Exploring the Parallel Postulate: Euclid’s 5^th Postulate Through History and Its Alternatives

Level of Students and Prerequisite Knowledge

• The lesson is designed so that any student with a sufficient understanding of Euclidean geometry can have success in understanding the concepts presented. With that said, I believe that “advanced” students will have more success as the lesson requires a “leap” from Euclidean to non-Euclidean thought (which can be very difficult given the struggle that many students have with Euclidean geometry alone).

Objectives

• The 1989 National Council of Teachers of Mathematics Curriculum and Evaluation Standards for School Mathematics state, “Students also could examine some of the history associated with attempts to prove Euclid’s famous fifth postulate from both a mathematical and a cultural perspective” (p. 180). Additionally, the Standards suggest that students “investigate properties of other geometries to see how the basic axioms and definitions lead to quite different—and often contradictory—results" (p. 180). With those in mind, this lesson has the following objectives:
• To gain an appreciation for:
• the axiomatic nature of geometry and its importance.
• the need for proof in mathematics.
• To learn that the parallel postulate cannot be proven and that changing it leads to other, non-Euclidean geometries.
• To learn some “key players” in the history of the parallel postulate .
• To evaluate the parallel postulate’s importance to geometry.
• To learn some of the basic tenants of non-Euclidean geometries.
• Others as needed.

Length of Lesson

• Two class meetings, minimum. Can be easily expanded.

Materials

• Computer with Internet access and Java capability.

Activities/Lesson

• You can present the lesson in many different ways, but if your textbook or other material has a lesson on non-Euclidean geometry, then you could use that. Otherwise, you could pose the following question: “How can geometry change if Euclid’s 5^th postulate is thrown out or reworded?” Additionally, you can have another question of your choice.
• Students will engage in a TrackStar© lesson. The Track can be accessed by the ID, 90597, or the link, http://trackstar.hprtec.org/main/display.php3?track_id=90597. Or, the links on the track are individually listed if you just want to go to one or some of the websites.
• I believe the lesson would work best in groups of 2 or 3 students, but you can do whatever fits your situation best.
• The Track navigates through 9 websites related to the parallel postulate and culminating with descriptions of Hyperbolic and Spherical geometries. I have organized the 9 sites as such (there are questions for nearly all the websites for the kids to either do some math or think about what is going on):

• Euclid’s 5^th Postulate: The actual text of the postulate and a java applet description.
• Why All the Fuss?: Site describing why mathematicians were so determined to prove the parallel postulate.
• Initial Success and Eventual Failure: Site giving information on who tried to prove the parallel postulate and that when they believed they had succeeded, it was determined that they used an equivalent to the parallel postulate.
• Those Pesky Equivalent Assumptions!: Site giving 26 equivalents to the parallel postulate itself.
• Revolution in Geometry! The Parallel Postulate is Scrapped. Or is it?: Great history of the parallel postulate and attempts to prove it. It leads up to the great revolution when the parallel postulate was shown improvable and new geometries emerged.
• New Geometry World #1: Description of Hyperbolic Geometry
• Hyperbolic Java Applet: Java Applet for students to explore Hyperbolic geometry.
• New Geometry World #2: Description of Spherical Geometry.
• Spherical Geometry Applet: Java Applet for students to explore Spherical geometry.

• To get a better idea (as I would recommend anyway), try the track yourself. While you cannot change the track itself, feel free to modify what you can to better achieve your goal. All I ask you to do is simply send me an email to greatmathmaniac@yahoo.com and list what you did differently or what suggestions you have for the track itself.
• If you use the track, email me (greatmathmaniac@yahoo.com) to tell me how it went. Include the other information (from the previous bullet as well). I would appreciate it greatly!

Assessment

· I have purposefully left out assessment for several reasons. The lesson should be assessed, but not necessarily with a quiz or test. One could easily make this into a presentation or some other type of nontraditional assessment. If I had to choose an assessment for most, I would have students give a presentation or write something to turn in.