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 again comes from my former student Maranda Edmonson. Her topic, from Geometry: deriving the Pythagorean theorem.

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D. History: What are the contributions of various cultures to this topic?

Legend has it that Pythagoras was so happy about the discovery of his most famous theorem that he offered a sacrifice of oxen. His theorem states that “the area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides.” It is likely, though, that the ancient Babylonians and Egyptians knew the result much earlier than Pythagoras, but it is uncertain how they originally demonstrated the proof. As for the Greeks, it is likely that methods similar to Euclid’s Elements were used. Also, though there are many proofs of the Pythagorean Theorem, one came from the contemporary Chinese civilization found in the Arithmetic Classic of the Gnoman and the Circular Paths of Heaven, a Chinese text containing formal mathematical theories.


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E. Technology: How can technology be used to effectively engage students with this topic?

The following link is for a video that not only engages students from the very beginning by playing the Mission: Impossible theme and giving students a mission – “should they choose to accept it” – but that has great information. It begins with a short engagement, as stated before, and goes into a little bit of history about Pythagoras and the Pythagoreans. It then briefly describes what the Pythagorean Theorem is before the commentator says, “Does it have applications in our lives today?” At this point (2:43 in the video), it would be beneficial to stop the video and let students discuss where they could use the theorem. The rest of the video simply shows some examples of how the Pythagorean Theorem is used on sailboats, inclined planes, and televisions. It would be up to the teacher whether or not to show the last five minutes of the video to show students these examples, but they could take notes on these examples as they are worked out on the screen.


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B. Applications: How can this topic be used in your students’ future courses in mathematics or science?

After students learn the Pythagorean Theorem in their Geometry classes, they will use it throughout their mathematical careers. They will use it specifically in Pre-Calculus when they are learning about the unit circle. The theorem is fundamental to proving the basic identities in Trigonometry. It is also used in some of the trigonometric identities, aptly named the Pythagorean Identities based on the nature of their derivation.

In Physics, the kinetic energy of an object is

\displaystyle \frac{1}{2} (\hbox{mass})(\hbox{velocity})^2.

But, in terms of energy, energy at 500 mph = energy at 300 mph + energy at 400 mph. This equation means that, with the energy used to accelerate something at 500 mph, two other objects could use that same energy to be accelerated to 300 mph and 400 mph. Looks like a Pythagorean triple, right? The theorem is also used in Computer Science with processing time. Other examples are found in the link below.


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