幾何学入門における興味深い問題: パート A

The purpose of these seminars is to explore the riches of geometry. The topics are designed to clarify and enhance the understanding you have of the subject. Essentially all of the familiar facts from high school geometry appear here. However, they are presented in ways that make connections to other geometric concepts and provide material for classroom use. We draw on the pictorial nature of geometry, for this is what attracts students at every level to the subject.

I draw great inspiration from the problem-based approach of distinguished mathematicians such as Moise and Downs (see their: Geometry) and Gelfand (see his: Method of Coordinates and Trigonometry). Thus, each class will consist of solving 5 to 10  interesting and/or challenging problems.  

Thus, each class will consist of solving 5 to 10  interesting or/and challenging problems. The problems are chosen so that the students will:
Develop their problem-solving and analytical skills
Understand how geometry fits with the broader structure of mathematics
Explore all the familiar facts of the subject (without memorizing algorithms)
Discover how to write proofs
Enjoy learning about geometry 
All of which will help to bridge the gap between high school and university math at an early stage. 

I have arranged the problems that we will consider in each class to match the content of the first half of a Introduction to Geometry course. These topics include: Lesson 1. Lines, Planes, and Separation, Angles and Triangles, Congruences, A Closer Look At Proof, Geometric Inequalities, Perpendicular Lines And Planes In Space, and Parallel Lines In A Plane.
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How is the class structured? 
Students will be presented with a problem. When faced with an easier problem, the student will have a limited amount of time, after which we will go over the solution and review relevant fundamentals. When faced with a particularly challenging problem, each student can work as a team or on their own. At such times, I will offer some suggestions and review the relevant fundamentals before allowing the class to complete the problem. At the end of class, we will discuss the bigger picture and establish deeper connections to Geometry.  

Students may join all of the following classes or may choose from the following topics according to their preference:

Week of September 13th: Common Sense and Exact Reasoning
Week of September 20th: Sets Real Numbers and Line
Week of September 27th: Lines, Planes, and Separation
Week of October 4th: Lines, Planes, and Separation
Week of October 11th: Angles and Triangles
Week of October 18th: Angles and Triangles
Week of October 25th: Congruences
Week of November 1st: A Closer Look At Proof
Week of November 8th: A Closer Look At Proof
Week of November 15th. Geometric Inequalities
Week of November 22nd. Perpendicular Lines And Planes In Space
Week of December 6th. Parallel Lines In A Plane