G.1.2.3- The Pythagorean Theorem

Historical Account:

Pythagoras, the namesake and supposed discoverer of the Pythagorean Theorem, was born on the Greek island of Samos in the early in the late 6th century. Not much is known about his early years of life, however, we do know that Pythagoras traveled through Egypt in the attempt to learn more about mathematics.

Besides his famous theorem, Pythagoras gained fame for founding a group, the Brotherhood of Pythagoreans, which was dedicated solely to study of mathematics and worship of numbers. Pythagoras passed on his belied that numbers are in fact the true "rulers of the universe".

While studying in Egypt, it is believed that Pythagoras studied with people known as the "rope-stretchers", the same people who engineered the pyramids. By using a special form of a rope tied in a circle with 12 evenly spaced knots, they discovered that if the rope was pegged to the ground in the dimensions of 3-4-5, the rope would create a right triangle. The rope stretchers used this principle to help accurately lay the foundations of for their buildings.

It was this fascination with the rope stretchers 3-4-5 triangle that ultimately led to the discovery of the Pythagorean theorem. While experimenting with this concept by drawing in the sand, Pythagoras found that if a square is drawn from each side of the 3-4-5 triangle, the area of the two smaller squares could be added together and equal the area of the large square.



3Ð'І+4Ð'І=5Ð'І

9 + 16 = 25

(image of proof barrowed from http://www.themathlab.com/pythagor.htm)

When he examined right triangles with different measurements he found that the this theory still held true. He revealed this idea to his followers and the brotherhood assigned the general terms of a & b for the shorter legs and c for the longer side which they gave the name "hypotenuse". Thus we have the what is now know as the Pythagorean Theorem: aÐ'І + bÐ'І = cÐ'І .

Although he is credited with the discovery of the famous theorem, it is not possible to tell if Pythagoras is the actual author. The Pythagoreans wrote many geometric proofs, but because they attempted to keep their findings so secretive is is hard to determined which brother authored which proof. Each member of the Brotherhood made a vow of secrecy which in turn prevented any of their important mathematical findings from being made known to the public.

Liu Hui, a famous Chinese Mathematician who lived during the 3rd Century, offered a different proof of the Pythagorean theorem. In his method, Hui cut up the squares on the legs of the right triangle and rearranging the pieces in what is known as "Tangram style". The pieces of the smaller squares fit together to make up the square largest square on the hypotenuse. Unfortunately, Liu Hui's original drawing of this concept did not survive, however, mathematicians have been able to recreate this "Tangram" proof of the Pythagorean theorem. Below is Encyclopedia Britannica's reconstruction of Liu Hui's method:



(image of proof barrowed from http://concise.britannica.com/ebc/art-60479)

Euclid was a Greek mathematician best known for writing The Elements. The Elements greatly influenced the development of Western mathematics and is the most successful mathematics book in history. Euclid was able to prove the converse of Pythagorean's theorem explaining that:

"For any three positive numbers a, b, and c such that a2 + b2 = c2, there exists a triangle with sides a, b and c, and every such triangle has a right angle between the sides of lengths a and b. Let ABC be a triangle with side lengths a, b, and c, with a2 + b2 = c2. We need to prove that the angle between the a and b sides is a right angle. We construct another triangle with a right angle between sides of lengths a and b. By the Pythagorean theorem, it follows that the hypotenuse of this triangle also has length c. Since both triangles have the same side lengths a, b and c, they are congruent, and so they must have the same angles. Therefore, the angle between the side of lengths a and b in our original triangle is a right angle."

(proof taken from: http://www.themathpage.com/aBookI/propI-47.htm)

Discussion K-8 Development:

The Michigan Department of Education provides a list of mathematical concepts that must be covered in the curriculum in order to enable students to graduate high school. The state's list of these requirements is provided for students, parents, and educators in order to establish clear expectations of what ideas and concepts should be learned while completing math coursework.

As with any subject matter, the material learned and mastered by students builds upon itself. It is necessary for elementary students to gain mastery with certain aspects of the curriculum in order to ensure continued success in upper-level classes. During grades K-5, students should be studying geometric figures, such as the triangle and the square, that are necessary in order to grasp the concept of the Pythagorean Theorem. In addition students should be able to assess similarities and differences in these shapes. They should be able "quantify" these figures, learning how to accurately measure lengths and angles, area and volume. Quantification, such as the area of a square, becomes essential in understanding Pythagoras's original proof. After mastery of these primary topic areas, 6-8 graders will be introduced to concepts that build off of these basics and will begin to study geometric concepts such as symmetry, similarity, congruence, and the Pythagorean Theorem as discussed in this paper.

K-4 Activity:

It would be quite unreasonable to introduce the complex idea of Pythagorean's Theorem to a kindergarten class. Without knowledge of triangle classification, squares, and radicals, it would be unrealistic to expect any real understanding of more complex topics. Every knowledge base must start somewhere, and the most basic understanding of the Pythagorean theorem must begin by understating the basics of triangle. The following lesson plan is intended kindergarteners and can be done in conjunction with the book Barney & Friends: Three Lines, Three Corners. This lesson plan provides a list of goals for that activity that make it worth while. By playing on visual and interactive aspects