[sorry for what is likley a sloppy write up – had to do it at 5:00 am because of some internet problems last night]
Saw a great problem for kids yesterday via a Steven Strogatz tweet:
Thought it would be a fun one to go through with the kids, and it turned out to be even better than I was expecting as the kids approached the problem completely differently than I did.
I started off with my younger son. We spent the first half of this year going slowly through Art of Problem Solving’s Introduction to Number Theory book, so he has seen problems similar to this one. Another reason that I thought this would be a great problem for him is that we just began talking about square roots last week. So, well-timed Wordplay blog!
His reasoning for how to make the number 256 out of 5’s and 7’s around 4:00 in the movie made me really happy.
Next up was my older son. The problem gave him quite a bit of trouble in the beginning, and in fact he told me at one point he was stuck. We’ve learned an extremely important lesson from James Tanton about what to do when you are stuck – try something.
When he tried something, he found the path to the solution. I think his struggle through this problem was really productive.
Next up we moved to a little “broken calculator” computer program I wrote on Khan Academy’s site to illustrate the problem:
The NY Times Wordplay Blog’s Broken Calculator Problem
Using this calculator we talked through an approach to solving the problem that didn’t require you to find 256 first. Sorry this part is broken into two videos, I accidentally turned off the camera when I was moving it – oops (and the noise in the background is our dishwasher).
Finally, we wrapped up by quickly talking through why you could make every positive integer other than 1. I was much more interested in having the kids think through the first part of this problem rather than this part, so that’s why we didn’t go into all of the details here. Still, they were pretty surprised to see (even informally) that you could make every positive integer other than one.
So, a great problem from the Wordplay blog led to some great math conversations with the kids. The only little issue was some internet problems at our house last night that meant I had to write this one up at 5:00 am – so sorry for the likely sloppy writing.
Mike's Math Page 
The pi observation was interesting. Here’s an extension question inspired by that: can I use the broken calculator to display a value within 0.01 of pi? To any desired degree of accuracy?
On all (almost all?) real calculators, you will eventually get 1 from repeatedly taking square root of any positive number as you exhaust the precision of the calculation.
I’m slightly surprised they didn’t see that every other natural number is possible once they recognized that 2, 3, 4, 5, and 6 were possible. Again, within the calculator’s memory constraints.
I think that pattern is a tough thing for kids to see.
I’ve been through a similar exercise once with the Chicken McNugget problem and I think it is a little easier there.
In this problem the solution that seemed natural to them was looking at 4th or 8th powers. With a little more experience, I think, they’d be able to recognize that all 4th powers of integers 3 or greater can be written as a sum of 5’s and 7’s, but that idea isn’t at the front of their mind just yet.
Also, they just got back from a 3 hour car trip when we started in on this exercise, so take that into consideration, too.
I was thinking 2 + 5n, 3+5n, 4+5n, 5+5n, 6+5n are enough to exhaust all naturals > 1.
No particular criticism meant, just find it interesting what is easy to see and what is hard.
That’s what I was trying to say about the Chicken McNugget problem. Once you see that you can do 44, 45,46, 47,48, and 49 McNuggets, it is natural to see that you can do the rest.
Here they were thinking about powers rather than about adding, so it wasn’t as easy (or natural) for them to see the adding piece. Connecting the powers and the addition, I think, is not the easiest step.