Solving Quadratics

Overview

This 45-minute lesson is designed for Year 10 students, focusing on solving quadratic equations. The lesson aligns with the National Curriculum for England (Key Stage 4 Mathematics) specifically addressing the area of algebra and equations. It incorporates inclusive strategies to support autistic students, ensuring accessibility through clear structure, visual aids, and sensory-friendly activities.

National Curriculum Reference

• Content domain: Number: Algebra
• Programme of Study (KS4): Solve quadratic equations algebraically by factorising and using the quadratic formula; graphically find roots of quadratic functions.
• Learning objectives:
  • Solve quadratic equations by factorising, completing the square, or using the quadratic formula.
  • Understand the nature of roots in the context of factorisation and the discriminant.
  • Represent quadratic equations graphically and interpret their solutions.

Learning Objectives

By the end of the lesson, students will be able to:

1. Solve simple quadratic equations by factorising (NC: Solve quadratic equations algebraically).
2. Apply the quadratic formula to find roots of quadratic equations.
3. Explain the meaning of the solutions graphically by identifying roots on a parabola.
4. Demonstrate understanding of discriminant to determine the nature of roots (real & distinct, real & equal, or complex).
5. Use clear, structured working methods to promote accessibility and confidence, especially for autistic learners.

Resources Needed

• Whiteboard and markers
• Visual aids: step-by-step flowcharts for each solving method
• Graph paper and coloured pencils
• Pre-prepared worksheets with worked examples (simple to challenging), including blank templates with scaffolds
• Mini whiteboards and pens for paired work
• Timer for activity pacing
• Quiet workspace area for sensory breaks (optional)

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Lesson Breakdown

1. Starter (5 minutes)

Activity: Quick recall quiz on factorisation and basic algebra manipulation.

• Use mini whiteboards for students to write answers.
• Example questions: Factorise expressions like x² + 5x + 6.
• Purpose: activate prior knowledge and build confidence.

Differentiation:

• Provide visual prompts on the board for autistic students—e.g., colour-coded terms in expressions.
• Use clear, simple language during questioning.

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2. Introduction to Solving Quadratics by Factorising (10 minutes)

Explanation:

• Teacher models solving a quadratic equation by factorising on the board, e.g., x² + 5x + 6 = 0.
• Use a step-by-step flowchart on a visual poster:
  1. Write equation in standard form.
  2. Factorise into (x + a)(x + b) = 0.
  3. Apply zero product principle: set each factor to zero.
  4. Solve for x.

Visual Aid: Colour code factors and solutions to enhance comprehension.

Student Task:

• Attempt a similar equation on whiteboards in paired work.
• Teacher circulates to support autistic students by asking guiding questions and maintaining a calm, clear tone.

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3. Introducing the Quadratic Formula (10 minutes)

Presentation:

• Reveal the quadratic formula: x = [-b ± √(b² - 4ac)] / 2a.
• Discuss components (a, b, c from standard form ax² + bx + c = 0).
• Model one example with clear annotation of each step.

Concept Explanation:

• Introduce the discriminant (b² - 4ac) and what it indicates about the roots.
• Show three cases: discriminant > 0, = 0, < 0 through simple examples and diagrammatic representation.

Visual & Sensory Support:

• Use coloured pens to underline a, b, and c.
• Show discriminant cases with symbols (✓ for two real roots, ∆ for one real root, × for no real roots).
• Pause after each step to check understanding, offering a quiet moment for reflection, helpful for autistic learners.

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4. Graphical Interpretation (10 minutes)

Activity:

• Draw a sample quadratic graph on the board, labelling roots clearly.
• Students sketch a basic parabola and mark roots corresponding to different discriminant values.

Group Task:

• In small groups, students match quadratic equations to graphs and explain why roots appear where they do.
• Use sentence starters to scaffold discussion:
  • “The roots are ___ because the graph touches/crosses the x-axis ___ times.”

Differentiation:

• Provide pre-drawn graphs for those who struggle with sketching.
• Allow use of graphing calculators or apps if available for visual support.

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5. Plenary and Formative Assessment (5 minutes)

Activity:

• Exit ticket: Students complete two questions—one factorising and one using the quadratic formula.
• Collect on exit or discuss answers orally.

Check:

• Teacher notes any misconceptions, especially from autistic students who may need follow-up support.
• Positive reinforcement for all attempts, emphasizing method and clarity rather than only correctness.

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Inclusion Strategies for Autistic Students

• Provide a written copy of instructions and key formulas.
• Use a calm, predictable lesson structure with clear transitions.
• Visual supports (colour coding, flowcharts, symbols).
• Allow quiet workspace if overwhelmed.
• Use short, clear sentences and minimise figurative language.
• Provide designated partner/buddy for peer support during group activities.
• Frequent, gentle check-ins to address anxiety or confusion.

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Extension and Homework Suggestions

• Challenge: Solve quadratic equations that require completing the square or have irrational roots.
• Homework: Create a mini-poster explaining one method of solving quadratics, using examples and diagrams.

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This lesson blends clear explanation, visual reinforcement, and scaffolded practice tailored to diverse learners, fulfilling the National Curriculum expectations while creating an accessible, engaging environment.