Vermont State Mathematics Coalition

EXPANDING HORIZONS XXV

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


Note: For the safety of all concerned presentations for the 2020-2121 school year will be via Zoom or other virtual platform. The Coalition will provide the appropriate link.

Recording of VCTM Zoom presentation on Expanding Horizons and slides from George Ashline, David Hathaway, and Mary Beth Ruskai from October 24, 2020.

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.

For more information contact:

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.


Josh Bongard

University of Vermont

Email: jbongard@uvm.edu

"What does math have to do with robots?"

We will explore the relationship between math and robots by performing two collaborative games. One will explore the mathematics of optimization: how to search very large spaces, filled with mostly useless patterns, to find the small minority of useful ones. In the second game, we will apply this idea to find useful brains for robots, so that they perform useful or entertaining tasks.

Level: Middle or high school. This presentation can be adapted given the age level and mathematical sophistication of the audience.


Joanna Ellis-Monaghan

Saint Michael’s College

E-mail: jellismonaghan@gmail.com

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.

Prerequisite: Grade 6 and up
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, IBM (retired)

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

"Where am I? - How GPS Works"

We start with an overview of the latitude / longitude / altitude coordinate system, and then do a little review of celestial navigation methods and history. We then talk about how a GPS receiver works (and why GPS satellites do not have to know where receivers are, or even if any are listening), and do a hands on exercise with tape measures to demonstrate the basic idea of triangulation. If there's time we'll wrap up by talking about some of the complications in real life, like non-uniform and non-spherical earth, relativistic time dilation due to the earth's gravity.


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

"And the Winner Is ..."

Most people think of voting as a choice between two alternatives, with the "winner" being the alternative with the most votes and the "majority rule". What happens if there are more than two alternatives - how is the winner chosen? It turns out there are a number of ways you can determine the winner of an election with more than two candidates. We will explore four of the most popular methods and talk about situations in which they are used. We will also talk about the benefits and downfalls of each.

This talk uses basic math (addition and multiplication) and is appropriate for students in Grade 5 through High School.

Gerard T. LaVarnway

Norwich University

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

Not available until January 2021.

"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.

Prerequisite: Grades 9 – 12
Length 40 – 50 minutes

"The Use of Linear Algebra in Cryptology"

Humankind is fascinated with message concealment. Cryptology – the art and science of secret writing enjoys a rich history of mystery, intrigue and suspense. For mathematicians, cryptology employs applications of mathematics from a variety of fields including linear algebra. Examples of matrix techniques for encryption and decryption will be discussed. In particular, the Hill cipher will be demonstrated. Techniques for decrypting secret messages will be demonstrated.

Prerequisite: Grades 9 – 12
Length 40 – 50 minutes


Daisy McCoy

Northwestern Vermont University - Lyndon

Phone: (802) 626-6260
E-mail: daisy.mccoy@northernvermont.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.

Grade Level: Adapted to 3 – 12


Bjoern Muetzel

Eckerd College, Department of Mathematics, Florida

E-mail: bjorn.mutzel@gmail.com
Web page: http://math.dartmouth.edu/lineargames

"Zoom presentation with activities suited for up to 60 students. Students should have a computer where they access the above webpage. This presentation is for students grade 5 or higher."

"Linear Games"

Linear Games teach algorithms from Linear Algebra in a fun and open setting as puzzle games. In the presentation students learn how to master these games and get an insight into the math behind them.


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"

Mathematics provides the answer

"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.

Grade Level: 10 – 12
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


Rob Poodiack

Norwich University

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

"Paradoxes in Probability"

In certain games, our intuition will tell us one thing, when probability calculations clearly tell us to do another. We will investigate the effect of human nature on probability using: “Let’s Make a Deal” and the Hershey’s Kiss Challenge. If time permits, we’ll engage in a series of three-way duels (“truels”).

Prerequisite: At least Algebra
Length of Presentation: 45 – 75 minutes


Mary Beth Ruskai

Retired mathematical physicist

Phone: (802) 489-4954

E-mail: mbruskai@gmail.com

"How Steep Can a Mountain Be? How High?"

In the book Innumeracy, John Paulus estimates the volume of Mount Fuji assuming that the diameter is approximately the same as the height, an assumption that was repeated in the text For All Practical Purposes. Is this a a reasonable assumption? How does it compare to the real Mount Fuji? How does it compare to other famous mountains, e.g., the Matterhorn? How steep would a mountain have to be for this assumption to hold? How would it feel to hike up a mountain this steep? One can accompany this talk with a simple classroom exercise to test the climber’s rule of thumb that if a slope is 60o. If you stand straight up and extend your arm, then your fingertip would touch the mountain.

Prerequisite: Middle school or above

"What is a Quantum Computer and Why Should You Care?"

Quantum computers can perform certain tasks much more efficiently than ordinary classical computers, including some that would make current methods of encryption insecure. However, quantum particles can also be used for new methods of cryptography. Although quantum theory is a deep subject, quantum information uses only 2-level systems which makes some of the key concepts accessible. The basics of quantum key distribution can be explained using arrows with minimal math needed (and even demonstrated using hiking poles). A talk on this topic can be adapted to almost any grade level starting with middle school.

One subtopic that can be discussed is the meaning of exponential growth by looking (without calling it that) thru a logarithmic lens. If you use of a computer for something that grows exponentially and buy one that is a million times as powerful, how many more items can you actually process?

Prerequisite: Middle school or above


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