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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 − 11x + 18 ((x − 11)(x + 1) / (x − 9)(x − 2) / (x + 18)(x + 1) / (x + 9)(x + 2)) → (x − 9)(x − 2)Prompt: Find two numbers that multiply to 18 and add to −11.
  2. Factorise: 9x² − 4 ((3x + 2)(3x − 2) / (3x + 2)² / 3(x + 2)(x − 2) / (3x − 2)²) → (3x + 2)(3x − 2)Prompt: This is a difference of two squares: A² − B² = (A + B)(A − B).
  3. Factorise fully: x² + 9x + 20 → (x+4)(x+5)Prompt: Two numbers: product 20, sum 9.

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 + 4x − 5)/(x² − 1) ((x + 5)/(x + 1) / (x + 4)/(x − 1) / (x + 5)/(x − 1) / (x − 5)/(x + 1)) → (x + 5)/(x + 1)Prompt: Factorise the top and the bottom.
  2. Chloe simplified (x + 7)/x by crossing out the x's and got 1 + 7 = 8. 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 + 7)/x. Do you get 8? 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 − 16)/(x + 4) ((x + 4) / x − 16 / 1/(x − 4) / (x − 4)) → (x − 4)Prompt: Factorise the top: it is a difference of two squares.
  2. Ines simplified (x + 8)/x by crossing out the x's and got 1 + 8 = 9. 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 + 8)/x. Do you get 9? 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

    Chloe simplified (x + 2)/x by crossing out the x's and got 1 + 2 = 3. 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 − 49)/(x + 7)

    • A. (x² − 7)
    • B. 1/(x − 7)
    • C. x − 49
    • D. (x − 7)

  3. Question 3

    Nia 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

  4. Question 4

    Simplify: (x² + 13x + 42)/(x + 7)

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

  5. Question 5

    Tariq 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 − x − 2)/(x + 1)

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

  7. Question 7

    Isla simplified (x + 8)/x by crossing out the x's and got 1 + 8 = 9. 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 + 15)/(x − 5)

    • A. (x + 15)
    • B. (x − 3)/(x − 5)
    • C. (x − 8)
    • D. (x − 3)

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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. D. (x − 7)

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

  4. 4. B. (x + 6)

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

  6. 6. D. (x − 2)

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

  8. 8. D. (x − 3)