Factoring Quadratics

Overview

Duration: 60 minutes
Class size: 1 student
Unit: Mastering Algebraic Concepts (Lesson 10 of 20)
National Curriculum reference:

• KS4 Mathematics Programme of Study (Years 10-11)
• A1: Use algebraic methods, including factorising quadratic expressions, to solve equations and inequalities (National Curriculum for England, Key Stage 4, Number and Algebra)
• Specifically focuses on the learning objective in content domain "Algebra: Manipulate algebraic expressions", developing fluency in factorising quadratic expressions of the form ax² + bx + c where a=1.

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Learning Objectives

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

• Recognise and identify quadratic expressions suitable for factorisation.
• Factorise simple quadratic expressions of the form x² + bx + c accurately.
• Apply the factorisation technique to solve related algebraic problems.

These align with the KS4 framework requirements of manipulating and factorising algebraic expressions.

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Success Criteria

• I can identify coefficients and constants in quadratic expressions.
• I can break down the middle term into two parts whose product equals the product of the coefficient of x² and the constant term.
• I can write the factored form of quadratic expressions as a product of two binomials.
• I can factor at least 3 quadratic expressions correctly in class.

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Resources Needed

• Notebook and pen for working through examples
• Whiteboard and marker (if teaching in a physical classroom or a digital whiteboard tool if online)
• Pre-prepared worksheet with 6 quadratic expressions of varying difficulty (all with a = 1)
• Factorisation grid template (optional visual aid)
• Mini whiteboard or scratch paper for quick working out
• Set of algebra tiles or visual manipulatives (optional but recommended for kinaesthetic connection)

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Lesson Structure

1. Starter (5 minutes)

• Diagnostic Question: Write an expression on the board — for example, x² + 7x + 10. Ask: "What kind of expression is this? What can we do with it?"
• Brief discussion: Recap what a quadratic expression is, focusing on the x² term, the linear x term, and the constant.
• Objective sharing: Explain today’s goal: learn to factor these quadratic expressions to help solve them quickly.

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2. Explicit Teaching (15 minutes)

• Explain the factoring method (decomposition method):

  • Identify a, b, c in ax² + bx + c (here a=1).
  • Find two numbers that multiply to c (constant term) and add to b (middle term).

• Use an example: x² + 5x + 6

  • Pairs of 6: (1,6), (2,3)
  • Check which pair sums to 5 → (2 + 3) = 5
  • Write: (x + 2)(x + 3)

• Visual aid: Use an algebra tiles or factor grid to demonstrate this visually.

• Write a second example for more clarity, including a case with negative terms: x² - x - 6.

• Emphasise vocabulary: factors, coefficient, constant term, sum, product.

• Check understanding: Ask the student to explain the process back in their own words.

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3. Guided Practice (15 minutes)

• Work through 3 quadratic expressions together from the worksheet.

• Student performs factorisation step-by-step with teacher support, guided questioning to scaffold.

• Examples:

  1. x² + 8x + 15
  2. x² - 3x + 2
  3. x² + 6x + 9

• Praise accuracy and process explanation.

• Address misconceptions immediately (e.g., mixing up sum and product).

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4. Independent Practice (15 minutes)

• Student attempts 3 more quadratic expressions independently to meet success criteria.

• Examples:

  1. x² + 4x + 3
  2. x² - 7x + 12
  3. x² + 2x - 8 (introducing a negative constant)

• Teacher observes, offers hints only if stuck, encouraging self-correction.

• Use mini whiteboard or paper for quick iteration and trial.

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5. Plenary & Assessment (10 minutes)

• Reflective questioning:
  • "How did you decide to split the middle term?"
  • "Which part do you find easiest/hardest?"
• Student explains factoring of one expression from today’s work.
• Provide mini-assessment: verbally factorise a quadratic presented by the teacher with increasing difficulty.
• Success criteria check: Confirm if student successfully factors at least 3 expressions accurately without help.

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Extension and Challenge (Optional)

• Challenge the student to factorise expressions where a ≠ 1 (e.g., 2x² + 7x + 3) linked to a preview of upcoming lessons.
• Explore difference of squares briefly if time permits (x² - 9).

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Differentiation

• For this one-to-one setting, pace adjusts instantly to student needs.
• Use concrete materials (algebra tiles) if abstract methods prove difficult.
• Extend with problem-solving questions involving quadratic factorisation.

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Links to Future Lessons

• Next lesson: solving quadratic equations by factorisation (Lesson 11).
• Later lessons: factorising quadratics where a ≠ 1, completing the square.

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National Curriculum Alignment Summary

• Develops fluency in manipulating algebraic expressions (KS4 Number and Algebra).
• Supports problem solving and reasoning with algebra in line with GCSE progression.
• Builds foundation required for solving quadratic equations by factorisation, covered in subsequent lessons.

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This detailed plan ensures the lesson is structured, engaging, and firmly rooted in the expectations of the national curriculum while offering personalised support and interactive elements to maximise learning for a single student.