This class is cross-listed as:
Euclidean and Transformational Geometry
by Shlomo Libeskind (Jones and Bartlett Publishers, INC 2008)
. Here are pdf copies of Chapter Zero
and upside down copies of the Appendix from our textbook for those of
you who do not have yet the book.
There are many other excellent introductory geometry books. Reading from other sources is always very valuable. For example: Continuous Symmetry. from Euclid to Klein by William Barker and Roger Howe (American Mathematical Society [AMS] 2007), or Complex Numbers & Geometry by UNM Emeritus Professor Liang-shin Hahn (Spectrum Series of the Mathematical Association of America [MAA] 1994). There are of course the classics, such as Euclid's Elements which you can find in a very affordable Dover edition. A personnal favorite is Geometry Revisited by H. S. M. Coxeter and S. L. Greitzer (The Mathematical Association of America; 1ST edition 1967).
Course Structure: There are 2 lectures per week, Mondays and Wednesdays, that will be devoted to lecturing, solving problems, group work, and student presentations.
"Geometry" goes back at least four thousand years,
the first attempts to formalize it go
back more than two thousand years, and a satisfactory understanding of the
basic principles behind
were only understood when
mathematicians discovered the
non-Euclidean geometries. There is a lot
of fascinating history that we will touch, in this journey we are about to
embark together. Euclidean geometry is an ideal ground for learning how to think
logically, how to deduce results from very basic principles and definitions.
It is also a source of fascinating problems that can capture anyones
imagination. We will try to exploit the playful side of geometry in the hope
that you will be able to enjoy and transmit the beauty and elegance of geometric proofs.
We will learn how to deduce useful results about triangles, quadrilaterals, polygons and circles given basic objects (points, lines and space), basic definitions (angles, segments, distance), and clear rules of the game (axioms or postulates). We will discuss the notions of area and volume and how to calculate them for specific objects. The notions of congruent and similar figures are very important. We will first work with notions of congruent and similar triangles based on first principles, and once we learn about transformations in the plane we will develop a notion of congruence for more general figures. We will study symmetries in the plane (rotations, translations, reflections and their compositions), and we will use them to solve geometric problems.
At the end of the course we expect the students to have adquired the basic skills of mathematical reasoning, a deeper understanding of Euclidean geometry and its applications.
This class is a requirement for the future mathematics teachers. Here is a link to the Common Core Standards in geometry for K8. Note that there is an emphasis on transformations and two figures are defined to be congruent if we can go from one to the other by a succession of translations, rotations and reflections.
Group work: We will do group work periodically. The materials we will use for the different activities will be posted here: the proposed activity and any updates and follow-ups from the class discussions. I expect a group report in writing and on the board. Sometimes different groups will tackle different problems, sometimes they will tackle the same problem. In the group report make sure the names of all the members of the group are included, if you have used sources other than the material I have provided, make sure you make proper references (books, online references, names of consultants, etc).
Homework: The problems and exercises in the textbook are an integral part of the course. The book has "investigation problems" and higlighted green paragraphs titled "Now solve this" which I encourage you to consider. Some of these problems will be attempted working in groups in class. Homework will be assigned periodically, the problems in the homework will be carefully graded, and returned to you with feedback that will help you correct any errors. You are encouraged to discuss the homework with each other, but you should do the writing separately. You learn mathematics by doing, and there is no way around this. It is not enough to see your teacher or your friends solving problems, you have to try it yourself. Difficult as it may seem at the beginning, if you persist you will learn how to write a proper mathematical proof, you will learn how to read and understand other's proofs, and you will learn to appreciate and enjoy the beauty of an elegant argument.
Exams: There will be one midterm and a final exam or group project.
Grades: The final grade will be determined by your performance on homeworks, group work, the midterm, and a final exam (official date for final exam is: Monday May 6, 5:30-7:30pm) or group project. The grading policies will be discussed in class.
Prerequisites: Math 215 or Math 162 (or permission from the instructor).
Americans with Disabilities Act: Qualified students with disabilities needing appropriate academic adjustments should contact me as soon as possible to ensure your needs are met in a timely manner. Handouts are available in alternative accessible formats upon request.
Return to: Department of Mathematics and Statistics, University of New Mexico
Last updated: Jan 11, 2013