Math lesson plan · 10th grade (Year 11 in England) · 30 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 · 4 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.
- What is the centre of the circle (x + 5)² + (y − 4)² = 9? Type it like (2, −1). → (−5, 4)Prompt: The form is (x − a)² + (y − b)² = r², with centre (a, b).
- The circle x² + y² = r² passes through (0, 5). What is r? → 5Prompt: Substitute the point into x² + y².
- What is the centre of the circle x² + (y − 6)² = 36? Type it like (2, −1). → (0, 6)Prompt: The form is (x − a)² + (y − b)² = r², with centre (a, b).
Hook · 3 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 · 7 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 · 6 min
- Do these together. Students try each step on whiteboards before you reveal it.
- Use the hints as prompts rather than giving the answer.
- Which is the equation of the circle with centre (5, −5) and radius 5? ((x + 5)² + (y − 5)² = 5 / (x − 5)² + (y + 5)² = 25 / (x + 5)² + (y − 5)² = 25 / (x − 5)² + (y + 5)² = 5) → (x − 5)² + (y + 5)² = 25Prompt: (x − a)² + (y − b)² = r² has centre (a, b).
- A circle has centre (0, 0) and passes through (−4, 6). Its equation is x² + y² = k. Find k. → 52Prompt: k is the radius squared.
You do: independent practice · 7 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 6 questions are on the student sheet below, with answers in the key.
Exit ticket · 3 min
- Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
- A circle has centre (−1, −1) and passes through (−6, 1). Its equation is (x + 1)² + (y + 1)² = k. Find k. → 29Prompt: k is the radius squared.
- Where is the point (−8, −6) relative to the circle (x + 2)² + (y + 5)² = 36? (Inside the circle / On the circle / Outside the circle) → Outside 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