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Summing Consecutive Numbers

Age 11 to 14
Challenge Level


Why do this problem?

This problem offers a simple context to begin exploration that naturally leads to some interesting conjectures about properties of numbers, and the possibility of developing some quite sophisticated algebraic arguments and proofs.

Possible approach

This printable worksheet may be useful: Summing Consecutive Numbers. 

One approach is to introduce the lesson in the way it is introduced on the video, with students suggesting numbers and then the teacher at first, and then the whole class, finding ways to express them as consecutive sums.

Here is an alternative approach:
"Can someone give me a set of two or more consecutive numbers?"
Write a few sets on the board.

"What totals do we get by adding the consecutive numbers in these sets?"

Write + signs in between the lists of numbers.

"These totals are all examples of numbers that can be written as the sum of consecutive numbers. I wonder if all numbers can be written in this way ”¦"

"How about trying to write the numbers from $1$ to $30$ as sums of consecutive numbers?"

Give students time to work in pairs on filling in the gaps from $1$ to $30$. While they are working, write the numbers from $1$ to $30$ on the board ready to collect together the sums the class have found.

If students ask about negative numbers, one possible answer is: "Stick to positive numbers for now, and then perhaps investigate negative numbers later."

Once most pairs have filled in most of the gaps, collect their results on the board.

"Spend a minute looking at these results and then be prepared to talk about anything interesting that you notice."

Give them time to think on their own at first and then share ideas with their partner, before discussion with the whole class.

Next, collect together any noticings, and write them on the board in the form of questions or conjectures. If such conjectures are not forthcoming, here are some suggested lines of enquiry:

• "I wonder if we could write every number as the sum of consecutive numbers?"
• "Some numbers can be written in more than one way! I wonder which ones?"
• "9, 12 and 15 can all be written using three consecutive numbers. I wonder if all multiples of 3 can be written in this way?"
• "Maybe you could write the multiples of 4 if you used four consecutive numbers ..."

Allow pairs time to work on the conjectures of their choosing, reminding them that they will need to provide convincing arguments to explain any of their conclusions.

If appropriate, bring the class together to spend some time discussing algebraic representations of consecutive numbers ($n, n+1, n+2...$) to give students the tools to create algebraic proofs.

Finish the lesson by challenging students to express numbers that you have chosen as a sum of consecutive numbers in as many ways as possible - e.g. 45 can be expressed as the sum of consecutive numbers in 5 different ways.

Key questions

What happens when you add together a pair of consecutive numbers? Three consecutive numbers? Four?
What do you notice about the numbers you can't make?
If the first of a set of consecutive numbers is $n$, how can you express the next numbers in the set, and hence the total, algebraically?

Possible support

When collecting together the class's results for the numbers from $1$ to $30$, they can be arranged on the board in ways that will make it easier for patterns to emerge. Charlie's results from the problem could be set out like this:
9 = 4+5 2+3+4    
10 =     1+2+3+4  
11 = 5+6      
12 =   3+4+5    
13 = 6+7      
14 =     2+3+4+5  
15 = 7+8 4+5+6   1+2+3+4+5

Possible extension

Challenge students to find an efficient way to calculate how many different ways $x$ can be expressed as a sum of consecutive numbers, for any $x$.

More challenging extension - prove that it is not possible to write $2^n$ as the sum of consecutive positive whole numbers, for any $n$.
Pair Products offers another good context for exploring properties of numbers and then using algebra to represent and explain the results.