I finally led a pretty good math circle there, with high school aged kids, using the Magic Pancake problem. (All the pieces magically change to your favorite kind of pancake, and grow to give you as many yummy bites as you want, perfectly filling your tummy. One cut makes two pieces, two cuts can make 4 pieces, how many pieces can you get with N cuts?) The first year I went, I told a story that used fractions, with kids who were too young for it. The second year, I did base 3 or 8, with junior high aged kids, who I couldn't get to talk. This year my circle finally went well, though I still have lots of room for improvement. My big goal is to learn how to ask more open questions, and let the students guide us along, keeping my mouth shut more, and giving less hints about where I think we should go.
Our morning math circles continued to be my favorite part of the day. I came in late on Monday, and got to hear Bob leading the group as they worked on finding the circumcenter of a triangle. I've never taught geometry, so I don't know this stuff - it was a blast to think about. Next came the incenter, and then the orthocenter. We worked on the topic a bit more on Tuesday, and again on Friday. I'm looking forward to playing with it on my own.
On Wednesday, Leo led us in thinking about divisibility by 3 and 7. I was very familiar with the first part, and got to watch people thinking again. That was lovely. (I'm not a very good observer, and it's a skill I really want to improve.) The second part was tantalizing. He showed us how you can find out whether 7 is a factor of a large number by removing the last digit, doubling it, and subtracting it from the number formed by the remaining digits.
Example: Is 36,666 divisible by 7? Double the last 6, getting 12. Subtract 12 from 3,666, getting 3654, still too big. Do it again. Double the 4, getting 8, subtract 8 from 365, getting 357, still too big. One more time. 2*7 is 14, 35-14=21. Yes, 7 goes into 21, so that means 7 goes into 36,666.Buy why? That's the question! I solved it on Wednesday evening. When I'd done that, Amanda showed me her proof which I liked much better, because it involved modulo 7. I've been playing with modular arithmetic lately, and I like the power it has. She challenged me to make up another way to break a big number up to see if it's divisible by 7. I did it, using mod 7. And then I got excited, thinking about how you could mix and match different techniques. I know that 7*11*13 = 1001, so any number over 1000 can have multiples of 1001 subtracted before using the other techniques. (30030 is a multiple of 7, subtract that from 36,666, getting 6636. Now subtract 6006, getting 630. Yep, multiple of 7. Oops! I didn't even need the other techniques for this one.)
On Thursday, Ellen led us in thinking about Euler's formula for polyhedra (V-E+F=2), as an example of something whose proof is very subtle. Once again, I got to watch people thinking about the beginning steps. If you're interested in this, Dave Richeson's book, Euler's Gem, is a great treatment of it. (And Michael Paul Goldenberg wrote about this session in his post about the Institute. Thanks, Michael, for the reminder that I wanted to write a post.)
What else did we do? My swimming buddy didn't make it back this year, so I only managed to get up early once to swim in the lovely pool. (Missed you, Ellen. Thanks for the cap, Linda.) I ate less than in previous years, but the food was still amazingly good for a big food services operation. And I brought games to share once or twice during our evening gatherings at the dorm lounge. My friend Linda had just given Dizio to my son, and I brought it along. We had a great time with it. We also enjoyed Kataminos, which I mentioned in a previous post.
Come join us next summer!