Axiom-Geom-S21
Axiomatic Geometry & Advanced Euclidean Geometry
MAT345 and MAT630 Spring 2021
Welcome to Axiomatic Geometry.
This is a course for future math teachers,
so we will be learning how to write proofs,
how to present proofs, and how to correct proofs.
Course Description: Geometric theory from an axiomatic viewpoint motivated by Euclidean geometries (plane= E2, solid= E3) and additional non-Euclidean examples (Hyperbolic=H2 and Spherical=S2) . Emphasis on the relationship between proof and intuition. This course is for future math teachers and is not recommended for other math majors.
Prerequisites: Linear Algebra and Calculus
Meeting Times: Online Asynchronous but must complete each week’s lesson lessons on time. The course starts on Feb 1 and you will find the lessons linked to within the schedule below.
Significant Deductions for Late Work even with a good excuse because late work hurts your teammates.
Professor: C. Sormani
Office Hours: Email questions in an email with subject line MAT345 Question and the professor will respond within 48 hours. Questions about homework must include a photo of the statement of the problem and your initial attempt. Email:sormanic (at) gmail.com
Professor's Webpage: https://sites.google.com/site/professorsormani/home
Resources: no text
Euclid's Elements, online
Projects which will be distributed via email
Wikipedia can be a useful resource for mathematics.
Engage NY website with complete info about the NYS Geometry Course
For those who like to buy textbooks: College Geometry: A discovery approach, by David C. Kay, 2nd Edition. ISBN-13: 978-0321046246 ISBN-10: 0321046242 (HW is not assigned from textbook. Only a few pages may be used from this textbook and those will be distributed.) Appendix F has a list of axioms for quick reference which will be distributed to the class.
Supplies:
Compasses, Ruler, Protractor, Rubber Bands, Blue Handball, Graph Paper Spiral Notebook
A tablet or whiteboard or some way to make a presentation of your homework to submit as a private video via youtube. Lehman College has a tablet and laptop loan program.
A gmail address you are willing to share with classmates and use to access team work on googledocs.
Homework Projects: will be completed in preassigned rotating teams of four-five students. All proofs must be written in two column format in black with a series of well labelled diagrams in black. Photos or screenshots of the work will be put together in a common googledoc following this template with the name GeomS21-ProjX-TeamY where X is the number of the project and Y is the number of the team. All students should try all the proofs alone or together in pairs and upload them into their teams document on Sunday. Then switch so that proofs can be checked by Tuesday. Corrections should be made neatly in red directly on the photo or typed in red below it. Below each proof the names of the students who completed the proof need to listed as well as the names of the students who checked the proof. If there are two versions of a proof and the team cannot agree on a correct proof or believe they have two correct proofs, both proofs may be submitted. Students will be graded both on their success at writing proofs and on checking them. My comments and corrections will always be in purple. No late work will be accepted. Unfinished work will be accepted. If a proof is not done or you are unsure, please write this clearly on the top of the proof. It is important to admit when you are unsure as a teacher.
Four Presentations: Each student will create four 10-15 minute video presentations presenting a proof from one of the projects. The presentations will be uploaded to youtube and shared with classmates. Students may choose the technology of their choice but must present a proof in two column format along with a diagram that is drawn as the proof progresses using the techniques taught in this course. It is best to take a video of yourself at a whiteboard or a screenshot recording writing on a tablet rather than typing the presentation up. This is meant to simulate spontaneous in person teaching. It should not take the student more than 30 minutes to create the video.
Oral Final Exam: Each student will meet with the professor in a video chat to verbally answer random questions from the course. Students should imagine they are teachers answering students’ questions.
Accommodating Disabilities: Lehman College is committed to providing access to all programs and curricula to all students. Students with disabilities who may need classroom accommodations are encouraged to register with the Office of Student Disability Services. For more info, please contact the Office of Student Disability Services, Shuster Hall, Room 238, phone number, 718-960-8441.
Accommodating Holidays: If you have a holiday during a lesson or extra lesson, let me know, and something will be arranged for you.
Names/Gender: We will use last surnames in this class. You may call me Sormani or Professor.
Respect: All students will treat each other with respect and dignity. Let me know if you have concerns.
Grading Policy: The 14 projects are worth 5% each, the 4 presentations are worth 5% each, and the final exam is worth 10%.
Course Objectives:
1. Prove theorems about open sets, unions and intersections in metric spaces (E, F & G)
2. Prove the Euclidean Geometry Theorems for similar, congruent and right triangles (A, E & F)
3. Prove statements about parallelograms, circles, and the coordinate plane (A, B, E, F & G)
4. Identify and describe the main properties of hyperbolic and spherical geometry. (E)
5. Prove theorems about symmetries and transformations. (B, E, F & G)
6. Identify the properties of solid Euclidean Geometry. (E)
Homework assignments and projects are available below on the course webpage. All can be accessed on a phone or tablet with the google docs app and the youtube app.
Schedule:
The course begins on Feb 1. Email me from a gmail address that you are willing to share with the class by Feb 1 and introduce yourself in a way that you are willing to share with the class. Why have you decided to be a teacher? Where did you go to high school? Did you take the NY State Geometry and Algebra 2 Regents Exam? Do you have experience teaching or tutoring? Have you checked in with the Department of Middle and High School Education about certification requirements? What dates and times would you like to work with teammates? Do you have all the supplies?
Week 1: Proofs, Set Theory, and Metric Spaces
Lecture Videos and Notes to watch and read Feb 1-4
Project 1 due Sun Feb 7 (no teammates for this one)
Week 2: Incidence Axioms of Euclidean, Hyperbolic, and Spherical Geometry
Lecture Videos and Notes to watch and read Feb 8-11
Project 2 due Sun Feb 14 (no teammates for this one)
Week 3: Rulers, Compasses, and Proof by Contradiction
Lecture Videos and Notes to watch and read Feb 15-17
Project 3 due Sun Feb 21 extended to Feb 28
Week 4: Angles, Perpendicular Lines, Convex Sets, and Half Planes
Lecture Videos and Notes to watch and read Feb 22-24
Project 4 due Sun Mar 7
Videos based on Weeks 2-3 due Thursday Mar 4
Week 5: Side Angle Side Hypothesis holds in Euclidean, Hyperbolic, and Spherical Geometry
Lecture Videos and Notes to watch and read Mar 1-3
Project 5 due Sun Mar 14
Week 6: Congruent Triangles
Lecture Videos and Notes to watch and read Mar 8-11
Project 6 due Sun Mar 21 (no team)
Videos based on Weeks 4-5 due on Thursday Mar 18
Week 7: Incenter, Circumcenter, and Orthocenter in Euclidean, Hyperbolic, and Spherical Spaces
Lecture Videos and Notes to watch and read Mar 15-18
Project 7 due Sun Mar 28 (no team)
Week 8: Parallel Postulate
Lecture Videos and Notes to watch and read Mar 22-24
Spring Recess March 27- April 4
Project 8 due Tuesday Apr 6 or later (no team)
Week 9: Similar Triangles in Euclidean Geometry
Lecture Videos and Notes to watch and read Apr 5-7
Project 9 due Sun Apr 11 (with teams)
Videos based on Weeks 6-8 due on Thursday Apr 15
Please see this page for info about the final so you can start to prepare.
Week 10: Pythagorean Theorem and Coordinate Plane for Euclidean Geometry
Lecture Videos and Notes to watch and read Apr 12-14
Project 10 due Sun Apr 18 (no teams)
Week 11: Symmetries and Isometries
Lecture Videos and Notes to watch and read Apr 19-21
Project 11 due Sun Apr 25
Week 12: Circles in Euclidean, Hyperbolic, and Spherical Geometry
Lecture Videos and Notes to watch and read Apr 26-28
Project 12 due Sun May 2
Videos based on Weeks 9-11 due Thursday Apr 29
Week 13: Review
Lecture Videos and Notes to watch and read May 3-5
Project 13 due Sun May 9
Week 14: Areas and Volumes in Euclidean Geometry
Lecture Videos and Notes to watch and read May 10-12
Project 14 due Th May 20 (no teammates)
Finals Week: Oral Finals will be scheduled individually with students. May 17-19 or May 24.
Please see this page for info about the final. Topics from Project 14 are not on the final.
You may schedule your final as soon as you have submitted Project 13.