A neat number theory problem for kids from Tracy Johnston Zager

I been traveling the past couple of days but did notice this post on twitter a few days ago:

Q from a 4th grade teacher I admire: "What do you think of questions like this: 'What is the sum of all the factors of 35?'" #elemmathchat

— Tracy Johnston Zager (@TracyZager) October 19, 2014

There was some debate about whether or not this is a good problem for kids (4th graders, in particular), but I thought it had some fun possibilities and looked forward to going through it with the kids when I got home.   By coincidence I’m in the middle of a Art of Problem Solving’s “Introduction to Number Theory” book with my younger son right now,  so I’m probably a little more primed than usual to be interested in this type of problem.   Part 3 of the post below – the video FamilyMath185c – is one of the best math talks that I’ve ever had with the boys, so I’m sort of double happy that we discussed this problem today.   Sorry this post is written up so quickly – I’m tired from the trip, but really excited about part 3.  Probably not the best combination for good writing 🙂

I started by simply introducing the problem and then asking each of the kids to come up with a new (but similar) problem that they would also be interested in solving.   It turned out that they both came up with a fairly neat twist, so we had a nice little math talk set up just from their ideas.

Next up the revised problem proposed by my younger son – what if instead of of adding up the factors, we alternate adding and subtracting.   I certainly wasn’t expecting this idea and didn’t really know what to do with it, so the direction I went was mainly arithmetic practice.  If there’s a easy to understand number theory ideas here, I’d love to hear about them.

My older son picked an alternate problem that does turn out to have some really interesting math hiding in it.   What made this part of our talk more fun than usual was that both kids picked up on the interesting math reasonably quickly – even noticing that the answer would be different if you were dealing with a perfect square!!  Yay!  This part of the discussion shows both why I think kids will find problems like the original one to be interesting and also why I love having the opportunity to talk about math ideas with my kids.  So much fun:

Finally, the extension of the problem I chose was counting the total number of divisors.  This is a topic that both kids have seen before, so I expected them both to remember at least some of the ideas.  I wanted to cover this angle of the problem originally more as a reminder of something that we’ve discussed before but when my older son wanted to look at the product of divisors, it turned out that this topic fit in pretty naturally.

I’m really happy to have seen the original problem on twitter.   Part of the attraction of problems like these is the opportunity to get in some arithmetic practice in the same way that you can with “number talks.”   But, in my mind this type of problem brings a little more than the basic number talks do because there is also some pretty interesting math hiding in and around the problem.  I have always believed that many kids will find the math in problems like this to be really interesting and the fun we had today – particularly with the product of the divisors part – makes me believe in this type of problem even more.

I’m not advocating for a book like Art of Problem Solving’s “Introduction to Number Theory” to become a part of a standard elementary school’s math curriculum, but I would advocate for kids to see problems like this one occasionally.  You never know what sort of surprising fun you’ll have talking through it!