Click on the image above to play the video clip!
Summary
According to the Scarecrow in this clip from The Wizard of Oz, “in an isosceles triangle, the sum of the square roots of two sides is the square root of the third side”. While this statement is not entirely accurate, this was a fair attempt at quoting the Pythagorean theorem. The Pythagorean theorem states that in any right angle triangle, the sum of the squares on the two sides is equal to the square on the hypotenuse. The Pythagorean Theorem gives rise to the following formula:
One common mistake is the belief that this is an algebraic theorem, since it has an easy-to-use formula. This is not the case. In fact, the Pythagorean theorem is a geometric relationship (which makes sense, since the Ancient Greeks had a vested interest in Geometry). Let’s put emphasis on what makes this theorem geometric:
“In any right angle triangle, the sum of the squares ON the two sides is equal to the square ON the hypotenuse.”
Essentially, this formula is referring to shapes, and the relationship between these shapes. Namely, that when you add the areas of the two square shapes lying on the legs of the right angle triangle, you obtain the area of the square lying on the hypotenuse. The following video demonstrates this idea perfectly:
Yes! The triangle is only right if the Pythagorean theorem holds. Similarly, the Pythagorean theorem only holds if the triangle is right. It goes both ways! (In other words, “Pythagorean theorem works” and “the triangle is right” are considered equivalent statements. They mean the same thing.)
…because you will be using it in every mathematics course you will ever take! It is THAT important.
Always remember to take the square root when you are solving for the length of the hypotenuse!
Additional Resources
- Practice using the Pythagorean Theorem
http://www.regentsprep.org/regents/math/algebra/at1/pracpyth.htm
http://www.transum.org/software/SW/Starter_of_the_day/Students/pythagoras.asp
These webpages provide a number of problems involving the Pythagorean Theorem. These are interactive quizzes: as such, answers can be checked for immediate feedback.
Tip for Teachers: The website called RegentsPrep covers a wide range of math topics. It has explanatory modules as well as an interactive quiz for each topic.
- “What was up with Pythagoras?”
Pythagoras, the Greek Mathematician and Philosopher, was a rather interesting guy. In her video, Youtube sensation Vi Hart explains a little more about Pythagoras and his thinking.
- Pythagorean Theorem (Math Song)
A song that might students remember the Pythagorean theorem.
- The Best Pythagorean Theorem Rap Ever
A song to explain the Pythagorean theorem and to help students remember it. The video also provides an excellent visual representation of the theorem!
- The Simplest Way to Prove the Pythagorean Theorem
A great video for those who are interested in a simple proof of this important theorem. Let it be known, however, that there are over 200 and counting potential proofs (some being more complicated than others). This video also provides an excellent visual geometric representation of the theorem.
FOR TEACHERS:
- Lesson Plan: Why was the scarecrow wrong?
http://robertkaplinsky.com/work/wizard-of-oz/
This webpage proposes a lesson plan for students to deconstruct the scarecrow’s statement and explain why it is incorrect. It also provides the students with an opportunity to make the statement mathematically precise.
- Lesson Plans and Assessment
http://jwilson.coe.uga.edu/emt668/emat6680.f99/challen/pythagorean/
The URL above links to a page with four lessons involving the Pythagorean theorem – History, Introduction, Connections and Applications – as well as an assessment module. This would make for an exceptionally thorough unit on the Pythagorean theorem.
Math in Popular Media 