MTH 322 Geometry Fall 2012

Department of Mathematics, University of Rhode Island


Orlando Merino,, 874-4442, Lippitt Hall 101C, Office Hrs. MWF 10-10:50 a.m. or by appt.


Lippitt Hall 204, MWF 11 a.m.


Euclidean and Non-Euclidean Geometries, by Marvin J. Greenberg, Fourth Edition. ISBN 978-0-7167-9948-0


MTH 215 or permission of the instructor


History of early geometry, Logic and Incidence Geometry, Hilbert's Axioms, Neutral Geometry, The Parallel Postulate, Non-Euclidean Geometriy.


Midterm 15%, Final Exam 25%, Written project 20%, oral presentation 20%, Homework 20%.

About the Course

This course is an introduction to Geometry. In this class you will do mathematical proofs, as well as oral and written exposition of mathematical topics. Students in Secondary Education in Mathematics must take MTH322, and we will pay special attention to the NCATE/NCTM Program Standards 2, 3, and 11, which are listed below.  MTH322 is also a good class for students of mathematics or other areas who are interested in learning about geometry.

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

All submitted work must be your own. If you consult other sources (class readings, articles or books from the library, articles available through internet databases, or websites) these MUST be properly documented, or you will be charged with plagiarism and will receive an F for the paper. In some cases, this may result in a failure of the course as well. In addition, the charge of academic dishonesty will go on your record in the Office of Student Life. If you have any doubt about what constitutes plagiarism, see the URI Student Handbook, and UNIVERSITY MANUAL sections on Plagiarism and Cheating at - cheating.

Late work

Late work is either not accepted, or accepted under certain conditions and with a penalty.  More details will be given in class.

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

Students are responsible for being familiar with and adhering to the published "Community Standards of Behavior: University Policies and Regulations” which can be accessed in the University Student Handbook. If you must come in late or leave early, please do not disrupt the class. Please turn off all cell phones (no texting!), pagers, or any electronic devices.

"Incomplete" grade

URI regulations concerning incomplete grades will be followed to the letter. The following paragraphs are 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.