MATH 305                              Modern Geometry

Dr. Fenton                               Spring 2014

Web page:  http://www.bellarmine.edu/faculty/fenton/

 

This course meets 3:05 – 4:20 on Tuesdays and Thursdays in Pasteur 104.

 

I.          INTRODUCTION

Geometry is a large subject with an extensive history. In fact, much of what we will discuss this semester has been known for decades, centuries, or even millennia. However, the field has had a renewal of research and applications, and geometric knowledge continues to grow. Geometry is once again a major branch of mathematics and a valuable part of a mathematical education.

            This is an unusual undergraduate course, for it does not have a set of topics that have to be covered to prepare you for a later course. In fact, the content of an advanced geometry course varies greatly from university to university. The course includes what I think are significant and interesting topics: advanced properties of triangles and circles, analytic geometry, taxicab geometry, transformations, symmetry, and hyperbolic geometry. Please realize, however, that this is not a review of high school geometry. The course uses concepts from high school geometry to take a deeper look at familiar topics, plus additional topics that may be unfamiliar.

            More important than any content, I hope to strengthen your abilities as mathematicians. This includes practice with exploring mathematical concepts, raising conjectures, proving and disproving geometric statements, and communicating your mathematical ideas to others. The course will include work with The Geometer's Sketchpad, software designed for exploring and experimenting with geometric objects. My hope is that you will discover many geometric concepts for yourself. There will be an emphasis on discussing the mathematics and writing your results.

            The pedagogy for this course is exploratory learning through small group work. Each chapter opens with exploration activities, to be done as a group of two or three people. These activities are introductions to the concepts. Getting the explorations right or wrong is not the point; what is important is that you just try them. By working the activities, you will gain insight into the basic issues and you will have a foundation on which to build a stronger understanding of the concepts. This will make our class discussions more valuable for everyone.

 

II.        CONTACTING ME

My office is Alumni Hall 202, phone 272-8059. My office hours are 2:00-3:00 Tuesdays and Thursdays, or by appointment. Messages or assignments may be left in my mailbox. I encourage you to contact me electronically either through the campus email system or at wfenton@bellarmine.edu. Feel free to call me at home 454-7855 (but not after 9:00 p.m. please).

            Though my schedule is often very erratic, I will be in my office a good deal and you are welcome whenever my door is open.

 

IV.             MATERIALS

College Geometry Using The Geometer’s Sketchpad by Fenton & Reynolds (blue cover)

            You will be creating computer files in class and out of class. You may wish to have a flash drive for saving these files.

            The software we will use, The Geometer’s Sketchpad, is available on campus in Pasteur 104, Pasteur 106, Horrigan 017, and Level B of the library.

 

IV.       PREREQUISITE

MATH 215 Linear Algebra. While the chapters I have chosen will not use matrices and vectors, the study of linear algebra provides mathematical maturity, including experience at developing and writing proofs.

 

V.        COURSE DESCRIPTION (from the University catalog)

“A survey of topics in advanced geometry from three historical perspectives: synthetic, analytic, and transformational. Topics include advanced results in Euclidean geometry, axiomatics of Euclidean geometry, axioms and results in non-Euclidean geometry, an introduction to projective geometry, the use of coordinates, and insights gained from transformations.”

 

VI.       OBJECTIVES

Process Objectives

At the conclusion of this course, a successful student will:

·         be able to use Geometer’s Sketchpad software as an exploratory tool;

·         demonstrate an ability to generalize and conjecture from geometric examples;

·         demonstrate improved skills at developing logical proofs of geometric statements;

·         exhibit familiarity with the vocabulary of geometry and use geometric language more precisely;

·         understand the role of assumptions (axioms) when drawing conclusions.

 

Content Objectives

At the conclusion of this course, a successful student will be able to:

·         construct various centers of a triangle and recognize their significance;

·         explain how the various definitions for power of a point are related;

·         relate algebra to geometry through the use of a coordinate system;

·         recognize and compose Euclidean motions visually;

·         use transformations to classify two-dimensional objects by their symmetries;

·         work with the Poincaré disk model of hyperbolic geometry;

·         understand basic results in hyperbolic geometry and the critical role of the Parallel Postulate;

 

In addition, MATH 305 addresses the first three goals of the Mathematics Department. There will be a strong emphasis on problem-solving. There will be much practice at communicating mathematical ideas, both orally and in writing. The reading and homework will include understanding and creating mathematical proofs.

 

VII.     ATTENDANCE

I will not grade on attendance. However, during class discussions I will call upon student teams for suggestions and explanations. Further, class time will include time to work on the exploratory computer assignments, and time to discuss Exercises with each other and with me. So it is to your advantage to attend class. (See also the Travel Policy in the section on University policies.) 

 

VIII.    ACADEMIC HONESTY

(Also see the more complete statement in the section on University policies.)

“… All members of our community have an obligation to themselves, to their peers, and to the institution to uphold the integrity of Bellarmine University. In the area of academic honesty, this means that one’s work should be one’s own and that the instructor’s evaluation should be based on the student’s own efforts and understanding …”

            While this may at first seem inconsistent with the notion of group work, the principle still applies: you are expected to contribute honestly to the intellectual work of your group and of the course. Copying the work of others does not contribute to your learning; you need to put in the time and effort yourself to really understand the concepts.

 

IX.       GROUP WORK

Much of the work you do in this course will be in cooperation with other people, both in and out of class. Here are some things to think about as you decide who you want to work with.

·         Is everyone comfortable with the other people in the group?

·         Will we be able to work together productively?

·         Are there definite times when the entire group can meet outside of class?
(This is very important!)

Your group must have two or three members. The decision deadline is Tuesday January 21st.

 

X.        COURSE REQUIREMENTS

 

LAB ACTIVITIES    Ten of these assignments at 10 points each = 100 points

Every chapter begins with a set of lab activities. Each group is to turn in one assignment together. These will be graded primarily on completeness and effort. Late homework will not be accepted. If your group has some difficulty with meeting a deadline, please talk to me before the assignment is due. These assignments will be posted on my web page.

 

EXERCISES              Nine of these assignments at 15 points each = 135 points

Every chapter closes with a set of exercises. Again, each group is to turn in one assignment together. The Exercises are intended to show what you have learned about the topic, so they will be graded on correctness and clarity. Late homework will not be accepted. Again, if your group has some difficulty with meeting a deadline, please talk to me before the assignment is due. These assignments also will be posted on my web page.

 

TEST 1            Tuesday February 18 and the preceding weekend     100 points

Chapters 1, 2, 3, 4

The exam is in two parts. The first part is a group exam worth 50 points, done out of class. Your group will work together and turn in one paper. All group members will get the same grade on the group part. The second part is an individual exam worth 50 points, done in class on February 18.

 

TEST 2            Tuesday April 1 and the preceding weekend             100 points

Chapters 5, 6, 8

The exam is in two parts. The first part is a group exam worth 25 points, done out of class. The second part is an individual exam worth 75 points, done in class on April 1.

FINAL EXAM                       Tuesday April 29, 3:00-6:00               150 points

This is solely an individual exam. It will be comprehensive, covering the entire semester but with extra emphasis on Chapters 10 and 11.

 

XI.       GRADING

Grades will be assigned as follows:

            A+       Impress me!

            A         92% or higher

A-                88 – 91%

            B+       84 – 87%

            B         79 – 83%

            B-        75 – 78%

            C+       70 – 74%

            C         63 – 69%

            C-        60 – 62%

            D+       58 – 59%

            D         52 – 57%

            D-        50 – 51%

            F          0 – 49%

Lab Activities             100 points

Exercises                     135 points

Test 1                          100 points

Test 2                          100 points

Final Exam                  150 points

 

Your course grade will be your point total as a percentage of the 585 possible points.

 

XII. UNIVERSITY POLICIES

 

Attendance Policy

The University requires students who will be absent from class while representing the University to inform their instructors in two steps.  During the first week of the course, students must meet with each instructor to discuss the attendance policy and arrangements for absences related to University-sponsored events.  Second, students must provide the instructor with a signed Student Absentee Notification Form, available via the student portal on the University intranet, at the earliest possible opportunity, but not later than the week prior to the anticipated absence.  The Student Absentee Notification Form does not serve as an excused absence from class.  Your instructor has the final say about excused and unexcused absences and it is the student’s responsibility to know and abide by the instructor’s policy.

 

Academic Honesty

I strongly endorse and will follow the academic honesty policy as published in the Bellarmine University Course Catalog, available on the university website.  Students and faculty must be fully aware of what constitutes academic dishonesty; claims of ignorance cannot be used to justify or rationalize dishonest acts.  Academic dishonesty can take a number of forms, including but not limited to cheating, plagiarism, fabrication, aiding and abetting, multiple submissions, obtaining unfair advantage, and unauthorized access to academic or administrative systems.  Definitions of each of these forms of academic dishonesty are provided in the academic honesty section of the Course Catalog.  All confirmed incidents of academic dishonesty will be reported to the Assistant Vice President for Academic Affairs, and sanctions will be imposed as dictated by the policy. The instructor’s choice of penalty ranges from a minimum penalty of failing the assignment or test to failing the course itself.  If the student has a record of one prior offense, he or she will be suspended for the semester subsequent to the one in which the second offense took place.  This sanction is in addition to the penalty imposed by the faculty member.  If the student has a record of two prior offenses, he or she will be immediately dismissed from the university upon the third offense. 

 

Disability Services

Students with disabilities who require accommodations (academic adjustments and/or auxiliary aids or services) for this course must contact the Disability Services Coordinator.  Please do not request accommodations directly from the professor.  The Disability Services Coordinator is located in the Counseling Center, phone 272-8480.

 


Suggestions for Working in Groups

 

 

           Let everyone know that their ideas are of value, that no question is too trivial to ask, and that everyone makes mistakes. Criticism should be directed at ideas, not at individuals.

 

           Cooperate with other group members. This means listening to the ideas of others, trying methods that may be different from yours, and then coming to an agreement on a group solution.

 

           Be flexible, especially about finding time for the group to meet. Cooperation includes compromise. Remember that you are expected to work 6 - 8 hours a week outside of class. Work together as much as possible.

 

           Make sure that everyone participates. If someone is not offering her/his ideas, stop and ask for their perspective on the problem.

 

           Make sure everyone understands the solutions that are submitted for grading and be sure that everyone has written up some of these solutions.

 

           The most important thing is that the work must be done cooperatively! Dividing up the set of problems without ever coming together to discuss them is not an effective way to learn. Everyone needs to understand the questions asked in all of the problems and the ideas involved in solving them.