Thursday, October 6, 2016

Math Circle

G consulting with the teacher after class
We have an enormously talented teacher facilitating a math circle for our group of friends this year.  And the topics!  Exciting:

1) Euler circuits: We covered historical development and discussed the abstraction of a practical situation that led to graph theory. Talked about a few applications to real life situations and then worked on a few connected graphs to gain insight. That led to the Euler's theorem's condition's necessary part for the existence of Euler circuit for a (connected) graph. We informally proved why it is necessary to have even degree for each node. (Didn't address that it is sufficient also since it is beyond the level of our class.) 

2) Introduced the rules for playing the game SET and compared it with the rules for tic-tac-toe. Played the game to develop familiarity and then talked about how remainder changes for each characteristic (mod3) as one removes "sets". With that knowledge, played the game of guessing the missing card for the end game. 
This game introduces some higher dimensional geometry and the notion of modulo arithmetic.

3) Introduced counting techniques for "n choose 2" without using this terminology. Talked about three real life applications to introduce this concept and used different counting techniques suitable for each situation. Showed the equivalence of 1+2+....+n  and  n(n+1)/2. Also introduced the notion of triangular numbers.

4) We talked about the ratios of weights around a pivot in one dimension. Checked it out by placing coins on cardboard strips. Then extended that to two dimensions. We worked out a problem of figuring the weights on the vertices of a triangle using two intersecting segments and the arm ratios for them. We also talked about strategy to avoid fractions. (Without mentioning the term, we basically used the least common multiple).  The goal here was to start a discussion that will lay the foundation for the notion of centroid.  Revisited SET game for the last 10 minutes.

5) Bases for numbers:  We talked about the abstract concept of numbers vs their representation in decimal system. Then compared their representations in different bases. Addressed historical reasons for base 10 and base 60 as well as contemporary reason for base 2. Introduced dot fusing for conversion from base 10 to base 2. 



G's notes from Week #5 (I'm impressed)



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