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solving linear equations

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4 detailed 50-minute lessons with teaching scripts, worked examples, parent guides, and assessment criteria.

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Lesson Overview

Total Lessons: 4
Tier: Foundation and Higher
Duration: 50 minutes per lesson (200 minutes total)
Exam Boards: AQA, Edexcel, OCR, Eduqas, CCEA

Learning Objectives

Prerequisites

Materials & Equipment

Lesson 1: Introduction: solving linear equations

Duration: 50 minutes

Starter Activity (5 minutes)

Quick Recall

Write down everything you already know about solving linear equations. Then check against the key terms: Linear equation, Goal. Use a mini-whiteboard or paper.

Main Content (35 minutes)

Parent/Teacher Guide:
Before lesson: Read the script below. Pre-teach key vocab: Linear equation, Goal.
If stuck: Re-read the revision notes (link above), then break the content into smaller steps.
Extension: See the Stretch & Challenge ideas in Lesson 4.
Teaching Script (35 mins):
Mins 0-5 - Hook: "Today: solving linear equations. By the end you will be able to answer exam questions on it unaided. It connects to the rest of Mathematics because the ideas here recur across the spec."
Mins 5-20 - Direct Instruction: Work through the core ideas below one at a time; after each, ask your student to explain it back in their own words.
Mins 20-30 - Guided Practice: Model the worked example together, then let your student attempt the first practice question with guidance.
Mins 30-35 - Independent Practice: 2-3 practice questions from Lesson 3 below, with immediate feedback.
First Look

Start with the revision notes summary, then attempt: Solve: x + 9 = 15

Plenary (5 minutes)

Check Out

Your student states one thing they learned and one question they still have about solving linear equations.

Lesson 2: Core Concepts: solving linear equations

Duration: 50 minutes

Starter Activity (5 minutes)

Review Previous Lesson

Quick recap: write 3 key points from Lesson 1 on solving linear equations. Check them against the notes below.

Main Content (35 minutes)

Linear equation: An equation where the highest power of the variable is 1.
Goal: Find the value of the unknown that makes the equation true. Use inverse operations to isolate the variable.

Practice (10 minutes)

Q: Solve: x + 9 = 15

Answer: x = 6

Plenary (5 minutes)

Explain Back

Your student teaches the key points back to you without looking. Fill any gaps immediately.

Lesson 3: Application: solving linear equations

Duration: 50 minutes

Starter Activity (5 minutes)

Quick Recall

Recall the key terms: Linear equation, Goal. Define each in one sentence.

Main Content (35 minutes)

Parent/Teacher Guide: Let your student attempt each question alone first, then compare with the model answer. Award method marks for correct working even if the final answer is wrong.

Q1: Solve: x + 9 = 15

Answer: x = 6

Q2: Solve: 5x = 35

Answer: x = 7

Q3: Solve: 3x - 7 = 11

Answer: x = 6

Q4: Solve: 4x + 3 = 2x + 13

Answer: x = 5

Q5: Solve: 2(x + 4) = 18

Answer: x = 5

Q6: Solve: x ⁄ 5 - 2 = 4

Answer: x = 30

Plenary (5 minutes)

Error Review

Review any questions answered incorrectly. Identify whether the error was knowledge, method, or reading the question.

Lesson 4: Exam Practice: solving linear equations

Duration: 50 minutes

Starter Activity (5 minutes)

Command Words

Review what these command words require: state (one point), describe (say what happens), explain (say why), compare (both sides), evaluate (judgement).

Main Content (35 minutes)

Extended Answer

Extended question: Extended Answer 6 marks: A rectangle has length (2x + 1) cm and width (x + 3) cm. The perimeter equals the area. (a) Write an equation. (b) Solve to find x. (c) Find the dimensions of the rectangle. <div class="

(a) Perimeter = 2(2x + 1 + x + 3) = 2(3x + 4) = 6x + 8. Area = (2x + 1)(x + 3) = 2x² + 7x + 3 Equation: 2x² + 7x + 3 = 6x + 8 → 2x² + x - 5 = 0 (b) Using the quadratic formula: x = -1 ± √(1+40) ⁄ 4 = -1 ± √41 ⁄ 4 . Since x > 0: x = -1 + 6.40 ⁄ 4 ≈ 1.35 (c) Length ≈ 2(1.35) + 1 = 3.70 cm, Width ≈ 1.35 + 3 = 4.35 cm Mark scheme: (a) 2 marks for both expressions and equation. (b) 2 marks. (c) 2 marks for dimensions.

Exam Tips: Do the same to both sides | Collect x terms on one side, numbers on the other | Expand brackets before solving | Multiply by denominator to remove fractions | Always check your answer by substituting back
Common Errors: Watch Out! 1. Wrong: Expanding -2(x + 4) as -2x + 4 Correct: -2(x + 4) = -2x - 8 (multiply BOTH terms by -2) 2. Wrong: Solving x ⁄ 3 + x ⁄ 4 = 7 by adding numerators: 2x ⁄ 7 = 7 Correct: Multiply all terms by 12 (LCM of 3, 4): 4x + 3x = 84 → 7x = 84 → x = 12 3. Wrong: 4x + 3 = 2x + 9 → 4x + 2x = 9 + 3 Correct: 4x - 2x = 9 - 3 (change signs when moving across the equals sign)
AO3 - Reasoning & Interpretation: Reasoning and Interpretation A plumber charges £45 call-out plus £30 per hour. A second plumber charges £20 call-out plus £40 per hour. (a) Write a formula for each plumber's charge C for h hours. (b) For how many hours are both plumbers the same cost? (c) A job takes 3 hours. Which plumber is cheaper and by how much? Answers: (a) Plumber 1: C = 45 + 30h. Plumber 2: C = 20 + 40h. (b) 45 + 30h = 20 + 40h → 25 = 10h → h = 2.5 hours. (c) P1: £135, P2: £140. Plumber 1 is £5 cheaper.
Stretch & Challenge (Grade 8-9):
  • Synoptic links: explain how solving linear equations connects to another Mathematics topic you have studied
  • Real-world: research one real-world use or example of solving linear equations
  • Critical: "What are the limitations of the models used in solving linear equations?"

Plenary (5 minutes)

Assessment Criteria
  • Got it: Confident explanation + correct worked examples
  • Getting there: Main points OK, needs support with detail
  • Not yet: Confused on key concepts - re-run Lesson 2

Homework & Consolidation

Recommended Resources

🎓 Smart Lesson (Guided)