Leadership Program:
Exploring Discrete Mathematics in the
Classroom
DIMACS  Leadership Program 
LP Web Pages  The Math Forum
K8 Teachers Leadership Institutes:
Exploring Discrete Mathematics in the Classroom
Week 2
Day 1  Systematic Listing and
Counting
 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.
 Tree Diagrams  Oswego City School District Regents Exam Prep Center
The use of tree diagrams explained very clearly for children.
 Working with Tree Diagrams  Oswego City School District Regents Exam Prep Center
Use of tree diagrams for simple counting and probability questions. Good practice for young students.
 Probability and Tree Diagrams  BBC Home UK
Tree diagrams and their use to find simple and compound probabilities.
 Basic Counting Rules  Russell Bruce Campbell, University of Northern Iowa
Clarification of addition rule ("or") vs. multiplication rule ("and").
Day 2  The Choose Numbers
 Pascal Web Unit  Math Forum
A Web unit designed to support workshops given by
the Math Forum for the Urban Systemic Initiative (Philadephia and San
Diego). Read about the history of Pascal's triangle and learn to
construct it; view illustrations of number patterns to be discovered;
carry out interactive investigations in Java script or the Geometer's
Sketchpad, and explore this famous triangle through lesson plans that
feature questions, answers, discussion, and student worksheets.
 Combinatorial Figures  Robert Dickau
Mathematica pictures that are interesting for elementary combinatorics. Catalan number diagrams; permutation diagrams; derangements; shortestpath diagrams; Stirling numbers of the first and second kind; Bell numbers, harmonic numbers and the bookstacking problem; and Fibonacci numbers. Includes some Mathematica code for generating the pictures.
 Explore Number Sets and Mathematical Patterns in Pascal's Triangle.  The Math Forum
Pascal's triangle is an arithmetical triangle made up of staggered rows of numbers.
 Advanced Anagramming  Wordsmith.org
A fun way to get your students interested in anagrams . . . let them create their own!
 Applied Binomial Theory
Scroll down to the “Cheese Puzzle” for more explanation of Counting Paths in today's Workshop.
 Combinatorics & Probability Up Close  MG Prep, Inc
Detailed discussion of introductory combinatorics.
 Counting: Permutations and Combinations  Keith G. Calkins
A nonintimidating textbooktype of explanation of combinations and permutations.
 Pascal's Triangle and Its Patterns
Great introduction to Pascal's Triangle and many of its mathematical patterns.
 Numbers & Patterns  Using Pascal's Triangle  Bruce Jacobs
Wonderful example of the connection between Pascal's Triangle and “choose numbers”.
 The Fabulous Pascal's Triangle  Kjartan Poskitt
More on Pascal and choosing.
 Number of Diagonals  Math Central, University of Regina & Imperial Oil Foundation
Nice discussion of counting the number of diagonals in an nsided polygon.
Day 3  Number Patterns and
Iteration
 Triangular Numbers  The Math Forum
Second and third graders ask Dr. Math why 1 is a triangular number.
 Triangular Numbers  The Math Forum
A Middle School student asks Dr. Math about triangular numbers.
 Turtle Tracks  Ivan Peterson
One way to describe a geometric figure is
in terms of the path generated by a moving point. Instead of defining a
square, for example, as a foursided polygon with equal sides and
angles, you can call it the path generated by the following rule: Go
straight for a distance s, turn 90 degrees right, and repeat until the
path returns to its starting point.
 The Art of Spirolaterals  Robert J. Krawczyk
Illustrated introductions to
spirolaterals and strange attractors. See a selfrunning demonstration
of a variety of spirolaterals, or generate one of your own and see all
the reversals. The link of current exhibits lists installations showing
strange attractors and spirolaterals on display around North America.
 Supreme Court Welcome: Beyond Handshakes  NCTM
Good NCTM lesson relating to the handshake problem
 Square and Triangular Numbers  The Math Froum
Good discussion of square and triangular #'s for middle schoolers (gets a little algebraic).
 Triangular Numbers  Radoslav Jovanovic
Triangular and other “figurate” numbers—good but gets a bit technical.
 Fascinating Triangular Numbers  Shyam Sunder Gupta, Jaipur, India
Everything one could ever want to know about Triangular Numbers.
 Spirolaterals  Mounds Park Academy, St Paul, MN
A brief writeup of the spirolaterals activity.
 Spirolaterals Creator  Robert J. Krawczyk, Illinois Institute of Technology, IL
Java applet to create your own spirolaterals.
Day 4  Patterns in Geometry
 Fibonacci Numbers and Nature  Ron Knott
Fibonacci and his original problem
about rabbits that gave the series its name, family trees of bees, the
golden ratio and the Fibonacci series, the Fibonacci Spiral and sea
shell shapes, branching plants, flower petal and seedheads and the leaf
arrangements around stems all involve the Fibonacci numbers.
 Sierpinski Gasket  Mega Math
The Sierpinski Gasket is named for the Polish
mathematician who first proposed it. It is a fractal image that is made
from equilateral triangles. A Sierpinski Gasket can be easily
constructed by anyone who can manage paper, scissors, and glue. The
concepts behind it are accessible to young children, yet challenging to
older students.
 Chaos in the Classroom  Robert L. Devaney
One of the most interesting
applications of technology in the mathematics classroom is the fact that
it allows teachers to bring many new and exciting topics into the
curriculum. In particular, technology lets teachers bring some topics of
contemporary interest in research mathematics into both middle school
and high school classrooms. The mathematical topics of chaos and
fractals are particularly appropriate in this regard. An internet paper,
with the following sections: The Chaos Game; The Sierpinski triangle;
Why does the Sierpinski triangle arise from the chaos game?; Playing the
chaos game in class and on the web; Selfsimilarity; Fractal Dimension;
Changing the rules in the chaos game; and Rotations and Animations, with
summary and references.
 SelfSimilar Structures  Jan Koster, University of Groningen, The Netherlands
Good introduction to selfsimilarity and iteration.
 What is a fractal?  Math Central, University of Regina & Imperial Oil Foundation
What, exactly, is a fractal? (A short introduction)
 Fractal's in Layman's Terms  Damien M. Jones
Fractals in layman's terms.
Day 5  Generating Fractals
 Fractals  Cynthia Lanius
This lesson plan for exploring fractals is designed so 4th
through 8th grade students can work independently and be assessed
innovatively. Conforms to 1989 NCTM standards; links to other fractal
sites. Contents: Why study fractals? Making fractals: Sierpinski
Triangle, Sierpinski Meets Pascal, Jurassic Park Fractal, Koch
Snowflake. Fractal Properties: Selfsimilarity, Fractional dimension,
Formation by iteration. TeachertoTeacher notes; Fractals on the Web.
 Fractals and Chaos Lesson: SelfSimilarity and Recursion  Shodor Foundation
The discussions and activities are designed to lead the students through
the discovery of the features common to all fractals: Selfsimilarity
and recursion.
 Exploring Fractals  Mary Ann Connors, University of Massachusetts Amherst
A wealth of good info. about fractals.
 Fractals Unleashed  Oracle ThinkQuest Educational Foundation
Very detailed site with stepbystep instructions for generating many famous fractals.
 Sprott's Fractal Gallery  Julien Clinton Sprott, University of Wisconsin
Tons of cool fractals. Of particular interest are the “natural fractals” near the bottom of the site.
DIMACS 
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LP Web Pages 
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Created by Judy Ann Brown, Brian Rollfinke, and Gail Holmes
Last Update: April 21, 2008
