## Consecutive Numbers

Well I wonder how often you have noticed that there are numbers around the place that follow one after another $1, 2, 3, \ldots$ 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.

You can start anywhere [ $3, 4, 5, 6, \ldots$ etc. or $165, 166, 167, 168, \ldots$ etc.] and they can be explored in a number of different ways, when they are not counting anything particular. This investigation is about using the idea of consecutive numbers and gives us other numbers that we can explore much further and find out all kinds of things. 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:

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. 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). Are you sure you've got them all?

If so, try other sets of four consecutive numbers and look carefully at the sets of answers that you get each time. It is probably a good idea to write down what you notice. 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 3 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?"

This problem was chosen as a favourite for the NRICH 10th Anniversary website by Bernard Bagnall. Find out why Bernard selected it in the Notes

### Why do this problem?

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.

### Possible approach

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.

### Key questions

Do you think you've found all the possibilities?
Can you explain why these things always happen?

### 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 [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?
... we were allowed to use fractions or decimals in between the whole numbers?
and for negatives look at Consecutive Negative Numbers.

### 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 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.