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 is a fantastic context in which to look for patterns, explain why these patterns occur, and as a result, to gain a deeper understanding of our number
system. It offers opportunities to work together by sharing results and making decisions as to which consecutive numbers to look at next.
To introduce the problem, discuss what consecutive means, and invite pupils to suggest a starting number. Let them select the three operations and take four or five examples, but don't discuss how many possibilities there are at this stage. Give children time to find other possibilities and encourage them to explain how they know they have them all.
Once all eight have been agreed on, let learners choose other sets of consecutive numbers to investigate. It is sometimes useful to suggest to the class that they are being detectives and seeking out links, relationships and reasons. Most children find some connections between the eight answers that they find in each case. This could be that all the answers are even or the fact
that some results appear with every group of four consecutive numbers. Encourage pupils to explain why in each case.
How do you know 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 ...?", as suggested at the end of the problem itself. For consideration of negative numbers as well, look at Consecutive Negative Numbers.
For more extension work
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 consecutive 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 helpful.
These notes are based on writings by Bernard Bagnall who has used this activity more than sixty times and chose it as his favourite problem on the NRICH site.