2. http://www.floridastandards.org/index.aspx

3. MA. 8.G. 2. 4 Next Generation Sunshine State Standards Subject Area: Grade Level: Supporting Idea/ Big Ideas: Benchmark: Mathematics

4. MA. 8. G. 2. 4 Validate and apply Pythagorean theorem to find distances in real world situations or between points in the coordinate plane.

5. MA. 8.A. 6. 2 Next Generation Sunshine State Standards Subject Area: Grade Level: Supporting Idea/ Big Ideas: Benchmark: Mathematics

6. MA. 8. A. 6. 2 Make reasonable approximations of square roots and mathematical expressions that include square roots, and use them to estimate solutions to problems and to compare mathematical expressions involving real number s and radical expressions.

7. This lesson will be divided into 3 parts Pythagorean’s Theorem and Square Roots by Hand (2 Hours) Calculators in the Middle School Classroom (2 Hours) Validate, Explore, Practice (2 Hours)

8. Please take the Content Pretest Content Pretest for Block 13.docx

9. Because triangles are seen everywhere! Why Do I Have to Learn This?

10. Located in New York One of the first skyscraper (1902) Sits on a triangular island http://www.prometheanplanet.com/server.php?show=ConResource.20665

11. San Francisco Bay Originally the longest suspension bridge in the world when it was completed during the year of 1937 Since its completion, the span length has been surpassed by eight other bridges http://www.prometheanplanet.com/server.php?show=ConResource.20665

12. The Louvre Pyramid Paris, France Glass and Metal Pyramid Main entrance to the museum Completed in 1989 Landmark for the city of Paris http://www.prometheanplanet.com/server.php?show=ConResource.20665

13. The Epcot A theme park at Walt Disney World Second part built In 2007, Epcot was ranked the third-most visited theme part in the United States, and the sixth-most visited in the world. http://www.prometheanplanet.com/server.php?show=ConResource.20665

14. http://users.ucom.net/~vegan/images/Pythagoras_6.jpg

15. Things You Will Need You will need a Partner 3 Sheets of Graph/Grid Paper Scissors Ruler Crayons

16. Start with any right triangle Draw any size right triangle in the middle of the page. You and your partner should have the same triangle. ½ab a b c

17. -b a Constructing Three Squares Draw three squares so that each square corresponds to each side of the triangle. The squares must be connected to the triangle. ½ab a b a a² c² b b² a -b c

18. Cut and Color Cut out the three squares and the triangle. ½ab a b a a² c c² b b² a -b

19. Construct, Cut and Color Take out another sheet of graph paper and construct three more triangles with area ½ab. Write ½ab on the three triangles. Cut them out. ½ab a b c a b c a b c

20. Construct, Cut and Color Finally construct a square whose length is (a-b). To do this, put you a² and b² on the graph paper. Then cut it out and color it. a² b² a-b (a-b)²

21. Fit the 4 triangles into c². ½ab c²

22. Validating Pythagorean’s Theorem Geometrically There is a square left in the middle to fill? Is it a²? Is it b²? Or is it (a-b)² a a² b² (a-b)² Yes (a-b)² fits!

23. Validating Pythagorean’s Theorem Geometrically Now take a² + b². Check to see if the same five objects fit into this configuration. a b² (a-b)² ½ab a² They do!!

24. What does this mean Geometrically? c² a² b² a a²

25. What does this mean Geometrically? c² a² b² a It means that c² has the same area as a² + b². We have now validated that a² + b² = c² !!

26. What does this mean algebraically? It means that c² = (a – b)² + ½(ab) 4 Simplifying we get, c² = a² + b² We have now validated Pythagorean’s Theorem algebraically.

27. http://users.ucom.net/~vegan/images/Pythagoras_6.jpg

28. Cut and Color Again begin with a², b², and c². ½ab a b a a² c c² b b² a -b

29. Validating Pythagorean’s Theorem Geometrically Construct a square with your four triangles and c². This time the triangles don’t go inside. Can you do it? ½ab c²

30. Validating Pythagorean’s Theorem Geometrically c²

31. Validating Pythagorean’s Theorem Geometrically How about with your four triangles,a² and b²? Can you do it? ½ab a² b²

32. Validating Pythagorean’s Theorem Geometrically a² b²

33. Validating Pythagorean’s Theorem Geometrically a² b² c² What does this mean geometrically?

34. Validating Pythagorean’s Theorem Geometrically a² b² c² It means that without the triangles, c² = a² + b², and geometrically validates Pythagorean’s Theorem.

35. What does it mean algebraically? c² a b b a b a It means that (a + b)² = ½ab4 + c² a² + 2ab + b² = 2ab + c² Subtracting 2ab from each side, we get a² + b² = c² This validates Pythagorean’s Theorem algebraically.

36. MA. 8. G. 2. 4 Validate and apply Pythagorean theorem to find distances in real world situations or between points in the coordinate plane.

37. Will you do it algebraically or geometrically or both? Discuss this with your partner. http://eppsnet.com/images/math- problems.gif

38. MA. 8. A. 6. 2 Make reasonable approximations of square roots and mathematical expressions that include square roots, and use them to estimate solutions to problems and to compare mathematical expressions involving real number s and radical expressions.

39. http://www.coverbrowser.com/image/bestselling-comics-2007/2502-1.jpg

40. There are several ways that you can approximating the square root by hand. 1. Using the number line to round to the nearest integer 2. Using a Percentage Approximation to round to the nearest tenths place.

41. How do you approximate to the nearest integer using the number line? 0 7 8 is between and. Which integer is it closer to? 7 7 8

42. How do you approximate to the nearest integer using the number line? 0 5 6 is between and. Which integer is it closer to? 6 5 6

43. Using percentage approximations, how do you approximate to the nearest tenths place. You know now that is closest to 7. Count how many spaces it is away from 7. 7 8 Now count how many integer square roots there are between 7 and 8. 3 15 Your approximation is 3/15 = 1/5 = 0.2. So your percentage approximation to the nearest tenth is 7.2.

44. Using percentage approximations, how do you approximate to the nearest tenths place. 5 6 You know now that is closest to 6. Count how many spaces it is away from 5. 8 Now count how many integer square roots there are between 5 and 6. 11 Your approximation is 8/11 ≈ 0.72… So your percentage approximation to the nearest tenth is 5.7.

45. Approximate to the nearest tenths place. Approximately 12.1 Approximate to the nearest tenths place. Approximately 4.8

46. 1.Using the number line to round to the nearest integer 2.Using a percentage approximation to round to the nearest tenths place. http://eppsnet.com/images/m ath-problems.gif There is still another method called the Classical Approach that you will learn in the next unit.