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 Julie Thompson. Her topic, from Geometry: deriving the Pythagorean theorem.

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

I believe the best way to convince students that a certain theorem is true is to model it visually. Luckily, the Pythagorean Theorem has several ways to derive it and show that it works. My favorite is showing it with squares. You ask students to consider the numbers 3, 4, and 5. Given paper, ask them to create three squares with each of those dimensions. Then, see if they can form a right triangle out of the three squares they made. Next, ask them if they can find a way to make two squares fit exactly into another square (cutting the squares up if necessary). Hopefully, they will get the squares with dimensions 3 and 4 to fit into the biggest square. Finally, ask them to write a conjecture about what they find. It turns out that the two smaller squares fit perfectly into the bigger square, or, more mathematically, 32+42=52. Generally, a2+b2=c2

I did the activity myself and it is pictured below:

 

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How does this topic extend what your students should have learned in previous courses?

Students learn how to derive the Pythagorean Theorem in Geometry. However, they should have prior knowledge on square numbers, finding the area of a square, and simple algebraic equations. Students should also be able to solve equations and evaluate expressions when given values for the variables. The students will then be able to use all of this prior knowledge and apply it to one fantastic theorem: The Pythagorean Theorem. They can then use the theorem to find missing side lengths of a triangle. This extends their prior knowledge because they are now using their mathematical skills and applying it to the real world.

An example of this extension would be assigning this problem to my students: Think about your rectangular room at home. We want to estimate the length of the diagonal from corner to corner. Estimate that length to 3 decimal places. Then create a model to show why it is true, using the area of squares proof (from my A2 activity). The students are using their prior knowledge of square numbers, area of squares, and solving equations for this problem.

 

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What interesting things can you say about the people who contributed to the discovery?

Pythagoras contributed greatly to the discovery of the Pythagorean Theorem (clearly it is named after him). ”It is sometimes claimed that we owe pure mathematics to Pythagoras, and he is often called the first “true” mathematician.” We think of him as having been a very logical man, but he had some very weird, illogical beliefs as well. According to the article, “Pythagoras imposed odd rules on all the members of his school (including bizarre and apparently random edicts about never urinating towards the sun, never marrying a woman who wears gold jewelry, never passing an ass lying in the street, never eating or even touching black fava beans, etc.”

The Pythagoreans (Pythagoras and his followers) discovered something very interesting about the number 10. Today, when we wonder why we use base 10, we attribute it to the simple fact that we have ten fingers and ten toes. Our ten fingers are what we use to count with. Pythagoras deemed 10 to be a very special number, but for a more abstract reason. You can form an equilateral triangle with rows of 4, 3, 2 and 1. Altogether this triangle contains 10 points. He called it the tetractys. And 10 was thought to be a very holy number. Of course, he is most known for this theorem. “it has become one of the best-known of all mathematical theorems, and as many as 400 different proofs now exist, some geometrical, some algebraic, some involving advanced differential equations, etc.”

REFERENCES:

https://www.storyofmathematics.com/greek_pythagoras.html

 

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