Half Hearted

Starter of The Day Transum Mathematics Topics

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.
Transum,
Friday, November 29, 2024
Thanks for your comments David. Hopefully this Starter encourages exploration and problem-solving, providing a rich opportunity for students to deepen their understanding of fractions. An important outcome of this activity is that students can discover a fundamental property of fractions: adding or subtracting the same number to both the numerator and denominator changes the value of the fraction, while multiplying or dividing both by the same number preserves its value. Highlighting this distinction through the investigation helps to address and correct potential misconceptions, reinforcing a key mathematical concept in a meaningful and engaging way.

How did you use this starter? Can you suggest how teachers could present or develop this resource? Do you have any comments? It is always useful to receive feedback and helps make this free resource even more useful for Maths teachers anywhere in the world.
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Answers

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?

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:

Transum.org/go/?Start=May22

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.

Transum.org/go/?to=vinculum

A glass ½ full = A glass ½ empty

Multiply both sides by two

A full glass = An empty glass

Curriculum Reference

See the National Curriculum page for links to related online activities and resources.

Half Hearted

Answers

Extension activities

Curriculum Reference

Example