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circles

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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: 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.
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.
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

✅ Success Criteria

You are ready for circles - GCSE Mathematics Lessons when you can:

  • [ ] Define the key terms in your own words
  • [ ] Explain the core concept with an example
  • [ ] Apply the method to an unfamiliar context
  • [ ] Score 80%+ on past-paper questions

Tick each item honestly — if any box is unticked, re-read the relevant section above.

Board Notes & Revision Links

This lesson is examined by: AQA 8300 | Edexcel 1MA1 | OCR J560 | Eduqas C300 | CCEA M-units. Tier and paper details vary - always check your board's current specification.

Follow-up revision: G17-circles-area-circumference on GCSE Revise (notes, flashcards and auto-graded target tests for this exact topic).

📐 Reference Diagram

x y y = x² Origin (0,0)

✏️ Draw-Along Instructions

Parent/teacher: Draw the shape using a ruler. Label all sides with their lengths. Mark equal angles with arcs. Add any parallel-line arrows. Write the formula you will use inside the shape.

Display the reference diagram (or draw it live) while pupils copy. Check each label as it is added.

Homework & Consolidation

Recommended Resources

🎓 Smart Lesson (Guided)