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Welcome to the

Welcome to the. This 6-Hour Module focuses two Concepts from the Next Generation Sunshine State Standards. http://www.floridastandards.org/index.aspx. Next Generation Sunshine State Standards. 8. Subject Area:. Mathematics. Big Idea #2 Geometry. Grade Level:. Supporting Idea/ Big Ideas:.

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Welcome to the

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  1. Welcome to the

  2. This 6-Hour Module focuses twoConcepts from the Next Generation Sunshine State Standards http://www.floridastandards.org/index.aspx

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

  4. Benchmark 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. Next Generation Sunshine State Standards 8 Subject Area: Mathematics Supporting Idea #6 Number and Operations Grade Level: Supporting Idea/ Big Ideas: • MA. 8 .A. 6. 2 Benchmark:

  6. Benchmark 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. 3 – Two Hour Parts • 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. Before continuing with this Power Point… Please take the Content Pretest Content Pretest for Block 13.docx

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

  10. The Flatiron Building • 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. The Golden Gate Bridge • 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. One Way to Validate Pythagorean’s Theorem 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. c b ½ab a

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

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

  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. c b ½ab a c b ½ab a c b ½ab a

  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. (a-b)² a-b a² b²

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

  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 Yes (a-b)²fits! (a-b)²

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

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

  25. What does this mean Geometrically? c² a² b² a b² 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. Another Way to Validate Pythagorean’s Theorem http://users.ucom.net/~vegan/images/Pythagoras_6.jpg

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

  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 ½ab ½ab 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 ½ab ½ab ½ab a² b²

  32. Validating Pythagorean’s Theorem Geometrically b² a²

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

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

  35. What does it mean algebraically? a b It means that (a + b)² = ½ab•4 + c² a² + 2ab + b² = 2ab + c² Subtracting 2ab from each side, we get a² + b² = c² a b c² b a a b This validates Pythagorean’s Theorem algebraically.

  36. Benchmark 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. How will you explainValidating Pythagorean’s Theorem to your eighth grade students? Will you do it algebraically or geometrically or both? Discuss this with your partner. http://eppsnet.com/images/math-problems.gif

  38. The Next Benchmark states 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. Making Reasonable Approximations of Square Roots http://www.coverbrowser.com/image/bestselling-comics-2007/2502-1.jpg

  40. Approximating the Square Root of a Number by Hand There are several ways that you can approximating the square root by hand. Using the number line to round to the nearest integer 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 8 7

  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 6 5

  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. 3 Now count how many integer square roots there are between 7 and 8. 15 8 7 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. • 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 6 5 • Your approximation is 8/11 ≈ 0.72… So your percentage approximation to the nearest tenth is 5.7.

  45. Now you try!! Approximate to the nearest tenths place. Approximately 12.1 • Approximate to the nearest tenths place. Approximately 4.8

  46. How will you teach your eighth students to approximate the square root of a number? Using the number line to round to the nearest integer Using a percentage approximationto round to the nearest tenths place. There is still another method called the Classical Approach that you will learn in the next unit. http://eppsnet.com/images/math-problems.gif

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