Half Hearted

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

Starter of The Day Transum Mathematics Topics


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.

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?

Christmas Present Ideas

It is often very difficult choosing Christmas presents for family and friends but so here are some seasonal, mathematics-related gifts chosen and recommended by Transum Mathematics.

Go Genius Maths

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How Not To Be Wrong

How Not To Be Wrong

The maths we learn in school can seem like an abstract set of rules, laid down by the ancients and not to be questioned. In fact, Jordan Ellenberg shows us, maths touches on everything we do, and a little mathematical knowledge reveals the hidden structures that lie beneath the world's messy and chaotic surface. In How Not to be Wrong, Ellenberg explores the mathematician's method of analyzing life, from the everyday to the cosmic, showing us which numbers to defend, which ones to ignore, and when to change the equation entirely. Along the way, he explains calculus in a single page, describes Gödel's theorem using only one-syllable words, and reveals how early you actually need to get to the airport.

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Graphic Display Calculator

This handheld device and companion software are designed to generate opportunities for classroom exploration and to promote greater understanding of core concepts in the mathematics and science classroom. TI-Nspire technology has been developed through sound classroom research which shows that "linked multiple representation are crucial in development of conceptual understanding and it is feasible only through use of a technology such as TI-Nspire, which provides simultaneous, dynamically linked representations of graphs, equations, data, and verbal explanations, such that a change in one representation is immediately reflected in the others.

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 Calculator

Apple iPad Pro

The analytics show that more and more people are accessing Transum Mathematics via an iPad as it is so portable and responsive. The iPad has so many other uses in addition to solving Transum's puzzles and challenges and it would make an excellent gift for anyone.

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Before giving an iPad as a Christmas gift you could add a link to iPad Maths to the home screen.

Craig Barton's Tips for Teachers

Teaching is complex. But there are simple ideas we can enact to help our teaching be more effective. This book contains over 400 such ideas." more... #ad

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 Another Craig Barton Book

The Story Of Maths [DVD]

The films in this ambitious series offer clear, accessible explanations of important mathematical ideas but are also packed with engaging anecdotes, fascinating biographical details, and pivotal episodes in the lives of the great mathematicians. Engaging, enlightening and entertaining, the series gives viewers new and often surprising insights into the central importance of mathematics, establishing this discipline to be one of humanity s greatest cultural achievements. This DVD contains all four programmes from the BBC series.

Marcus du Sautoy's wonderful programmes make a perfect Christmas gift more... #ad

Christmas Maths

This book provides a wealth of fun activities with a Christmas theme. Each photocopiable worksheet is matched to the Numeracy Strategy and compatible with the Scottish 5-14 Guidelines. This series is designed for busy teachers in the late Autumn term who are desperate for materials that are relevant and interesting and that can be completed with minimun supervision.

All the activities are suitable for use by class teachers, supply teachers, SEN teachers and classroom assistants and cover topics such as 'How many partridges did the true love give all together?' and 'Filling a sleigh with presents by rolling a dice!'. Children will have lots of fun working through the Christmas Maths themes but also gain valuable skills along the way.

A great source of ideas and another reasonably priced stocking filler. more... #ad
A Compendium Of Mathematical Methods

A Compendium Of Mathematical Methods

How many different methods do you know to solve simultaneous equations? To multiply decimals? To find the nth term of a sequence?

A Compendium of Mathematical Methods brings together over one hundred different approaches from classrooms all over the world, giving curious mathematicians the opportunity to explore fascinating methods that they've never before encountered.

If you teach mathematics to any age group in any country, you are guaranteed to learn lots of new things from this delightful book. It will deepen your subject knowledge and enhance your teaching, whatever your existing level of expertise. It will inspire you to explore new approaches with your pupils and provide valuable guidance on explanations and misconceptions. more... #ad

Math with Bad Drawings

I had been tutoring the wonderful Betsy for five years. When the day came for our last ever session together before the end of her Year 13, I received this beautiful book as a gift of appreciation.

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Ben Orlin answers maths' three big questions: Why do I need to learn this? When am I ever going to use it? Why is it so hard? The answers come in various forms-cartoons, drawings, jokes, and the stories and insights of an empathetic teacher who believes that mathematics should belong to everyone. more... #ad

Click the images above to see all the details of these gift ideas and to buy them online.

Maths T-Shirts

Maths T-shirts on Amazon
Maths T-Shirts on Amazon

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Laptops In Lessons

Teacher, do your students have access to computers?
Do they have iPads or Laptops in Lessons?

Whether your students each have a TabletPC, a Surface or a Mac, this activity lends itself to eLearning (Engaged Learning).

Laptops In Lessons

Here a concise URL for a version of this page without the comments.

Transum.org/go/?Start=May22

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

Transum.org/go/?to=vinculum

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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Example

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

 

Half Hearted Examplde

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