Math 445 and 487 Autumn 2000

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This is a brief collection of information about Math 445 and Math 487. It will be expanded as the quarter progresses.


Purpose of 444-445

While everyone is welcome in the course, the primary purpose of the sequence Math 444-5 is to prepare students for teaching geometry in middle and high school. Thus on the one hand, the courses aim to instill a broad understanding and appreciation for geometry and on the other hand they aim to help students to acquire the mastery of geometry they will need as mathematics professionals. In addition, it is important that students experience a variety of ways of learning geometry so that they can make informed choices for methods of instruction when they become teachers. Thus the course provides experiences in working in groups, problem solving, mathematical writing, making and experimenting with physical models and manipulatives.

Plus, the required Math 487 Computer Lab provides an extensive experience using technology, mostly The Geometer's Sketchpad in 444 and also other software such as Cinderella in 445.

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Content of 445

This is a rough outline of the topics in Math 445.

Conic Sections
Conic sections will appear in many guises, as plane cross-sections of cones, as loci defined in terms of distance, and as zero sets of quadratic polynomials. These different points of view will be linked and the geometry of the curves will be explored.
Affine Geometry: Vector and Coordinate Geometry.
The approach will be synthetic (no coordinates) at the beginning, but will later include coordinates and algebraic tools. Theorems will include the famous theorems of Ceva and Menelaus and a revisiting of theorems about midpoints and medians. Applications to computer graphics will be mentioned.
Beginning Polyhedra.
The Platonic solids and derived polyhedra will be studied using physical models.
Projective Geometry.
The elements of projective geometry of points and lines will be established and then the theory of conics will be surveyed from the projective point of view.
Inversive Geometry.
Inversive geometry is the geometry of reflection in circles. The inversion transformation is used t0 construct orthogonal circles, which are interesting in themselves but especially as the building blocks for models of non-Euclidean geometry.
Non-Euclidean Geometry.
Non-euclidean geometry will be examined using the disk model and a bit of history .
Spherical Geometry.
Spherical geometry will be studied as another example of non-euclidean geometry and as an interesting geometry for its applications.

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Structure of 445

The course is organized fairly conventionally in many respects, with this work.


Classroom activities and assessments


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Math 487

This is a computer lab required of all 445 students. The two-hour labs will generally consist an exploratory geometry investigation with software such as The Geometer's Sketchpad. There may be a small amount of additional work outside the lab to finish the 487 assignment for the week. However, the 487 work will often segue into a homework activity in Math 445. It is probably most useful to think of 445 and 487 as two parts of a single course. It is misleading to think of every computer-based assignment as part of 487 just because it uses computers.

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Textbooks for 445 and 487

  1. Author:  Brannan, Esplen, and Gray
    Title:  Geometry
    Publisher:  Cambridge University Press
    (required for 445)
  2. Author: Ogilvy, Stanley
    Title: Excursions in Geometry
    Publisher: Dover
     (required for 445)
  3. Author: Sved, Marta
    Title: Journey into Geometries
    Publisher: Mathematical Association of America
     (required for 445)
  4. Author: Birkhoff, G. D., and Beatley, R.
    Title:  Basic Geometry
    Publisher:  American Math Society (Chelsea)
    (a Math 444 book useful as reference for 445)
  5. Author: James King
    Title: Geometry Through the Circle with The Geometer's Sketchpad
    Publisher: Key Curriculum Press
    ISBN:  1-55953-047-2
    (required for 445 and 487)

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Additional Materials

Students in Math 445 should bring the following materials in class each day:

A good source of geometry manipulatives is Math 'n' Stuff.

Finally, students who have home computers may find it convenient to download a Free Demo Version or purchase a copy of the student version.

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The prerequisites are listed in the catalog. The main prerequisite is Math 444 or equivalent. Also, some linear algebra will be used and some knowledge of equations of surfaces in 3-space will also be assumed.

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