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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 − 1) / (x − 1) (x + 1 / x / x − 1) → x + 1Prompt: Factorise the numerator first.
  2. Write as a single fraction: 1/(x + 1) + 1/(x + 6). Type the numerator over the factorised denominator, e.g. (3x+5)/((x+1)(x+2)). → (2x+7)/((x+1)(x+6))Prompt: The common denominator is the product of the two brackets.
  3. Simplify: (x² − 6x + 5) / (x − 1) (x − 6 / x + 5 / x − 5 / x − 1) → x − 5Prompt: 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: 4/(x + 3) − 1/(x + 5) ((3x + 23)/((x + 3)(x + 5)) / (3x + 17)/((x + 3)(x + 5)) / 3/((x + 3)(x + 5)) / (3x + 17)/((x + 3) + (x + 5))) → (3x + 17)/((x + 3)(x + 5))Prompt: Use the common denominator (x + 3)(x + 5).
  2. Simplify: 2/(x + 3) + (x − 3)/(x² − 9) (3/(x + 3) / 2/(x + 3) / 3/(x − 3) / 3/(x² − 9)) → 3/(x + 3)Prompt: Factorise x² − 9 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: 5/(x − 3) + 5/(x + 1) ((10x − 10)/((x − 3)(x + 1)) / 10/((x − 3)(x + 1)) / (10x − 10)/((x − 3) + (x + 1)) / (10x + 20)/((x − 3)(x + 1))) → (10x − 10)/((x − 3)(x + 1))Prompt: Use the common denominator (x − 3)(x + 1).
  2. Simplify: 2/(x + 4) + (x − 4)/(x² − 16) (3/(x² − 16) / 3/(x − 4) / 3/(x + 4) / (x − 6)/(x² − 16)) → 3/(x + 4)Prompt: Factorise x² − 16 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

    Write as a single fraction: 1/(x + 5) + 4/(x + 3)

    • A. 5/((x + 5)(x + 3))
    • B. (5x + 23)/((x + 5) + (x + 3))
    • C. (5x − 17)/((x + 5)(x + 3))
    • D. (5x + 23)/((x + 5)(x + 3))

  2. Question 2

    Simplify: 5/(x + 6) + (x − 6)/(x² − 36)

    • A. 6/(x − 6)
    • B. 6/(x + 6)
    • C. 5/(x + 6)
    • D. (x − 11)/(x² − 36)

  3. Question 3

    Write as a single fraction: 4/(x − 3) − 5/(x + 6)

    • A. (−x + 39)/((x − 3)(x + 6))
    • B. (−x + 9)/((x − 3)(x + 6))
    • C. (−x + 39)/((x − 3) + (x + 6))
    • D. -1/((x − 3)(x + 6))

  4. Question 4

    Simplify: 6/(x + 2) + (x − 2)/(x² − 4)

    • A. 6/(x + 2)
    • B. 7/(x − 2)
    • C. 7/(x + 2)
    • D. (x − 8)/(x² − 4)

  5. Question 5

    Write as a single fraction: 3/(x − 6) + 5/(x − 2)

    • A. (8x − 36)/((x − 6)(x − 2))
    • B. 8/((x − 6)(x − 2))
    • C. (8x + 24)/((x − 6)(x − 2))
    • D. (8x − 36)/((x − 6) + (x − 2))

  6. Question 6

    Simplify: 4/(x + 4) + (x − 4)/(x² − 16)

    • A. 5/(x + 4)
    • B. 4/(x + 4)
    • C. 5/(x − 4)
    • D. (x − 8)/(x² − 16)

  7. Question 7

    Write as a single fraction: 5/(x + 2) + 3/(x + 5)

    • A. (8x + 19)/((x + 2)(x + 5))
    • B. (8x + 31)/((x + 2)(x + 5))
    • C. 8/((x + 2)(x + 5))
    • D. (8x + 31)/((x + 2) + (x + 5))

  8. Question 8

    Simplify: 1/(x + 4) + (x − 4)/(x² − 16)

    • A. 2/(x − 4)
    • B. 1/(x + 4)
    • C. 2/(x² − 16)
    • D. 2/(x + 4)

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Answer key: Add and subtract algebraic fractions

  1. 1. D. (5x + 23)/((x + 5)(x + 3))

  2. 2. B. 6/(x + 6)

  3. 3. A. (−x + 39)/((x − 3)(x + 6))

  4. 4. C. 7/(x + 2)

  5. 5. A. (8x − 36)/((x − 6)(x − 2))

  6. 6. A. 5/(x + 4)

  7. 7. B. (8x + 31)/((x + 2)(x + 5))

  8. 8. D. 2/(x + 4)