Math lesson plan · 10th grade (Year 11 in England) · 45 minutes
Find the centre and radius from x² + y² + Dx + Ey + F = 0
Geometry
CCSS HSG.GPE.A.1England: A-level Pure Mathematics
Objective and success criteria
Students will: Find the centre and radius from x² + y² + Dx + Ey + F = 0.
- 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: Write the equation of a circle from its centre and radius; Completing the square
Leads to: the next skill in this area.
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 (Write the equation of a circle from its centre and radius; Completing the square). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- Which is the equation of the circle with centre (−2, 3) and radius 5? ((x − 2)² + (y + 3)² = 25 / (x − 3)² + (y + 2)² = 25 / (x + 2)² + (y − 3)² = 5 / (x + 2)² + (y − 3)² = 25) → (x + 2)² + (y − 3)² = 25Prompt: (x − a)² + (y − b)² = r² has centre (a, b).
- Write x² + 2x + 6 in the form (x + p)² + q. What is p? → 1Prompt: p is half of the coefficient of x.
- A circle has centre (−1, −4) and passes through (−3, −3). Its equation is (x + 1)² + (y + 4)² = k. Find k. → 5Prompt: k is the radius squared.
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: Find the centre and radius from x² + y² + Dx + Ey + F = 0. 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: Find the centre of the circle x² + y² + 6x − 12y + 36 = 0. Type it like (2, −1).
- Step 1: (x + 3)² + (y − 6)² = 9
- Step 2: (−3, 6)
- Answer: (−3, 6)
- 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.
- Find the centre of the circle x² + y² − 12x − 8y + 27 = 0. Type it like (2, −1). → (6, 4)Prompt: Complete the square for the x terms and for the y terms.
- Find the radius of the circle x² + y² + 10x − 2y − 23 = 0. → 7Prompt: Complete the square in x and in y.
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.
- Find the radius of the circle x² + y² + 6x − 8y + 24 = 0. → 1Prompt: Complete the square in x and in y.
- Find the centre of the circle x² + y² − 10x − 12y + 45 = 0. Type it like (2, −1). → (5, 6)Prompt: Complete the square for the x terms and for the y terms.
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: Write the equation of a circle from its centre and radius.
- 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
- 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