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**EDUC 528: Cultural and Historical Significance of Mathematics**
 * The Pythagoras Group**
 * Members: Stephanie Bedard, Terry Gastauer, Michael Molinaro, and Diane Zulli**
 * May 2010**

"One of the most celebrated relationships in mathematics is the Pythagorean theorem. Why is this so much in the minds of adults, who usually remember this above all else learned in school mathematics? Could this be because we usually refer to the theorem with the first three letters of the alphabet, and it is like learning your ABCs? Whatever makes it popular, it still requires a proof for us to be able to accept it as a theorem (Posamentier, 2003, p. 170)." media type="youtube" key="CAkMUdeB06o" height="385" width="480"

Even Hollywood has heard of it: "[t]he hypotenuse even makes a cameo appearance in Gilbert and Sullivan's 1879 comic operetta, //The Pirates of Penzance// (Berlinghoff & Gouvea, 2004, p. 139)." It also appeared in the Wizard of Oz proving anyone can recite the theorem, albeit not correctly. media type="youtube" key="DUCZXn9RZ9s" height="385" width="480" What he said: "The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side ." What it is: "The sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse ."

The Pythagorean Theorem is a fundamental mathematical topic. The formula is used to find the lengths of right triangles, to compute the distance between any two points, and to determine the equation of a circle. It is the very basis for trigonometry.

Pythagoras was a charismatic figure and a genius, but he was also a good self-promoter (Mlodinow, L., 2002, p. 23). There is little evidence that he himself was interested in mathematics. It is known, however, that he was the founder of a society, a group for learning and contemplation called the Pythagorean Brotherhood (Berlinghoff, W. & Gouvea, F., 2004, p. 139).

Pythagoras and his followers, the Pythagoreans, believed that there was an order to the world and that it could be described through numbers. They further believed that exact values for calculations were possible -- that there were no irrational numbers. Ironically, the theorem bearing his name proved him wrong. Given a right isosceles triangle with legs of one unit, the hypotenuse is not a rational number. The Pythagoreans discovered the truth of that and decided to keep mum. Lengths that could not be represented as rational numbers were, to the Pythagoreans, "alogon" which means both "not a ratio" and "not to be spoken." That didn't solve the problem. According to legend, one follower, a fellow by the name of Hippasus, did not keep mum. Politics trumps truth and Hippasus, according to some stories, died within a couple of weeks in a boating accident (Mlodinow, L., 2002, p. 27).

The lesson for students (aside from politics and survival) is that the Pythagorean Theorem works for all numbers, not just the rational. As in the case of the isosceles right triangle mentioned above, many solutions are irrational numbers.

There are over 400 proofs of the Pythagorean Theorem. Several of them, chosen for cultural significance, can be viewed on the pages at left. Each proof is a visual display allowing interaction, offering cultural relevance, and occasionally, as in the sample below, something unexpected. media type="custom" key="6229225" Created using GeoGebra 3.2.40.0 GeoGebra - Dynamic Mathematics for Everyone []

**__Tasks:__** In the various activities on this site, we'll explore: 1. The theorem linked to his name (the contribution by various cultures and its almost universal appeal) 2. An exercise in which numbers work for the theorem and why (Pythagorean triples) 3. An extension of the theorem (in the Law of Cosines). 4. One of the contributions truly made by him (the Golden Ratio so common in art and architecture) Finally, after all activities (1 - 4), go back and complete the assessment (page 5 - The Least You Should Know). Post your score and any questions you have to the discussion board by Saturday night.
 * Read and interact with the entire "home" page.
 * Next, choose at least two dynamic constructions (pages 1a - 1f). Complete the dynamic worksheet associated with each and post your two proofs of the Pythagorean Theorem to the class discussion board by Saturday at midnight.
 * Lastly, by Tuesday, respond at least to the post above and below yours.
 * Lesson and activities in presentation on page 2
 * See page 3 for the concepts and page 3a for the activities
 * See page 4 for the concepts and activities.

Enjoy!