Leadership Program:

Exploring Discrete Mathematics in the Classroom

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DIMACS || Leadership Program || LP Web Pages || The Math Forum
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K-8 Teachers Leadership Institutes:
Exploring Discrete Mathematics in the Classroom

Week 3

During the second summer institute, program participants learn about discrete mathematics and review and prepare materials that they can use to introduce these topics in their classes. The following web sites will help teachers to explore the topics farther.

Day 1 - Review of Graphs and Paths

Day 2 - Matching and Bipartite Graphs

Day 3 - Games and Winning Strategies

Day 4 - Counting and Probability Problems

Day 5 - Casino Games

Day 1 - Review of Graphs and Paths

  • The Mathematics of Graphs and their Games - Mega Math
    Most of the special terminology of graph theory consists of familiar words that have special meaning in the context of graphs. The concept of a graph is very simple to grasp, yet these mathematical objects are extremely varied. The specialized vocabulary for talking about graphs is most useful for trying to describe the various graphs and their properties.
  • Numb3rs Blog - Northeastern Univeristy
    Graph theory and its use in the tv hit Numb3rs!
  • The Konigsberg Bridge Problem - Professor V.P.N. Nampoori, International School of Photonics at Cochin, India
    A thorough review of the first week of Graph Theory
  • Introduction to Graph Theory - Rashid Bin Muhammad, Kent State Univeristy, Ohio
    Reviews Week 1 nicely and begins to move into bipartite graphs.
  • Sudoku and Graph Theory - Eytan Suchard, Raviv Yatom, and Eitan Shapir
    Pretty technical, but a neat connection between Sudoku and graph theory matchings!
  • Matching - Eric W. Weisstein, MathWorld - A Wolfram Web Resource.
    Clear, brief explanations of matching, perfect matching, and bipartite graphs.

Day 2 - Matching and Bipartite Graphs

  • Puzzles on Graphs
    Bipartite graphs often appear as a description of mapping (or matches) between two sets. It is more or less obvious that the board of Sam Loyd's 15 puzzle might be abstracted to a 4x4 graph every counter position corresponding to a node. Edges indicate possible puzzle moves, i.e. moves of the empty square. Less obvious is that the graph is bipartite.
  • Interactive Nim Game - Jill Britton,Camosun College, Victoria, BC, Canada
    Interactive site for playing Nim.
  • Nim Skulls - Timothy J. Rogers
    A fun, albeit macabre, version of Nim that never loses!
  • Northcott's Game - Alexander Bogomolny
    Easily change the game board dimensions as you explore Northcott's Game.
  • Dots and Boxes - Wikipedia Foundation, Inc.
    An introduction to Dots and Boxes.

Day 3 - Games and Winning Strategies

  • The Game of Nim - Alexander Bogomolny, Interactive Mathematics Miscellany and Puzzles
    A Java simulation of the famous game, with a theoretical explanation and extension to other similar games: see also similar Java games: Nimble, Northcott's game, Plainim, the Scoring game, and Turning Turtles.
  • The Fruit Game
    An interactive game in which players take turns removing fruit from the table according to the rules. The player who removes the last fruit from the table loses. Based on the mathematical puzzler known as the game of Nim.
  • Winning at NIM - The Math Forum
    In a game of NIM, there are three rows of 5, 4, and 3 sticks respectively. Picking up as many as you want in a row, how do you win?
  • Combinatorial Game Theory - David Eppstein, Theory Group, ICS, UC Irvine
    A list of links to materials on the Web. Combinatorial Game Theory studies strategies and mathematics of two-player games of perfect knowledge such as chess or go (but often either concentrating instead on simpler games such as nim, or solving endgames and other special cases). An important distinction between this subject and classical game theory (a branch of economics) is that game players are assumed to move in sequence rather than simultaneously, so there is no point in randomization or other information-hiding strategies.
  • Introductory Combinatorial Game Theory - Lim Chu Wee, National University of Singapore
    Theoretical analysis of many well-known games, including Nim (with binary representation).
  • First Bites - Brian Hayes, Bit Player
    Winning strategy for Chomp.
  • Game Theory - Mike Shor, Game Theory .net
    The gateway to loads of information about Game Theory.
  • Erich's Undergraduate Research Problems - Erich Friedman, Stetson University, DeLand, FL
    Interesting food for thought in Game Theory and other areas of Discrete Math.
  • Sprouts - Danny Purvis, World Game of Sprouts Association
    Rules for the game of Sprouts.

Day 4 - Counting and Probability Problems

  • Permutations and Combinations - The Math Forum
    An introduction to permutations and combinations from the Dr. Math Frequently Asked Questions Archive.
  • Introduction to Probability - The Math Forum
    An introduction to probability from the Dr. Math Frequently Asked Questions Archive.
  • Elementary Combinatorics - University of Cincinnati, Department of Mathematical Sciences
    The art of counting is called combinatorics. Here is a short listing of the formulas. All are the consequences of the product rule of counting.
  • Fun with Probability! The Probable Pen in the Cereal Box - Michael Cornell; College of Education, Univ. of Illinois at Urbana-Champaign
    A simple probability simulation in which one attempts to get an entire set of “trinkets” offered in cereal boxes.
  • Probability - Mrs. Glosser's Math Goodies
    Very basic introduction to probability of simple events.
  • Independent Events - Mrs. Glosser's Math Goodies
    Probability of compound events (independent events happening in succession).
  • Chances - National Center for Education Statistics, U.S. Department of Education
    Two-dice sums . . . nicely illustrates how experimental probability approaches theoretical with more and more trials.
  • Ken White's Coin Flipping Page - SHAZAM Econometrics Software
    Online coin flipper.
  • Applied Binomial Theory
    Pascal's Triangle and its use in determining “binary” probabilities.
  • How Many Ways Can a Team Win a Best-of-7 Series? - Kent Anderson, Kings County Office of Education, California
    Students will discover how many ways a team can win a 7-game series.
  • The Cereal Box Problem - George Reese, Office for Mathematics, Science, and Technology Education, University of Illinois at Urbana-Champaign
    Another look at the cereal box “trinket” simulation.
  • The Birthday Problem - Nicholas Exner, Office for Mathematics, Science, and Technology Education, University of Illinois at Urbana-Champaign
    Duplicate birthday simulation.

Day 5 - Casino Games

  • Baccarat; A Casino Game - The Math Forum
    Dr. Math explains the game of Baccarat.
  • Is It Fair? - Illuminations, National Council of Teachers of Mathematics
    Fair or unfair?
  • She Always Wins. It's Not Fair! - National Council of Teachers of Mathematics
    More practice analyzing fair and unfair games.
  • A Fair Hopper - Illuminations, National Council of Teachers of Mathematics
    The game of “Hopper”: fair or unfair? How to reassign points?
  • Can A Happy Hopper Escape? - Illuminations, National Council of Teachers of Mathematics
    Extension to a related game, the Happy Hopper.
  • What Are Your Chances to Win? - LearningWave Online, a division of HRM Video
    Introduction to lotteries and basic probabilities.
  • Lottery Math - Paul Cox, Math Mistakes Website
    Probability and lotteries: a bit more advanced.
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DIMACS || Leadership Program || LP Web Pages || The Math Forum
_________________________________

Created by Judy Ann Brown, Brian Rollfinke, and Gail Holmes
Last Update: April 21, 2008