circles
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 circles
- Apply circles to exam-style questions
- Develop independent study skills
- Build cross-topic connections
Prerequisites
- Read the revision notes for circles first (GCSE Revise summary)
- Key vocab to pre-teach: Equation of a circle, General form
- 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: circles
Duration: 50 minutes
Starter Activity (5 minutes)
Quick Recall
Write down everything you already know about circles. Then check against the key terms: Equation of a circle, General form. Use a mini-whiteboard or paper.
Main Content (35 minutes)
Parent/Teacher Guide:
Before lesson: Read the script below. Pre-teach key vocab: Equation of a circle, General form.
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: Equation of a circle, General form.
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: circles. 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: circles. 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: Write the equation of a circle with centre (0, 0) and radius 7.
Plenary (5 minutes)
Check Out
Your student states one thing they learned and one question they still have about circles.
Lesson 2: Core Concepts: circles
Duration: 50 minutes
Starter Activity (5 minutes)
Review Previous Lesson
Quick recap: write 3 key points from Lesson 1 on circles. Check them against the notes below.
Main Content (35 minutes)
Equation of a circle: A circle with centre (0, 0) and radius r has equation: x² + y² = r²
General form: For a circle with centre (a, b) and radius r: (x - a)² + (y - b)² = r²
Practice (10 minutes)
Q: Write the equation of a circle with centre (0, 0) and radius 7.
Answer: x² + y² = 49
Plenary (5 minutes)
Explain Back
Your student teaches the key points back to you without looking. Fill any gaps immediately.
Lesson 3: Application: circles
Duration: 50 minutes
Starter Activity (5 minutes)
Quick Recall
Recall the key terms: Equation of a circle, General form. 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: Write the equation of a circle with centre (0, 0) and radius 7.
Answer: x² + y² = 49
Q2: Find the radius of x² + y² = 36.
Answer: 6
Q3: Does (5, 12) lie on x² + y² = 169?
Answer: Yes: 5² + 12² = 25 + 144 = 169 ✓
Q4: Find the centre and radius of (x - 4)² + (y + 1)² = 9.
Answer: Centre (4, -1), Radius 3
Q5: Find the gradient of the tangent to x² + y² = 25 at point (4, 3).
Answer: m_radius = 3 ⁄ 4 , so m_tangent = - 4 ⁄ 3
Q6: Find the centre of x² + y² + 8x - 2y - 8 = 0.
Answer: Centre (-4, 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: circles
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) Find the centre and radius of the circle x² + y² + 4x - 6y - 3 = 0. (b) Does (1, 4) lie inside, on or outside the circle? (c) Find the equation of the tangent at point (2, 3+√3) on the circle. <div class="
(a) (x² + 4x) + (y² - 6y) = 3. (x + 2)² - 4 + (y - 3)² - 9 = 3. (x + 2)² + (y - 3)² = 16. Centre (-2, 3), radius 4. (b) Distance from (1, 4) to (-2, 3) = √(9 + 1) = √10 ≈ 3.16. Since 3.16 < 4, point is inside the circle. (c) Gradient from centre to point: (3+√3 - 3)/(2-(-2)) = √3/4. Tangent gradient = -4/√3. Equation: y - (3+√3) = -4/√3(x - 2). Mark scheme: (a) 2 marks. (b) 2 marks for distance and conclusion. (c) 2 marks.
Exam Tips: x² + y² = r² for circle at origin | (x - a)² + (y - b)² = r² for centre (a, b) | Tangent is perpendicular to radius | Use negative reciprocal for perpendicular gradient | Complete the square to find centre and radius from expanded form
Common Errors: Watch Out! 1. Wrong: (x + 3)² + (y - 2)² = 25 has centre (3, -2) Correct: Centre is (-3, 2) — the signs in the equation are opposite to the coordinates 2. Wrong: The tangent has the same gradient as the radius Correct: The tangent is perpendicular to the radius — use the negative reciprocal 3. Wrong: x² + y² - 6x = 0 has radius 6 Correct: Complete the square: (x-3)² + y² = 9, so radius = 3
AO3 - Reasoning & Interpretation: Reasoning and Interpretation A circular fountain has equation x² + y² = 100 (units in metres). A path has equation 3x + 4y = 50. (a) Find the shortest distance from the centre to the path. (b) Does the path cross the fountain? (c) If the fountain radius increases by 2m, will the path now cross it? Answers: (a) Distance from (0,0) to line = |50|/√(9+16) = 50/5 = 10 m. (b) No — the distance equals the radius exactly, so the path is a tangent. (c) New radius = 12 m > 10 m, so yes, the path would now cross the fountain at two points.
Stretch & Challenge (Grade 8-9):
- Synoptic links: explain how circles connects to another Mathematics topic you have studied
- Real-world: research one real-world use or example of circles
- Critical: "What are the limitations of the models used in circles?"
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: Equation of a circle, General form (10 mins)
- Exam practice: One past-paper question on circles from the board websites (15 mins)
- Extension: Explain circles to someone else in your own words (10 mins)