HIGH
SCHOOL VISITING LECTURE PROGRAM 2007-2008
Sponsored by the
Michigan
Section of the
Mathematical Association of
America
List of Speakers (ordered by ZIP code)
Lawrence Technological University, Southfield 48075
Ruth Favro
1.
The chaos game (Students play
the Chaos Game and display the results. Analysis covers generating fractals,
some discussion of probability, and a computer program. Grades 9-10 and 11-12.)
2.
Matrices, computer graphics,
and fractals (Simple matrix operations form the basis of computer graphics.
Systems of matrices can generate fractals, using iteration. Grades 10-12.)
Christopher Cartwright
1.
How do calculators work?
(the mathematics behind calculators) Topics would vary depending on grade
level. Emphasis would be on the types of calculators students are currently
using in class. Topics would be selected from: how does the calculator compute
trig functions, logs, exponents, reciprocals, roots, fraction conversion.
Additional topics for students using graphing calculators include: how does it
draw a graph? Maxes and mins?
2.
Solve this math problem and
you could win a million dollars. The
millennium problems in Mathematics are problems that no one has been able to
solve for decades and carry a prize of a million dollars each. This talk will discuss
one of those problems, the P=NP problem. This talk will discuss how much time
it takes to solve a problem and which problems take more time than others. This
is known as complexity theory. The growth rates of polynomials and exponentials
will be compared. (Grades 9-12)
Oakland
University
,
Rochester
48309
Eddie Cheng
1.
Greedy Algorithms (Many
operational problems such as scheduling problems require the manager to look
for optimal solutions. These types of problems are usually very difficult.
However a simple procedure called greedy algorithm will give an optimal
solution for a special type of optimization problems. In this talk, we will
look at examples of this type of problems. Grades 9-12.)
Jerrold Grossman
1.
How to create more numbers
than you ever thought existed out of absolutely nothing (Starting from the
empty set, we will construct the surreal numbers and look at some of their
fascinating properties. These numbers include all the rational and irrational
numbers, infinities, infinitesimals, and much, much more. Grades 9-12.)
2.
What day of the week is
that? (Can you figure out quickly, in your head, what day of the week you were
born on, or what day of the week any given date falls on? We will explore some
easy algorithms for carrying out these calculations. You'll never need to carry
around a calendar again. In the process, we'll learn about the fascinating
branch of mathematics called number theory. Grades 7-12.)
3.
A panoply of partition
problems (How many different ways are there to distribute 50 cookies among your
12 friends? We'll look at some fascinating patterns that emerge when one tries
to count such things, in this branch of mathematics called combinatorics.
Grades 9-12.)
4.
Mathematical games for fun
and profit (Almost everyone has analyzed the game of tic-tac-toe and realized
that if both players use perfect strategies, then the game will end in a tie.
We will study some more complex games, such as Nim,
Chomp,
Split
, and Sprouts, and the fascinating mathematical
patterns that emerge in trying to perform a similar analysis for them. In the
process, we'll encounter many unsolved problems. Grades 10-12.)
5.
What's the area? (There is
more to calculating areas than "one-half base times height" or
"pi r squared." We'll look at Pick's formula, Heron's formula, problems of round-off error, infinite areas, and higher
dimensional analogs of area. Grades 9-12, especially appropriate for geometry
or trigonometry classes.)
6.
Dividing a cake: Beyond
"I cut, you choose." (How can three or more people divide a cake
among them so that all of them believe that they got a fair shake? We will look
at how to define "fair" in this context and what algorithms might
accomplish this efficiently. An entire book has been written on this subject.
Grades 8-12.)
7.
The mathematics of infinity.
(Infinity comes up in mathematics in many contexts, such as decimals that go on
forever, calculus, geometry, and set theory. It's an appealing topic that
almost everyone thinks about at some
point. This talk explores various aspects of infinity, appropriate to grade
level. Grades 7-12.)
University
of Michigan-Flint
48502
Robert Bix
1.
When parallel lines meet
(Projective geometry is introduced by projecting between planes. Grades 10-12.)
2.
Frieze Patterns (Symmetries
of designs repeating horizontally. Grades 10-12.)
Mehrdad Simkani
1.
A computer workshop on
ruler-compass geometry. (Using modern computers, students will explore geometry
the way the masters explored it over two millennia ago. Using virtual ruler and
compass, provided by a Java applet that is available on the World Wide Web,
students will construct solutions to the geometric problems of
Euclid
's Elements. This workshop
requires a modern computer lab. For the details on necessary hardware/software
please contact the speaker simkani@umich.edu. In the event such a facility is
unavailable, the speaker and the coordinator will explore options to find a
nearby site for the workshop. Grades 10 - 12)
Kettering
University
,
Flint
48504
Brian J. McCartin
1.
Math Makes the World Go
'Round (Find out: Why the wind is always blowing somewhere on Earth. Can you
hear the shape of a drum? Why a striped animal cannot have a spotted tail. Did
Romeo and Juliet ever really have a chance? About the math of a human
heartbeat. What do Mozart and the Beatles have in common?) Grades 10-12. Please allow at least 45 minutes for this
presentation.
Northwood
University
,
Midland 48640
Melvin Billik
1.
Coupons, duels, and
encounters (Some interesting problems in probability. Grades 11-12.)
2.
The green monster (The green
monster is the left-field fence at
Fenway
Park
. We will simulate a batted
ball with a graphing calculator and see if it will be a home run. This is an
intriguing, interesting, and enjoyable problem. Grades 11-12.)
Michigan
State
University
,
East Lansing
48824
Charles R. MacCluer
1.
Several Industrial
Mathematics Problems (MSU provides a problem-solving service to Michigan
industry) --- teams of three mathematics graduate students obtain class credit
by donating several hours per week for 4 months to resolve questions posed by
industry or government. Many of these projects have proved to be extremely
interesting and successful, where mathematics has resolved significant
business, regulatory, and production problems. The speaker will describe
several of the past and on-going projects described at the site
http://www.math.msu.edu/Academic_Programs/graduate/msim/ProjectPage.aspx
The talk will demonstrate the
wide range of problems faced by a mathematician working in industry. Suitable
for a high school math club meeting or grades 11-12.
2.
The Classical
Problems of the Calculus of Variations. One of the earliest uses of the
calculus was to attack "variational problems," where the objective is to minimize certain path integrals.
These first problems were proposed by Johann Bernoulli,
Newton
, von Leibniz, and others --- in certain cases as
challenges to smoke out their competition. We will tour (but not solve)
a collection of these early problems on least time of descent, geodesics, bluff
bodies, isoperimetric problems, hanging cable, etc, as well as the modern
Nobel-winning Mirrlees formulation of optimal
tax structure. (Reference: C. R. MacCluer,
Calculus of Variations, PrenHall, 2004.) Prerequisites:
An exposure to integral calculus.
Central
Michigan
University
,
Mount Pleasant
48859
Tom Miles
1.
Careers in mathematics and
mathematics in careers (This talk can be directed toward either part of the
title. The second part, mathematics in careers, lists a number of reasons for
continuing with high school mathematics. Grades 8-12.)
2.
The
Alabama
paradox (This talk examines
how many representatives each unit (state) in a representative body (House of
Representatives) should have. After the 1880 census, Congress found that in a
House with 299
members,
Alabama
would have 8 members, while in a House with 300 members,
Alabama
would have only 7 seats.
The talk uses basic arithmetic and, perhaps, inequalities to explore various
historical methods of
apportioning. It is particularly relevant since
Michigan
has just lost a seat after
the 2000 census. Grades 9-12.)
3.
The Marriage Problem: Given a set
of men and a set of women, all unmarried, is it possible to arrange marriages
so that all the marriages are stable? What does stable mean? Can it
be done? In what other contexts does the problem arise?
Western
Michigan
University
,
Kalamazoo
49008
John Petro
1.
The Intriguing Sierpinski Triangle. Ever since Waclaw Sierpinski introduced in 1916 a new triangular
construction, now called the Sierpinski gasket or the Sierpinski triangle, the mathematical world has been
fascinated by this most intriguing fractal and its many generalizations. In
this talk we shall discuss three quite different constructions of this fractal
and show how they are related to each other. Moreover, we will show that the Sierpinski Triangle also has a rather surprising
association with the classical
Tower
of
Hanoi
puzzle. We shall briefly discuss some of the many
generalizations of the Sierpinski Triangle.
2.
Old and new
questions about triangles. It is well known that a triangle is completely
determined, up to congruence, by the lengths of its three sides. Moreover, the
classical Hero formula gives us the area of the triangle in terms of the
lengths of the three sides. The proof of Hero's formula involves some interesting
elementary algebra and highlights the well known requirement that no one of the
sides may be longer than the sum of the lengths of the other two sides -- a
restatement of the well known principal that a straight line is the shortest
distance between two points. Now, let us pose two related questions. (1) Do the
lengths of a triangle's three medians determine the triangle? (2) Do the
lengths of a triangle's three altitudes determine the triangle? The
answer in both cases is yes and we encounter some interesting surprises
concerning the conditions on the numbers and how one would go about
constructing the triangle from the given numbers. We shall discuss a variety of
generalizations.
Grand
Valley
State
University
,
Allendale 49401
Matt Boelkins
1.
Fibonacci's Garden: By 300 BC, Euclid and other Greek
mathematicians were aware of a number with special properties linked to
proportion in geometric figures. Specially divided line segments, aesthetic
rectangles, and regular pentagons all exhibited an amazing number that later
came to be known as "the Golden Ratio." The Golden Ratio has since
made many appearances in surprising places in mathematics, including rather
recently in symmetries in Penrose tilings, and has
also manifested itself in curious ways in art, architecture, and the natural
world.
Around 1200 AD, Leonardo of Pisa (known to us today
as Fibonacci) began experimenting with a sequence of numbers that has since
come to bear his name. The list of numbers generated by starting with 0 and 1
and then adding the two previous numbers to find the next term results in
0,1,1,2,3,5,8,13,21,34,55,89,...
and is called "the
Fibonacci Sequence." This collection of numbers has been discovered to
have a seemingly unlimited list of interesting properties that fascinate
mathematicians to this day. Like the Golden Ratio, Fibonacci numbers arise
naturally in some startling places. One example is seen in pine cones, where
the numbers of spirals exhibited on the pine cone in opposing directions
normally turn out to be consecutive Fibonacci numbers.
Perhaps even more remarkably, the Golden Ratio and
the Fibonacci Sequence are inextricably linked with
each other. After an introduction to some of the history and mathematical ideas
surrounding each of these concepts separately, we will explore how the
development of seeds in flowers demonstrates some of these connections between
the Golden Ratio and the Fibonacci Sequence. (This
talk is accessible to any student in grades 9-12.)
2.
Student Research with
Polynomial Functions: Polynomials are
among the most fundamental functions in all of mathematics. While students
typically begin to encounter such functions as early as junior high, and
mathematicians have proved a remarkable number of sophisticated results on
them, new results continue to emerge and a surprising number of open questions
that involve polynomials persist. Many of these are accessible to
undergraduates.
A relatively recent result is the Polynomial
Root-Dragging Theorem, proved by B. Anderson in 1993, which states
If one or more roots of a
polynomial with all real zeros are shifted to the right, then all of the
critical points of the new function will lie to the right of the respective
critical points of the original function.
Over the past five years,
several different students and I have used the Polynomial Root-Dragging Theorem
as a central tool to investigate patterns in polynomial functions. In this talk
I will summarize some fundamental results and ideas share some of the many new
results these undergraduates have proved. (This talk is ideally suited for
students in a calculus course, though students with experience in precalculus can appreciate the ideas present)
3.
Careers in Mathematics (this
talk can be delivered jointly with Paul Fishback): What types of careers do college mathematics
majors pursue? What important job-related skills does the study of mathematics
promote? Why do employers value having mathematics majors as employees? The
goal of this talk is to answer these questions and to address common student
misconceptions regarding career possibilities for those who study
mathematics. Examples discussed include
specific mathematics-related careers, particular companies that hire
mathematics majors, and names of individuals ranging from doctors to actuaries
to economists to lawyers who majored in mathematics while in college. A list of
Internet resources will also be provided. (Best suited to grades 11-12)
Jon Hodge
1.
Surprises and Paradoxes in Mathematical Voting Theory
(Have you ever wondered why elections sometimes produce results that are
displeasing to many of the voters involved? Would you be surprised to learn that a perfectly fair election can
produce a result that is the last choice of every single voter? In this talk, we'll use mathematics to study
a variety of different systems for deciding the winner of an election. We'll see how these systems can produce
vastly different outcomes, and we'll try to understand what these differences
tell us about the search for a perfect voting system.)
Hope
College
,
Holland
49423
Tim Pennings
1.
Do dogs know calculus? (A standard calculus modeling problem is
to find the quickest path from a point on shore to a point in a lake, given
that running speed is greater than swimming speed. Elvis, my Welsh Corgi, has never had a calculus course. But when we play
"fetch" on the
shore
of
Lake Michigan
, he appears to choose paths close to the optimal one.
In this talk we reveal what was found when we experimentally tested this
ability. (Elvis will be available for follow-up questions.)
Lake Superior
State
University
,
Sault Sainte Marie
49783
Randy Suggitt
1.
The mathematics
of the
Mackinac
Bridge
. (The
Mackinac
Bridge
is not a static structure; it swings, it sways, it actually
changes shape under the influence of such factors as winds and varying traffic
loads. This talk investigates one of the changes that can occur in the shape of
large steel suspension bridges in response to temperature changes. The material
presented will be accessible to anyone with an understanding of algebra,
geometry, and quadratic equations.)