We have a skill we would like our students to practise. Option 1 is to set a back-breaking set of exercises. Option 2 is to set an extended slow-burn exercise, generated by the student, which works towards a “mini theorem”. The idea here is to pull off a magic trick where the student learns without realising they are learning.
The exercise for GCSE and A-level students goes like this: pick three different non-zero whole numbers that add up to 0 and put them in a bag - for example, 5, 2 and -7. No pair of numbers should have a common factor.
Now pick two numbers from the bag and use them to make a fraction. How many different ones can you make from the three numbers? Write them down.
Put the fractions you have in order. What do you get if you multiply all the fractions? What do you get if you add all the fractions? Compare notes with your classmates. Can you prove that your finding will always work?
Note that the students start by choosing the numbers; they create the question, so they have a vested interest in making it work. The exercise is differentiated - not everyone will make it to the final question, but hopefully everyone will be able to create the starting fractions and most will be able to order them.
Multiplying the fractions to get the result of 1 is not a great surprise, but adding the six fractions to get -3, apparently every time, can be. Challenging your students to produce a proof may be a new experience for them, and a rewarding one.
How might a bright student do the last part? Say the bag contains a, b and c where a + b + c = 0. We can make six different fractions:
Adding them gives:
And if you are wondering how this exercise was constructed, it might interest you to know that specifying that the three numbers should add up to 0 was in fact the very last part of the problem to be written.
Jonny Griffiths teaches at a sixth-form college in Norfolk
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