Minicourse Descriptions 2001


Session A: (August 13 - 14)


Fractals: an Introduction

Instructors: Will Webber and Ronda Webber

In this minicourse we will look at the basic concepts in fractal generation. The topics will include fractal curves such as the Koch, Hilbert, and Peano curves, "Mandelbrot" sets, Julia sets, fractal dimension, iterated function systems, chaos and others. An emphasis will be placed on creating fractals by repeating simple sets of instructions. The activities will be hands-on and include stamping, folding strips of paper, cutting pop-up fractals, building 3-dimensional models, creating computer generations of fractals, and others.


Hands-on Spherical Geometry

Instructors: Philip Mallinson with Carol Hattan

You know very well what an angle is, what a straight line is, and how to measure angles and segments and areas in the plane. But suppose your universe is not a plane but a cone or a cylinder or a cube or a sphere? This minicourse invites you to extend the these basic notions to surfaces that don't look like a flat sheet of paper. You will build your own models and also use commercial products like the Lenart sphere. As well as opening a new world of geometry, geometry on the sphere is also very practical. After all, map makers need to flatten out the world for the pages of an atlas and navigators need to sail in straight lines. How do they do it?


Session B: (August 15 - 16)


In the Footsteps of Escher: 3-D Puzzles that Tessellate Space

Instructors: Joyce Frost with Karen Hall

Who can resist the allure of 3-D puzzles? Examine the mathematics of polyhedra that pack space: cube, triangular prism, hexagonal prism, rhombic dodecahedron, stellated rhombic dodecahedron, and truncated octahedron. Learn how to make 3-dimensional geometry come alive for your students by involving them in the entire process: discovering puzzles, designing nets, building puzzles, and creating new puzzles. From card stock and contact paper, you will make a variety of puzzles, some old, some obscure, and some recent discoveries. You will leave with numerous high quality puzzles to display, share, or amuse others.

Graph Theory: The Mathematics of Connection

Instructor: Brian Hopkins

Graph theory is a very accessible and relatively young area of mathematics. It offers a good venue for experiencing and developing mathematical proofs (often constructive). In its general study of connection, it is widely applicable. For current high school teachers, beyond general interest and the experience of interactive learning, some of this material is now in some secondary curricula.

Emphasis will be given to historical development and applications. The area is now considered to begin with Euler's bridges of Koenigsberg problem in 1736, although there are older treatments of the knight's tour problem in chess. Hamilton's "around the world" puzzle is the 1800's contribution, but the field doesn't really get going until the mid-1900's with the availability of computers. Graph theory topics will include types of paths and cycles, graph & map coloring, planarity. Related applications, in addition to the historical examples, include the structure of semiconductors, the travelling salesman problem, and scheduling. We'll spend a lot of time on understanding the notion of "six degrees of separation" in this context, using Malcolm Gladwell's book The Tipping Point and the University of Virginia's "Oracle of Bacon" (a web resource) which uses a movie database to fully answer the "Six Degrees of Kevin Bacon" question.

Instruction will include a great deal of collaborative student work. Proofs will be developed from investigation and particular solutions.


Integrating Geometer's Sketchpad across the Curriculum

Instructors: James King, Art Mabbott, Linda Thornberry

The dynamic software The Geometer's Sketchpad is a powerful tool in any course named "Geometry", but this minicourse will focus how Sketchpad can be used in other math courses as well, including Integrated Math courses. Sketchpad can be used for visualization, experiment, modeling, and dynamic graphing. This course will demonstrate how Sketchpad can be used to study such topics as trigonometry, conics and projectile motion, centers of mass, complex numbers, max-min problems, nets for polyhedra, dynamic graphs linked to models, simulation of physical devices.

The minicourse will highlight the new graphing and tool features in the upcoming Sketchpad 4.0. This is an Intermediate Sketchpad course; participants will be expected to know basic Sketchpad operations. (If there are questions about the level of Sketchpad knowledge needed and/or how to acquire this knowledge, please inquire of the instructors.)


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