Park City Mathematics Institute
Exploring Discrete Mathematics
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- Vicki Clancy, Brian Hamel, Armando Madrigal, Brian Hopkins*
- This three day project uses operations on polynomials to answer a question from basic probability about dice. The motivating question asks whether there are two six-sided dice with non-standard labels whose sum data match the results of two standard dice. Students address the problem first in an ad hoc way using logic and reasoning. For the second part of the project, a polynomial representation is introduced which allows students to solve similar problems more easily and prove the uniqueness of their solutions using factoring, multiplication, and synthetic division.
- Catalan Numbers Unit
- Allen Jacobson, Ted Linnenbringer, Breedeen Murray, Andrew Richardson, Brian Hopkins*
- The Catalan numbers play an important role in combinatorics and count a variety of objects accessible to students. We revised the Catalan Numbers Project written by the 2004 working group, incorporating reviewer suggestions and our own ideas. The current project is a unit where students explore five objects counted by Catalan numbers, develop four bijections connecting the various objects, and derive a formula using binomial coefficieints.
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This material is based upon work supported by the National Science Foundation under DMS-0940733 and DMS-1441467. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.