Concepts used:

Deductive Reasoning

Geometry


Grade level:

Grades 8-12


Subject area:

Math


Standards

NCTM

- Geometry

- Reasoning & Proof

- Communication

- Connections

  1. -Representation

  2. -Technology


NETS*S

3a, 5a, 5b, 6a, 6b

The Flat Earth (Kmz file)

Objectives

  1. •Examine geometric truths for a plane and a sphere.

  2. •Follow a logical progression of geometric ideas.

  3. •Identify the basic spherical geometry terms of Great Circle, Geodesic, and Lune.

  4. •Create a progression of rules for Spherical Geometry


Lesson Description

The Spherical Geometry exercises included here will cover what some may consider Non-Euclidean Geometry or Differential Geometry.  The primary goal is to demonstrate a logical progression of geometric ideas, much as Euclid did in his Elements.  The students will start with assumed truths (postulates) and definitions, and then use those to form logical truths (theorems).  At the end of the activity, the students will be asked to write a logical progression of geometric ideas as they relate to the curved surface of sphere. 


Google Earth provides students with computer-generated models in 3-dimensional space that they can manipulate.  This is helpful because, even though we live in a 3-dimensional world, some students may find these concepts abstract, especially after years of exposure to Euclidean ideas.


Consider the image above.  Most students can dutifully recite that the sum of a triangle’s angles is 180 degrees.  But what about on a sphere?  This image shows a triangle with an angle sum of 270 degrees with two right angles on the equator and one at the pole.  Like this example, it is hoped that the included exercises will allow students to explore dimension and space in a new way.


The lesson begins with students examining geometric ideas for a plane and sphere on paper.  The Geometric Ideas worksheet should be completed by the students in pairs to encourage dialogue.  Once they have completed the worksheet, distribute the Google Earth file.  Again dialogue and debate is encourag-ed, so have the students work in pairs for this portion also.  When the students have finished in Google Earth, ask them to write a list of postulates or theorems for Spherical Geometry.  Each pair could present these to the class, and a master version could evolve.



Downloads:

The Flat Earth - Google Earth Kmz file

Geometric Ideas: Student Wkst. - Word.doc

Geometric Ideas: Teacher Wkst. - Word.doc



You may find these links helpful:

  1. •The Geometry of a Sphere by John C. Polking (link)

  2. •Postulates, Theorems, and Corollaries from Geometry from Math Homework Help website (link)

  3. •Euclid’s 13 Books of Elements  courtesy of D.E. Joyce at Clark University (link)



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