quadratic graphs
4 detailed 50-minute lessons with teaching scripts, worked examples, parent guides, and assessment criteria.
Lesson Overview
Total Lessons: 4
Tier: Foundation and Higher
Duration: 50 minutes per lesson (200 minutes total)
Exam Boards: AQA, Edexcel, OCR, Eduqas, CCEA
Tier: Foundation and Higher
Duration: 50 minutes per lesson (200 minutes total)
Exam Boards: AQA, Edexcel, OCR, Eduqas, CCEA
Learning Objectives
- Explain the key ideas of quadratic graphs
- Apply quadratic graphs to exam-style questions
- Develop independent study skills
- Build cross-topic connections
Prerequisites
- Read the revision notes for quadratic graphs first (GCSE Revise summary)
- Key vocab to pre-teach: Quadratic graph, Key features
- Basic skills: reading the summary notes and answering the practice questions there
Materials & Equipment
- Exercise book, pencil, ruler
- Scientific calculator
- Pair of compasses and protractor (geometry topics)
- Graph paper
- Printed revision notes (link below)
Lesson 1: Introduction: quadratic graphs
Duration: 50 minutes
Starter Activity (5 minutes)
Quick Recall
Write down everything you already know about quadratic graphs. Then check against the key terms: Quadratic graph, Key features. Use a mini-whiteboard or paper.
Main Content (35 minutes)
Parent/Teacher Guide:
Before lesson: Read the script below. Pre-teach key vocab: Quadratic graph, Key features.
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.
Before lesson: Read the script below. Pre-teach key vocab: Quadratic graph, Key features.
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: quadratic graphs. 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.
Mins 0-5 - Hook: "Today: quadratic graphs. 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: Find the y-intercept of y = x² - 4x + 3.
Plenary (5 minutes)
Check Out
Your student states one thing they learned and one question they still have about quadratic graphs.
Lesson 2: Core Concepts: quadratic graphs
Duration: 50 minutes
Starter Activity (5 minutes)
Review Previous Lesson
Quick recap: write 3 key points from Lesson 1 on quadratic graphs. Check them against the notes below.
Main Content (35 minutes)
Quadratic graph: A U-shaped curve called a parabola. The equation is y = ax² + bx + c.
Key features: Roots (x-intercepts): Where the curve crosses the x-axis (y = 0) y-intercept: Where the curve crosses the y-axis (x = 0) Turning point: The vertex (minimum or maximum point) Line of symmetry: Vertical line through the turning point
| Term | Meaning | Example |
|---|---|---|
| y | 6 | 0 |
Practice (10 minutes)
Q: Find the y-intercept of y = x² - 4x + 3.
Answer: (0, 3)
Plenary (5 minutes)
Explain Back
Your student teaches the key points back to you without looking. Fill any gaps immediately.
Lesson 3: Application: quadratic graphs
Duration: 50 minutes
Starter Activity (5 minutes)
Quick Recall
Recall the key terms: Quadratic graph, Key features. 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: Find the y-intercept of y = x² - 4x + 3.
Answer: (0, 3)
Q2: Find the roots of y = x² - 9.
Answer: x = 3 and x = -3, so roots are (3, 0) and (-3, 0)
Q3: Find the roots of y = x² - 5x + 6.
Answer: x = 2 and x = 3, so roots are (2, 0) and (3, 0)
Q4: A quadratic has roots at x = 1 and x = 5. What is the x-coordinate of the turning point?
Answer: x = 3
Q5: Find the turning point of y = (x - 2)² + 3.
Answer: (2, 3)
Q6: Find the line of symmetry of y = x² + 4x + 1.
Answer: x = -2
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: quadratic graphs
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: y = x² - 6x + 5. (a) Find the y-intercept. (b) Find the roots. (c) Find the coordinates of the turning point. (d) Sketch the curve, labelling all key points. <div class="
(a) When x = 0: y = 5. y-intercept = (0, 5) (b) x² - 6x + 5 = 0 → (x - 1)(x - 5) = 0 → x = 1 or x = 5. Roots: (1, 0) and (5, 0) (c) By completing the square: (x - 3)² - 9 + 5 = (x - 3)² - 4. Turning point: (3, -4) (d) U-shaped parabola passing through (0, 5), (1, 0), (3, -4), (5, 0). Minimum at (3, -4). Line of symmetry x = 3. Mark scheme: (a) 1 mark. (b) 1 mark. (c) 2 marks. (d) 2 marks for sketch with labelled points.
Exam Tips: y-intercept is always (0, c) - just read off the constant | Find roots by factorising or using the quadratic formula | Turning point x-coordinate is between the two roots | Use completing the square for turning point (Higher) | Check if parabola opens up (a > 0) or down (a < 0)
Common Errors: Watch Out! 1. Wrong: The y-intercept of y = x² - 5x + 6 is 6 at point (6, 0) Correct: The y-intercept is (0, 6) — it's on the y-axis where x = 0 2. Wrong: The turning point is always a minimum Correct: It's a minimum when a > 0 (opens up), a maximum when a 3. Wrong: Completing the square: x² + 6x + 5 = (x + 6)² + 5 Correct: x² + 6x + 5 = (x + 3)² - 9 + 5 = (x + 3)² - 4 (halve the coefficient of x)
AO3 - Reasoning & Interpretation: Reasoning and Interpretation The profit P (in £1000s) from selling x items is P = -x² + 20x - 64. (a) Find the profit when 10 items are sold. (b) How many items give zero profit (break-even)? (c) What is the maximum profit and how many items achieve it? Answers: (a) P = -100 + 200 - 64 = 36 (£36,000). (b) -x² + 20x - 64 = 0 → (x-4)(x-16) = 0 → x = 4 or x = 16 items. (c) Turning point at x = 10: P = 36, so max profit is £36,000 at 10 items.
Stretch & Challenge (Grade 8-9):
- Synoptic links: explain how quadratic graphs connects to another Mathematics topic you have studied
- Real-world: research one real-world use or example of quadratic graphs
- Critical: "What are the limitations of the models used in quadratic graphs?"
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
- Consolidation: Re-answer any Lesson 3 practice questions answered incorrectly (20 mins)
- Retrieval: Write flashcards for the key terms: Quadratic graph, Key features (10 mins)
- Exam practice: One past-paper question on quadratic graphs from the board websites (15 mins)
- Extension: Explain quadratic graphs to someone else in your own words (10 mins)