Factoring Quadratics

Overview

Year Group: 10
Duration: 60 minutes
Class Size: 1 student
Unit: Mastering Algebraic Concepts (Lesson 11 of 20)
Topic: Solving Quadratic Equations by Factoring
National Curriculum Reference:

Mathematics Programmes of Study for Key Stage 4 (Years 10-11), Number and Algebra:
- Solve quadratic equations algebraically by factorising, completing the square or using the quadratic formula (NC ref: 9-10 Algebra - solving quadratic equations)
- Use algebraic methods to solve problems (NC ref: 9-10 Algebra - apply knowledge of algebra to solve equations and inequalities)

Learning Objectives

By the end of this lesson, the student will:

Understand the process of solving quadratic equations by factorising.
Identify when a quadratic equation is factorable.
Solve at least 3 quadratic equations correctly using factoring methods.
Verify the solutions by substitution.
Apply factoring skills to algebraic problems involving quadratics.

Success Criteria

The student can:

Correctly factorise quadratic expressions of the form ( ax^2 + bx + c ).
Solve quadratic equations by setting factors equal to zero and solving for ( x ).
Demonstrate understanding by checking the solutions satisfy the original equation.
Explain each step clearly and justify methods chosen.

Resources

Whiteboard and marker
Worked examples handout
Notebook and pencil
Algebra tiles or online virtual manipulatives (optional for visualising factorisation)
Calculator (for verification)

Lesson Structure

1. Starter (5 mins)

Focus: Review of key skills from previous lessons

Quick recap of what factorising polynomials means, using simple quadratics.
Ask: “What does it mean to factorise a quadratic expression?”
Discuss the zero-product property: If ( AB=0 ), then either ( A=0 ) or ( B=0 ).
Write a simple quadratic, e.g., ( x^2 + 5x + 6 ) and ask to factor it mentally or by inspection.

Note: This primes understanding for solving equations by factoring.

2. Introduction / Input (10 mins)

Explain the step-by-step method for solving quadratic equations by factoring:

Write a general quadratic equation, e.g., ( ax^2 + bx + c = 0 ).
Demonstrate factorising a quadratic, e.g., ( x^2 + 7x + 12 = 0 ), by finding two numbers that multiply to ( c ) and add to ( b ).
Show how to rewrite the quadratic as a product of binomials, e.g., ( (x + 3)(x + 4) = 0 ).
Solve the equation by setting each factor equal to zero: ( x + 3 = 0 ) or ( x + 4 = 0 ).
Find the roots and verify by substituting back into the original equation.
Emphasise the importance of rearranging into standard form ( = 0 ) before factorising.

3. Guided Practice (15 mins)

Work through 3 progressively challenging examples together:

( x^2 - 5x + 6 = 0 )
( 2x^2 + 7x + 3 = 0 ) (factorising with ( a \neq 1 ))
( x^2 - 9 = 0 ) (difference of squares)

Use questioning to encourage student to identify factors.
Encourage explaining each step aloud to reinforce understanding.
Use algebra tiles or digital tools if student benefits from visualisation.

4. Independent Task (20 mins)

The student solves at least 3 quadratic equations independently:

Provide these equations:

( x^2 + 4x - 5 = 0 )
( 3x^2 - 2x - 8 = 0 )
( x^2 - 16 = 0 )

Ask the student to solve by factorising.
Have them write out full working and check their answers by substitution.
Monitor and give hints/prompts as needed, encouraging metacognitive reflection on processes used.

5. Plenary / Assessment (10 mins)

Discuss solutions together:

Review answers to independent tasks.
Ask student to reflect: "Which step did you find easiest? Which was challenging?"
Pose a short, extension challenge:
“Can you create your own quadratic equation that factorises and solve it?”
Identify any misconceptions and clarify.

6. Homework / Extension

Provide 3 additional quadratic equations to solve by factoring, encouraging practice at home.
Extension: Investigate cases where quadratics do not factorise neatly and note what to do next (prepare for next lessons on completing the square and quadratic formula).

Differentiation

As this is a 1:1 lesson, adapt pace and complexity dynamically.
Use manipulatives and visual aids to support concrete understanding if needed.
Challenge the student with harder factorising tasks or problem-solving where relevant.

National Curriculum Links Recap

Number and Algebra (Years 9-10): Solve quadratic equations algebraically by factorising (9-10).
Develop fluency in algebraic manipulation and solving equations.
Solve problems involving quadratic functions or equations, interpreting solutions.

Final Notes to Teacher

Keep the session interactive with continuous questioning to maintain engagement.
Encourage the student to think about reasoning, not just procedures.
Celebrate successes to build confidence, especially for individually paced learning.
Use formative assessment through questioning and observing working to tailor support instantly.

This highly focused, active, and student-centred lesson should ensure the learner masters quadratic factoring concepts ready for subsequent algebraic methods.

Teaching Instructions