Lesson Plan — EMAT 4680

Chris Romano

Introductory Lesson to The Pythagorean Theorem

Lesson 1: History / Introduction of Pythagorean Theorem

Objectives:

1. Students should be familiar with Pythagoras
2. Students should be familiar with contributions made by Pythagoras and his followers
3. Students should be familiar with geometric proof of Pythagorean Theorem

Overview:

1. Briefly discuss Pythagoras’ life
2. Briefly discuss actions and contributions of Pythagoras and his followers
3. Run through geometric proof of Pythagorean Theorem

Lesson:

1. Who was Pythagoras?

• Born about 569 B.C. on the island of Samos in Greece, he is often described as the first pure mathematician.
• As a child he spent his early years on the island, but traveled frequently with his father Mnesarchus.
• Among his teachers were 3 philosophers who influenced him the most:
  • Pherekydes — many describe him as the teacher of Pythagoras
  • Thales and his pupil Anaximander
  • The latter 2 introduced him to mathematical ideas
• Following Thales’ advice, Pythagoras travelled to Egypt where he was accepted into priesthood and studied geometry, among other things
• About 10 to 15 years later, Egypt was invaded by the King of Persia, Cambyses II, and Pythagoras was taken prisoner and brought to Babylon
• While in Babylon he associated with the Magoi and studied their sacred rites and a very mystical worship of the gods, while also perfecting arithmetic, music and other mathematics under the instruction of the Babylonians.
• Pythagoras would later return to Samos, then leave for southeastern Italy.
• It was there in Croton that he founded a philosophical and religious school, often called the Brotherhood of Pythagoreans, which was devoted to the study of mathematics.

1. What did the Brotherhood of Pythagoreans accomplish?

• This school was divided into 2 parts:
  • Inner circle — whose members were known as mathematikoi and lived permanently with the society and were allowed no personal possessions
  • Outer circle — whose members were known as the askousmatics and were allowed to live in their own houses with their own possessions
• Among other things, the members of this society believed that reality is mathematical in nature, and observed strict loyalty and secrecy
• Pythagoras and his followers were interested in the principles of mathematics, the concept of number , the concept of a triangle or other mathematical figure and the idea of an abstract proof
• Pythagoras and his followers discovered many mathematical ideas and proofs, such as:
  • Contributions to the mathematical theory of music. Pythagoras was a musician as well, playing the Lyre. He noticed that vibrating strings produce harmonious tones when the ratios of the lengths of the strings are whole numbers, and that these ratios could be used in other instruments as well
  • The sum of the angles of a triangle is equal to 2 right angles, or 180 degrees.
  • In astronomy, he taught that the Earth was a sphere at the center of the universe.
  • He also taught the orbit of the Moon was inclined to the equator
  • The discovery of irrational numbers. Since Pythagoras believed the philosophy that all things are numbers, the fact that a number which could not be expressed as a ration of two whole numbers did not sit well with him. Legend has it that the Brotherhood drowned a man at sea for trying to announce the discovery of irrational numbers
• Just before 500 B.C., the Society was attacked by Cylon, a native of Croton who may have been denied entrance into the Brotherhood. Pythagoras escaped to Metapontium, where many believed he died, possibly of suicide.

1. The Pythagorean Theorem

Possibly the Brotherhoods most famous proof, the Pythagorean Theorem, was already known by the Babylonians 1000 years earlier, however it is believed that the Pythagoreans were the first to prove it.

Legend has it that when finally completing the proof, the Pythagoreans sacrificed 100 oxen.

Theorem:

The square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the legs.

Click here for proof