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Math lesson plan · 10th grade (Year 11 in England) · 45 minutes

Simplify algebraic fractions by factorising

Algebra

CCSS HSA.APR.D.6CCSS HSA.SSE.A.2England: GCSE Higher Algebra

Objective and success criteria

Students will: Simplify algebraic fractions by factorising.

  • 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: Factorise quadratics x² + bx + c; Difference of two squares

Leads to: Multiply and divide algebraic fractions; Evaluate limits by substitution or by factorising

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 (Factorise quadratics x² + bx + c; Difference of two squares). Students answer on mini whiteboards.
  • Show answers together and note who needs support in the main task.
  1. Factorise: x² − 8x − 9 ((x − 1)(x + 9) / (x − 8)(x + 1) / (x + 1)(x + 9) / (x + 1)(x − 9)) → (x + 1)(x − 9)Prompt: Find two numbers that multiply to −9 and add to −8.
  2. Factorise: x² − 16 ((x + 4)² / (x + 4)(x − 4) / (x + 16)(x − 1) / (x − 4)²) → (x + 4)(x − 4)Prompt: This is a difference of two squares: A² − B² = (A + B)(A − B).
  3. Factorise fully: x² + 14x + 48 → (x+6)(x+8)Prompt: Two numbers: product 48, sum 14.

Hook · 5 min

  • Show a "think of a number" puzzle and ask the class to work out the starting number.
  • Share the objective: Simplify algebraic fractions by factorising. 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: Simplify: (x² + 7x + 12)/(x + 3)
  • Step 1: (x + 3)(x + 4)/(x + 3)
  • Step 2: (x + 4)
  • Answer: (x + 4)
  • 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. Simplify: (x² + 11x + 28)/(x + 7) ((x + 4) / (x + 4)/(x + 7) / (x − 4) / (x + 11)) → (x + 4)Prompt: Factorise the numerator.
  2. Mateo simplified (x + 6)/x by crossing out the x's and got 1 + 6 = 7. Is this right? (Right / Wrong: only common factors can be cancelled, not terms / Right, but only when x is positive) → Wrong: only common factors can be cancelled, not termsPrompt: Test it: put x = 1 into (x + 6)/x. Do you get 7? Try x = 2.

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. Simplify: (x² + x − 6)/(x − 2) ((x − 6) / (x + 1) / (x + 3) / (x − 3)) → (x + 3)Prompt: Factorise the numerator.
  2. Ben simplified (x + 9)/x by crossing out the x's and got 1 + 9 = 10. Is this right? (Right / Wrong: only common factors can be cancelled, not terms / Right, but only when x is positive) → Wrong: only common factors can be cancelled, not termsPrompt: Test it: put x = 1 into (x + 9)/x. Do you get 10? Try x = 2.

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: Factorise quadratics x² + bx + c.
  • 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

  • Move on to: Multiply and divide algebraic fractions; Evaluate limits by substitution or by factorising.
  • 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

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Simplify algebraic fractions by factorising

Name

Date

Score/ 8

  1. Question 1

    Luis simplified (x + 7)/x by crossing out the x's and got 1 + 7 = 8. Is this right?

    • A. Right
    • B. Wrong: only common factors can be cancelled, not terms
    • C. Right, but only when x is positive

  2. Question 2

    Simplify: (x² − 3x − 18)/(x² − 36)

    • A. (x − 3)/(x + 6)
    • B. (x − 3)/(x − 36)
    • C. (x + 3)/(x + 6)
    • D. (x − 6)/(x + 6)

  3. Question 3

    Kwame simplified (x + 3)/x by crossing out the x's and got 1 + 3 = 4. Is this right?

    • A. Right
    • B. Wrong: only common factors can be cancelled, not terms
    • C. Right, but only when x is positive

  4. Question 4

    Simplify: (x² − 13x + 42)/(x − 6)

    • A. (x + 7)
    • B. (x + 42)
    • C. (x − 7)
    • D. (x − 13)

  5. Question 5

    Grace simplified (x + 9)/x by crossing out the x's and got 1 + 9 = 10. Is this right?

    • A. Right
    • B. Wrong: only common factors can be cancelled, not terms
    • C. Right, but only when x is positive

  6. Question 6

    Simplify: (x² + 13x + 42)/(x² − 49)

    • A. (x + 13)/(x − 49)
    • B. (x + 6)/(x − 7)
    • C. (x + 6)/(x + 7)
    • D. (x + 7)/(x − 7)

  7. Question 7

    Ana simplified (x + 5)/x by crossing out the x's and got 1 + 5 = 6. Is this right?

    • A. Right
    • B. Wrong: only common factors can be cancelled, not terms
    • C. Right, but only when x is positive

  8. Question 8

    Simplify: (x² + 8x + 7)/(x² − 49)

    • A. (x + 1)/(x + 7)
    • B. (x + 7)/(x − 7)
    • C. (x + 1)/(x − 7)
    • D. (x + 8)/(x − 49)

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Answer key: Simplify algebraic fractions by factorising

  1. 1. B. Wrong: only common factors can be cancelled, not terms

  2. 2. C. (x + 3)/(x + 6)

  3. 3. B. Wrong: only common factors can be cancelled, not terms

  4. 4. C. (x − 7)

  5. 5. B. Wrong: only common factors can be cancelled, not terms

  6. 6. B. (x + 6)/(x − 7)

  7. 7. B. Wrong: only common factors can be cancelled, not terms

  8. 8. C. (x + 1)/(x − 7)