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Consecutive Numbers

Age 7 to 14
Challenge Level

Consecutive Numbers

Well I wonder how often you have noticed that there are numbers around the place that follow one after another 1, 2, 3 ... etc.? Sometimes they appear in reverse order when a countdown is happening for a launch of a rocket. But usually they happen in an order going up, like when you read through a book and notice the page numbers. These kinds of numbers are called consecutive numbers, you may have heard of the word before - it simply means that they are whole numbers that follow one after another.

This investigation uses the idea of consecutive numbers and gives us other numbers to explore.  You may very well discover things that NO ONE else has discovered or written about before, and that's GREAT!

So this is how it starts. You need to choose any four consecutive numbers and place them in a row with a bit of a space between them, like this:

4, 5, 6, 7

When you've chosen your consecutive numbers, stick with those same ones for quite a while, exploring ideas before you change them in any way. Now place $+$ and $-$ signs in between them, something like this :

4 + 5 - 6 + 7
4 - 5 + 6 + 7

and so on until you have found all the possibilities. Are you sure you've got them all?  You should include one using all $+$'s and one that includes all $-$'s.

Now work out the answers to all your calculations (e.g. 4 - 5 + 6 + 7 = 12 and so on).

Now try other sets of four consecutive numbers and look carefully at the sets of answers that you get each time.

Are you surprised by anything you notice?

It is probably a good idea to write down your 'noticings'. This can lead you to test some ideas out by starting with new sets of consecutive numbers and seeing if the same things happen in the same way.

You might now be doing some predictions that you can test out...

FINALLY, it is good to ask the question "I wonder what would happen if I ... ?"
You may have thought up your own questions to explore further. Here are some we thought of:

"What would happen if I took the consecutive numbers in an order going down, instead of up?"
"What would happen if I only used sets of three consecutive numbers?"
"What would happen if I used more consecutive numbers?"
"What would happen if I changed the rule and allowed consecutive numbers to include fractions or decimals?"
"What would happen if I allowed a $+$ or $-$ sign before the first number?"


Why do this problem?

This problem has the capacity to interest young and old alike. It offers an element of surprise which makes learners curious to investigate further and want to explain. 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.

Possible approach

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. Listen out for expressions of surprise, which you could proble a little by asking learners why they are surprised and how that makes them feel towards the task.

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. 

Key questions

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?

Possible support

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.

Possible extension

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 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, for example between using four consecutive 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.
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.