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Math lesson plan · 10th grade (Year 11 in England) · 45 minutes

Write the equation of a circle from its centre and radius

Geometry

CCSS HSG.GPE.A.1England: GCSE Higher Algebra

Objective and success criteria

Students will: Write the equation of a circle from its centre and radius.

  • I can answer questions on today's skill on my own, without help.
  • I can explain each step and say why it works.
  • I can check my answer and spot a mistake.

Before and after

Needs first: Equation of a circle

Leads to: Find the centre and radius from x² + y² + Dx + Ey + F = 0

Materials: Rulers and protractors, Squared or dotted paper, Shape cut-outs, Mini whiteboards and pens, Printed student sheet (below).

Warm-up · 5 min

  • Quick retrieval from earlier learning (Equation of a circle). Students answer on mini whiteboards.
  • Show answers together and note who needs support in the main task.
  1. What is the radius of the circle (x − 5)² + (y − 4)² = 81? → 9Prompt: The form is (x − a)² + (y − b)² = r².
  2. The circle x² + y² = r² passes through (−15, 20). What is r? → 25Prompt: Substitute the point into x² + y².
  3. What is the radius of the circle (x + 3)² + (y − 4)² = 1? → 1Prompt: The form is (x − a)² + (y − b)² = r².

Hook · 5 min

  • Show a photo of a building, a tile pattern or a road sign and ask what shapes and angles students can spot.
  • Share the objective: Write the equation of a circle from its centre and radius. Read the success criteria together.

I do: model it · 10 min

  • Work through the example out loud, saying why you do each step, not just what you do.
  • Example: Which is the equation of the circle with centre (−1, −1) and radius 4?
  • Step 1: (x + 1)² + (y + 1)² = 16
  • Answer: (x + 1)² + (y + 1)² = 16
  • Pause after each step and ask a student to say what comes next.

We do: guided practice · 10 min

  • Do these together. Students try each step on whiteboards before you reveal it.
  • Use the hints as prompts rather than giving the answer.
  1. Which is the equation of the circle with centre (−5, −4) and radius 4? ((x + 4)² + (y + 5)² = 16 / (x − 5)² + (y − 4)² = 16 / (x − 5)² + (y − 4)² = 4 / (x + 5)² + (y + 4)² = 16) → (x + 5)² + (y + 4)² = 16Prompt: (x − a)² + (y − b)² = r² has centre (a, b).
  2. A circle has centre (0, 4) and passes through (−6, −1). Its equation is x² + (y − 4)² = k. Find k. → 61Prompt: k is the radius squared.

You do: independent practice · 10 min

  • Students work on the student sheet. Circulate and check the first two answers of every student early.
  • Students who finish: write a question of their own for a partner, with the answer.

The 8 questions are on the student sheet below, with answers in the key.

Exit ticket · 5 min

  • Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
  1. A circle has centre (−4, 5) and passes through (5, 0). Its equation is (x + 4)² + (y − 5)² = k. Find k. → 106Prompt: k is the radius squared.
  2. Where is the point (−1, 5) relative to the circle (x + 2)² + (y − 5)² = 9? (Inside the circle / On the circle / Outside the circle) → Inside the circlePrompt: Substitute the point into the left-hand side.

Watch out for

  • Thinking a square turned on its corner is no longer a square.
  • Reading the wrong scale on a protractor.

Support and stretch

Support

  • Review first: Equation of a circle.
  • Use concrete materials: rulers and protractors.
  • Give the first step of each question, or a partly worked example to finish.
  • Pair with a confident partner for the guided practice.

Stretch

  • Move on to: Find the centre and radius from x² + y² + Dx + Ey + F = 0.
  • Explain a common mistake in writing and how to fix it.
  • Write a word problem that uses today's skill, then solve a partner's.

Homework and more practice

Free online practice with hints and printable worksheets for this skill: eurekami.com/math/grade-10.

Free lesson plan from eurekami.com/lesson-plan-builder

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Write the equation of a circle from its centre and radius

Name

Date

Score/ 8

  1. Question 1

    Which is the equation of the circle with centre (−6, −4) and radius 8?

    • A. (x − 6)² + (y − 4)² = 8
    • B. (x + 6)² + (y + 4)² = 8
    • C. (x − 6)² + (y − 4)² = 64
    • D. (x + 6)² + (y + 4)² = 64

  2. Question 2

    Where is the point (6, −1) relative to the circle (x − 1)² + (y + 2)² = 9?

    • A. Inside the circle
    • B. On the circle
    • C. Outside the circle

  3. Question 3

    A circle has centre (1, 1) and passes through (−2, 3). Its equation is (x − 1)² + (y − 1)² = k. Find k.

    Answer

  4. Question 4

    Which is the equation of the circle with centre (−3, 3) and radius 5?

    • A. (x − 3)² + (y + 3)² = 5
    • B. (x − 3)² + (y + 3)² = 25
    • C. (x + 3)² + (y − 3)² = 25
    • D. (x + 3)² + (y − 3)² = 5

  5. Question 5

    Where is the point (7, −5) relative to the circle (x − 3)² + (y + 5)² = 9?

    • A. Inside the circle
    • B. On the circle
    • C. Outside the circle

  6. Question 6

    A circle has centre (−2, 4) and passes through (−6, −1). Its equation is (x + 2)² + (y − 4)² = k. Find k.

    Answer

  7. Question 7

    Which is the equation of the circle with centre (−1, 5) and radius 5?

    • A. (x + 1)² + (y − 5)² = 25
    • B. (x − 1)² + (y + 5)² = 25
    • C. (x − 5)² + (y + 1)² = 25
    • D. (x + 1)² + (y − 5)² = 5

  8. Question 8

    Where is the point (−9, 3) relative to the circle (x + 3)² + (y − 3)² = 36?

    • A. Inside the circle
    • B. On the circle
    • C. Outside the circle

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Answer key: Write the equation of a circle from its centre and radius

  1. 1. D. (x + 6)² + (y + 4)² = 64

  2. 2. C. Outside the circle

  3. 3. 13

  4. 4. C. (x + 3)² + (y − 3)² = 25

  5. 5. C. Outside the circle

  6. 6. 41

  7. 7. A. (x + 1)² + (y − 5)² = 25

  8. 8. B. On the circle