Multiply Sum by Difference then Divide

This is a difficult Starter so there is a prize of a watch if you can complete it.

For each pair of numbers multiply the sum by the difference then divide the answer by 5 without using a calculator.

3, 2
18, 17
13, 11
21, 18
22, 18
25, 16
15.5, 15
18.5, 18.5

No Calculator No Calculator No Calculator



At least one of your answers is wrong.

Congratulations, all of the answers are correct.

You have won a watch. You can watch this sheep for as long as you want!


A Mathematics Lesson Starter Of The Day

A Starter which is similar to this one but slightly harder is Multiply, Add, Subtract and Divide.

Topics: Starter | Mental Methods

  • 7 Mandelbrot, Admiral Lord Nelson School
  • When calculating difference the order is usually taken as important so some of the answers should be negative. Or are we mistaken?
  • Laura, Minnesota
  • I think this is a good way to get your students to do mental math, however if I were to use this with students who knew about negative numbers it wouldn't work. With the order the numbers are set up in 4 of the answers should have actually been the negatives of what the 'correct' answer on the site gives.
  • Transum,
  • Thanks so much for your comments. I think we have an ambiguous situation here. Should the difference between two numbers always be positive? This topic was discussed on the Math Forum - Ask Dr. Math. In support of this idea is the following:
    In any real situation I can think of, if you were to ask someone for the difference between, say, 3 and 5, the answer would be 2 not -2! We tend to think of differences as positive numbers, and thus the proper rendition of that expression algebraically would be |3-5| (or |5-3|). This is the same idea as the "distance" between two numbers on the number line, which is always positive.
    In textbooks, as you've observed, it seems common in 'word problems' to make a different convention, that 'the difference between a and b' means a - b.

    The full discussion can be seen here.

    Perhaps we'll wait for a few more comments on this page and then adapt this starter to conform with the opinions of the majority. In the meantime you'll see that all the pairs of numbers above have the largest first to take out the ambiguity.

    Thanks once again for your feedback 7 Mandelbrot and Laura.
  • Don, Loughborough
  • I disagree with Mandlebrot and Laura. I think difference means the positive difference and not a rigid formula of the first number take away the second - in this regard it's disappointing that you've changed the starter (it looks as though you have). If you have students provide you with a negative answer then perhaps it will lead to an interesting discussion about word problems and the wording of questions, which they'll learn more from than just a correct answer.
  • Y. Bonter, Cardinal Newman School
  • Isnt the definition of the difference the highest number take away the lowest number.
  • 11S Coln House,
  • We enjoyed the challenge and thank you for giving us a free lesson!!!!!
    But we thought the prize was misleading.
  • Benjamin, Sydney
  • Argh, I checked my answers and it said they were wrong so I pressed the back button and it took me back to the home menu of starter of the day (the calendar). I pressed forward and it took me back to the page saying my answers were wrong! In other words it deleted my answers! It took me so long! Please get rid of this bug!
    [Transum: Sorry about that Benjamin, you must have accidentally pressed the back button twice. As a result of your feedback the page has been rewritten so that the use of the back button is not required.]
  • Benjamin, Sydney
  • Ok, thanks! Anyway, I finished it and the prize was very funny!
  • Samuel, Hornsby
  • Wow what a funny prize hope you have more prizes like this again!
  • Jackie, South Australia
  • Please could we have the answers? Every way we have written this it comes up as wrong. Thank you.

    [Transum: Sorry Jackie. The answers are only available to teachers when they are signed in.]

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.
Click here to enter your comments.

Previous Day | This starter is for 27 October | Next Day



1 7 9.6 23.4
32 73.8 3.05 0

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.


Dividing by five is a great deal to ask,

But an easier method will help you.

Dividing by ten is a much easier task,

Then multiply your answer by two!


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.

Equate board game

Here's a great board game that will give any family with school-aged kids hours of worthwhile fun. Christmas is a time for board games but this one will still be useful at any time of year. Games can be adapted to suit many levels of Mathematical ability.

For Maths tutors working with just one or small groups of pupils this game has proved to be an excellent activity for a tutorial. Deciding on the best moves can spark pertinent discussions about mathematical concepts.

Equate looks a bit like Scrabble--for aspiring mathematicians, that is. Designed by a real mathematician, it works like this: You put down tiles on a board and make points by correctly completing simple equations. Your nine tiles include both numbers and mathematical symbols; you can add on to previous plays both vertically and horizontally. more...

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.

For the young people in your life it is a great investment. Bought as a Christmas present but useful for many years to come as the young person turns into an A-level candidate then works their way through university. more...

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.

The redesigned Retina display is as stunning to look at as it is to touch. It all comes with iOS, the world's most advanced mobile operating system. iPad Pro. Everything you want modern computing to be. more...

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

Aristotle's Number Puzzle

It’s a bit of a tradition to give puzzles as Christmas Gifts to nieces and nephews. This puzzle is ideal for the keen puzzle solver who would like a challenge that will continue over the festive period (at least!).

This number puzzle involves nineteen numbers arranged into a hexagon. The goal of the puzzle is to rearrange the numbers so each of the fifteen rows add up to 38. It comes in a wooden style with an antique, aged look.

Keep the Maths in Christmaths with this reasonably priced stocking filler. more...

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...

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...

A Compendium Of 	Mathematical Methods

A Compendium Of Mathematical Methods

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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...

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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.

This a very readable book by Ben Orlin. I'm really enjoying the humour in the writing and the drawings are great.

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...

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 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


Curriculum Reference

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


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