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Why do this problem?
has the capacity to interest young and old alike. I have used it with a wide range of attainment levels, and new things keep on being found out. It offers opportunities to work together when sharing results and making decisions as to which consecutive numbers to look at next.
It is sometimes useful to suggest to the students that they are being detectives and seeking out links, relations and reasons.
To introduce the problem, go through what consecutive means, getting suggestions from the pupils for the starting number. It is good to let the pupils select the three operations and to take four or five examples, but not to discuss how many possibilities there are at this stage.
Most children find some connections between the eight answers that they find. The first finding is usually that all the answers are even. The fact that $0$, $-2$, and $-4$ appear with every group of four consecutive numbers suggests the question "why?" leading to interesting discussions about the occurrence of negative numbers.
Do you think you've found all the possibilities?
Tell me about your answers.
Do you notice anything about your answers?
Can you explain why these things always happen?
I have found that all the students who have been involved in this investigation have got very excited as various observations are made, patterns seen and questions asked. The most enjoyable times for me have been hearing ten year olds using their own form of algebra and coming to some powerful [for them] realisations about why every one has a $0$, $-2$ and $-4$.
The problem has also been the starting point for some pupils to be able to ask "I wonder what would happen if ...?" And in this case it's been:
... we used more consecutive numbers each time?
... we had a starting point in the negative numbers?
... we took consecutive to mean going up in $2$s?
For the exceptionally mathematically able
These pupils would be encouraged to work on proofs. They could also begin to make comparisons - say between using four consecutuve numbers and six consecutive numbers. Some learners may want to examine other properties of the answers for any set of four consectuive numbers and this could lead on to generalisations.
On the odd occasions that pupils needed support I have found that putting a number of pupils together to work as a sharing group is all that has been necessary.
These notes are taken from writings by Bernard Bagnall who has used this activity more than sixty times and chose it as his favourite problem on the NRICH site.