Mt 322 Topics in Geometry - Fall 2003

Department of Mathematics & Computer Science
Sr. Barbara E. Reynolds, Ph.D.

Cardinal Stritch University is a learning community in which the elements of scholarship and learning -- discovery, application, integration, and teaching -- are embraced by faculty, staff, and students. Stritch graduates are critical thinkers, ethical decision makers, and life-long learners.

Our mission is to transform lives. We do this through value-centered education. This is our business.

Course Description

(as found in the Cardinal Stritch University Catalog for Undergraduate Studies 2000 - 2002)

This course offers a variety of geometrical topics which may include taxicab geometry, conic sections, four-dimensional space, trigonometry in the unit circle, the geometry of the sphere, and geometric patterns in art. The topics will be determined by the instructor and the needs of the students.

Mt 322 Topics in Geometry Weekly Course Assignments.

Table of Contents:

[ Course Description | Texts and Required Materials | Course Objectives (indicating methods of assessment) | Franciscan Intellectual Tradition | Calculators and Computers | Course Content | Prerequisites | Cooperative Learning Groups | Writing and Speaking across the Curriculum | Requirements (including grading criteria) | Administrative Policies | Office Hours ]

Texts and Required Materials

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Course Objectives

The most important objective of this course is to introduce students to mathematical thinking and reasoning through a hands-on exploration of interesting and challenging topics in geometry. The emphasis in this course is on conjecture, exploration, and articulation of geometric ideas, leading to the development of a robust proof or refutation. The goals of this course can be roughly divided into the broad areas of process goals (exploration, proof, and communication) and content goals (in synthetic, analytic, and transformational geometry).

Process Goals

By the end of the course, the successful student will be able to

Broad Content Goals

By the end of the course, the successful student will be able to:

In this course, benchmark tests give the student an opportunity to demonstrate fluency with basic vocabulary and computational strategies. In-class tests give the student an opportunity to demonstrate mastery of the various mathematical strategies and techniques -- particularly the construction of sound mathematical proofs. The Proof Book will provide a record of the student's growth in the ability to write coherent geometric proofs. The paper will give the student an opportunity to demonstrate an ability to make application of the material studied in this course to situations in the wider mathematical world.

Beyond the mathematical content of individual courses, the program of courses for a major in mathematics is designed to prepare students for the 21st century by helping students to become problem solvers, effective communicators, users of appropriate technology, and team players. In this course, students will be engaged in a variety of activities which will help them to move toward achieving these goals.

  • As problems solvers, students will be learning to:

  • As effective communicators, students will be learning to:

  • As users of appropriate technology, students will be learning to:

  • As team players, students will be learning to:

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    Franciscan Intellectual Tradition

    Cardinal Stritch University is a Catholic institution of higher education, founded and sponsored by the Congregation of the Sisters of Saint Francis of Assissi. A Stritch education is about more than simply earning a degree. The history of the University is rooted in Franciscan values. You can find out more about Franciscan values by going to the Cardinal Stritch University homepage, and choosing Franciscan Values on the Choose a Topic link.

    While neither Saint Francis nor Saint Clare actually taught mathematics (or any other subject) in an established university, their lives do offer for us a model of cooperation, respect for diversity and inclusivity, and reverence for creation that we strive to reflect throughout the university community. Creating a caring community, showing compassion for others, reverencing all of creation, and living lives committed to peace-making are life-long habits of the heart, which -- while not part of the mathematical content of a mathematics course -- shape the atmosphere of this classroom.

    The cooperative learning environment of this classroom offers an experience of working with others in a caring community. Students are expected to show respect for each other and to extend ordinary human courtesy to their classmates. The instructor is committed to working with students to foster a safe and comfortable environment in which to engage in intellectual argumentation. Occasional conflicts in the cooperative learning groups offer opportunities for individuals to practice conflict resolution in a real life setting.

    Students are asked to exercise responsible stewardship of the equipment in the classroom computer lab, treating the computers, printers, and other equipment with gentle care. By switching to a smaller font size before printing, making use of the Print Preview feature of the software, and recyling paper when this is practical, students are encouraged to avoid excessive paper wastage.

    Critical thinking skills, which form an intellectual framework for developing good mathematical proofs, are necessary skills for persons committed to taking responsible social action and working for justice in our world.

    (These paragraphs draw heavily from the booklet, "Franciscan Values at Cardinal Stritch University," (Stritch Publications, 2002), Cardinal Stritch University Archives, Milwaukee, Wisconsin. Print versions of this booklet are available through the Franciscan Mission Office. (Phone: 414-410-4151)

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    Calculators and Computers

    Calculators and computers are legitimate tools for doing mathematics. One of the goals of the Department of Mathematics & Computer Science is that our students develop a facility with various forms of technology and learn to use these effectively to explore and solve problems. One of the specific goals of this course is to provide an opportunity for students to become proficient in the use The Geometer's Sketchpad, a dynamic geometry tool.

    Throughout the semester, students will be given opportunities to use electronic communication tools (such as email and a graphical web browser). Class assignments will be regularly posted to the internet.

    Each student will need to have access to computers and/or to work in the computer lab on some of the homework assignments and projects. Computer lab schedules are posted on the doors of the computer lab. Although other classes also meet regularly or occasionally in the labs, there is one lab which is always reserved for student use. You will need to plan to hold some of your outside-of-class group meetings in the computer labs (or in another location where you have access to a computer with The Geometer's Sketchpad).

    The Geometer's Sketchpad is available on computers in various campus computer labs (CH 31-A, Macs in BH 35, PCs in BH 46). Students who wish to purchase a Student Edition of Sketchpad (for approximately $40) can find ordering information on the Key College Publishing website.

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    Course Content

    We will be studying the material in seven of the chapters of our textbook:

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    To be successful in this course, the student should have successfully completed Mt 209, or at least two years of high school mathematics, including both algebra and geometry. Students are expected to have familiarity with most of the topics listed on the study guide for the first benchmark test. While these topics will be reviewed in the first weeks of the semester, the assumption is that students have learned this material in previous courses.

    Please see the instructor if you have any concerns about your preparation for this course.
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    Cooperative Learning Groups

    Students will do much of the work of this course in cooperative learning groups. These groups will work together on some homework problems as well as on in-class problems.

    The objective of group work in this course is primarily to engage the students in thinking more deeply about important mathematical issues. Even though a lot of the work of the course is done in small groups, students are expected to learn to develop proofs and to solve problems on their own. Working with colleagues in this class and talking about problems in small groups are strategies for developing an understanding of a problem situation from several points of view. Learning theory research has shown that cooperative learning leads to deeper understanding and longer retention of material that is studied.

    Working well in a group is an important skill. Some students enjoy the group work more than others, and all will benefit from further developing this skill. Most of our graduates are employed in workplace settings where they are expected to function as a member of a project team. Many of our graduates tell us that the skill they developed by working in cooperative learning groups is very beneficial in the workplace.

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    Writing and Speaking across the Curriculum

    The faculty of Cardinal Stritch University are committed to developing writing and speaking skills for all students across all disciplines. Students will have opportunities in every class session to speak with peers informally in small-group and full-class discussions. Students will also have opportunities to give short presentations to their colleagues -- usually with respect to a proof that she/he is developing. This provides some experience in presenting work-in-progress (that is, discussing ideas about problems that have not yet been solved, and progress on work which is not yet complete). Any written work that is turned in -- graded or ungraded homework assignments, tests, papers, and projects -- is to be written in standard English using complete sentences. Throughout the semester, students will receive feedback from the instructor (and from their peers) which is intended to assist in developing good speaking and writing skills.

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    Regular attendance, participation, lab activities, quizzes: 10%

    Regular attendance is expected. Class discussion is dynamic, and hands-on computer activities are designed to stimulate discussion among students working in small groups. Getting someone's notes is a poor substitute for being present and involved in class discussion.

    However, if you must miss a class, it is your responsibility to find out what you missed.

    Problem solving is not a spectator sport. During most class periods there will be time for large and/or small group discussions about selected problems. It is important to learn to ask helpful questions and to listen constructively to each other. Constructive participation sometimes means allowing others time and space to think about the problem.

    Homework assignments will include reading the text, checking the library and other sources for information beyond that given in the text, and doing problems from each chapter. If the discussion is lively, with everyone contributing, the instructor will collect and grade fewer problems and have fewer quizzes; if the discussion lags or is dominated by a few individuals, it may become necessary to collect more homework and/or give (unannounced) quizzes.

    At the end of each class period, each student will fill out a Class Participation Form. These forms are used to check attendance, to respond to the student's self evaluation, and to give a score for class participation.

    If you must miss a class for any reason (excused or unexcused absence), your participation score for that day will be recorded as 0. However, if you wish to make up these absences, you may turn in written evidence that you have done some work to make up missed lab/class activities. This make-up work must be turned in within two class periods of the missed classes.

    Graded Homework and the Proof Book: 25%

    One of the principal goals of this course is for students to learn to develop and write robust mathematical proofs. Throughout the semester you will be asked to write a lot of proofs. Some of these proofs will be turned in and graded individually. You are to collect these and other proofs that you develop into a Proof Book, a portfolio of your best geometric proofs. By the end of the semester, you should have 16 - 20 proofs in your Proof Book. These proofs should represent the entire scope of the course; that is, you should have many different styles of proofs, and you must include proofs from each chapter of the course.

    Each proof that you write and turn in will be marked according to a rubric for grading proofs, which will be handed out in class. Each proof will be graded on logic, appropriate diagrams/tables, completeness, and clarity of explanation. These grading criteria are designed to give you constructive feedback so that you learn to develop more correct and robust proofs. Writing proofs is one important form of written communication in mathematics.

    Several individual proof problems will be collected and evaluated early in the semester to give students an idea of how the proofs will be graded. The Proof Book will be collected and graded October 23. At that time, the Proof Book must have at least eight proofs, including corrected versions of previously graded proofs. The completed Proof Book will be due on December 4, at the last class meeting for this course.

    As a portfolio, the Proof Book should document the student's improved skill in developing good proofs over the semester. This kind of portfolio can be used as a learning tool. However, the key to learning is regular reflection on the learning process.

    • If you receive a low grade on a proof (perhaps early in the semester), you may include the original proof along with a revised proof. You should include a reflection on what you learned in developing the revision. In grading the Proof Book, I will consider only the grade on the revision with the reflection. This will count as one proof.
    • In each chapter we will be developing additional proof strategies. For example, you might learn one strategy for proving a certain theorem about triangles in Chapter 2, a second proof strategy for the same theorem in Chapter 4, and yet another strategy for proving this theorem in Chapter 7. You may include all three proofs in your Proof Book, along with a reflection on how the three proofs are related. This would count as three proofs in your Proof Book, and the individual grades on all three proofs will be incorporated into the overall grade on the Proof Book.
    • The grade on the Proof Book will be based on the following elements:
      1. average of the individual grades on the proofs (or revisions) included in the Proof Book

        The original attempt should be included along with the revision and a reflection on what was learned.

      2. inclusion of the required number (16 - 20 proofs) and distribution of proofs (at least two proofs from each chapter)

        You will very likely write many more than 16 proofs over the course of the semester. Collect the best of your proofs into the Proof Book.

      3. demonstration that the student has mastered a variety of proof strategies

        Various proof strategies will be identified throughout the semester. For example, in Chapters 2 and 3 proofs will be based on constructions which have their roots in Eulcid's postulates. In Chapter 4, proofs will be based on algebraic calculations. In Chapters 5 and 7, proofs will use properties of transformations.

      4. the quality of the reflections

        Most of the reflections in the Proof Book will refer to a cluster of proofs. For example, if you have corrected a proof, you should include the revision, the original (incorrect or weak) proof, and one reflection on what you learned in doing the correction. If you are presenting three different proofs of the same theorem, include one reflection comparing and/or contrasting the various proof strategies. You might have proved several different theorems using a similar strategy; you could group these together with one reflection. These reflections are what make the Proof Book a portfolio, and not merely a notebook with lots of proofs. Reflections should be thoughtfully written (using standaard English and orthodox spellings, of course!).

      5. professional presentation of the Proof Book as a portfolio of geometric proofs

        Your Proof Book should be something that you can take with you to a job interview. You should be proud to show a principal or department chair how you have grown in being able to write correct proofs throughout the semester. With this in mind, each proof should be neatly written. You are encouraged to include Sketchpad diagrams with each proof. One option is to develop your Proof Book as a multi-page Sketchpad document (a Sketch Book). (You will see examples of this as the course progresses.)

    Paper on NCTM Principles & Standards: 5%
    Most students who take this course are planning careers in the teaching profession. Future teachers should make themselves familiar with mathematics curriculum guidelines published by the National Council of Teachers of Mathematics (NCTM), the professional organization for elementary and secondary teachers of mathematics. The NCTM Principles & Standards for School Mathematics can be found on-line. Students will be asked to study a portion of that document, and write a paper which makes connections to what we are studying in this course. More information about this paper will be available after we have studied several chapters of our text. The paper will be due about October 9. (Students who are not in a teacher certification program may request an alternative to this assignment.)

    This paper wil be graded on format (professional presentation) 10%, writing style (spelling and grammer) 15%, content 60%, and synthesis/application 15%.

    Tests: 25%
    There will be three tests. In calculating your final course grade, the first test will be weighted 5%, and each of the other two will be weighted 10%. Tests are tentatively scheduled for the following dates:
    • Test 1: Tuesday, September 30
    • Test 2: Tuesday, October 21
    • Test 3: Thursday, November 20

    A study guide for each test will be available about one week before the scheduled test date.

    Ordinarily, I do not give make-up tests; exceptions to this policy will be considered on a case-by-case basis.

    Benchmark Tests: 10%
    Benchmark testing is the department's way of assuring that students have achieved minimum levels of calculational competency. Although students are encouraged to use computers throughout this course, they are expected to be able to do basic computations by hand. These basic computations and vocabulary are covered in the benchmark tests.

    There will be two benchmark tests in this course. These are offered outside of regular class meetings. Each student should make an appointment with the instructor to take the benchmark test during the scheduled window-of-opportunity.

    Procedures for Benchmark Tests:

    • To pass the benchmark test, a student must get nine or ten of ten problems completely correct; there will be no partial credits. If a student passes on the first attempt, the score will be recorded as 100%.

    • A student who does not pass a benchmark test on the first attempt, may demonstrate that that he/she has done some additional practice, and make an appointment with the instructor to try the test up to two more times.

    • If a student passes a benchmark test on the first or second re-test, the score will be recorded as the average of the scores made on each attempt.

    • If a student has not passed a benchmark after two re-tests or by the specified date, the score will be recorded as 0%.

    Note: Midterm is October 16, and the last day to withdraw from fall semester courses is Friday, November 7. Each student's midterm grade will be based on several graded proofs, one benchmark, the first in-class test, the paper, and half the semester of participation. This represents approximately 25% of the final course grade. Additional graded work will be due shortly after midterm (the second test, the first part of the Proof Book), so that by the end of October 45% of the required course material will have been graded and returned to each student. Any student who has concerns about her/his progress or ability to keep up with course assignments should discuss these concerns with the instructor.

    Final Exam: 25%

    The cumulative final exam is scheduled for Thursday, December 11, 10:30 am - 12:30 pm.

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    Administrative Policies

    Academic Integrity Policy

    Inherent in the mission of Cardinal Stritch College is the strong belief in the principle of academic integrity. Students who cheat violate their own integrity and the integrity of the College by claiming credit for work they have not done and knowledge they do not possess. All students are expected to recognize and to abide by the policy on academic integrity found in the Student Handbook.

    Always give credit to your sources. If you find a proof you are trying to construct in another book, or if you develop a solution to a problem while you are working with members of your small group, you are expected to mention this in writing on the paper you turn in.

    Because you will be asked to do a lot of work in small groups, whenever I give a take-home assignment or test on which I expect you to work on your own, I will make this very explicit.

    Compliance with the Rehabilitation Act of 1973 (Rehabilitation Act 504)

    If you have any special needs for alternative instruction and/or evaluation procedures, please feel free to discuss these needs with me so that appropriate arrangements can be made.

    Cell Phones and Pagers

    As a matter of courtesy, students are expected to turn off cell phones and pagers during class. If extraordinary circumstances require an exception to this policy, the student is expected to discuss this with the instructor before class begins.

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    Office Hours

    My office is located in CH 34, just across the hall from the classroom where this course meets. I will be on-campus and regularly available for students most weekdays. If you wish to make an appointment with me, you may sign up on the sheets which are posted on the in the hallway just outside the door of my office. The best way to contact me is via email, which I check regularly (several times a day).

    If you need to reach me between classes:

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    Return to Sr. Barbara E. Reynolds Home Page.
    Return to course list for 2003 -- 2004.
    Go to Mt 322 Topics in Geometry Syllabus.
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    The easiest way to contact me is to send an email message to Sr. Barbara E. Reynolds.
    This page was updated on August 22, 2003.