Exploring Discrete Mathematics in the Classroom

Exploring Discrete Mathematics in the Classroom

Week 2

During the 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 - Systematic Listing and Counting

Day 2 - The Choose Numbers

Day 3 - Number Patterns and Iteration

Day 4 - Patterns in Geometry

Day 5 - Generating Fractals

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; shortest-path diagrams; Stirling numbers of the first and second kind; Bell numbers, harmonic numbers and the book-stacking 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.
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 non-intimidating textbook-type 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 n-sided 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
• 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 four-sided 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 self-running 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 write-up 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 seed-heads 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; Self-similarity; Fractal Dimension; Changing the rules in the chaos game; and Rotations and Animations, with summary and references.
• Self-Similar Structures - Jan Koster, University of Groningen, The Netherlands
Good introduction to self-similarity 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: Self-similarity, Fractional dimension, Formation by iteration. Teacher-to-Teacher notes; Fractals on the Web.
• Fractals and Chaos Lesson: Self-Similarity and Recursion - Shodor Foundation
The discussions and activities are designed to lead the students through the discovery of the features common to all fractals: Self-similarity 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 step-by-step 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.

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