## EXPANDING HORIZONS XXI

A project of the Vermont State Mathematics Coalition
Bring college and university mathematicians to your classroom!

### DIRECTIONS

1. Choose a topic which would be of interest to your students.

2. Contact the presenter directly by phone, fax, e-mail, or postal mail.

3. Check for prerequisites, if any.

There is no charge for this service. The presenters give of their own time, and they cover their own travel expenses.

John Devino
Phone: 802-863-5403
E-mail: devino13@comcast.net

George Ashline

Saint Michael’s College

Phone: (802) 654-2434
E-mail: gashline@smcvt.edu

I am willing to present to multiple classes at once, as well as visit different classes on the same day. I have given versions of these talks at various levels, from high school to elementary school. I have much faculty consultant experience in grading AP Calculus Free Response questions, and I would be willing to answer questions that any AP Calculus teachers may have about that.

Correlation Properties and Applications”

Through an activity and examples, we investigate properties of scatter plots and correlation in context, leading to a discussion of the correlation coefficient and challenges inherent in attempting to find causal links between variables.  If time and technology permit, students can explore the online Correlation Guessing Game.

Prerequisite: Familiarity with the concepts of the mean and standard deviation of a variable (also, two-variable statistics calculators are helpful)

An Introduction to Bias and Margin of Error”

Through an initial activity, we explore the potential impact of bias in statistical analysis. We can also consider how bias may arise in survey questions and ways that it can be reduced. In another activity, we can consider different types of error that may impact a survey or experiment and the meaning of margin of error.

Prerequisite: Familiarity with averages, percentages, and surveys

Exponential Functions in Snowflakes, Carpets, and Paper Folding”

Through constructions of initial stages of several fractals, students can explore and represent underlying patterns using exponential functions. Other examples of exponential functions and their properties can be discussed. If time permits, students can play the Chaos Game to “create” the Sierpinski Triangle.

Prerequisite: Familiarity with exponents and functions

Number Pattern Challenges”

How can you predict the value of a secret number based on its location on some "magical" cards? How can you advise a game show host as to how to best award prizes from one dollar up to one thousand dollars using only dollar bills filling a mere ten envelopes? How can we guide a local farmer about using an amazing forty pound broken rock to measure various weights from one pound up to forty pounds? These challenges and more reveal fascinating patterns of numbers, and strategies for solving problems.

Framing the Proof of the Pythagorean Theorem and Investigating Some Interesting Pythagorean Triple Properties”

We will begin this session with some hands-on proofs of the Pythagorean using sets of congruent right triangles and other famous methods, with some interesting historical connections to some ancient mathematicians and civilizations. We will then discuss Pythagorean triples and some of their properties, including some neat connections that they have with Fibonacci numbers.

Encountering the Great Problems from Antiquity: Hands-On Trisection, Duplication, and Quadrature”

The Ancient Greeks grappled with the three classical problems of trisecting an angle, doubling the volume of a cube, and squaring a circle using only straightedge and compass constructions. These constructions were shown to be impossible millennia later with the evolution of abstract algebra and analysis in the nineteenth century.

We will consider some of the rich approaches that have arisen to solve these problems using additional techniques and tools, including origami. Along the way, we will encounter some interesting work of such mathematicians as Archimedes and Eratosthenes and more recent scholars.

Estimating the Circumference of the Earth – Following in the Shadow of Eratosthenes”

The goal of this activity is to recreate to a certain degree the remarkable estimate of the circumference of the earth done by the Greek mathematician Eratosthenes over two millennia ago. Using the length of the sun’s shadow at high noon (“sun transit”) at two locations, groups will estimate the “sun” angle (the angle between the sun’s rays and a vertical stick) at these two locations. Knowing the “sun” angle at two different locations will allow us to estimate the circumference of the earth.

James R. Bozeman

Lyndon State College

Phone: (802) 626-6489
Fax: (802) 626-9770
E-mail: james.bozeman@lsc.vsc.edu

"What Your Math Teacher Never Told You"

The talk introduces topics not normally taught in high school which could be. Ex: (a+b)2 = a2 + b2; triangles whose angle sum does not equal 180 degrees; pieces of paper with only one side; bottles with no inside/outside.

Prerequisite: A little algebra and geometry.

"The Mathematics of DNA"

Through hands-on demonstrations and technology, students will discover the formula which describes DNA’s three-dimensional conformation.

Applied Geometry - Nearly Convex Sets and the Shape of Legislative Districts”

In this talk we develop a method for deciding how close a polygonal planar set is to being convex. We introduce 'nearly convex' sets and then 'nicely shaped' legislative districts. Those districts which are not nicely shaped may indicate improper gerrymandering in the redistricting plan.

Priscilla Bremser

Middlebury College

Phone: (802) 443-5555
E-mail: Bremser@middlebury.edu

William R. Hamilton and the Quaternions”

In 1843, while strolling across a bridge in Dublin, Hamilton had a flash of insight and discovered the Quaternions, a new number system he had been searching for 15 years. What did he hope his discovery would do, why did it fail, and why was it nevertheless a major contribution in mathematical history?

Prerequisites: None
Length 40 or 80 minutes

Symmetry”

What do wallpaper patterns, prints of M.C. Escher, and molecular structure have to do with mathematics? We will discuss what symmetry means to mathematicians.

The Mathematics of Change Ringing”

Change Ringing is the art of ringing a set of bells in specific sequences. It started in England, where it is still popular, and is practiced in various places around the world. In this session we will discuss the history of this art as well as the mathematical principles involved. Students will get a chance to practice on handbells. This presentation is appropriate for middle school and high school.

Joanna Ellis-Monaghan

Saint Michael’s College

Phone: (802) 654-2660
E-mail: jellis-monaghan@smcvt.edu

Willing to do multiple classes at one location.

"Instant Insanity"

A hands-on introduction to mathematical modeling with graph theory.

"Networks and Graphs"

The above model intercommunications, relationships, and conflicts. We will explore a variety of applications from: the internet, the stock market, classroom scheduling, power grids, the Kevin Bacon game, computer chips, social circles, and DNA.

"To Knot or Not"

Is your shoelace really knotted? How can you tell? A gentle introduction to knot theory.

"Graph Theory in the Real World"

Where does math come from”? We will see some of the new math in network theory being developed today as well as some of the critical applications driving its creation. In particular, we will see new mathematical theory created for DNA origami and tile assembly used for biomolecular computing, nanoelectronics, and cutting-edge medicine. We conclude the talk by showcasing examples of what mathematicians do in real life, and how some of the top jobs use mathematical skills.

Length of Presentation: 20 min to 2 hours (longer versions may have some hands on activities).

David Hathaway

Past instructor in CS & EE at University of Vermont

Phone: 802-899-9982
E-mail: david.hathaway.78@gmail.com

"3D Printing, in Space and at Home"

When they needed a wrench on the international space station, instead of waiting for the next supply flight, they beamed up the design and used a 3D printer to make one on the spot. We talk about how they work, watch a printer I made printing something, and pass around some things I've printed with it. We'll also talk about how you can design objects to be printed by adding, subtracting, intersecting, rotating, moving, and scaling a few basic 3D geometric figures.

Level: Middle School or High School

Length: 60 to 80 minutes (80 preferred)
Prerequisites: None (This is mostly to interest students in possible applications of math and technology. The math discussion is optional, and can include some algebra, basic planar geometry, and graphing points in 2 or 3 dimensions).
Other requirements:
An electrical outlet (for the printer and the laptop to drive it).
A table on which to set up the printer (about 3 feet by 2 feet).
A VGA projector

Karla Karstens

University of Vermont

Phone: (802) 878-7322
E-mail: karla.karstens@uvm.edu

The Mathematics of Sharing: Getting Your Piece of the Pie”

Your family inherits some artifacts that need to be distributed among all the relatives, or you want to divide a pizza among friends. How can you accomplish this so everyone involved gets a fair share? Principles of fair division lead to the solution of this class of problems.

Prerequisite: Middle School level or above
Length 40 – 50 minutes

Gerard T. LaVarnway

Norwich University

Phone: (802) 485-2325
Fax: (802) 485-2333
E-mail: lavarnwa@norwich.edu

Cryptology: The Art and Science of Secret Writing”

An introduction to cryptology will be given. The history of cryptology will be discussed from the time of Caesar to the present. Various ciphers will be demonstrated. The mathematical foundations of ciphers will be discussed.

Length 40 – 50 minutes

The Use of Linear Algebra in Cryptology”

Daisy McCoy

Lyndon State College

Phone: (802) 626-6260
E-mail:
daisy.mccoy@lsc.vsc.edu

Travel Limitations: Northeast Vermont

"The Magic of Nine"

There are a number of special properties of the number 9. This session will look at these properties and other properties of our number system such as "casting out nines" and doing number tricks.

Prerequisite: Can be adapted to various levels.
Length of Presentation: 40 – 80 minutes

"Math Like an Egyptian"

Four thousand years ago the Egyptians were writing numbers and doing mathematics. Try out some of the computational methods they used and look at some of the problems they did.

Prerequisite: Multiplication and Fractions
Length of Presentation: 40 to 80 minutes
Travel Limitations: Northern or Eastern Vermont

Mayan Mathematics”

The Mayan numeration system, the first to develop the concept of zero, will be investigated. Pictures of monuments will be used to identify the numerals. Mayan arithmetic and the development of a calendar will also be covered.

Michael Olinick

Middlebury College

Phone: (802) 443-5559
Fax: (802) 443-2080
E-mail: molinick@middlebury.edu

"Cryptology: The Mathematics of Making and Breaking Secret Codes"

"The Near-Sighted Fly: A Topological View of the Universe"

Length of Presentation: 40 – 80 minutes

I See It but I Don’t Believe It: Some Surprising Facts About Infinite Sets”

For much of the history of mathematics and Western thought, “infinity” was viewed as an unknowable subject, not susceptible to rational thought and investigation. Georg Cantor changed all this with a seemingly simple, but revolutionary breakthrough in the late 19th Century. Cantor proved a number of results about infinite sets, many of which challenge our intuitions and startled the mathematicians of his time. Even Cantor himself found it hard to believe some of his own theorems. We will examine Cantor’s controversial breakthrough and see why one leading mathematician labeled it “a disease from which mathematics will one day recover”, while another boasted that “No one shall expel us from the paradise that Cantor has created.”

Darlene M. Olsen

Norwich University

Phone (802) 485-2875
E-mail: dolsen1@norwich.edu

"Maximizing the Flight Time of a Paper Helicopter"

The mission is to design a paper helicopter that remains aloft the longest when dropped from a certain height. Various combinations of design factors contribute to the flight time.

Response surface methodology (RSM) is a statistical technique that explores optimization through experimentation. Three tools in RSM are design of experiments, multiple regression, and optimization. These tools will be used to explore efficiently the combination of design factors that will improve the performance of the paper helicopter.

Length 30 – 45 minutes

Mathematical Ties to Tying Neckties”

Did you ever ask the question of how many possible ways there are to tie a necktie? Furthermore, what factors determine an aesthetic tie knot? This problem can be answered using mathematics. We will discover the mathematical ways for describing how to tie necktie knots. We will also classify knots according to their size and shape.

High School level
Length 45 minutes

Bill Peterson

Middlebury College

Phone: (802) 443-5417
Fax: (802) 443-2080
E-mail: wpeterso@middlebury.edu

Benford‘s Law”

In 1938 a physicist named Frank Benford observed that the earlier pages of logarithmic tables showed more wear. There is a message here about the distribution of “naturally occurring” numbers. This property will be explored along with some applications such as detecting fraud in financial statements.

The Cars and the Goats”

This “game-show” puzzle is a variant of a famous problem in conditional probability. Some years ago, Marilyn von Savant's solution in her Sunday column in Parade generated a lot of irate mail from professional mathematicians—all of whom turned out to be mistaken. More recently, the problem has appeared in the novel The Curious Incident of the Dog in the Night-Time and the movie 21. What makes this problem so intriguing? And why won’t it stay “solved”?

Length of Presentation: 40 – 80 minutes (80 preferred)

Great Expectations: From Huygens to Hedging”

Probability emerged with the first book published in this field by Christian Huygens. This presentation will begin with Huygens’ “expected value” of a wager and trace some elementary ideas leading to applications in the modern world of mathematical finance.

How Many Times Should You Shuffle?”

In 1991 Harvard mathematician Persi Diaconis announced that, to insure that the cards were well mixed, seven was the answer. Simple models of card shuffling will be presented in order to motivate Diaconis’ result, and give an elementary introduction to the mathematics involved in the analysis.

e: The Miniseries”

One cannot study probability theory for long without being struck by the many occurrences of e (or its reciprocal) as the answer to questions that at first glance appear unrelated. In this talk, we will meet four examples. Each can be solved by applying single-variable calculus involving the natural log and exponential function.

Prerequisite: Enrollment in Calculus

Rob Poodiack

Norwich University

Phone: (802) 485-2339
E-mail: rpoodiac@norwich.edu