"Math Magic and Probability and Game Theory, oh my!"
Our class of January 21 had us guessing and strategizing! I performed three different "magic tricks" involving consecutive numbers and patterns and challenged the children to discover the method behind the madness. The first two involved averaging and algebra (!), although we did not get bogged down in formulas, the students were able to recognize and understand that we were applying a formula to solve the trick. The last was truly a "trick" that required them to spot patterns -- they were able to see the solution within about three or four minutes of studying the problem. I think after 14 classes they have just about figured me out! If your kids would like to see some math magic performed by a real "mathemagician" you can visit young Ethan Brown's website, it's truly incredible stuff!
We then played three different games of chance which introduced the students to the concept of probability, and while discussion of probability calculations would have been out of place here, we enjoyed trying to predict what might happen given certain scenarios. Mostly we were wrong, and that's ok!
We wrapped up the day with a little discussion of game theory. We heard a scenario and then played a game, in pairs, that illustrated the story. It was great watching the kids figure out what the strategy should be after a couple of rounds! In our final game, tootsie rolls were on the line, and thankfully won and not lost.
Lastly, here's a fun game for you -- have you ever played "Tower of Hanoi?" A good game of strategy and elimination. Enjoy Hanoi!
The Study of TopologyClick photo to learn more about Mobius Strips
Our last class! Hard to believe that our time together is at an end for now, but here we are. Before today's class began, I asked the students to tell me something they had learned this semester that they didn't know before. Not surprisingly, they gave very concrete answers such as Pascal's Triangle and paradoxes. I told them what I hoped they would take with them was a sense of exploration -- of not being afraid to spend time thinking through all possible answers, and that sometimes we can be presented with problems that have multiple solutions. The problem is not considered solved until all solutions are found!
We spent much of today discussing surfaces, vertices, sides and edges, and playing with Mobius Strips. Many children are familiar with these strips that seemingly have just one side, but today we attempted to predict what would happen if we cut the strips into certain configurations. For example, if we cut a Mobius strip down the middle, what happens? (You have one loop with four half twists, the new loop is twice as long and half as wide!) Next, what happens if you make another loop and cut it 1/3 from the edge? We had some technical difficulties with this one, which I later realized resulted from the ends not being securely fastened (big problems, simple answers!). If your student would like to try more at home, here are the problems and expected results:
All strips should be 11" long and 1" wide:
Make a Mobius Strip (a strip of paper with a half twist securely fastened at the ends with tape) and cut parallel to the edges about 1/3 of the way from one edge. What will happen? (You get two intertwined loops, one is the same length as the original but only 2/3 as wide, the second has four half-twists and is twice as long and only 1/3 as wide as the original).
Make a loop with two half-twists and cut parallel to the edge in the middle -- what happens? (You get two intertwined loops half as wide and same length as original, both with two half-twists. Now, for the really brave, cut each of those loops in half again -- what happens now?)
Here we are with our collection of Mobius Strips!
Our last challenge for the day was to make five separate objects with a different number of surfaces or "faces;" including 6 faces, 4 faces, 3 faces, 2 faces, and 1 face.
One or two of them stumped us, but we came up with some good solutions that I share with you here.
I look forward to continuing to watch your children learn and grow each Monday! Thank you for a great semester!