Sunday, September 8, 2013

A number theory problem that I modified a bit for primary schoolers

This is a problem that was a huge hit with some 5th and 6th graders, and an OK day with some 4th grade kids.

It's a straightforward rip-off this problem from PCMI:

The sum of 4's divisors are 7. The sum of 5's divisors are 6. Etc.

Except that a lot of cool stuff happens along the way. I don't want to dish out any spoilers, but there is a lot of cool stuff to dig into.

We're talking about divisors, summing and multiplication here, so it's a good workout for the elementary school brain. There are also ample opportunities for spotting patterns, and a few good chances to offer justifications. It's a ton of fun when it works.

The only thing you need to work around is function notation, but that's not a big deal. 

Anyway, I do love this problem, and until you dig into the math you're just going to have to trust me that it has connections with exponents, adding fractions, .9999... = 1 and all sorts of other rich elementary school (and high school and number theory) topics. It's absolutely and truly one of my favorite functions and problems.

Only issue: it was definitely not a hit on the first day of class with my 4th graders. They struggled with finding divisors and their multiplication isn't strong enough to make the sorts of connections that make this a blast. It was OK, and there were a few cool moments when I pointed some things out, but overall it was a bit disappointing.

But that's more an issue with my taking an unnecessary risk on the first day than with the actual problem. I imagine that I'll bring this back into class once the kids are up to snuff, operations-wise. Good practice for all sorts of operations, and a lot of patterns to sniff out too. 

Let's hang out a bit in the comments. I'll give you two prompts: (1) Would this work with your elementary school students? Why, why not, etc. (2) Can you think of more "higher" math topics that would do well in a small-person classroom?


  1. I think that compositions of numbers could work with elementary school students. How many ways can you write N as a sum of positive integers, if order matters? For example, 4 can be written in 7 ways: 3+1, 2+2, 1+3, 2+1+1, 1+2+1, 1+1+2, and 1+1+1+1.

    I am also fond of unit fractions. How many ways can 1 be written as a sum of three unit fractions (i.e. fractions whose denominator is 1)?

  2. I always liked doing factor trees with elementary students.(big tie-ins with roots for high schoolers)

    multiplying big numbers with distribution is also a hit. (I did it last year with 3rd graders using a 'picture-frame box' and it was fun. The teacher asked me to help them work on 2 digit multiplication, I didn't realize she meant 1 digit times 2 digit as my son was well beyond that. We went much further than that and she said they all did well. **I always go into my son's classes and teach 1-2 days a school year**

  3. I love the compositions problem that Dave Radcliffe suggests, because you can uncover so many different kinds of patterns by organizing your list in different ways (by number of 1s, by number of addends, by largest addend ...) and yet more by adding restrictions (no 1s allowed, only odd numbers allowed, ...)

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    1. Great find, thanks for sharing! It worked great with my 6th graders! It's coming towards the middle of our number sense unit but I would have liked to have done it a lot earlier without giving them as much info to get started. Also great call on losing the function notation, it makes the task way more accessible! I tweaked it a little bit also, and wrote it up on my 180 blog over at

      Thanks again for posting!