As I continue to think about my new job in September and plan for what I want to hang in my classroom, I am exploring the Pythagorean Theorem. The Pythagorean Theorem is probably one of the most well-known or well-remembered theorem in math. It is often taught in both algebra and geometry. In algebra it lends to working with exponents and roots and in geometry with triangles. I have seen memes saying how people did not use the Pythagorean theorem today, but I have also been told by many people that they have used it in their lives from building a new deck and woodwork to programming and more. Although math has real life applications and was mostly discovered to explain the world, much of math is taught to help develop the brain of our children. In high school the brain is just beginning to truly develop its logic skills and math is huge in helping with this. The Pythagorean theorem also is mentioned (incorrectly) in the Wizard of Oz. Yes, it is this famous!
Although it is named for the Greek mathematician, Pythagoras, it was known throughout the world before his time. It is referenced in Ancient Egypt and Babylon (around 1900 BC). Apparently, it did not become as well known until Pythagoras stated it. There are many proofs of this theorem and some of them like the one below is a visual proof.
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| Pythagorean Theorem Proof by AmericanXplorer13 at English Wikipedia, CC BY-SA 3.0, via Wikimedia Commons |
Now Pythagoras was a teacher and philosopher in the 6th century BC. He had a group of followers known as the Pythagoreans. The Pythagorean were a secretive group and many of their discoveries were stated under the name of Pythagoras and not an individual. In Mathematical Scandals, Theoni Pappas shares the story of the death or at least expulsion from the Pythagoreans of Hippasus of Metapontum. It is believed by some that he was pushed overboard or put to death at sea for his discovery (and proof) of the square root of 2 as the length of the diagonal of a square with side length of 1 and that the square root of 2 is not a rational number. The Pythagoreans believed all numbers were whole or could be written as a ratio of whole numbers. It was scandalous that there was an irrational number. For more about the Pythagoreans check out here.
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| Proof of the Square Root of 2 by Stephan Kulla (Stephan Kulla), CC BY 3.0, via Wikimedia Commons |
Sources:
- Pappas, Theoni. Mathematical Scandals. Wide World Publishing. 1997.
- UCLA. "Right Triangles - Pythagorean Theorem." http://web.cs.ucla.edu/~klinger/dorene/math1.htm
Mathematical Scandals is a fun book to add some scandalous history to your math classes! There are many fun stories that relate to different areas of math. Now I have been focusing on mathematical art and things I can make for my classroom as well as projects I can have my students make. In my search for mathematical art, I discovered the Spiral (or Wheel) of Theodorus. It is also called the Square Root Spiral and Pythagorean Spiral. It gives a visual of the square roots in numerical order.
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| My drawing of the Spiral of Theodorus 1 unit = 1 cm with Square Roots Labeled |
It is actually pretty easy to make on your own and involves some great Pythagorean theorem use!! I made this project sheet for kids to calculate the square roots using the Pythagorean theorem and then to create their own spiral by starting with a different sized triangle rather than the legs being 1 unit.
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| My Drawing of the Spiral of Theodorus 1 unit - 1 inch |
I however decided to make mine as true spirals of Theodorus. To create it you start with an isosceles triangle with legs of one unit. The hypotenuse of the triangle will be the square root of 2. Using the hypotenuse as one leg of the next triangle you draw a second leg perpendicular to the first that is one unit long. Then continue this as you add triangles to your spiral.
I found a clear ruler eased this since you could use the ruler lines to get perpendicular lines without a protractor.
I drew out several of the Spiral of Theodorus and tried different coloring techniques to make it beautiful. (By the way be sure to check out the images when you google Spiral of Theodorus there are some great student projects that use the spiral as part of drawings!) You can also check out this website for a project drawing it and calculating the hypotenuses.
While drawing it I played with whether to show the full lines of the overlapping pieces or not. I liked the idea of it but did not like how it looked, so in the one above I did not. (In the one colored with pencil, I did.) I had pulled out my fancy papers for inspiration for some of these projects and came up with the idea to make one out of a sheet of paper with every other piece being black. I needed some black poster board and didn't feel like running out and I came up with the idea of painting a canvas black and using it as the background for my new idea. I painted a canvas that I had painted previously but didn't want anymore. I picked a flowered piece of paper (actually was the "picture" of a Paper Source calendar for 2017). I put one of my hand drawn spirals on top of it and taped it down and started cutting it out. When I could I kept the pieces connected so there would be less putting it back together like a puzzle. Then I glued it onto my black canvas. My idea was to outline it in silver. It took a couple of tries to find the right silver pen to do this. I am very happy with how it turned out. I think I will put a coat of Mod Podge on it to seal the paper though.
So this art is literally recycled. I reused a canvas and used paper from a calendar. Plus it is pretty and will look great in my classroom!
Now around this time I got my hands on a copy of Mathematical Quilts: No Sewing Required by Diana Venters and Elaine Krajenke Ellison. This is an old book that is no longer published but a great extension book for math teachers. It has an entire section on Right Triangle Quilts but also has a section titled The Spiral Quilts which includes the Wheel of Theodorus. It includes several activities involving the Wheel of Theodorus as well as designing your own Wheel of Theodorus Quilt. I should add it has a chapter on the golden ratio as well and has several Fibonacci quilt projects including the Fibonacci Spiral "Quilt" like my last mathematical art project.
In The Right Triangle Quilts chapter, there are different activities that include working with a proof of the Pythagorean theorem, Pythagorean triples, relationship with the Fibonacci Sequence and more. Then there are three right triangle "quilt" projects. I decided to try the Pythagorean Triples Quilt. Pythagorean triples are three whole numbers that create the sides of a right triangle. The first being 3, 4, 5. This is probably the most well-known one.
I found this project on-line that helps ask more questions for the kids and combines both "quilts". I found the table for the Spiraling Pythagorean Triples very useful! I love that it has kids look for patterns in the Pythagorean triples used in the quilt. Now my Pythagorean Triple "Quilt" I used 3/4", 1", 1 1/4" for my first triangle or my scale is 1/4" : 1". I literally divided all the lengths of the triples by 4 inches. With my scale, my largest triangle has a leg that is 28 inches! I used poster board for the black, blue and purple since the triangles were too long for the paper I had. (The Dollar Tree sells colored poster board for 89 cents!)
Now I have some beautiful mathematical artwork to hang in my room and I am definitely going to have my students try some of these projects!
More Pythagorean Theorem Ideas:
Looking for more to do with the Pythagorean Theorem? Check out these ideas:
Fun Pythagorean Theorem Activities and Lessons--I love the water proof!
Hands-On Explanations of the Pythagorean Theorem
8th Grade Lesson Plan for Pythagorean Theorem
