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
13, 12
14, 12
21, 16
15, 11
23, 17
14.5, 14
11.5, 11.5

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 5 10.4 37
20.8 48 2.85 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!


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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


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