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

Add and subtract algebraic fractions

Algebra

CCSS HSA.APR.D.7England: GCSE Higher Algebra

Objective and success criteria

Students will: Add and subtract algebraic fractions.

  • 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: Algebraic fractions

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 (Algebraic fractions). Students answer on mini whiteboards.
  • Show answers together and note who needs support in the main task.
  1. Simplify: (x² − x − 20) / (x − 5) (x + 4 / x − 4 / x − 1 / x − 20) → x + 4Prompt: Factorise the numerator first.
  2. Write as a single fraction: 1/(x + 5) + 1/(x + 3). Type the numerator over the factorised denominator, e.g. (3x+5)/((x+1)(x+2)). → (2x+8)/((x+5)(x+3))Prompt: The common denominator is the product of the two brackets.
  3. Simplify: (x² + 8x + 15) / (x + 5) (x + 8 / x + 5 / x + 15 / x + 3) → x + 3Prompt: Factorise the numerator first.

Hook · 5 min

  • Show a "think of a number" puzzle and ask the class to work out the starting number.
  • Share the objective: Add and subtract algebraic fractions. 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 as a single fraction: 4/(x − 2) + 4/(x + 1)
  • Step 1: (4(x + 1) + 4(x − 2))/((x − 2)(x + 1))
  • Step 2: (8x − 4)/((x − 2)(x + 1))
  • Answer: (8x − 4)/((x − 2)(x + 1))
  • 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 as a single fraction: 3/(x − 4) − 3/(x + 3) ((21)/((x − 4) + (x + 3)) / (−3)/((x − 4)(x + 3)) / (21)/((x − 4)(x + 3)) / 0/((x − 4)(x + 3))) → (21)/((x − 4)(x + 3))Prompt: Use the common denominator (x − 4)(x + 3).
  2. Simplify: 1/(x + 4) + (x − 4)/(x² − 16) (2/(x + 4) / 2/(x² − 16) / (x − 5)/(x² − 16) / 2/(x − 4)) → 2/(x + 4)Prompt: Factorise x² − 16 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 as a single fraction: 4/(x + 2) − 2/(x − 3) ((2x − 16)/((x + 2)(x − 3)) / 2/((x + 2)(x − 3)) / (2x − 16)/((x + 2) + (x − 3)) / (2x − 8)/((x + 2)(x − 3))) → (2x − 16)/((x + 2)(x − 3))Prompt: Use the common denominator (x + 2)(x − 3).
  2. Simplify: 6/(x + 3) + (x − 3)/(x² − 9) (7/(x² − 9) / 7/(x + 3) / (x − 9)/(x² − 9) / 6/(x + 3)) → 7/(x + 3)Prompt: Factorise x² − 9 first.

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: Algebraic fractions.
  • 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

eurekami

Add and subtract algebraic fractions

Name

Date

Score/ 8

  1. Question 1

    Simplify: 2/(x + 3) + (x − 3)/(x² − 9)

    • A. 3/(x − 3)
    • B. 3/(x + 3)
    • C. 2/(x + 3)
    • D. (x − 5)/(x² − 9)

  2. Question 2

    Write as a single fraction: 2/(x − 5) + 4/(x − 6)

    • A. (6x − 32)/((x − 5)(x − 6))
    • B. 6/((x − 5)(x − 6))
    • C. (6x − 32)/((x − 5) + (x − 6))
    • D. (6x + 8)/((x − 5)(x − 6))

  3. Question 3

    Simplify: 3/(x + 6) + (x − 6)/(x² − 36)

    • A. 4/(x − 6)
    • B. 4/(x + 6)
    • C. 3/(x + 6)
    • D. 4/(x² − 36)

  4. Question 4

    Write as a single fraction: 3/(x − 4) + 4/(x + 1)

    • A. (7x − 13)/((x − 4)(x + 1))
    • B. 7/((x − 4)(x + 1))
    • C. (7x + 19)/((x − 4)(x + 1))
    • D. (7x − 13)/((x − 4) + (x + 1))

  5. Question 5

    Simplify: 3/(x + 5) + (x − 5)/(x² − 25)

    • A. 4/(x² − 25)
    • B. (x − 8)/(x² − 25)
    • C. 4/(x + 5)
    • D. 3/(x + 5)

  6. Question 6

    Write as a single fraction: 5/(x − 1) + 4/(x + 1)

    • A. (9x + 1)/((x − 1) + (x + 1))
    • B. (9x + 9)/((x − 1)(x + 1))
    • C. 9/((x − 1)(x + 1))
    • D. (9x + 1)/((x − 1)(x + 1))

  7. Question 7

    Simplify: 4/(x + 7) + (x − 7)/(x² − 49)

    • A. 4/(x + 7)
    • B. 5/(x − 7)
    • C. (x − 11)/(x² − 49)
    • D. 5/(x + 7)

  8. Question 8

    Write as a single fraction: 3/(x + 5) − 3/(x − 4)

    • A. 0/((x + 5)(x − 4))
    • B. (3)/((x + 5)(x − 4))
    • C. (−27)/((x + 5)(x − 4))
    • D. (−27)/((x + 5) + (x − 4))

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Answer key: Add and subtract algebraic fractions

  1. 1. B. 3/(x + 3)

  2. 2. A. (6x − 32)/((x − 5)(x − 6))

  3. 3. B. 4/(x + 6)

  4. 4. A. (7x − 13)/((x − 4)(x + 1))

  5. 5. C. 4/(x + 5)

  6. 6. D. (9x + 1)/((x − 1)(x + 1))

  7. 7. D. 5/(x + 7)

  8. 8. C. (−27)/((x + 5)(x − 4))