1. By Irma Crespo

2. What do you see? ISZBCrespo

3. Meet Pythagoras Known as Pythagoras of Samos. Often described as the first Pure Mathematician. Studied properties of numbers which would be familiar to mathematicians today, such as even and odd numbers, triangular numbers, perfect numbers etc. Named for his geometric Pythagorean Theorem. ISZBCrespo www.sciencecastle.com http://www.gap-system.org/~history/Biographies/Pythagoras.html

4. Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the legs (a and b). Larson et. al. Geometry. 2001. ISZBCrespo a b c c 2 = a 2 + b 2 hypotenuse leg

5. A Casual Inquiry Informal Proof is a loose style of proof that outlines the main ideas of a formal proof with less detail and more clarity. http://faculty.matcmadison.edu/alehnen/weblogic/logproof.htm#InformalProof It is represented in high level sketches from which formal proofs can be reconstructed. http://en.wikipedia.org/wiki/Proof_theory#Formal_and_informal_proof ISZBCrespo

6. Do the Informal Proof ISZBCrespo OR Compass and Straightedge

7. The Informal Proof (Compass and Straightedge) Construct the triangle. ISZBCrespo Create two points. Connect the two points to make a line segment. Make the circle. Draw a perpendicular line that intersects the point on the circle. Put a point either above or below the point on the perpendicular line. Connect the points with line segments.

8. The Informal Proof Label the triangle. ISZBCrespo C B A Measure the angles. ABC Measure the segments. ABCBAC

9. The Informal Proof With your angle measurements, do you have a 90 0 angle? What type of triangle do you have? ISZBCrespo With your segment measurements in place, follow the worksheet directions on the computational part to answer each question. Which one demonstrates the Pythagorean Theorem?

10. You’ve proven Pythagorean Theorem visually. Now what? How do we use it? ISZBCrespo On integers. Finding lengths of sides. Finding areas.

11. Applying Pythagorean Theorem Its Triple Pythagorean triple is a set of three positive integers a, b, and c that satisfy the equation c 2 = a 2 + b 2. ISZBCrespo These are not your regular triples. Try a = 1, b = 3, c = 8 8 2 = 1 2 + 3 2 Try a = 4, b = 3, c = 5 5 2 = 4 2 + 3 2 Let’s try a few more on the whiteboard. Is 8 2 equal to 1 2 + 3 2 ? Is 5 2 equal to 4 2 + 3 2 ?

12. Applying Pythagorean Theorem Its Triple Take this advice. Get any positive integers m and n such that m > n and then, find 2mn, m 2 – n 2, m 2 + n 2. ISZBCrespo Example Let m = 3, n = 2 since 3 > 2. Then 2mn = 2*3*2 = 12 m 2 – n 2 = 3 2 – 2 2 = 9 – 4 = 5 m 2 + n 2 = 3 2 + 2 2 = 9 + 4 = 13 Plug in to c 2 = a 2 + b 2 So, (13) 2 = 12 2 + 5 2 Check. 169 = 144 + 25 169 = 169

13. Applying Pythagorean Theorem Finding the Length of a Hypotenuse Find the length of the hypotenuse of the right triangle. ISZBCrespo X 20 21 (hypotenuse) 2 = (leg) 2 + (leg) 2 x 2 = (20) 2 + (21) 2 x 2 = 400 + 441 x 2 = 841 √ (x 2) = √ (841) x = 29 29 Are the side lengths Pythagorean Triple?

14. Applying Pythagorean Theorem Finding the Length of a Leg Find the length of the leg of the right triangle. ISZBCrespo X 10 6 (hypotenuse) 2 = (leg) 2 + (leg) 2 10 2 = 6 2 + x 2 100 = 36 + x 2 100 – 36 = x 2 64 = x 2 8 = x 8 Are the side lengths Pythagorean Triple? √(64) = √ (x 2 )

15. Applying Pythagorean Theorem Finding the Area Find the area of the right triangle. ISZBCrespo 9 m 3 m h Use Pythagorean Theorem 9 2 = 3 2 + h 2 81 = 9 + h 2 Get the Area 81 - 9 = h 2 72 = h 2 √(72) = √ (h 2 ) √(72) = h √ (72) m Area = ½ bh = ½ (3* √ (72)) = ½ (25.456) = 12.728 m 2 Are the side lengths Pythagorean Triple?

16. Jigsaw Time Join your teacher assigned groups. Every group has 4 color-coded worksheets with each member designated to work on a specific color : red (integers), blue (length of a hypotenuse), yellow (length of a leg), green (area). Next, breakaway from the group and meet up with students who have the same color of worksheet to discuss a resolution on the problem assigned for that specific color. Go back to your assigned groups. Teach them your solutions. Learn from their solutions. Then, staple all your worksheets together as a group. Submit. ISZBCrespo

17. Jeopardy Game ISZBCrespo

18. Exit Slip Bring out your Perfect Square worksheet. Think about how you can represent the diagram into an algebraic equation. When done, turn in your worksheet before leaving the class. Tomorrow, the solution will be discussed. ISZBCrespo

19. Acknowledgement Larson, Boswell, and Stiff. McDougall Littell : Geometry. 2001. ISZBCrespo

20. Thank you. ISZBCrespo