MTH 322 Geometry Fall 2010

Department of Mathematics, University of Rhode Island


Orlando Merino,, 874-4442, Lippitt Hall 101C


Lippitt Hall 204, MWF 11 a.m.


Revolutions of Geometry, by Michael O’Leary, Wiley 2010. ISBN 978-0-470-16755-7


MTH 215 or permission of the instructor


Geometry in Egypt, Babylon and China. Thales and mathematical proofs. Pythagoras. Euclid's Elements, Triangles, Parallel Lines, Circles, Archimedes, Analytic Geometry, Projective Plane, The Fifth Postulate, Non-Euclidean geometries.


Midterm 25%, Final Exam 25%, Written project and oral presentation 25%, Homework 25%.

About the Course

This course is an introduction to Geometry. We will study the subject by following its development in history. In this class you will do mathematical proofs, as well as oral and written exposition of mathematical topics. We will pay special attention to the NCATE/NCTM Program Standards 2, 3, and 11, which are listed below.

NCATE/NCTM Program Standards (2003)

Standard 2: Knowledge of Reasoning and Proof

Candidates reason, construct, and evaluate mathematical arguments and develop an appreciation for mathematical rigor and inquiry.

  1. Recognize reasoning and proof as fundamental aspects of mathematics.
  2. Make and investigate mathematical conjectures.
  3. Develop and evaluate mathematical arguments and proofs.
  4. Select and use various types of reasoning and methods of proof.

Standard 3: Knowledge of Mathematical Communication

Candidates communicate their mathematical thinking orally and in writing to peers, faculty, and others.


  1. Communicate their mathematical thinking coherently and clearly to peers, faculty, and others.
  2. Use the language of mathematics to express ideas precisely.
  3. Organize mathematical thinking through communication.
  4. Analyze and evaluate the mathematical thinking and strategies of others.

Standard 11: Knowledge of Geometries

Candidates use spatial visualization and geometric modeling to explore and analyze geometric shapes, structures, and their properties.


  1. Demonstrate knowledge of core concepts and principles of Euclidean and non- Euclidean geometries in two and three dimensions from both formal and informal perspectives.
  2. Exhibit knowledge of the role of axiomatic systems and proofs in geometry.
  3. Analyze characteristics and relationships of geometric shapes and structures.
  4. Build and manipulate representations of two- and three- dimensional objects and visualize objects from different perspectives.
  5. Specify locations and describe spatial relationships using coordinate geometry, vectors, and other representational systems.
  6. Apply transformations and use symmetry, similarity, and congruence to analyze mathematical situations.
  7. Use concrete models, drawings, and dynamic geometric software to explore geometric ideas and their applications in real-world contexts.
  8. Demonstrate knowledge of the historical development of Euclidean and non- Euclidean geometries including contributions from diverse cultures.


We will use the software GEOGEBRA, which is available free for download.

Instructor's expectations

  • IN THE CLASSROOM Lecture time is at a premium, so it must be used efficiently. Expect to have material covered at a fast pace. We expect you to come prepared to class as detailed below.
  • OUTSIDE THE CLASSROOM You cannot be taught everything in the classroom. Much of your learning must take place outside the classroom. At a minimum you should plan on studying two or more hours outside the classroom for each hour in class. You should attempt all the homework that is assigned and try additional problems in areas where you feel weak.
  • THE TEXTBOOK You are expected to read the textbook for comprehension. It gives a detailed account of the material of the course. It also contains many examples of problems worked out, and these should be used to supplement those you see in the lecture. Use pencil and paper to work through the material and to fill in omitted steps. Read the appropriate section(s) of the book before the material is presented in lecture. Then the faster-pace lecture will make more sense. After the lecture carefully reread the textbook along with your lecture notes to cement your understanding of the material.
  • EXAMS Our intent is to determine how well you understand the basic principles underlying the methods and if you are able to apply these principles to novel as well as routine situations. Some problems on an exam may seem new, but all will be solvable using principles from the material on which you are being tested.
  • SOLUTIONS TO PROBLEMS It is your responsibility to communicate clearly in writing up solutions for homework, quizzes, and exams. Your results must display your understanding well and be written in a correct, complete, coherent, and well organized fashion. The rules of language still apply in mathematics, and apply even when symbols are used in formulas, equations, etc. Neatness counts!

[Based on: Zucker, S., Teaching at the University Level, AMS Notices (43), 1996, pp 863-865.]

Special Needs

Any student with a documented disability is welcome to contact me early in the semester so that we may work out reasonable accomodations to support your success in this course. Students should also contact Disability Services for Student, Office of Student Life, 330 Memorial Union, Kingston, 874-2098.

Academic Honesty

Students are expected to be honest in all academic work. A student's name on any written work shall be regarded as assurance that the work is the result of the student's own thought and study. Work should be stated in the student's own words, properly attributed to its source. Students have an obligation to know how to quote, paraphrase, summarize, or reference the work of others with integrity. The following are examples of academic dishonesty.

  • Using material from published sources (print or electronic) without appropriate citation
  • Claiming disproportionate credit for work not done independently
  • Unauthorized possession or access to exams
  • Unauthorized communication during exams
  • Unauthorized use of another's work or preparing work for another student
  • Taking an exam for another student
  • Altering or attempting to alter grades
  • The use of notes or electronic devices to gain an unauthorized advantage during exams
  • Fabricating or falsifying facts, data or references
  • Facilitating or aiding another's academic dishonesty
  • Submitting the same paper for more than one course without prior approval from the instructors.

H1N1 Flu

The H1N1 Flu Pandemic may impact classes this semester. If any of us develop flu-like symptoms, we are being advised to stay home until the fever has subsided for 24 hours. So, if you exhibit such symptoms, please do not come to class. Notify me at 874-4442 or of your status, and we will communicate through the medium we have established for the class. We will work together to ensure that course instruction and work is completed for the semester.

The Centers for Disease Control and Prevention have posted simple methods to avoid transmission of illness. These include: covering your mouth and nose with a tissue when coughing or sneezing; frequently washing your hands to protect from germs; avoiding touching your eyes, nose and mouth; and staying home when you are sick. For more information, please view . URI information on the H1N1 is posted on the URI website at , with links to the CDC at .

Additional Information

The University Manual (See ) contains useful information: 8.39.10-12 (attendance); 8.51.11-14 (excused absences); 8.51.15 (examinations during the semester); 8.51.16 (final examinations); 8.27.16-19, 8.27.17-19, 8.27.10-15 (plagiarism-instructor's responsibilities, judicial action, and student's responsibilities); and 8.52.10 (grading criteria).

Civility Policy

The University of Rhode Island has adopted a civility policy regarding disruptive classroom behaviors. Disruptive behaviors are defined as behaviors that interfere with the learning and/or teaching process. Disruptive behaviors in the classroom include inappropriate talking during lectures or class discussions or in any manner interfering with other student's ability to have a quality learning experience. Students who engage in disruptive behavior will receive one warning without penalty. Continued incidents of disrupting the class will result in the initiation of removal procedures or the loss of a letter grade. Disruptive behaviors include cell phone and pager use. Cell phones and pagers must be turned off, silent, and out of sight during classes, and you should not be checking for calls or messages during class (including "texting"). Common sense and common courtesy should govern classroom civility.

"Incomplete" grade

I follow to the letter the URI regulations concerning incomplete grades, namely the following paragraphs taken from the university manual:

  • 8.53.20. A student shall receive a report of "Incomplete" in any course in which the course work has been passing up until the time of a documented precipitating incident or condition, but has not been completed because of illness or another reason which in the opinion of the instructor justifies the report. An instructor who issues a grade of "Incomplete" shall forward a written explanation to the student's academic dean.
  • 8.53.21. The student receiving "Incomplete" shall make necessary arrangement with the instructor or, in the instructor's absence, with the instructor's chairperson to remove the deficiency. This arrangement shall be made prior to the following midsemester for the undergraduate student and within one calendar year for the graduate student.