The Pythegorean Theorem

Introduction

The Pythagorean theorem: The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).

In mathematics, the Pythagorean theorem—or Pythagoras' theorem—is a relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The theorem can be written as an equation relating the lengths of the sides a, b and c, often called the Pythagorean equation:[1]

a^2 + b^2 = c^2\!\,

where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.

The Pythagorean theorem is named after the Greek mathematician Pythagoras (ca. 570 BC—ca. 495 BC), who by tradition is credited with its proof,[2][3] although it is often argued that knowledge of the theorem predates him. There is evidence that Babylonian mathematicians understood the formula, although there is little surviving evidence that they used it in a mathematical framework.[4][5] Also, Mesopotamian, Indian and Chinese mathematicians have all been known for independently discovering the result, some even providing proofs of special cases.

The theorem has numerous proofs, possibly the most of any mathematical theorem. These are very diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. The theorem can be generalized in various ways, including higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, and indeed, to objects that are not triangles at all, but n-dimensional solids. The Pythagorean theorem has attracted interest outside mathematics as a symbol of mathematical abstruseness, mystique, or intellectual power; popular references in literature, plays, musicals, songs, stamps and cartoons abound.

Task

The goal of this assignment is for each student to gain knowledge about the history behind the Pythagorean Theroem and some knowledge about the many ways mathematicians have proved its exsistence.  This knowledge will be obtained through the Internet using the Websites provided. 

The student will be expected to prove the Pythagorean Theroem on the chalkboard in front of the class as well as turn in a brief written summary of the proof and some history behind it.  A worksheet compiled of sample problems using the Pythagorean Theroem will also be given to each student to solve.   

                                       

Process

To be able to complete this assignment, you must gain knowledge about the history of the Pythagorean Theroem.  Also, each of you will demonstrate proof of the Theroem in front of the class.  Don't worry, I don't expect you to invent your own proof, but in accordance with Indiana's Academic Standards you will recite a prior-known proof.  The Websites below have been provided for you to use in accomplishing your task. 

http://mathforum.org/isaac/problems/pythagthm.html http://www.arcytech.org/java/pythagoras/history.html http://jwilson.coe.uga.edu/EMT668/emt668.student.folders/HeadAngela/essay1/Pythagorean.html http://www.cut-the-knot.org/pythagoras/index.shtml http://www.hitxp.com/math/geo/euclid/051202.htm

Throughout these websites you will view dozens of interesting Proofs of the Theorem as well as different accounts of the history of the Theorem and Pythagoras himself. The varied accounts exist because historians agrue who and when knew about the Theorem before Pythagoras.         

Step by Step Approach

1)  Look at all five of the Websites listed.  Take some notes on the history of the Theorem and Pythagoras.  Look at several of the Proofs given and try to follow them step by step.

2)  Choose a Proof that you will present to the class.  If you are having trouble understanding the Proofs, I would recomend that you choose one of the following: Pythagoras's Proof, Bhaskara's 2nd Proof, or President Garfield's Proof.  These three are among the easiest to comprehend and explain.  After you have chosen one, take very good notes on the Proof.  When you present the Proof to the class, be sure you can EXPLAIN how the Proof works.  Do not simply write the Proof on the chalkboard.

3)  Based on your notes, write TWO brief paragraphs summarizing the history behind the Pythagorean Theorem.  This summary can be either hand written or typed. 

4)  Complete the worksheet given to you in class containing sample problems.   

Evaluation

 

Developing

5

Accomplished

10

Exemplary

15

Score

 

 

Written Paper

Student    provides a paper which shows little or no understanding of the Theorem.  The paper has many grammatical and mathematical errors.

Student provides a well written paper which shows some understanding of the Theorem. The paper has several gramatical and mathematical errors. 

Student provides an excellent paper which shows an exemplary understanding of the Theorem.  The paper has few gramatical and mathematical errors.   

 

 

Presentation to the Class

Student's mathematical techniques are unsatisfactory and provides little to no explanation. 

Student uses good mathematical techniques and supports them with some explanation. Student uses excellent mathematical techniques and supports them with clear, concise explanation.

 

 

Example Problems Worksheet

Student answers questions with less than 70% accuracy.

Student answers questions with 70%-89% accuracy.

Student answers questions with 90% - 100 % accuracy.

Conclusion

The Pythagorean Theorem is one of the most famous and fundamental mathematical concepts in history.  You, as a student, will use this formula countless times throughout your academic career.  Every time you see a right triangle in this course, or a unit circle in pre-calculus, or vectors in calculus, you will have the Pythagorean Theorem at your side ready to be applied to any or all of the right triangles you encounter.  

                                                                                                                                                                    
                                   

Credits

Websites:

Images:

YouTube Videos:

Teacher Page

The primary purpose of this WebQuest is to teach students how to prove the Pythagorean Theorem. The secondary purpose is for them to learn some background information on Pythagoras.It is also structured so that students may work individually or collaboratively as they learn about this famous theorem.