Connecting with Mathematics

What is Mathematical Investigation?
Dissections and Area
Linearity and Proportional Reasoning
Pythagoras and Cousins
Pascal's Revenge: Combinatorial Algebra
Games and Number Theory

Combinatorial Algebra

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In Pascal's Revenge, participants engage in combinatorial investigations while they work on concrete problems that encourage them to search for patterns and to create and confirm/disprove conjectures about more general situations. Through hands-on activities, teachers reflect on and discuss a variety of mathematical contexts all tied together by Pascal's Triangle and binomial coefficients. In the first session, participants work with partitions of a positive integer n (the number of different ways to express n as a sum of positive integers) through activities with Cuisenaire rods. Session 2 uncovers facts about graphs on the coordinate plane (specifically taxicab geometry) by asking questions like, "How many ways are there to go a total of 5 blocks north and 8 blocks west?" In Sessions 3 and 4, participants will discover explicit connections between Pascal's Triangle, binomial coefficients, and the Binomial Theorem, while Session 5 introduces the capstone notion of generating functions.

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