**Expanding Horizons XXIX**

**Presenters & Topics**

**George Ashline**

Saint Michael’s College

**George Ashline **is a Professor of Mathematics at Saint Michael’s College. He has also taught in the Vermont Mathematics Initiative for many years. He is a participant and faculty consultant in Project NexT (New Experiences in Teaching), a Mathematical Association of America program created for new or recent doctorates in the mathematical sciences who are interested in improving the teaching and learning of undergraduate mathematics. He has written and co-written a number of articles concerning mathematics education and pedagogy. For many years, he has served as a faculty consultant, including several years as a Table Leader and also Question Leader for the Advanced Placement Calculus Reading.

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.

**Video introduction** to talks.

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

**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.

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

CC connections:

- 8.SP) Statistics and Probability
- Investigate patterns of association in bivariate data

**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.

**Prerequisites**: Familiarity with averages, percentages, and surveys

CC connections:

- 6.SP) Statistics and Probability
- Develop understanding of statistical variability

- S-IC) Statistics and Probability
- Make inferences and justify conclusions from sample surveys, experiments and observational studies

**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.

**Prerequisites**: Familiarity with exponents and functions

CC connections:

- 8.F) Functions
- Define, evaluate, and compare functions
- Use functions to model relationships between quantities

- F-LE) Functions: Linear, Quadratic, and Exponential Models
- Construct and compare linear, quadratic, and exponential models and solve problems
- Interpret expressions for functions in terms of the situation they model

**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.

CC connections:

- 4.NBT) Numbers and Operations in Base Ten
- Generalize place value understanding for multi-digit whole numbers
- Use place value understanding and properties of operations to perform multi-digit arithmetic

- 5.NBT) Numbers and Operations in Base Ten
- Understand the place value system

**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.

CC connections:

- 8.G) Geometry
- Understand and apply the Pythagorean Theorem
- Understand congruence and similarity using physical models, transparencies, or geometry software

- G-CO) Geometry: Congruence
- Prove geometric theorems

**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.

CC connections:

- 7.G) Geometry
- Draw, construct, and describe geometrical figures and describe the relationships between them

- 8.G) Geometry
- Understand congruence and similarity using physical models, transparencies, or geometry software

- G-CO) Geometry: Congruence
- Make geometric constructions

**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.

CC connections:

- 7.G) Geometry
- Draw, construct, and describe geometrical figures and describe the relationships between them

- 8.G) Geometry
- Understand congruence and similarity using physical models, transparencies, or geometry software

- G-CO) Geometry: Congruence
- Make geometric constructions

- G-SRT) Similarity, Right Triangles, and Trigonometry
- Define trigonometric ratios and solve problems involving right triangles

**Exploring Ancient Number Systems**

We investigate how numbers are written in three ancient number systems, namely the Egyptian Hieroglyphic (ca. 3400 B.C.), the Attic-Greek Herodianic (ca. 600 B.C.), and the Mayan (ca. 400 A.D.). These systems not only provide useful mathematical information about properties of number systems, but also offer interesting historical connections to cultural beliefs and characteristics of several important ancient civilizations. Experience with these number systems helps to provide deeper understanding of our own decimal number system.

CC connections:

- 4.NBT) Numbers and Operations in Base Ten
- Generalize place value understanding for multi-digit whole numbers
- Use place value understanding and properties of operations to perform multi-digit arithmetic

- 5.NBT) Numbers and Operations in Base Ten
- Understand the place value system

**Josh Bongard**

University of Vermont

Professor Josh Bongard’s research centers on evolutionary robotics, evolutionary computation and physical simulation. He runs the Morphology, Evolution & Cognition Laboratory, whose work focuses on the role that morphology and evolution play in cognition. In 2007, he was awarded a prestigious Microsoft Research New Faculty Fellowship and was named one of MIT Technology Review’s top 35 young innovators under 35. In 2010 he was awarded a Presidential Early Career Award for Scientists and Engineers (PECASE) by Barack Obama at a White House ceremony.

**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.

**Using Math to Create Robots … and Xenobots**

Look around your house: how many computers do you see? Now look again: how many robots do you see? With the exception of maybe a Roomba vacuum cleaner, you probably don’t see any: It turns out that making computers is much easier than making robots. Why? We’ll explore this question by looking at some of the math that lies behind robots, computers and AI. We’ll also look at the math behind a brand new kind of robot — the xenobot — that was invented right here in Vermont, and is already revolutionizing the way we think about cleaning micro plastics from the oceans, or cancer cells from our arteries.

**Joanna Ellis-Monaghan**

Saint Michael’s College

Joanna is currently in the Netherlands, and so is available for Zoom presentations only.

**E-mail**: jellismonaghan@gmail.com

**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.

**Level**: Grade 6 and up

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

**David Hathaway**

Vermont State Mathematics Coalition

David spent 32 years at IBM developing electronic design automation software, retiring in 2014. He taught a graduate course on Computer-Aided Design algorithms for Logic Design and Analysis in the late 1990s and early 2000s and Computer Architecture in 2015 and 2016, and in 2014 built his own 3D printer. He is a trip leader and map and compass instructor for and general director of the Green Mountain Club. He started and ran the VSMC tutoring program in the 1990s, is a past VSMC Co-Director / Business, and has served as the VSMC executive director since 2020.

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

**3D Printing**

In 2014 I became interested in the then-new field of 3D printing, and built my own 3D printer. I’ll talk about how they work, watch my printer making 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. The goal of the talk is 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).

**Level**: Middle School or High School

**Length**: 60 to 80 minutes (80 preferred)

**Math content:** Coordinate systems, calculating area and volume, building objects from transformations (scale, rotate, translate) and set operations (union, intersection, difference) on geometric shapes

**Prerequisites**: None

**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 projector to connect to my laptop

CC connections:

- 7.G) Geometry
- Solve real-life and mathematical problems involving angle measure, area, surface area, and volume

- 8.G) Geometry
- Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres

**Where am I? – How GPS Works**

We review some of the math tools we need and 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). We then do a hands on exercise with tape measures to demonstrate the basic idea of trilateration. If there’s time we’ll talk about conversion from Cartesian (x,y,z) coordinates to the latitude / longitude / altitude coordinate system, do a little review of celestial navigation methods and history, and 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. The hands on exercise can be skipped for a larger group, and content and emphasis can be adjusted depending on your needs.

**Level**: Middle School or High School

**Length**: 60 minutes

**Math content:** Pythagorean theorem, expansion of squares of binomials, Cartesian coordinates, solving simultaneous linear equations using elimination, maybe a little trigonometry and triangle congruence

**Other requirements**: A projector to connect to my laptop

CC connections:

- 8.EE) Expressions and Equations
- Analyze and solve linear equations and pairs of simultaneous linear equations

- A-APR) Arithmetic with Polynomials and Rational Expressions
- Perform arithmetic operations on polynomials

- A-REI) Reasoning with Equations and Inequalities
- Solve systems of equations

- G-GMD) Geometric Measurement and Dimension
- Visualize relationships between two-dimensional and three-dimensional objects

- G-MG) Modeling with Geometry
- Apply geometric concepts in modeling situations

**Gerard T. LaVarnway**

Norwich University

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

**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.

**Level**: 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.

**Level**: Grades 9 – 12

**Length**: 40 – 50 minutes

**Barbara O’Donovan**

St. Michael’s College

Barbara O’Donovan teaches Mathematics and Engineering, and is the Engineering Program Co-Director at Saint Michael’s College. She’s been teaching mathematics and engineering for over twenty-five years. She loves to explore real-world applications with students so they can connect the math they’re learning in a classroom to things they can observe in the world around them. She is passionate about STEM education and equipping students with the critical thinking skills necessary to make a valuable contribution in our ever changing and high-tech world.

**E-mail**: bodonovan@smcvt.edu

**The Power in the Wind**

This presentation discusses the maximum amount of power that can be extracted from the wind by a wind turbine rotor. Using principles of fluid mechanics and physics with concepts such as control volumes, conservation of momentum, Bernoulli’s equation, thrust, power and simplifying assumptions an expression for rotor power is developed. The rotor power expression is optimized using derivatives to maximize the axial induction factor and thus the power extracted. This is followed by discussion of the simplifying assumptions made in developing the rotor power expression. And finally, a brief discussion of the current state of the wind power industry and potential locally, nationally, and internationally.**Prerequisites: **Calculus applications of derivatives – power rule and product rule, experience with physics would be helpful but is not necessary

**Michael Olinick**

Middlebury College

After earning a BA in mathematics and philosophy at the University of Michigan and an MA and PhD from the University of Wisconsin (Madison), Michael Olinick moved from the Midwest to New England where he joined the Middlebury College. Dr. Olinick has held visiting positions at University College Nairobi, University of California at Berkeley, Wesleyan University, and Lancaster University in Great Britain. He is the author or co-author of a number of books on single and multivariable calculus, mathematical modeling, probability, topology, and principles and practice of mathematics.

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

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

Mathematics provides the answer.

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

**Length**: 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

**Darlene Olsen**, Ph.D., is a Charles A. Dana Professor of Mathematics and Norwich coordinator for the Vermont Biomedical Research Network. She is the 2013 Homer L. Dodge Award winner for Excellence in Teaching.

She joined the Norwich faculty in 2006 and routinely teaches statistic courses, such as Introductory Statistics, Statistics for Health Science majors, and Statistical Methodology for STEM majors. She also teaches other general mathematics courses, including Linear Algebra and Liberal Arts Mathematics.

Her current research areas are biostatistics and pedagogy in mathematics and statistics. Olsen has received research grants through the Vermont Genetics Network, served as a statistical consultant, and published work in several research journals.

She received her doctorate in mathematics from the University at Albany in 2003. She also holds an MS in biometry and statistics (2001) and an MA in mathematics (1997) from the University at Albany and a BA in mathematics (1994) from SUNY Geneseo.

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

Prefers to give in person presentations.

**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.

**Level**: Grades 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.

**Level**: High school

**Length**: 45 minutes