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I wanted to give my Year 9s a problem solving activity to bed in their learning on recurring decimals. I gave them this question:

‘I divided two two-digit integers on my calculator and it gives the answer

How many different possible pairs of integers are there?’

I expected the questions as to what integers are and exactly what the dots represent and fairly quickly some students came up with

Some then realised they were looking at equivalent fractions and suggested

as the third and final solution and I praised them for solving a challenging question. I was surprised therefore when Zoe and Abi said they had six solutions!!

Take a minute or two now to see if you can find any more solutions.

It does not take too much thinking to revisit the definition of integers as whole numbers that are both positive and negative leading to negative thirty over negative ninety nine etc.

Well done Zoe and Abi, you did better than me and in fact better than the rest of my department. I wonder how many readers found all six solutions.

Many thanks,

Steve Cavill

Steve Cavill BSc(Hons) PGCE FCIEA has taught maths in both state and independent schools. He spent a few years as an Associate Lecturer for the OU and has written a number of GCSE maths books, workbooks and revision guides as well as being a senior examiner and moderator for GCSE and IGCSE.