Math lesson plan · 10th grade (Year 11 in England) · 30 minutes
Solve quadratics by completing the square (exact answers)
Algebra
CCSS HSA.REI.B.4.ACCSS HSA.REI.B.4.BEngland: GCSE Higher Algebra
Objective and success criteria
Students will: Solve quadratics by completing the square (exact answers).
- 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: Completing the square; Solve quadratics by taking square roots
Leads to: the next skill in this area.
Materials: Balance scales or a balance drawing, Algebra tiles, Function machine cards, Mini whiteboards and pens, Printed student sheet (below).
Warm-up · 4 min
- Quick retrieval from earlier learning (Completing the square; Solve quadratics by taking square roots). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- Write x² − 8x + 23 in the form (x + p)² + q. What is q? → 7Prompt: p is half of the coefficient of x.
- Solve: x² − 18 = 31. Give both solutions, separated by a comma. → x = -7 or x = 7Prompt: Get x² on its own first.
- Find the turning point of y = x² + 6x + 18 by completing the square. Type it like (2, −1). → (−3, 9)Prompt: Write it as (x − h)² + k.
Hook · 3 min
- Show a "think of a number" puzzle and ask the class to work out the starting number.
- Share the objective: Solve quadratics by completing the square (exact answers). 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: Rewrite x² + 8x − 7 = 0 as (x + 4)² = k. Find k.
- Step 1: (x + 4)² − 16 − 7 = 0
- Step 2: k = 23
- Answer: 23
- 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.
- Rewrite x² + 6x − 12 = 0 as (x + 3)² = k. Find k. → 21Prompt: (x + 3)² = x² + 6x + 9.
- Solve x² + 8x − 7 = 0 by completing the square. Give exact answers. (x = −8 ± √23 / x = −4 ± √23 / x = −4 ± √39 / x = 4 ± √23) → x = −4 ± √23Prompt: Write the left side as (x + p)² minus a number.
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.
- Solve x² + 2x − 22 = 0 by completing the square. Give the larger solution to 2 decimal places. → 3.8Prompt: Halve the x coefficient: (x + 1)².
- Solve x² + 8x − 14 = 0 by completing the square. Give exact answers. (x = −4 ± √46 / x = −4 ± √30 / x = −4 ± 30 / x = 4 ± √30) → x = −4 ± √30Prompt: Write the left side as (x + p)² minus a number.
Watch out for
- Doing an operation to one side of an equation but not the other.
- Reading 3x as 3 + x, or thinking x always stands for one particular number.
Support and stretch
Support
- Review first: Completing the square.
- Use concrete materials: balance scales or a balance drawing.
- 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