Freshman Seminar
Fall 2001
MATH 171.01 (1 credit), W, 10:00AM-10:50AM, Wickersham 124

Corequisites:

Mathematics majors taking Calculus I.

Instructor:

Dr. Buchanan
Office: Wickersham 112, Phone: 872-3659, FAX: 871-2320
Office Hours: 1:30PM-2:20PM (M-F), or by appointment
Email: Robert.Buchanan@millersville.edu
URL: http://banach.millersville.edu/~bob

Textbook:

In place of a textbook, we will have printed handouts and assigned readings from a variety of sources.

Objectives:

This seminar will introduce students to a mathematical way of thinking through a sequence of exploratory problem assignments. Students will experience problem solving and mathematical inquiry in a controlled environment using discussion, collaboration, abstraction, and technologies. The objectives of the seminar include prodding the students to critically analyze their use of mathematics, mathematical procedures, and technologies with a goal of having the students pierce the ``black box'' approach to mathematics.

Course Contents:

The semester activities may include exposure to and exploration of the following topics and activities. The topics may not be covered in precisely the order in which they are described below. Related topics from the list below will be covered together. Whenever possible, topics from the list below will be synchronized with the topics in the Calculus I syllabus.

• Mathematica Basics -- kernel and front-end, syntax, commonly used facilities.
• Cryptanalysis -- students will be introduced to the elementary number theory, abstract algebra, and statistics which play a role in enciphering and deciphering messages.
• Identification numbers -- students will be introduced to check digit schemes used to transmit, retrieve, and store data.
• Bertrand's paradox -- students will explore the probability that a randomly selected chord of a circle is longer than a side of an inscribed equilateral triangle.
• Complex numbers -- students will explore representations of complex numbers, complex number arithmetic, roots of unity, etc.
• Divisibility/indivisibility of natural numbers by 2, 3, 7, or 11 and why the familiar tests for divisibility work.
• Taylor polynomial approximations to functions -- students will extend the concept of linear approximations of a function to higher order polynomial approximations.
• Elementary matrix algebra -- affine transformations of the cartesian plane, iterated function systems.
• Buffon needle problem -- students will understand how a probability experiment can lead to an approximation to the transcendental number $\pi$.
• Quadratic forms -- students will explore some of the properties of the level curves of expressions of the form $\mathbf{x}^{\top} A \mathbf{x}$ where $\mathbf{x} \in \mathbb{R}^{2}$ and $A$ is a $2 \times 2$ hermitian matrix.
• Conic sections -- the circle, parabola, ellipse, and hyperbola and their geometric properties.
• Navigation -- an activity in which students will apply some of what they have learned so far to understanding the principles of operation of the Long Range Navigation (LORAN) system commonly used for coastal navigation.
• Envelopes of tangent lines -- students will determine the curves formed by envelopes of tangent lines of the forms $x \cos \alpha + y \sin \alpha = 1$ and ${{x} \over {\cos \alpha}} + {{y} \over {\sin \alpha}} = 1$ where $\alpha \in [0,2 \pi]$.
• Computer arithmetic -- fixed and floating point representations, round off errors, numerical precision, numerical approximation, and numerical errors.
• Root finding methods -- Newton's method and the secant method.
• Optimization -- an activity in which students will optimize a function whose critical points can only be numerically approximated.
• Binomial series -- students will explore through the notion of area under a curve the infinite series expansion of $(a + b)^{k}$ where $k \in \mathbb{Q}$.
• Mean Value Theorem for Definite Integrals -- students will explore the relationship between the mean of a set of numbers and the mean value of a function on an interval.
• Probability distributions -- binomial distribution and its approximation by the normal distribution.

If time permits other topics may be covered as well.

Attendance:

Students are expected to attend all class meetings. Since an objective of the seminar is to demonstrate that mathematical inquiry is often a collaborative activity, students must be present to participate and benefit from the seminar. Every absence in excess of two will result in the lowering of the students course grade by a letter grade.

Homework:

Since this is a single credit course, the workload will be proportionate. The assignments of the course will be designed so that they can be completed in the hour of class meeting and an estimated single hour of homework per week. Student work will be examined for correctness, generality, and evidence of insight into mathematical ideas.

Grades:

Course grades will be based on the student's participation in the seminar and their graded written homework. I keep a record of students' homework scores. Students should also keep an individual record of graded assignments. The course letter grades will be calculated as follows.

90-100	A
80-89	B
70-79	C
60-69	D
0-59	F

Page maintained by: Robert.Buchanan
Robert.Buchanan@millersville.edu

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