-Share: The Substitution Strategy

Provide students with a linear and quadratic pair. Individually, they identify which variable is easiest to isolate; in pairs, they perform the substitution and discuss why isolating 'y' might be simpler than 'x' in specific cases.

Analyze how factorising a quadratic expression reveals its roots.

Facilitation Tip

During Think-Pair-Share, assign specific roles so students verbalise each step of the substitution process to their partner before solving together.

What to look for

Provide students with three quadratic equations: x^2 + 5x + 6 = 0, 2x^2 - 7x + 3 = 0, and x^2 + x + 1 = 0. Ask them to factorise and solve the first two, and explain why the third cannot be factorised into real linear factors.

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

Activity 02

• Inquiry Circle
• 30 min · Small Groups

Inquiry Circle: Intersection Hunt

Give small groups sets of equations and a large coordinate grid. They must algebraically solve the systems and then plot them to verify if their solutions match the physical intersections on the graph.

Compare the efficiency of factorising versus other methods for specific quadratic forms.

What to look for

Display a partially factorised equation, for example, (x + 3)(x - 5) = 0. Ask students to write down the roots of the equation and then write the expanded quadratic expression. This checks understanding of the link between factors and roots.

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Activity 03

• Peer Teaching
• 20 min · Pairs

Peer Teaching: Error Detectives

Present a pre-written 'failed' solution containing common expansion or sign errors. Students work in pairs to find the mistake, explain the correct step to their partner, and present the fix to the class.

Explain why a quadratic equation can have two, one, or zero real solutions.

What to look for

Pose the question: 'When is factorising the most efficient method for solving a quadratic equation, and when might another method, like completing the square or the quadratic formula, be preferable?' Facilitate a class discussion where students justify their reasoning with examples.

• Understand
• Apply
• Analyze
• Create
• Self-Management
• Relationship Skills

A few notes on teaching this unit

Teachers should model the substitution process slowly, writing each step visibly and speaking the reasoning aloud. Research shows that students benefit from seeing errors corrected in real time, so deliberately introduce a common mistake and work through its resolution as a class. Avoid rushing to the final answer; emphasise the importance of checking that found solutions satisfy both original equations.

Successful students will confidently substitute a linear equation into a quadratic, factorise correctly, solve for both variables, and state intersection points with coordinates. They will also recognise when factorising is appropriate and when other methods are more efficient.

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Watch Out for These Misconceptions

• During Think-Pair-Share, watch for students who solve the quadratic equation but stop before calculating the corresponding y-coordinates.

  Provide a printed checklist with the steps ‘Substitute, Expand, Factorise, Solve for x, Find y, Write coordinates’ and have partners check off each step as they work.

• During Collaborative Investigation, watch for students who incorrectly expand binomials during substitution, such as treating (x + 3)^2 as x^2 + 9.

  Hand out algebra tiles or grid method templates so students can model the expansion physically before recording the algebraic form, reinforcing the need for the middle term.

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Methods used in this brief

• Think-Pair-Share
• Inquiry Circle
• Peer Teaching