other 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 other graphs
- Apply other graphs to exam-style questions
- Develop independent study skills
- Build cross-topic connections
Prerequisites
- Read the revision notes for other graphs first (GCSE Revise summary)
- Key vocab to pre-teach: Types of graphs
- 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: other graphs
Duration: 50 minutes
Starter Activity (5 minutes)
Quick Recall
Write down everything you already know about other graphs. Then check against the key terms: Types of graphs. Use a mini-whiteboard or paper.
Main Content (35 minutes)
Parent/Teacher Guide:
Before lesson: Read the script below. Pre-teach key vocab: Types of graphs.
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: Types of graphs.
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: other 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: other 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: What shape is the graph of y = x³ - 2x?
Plenary (5 minutes)
Check Out
Your student states one thing they learned and one question they still have about other graphs.
Lesson 2: Core Concepts: other graphs
Duration: 50 minutes
Starter Activity (5 minutes)
Review Previous Lesson
Quick recap: write 3 key points from Lesson 1 on other graphs. Check them against the notes below.
Main Content (35 minutes)
Types of graphs: Different equations produce different shaped graphs. Being able to recognise these shapes is essential.
| Term | Meaning | Example |
|---|---|---|
| Linear | y = mx + c | Straight line |
| Quadratic | y = ax² + bx + c | Parabola (U or ∩) |
| Cubic | y = ax³ + ... | S-curve |
| Reciprocal | y = a ⁄ x | Two curved branches |
| Exponential | y = a x or y = ka x | Increasing/decreasing curve |
| Trigonometric | y = sin x, cos x, tan x | Wave patterns |
Practice (10 minutes)
Q: What shape is the graph of y = x³ - 2x?
Answer: S-shaped cubic curve
Plenary (5 minutes)
Explain Back
Your student teaches the key points back to you without looking. Fill any gaps immediately.
Lesson 3: Application: other graphs
Duration: 50 minutes
Starter Activity (5 minutes)
Quick Recall
Recall the key terms: Types of graphs. 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: What shape is the graph of y = x³ - 2x?
Answer: S-shaped cubic curve
Q2: Where are the asymptotes on y = 2 ⁄ x ?
Answer: x = 0 and y = 0 (the axes)
Q3: What point does y = 3 x pass through?
Answer: (0, 1)
Q4: Sketch y = sin x for 0° ≤ x ≤ 360°. Where does it cross the x-axis?
Answer: x = 0°, 180°, 360°
Q5: Which graph has branches in quadrants 1 and 2: y = 1 ⁄ x or y = 1 ⁄ x² ?
Answer: y = 1 ⁄ x² (both branches are above x-axis)
Q6: What is the maximum value of y = cos x?
Answer: 1
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: other 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: Match each equation to its graph description: (i) y = x³ - 3x, (ii) y = 4 ⁄ x , (iii) y = 3 x , (iv) y = sin x. A: Wave between -1 and 1, B: Two branches in opposite quadrants, C: S-curve crossing x-axis 3 times, D: Curve passing through (0,1) increasing rapidly. <div class="
(i) y = x³ - 3x → C (cubic, factorises to x(x²-3) = x(x-√3)(x+√3), 3 x-intercepts) (ii) y = 4 ⁄ x → B (reciprocal, branches in 1st and 3rd quadrants, asymptotes at axes) (iii) y = 3 x → D (exponential, passes through (0,1), increases rapidly) (iv) y = sin x → A (trigonometric wave, oscillates between -1 and 1) Mark scheme: 1.5 marks each for correct matching with reasoning (6 total).
Exam Tips: Learn the shapes: linear (straight), quadratic (U), cubic (S), reciprocal (two branches) | Reciprocal graphs never touch the axes | Exponential graphs pass through (0, 1) | Trig graphs repeat periodically | Always label key points: intercepts, asymptotes
Common Errors: Watch Out! 1. Wrong: A reciprocal graph touches the axes Correct: A reciprocal graph has asymptotes at the axes — it never touches them 2. Wrong: y = 2 x passes through (1, 0) Correct: y = 2 x passes through (0, 1) — when x = 0, y = 2⁰ = 1 3. Wrong: A cubic graph always passes through the origin Correct: Only y = x³ passes through the origin; y = x³ + 2 is shifted up and doesn't
AO3 - Reasoning & Interpretation: Reasoning and Interpretation Drug concentration in blood is modelled by C = 10 × (0.5) t mg/L, where t is hours after injection. (a) What is the initial concentration? (b) What type of graph does this produce? (c) The drug is effective above 1 mg/L. For how many hours is it effective? Answers: (a) C = 10 × 0.5⁰ = 10 mg/L. (b) Exponential decay — decreasing towards 0. (c) 10 × 0.5 t > 1 → 0.5 t > 0.1 → t 0.5 (0.1) ≈ 3.32 hours.
Stretch & Challenge (Grade 8-9):
- Synoptic links: explain how other graphs connects to another Mathematics topic you have studied
- Real-world: research one real-world use or example of other graphs
- Critical: "What are the limitations of the models used in other 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: Types of graphs (10 mins)
- Exam practice: One past-paper question on other graphs from the board websites (15 mins)
- Extension: Explain other graphs to someone else in your own words (10 mins)