Resources for Teaching Discrete Mathematics: Classroom Projects, History Modules, and Articles

Resources for Teaching Discrete Mathematics: Classroom Projects, History Modules, and Articles

Brian Hopkins

Language: English

Pages: 338

ISBN: 2:00227594

Format: PDF / Kindle (mobi) / ePub

This book collects the work of 35 instructors who share their innovations and insights about teaching discrete mathematics. Whether you teach at the college or high school level; whether your students are from mathematics, computer science, or engineering; whether you emphasize logic, proof, counting, graph theory, or applications, you will find resources in this book to supplement your discrete mathematics course.

19 classroom-tested projects, eleven history modules drawing on original sources, and five articles address a wide range of topics reflecting the breadth of content and audience in various discrete mathematics courses. A sampling of topics includes building a geodesic dome, working with excerpts of Pascal’s “Treatise on the Arithmetical Triangle,” applying discrete mathematics to biology and chemistry, and using logic in encouraging students to construct proofs.

Table of Contents
I. Classroom-tested Projects
II. Historical Projects in Discrete Mathematics and Computer Science
III. Articles Extending Discrete Mathematics Content
IV. Articles on Discrete Mathematics Pedagogy
About the Editor

About the Editor
Brian Hopkins received his BS in mathematics and a BA in philosophy from the University of Texas in 1990, and earned his PhD in mathematics from the University of Washington in 1997. He teaches at Saint Peter’s College, a Jesuit liberal arts institution in Jersey City, where he has led several undergraduate research projects and has been the recipient of the Varrichio Award for Teaching Excellence (awarded by the SPC Pi Mu Epsilon chapter) two times, in 2004 and 2007. Brian is the author, with Carl Swenson, of Getting Started with the TI-92 in Calculus (1998, John Wiley & Sons). Also, he has published several research articles in combinatorics and, with Robin J. Wilson, won the MAA’s 2005 George Pólya Award for excellent expository writing in The College Mathematics Journal for “The Truth About Königsberg.” Brian works with secondary school teachers in professional development projects with various organizations including the Institute for Advanced Study’s Park City Mathematics Institute, the Northwest Math Interaction, the New Jersey Professional Development and Outreach group, the Institute for New Jersey Mathematics Teachers, and the Pikes Peak Math Teacher Circle Academy. He is a member of the Mathematical Association of America, the American Mathematical Society, the National Council of Teachers of Mathematics, and the National Association of Recording Arts and Sciences. Brian plays piano, sings with Cantori New York, and enjoys New York City with his partner Michael.

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where students first learn to write proofs? Are the students mathematics majors, computer science majors, or is the course offered for a general education requirement? This book does not address those questions. The projects and articles here reflect the wide breadth of topics taught in the diverse discrete mathematics courses offered in universities, colleges, and (increasingly) high schools. I hope that every instructor of discrete mathematics will find projects and articles relevant to the

Frobenius counting functions n o # .m1 ; m2 ; : : : ; md / 2 Zd W all mj 0; m1 a1 C C md ad D k : Now the line segments get replaced by triangles (d D 3), tetrahedra (d D 4), and higher-dimensional simplices, but the general picture, namely that these counting functions enumerate integer points in Zd in dilates of nice geometric 1 However, one should be careful with such a statement—we invite the reader to prove that if a and b are not relatively prime, there are infinitely many line segments

the glued face. What is V E C F ? (c) Take three of the polyhedral tori, and glue them along two of the outer rectangles. You have now created a surface with three holes, called a three-holed torus. Count the number of vertices, edges and faces, ignoring the glued faces. What is V E C F ? (d) If you were to create a four-holed torus, what do you predict for the value of V E C F? (e) What about an n-holed torus? Exercise 2. Consider a solid cube with holes punched out of it in the following way:

to G¨ottingen in 1926, after which he was offered the position of a 2 The sequence 1, 3, 6, 10, 15, : : : , giving the number of dots in certain triangles [12, p. 49] forms the triangular numbers. sequence 1, 4, 10, 20, 35, : : : , giving the number of dots in certain pyramids [19, p. 76] forms the pyramidal numbers. 4 The mythical first Emperor of China. 3 The Lodder: Binary Arithmetic: From Leibniz to von Neumann 175 Privatdozent (an un-salaried lecturer) at the University of Berlin and

appear in the row immediately following. Recall: .1 C x/0 .1 C x/1 .1 C x/2 .1 C x/3 .1 C x/4 .1 C x/5 1 1 1 1 1 1 1 2 3 4 5 1 3 6 1 4 10 10 1 5 1 Figure 1. Pascal’s Triangle showing the binomial coefficients up to the 5th power. The coefficients can then be used to write out the expansion. For example, using the triangle above: .1 C x/4 D 1 C 4x C 6x 2 C 4x 3 C 1x 4 : The coefficients can also be determined by counting the numbers of paths from the apex of the triangle to the entry

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