Park City Mathematics Institute
Project Abstracts

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Applet Based Combinatoric Activities
Cal Armstrong, Kyle Barriger, Holly Farnsworth, John F. Mahoney

This project consists of four activities that are based on applets. Teachers and students can explore these activities both by using these applets and by hand. Bobbie Bear explores elementary combinatorics and is appropriate for upper elementary and middle school students. The following activities are suitable for both middle and upper school students. Knight's Tour and Instant Insanity are challenges that lend themselves to applications of both graph theory and combinatorics. These two activities are appropriate for use as Problems of the Week that students could work on over 1 - 2 weeks. The Date Game looks at combinatorics and game theory in a series of Nim-like games of increasing difficulty. Fill 'n Pour relates aspects of combinatorics and number theory so that students can discover ways to make quantities of liquid from using two containers with relatively prime capacities.

Investigating Catalan Numbers
Jason Bershatsky, Jerry Gribble, Peter Herreschoff, Michael Holsten, Reynaldo Jope, Mark Kammrath, and Jing Shiau

Catalan numbers are a set of numbers that play an important role in combinatorics, but can also be found in a variety of situations easily accessible to the student. We present a variety of activities from which the Catalan numbers arise. Each activity can stand alone as a good problem solving experience. More in depth investigations follow for the student to make connections between several of these representations by using direct one-to-one correspondences. For the most advanced student, activities are provided that leads the student to conjecture a formula for Catalan numbers and justify his/her conjecure. Through these activities, the student will experience problem-solving in a number of areas, exposing him/her to an important set of numbers not normally encountered in school mathematics.

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IAS/Park City Mathematics Institute is an outreach program of the Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540
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With program support provided by Math for America

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.