Pythagoras and the Pythagorean Theorem

1. Pythagoras and the Pythagorean Theorem Grade 8-9 Lesson By Lindsay Kallish

2. Biography of Pythagoras Pythagoras was a Greek mathematician and a philosopher, but was best known for his Pythagorean Theorem. He was born around 572 B.C. on the island of Samos. For about 22 years, Pythagoras spent time traveling though Egypt and Babylonia to educate himself. At about 530 B.C., he settled in a Greek town in southern Italy called Crotona. Pythagoras formed a brotherhood that was an exclusive society devoted to moral, political and social life. This society was known as Pythagoreans.

3. Biography of Pythagoras • The Pythagorean School excelled in many subjects, such as music, medicine and mathematics. • In the society, members were known as mathematikoi. • History tells us that this theorem has been introduced through drawings, texts, legends, and stories from Babylon, Egypt, and China, dating back to 1800-1500 B.C. • Unfortunately, no one is sure who the true founder of the Pythagorean Theorem is. But it does seem certain through many history books that some time in the sixth century B.C., Pythagoras derives a proof for the Pythagorean Theorem.

4. Venn Diagram Homework Assignment • http://www.arcytech.org/java/pythagoras/history.html • http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Pythagoras.html 1st Website 2nd Website

5. Can Any Three Numbers Make A Triangle???? Write a sentence to describe the relationship between the sum of the areas of smaller and middle size squares compared to the area of the largest squares for: 1) An acute triangle 2) A right triangle 3) An obtuse triangle

6. The Pythagorean Theorem • The sum of the squares of each leg of a right angled triangle equals to the square of the hypotenuse a² + b² = c²

7. Many Proofs of the Pythagorean Theorem Euclidean Proof • First of all, ΔABF = ΔAEC by SAS. This is because, AE = AB, AF = AC, and BAF = BAC + CAF = CAB + BAE = CAE. • ΔABF has base AF and the altitude from B equal to AC. Its area therefore equals half that of square on the side AC. • On the other hand, ΔAEC has AE and the altitude from C equal to AM, where M is the point of intersection of AB with the line CL parallel to AE. • Thus the area of ΔAEC equals half that of the rectangle AELM. Which says that the area AC² of the square on side AC equals the area of the rectangle AELM. • Similarly, the are BC² of the square on side BC equals that of rectangle BMLD. Finally, the two rectangles AELM and BMLD make up the square on the hypotenuse AB. • QED • http://www.sunsite.ubc.ca/LivingMathematics/V001N01/UBCExamples/Pythagoras/pythagoras.html

8. Many Proofs of the Pythagorean Theorem Indian Proof • Area of the original square is A = c² • Looking at the first figure, the area of the large triangles is 4 (1/2)ab • The area of the inner square is (b-a) ² • Therefore the area of the original square is A=4(1/2)ab + (b-a) ² • This equation can be worked out as 2ab + b² - 2ab + a² = b² + a² • Since the square has the same area no matter how you find it, we conclude that A = c² = a² + b²

9. Many Proofs of the Pythagorean Theorem • Throughout many texts, there are about 400 possible proofs of the Pythagorean Theorem known today. • It is not a wonder that there is an abundance of proofs due to the fact that there are numerous claims of different authors to this significant geometric formula. • Specifically looking at the Pythagorean Theorem, this unique mathematical discovery proves that there is a limitless amount of possibilities of algebraic and geometric associations with the single theorem. • http://www.cut-the-knot.org/pythagoras/index.shtml

10. Connection to Technology • Geometer’s SketchPad • Students can see the Pythagorean Theorem work using special triangles with 45-45-90 degree angles and 30-60-90 degree angles

12. Pythagoras Board Game Rules: • To begin, roll 2 dice. The person with the highest sum goes first. • To move on the board, roll both dice. Substitute the numbers on the dice into the Pythagorean Theorem for the lengths of the legs to find the value of the length of the hypotenuse. • Using the Pythagorean Theorem a²+b²=c², a player moves around the board a distance that is the integral part of c. • For example, if a 1 and a 2 were rolled, 1²+2²=c²; 1+4=c²; 5=c²; Since c = √5 or approximately 2.236, the play moves two spaces. Always round the value down. • When the player lands on a ‘?’ space, a question card is drawn. If the player answers the question correctly, he or she can roll one die and advance the resulting number of places. • Each player must go around the board twice to complete the game. A play must answer a ‘?’ card correctly to complete the game and become a Pythagorean

13. Pythagoras Board Game

14. References: DeLacy, E. A. (1963). Euclid and geometry (2nd ed.). USA: Franklin Watts, Inc. Ericksen, D., Stasiuk, J., & Frank, M. (1995). Bringing pythagoras to life. The Mathematics Teacher, 88(9), 744. Gow, J. (1968). A short story of greek mathematics. New York: Chelsea Publishing Company. Katz, V. (1993). A history of mathematics (2nd ed.). USA: Addison Wesley Longman, Inc. Swetz, F. J., & Kao, T. I. (1977). Was pythagoras chinese? an examination of right triangle theory in ancient china. USA: The Pennsylvania State University. Veljan, D. (2000). The 2500-year-old pythagorean theorem. Mathematics Magazine, 73(4), 259.