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functions

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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: functions

Duration: 50 minutes

Starter Activity (5 minutes)

Quick Recall

Write down everything you already know about functions. Then check against the key terms: Function, Key terms. Use a mini-whiteboard or paper.

Main Content (35 minutes)

Parent/Teacher Guide:
Before lesson: Read the script below. Pre-teach key vocab: Function, Key terms.
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: functions. 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: Given f(x) = 4x - 3, find f(5) and f(-2).

Plenary (5 minutes)

Check Out

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

Lesson 2: Core Concepts: functions

Duration: 50 minutes

Starter Activity (5 minutes)

Review Previous Lesson

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

Main Content (35 minutes)

Function: A rule that takes an input (x) and gives exactly one output. Written as f(x), g(x), etc.
Key terms: Domain: All possible input values Range: All possible output values

Practice (10 minutes)

Q: Given f(x) = 4x - 3, find f(5) and f(-2).

Answer: f(5) = 17, f(-2) = -11

Plenary (5 minutes)

Explain Back

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

Lesson 3: Application: functions

Duration: 50 minutes

Starter Activity (5 minutes)

Quick Recall

Recall the key terms: Function, Key terms. 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: Given f(x) = 4x - 3, find f(5) and f(-2).

Answer: f(5) = 17, f(-2) = -11

Q2: Given g(x) = x² - 1, find g(3) and g(x + 1).

Answer: g(3) = 8, g(x + 1) = (x + 1)² - 1 = x² + 2x

Q3: Find the inverse of f(x) = 2x - 7.

Answer: f⁻¹(x) = x + 7 ⁄ 2

Q4: Find the inverse of g(x) = x ⁄ 4 + 1.

Answer: g⁻¹(x) = 4(x - 1) = 4x - 4

Q5: Given f(x) = x + 3 and g(x) = 2x, find fg(x) and gf(x).

Answer: fg(x) = 2x + 3, gf(x) = 2(x + 3) = 2x + 6

Q6: Given f(x) = x² and g(x) = x + 1, find fg(2) and gf(2).

Answer: fg(2) = f(g(2)) = f(3) = 9, gf(2) = g(f(2)) = g(4) = 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: functions

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: f(x) = 3x - 5 and g(x) = x + 2 ⁄ 4 . (a) Find f⁻¹(x). (b) Find gf(3). (c) Show that fg(x) ≠ gf(x) by finding both expressions. <div class="

(a) y = 3x - 5 → y + 5 = 3x → x = y + 5 ⁄ 3 , so f⁻¹(x) = x + 5 ⁄ 3 (b) f(3) = 3(3) - 5 = 4. g(4) = 4 + 2 ⁄ 4 = 6 ⁄ 4 = 1.5 (c) fg(x) = f( x + 2 ⁄ 4 ) = 3( x + 2 ⁄ 4 ) - 5 = 3x + 6 ⁄ 4 - 5 = 3x - 14 ⁄ 4 gf(x) = g(3x - 5) = 3x - 5 + 2 ⁄ 4 = 3x - 3 ⁄ 4 These are different: 3x - 14 ⁄ 4 ≠ 3x - 3 ⁄ 4 Mark scheme: (a) 2 marks. (b) 1 mark for f(3), 1 mark for final answer. (c) 1 mark for fg(x), 1 mark for gf(x).

Exam Tips: For composite functions, work from inside out: fg(x) means f(g(x)) | To find inverse, swap x and y, then make y the subject | Check your inverse: f(f⁻¹(x)) should equal x | Domain restrictions may apply to inverse functions | Read carefully: fg(x) ≠ gf(x)
Common Errors: Watch Out! 1. Wrong: fg(x) means f × g(x) Correct: fg(x) = f(g(x)) — apply g first, then f to the result 2. Wrong: f⁻¹(x) means 1 ⁄ f(x) Correct: f⁻¹(x) is the inverse function, not the reciprocal 3. Wrong: fg(x) = gf(x) always Correct: Order matters! fg(x) ≠ gf(x) in general
AO3 - Reasoning & Interpretation: Reasoning and Interpretation f(x) = 2x + 3 converts a temperature from °C to an adjusted scale. g(x) = x - 3 ⁄ 2 is its inverse. (a) What does f(0) represent? (b) If the output of f is 15, what was the input? (c) Explain why g undoes f. Answers: (a) f(0) = 3 — the adjusted value when the input is 0°C. (b) f⁻¹(15) = 15 - 3 ⁄ 2 = 6. (c) gf(x) = g(2x+3) = 2x+3-3 ⁄ 2 = x. So applying g after f returns the original value.
Stretch & Challenge (Grade 8-9):
  • Synoptic links: explain how functions connects to another Mathematics topic you have studied
  • Real-world: research one real-world use or example of functions
  • Critical: "What are the limitations of the models used in functions?"

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)