Half Hearted


Find the number which when added to the top (numerator) and bottom (denominator) of each fraction make it equivalent to one half. more...

Topics: Starter | Algebra | Fractions

  • Transum,
  • It must seem strange to so many people that you can multiply the top and bottom of a fraction by the same number and the fraction's value remains the same but when you add the same number to top and bottom the fraction's value changes.

    Then we think of equations and the rules are different. You can multiply or add to both sides and the equation remains balanced.

    What is your best explanation for that?
  • David Payne, Secondary Maths Teacher
  • I use your starters a lot. Thank you for such a thought provoking, varied and rich set of activities.

    We did the May 22nd “Half hearted” starter today and one of the pupils pointed out that d-n = the new numerator. Gasps all around. Why? How can it be? And that was just me. I have now worked it out using your extension activity but it seemed like a really surprising result at first.

    I just hope it doesn’t encourage the students to add a number to the numerator and denominator in any other contexts. I did tell them not to try this at home.

    Anyway, thanks, and keep up the good work.

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Note to teacher: Doing this activity once with a class helps students develop strategies. It is only when they do this activity a second time that they will have the opportunity to practise those strategies. That is when the learning is consolidated. Click the button above to regenerate another version of this starter from random numbers.



Extension activities

Find the number which when added to the numerator and denominator of each of the fractions above make them equivalent to one third.

What should be added to the numerator and denominator of the fraction whose numerator is n and denominator is d to make it equivalent to one half?

What should be added to the numerator and denominator of the fraction whose numerator is n and denominator is d to make it equivalent to one third?

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Laptops In Lessons

Teacher, do your students have access to computers such as tablets, iPads or Laptops?  This page was really designed for projection on a whiteboard but if you really want the students to have access to it here is a concise URL for a version of this page without the comments:


However it would be better to assign one of the student interactive activities below.

Laptops In Lessons

Here is the URL which will take them to a related student activity.


Student Activity

A glass ½ full = A glass ½ empty

Multiply both sides by two

A full glass = An empty glass



Extension Answers


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Adding two to the numerator and denominator of one quarter produces a fraction equivalent to one half.


Half Hearted Examplde