Solving Quadratic Equations

Overview

This 30-minute lesson targets a Year 11 student, focusing on solving quadratic equations by factoring, as part of the unit "Mastering Algebraic Concepts" (lesson 8/10). It aligns specifically with the National Curriculum for England's Key Stage 4 content for mathematics, ensuring the student develops fluency and reasoning skills essential for GCSE-level algebra.

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National Curriculum Links

• Content domain: Algebra
• Key objective: Solve quadratic equations algebraically by factorisation (Key Stage 4 Statutory Programme of Study)
• Focus statements:
  • Use algebraic methods (including factorising quadratics) to find exact solutions of equations in one variable (including quadratic equations).
  • Understand the relationship between factors and roots of quadratic equations.
  • Check solutions by substitution into the original equation.

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Learning Objectives

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

1. Factorise quadratic expressions into binomial factors correctly.
2. Solve quadratic equations by setting each factor equal to zero and finding roots.
3. Verify solutions by substituting values back into the original quadratic equation.

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Success Criteria

• Correctly factorise given quadratic expressions.
• Accurately solve at least 2 quadratic equations by factoring.
• Successfully check solutions by substitution to confirm correctness.

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Resources

• Whiteboard and marker or paper and pen
• Worked examples prepared beforehand
• Calculator (optional for verification)
• Concrete manipulatives (optional: algebra tiles) for kinesthetic learning
• Success criteria checklist

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Lesson Structure

1. Engage & Reactivate Prior Knowledge (5 minutes)

• Teacher prompts: Quick recall of factorising simple quadratics (e.g., x² + 5x + 6)
• Ask student to factorise a simple quadratic, e.g. x² + 7x + 10, verbally explaining each step.
• Discuss the significance of factors in zero-product property for solving equations.
• Use algebra tiles or draw rectangles to visualise the factorisation (e.g., creating arrays representing terms).

2. Introduction to Solving Quadratics by Factoring (7 minutes)

• Explain that to solve quadratic equations like ax² + bx + c = 0 by factoring:
  • First, rewrite equation setting rhs = 0.
  • Factorise into two binomials.
  • Apply zero-product property: set each factor to zero.
  • Solve for x.
• Show teacher-led worked example:
  • Example 1: Solve x² + 5x + 6 = 0
  • Factorise → (x + 2)(x + 3) = 0
  • Set each factor to zero → x + 2 = 0 or x + 3 = 0
  • Find roots → x = -2 or x = -3
• Model checking by substitution explicitly: substitute roots back into original equation and verify 0 is obtained.

3. Guided Practice (10 minutes)

• Provide two quadratic equations for student to solve by factoring:
  1. x² - 3x - 10 = 0
  2. 2x² + 7x + 3 = 0
• Support student through:
  • Rearranging if needed, factorisation (possibly by grouping for the second example).
  • Setting factors to zero and solving equations.
  • Checking answers by substitution.
• Encourage verbal explanation of steps for reinforcement.
• Use success criteria to self-assess after each problem.

4. Independent Application and Reasoning (5 minutes)

• Present a slightly more challenging quadratic (e.g., 3x² - 2x - 8 = 0) or one with a coefficient of x² > 1 requiring more careful factorisation.
• Ask student to solve and justify their method and answers out loud.
• Reflect briefly: "How do you know your answers are correct?"

5. Plenary and Reflection (3 minutes)

• Recap key steps using success criteria:
  • Factorises correctly
  • Applies zero product property
  • Checks answers accurately
• Ask student two reflection questions:
  • What was easiest or most difficult part of solving by factoring?
  • How can checking solutions help avoid mistakes?
• Provide positive feedback and set one improvement focus for the next lesson.

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Assessment

• Formative throughout via questioning, discussion, and verbal explanations.
• Success criterion checklist for factoring and solving steps.
• Review substitution checks for accuracy.
• Observation of reasoning and clarity during independent practice.

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Differentiation & Extension

• Support: Use algebra tiles or colour-coded steps to scaffold factorisation visually.
• Challenge: Explore solving quadratic equations where factorisation is less obvious or introduce completing the square as a preview for next lesson.
• Encourage articulating solution strategies for meta-cognitive development.

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Reflection for Next Lesson

• Note areas needing revisiting (e.g., factorisation patterns or zero-product property).
• Plan to introduce solving quadratics by completing the square or quadratic formula (lesson 9).

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This precise and engaging lesson plan bridges knowledge and procedure while ensuring adherence to the National Curriculum, promoting deep understanding and confidence in algebraic problem-solving for Year 11.