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Math lesson plan · 11th grade (Year 12 in England) · 45 minutes

Convert a quadratic to vertex form

Algebra

CCSS HSF.IF.C.8.ACCSS HSA.SSE.B.3.BEngland: A-level Pure Mathematics

Objective and success criteria

Students will: Convert a quadratic to vertex form.

  • 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; Quadratic functions in vertex form

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 · 5 min

  • Quick retrieval from earlier learning (Completing the square; Quadratic functions in vertex form). Students answer on mini whiteboards.
  • Show answers together and note who needs support in the main task.
  1. Write x² + 10x + 18 in the form (x + p)² + q. What is q? → -7Prompt: p is half of the coefficient of x.
  2. What is the minimum value of f(x) = (x + 2)² + 4? → 4Prompt: The squared term is never negative.
  3. Find the turning point of y = x² + 12x + 45 by completing the square. Type it like (2, −1). → (−6, 9)Prompt: Write it as (x − h)² + k.

Hook · 5 min

  • Show a "think of a number" puzzle and ask the class to work out the starting number.
  • Share the objective: Convert a quadratic to vertex form. 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: Write y = x² − 2x − 8 in vertex form.
  • Step 1: (x − 1)² = x² − 2x + 1
  • Step 2: y = (x − 1)² − 9
  • Answer: y = (x − 1)² − 9
  • 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. Write y = x² − 12x + 30 in vertex form. (y = (x + 6)² − 6 / y = (x − 6)² + 30 / y = (x − 6)² − 6 / y = (x − 12)² − 6) → y = (x − 6)² − 6Prompt: Halve the x coefficient to get the bracket.
  2. Write y = −x² − 6x − 18 in vertex form. (y = −(x − 3)² − 9 / y = (x − 3)² − 9 / y = −(x + 3)² − 9 / y = −(x + 3)² − 18) → y = −(x + 3)² − 9Prompt: Factor −1 out of the x² and x terms first.

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. Write y = −x² + 2x − 9 in vertex form. (y = −(x − 1)² − 8 / y = (x + 1)² − 8 / y = −(x − 1)² − 9 / y = −(x + 1)² − 8) → y = −(x − 1)² − 8Prompt: Factor −1 out of the x² and x terms first.
  2. Write y = x² + 6x − 3 in vertex form. (y = (x − 3)² − 12 / y = (x + 3)² − 3 / y = (x + 3)² − 12 / y = (x + 6)² − 12) → y = (x + 3)² − 12Prompt: Halve the x coefficient to get the bracket.

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-11.

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

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Convert a quadratic to vertex form

Name

Date

Score/ 8

  1. Question 1

    Write y = x² + 8x + 6 in vertex form.

    • A. y = (x + 4)² + 6
    • B. y = (x − 4)² − 10
    • C. y = (x + 4)² − 10
    • D. y = (x + 8)² − 10

  2. Question 2

    Write y = −2x² − 12x − 17 in vertex form.

    • A. y = (x − 6)² + 1
    • B. y = -2(x + 3)² − 17
    • C. y = -2(x + 3)² + 1
    • D. y = -2(x − 3)² + 1

  3. Question 3

    Write y = x² − 2x + 5 in vertex form.

    • A. y = (x − 1)² + 5
    • B. y = (x − 2)² + 4
    • C. y = (x − 1)² + 4
    • D. y = (x + 1)² + 4

  4. Question 4

    Write y = 3x² − 18x + 23 in vertex form.

    • A. y = 3(x + 3)² − 4
    • B. y = 3(x − 3)² − 4
    • C. y = (x − 9)² − 4
    • D. y = 3(x − 3)² + 23

  5. Question 5

    Write y = x² − 12x + 44 in vertex form.

    • A. y = (x + 6)² + 8
    • B. y = (x − 12)² + 8
    • C. y = (x − 6)² + 8
    • D. y = (x − 6)² + 44

  6. Question 6

    Write y = −x² − 2x − 2 in vertex form.

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

  7. Question 7

    Write y = −2x² + 4x + 8 in vertex form.

    • A. y = -2(x − 1)² + 10
    • B. y = (x + 2)² + 10
    • C. y = -2(x + 1)² + 10
    • D. y = -2(x − 1)² + 8

  8. Question 8

    Write y = x² − 4x + 1 in vertex form.

    • A. y = (x + 2)² − 3
    • B. y = (x − 2)² + 1
    • C. y = (x − 4)² − 3
    • D. y = (x − 2)² − 3

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Answer key: Convert a quadratic to vertex form

  1. 1. C. y = (x + 4)² − 10

  2. 2. C. y = -2(x + 3)² + 1

  3. 3. C. y = (x − 1)² + 4

  4. 4. B. y = 3(x − 3)² − 4

  5. 5. C. y = (x − 6)² + 8

  6. 6. B. y = −(x + 1)² − 1

  7. 7. A. y = -2(x − 1)² + 10

  8. 8. D. y = (x − 2)² − 3