# Modern Geometry

Professor Hlas

hlascs (@) uwec.edu

Drop-in times

*Each problem I solved became a rule, which served afterwards to solve other problems.*

—Rene Descartes

## Course Information

This course focuses on axiomatic thinking in Euclidean and non-Euclidean geometries. Emphasis will be on proof techniques, finite geometries, Euclidean constructions, transformations, spherical geometry, hyperbolic geometry, and geometry software.

More course information is posted on Canvas.

### Goals

- Use interactive geometry software to make geometric conjectures.
- Understand axiomatic thinking.
- Write geometric proofs.
- Create a wikibook of important axioms, theorems, and proofs.

### Structure

- Updates to the wikibook
- Weekly group quizzes
- Two projects
- Two in-class exams
- Final exam

### Required

*Geometry & Symmetry*by Kinsey, Moore, & Prassidis, 2011- Euclid Elements (with applets)
- Geometry software
- Geometer's Sketchpad (via UWEC virtual lab)
- GeoGebra
- Spherical Easel
- NonEuclid

### Research background

- Lakatos, I. (1976).
*Proofs and refutations: The logic of mathematical discovery.*Cambridge, United Kingdom: Cambridge University Press. - Fawcett, H.P. (1938). The nature of proof: A description and evaluation fo certain procedures used in a senior high school to develop an understanding of the nature of proof.
*National Council of Teachers of Mathematics, Yearbook 13.*Reston, VA: NCTM. - Hyperbolic geometry in a high school geometry classroom (Donald, 2005)
- Finite geometries and axiomatics systems
- Isometries
- Triangle congruence and similarity: A Common-Core-compatible approach
- Connecting research to teaching: Geometry and proof (Battista and Clements)

## Grading

It is important to accurately show your mathematical thinking and to communicate clearly. If any concerns arise regarding grading, contact the instructor outside of class time. Students earn "math points" (MP) for demonstration of mathematical thinking in their solutions.

### Warm-up activities

Warm-up activities is often assigned to prepare students for in-class activities. These activities are more effective when everyone attends class fully prepared. "Eyeglasses" in the calendar will indicate warm-up activities.

### Recommended practice

Explicit homework is usually not given to practice course material. Instead it is recommended to:

- try class activities on own
- handwrite proofs
- write proof outlines that summarize important proof ideas (beginning, key steps, etc.)
- use flash cards for vocabulary

### Wikibook author (3 × 10 MP)

Most class activities will require updating the wikibook. A student from each group will volunteer for that class's updates. See the score sheet for more information.

### Group quizzes (8 × 10 MP)

Each group will submit one quiz and each member of that group will receive the same score. Students are expected to fully contribute to quiz solutions and quiz items may appear on examinations. Quizzes will typically have an application problem and require a proof.

### Projects (2 × 20 MP)

Student projects will extend what is discussed in class. The first project will be as a group, the second project will be individual. More details will be discussed when appropriate.

### Exams (2 × 25 MP)

"To assess conceptual knowledge, researchers often use novel tasks … Because children do not already know a procedure for solving the task, they must rely on their knowledge of relevant concepts to generate methods for solving the problems." (Rittle-Johnson, Seigler, Alibali, 2001, p. 347). A such, assessments are a part of the learning experience and will not only require mastery of class material, but will also require the ability to apply class material to new situations.

For each exam, one page of notes (1-sided, handwritten) is allowed.

### Final exam (30 MP)

Cumulative final exam following same structure as in-class exams.

### Bonus (? MP) ~at most 25

Homework, quizzes, or examination points may be assigned beyond those indicated above.

#### Fine print

**Attendance** A record of attendance will be periodically collected. This is done to maintain accurate class rosters and to assess the impact of attendance on student achievement. Poor attendance may impact group activities.

If you will be absent, it is your responsibility to find out what was missed by checking D2L or contacting fellow classmates. Authorized absences (school functions or emergencies) may be made up for full credit. Assignments for non-authorized absences may be completed late for 75% credit if the assignment has not been returned to the class yet. All other make-ups receive 50% credit and must be completed within two weeks of the original due date or the last day of classes, whichever occurs first.

**Entry-level switching** The Department of Mathematics allows students within entry level mathematics courses (i.e., 010, 020, 104, 106, 108, 109, 111, 112, 113, 114, or 246) to move up to a higher numbered course during the first *two* weeks of a semester or move down during the first *three* weeks. Please contact the instructor for more details.

^{*}**Midterm grades** will be based on percentage of points completed at the time of midterm submission. For **final grades**, total points will be rounded up to the nearest whole number to determine a letter grade. Individual scores or grades will not modified because they represent a student's progress in the class throughout the semester.

The **UW-Eau Claire Liberal Education (LE) Core** curriculum serves as a strong foundation for all of our academic programs. Our LE Core embodies the Power of [AND] in its design. It has been developed to ensure that you acquire the knowledge AND skills AND responsibility that you will need to actively engage in a global society. Through meeting the requirements of the LE Core you will develop the ability to think critically, creatively and independently. You will learn to integrate and apply your knowledge and develop values essential to becoming a constructive global citizen. The outcomes will empower you and prepare you to deal with complexity, diversity, and change in multiple settings. They will also develop highly marketable skills and lead to life-long learning and civic engagement (see https://www.uwec.edu/ULEC/Liberal-Education-Framework-Learning-Outcome-and-Rubrics.htm).

**Student Accommodations** Any student who has a disability and is in need of classroom accommodations, please contact the instructor and the Services for Students with Disabilities Office in Centennial Hall 2106 to determine accommodations before contacting the instructor.

**Academic Integrity** Any academic misconduct in this course as a serious offense. The disciplinary procedures and penalties for academic misconduct are described on the UW-Eau Claire Dean of Students web site.

**Civility** As members of this class, we are members of a larger learning community where excellence is achieved through civility. Our actions affect everyone in our community. Courtesy is reciprocated and extends beyond our local setting, whether in future jobs, classes, or communities. Civility is not learned individually, it is practiced as a community.