## Pathways Presentations

DYNAMICAL SYSTEMS and CHAOS THEORY
Jon Jacobsen

Abstract: This talk explores how simple mathematical models can lead to very complex behavior. Starting from simple calculator games we work towards the logistic map and study it's behavior. Along the way we introduce the creative art of mathematical modeling. The underling theme is to expose students to the modern paradigm that simple rules can lead to complex behaviors and that mathematics can be used to understand this complexity. Students also see how important ideas from calculus lend insight into this complicated behavior.

Audience : Grades 9 - 12, Calculus Class(es), Math Club
Prerequisites : familiarity with basic derivatives
Keywords : fixed points, stability, bifurcations, cobweb plots, chaos, butterfly effect

FRACTALS
Jon Jacobsen, Michael Orrison

Abstract : Fractals are infinitely repeating geometric figures. This talk explores these shapes and teaches students how to make their own fractals. Along the way we touch on interesting ideas such as infinity and fractals in nature.

Audience : Grades 2 - 6.
Prerequisites : Ability to draw reasonably straight lines!
Keywords : fractals, the chaos game

FRACTALS and DYNAMICS
Jon Jacobsen

Abstract : This talk explores the process of iteration. Starting from simple calculator games we work towards the logistic map and explore its behavior. Then we introduce complex numbers and study how the picture changes when we play the same calculator games, now with complex numbers. This leads to a beautiful class of fractals known as Julia Sets.

Audience : Grades 8 - 12, Algebra II & Trig, Calculus Class(es), Math Club
Prerequisites : familiarity with basic algebra 2 and trigonometry. No calculus required!
Keywords: fractals, fixed points, complex numbers, Julia Sets, Mandelbrot Set

MATHEMATICAL PERSPECTIVES
Michael Orrison

Abstract: Being able to to look at a problem in the right way is often the first and most important step toward finding a solution. It can also be the most subtle step. The focus of this talk is a set of four brain teasers and what their solutions tell us about finding the right perspective when it comes to mathematical and non-mathematical problem solving.

Prerequisites: willingness to try solving some brain teasers
Keywords: problem solving, perspectives, brain teasers

VOTING and MATHEMATICS
Michael Orrison

Abstract: Voting is something that many of us do in a multitude of settings, but the way in which we vote is often seldom questioned. If you have ever wondered if different voting procedures can lead to different results, then this talk is for you! We'll explore a few different voting procedures from a mathematical perspective as we try to make sense of some mind-numbing paradoxical results that occur when we allow ourselves to vote in more than one way.

Prerequisites: familiarity with matrix multipliation is useful but not necessary

Susan Martonosi

Abstract: Sometimes the probability of something happening can change in an unpredicatable way depending on how the problem is phrased. This interactive workshop will look at some fun problems with "conditional probabilities" and show that trusting your instincts can be dangerous when probabilities are involved!

Prerequisites: basic understanding of probabilities
Keywords: probability, memorylessness, The Monty Hall problem

MAKING DECISIONS USING MATH
Susan Martonosi

Abstract: Relatively simple math can be used to solve some difficult real-world problems. The field of operations research is devoted to making optimal decisions about the use of scarce resources. What products should a company make to maximize profit? How does Mapquest know the shortest route between two addresses? In this workshop, we'll dabble in optimization, using linear programming and graph theory.

Prerequisites: graphing inequalities
Keywords: operations research, linear programming, graph theory

PATTERNS IN PASCAL'S TRIANGLE
Alissa Crans, Michael Orrison

Abstract: Blaise Pascal is a 17th century French philosopher, scientist, and mathematician who is perhaps best known as the first person to discover and record all of the patterns contained in the famous triangle which now bears his name. We will discuss the history of this triangle and then find for ourselves the numerous patterns hiding in it such as the hockeystick pattern, the triangular numbers, the Fibonacci numbers, the binomial coefficients, and many more!

Prerequisites: willingness to hunt for patterns
Keywords: Pascal's triangle, patterns, bionomial coefficients, sequences of numbers

KNOTS, LINKS, AND BRAIDS, OH MY!
Alissa Crans

Abstract: If you know how to tie your shoes then you're more than ready to become further acquainted with the mathematical field of knot theory! While knot theory has applications to biology, chemistry, and physics, it's an interesting mathematical field in its own right. Our discussion will include an introduction to knots, links, and braids, and will focus mainly on the problem of determining whether two given knots are the same.

Prerequisites: willingness to think about mathematics geometrically rather than algebraically
Keywords: knots, braids, topology

THE SECRETS OF ALGEBRAIC CODING THEORY
Alissa Crans

Abstract: Successfully encrypting information has become increasing important as technology plays a more prominent role in our lives. We encrypt data for cd players, cell phones, modems, internet purchases, and much more! We will discuss the strengths and weaknesses of different mathematical methods used to encode information and learn some new mathematics along the way!

Prerequisities: Algebra II and some knowledge of matrices would be helpful, but neither is absolutely necessary
Keywords: coding theory, modular arithmetic

MATH CURSE
Alissa Crans, Michael Orrison

Abstract: This children's book by Jon Scieszka and Lane Smith tells the story of a child who is "cursed" by finding mathematics everywhere in daily life! After reading the story, we will discuss questions posed by the main character such as, " If I have 1 white shirt, 3 blue shirts, 3 striped shirts and that 1 ugly plaid shirt my Uncle Zeno sent me, how many shirts do I have all together?" and "There are 24 kids in my class. The new girl, Kelly, sticks her tongue out at me. How many tongues are in our class?"

Prerequisites: enjoyment of story time!
Keywords: addition, subtraction, fractions

G IS FOR GOOGOL
Alissa Crans

Abstract: This children's book by David Schwartz provides a different mathematical idea for each letter of the alphabet. Prior to reading the book, students can be asked to each choose a letter and list all of the mathematical words that start with that letter. Sample words include "abacus," Fibonacci," "Mobius Strip," and "Venn Diagram." Depending on the background of the class, we can discuss one or more of the letters in depth.