Area: Final Report
It is now time to organize the experiences and observations of the past few weeks
into a final report. The form the report should take is described below; this
is essentially the standard definition-axiom-theorem presentation of the material
that is used in most mathematical texts with a couple of variations. The first
variation is that the report includes a section on areas for future investigation
and apparent gaps in the arguments. The second is that the report includes a section
on how well (or not) your group worked together. This is an important section.
Everyone in the class will be working in groups for a large portion of the course
and the success of the course depends upon each class member being able to work
effectively in a group.
Mathematical Content
The mathematical component of the report should be divided into two sections.
The first should deal with the planar area axioms and the second should deal with
finding the area of regions on a sphere.
I. Axioms of area for planar regions
The first part of the report should be based on the following lists of primitive
terms and axioms:
Undefined terms
point, line, line segment, interior of triangle, polygonal region, real number,
length of a line segment, perpendicular
Axioms for Area
- The area of a polygonal region can be described by a nonnegative real number
and each polygonal region has exactly one area.
- The area of the union of a finite collection of nonoverlapping triangular
regions is equal to the sum of the areas of the triangular regions.
- The area of a trianglular region is equal to one half the product of the
length of a base and its corresponding height.
- Any polygonal region can be partitioned into nonoverlapping triangular
regions.
In your report, answer the following questions.
- Using the above list of undefined terms, define the following terms: triangle,
square, rectangle, trapezoid, parallelogram, base of a triangle, height of
a triangle(corresponding to a given base), nonoverlapping.
- The formulas for the area of triangles, rectangles, parallelograms, trapezoids
and their proofs. The formulas should be stated as theorems with the proofs
immediately following their statement. Include any other theorems and their
proofs (or even conjectures) observed during the time spent on the project.
- Topics that need further work, gaps in proofs, and so forth.
II. Area on the sphere.
The second part of the report should include the following:
- A discussion of the validity of each of the planar axioms for area on the
sphere. Include any axioms that you need to add to compute area on a sphere.
(You may assume axiom 1 holds on a sphere.)
- Derive a formula for the area of triangle on the sphere.
- Topics that need further work, gaps in proofs, and so forth.
Group Process
As you entered into work with this group, were you concerned about any particular
problems that might arise? Did these problems arise? Were there any difficulties
that arose that you had not anticipated?
Format or Style
The report should be either typed or written in clear legible handwriting and
written with enough detail that a fellow student could follow the discussion.
Of course, the spelling and grammar should be correct. Any figures that you
use should be both clearly drawn and labeled.
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