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Math lesson plan · 10th grade (Year 11 in England) · 30 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 · 4 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 + 6x + 5 ((x + 1)(x + 5) / (x + 6)(x + 1) / (x + 5)(x + 1) / (x − 1)(x − 5)) → (x + 1)(x + 5)Prompt: Find two numbers that multiply to 5 and add to 6.
  2. Factorise: 16x² − 25 (4(x + 5)(x − 5) / (4x + 5)(4x − 5) / (4x + 5)² / (4x − 5)²) → (4x + 5)(4x − 5)Prompt: This is a difference of two squares: A² − B² = (A + B)(A − B).
  3. Factorise fully: x² + 10x + 24 → (x+4)(x+6)Prompt: Two numbers: product 24, sum 10.

Hook · 3 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 · 7 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 · 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.
  1. Simplify: (x² − 9x + 14)/(x² − 4) ((x + 7)/(x + 2) / (x − 9)/(x − 4) / (x − 7)/(x + 2) / (x − 2)/(x + 2)) → (x − 7)/(x + 2)Prompt: Factorise the top and the bottom.
  2. Ines 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.

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.
  1. Ben simplified (x + 2)/x by crossing out the x's and got 1 + 2 = 3. 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 + 2)/x. Do you get 3? Try x = 2.
  2. Simplify: (x² − 11x + 28)/(x² − 16) ((x − 11)/(x − 16) / (x − 7)/(x − 4) / (x − 4)/(x + 4) / (x − 7)/(x + 4)) → (x − 7)/(x + 4)Prompt: Factorise the top and the bottom.

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

eurekami

Simplify algebraic fractions by factorising

Name

Date

Score/ 6

  1. Question 1

    Simplify: (x² − x − 2)/(x² − 4)

    • A. (x + 1)/(x + 2)
    • B. (x − 1)/(x − 4)
    • C. (x − 2)/(x + 2)
    • D. (x + 1)/(x − 2)

  2. Question 2

    Lucia 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

  3. Question 3

    Simplify: (x² − 4)/(x + 2)

    • A. 1/(x − 2)
    • B. (x − 2)
    • C. (x² − 2)
    • D. (x + 2)

  4. Question 4

    Jamal 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

  5. Question 5

    Simplify: (x² + 3x − 10)/(x − 2)

    • A. (x + 5)/(x − 2)
    • B. (x − 5)
    • C. (x − 10)
    • D. (x + 5)

  6. Question 6

    Luis simplified (x + 4)/x by crossing out the x's and got 1 + 4 = 5. Is this right?

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

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Answer key: Simplify algebraic fractions by factorising

  1. 1. A. (x + 1)/(x + 2)

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

  3. 3. B. (x − 2)

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

  5. 5. D. (x + 5)

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