first day math problem 2017–18: interesting numbers

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:

pea-1-92-3.png

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:

  1. Which numbers up through 20 (or so) are interesting?
  2. Why are powers of 2 interesting?
  3. Are powers of 2 the only interesting numbers?
  4. Are there any interesting odd numbers?
  5. What happens when I sum any two consecutive positive integers?
  6. What happens when I sum any three consecutive positive integers?
  7. If is odd, what happens when I sum any n consecutive positive integers?
  8. If is even, what happens when I sum any n consecutive positive integers?
  9. How can I decompose any odd number?
  10. How can I decompose any multiple of 3?
  11. If is odd, how can I decompose any multiple of n?
  12. How can I decompose any even number?
  13. Is there a general algorithm for decomposing any number?
  14. How many ways are there to decompose a given number?

Realizations:

  1. All powers of 2 are interesting.
  2. Only powers of 2 are interesting.
  3. No odd numbers are interesting.
  4. The sum of two consecutive positive integers is odd.
  5. The sum of three consecutive positive integers is a multiple of 3.
  6. If n is odd, the sum of n consecutive positive integers is a multiple of n.
  7. If n is even, the sum of n consecutive positive integers is n/2 more than a multiple of n.
  8. There is an algorithm for decomposing even numbers.
  9. There is exactly one way to decompose a prime number greater than 2.
  10. 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.

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