Engaging students: Deriving the Pythagorean theorem

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Belle Duran. Her topic, from Geometry: deriving the Pythagorean theorem.

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How can technology be used to effectively engage students with this topic?

Using the video in which the scarecrow from The Wizard of Oz “explains” the Pythagorean theorem, I can get the students to review what the definition of it is. Since the scarecrow’s definition was wrong, I can ask the students what was wrong with his phrasing (he said isosceles, when the Pythagorean theorem pertains to right triangles). Thus, I can ask why it only relates to right triangles, starting the proof for the Pythagorean theorem.

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What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

While Pythagoras is an important figure in the development of mathematics, little is truly known about him since he was the leader of a half religious, half scientific cult-like society who followed a code of secrecy and often presented Pythagoras as a god-like figure. These Pythagoreans believed that “number rules the universe” and thus gave numerical values to many objects and ideas; these numerical values were endowed with mystical and spiritual qualities. Numbers were an obsession for these people, so much so that they put to death a member of the cult who founded the idea of irrational numbers through finding that if we take the legs of measure 1 of an isosceles right triangle, then the hypotenuse would be equal to sqrt(2). The most interesting of all, is the manner in which Pythagoras died. It all roots back to Pythagoras’ vegetarian diet. He had a strong belief in the transmigration of souls after death, so he obliged to become a vegetarian to avoid the chance of eating a relative or a friend. However, not only did he abstain from eating meat, but also beans since he believed that humans and beans were spawned from the same source, hence the human fetal shape of the bean. In a nutshell, he refused access to the Pythagorean Brotherhood to a wealthy man who grew vengeful and thus, unleashed a mob to go after the Brotherhood. Most of the members were killed, save for a few including Pythagoras (his followers created a human bridge to help him out of a burning building). He was meters ahead from the mob, and was about to run into safety when he froze, for before him stretched a vast bean field. Refusing to trample over a single bean, his pursuers caught up and immediately ended his life.


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How has this topic appeared in the news?

Dallas Cowboys coach, Jason Garrett recently made it mandatory for his players to know the Pythagorean theorem. He wants his players to understand that “’if you’re running straight from the line of scrimmage, six yards deep…it takes you a certain amount of time…If you’re doing it from ten yards inside and running to that same six yards, that’s the hypotenuse of the right triangle’” (NBC Sports). Also, recently the Museum of Mathematics (MoMath) and about 500 participants recently proved that New York’s iconic Flatiron building is indeed a right triangle. They measured the sides of the building by first handing out glow sticks for the participants to hold from end to end, then by counting while handing out the glow sticks, MoMath was able to estimate the length of the building in terms of glow sticks.

The lengths came out to be 75^2 + 180^2 = 38,025. After showing their Pythagorean relationship, MoMath projected geometric proofs on the side of the Flatiron building.








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