I decided to re-sequence my start-of-the-year activities and to lead with a low-floor, high-ceiling problem in assigned random groups of three or four students.

Here is the problem, which comes from Phillips Exeter Academy’s Math 1 curriculum:

I told the groups to figure out everything they could about this situation with prompts like, “What do you notice about interesting numbers? What do you wonder about them?”

As I watched twelve groups of students explore this problem over three classes, I began to see students latch onto different aspects of this problem. All of these questions and discoveries are inter-related, so I’m writing them down now so that I can map them out in the future.

**Questions:**

- Which numbers up through 20 (or so) are interesting?
- Why are powers of 2 interesting?
- Are powers of 2 the only interesting numbers?
- Are there any interesting odd numbers?
- What happens when I sum any two consecutive positive integers?
- What happens when I sum any three consecutive positive integers?
- If
*n*is odd, what happens when I sum any*n*consecutive positive integers? - If
*n*is even, what happens when I sum any*n*consecutive positive integers? - How can I decompose any odd number?
- How can I decompose any multiple of 3?
- If
*n*is odd, how can I decompose any multiple of*n?* - How can I decompose any even number?
- Is there a general algorithm for decomposing any number?
- How many ways are there to decompose a given number?

**Realizations:**

- All powers of 2 are interesting.
- Only powers of 2 are interesting.
- No odd numbers are interesting.
- The sum of two consecutive positive integers is odd.
- The sum of three consecutive positive integers is a multiple of 3.
- If
*n*is odd, the sum of*n*consecutive positive integers is a multiple of*n.* - If
*n*is even, the sum of*n*consecutive positive integers is*n/2*more than a multiple of*n.* - There is an algorithm for decomposing even numbers.
- There is exactly one way to decompose a prime number greater than 2.
- The powers of 2 are exactly the whole numbers without odd factors.

There was a split between groups that started by trying to answer (the very natural) question #1 (and thus getting to realizations #1 and #2) and those that started by generating and then trying to answer questions #5 and 6 (and thus getting to realizations #4 and #5). There was also one group in one class that decided to explore the sum of the first *n* consecutive integers (i.e., they wanted to know about the triangular numbers).

I think I will definitely use this problem again, with perhaps a bit more structure and guided mini-explorations along the way as groups arrive at various questions and realizations. It would probably be worth making a checklist for each group to help keep me organized as I keep tabs on each group’s progress.

Related:

- https://mathblag.wordpress.com/2011/11/13/sums-of-consecutive-integers/
- https://nrich.maths.org/507
- https://blogs.adelaide.edu.au/maths-learning/2015/07/28/the-sausage-stacking-theorem/