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Math lesson plan · 9th grade (Year 10 in England) · 45 minutes

Factorise fully: common factor first, then the bracket

Algebra

CCSS HSA.SSE.A.2England: GCSE Higher Algebra

Objective and success criteria

Students will: Factorise fully: common factor first, then the bracket.

  • 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: Take out the greatest common factor of a polynomial; Difference of two squares; Factorise quadratics x² + bx + c

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 (Take out the greatest common factor of a polynomial; Difference of two squares; Factorise quadratics x² + bx + c). Students answer on mini whiteboards.
  • Show answers together and note who needs support in the main task.
  1. Complete the factorisation: 6x⁴ + 4x³ + 4x² = 2x²( … ). Type what goes inside the bracket. → 3x^2+2x+2Prompt: Divide each term by 2x².
  2. Factorise: x² − 36 ((x + 6)² / (x − 6)² / (x + 36)(x − 1) / (x + 6)(x − 6)) → (x + 6)(x − 6)Prompt: This is a difference of two squares: A² − B² = (A + B)(A − B).
  3. Factorise: x² + 4x − 21 ((x − 3)(x − 7) / (x − 21)(x + 1) / (x + 4)(x + 1) / (x − 3)(x + 7)) → (x − 3)(x + 7)Prompt: Find two numbers that multiply to −21 and add to 4.

Hook · 5 min

  • Show a "think of a number" puzzle and ask the class to work out the starting number.
  • Share the objective: Factorise fully: common factor first, then the bracket. 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: Factorise fully: 2x² + 8x + 6
  • Step 1: 2(x² + 4x + 3)
  • Step 2: 2(x + 1)(x + 3)
  • Answer: 2(x + 1)(x + 3)
  • 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. Factorise fully: 3x² − 12x − 15 (3(x² − 4x − 5) / (3x + 1)(x − 5) / 3(x − 1)(x + 5) / 3(x + 1)(x − 5)) → 3(x + 1)(x − 5)Prompt: Every coefficient is a multiple of 3: take it out first.
  2. Factorise fully: 2x² − 162 (2(x + 9)(x − 9) / 2(x² − 81) / 2(x − 9)² / (x + 18)(x − 9)) → 2(x + 9)(x − 9)Prompt: Take out the common factor.

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. Factorise fully: 2x² − 2 ((2x − 1)(x + 1) / 2(x² − 1) / 2(x − 1)(x + 1) / 2(x + 1)(x − 1)) → 2(x − 1)(x + 1)Prompt: Every coefficient is a multiple of 2: take it out first.
  2. Ana says x³ − 100x = x(x² − 100) is fully factorised. What is the full factorisation? (x(x + 10)(x − 10) / x(x² − 100) / x(x − 10)² / x²(x − 100)) → x(x + 10)(x − 10)Prompt: A bracket can often be factorised further.

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: Take out the greatest common factor of a polynomial.
  • 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-9.

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

eurekami

Factorise fully: common factor first, then the bracket

Name

Date

Score/ 8

  1. Question 1

    Factorise fully: 3x² − 18x + 15

    • A. 3(x − 5)(x − 1)
    • B. 3(x + 5)(x + 1)
    • C. (3x − 5)(x − 1)
    • D. (x − 5)(x − 1)

  2. Question 2

    Hiro says x³ − 49x = x(x² − 49) is fully factorised. What is the full factorisation?

    • A. x²(x − 49)
    • B. x(x + 7)(x − 7)
    • C. x(x − 7)²
    • D. (x + 7)(x − 7)

  3. Question 3

    Factorise fully: 4x² − 144

    • A. 4(x + 6)(x − 6)
    • B. (4x + 6)(x − 6)
    • C. 4(x² − 36)
    • D. (x + 24)(x − 6)

  4. Question 4

    Factorise fully: 4x² + 8x − 60

    • A. (x − 3)(x + 5)
    • B. 4(x − 3)(x + 5)
    • C. 4(x² + 2x − 15)
    • D. 4(x + 3)(x − 5)

  5. Question 5

    Jamal says x³ − 4x = x(x² − 4) is fully factorised. What is the full factorisation?

    • A. x(x − 2)²
    • B. x²(x − 4)
    • C. x(x² − 4)
    • D. x(x + 2)(x − 2)

  6. Question 6

    Factorise fully: 5x² − 125

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

  7. Question 7

    Factorise fully: 4x² − 8x − 32

    • A. 4(x + 2)(x − 4)
    • B. (x + 2)(x − 4)
    • C. (4x + 2)(x − 4)
    • D. 4(x − 2)(x + 4)

  8. Question 8

    Liam says x³ − 4x = x(x² − 4) is fully factorised. What is the full factorisation?

    • A. (x + 2)(x − 2)
    • B. x(x² − 4)
    • C. x(x + 2)(x − 2)
    • D. x²(x − 4)

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Answer key: Factorise fully: common factor first, then the bracket

  1. 1. A. 3(x − 5)(x − 1)

  2. 2. B. x(x + 7)(x − 7)

  3. 3. A. 4(x + 6)(x − 6)

  4. 4. B. 4(x − 3)(x + 5)

  5. 5. D. x(x + 2)(x − 2)

  6. 6. B. 5(x + 5)(x − 5)

  7. 7. A. 4(x + 2)(x − 4)

  8. 8. C. x(x + 2)(x − 2)