## Geometry

- Course ID:MATH 307/308
- Semesters:2
- Department:Mathematics
- Course Rank:Required
- Teachers:Steve Coyne

## Description and Objectives

The Heights School

*Geometry*

Instructor: Stephen J. Coyne

Email: scoyne@heigths.edu

Phone: (301) 365-4300 (ext. 108)

** Course Description and Objectives: **Reason is the divine spark in man, and it manifests itself in two primary ways, both of which can be trained and honed through the study of geometry. It is the primary goal of this class to train your mind in both of these modes of reasoning.

- Synthetic reasoning leads to science and logic. The reasoning of lawyers, computer scientists, judges, and Euclidian geometry is primarily synthetic. This type of reasoning uses already known truths as premises to arrive at other truths as conclusions.
- Analytic reasoning leads to invention and discovery. The reasoning of engineers, detectives, and problem solving geometry is primarily analytic. This type of reasoning is able to look at a body of information and discover the underlying truths and basic principles at work.

*Texts:*

*Elements of Geometry,*by Euclid, (http://farside.ph.utexas.edu/euclid/Elements.pdf)- This is a masterpiece of synthetic reasoning and one of the greatest mathematical texts ever written. This is geometry in its purest and most perfect sense. Assigned propositions should be prepared well enough that you can demonstrate the synthetic reasoning on the board in front of the class if called upon to do so.

*Geometry*, by Jurgensen, Brown, and Jurgensen. Houghton Mifflin. ISBN 0-395-46146-4- While this text presents a mixture of synthetic and analytic geometry, we will be using it primarily for its analytic aspect. We will compare and discuss its synthetic components with Euclid, but homework and problems will focus on the analytic geometry it presents.

*Outline of Topics Covered:*

- Synthetic Geometry (Proposition Proofs from Euclid’s
*Elements*)- Book I: Plane Geometry Involving Straight-Lines (48 Props)
- Book II: Geometric Algebra (14 Props)
- Book III: Plane Geometry Involving Circles (37 Props)
- Book IV: Construction of Rectilinear Figures In and Around Circles (16 Props)
- Book V: Proportion (25 Props) (time permitting)
- Book VI: Similar Figures (33 Props) (time permitting)

- Analytic Geometry (from Houghton Mifflin’s
*Geometry*text)- Chapter 1: Points, Lines, Planes, and Angles
- Chapter 2: Difference Between Deductive and Inductive Reasoning
- Chapter 3: Parallel Lines and Planes
- Chapter 4: Congruent Triangles
- Chapter 5: Quadrilaterals
- Chapter 6: Inequalities
- Chapter 7: Similar Polygons
- Chapter 8: Right Triangles (and Trigonometry)
- Chapter 9: Circles
- Chapter 11: Areas of Plane Figures
- Chapter 12: Areas and Volumes of Solids
- Chapter 13: Coordinate Geometry (time permitting)

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*Course Requirements:*

After reviewing some prerequisites during the first quarter, we will cover the first few chapters of the Houghton Mifflin Geometry textbook. Each subsequent quarter will be divided into two primary sections: analytic geometry and synthetic geometry. We will focus on synthetic geometry by studying Euclid. After finishing up to 50 Euclidian Proofs, we will then return to analytic geometry for the rest of the 2^{nd} quarter. We will usually cover 2-4 chapters in the Houghton Mifflin *Geometry* book each quarter.

**Synthetic Geometry**– Euclidian propositions are for synthetic practice. During the synthetic portions of the class, 3 or 4 propositions from Euclid will be assigned daily to be read and summarized as two column proofs. These summaries will be turned in as homework. Each assigned proposition will also be assigned to individual students as a presentation proposition. Presentation propositions should be mastered by the individual student so that they can present the reasoning and diagram of the proposition in front of the class without any assistance during the next class. Presentation propositions will count as quiz grades.**Analytic Geometry**– Problems from the Houghton Mifflin*Geometry*textbook are for analytic practice. During the analytic portions of the class, homework will be assigned each night from the textbook and will be checked and/or collected during the following class. Analytic quizzes are for displaying knowledge and skill gained from the homework and will be given about every 2 or 3 classes. A typical quiz has 4 or more problems covering 2 topics and is graded on a 10 point scale. Analytic tests will be given at the end of each chapter. They are graded on a 100 point scale and are for displaying mastery, so there may be a few difficult extra credit questions.

*Successful Students:*

- Most people find that one type of reasoning, either synthetic or analytic, comes naturally to them, while the other type is more difficult. While this could be used as an indicator for the type of career you should pursue, by no means should you give up on training your mind to do well at the type of reasoning you find difficult. The best lawyers and computer scientists are the ones who weave analytic insights into their masterful synthetic courtroom arguments and computer programs. The best engineers and inventors are also able to convince people of their brilliant analytic solutions to problems by presenting those solutions to other people using synthetic arguments.
- Successful students should spend enough time to master their presentation propositions during the synthetic portions of the class, and should practice explaining them without reference to the text prior to their classroom presentation of the proposition. The two column homework propositions should be written clearly and neatly with an accurate diagram to accompany them.
- Successful students should always do their analytic homework assignments and make sure to ask questions about any difficult homework problems during the following class, as quiz problems are frequently drawn from the homework.
- If needed, I am available for extra help outside of class in room 16 after 5
^{th}period, during some Mass study halls (either in my office: 226, or in classroom covering a Mass Study Hall), and on some days after school when I do not have other meetings. I encourage you to contact me with any questions either by email or phone.

**Approximate Grade Calculation: 50% from test grades, 30% from quiz grades, and 20% from H.W.**

## Textbooks

The Houghton Mifflin *Geometry* textbook (copyright 1990) by Jurgensen, Brown, and Jurgensen (ISBN : 0-395-46146-4).

Euclid’s Elements of Geometry, The Greek text of J.L. Heiberg (1883-1885), edited and provided with a modern English translation, by Richard Fitzpatrick.

## Course Requirements

Prerequisites: Required: Algebra I ; Recommended: Algebra 2

## Successful Students

- Most people find that one type of reasoning, either synthetic or analytic, comes naturally to them, while the other type is more difficult. While this could be used as an indicator for the type of career you should pursue, by no means should you give up on training your mind to do well at the type of reasoning you find difficult. The best lawyers and computer scientists are the ones who weave analytic insights into their masterful synthetic courtroom arguments and computer programs. The best engineers and inventors are also able to convince people of their brilliant analytic solutions to problems by presenting those solutions to other people using synthetic arguments.
- Successful students should spend enough time to master their presentation propositions during the synthetic portions of the class, and should practice explaining them without reference to the text prior to their classroom presentation of the proposition. The two column homework propositions should be written clearly and neatly with an accurate diagram to accompany them.
- Successful students should always do their analytic homework assignments showing supporting work/calculations (and checking/fixing/correcting odd-numbered problems as they go using odd-numbered answers in the back of text) and making sure to correct/fix remaining problems in class and ask questions about any difficult homework problems, as quiz problems are frequently drawn from the homework.
- If needed, I am available for extra help outside of class in room 16 after 5
^{th}period, during some Mass study halls (either in my office: 226, or in classroom covering a Mass Study Hall), and on some days after school when I do not have other meetings. I encourage you to contact me with any questions either by email or phone.