# 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 Jillian Greene. Her topic, from Geometry: deriving the Pythagorean theorem.

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

Geometers Sketchpad is a fantastic resource to be able to more intuitively explore aspects of geometry without the approximation that often comes from using a graphing calculator or a pencil and paper. There is an exploratory activity that can either allow students to discover the Pythagorean Theorem in a different way, or just to reinforce the relationships between the sides. Have students create of a line segment AB with a length of one unit, whatever the measurement might be. Then create a right isosceles triangle using AB as the two equal sides. Now the students will build off of this triangle, making more right triangle (not necessarily isosceles) using the hypotenuse as one of the legs of the next triangle, and the other leg having the same length as AB.  Do this 6 times and find the length of final triangle’s hypotenuse. Now explain what the pattern is, and how the relationships work. The final product should look like this:

The final side should be sqrt(7), and the hypotenuses should go sqrt(2), sqrt(3), sqrt(4)…all the way up to x. Hopefully students will be fascinated by the relationship!

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

The Pythagorean Theorem was first theorized by Pythagoras, right? Wrong! There’s a very rich history that comes with this theorem that finds a relation in the sides on right triangles. Actually, there were clay tablets indicating an understanding of this theorem found in Babylonian settlements from more than 1000 years before Pythagoras. The Yale tablet, depicted below, has numbers written out in the Babylonian system that give the number “1.414212963” which is very close to √2 = 1.414213562, indicating an understanding of the 1-1-√2 relationship.

Similarly, there are relics from the Chinese and the Egyptian people having either the relationship between the legs figured out, or the existence of 3-4-5 triangles, or a “Pythagorean triple.” The Egyptians made sure their corners on their buildings were 90 degrees by using a rope with 12 evenly spaced notches to make a 3-4-5 triangle. So where does Pythagoras come in? Pythagoras was the first one to formulate a proof in regards to this theorem. So where are his proofs? Well, Pythagoras felt strongly against allowing anyone to record his teachings in any way, so there is no physical proof left behind. However, from what we know about Pythagoras, it is safe to assume that he approached it geometrically.

How could you as a teacher create an activity or project that involves your topic?

Hello Detective, thank you for coming in to help today. Scar Tellub, 24 year old male, brown hair, green eyes, was found shot early this morning. He was shot for an unknown purpose, but is luckily recovering now.  However, we are determined to find this shooter. We know from eye witness testimonies that the gunshot came from overhead, from the top of a nearby building. We know from where the bodies were found, Mr. Tellub was standing perfectly in the center of three buildings, specifically he was 9 feet away from each building. From the entry and exit of the bullet, we can tell the gun was shot from 15 feet away. We have three possible suspects that could be the culprit, but we need your mathematical prowess to help us nail the bad person.

These are the possible shooters:

1. Madison Bloodi: 19 years old, blonde hair/blue eyes, babysitter. Spotted atop the first building, Trump Tower (20 feet tall), at the time of the shooting.
2. Hunter Kilt: 34 years old, brown hair/brown eyes, landscaper. Spotted atop the second building, the Eiffel Tower (6 feet tall), at the time of the shooting.
3. Winston Payne: 26 years old, black hair/green eyes, lawyer. Spotted atop the third building, the Leaning Tower of Pisa (12 feet tall), at the time of the shooting.

Again, thank you for your time, Detective. We know full well that you won’t let us down. Please draw us a photo and show us your work for all three suspects so we can provide them to the judge. Happy mathing!

References:

http://jwilson.coe.uga.edu/emt668/emat6680.f99/challen/pythagorean/lesson4/lesson4.html

http://www.ualr.edu/lasmoller/pythag.html

(I did a similar activity to the murder one with students before, but I cannot find it online again, so I wrote a new one kind of similar to what I remember)

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