# 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 Michelle McKay. Her topic, from Geometry: deriving the Pythagorean Theorem.

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

Below I have attached an activity that I like to call “Being Pythagoras for a Day”. To summarize the activity, students are given instructions (with a few guiding images) that leads them to physically manipulate various shapes that demonstrate the relationship between the sides of a right triangle. By the instructions, students will derive the Pythagorean Theorem on their own and come to understand why each side in the equation is squared. Let it be noted that the title of this activity is not just a gimmick. The proof the students will work on in this activity is the same as the one Pythagoras was given credit for using.

1. How has this topic appeared in the news?

Not even a year ago to this day, Coach Jason Garrett of the Dallas Cowboys made a splash in the world of sports and math with his unusual demands of his players: they needed to have a sound understanding of Geometry, including the Pythagorean Theorem. Garrett fully believes that players must understand the Pythagorean Theorem to make better decisions out on the field. The following quote was taken from an interview where Garrett discusses why he feels being familiar with the Pythagorean Theorem can prevent a poor decision:

“If you’re running straight from the line of scrimmage, six yards deep, that’s a certain depth, right? It takes you a certain amount of time. But if you’re doing it from 10 yards inside and running to that same six yards, that’s the hypotenuse of that right triangle. It’s longer, right? So they have to understand that, that it takes longer to do that. That’s an important thing. Quarterbacks need to understand that, too. If you’re running a route from here to get to that spot, it’s going to be a little longer, you might need to be a little fuller in your drop.”

Let this be a wakeup call for everyone who wants to become a professional football player and never thought they would have to use the Pythagorean Theorem outside of high school!

What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?
People can easily recognize the Egyptian pyramids as one of the wonders of the world. What is not often discussed is how the engineers and architects of the day used the Pythagorean Theorem to lay the pyramids’ foundations correctly. Those primarily responsible for the pyramids’ construction were called “rope-stretchers”. This name came from the inventive method of tying thirteen, evenly spaced knots into a rope. When the rope was pegged to the ground, a 3-4-5 triangle was produced. This allowed them to accurately and consistently map out the bases of the pyramids.

Some argue that the rope-stretchers fully understood the Pythagorean Theorem and used that knowledge to manipulate the ropes, while others argue that they were intuitively using the properties of a right triangle. Due to this area of ambiguity, it is unclear whether Pythagoras was taught the theorem by the Egyptians first, or if, through watching the process, he was able to discover the relationship of a right triangle’s sides on his own.

Interestingly enough, there exist various pieces of artwork depicting Egyptians holding ropes and using them for measurement. Just by looking at the images, it is not clear if the ropes are being used for the construction of the pyramids or for dividing land (another event where the knotted ropes were used to fairly distribute plots of land).

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