Unit on Spherical Trigonometry

by Shawn D. Broderick

Day 1 Lesson Plan

Lesson Plan Title:

Introduction to Spherical Geometry

Concept / Topic To Teach:

Exploration of Spherical Geometry

Standards Addressed:

Principles & Standards for School Mathematics Geometry Standard for Grades 9-12: “Analyze characteristics and properties of two-and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships” (NCTM, 2000, p. 309).

General Goal:

Students will be able to use the Spherical Easel technology tool to form different geometric objects on the surface of a sphere. Through exploration, students will gain knowledge of the properties of some objects in spherical geometry. They will be able to compare and contrast the two-dimensional objects to the three-dimensional objects.

Specific Objectives:

1. To recall that a sphere is a three-dimensional surface in which all points are equidistant from a fixed point.

2. To construct a spherical point and to determine that a spherical point is analogous to a point in Euclidean geometry. Each spherical point determines an opposite point on the sphere.

3. To construct a great circle and to determine that it is the simplest line in spherical geometry. It is analogous to a line in planar geometry.

4. To determine that parallel great circles do not exist.

5. To construct a two sided polygon on the sphere and to determine that although a two sided polygon does not exist in planar geometry, it does exist in spherical geometry.

6. To construct a spherical triangle and to notice the differences in the properties of planar and spherical triangles.

7. To utilize the spherical easel tool to make other explorations and discoveries.

Required Materials:

One computer is needed for each group of three students. The computer should be equipped with “Spherical Easel” which can be downloaded from http://merganser.math.gvsu.edu/easel/download.html. Pencils are also needed.

Anticipatory Set (Lead-In):

Let students watch the video “Assignment Discovery: Spherical Geometry” by the Discovery Channel (1 min 15 sec.) which can be found at the following address: http://videos.howstuffworks.com/discovery/28013-assignment-discovery-spherical-geometry-video.htm.

Open the class with a discussion. Ask the following questions:

1) What are the limits of planar geometry? Possible answers: lines are straight, surfaces are flat

2) If we want to travel by air, what type of geometry do you think we should use? Possible answers: non-Euclidean geometry, spherical geometry Since the earth is a sphere, there is a branch of geometry known as spherical geometry.

3) When do you think that spherical geometry was discovered? Theodosius of Bithynia wrote Sphaerics as a supplement to Euclid’s Elements in 90 B.C.

4) What are some of the applications of spherical geometry? Possible answers: navigation, astronomy, GPS

Step-By-Step Procedures:

1) Have the students form groups of three. Each group should relocate to a computer and complete the attached worksheet.

2) Tell the students that the whole class will discuss the answers to the worksheet and each group will present any discoveries that they have found. Each group will participate in discussing answers.

Plan For Independent Practice:

For homework, let students write half a page of information about spherical geometry. They may consult any sources of information including the Internet.

Closure:

Using a whole class discussion format, students will discuss the answers to questions 1-10 on the worksheet. Then, each group will present their discoveries from question 11.

Assessment Based On Objectives:

Informal assessment will be performed by the educator. The educator will assess groups on the answers to the worksheet questions and the discoveries provided. Also, individuals will be assessed on participation in groups and whole class discussions.

Adaptations (For Students With Learning Disabilities):

Provide more time for the students with learning disabilities to complete the worksheet and to make discoveries. Provide all students with the Internet address for the Spherical Easel application so that they can have access at home.

Extensions (For Gifted Students):

Students are encouraged to explore and discover with Spherical Easel and with independent practice.

Possible Connections To Other Subjects:

History, Physical Science

Comments:

This lesson plan uses technology as a tool to encourage independent exploration and discovery of mathematics. NCTM advocates the use of technology in teaching mathematics in the Technology Principle. “Technology enriches the range and quality of investigations by providing a means of viewing mathematical ideas from multiple perspectives” (NCTM, 2000, p. 25). By including information from history and examples using the world as the sphere, important connections are made between the mathematics and real world concepts. The lesson also involves communication both verbally and written. The importance of this concept is stressed in NCTM’s communication standard. “To be prepared for the future, high school students must be able to exchange mathematical ideas effectively with others” (NCTM, 2000, p. 348). Cooperative learning is used in this lesson to help students learn from each other as well as to learn how to communicate effectively.

Reference:

National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston, VA: NCTM.