Unit on Spherical Trigonometry

by Shawn D. Broderick

Day 3 Lesson Plan

Lesson Plan Title:

Exploring Spherical Trigonometry

Concept / Topic To Teach:

Spherical Distances

Standards Addressed:

Principles & Standards for School Mathematics Connections Standard for Grades 9-12: “Instructional programs from prekindergarten through grade 12 should enable all students to recognize and apply mathematics in contexts outside of mathematics” (NCTM, 2000, p. 354). Principles & Standards for School Mathematics Geometry Standard for Grades 9-12: “In grades 9-12 all students should use trigonometric relationships to determine lengths and angle measures” (NCTM, 2000, p. 309). Principles & Standards for School Mathematics Communication Standard for Grades 9-12: “Instructional programs from prekindergarten through grade 12 should enable all students to communicate their mathematical thinking coherently and clearly to peers, teachers, and others” (NCTM, 2000, p. 60).

General Goal:

Students will gain insight into the formula used to determine spherical distances. Students will communicate discoveries to the class.

Specific Objectives:

1. Students will become familiar with the equation for calculating spherical distances.

2. Students will use technology to help explore various spherical distance calculations.

Required Materials:

One computer is needed for half of the groups of three students. The computer should be equipped with Excel and the spherical trigonometry calculator by James Q. Jacobs which can be downloaded from http://www.jqjacobs.net/astro/arc_form.html. Pencils and calculators are also needed.

Anticipatory Set (Lead-In):

Explain to the class that spherical trigonometry is used in a variety of ways. Today’s class will discover one important application of spherical trigonometry and a mnemonic tool that is used to help recall the relationships between angles in a spherical right triangle.

Step-By-Step Procedures:

1) Have the students form groups of three. The groups will work on the spherical distance exploration. The spherical distance groups should relocate to a computer. Each group should complete the attached worksheet.

2) Each group should designate a spokesperson to present the topic to the class. All students should participate and volunteer any discoveries that have been made.

Plan For Independent Practice:

For homework, assign the students to find the distance between two more sets of cities. Have students use spherical trigonometry to find both distances. Encourage them to find other methods and prepare them for demonstration. Have the students show their work. Encourage the students to use the Internet to assist them in any way possible.

Closure:

Each group will present the information that they have learned. Students will have an opportunity to ask questions.

Assessment Based On Objectives:

Informal assessment will be performed by the educator. Students 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 trigonometry calculator so that they can have access at home.

Extensions (For Gifted Students):

Students are encouraged to explore and discover with technology and with independent practice. They are encouraged to bring any additional information on the subject into the class for discussion.

Possible Connections To Other Subjects:

History, Physical Science, Navigation

Comments:

This lesson plan builds connections between mathematics and other topics such as navigation. By presenting the information that they have learned, students can learn the concepts on a deeper level. In their Communication Standard in the Principles & Standards for School Mathematics, NCTM states, “Students who have opportunities, encouragement, and support for speaking, writing, reading, and listening in mathematics classes reap dual benefits: they communicate to learn mathematics, and they learn to communicate mathematically” (NCTM, 2000, p. 60).

Reference:

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