Park City Mathematics Institute
Exploring Discrete Mathematics
Project Abstract

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*Aziz Jamash, Carol Kinney, Shafiq Welji

This collection of three modules treats dissections as a technique for understanding area and volume. The interdependent modules are designed to be implemented at different points throughout the year and include curriculum hooks through which the modules can be connected to the geometry strand. The first module reinforces the concept of area, grounds students in dissection techniques and strategies, and reveals the relationships among rectangles, triangles, parallelograms and trapezoids from which the area formulas stem. Students engage in experiential learning by dissecting plane geometrical shapes, using the idea of scissors congruence and extrapolating their experience to generalizable algorithms. Tangrams serve as an extension during the first module.

The second module builds upon the relationships derived in the first lesson and uses dissection techniques which were developed in the first module. Students derive or work toward the derivation of the Pythagorean Theorem and deepen their understanding of proofs. Various techniques are suggested to allow students to share their proofs and ideas with each other. The extension to the module considers whether any polygon can be dissected into any other polygon with the same area.

The third module considers volume and three-dimensional dissections and shows the volume formula for certain pyramids.tudents develop an intuitive understanding that the volume of a cone or pyramid is one-third the volume of a prism with the same base area and height. In the module, the students physically construct the dissection to help reinforce the understanding of the meaning behind volume formulas for a general cone (including a pyramid and circular cone).

Map Coloring
Rachelle Ashley, *Amanda Painter, Nadine Williams

Students will find strategies for deciding the best way to color a map using the minimum number of colors. This will be in the form of exploring maps and establishing that there are reasons for coloring maps the way we do. Further, if there are constraints, it may be favorable to use the fewest number of colors.

After introducing map coloring, teachers have the option to take a number of paths. One such path is to explore map coloring algorithms with students devising their own algorithms and compare with others students. Another path is to apply map coloring techniques to problems with constraints, such as scheduling or arranging fish into tanks. Lastly, teachers have the option to explore fairness using some principles of map coloring.

These lessons are designed to be used in a flexible manner. Teachers can use them all as a cohesive unit or spaced out as individual activities that build slowly over time.

Voting Theory
Debbie Lenz, Carl Oliver, *Peter Sell
We hold elections/votes all the time: to vote for a new president, to elect student council members, or even to decide where our family will go out to dinner tonight! So what is the best way to decide who or what wins a vote? The current method we are most familiar with here in the US is a simple majority rules vote. However, there are many other options to choose from and each one has different benefits. Voting theory (part of social choice theory) is one area of discrete mathematics with many important applications here at home and internationally. This project will take students through an investigation of top choice versus ranked preference ballots and of four main vote-counting methods to determine the winner of a vote. Students will look at the pros and cons of each method and complete a project analyzing the best voting method to use for a real world scenario of their choice (ex: sports rankings and awards, school votes, consumer products, contests and awards . . . ). Mathematically, students need only know about percents and permutations, but mathematical reasoning is a strong component of this unit.

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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.