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Questions that may be asked:
Questions that may be asked:


“If you add up consecutive integers what answers can you make?”
“If you add up consecutive integers what answers can you make?”


With some groups I put this on the board and ask the class whether there are any words they think someone else might not understand. This allows us to discuss the meaning of ‘consecutive’ and ‘integer’. I then ask if there are things they want to ask me. This might involve finding out whether negatives are allowed, or requesting an example.


With some groups I put this on the board and ask the class whether there are any words they think someone else might not understand. This allows us to discuss the meaning of ‘consecutive’ and ‘integer’. I then ask if there are any things they want to ask me. This might involve finding out whether negatives are allowed, or to request an example.
With other groups I might start directly with some examples such as:
 
 
With other groups I might start directly with some examples.
 
 
Eg: 9 <nowiki>=</nowiki> 2+3+4, 11 <nowiki>=</nowiki> 5+6, 12 <nowiki>=</nowiki> 3+4+5, 20 <nowiki>=</nowiki> 2+3+4+5+6 “What is special about the numbers I am adding up?”


9 <nowiki>=</nowiki> 2+3+4, 11 <nowiki>=</nowiki> 5+6, 12 <nowiki>=</nowiki> 3+4+5, 20 <nowiki>=</nowiki> 2+3+4+5+6 “What is special about the numbers I am adding up?”


Having established the ‘rules’, we then look at formulating a question.  
Having established the ‘rules’, we then look at formulating a question.  


 
With some classes I ask a question, with others they create it for themselves, either as a class or individually. Examples might be: “Can every number be made by adding consecutive numbers?”, “What is special about the numbers that can be made by adding 5 consecutive numbers?”.
With some classes I ask them a question, with others they create it for themselves, either as a class or individually. Examples might be: “Can every number be made by adding consecutive numbers?”, “What is special about the numbers that can be made by adding 5 consecutive numbers?”.
 


I often find it useful for pupils to be able to try a few sets of numbers for themselves (what the pupils describe as “random numbers”). Having tried a few out we can then discuss how to work more systematically.
I often find it useful for pupils to be able to try a few sets of numbers for themselves (what the pupils describe as “random numbers”). Having tried a few out we can then discuss how to work more systematically.


Some pupils want to make a list and try to find ways to make each number, starting from 1 …
Some pupils want to make a list and try to find ways to make each number, starting from 1 …


Others have the idea of narrowing the task to adding up two consecutive numbers. This one is often more fruitful and the idea of simplifying a task to gain more insights is an important one.
Others have the idea of narrowing the task to adding up two consecutive numbers. This one is often more fruitful and the idea of simplifying a task to gain more insights is an important one.


'''Extension/Further work'''
'''Extension/Further work'''


I like to encourage pupils to explain not only what is going on but also why.  
I like to encourage pupils to explain not only what is going on but also the reasons.  
 


Two consecutive numbers always give an odd answer. This can be explaining using ‘odd + even is odd’, or using some algebra (''n'' + ''n'' + 1 <nowiki>=</nowiki> 2''n'' + 1. ''n'' is an integer, so 2''n'' is even, which makes 2''n'' + 1 odd). Turning this round, if I tell you an odd number you can tell me the two numbers to add by halving it and taking the two integers either side of it. Eg: 27 ÷ 2 <nowiki>=</nowiki> 13.5, so we use 13+14.
Two consecutive numbers always give an odd answer. This can be explaining using ‘odd + even is odd’, or using some algebra (''n'' + ''n'' + 1 <nowiki>=</nowiki> 2''n'' + 1. ''n'' is an integer, so 2''n'' is even, which makes 2''n'' + 1 odd). Turning this round, if I tell you an odd number you can tell me the two numbers to add by halving it and taking the two integers either side of it. Eg: 27 ÷ 2 <nowiki>=</nowiki> 13.5, so we use 13+14.


Three consecutive numbers throw up some interesting things. Some pupils miss out some of the possibilities. They start out with 1+2+3 and then try 3+4+5 or 4+5+6, missing out 2+3+4. Some, when they include every possibility, will remark that the answers alternate between being even and odd. Of more use is that the answers are always multiples of 3. Every multiple of three (above 3) can be made in this way. If you divide a multiple of 3 by 3 you can then take the numbers either side of the answer and the three numbers you have add up to the original number. Eg: 33÷3 <nowiki>=</nowiki> 11, so we use 10+11+12. This can be explained in lots of ways, including referring to working out the mean of a set of numbers. Algebra can also help: the three numbers can either be treated as ''m'', (''m''+1) and (''m''+2), which have a total of 3''m''+3, or as (''n''-1), ''n'' and (''n''+1), which have a total of 3''n''. An alternative, geometric way of exploring and explaining this is shown later.
Three consecutive numbers throw up some interesting things. Some pupils miss out some of the possibilities. They start out with 1+2+3 and then try 3+4+5 or 4+5+6, missing out 2+3+4. Some, when they include every possibility, will remark that the answers alternate between being even and odd. Of more use is that the answers are always multiples of 3. Every multiple of three (above 3) can be made in this way. If you divide a multiple of 3 by 3 you can then take the numbers either side of the answer and the three numbers you have add up to the original number. Eg: 33÷3 <nowiki>=</nowiki> 11, so we use 10+11+12. This can be explained in lots of ways, including referring to working out the mean of a set of numbers. Algebra can also help: the three numbers can either be treated as ''m'', (''m''+1) and (''m''+2), which have a total of 3''m''+3, or as (''n''-1), ''n'' and (''n''+1), which have a total of 3''n''. An alternative, geometric way of exploring and explaining this is shown later.


Four consecutive numbers have an even total, but are not multiple of four, whereas five consecutive numbers give a multiple of five. This can lead to hypotheses about odd and even numbers.
Four consecutive numbers have an even total, but are not multiple of four, whereas five consecutive numbers give a multiple of five. This can lead to hypotheses about odd and even numbers.


'''Further work/other ways of teaching this topic'''
'''Further work/other ways of teaching this topic'''
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