## Exploring Discrete Mathematics Summary

### Monday - Friday, July 2 - 6, 2012

The Discrete Mathematics group started the week by discussing what discrete mathematics was to them. We then looked at a few textbooks and course outlines to get ideas on what topics we'd be interested in. Then we solved a set of problems from the Exeter course. There was discussion on elections, apportionment, fairness, graph theory, and geometry. Later in week one, we split into groups to develop ideas that resonated with individuals.

The Voters (Carl, Debbie, and Peter) are looking to produce a unit that will tie together fairness, apportionment, and election theory. This unit long project will take students and teachers through the traditional methods available to determine a winner while also looking at the pros/cons of each method and how "fair" each method is according to certain criteria. They anticipate having a set of data that can be used for all voting methods. They also hope to have templates set up for teachers to use in the classroom with data generated by their students.

The Geometers (Shaffiq, Aziz, and Carol) are working on a 3-module unit to cut across different parts of a geometry curriculum (probably 9th or 10th grade). Their goal is to use dissection as a theme to help students understand area and polygons, tessellations, and volume. Carol is mainly working on the dissections of polygons and writing two to four lessons that relate these to concepts of area. Shaffiq is writing the end of this module with a proof of the Pythagorean theorem through dissection. He will then work on tessellations. Aziz is working on volume, currently looking at the dissection of a large cube into smaller cubes and/or into rectangular prisms.

The Mappers (Nadine, Amanda, and Rachell) have created an outline for a collection of lessons on map coloring. Students will learn how to color a map with the fewest number of colors, write their discoveries as algorithms, and apply their understanding of map coloring to real-world problems. These lessons will be able to be used independently of the whole 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.