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simultaneous 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: simultaneous equations

Duration: 50 minutes

Starter Activity (5 minutes)

Quick Recall

Write down everything you already know about simultaneous equations. Then check against the key terms: Simultaneous equations, Methods. Use a mini-whiteboard or paper.

Main Content (35 minutes)

Parent/Teacher Guide:
Before lesson: Read the script below. Pre-teach key vocab: Simultaneous equations, Methods.
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: simultaneous 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 + y = 10 and x - y = 4

Plenary (5 minutes)

Check Out

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

Lesson 2: Core Concepts: simultaneous equations

Duration: 50 minutes

Starter Activity (5 minutes)

Review Previous Lesson

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

Main Content (35 minutes)

Simultaneous equations: Two or more equations with the same variables. Find values that satisfy ALL equations.
Methods: Elimination: Add or subtract equations to remove one variable Substitution: Express one variable in terms of another Graphically: Find where lines/curves intersect

Practice (10 minutes)

Q: Solve: x + y = 10 and x - y = 4

Answer: x = 7, y = 3

Plenary (5 minutes)

Explain Back

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

Lesson 3: Application: simultaneous equations

Duration: 50 minutes

Starter Activity (5 minutes)

Quick Recall

Recall the key terms: Simultaneous equations, Methods. 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 + y = 10 and x - y = 4

Answer: x = 7, y = 3

Q2: Solve: 2x + 3y = 12 and 2x + y = 8

Answer: x = 3, y = 2

Q3: Solve: 3x - 2y = 7 and x + 2y = 9

Answer: x = 4, y = 2.5

Q4: Solve by substitution: y = 3x - 1 and 2x + y = 9

Answer: x = 2, y = 5

Q5: Solve: y = x² and y = 4

Answer: x = 2 or x = -2, y = 4

Q6: Solve: y = 2x - 1 and y = x² - 4x + 5

Answer: x = 2, y = 3 and x = 3, y = 5

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: simultaneous 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 cinema sells adult tickets for £10 and child tickets for £6. On Saturday, 250 tickets were sold and the total revenue was £1980. (a) Write two simultaneous equations. (b) Solve to find how many adults and children attended. (c) On Sunday, adult prices increase to £12 and 200 tickets sell for £1920. How many adults now? <div class="

(a) a + c = 250 ... (1), 10a + 6c = 1980 ... (2) (b) From (1): c = 250 - a. Substitute: 10a + 6(250 - a) = 1980 → 10a + 1500 - 6a = 1980 → 4a = 480 → a = 120, c = 130 120 adults, 130 children (c) 12a + 6(200 - a) = 1920 → 12a + 1200 - 6a = 1920 → 6a = 720 → a = 120 adults Mark scheme: (a) 2 marks for both equations. (b) 2 marks for solving. (c) 2 marks.

Exam Tips: Use elimination when coefficients can be easily matched | Use substitution when one equation has y = ... form | Always check your answer in both equations | For linear/quadratic, set the y values equal | Graphically: solutions are where the graphs intersect
Common Errors: Watch Out! 1. Wrong: Adding equations when signs are the same and you should subtract Correct: If both have +3y, SUBTRACT to eliminate. If one has +3y and other -3y, ADD. 2. Wrong: Finding x but forgetting to find y Correct: Always substitute back to find the other variable 3. Wrong: For y = x² and y = 2x, writing x² + 2x = 0 Correct: Set them equal: x² = 2x → x² - 2x = 0 → x(x-2) = 0
AO3 - Reasoning & Interpretation: Reasoning and Interpretation A line y = 2x + 1 and curve y = x² - x + 3 intersect. (a) Find the coordinates of the intersection points. (b) Is the line above or below the curve between the intersection points? (c) Sketch both graphs on the same axes. Answers: (a) 2x + 1 = x² - x + 3 → x² - 3x + 2 = 0 → (x-1)(x-2) = 0 → x = 1 or 2. Points: (1, 3) and (2, 5). (b) At x = 1.5: line gives y = 4, curve gives y = 2.25 - 1.5 + 3 = 3.75. Line is above the curve. (c) Line and parabola intersect at (1,3) and (2,5), with the line above between these points.
Stretch & Challenge (Grade 8-9):
  • Synoptic links: explain how simultaneous equations connects to another Mathematics topic you have studied
  • Real-world: research one real-world use or example of simultaneous equations
  • Critical: "What are the limitations of the models used in simultaneous 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)