An Introduction to Non-Euclidean Geometry

This unit is designed to give students insight into non-Euclidean geometry. By doing so, they intuitively appreciate Euclidean geometry more. The students will research the formation of non-Euclidean geometry and its applications. They will then formulate a distance formula and midpoint formula using the taxicab geometry. Finally, the students will apply the knowledge to a real-life application

Understandings: • Students will understand that many types of geometries have evolved from Euclidean geometry. • Theoretical math does not always make sense in the real world. Essential Questions: • Was Geometry invented or discovered? • Do we travel in straight lines? Unit Questions: • How was non-Euclidean geometry discovered? • What are the differences in Euclidean and non-Euclidean geometry? • Does the Euclidean distance formula always make sense? • How are the distance formula and midpoint formula defined in Taxicab Geometry? Knowledge and Skills: The student should be able to: • Analyze the differences in Euclidean and non-Euclidean geometry • Formulate the distance formula and midpoint formula in taxicab geometry • Apply taxicab geometry to a real-life problem

Computer with Internet capabilities. Email account.

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Learning Activities: Activity One: After having the students complete the first task, the teacher will review the e-mail answers to the questions and then direct a discussion on non-Euclidean geometry. Students should recognize that mathematics are discovered not invented. Activity Two: The teacher will introduce taxicab geometry, a type of non-Euclidean geometry. The class will discuss its benefits versus those of Euclidean geometry, but the students should also see its downfalls. After a discussion of the principles of taxicab geometry, the class will play a treasure hunt game in which the theory of taxicab geometry is stressed (http://www.learner.org/teacherslab/math/geometry/shape/taxicab/). Activity Three: Students will begin the Geometer’s Sketchpad exercises using taxicab geometry. Once completed, they will answer questions on the handout. Activity Four: The teacher will introduce the students to the city of Mobile mapping system. The teacher will show the students how to search for property using names and addresses. The students will then begin the Internet Taxicab Activity. Activity Five: The class will lead a follow-up discussion in which we discuss the pros and cons of Euclidean geometry. The teacher will explain why we study Euclidean geometry versus non-Euclidean geometry.

online activity

see attached rubric.