To get you started, try adding some different sets of consecutive numbers.
Which of the numbers between 1 and 20 are polite numbers? Which are impolite?
What happens if you add two consecutive numbers?
Can you find an expression for the sum of three consecutive numbers? Can you find an expression for the sum of four consecutive numbers? What about five consecutive numbers?
When you are explaining why your rule works, you may find the questions below helpful.
If your set of consecutive numbers is $a + (a+1) + (a+2) + \cdots + (a+k)$, can you prove that the sum has an odd factor?
Can you show that if a number has a factor of 7 (i.e. it has the form 7n), then it can be split up into 7 consecutive numbers?
Can you show that if a number has a factor of the form $2k+1$, then it can be split up into the sum of $2k+1$ consecutive numbers?
Can you use these ideas to write 56 as the sum of 7 consecutive numbers?
Can use these ideas to write 44 as the sum of 11 consecutive numbers (where some will be negative)? Can you use this to write 44 as the sum of consecutive positive numbers?
The problem Impossible Sums explores the same ideas, but has some proof sorters you can use to generate a proof to this problem.