MathFest 2003 Workshop Titles

  1. Computer Vision and Image Processing
  2. Financial Decision Making: Making Money with Lines
  3. Fractals are Hiding Everywhere
  4. Game Theory
  5. Geometry Logo Style
  6. Is God a Mathematician?
  7. Mathemagic in Polygon Land
  8. Mathematics in Biology - Modeling Population Growth and Decline
  9. Parallel Knockouts
  10. Puzzle Solving Strategies
  11. Solving Third and Fourth Degree Polynomial Equations
  12. Some Strategies of Problem Solving
  13. Taking the Mystery out of i
  14. The Mathematics of Kicking a Football
  15. The SMART Board in the Mathematics Classroom

Descriptions of Workshops

Computer Vision and Image Processing
by Andy Chang, Youngstown State University

How can a computer see things? Mathematics! Statistics! Matrix! Algebra! That's right. In fact, with mathematics and statistics, a computer can do lots of wonderful things that humans cannot do. In this workshop, basic ideas of computer vision will be introduced, and the spreadsheet software EXCEL will be used to simulate Computer Vision and Image Processing.

Pre-Workshop Activities  for Computer Vision and Image Processing.

Financial Decision Making: Making Money with Lines
by Ward Shaffer, Youngstown State University

Everyone--individuals, businesses, organizations, and governments-- would like to make as much money as they can. The question is "How do I choose between competing options to maximize the amount I make?" This session will investigate how to do this using linear inequalities and their graphs.

Pre-Workshop Activities   for Financial Decision Making: Making Money with Lines.

Fractals are Hiding Everywhere
by Roy Mimna, Youngstown State University

We will begin with a short discussion of what fractals are and how they appear in nature. We will show that there are fractals which can be constructed by hand. i.e., We will construct fractals without the aid of a computer. Pascal's triangle will be constructed among others. There are also games and puzzles which have interesting connections to fractals, and we will look at some of these. There are many fractals which are best constructed by a computer program, so we will conclude with a look at some of the more exotic ones on the computer.

Pre-Workshop Activities   for Fractals are Hiding Everywhere.

Game Theory
by Nathan P. Ritchey, Youngstown State University

Although the branch of mathematics known as Game theory was first introduced by John von Neumann in 1928, it took the movie, “A Beautiful Mind,” which is a story about Nobel Prize winner John Nash, to bring this powerful subject into public view. Applications of game theory are quite extensive and can be found in anthropology, communications, social psychology, economics, politics, business, biology, and philosophy.

In this workshop students will be introduced to game theory and its uses. In particular, students will study and play matrix games, develop winning strategies using the ideas of dominance, saddle points, and the “minimax” principle. Students will also be introduced to non-cooperative games and the ideas of John Nash and his “Nash Equilibria,” for which he won the Nobel Prize for in 1994.

Pre-Workshop Activities   for Game Theory.

Geometry Logo Style
by Eric Wingler, Youngstown State University

Several theorems from geometry will be examined using the Logo programming language. If time permits, students will also see a demonstration of how to produce a fractal curve using recursion. A prior knowledge of Logo is not required as the few commands that are necessary will be presented.

Pre-Workshop Activities   for Geometry Logo Style.

Is God a Mathematician?
by Angela Spalsbury, Youngstown State University

In a letter written in 1818, the English Romantic poet John Keats wrote: "Poetry should surprise by a fine excess and not by Singularity - it should strike the Reader as a wording of his own highest thoughts, and appear almost a Remembrance." Unlike poetry, however mathematics often tends to delight when it exhibits an unanticipated result rather than when conforming to the reader's own expectations. The pleasure derived from mathematics is related in many cases to the surprise felt upon perception of totally unexpected relationships! In this workshop, we will investigate and discover a few special numbers and phenomena that are so common that they never cease to amaze us!

square

Mathemagic in Polygon Land
Floyd Barger, Youngstown State University

A Magic Square is a square of numbers such that the sum of every row equals the sum of every column equals the sum of the diagonals. (See the figure at the right.) In this workshop we will investigate Magic Squares and then the idea of Magic Squares will be generalized to regular polygons with an even number of sides. In how many ways can we label these polygons (by numbering the vertices, the center, and the midpoints) so that the sum of the numbers on any edge is the same as the sum of the numbers on any diagonal through the center and the same as the sum of the numbers on any line joining the midpoints of opposite sides?

Mathematics in Biology - Modeling Population Growth and Decline
by George Yates, Youngstown State University

Mathematics plays an important and often under-appreciated role in biology. It has played an important part in advances in epidemiology, genetics, ecology, physiology and other areas of biology. Due to the complexity of biological systems and advances in analytic methods mathematics will continue to expand its utility in biology. A growing number of schools recognize the need for quantitative skills in biologists and are requiring more mathematics in their programs. Opportunities for science and mathematics majors are likewise expanding, and a wide variety of mathematics is now being applied to problems with important biological, environmental and medical implications. The present workshop will explore a small corner of these expanding fields. The workshop participants will explore and model the dynamics of population growth and spread of deadly diseases.

Example. Suppose a culture of bacteria grows under ideal conditions, and the number of bacteria doubles every hour. If the initial population was 500, then the population after one hour would be 1,000. At the end a second hour, the population would double again to give 2,000 bacteria. Table 1 gives the expected population if sufficient nutrients are provided and if no diseases are introduced into the culture. This population can be modeled with a mathematical formula, which will be derived during the workshop.

Time Interval Total Population
0 500
1 1,000
2 2,000
3 4,000
4 8,000

In real populations, the recorded data are usually not such "nice" numbers, and the rate of growth is rarely exactly two and may even vary over the lifetime of the population. Several activities will be conducted to model the growth of a population and the spread of a disease through a population. The workshop participants will develop models for these populations and compare their model to the actual data observed during the workshop activities. A graphing calculator may be useful for this workshop.

Some Strategies of Problem Solving
by Jacek Fabrykowski, Youngstown State University

There are many sources that introduce students to solving techniques of competition type problems. Most of them however classify the problems according to the topic they belong, i.e., polynomials, inequalities, elementary number theory, Euclidean geometry, etc. In this presentation we will introduce students to various principles of problem solving. These methods will introduce methodological, rather than topical classification of mathematical problems. We will illustrate some of the common principles of the problem solving including: Invariant and Semi-invariant Principles, Extremal Principle, Box (Pigeonhole) Principle, Principle of Specialization and Generalization and Interpretation Principle. These strategies often reveal the real nature of the problem.

Example 1: Does there exist an integer that is divisible by 2003 and whose all digits are one?

Example 2: A dragon has 100 heads. A knight can cut off 15, 17, 20 or 5 heads respectively with one blow of his sword. In each of these cases 24, 2, 14 or 17 new heads grow back on his shoulders. If all heads are blown off, the dragon dies. Can the dragon ever die?

Solutions to Examples

Pre-Workshop Activities   for Some Strategies of Problem Solving.

Parallel Knockouts
by Anita Burris, Youngstown State University

In this Graph Theory talk we consider a situation where at each stage of the experiment every vertex of a graph randomly knocks out an adjacent vertex. We will compute the expected number of surviving vertices, and discuss some unsolved problems. Bungee cords and dart guns will be provided for audience participation.

 

Puzzle Solving Strategies
by Doug Faires, Youngstown State University

Puzzles!

If you ask people what they think mathematics is or what mathematicians do, they will likely tell you that mathematics involves calculations and that mathematicians spend much of their time using arithmetic, algebra, and, perhaps, calculus, to do these rather boring calculations. But while calculations are certainly connected with many applications of mathematics, what mathematicians really do is solve problems using logic, and, often, a great deal of clever thinking.

To introduce you to this greater world of mathematics, we will look at some common puzzles that can be solved systematically, but require a bit of ingenuity. Some of these you will likely find have solutions that are quite evident, but we have some that will challenge the best of you!

Pre-Workshop Activities   for Puzzle Solving Strategies.

 

Solving Third and Fourth Degree Polynomial Equations
by John Buoni, Youngstown State University

One of the more prominent mileposts in mathematics is the quadratic formula which solves the quadratic equation:

a x 2 + b x + c = 0.


Key in this formula is the computation of the square root of a number. It should not be surprising that the corresponding formulas for the third and fourth degree polynomial equations:

a x3 + b x2 + cx + d = 0

and

a x4 + b x3 + cx2 + dx + e = 0


requires the computation of a cube and fourth root. This workshop is aimed at high school students who have completed a study of quadratic equations and are interested in using the elementary capabilities of the TI-83 PLUS calculators to explore the solution methods of both the cubic and fourth degree polynomials.

Pre-Workshop Activities   for Solving Third and Fourth Degree Polynomial Equations.

Taking the Mystery out of i
by David Pollack, Youngstown State University

At one point in history, when people thought of numbers as magnitudes, the equation

must have seemed impossible to solve. Someone may have declared that its solution is -1, but most people would have wondered what that meant. How could you add 1 to an amount and end up with nothing? The number line enhanced our ability to reason by providing a visual model for numbers and their operations that made sense for negatives as well as for positives.

At a later point in history, the equation


was considered, and it too seemed impossible to solve. Your algebra book declared that the solution is i, but you must have wondered what that means. How can you multiply a number by itself and end up with -1? Again the difficulty is resolved with an extended visual model; the (complex) number plane takes the “imaginary” out of imaginary numbers and their operations.

The purpose of this session is to discuss a visual interpretation of numbers and their operations in which the equation

makes just as much sense as the equation -1+1=0.

The Mathematics of Kicking a Football
by Richard Rappach

A thorough discussion of why a football should be kicked at a 45 degree angle to get the most distance and where/how some of our formulas that we use were discovered. The presentation is geared for the algebra II or precalculus student. The workshop deals with parametric equations, trigonometry, acceleration, and motion. Students will need a TI-83 plus calculator.

Pre-Workshop Activities   for The Mathematics of Kicking a Football.

The SMART Board in the Mathematics Classroom
by Tom Reardon, Fitch High School & Youngstown State University

Students will be introduced to an exciting way of learning mathematics visually with the SMART Board, an electronic interactive white board. Students will also be entertained by Virtual TI -a powerful graphing calculator emulator. As an added bonus, we will solve the "paper folding" problem.

Pre-Workshop Activities   for The SMART Board in the Mathematics Classroom.




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