types of sequences
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 types of sequences
- Apply types of sequences to exam-style questions
- Develop independent study skills
- Build cross-topic connections
Prerequisites
- Read the revision notes for types of sequences first (GCSE Revise summary)
- Key vocab to pre-teach: key terms from types of sequences
- 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: types of sequences
Duration: 50 minutes
Starter Activity (5 minutes)
Quick Recall
Write down everything you already know about types of sequences. Then check against the key terms: key terms from types of sequences. Use a mini-whiteboard or paper.
Main Content (35 minutes)
Parent/Teacher Guide:
Before lesson: Read the script below. Pre-teach key vocab: key terms from types of sequences.
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: key terms from types of sequences.
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: types of sequences. 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: types of sequences. 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 nth term of: 4, 9, 14, 19, 24, ...
Plenary (5 minutes)
Check Out
Your student states one thing they learned and one question they still have about types of sequences.
Lesson 2: Core Concepts: types of sequences
Duration: 50 minutes
Starter Activity (5 minutes)
Review Previous Lesson
Quick recap: write 3 key points from Lesson 1 on types of sequences. Check them against the notes below.
Main Content (35 minutes)
Work through the revision notes for types of sequences section by section, asking your student to explain each idea back.
| Term | Meaning | Example |
|---|---|---|
| Arithmetic | Add/subtract same amount | 3, 7, 11, 15, 19, ... |
| Geometric | Multiply by same amount | 2, 6, 18, 54, ... |
| Square numbers | n² | 1, 4, 9, 16, 25, ... |
| Cube numbers | n³ | 1, 8, 27, 64, 125, ... |
| Triangular numbers | Sum of first n integers | 1, 3, 6, 10, 15, ... |
| Fibonacci | Add previous two terms | 1, 1, 2, 3, 5, 8, 13, ... |
Practice (10 minutes)
Q: Find the nth term of: 4, 9, 14, 19, 24, ...
Answer: 5n - 1
Plenary (5 minutes)
Explain Back
Your student teaches the key points back to you without looking. Fill any gaps immediately.
Lesson 3: Application: types of sequences
Duration: 50 minutes
Starter Activity (5 minutes)
Quick Recall
Recall the key terms: key terms from types of sequences. 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 nth term of: 4, 9, 14, 19, 24, ...
Answer: 5n - 1
Q2: Find the nth term of: 2, 6, 18, 54, ...
Answer: 2 × 3 n-1
Q3: What is the 8th triangular number?
Answer: 36
Q4: Find the next 3 terms: 1, 1, 2, 3, 5, 8, ...
Answer: 13, 21, 34
Q5: Find the nth term of: 2, 5, 10, 17, 26, ...
Answer: n² + 1
Q6: What type of sequence is: 100, 90, 81, 72.9, ...?
Answer: Geometric (ratio = 0.9)
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: types of sequences
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 ball is dropped from 80 cm. Each bounce reaches 3/4 of the previous height. (a) Write the first 5 heights as a sequence. (b) What type of sequence is this? Find the nth term. (c) After which bounce is the height below 10 cm? <div class="
(a) 80, 60, 45, 33.75, 25.3125 (b) Geometric sequence with a = 80, r = 3/4. nth term = 80 × (3/4)^(n-1) (c) 80 × (3/4)^(n-1) log(1/8)/log(3/4) ≈ 8.00. So n > 9. After the 9th bounce the height is below 10 cm. Mark scheme: (a) 2 marks. (b) 2 marks for type and formula. (c) 2 marks for solving inequality.
Exam Tips: Arithmetic: constant difference → an + b form | Geometric: constant ratio → a × r n-1 form | Fibonacci: each term = sum of previous two | For quadratic: check 2nd difference is constant | Triangular: n(n+1)/2
Common Errors: Watch Out! 1. Wrong: Treating a geometric sequence as arithmetic Correct: Check whether you ADD or MULTIPLY to get the next term — arithmetic adds, geometric multiplies 2. Wrong: The nth term of a geometric sequence is a × r^n Correct: It is a × r^(n-1) — the first term has n = 1, so r^(1-1) = r⁰ = 1, giving just a 3. Wrong: The 10th triangular number is 10 × 11 = 110 Correct: The 10th triangular number is (10 × 11)/2 = 55 (must divide by 2)
AO3 - Reasoning & Interpretation: Reasoning and Interpretation A salary starts at £24,000 and increases by £1,500 each year. (a) Write the nth term for the salary in year n. (b) After how many years does the salary exceed £40,000? (c) A different job starts at £24,000 and increases by 5% each year. Which job pays more in year 10? Answers: (a) Arithmetic: S = 24000 + 1500(n-1) = 1500n + 22500. (b) 1500n + 22500 > 40000 → n > 11.67, so after 12 years. (c) Job 1 year 10: £37,500. Job 2 year 10: 24000 × 1.05^9 ≈ £37,218. Job 1 pays slightly more in year 10.
Stretch & Challenge (Grade 8-9):
- Synoptic links: explain how types of sequences connects to another Mathematics topic you have studied
- Real-world: research one real-world use or example of types of sequences
- Critical: "What are the limitations of the models used in types of sequences?"
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: key terms from types of sequences (10 mins)
- Exam practice: One past-paper question on types of sequences from the board websites (15 mins)
- Extension: Explain types of sequences to someone else in your own words (10 mins)