1. SHADOWS SIMILARITY PROPORTIONS TRIGONOMETRY

2. HOW LONG IS A SHADOW?

3. EXPERIMENTING WITH SHADOWS

4. SHADOW DATA GATHERING

5. LOOKING FOR EQUATIONS

6. AN N-BY-N WINDOW

7. MORE ABOUT WINDOWS

8. DRAW THE SAME SHAPE

9. HOW TO SHRINK IT?

10. THE STATUE OF LIBERTY

11. MAKE IT SIMILAR

12. IS THERE A COUNTEREXAMPLE?

13. TRIANGULAR COUNTEREXAMPLES

14. WHY ARE TRIANGLES SPECIAL?

15. ARE ANGLES ENOUGH?

16. WHATS POSSIBLE?

17. SIMILAR PROBLEMS

18. VERY SPECIAL TRIANGLES

19. INVENTING RULES

20. WHAT’S THE ANGLE?

21. MORE ABOUT ANGLES

22. INSIDE SIMILARITY

23. A PARALLEL PROOF

24. INS AND OUTS OF PROPORTION

25. BOUNCING LIGHT

26. NOW YOU SEE IT, NOW YOU DON’T

27. MIRROR MAGIC

28. MIRROR MADNESS

29. TO MEASURE A TREE

30. A SHADOW OF A DOUBT

31. MORE TRIANGLES FOR SHADOWS

32. THE RETURN OF THE TREE

33. RIGHT TRIANGLE RATIOS

34. REFERENCE: SIN, COS, AND TAN...

35. YOUR OPPOSITE IS MY ADJACENT

36. THE TREE AND THE PENDULUM

37. SMOKEY AND THE DUDE

38. THE SUN SHADOW PROBLEM

39. THE SUN SHADOW

40. BEGINNING PORTFOLIO SELECTION

41. HOW LONG IS A SHADOW? THINK OF VARIABLES THAT WOULD AFFECT THE LENGTH OF A SHADOW MAKE A LIST

42. HOW LONG IS A SHADOW? IN YOUR GROUP, CHOOSE ONE VARIABLE TO EXPERIMENT WITH AT HOME KEEP EVERYTHING ELSE CONSTANT

43. EXPERIMENTING WITH SHADOWS EXPERIMENT AT HOME DESCRIBE RESULTS –INCLUDE NUMERCIAL DATA –DIAGRAM

44. SHADOW DATA GATHERING THE HEIGHT OF THE LIGHT SOURCE THE DISTANCE FROM THE OBJECT TO THE LIGTH SOURCE THE HEIGHT OF THE OBJECT

45. SHADOW DATA GATHERING WHAT FORMULA CAN BE USED TO EXPRESS S AS A FUNCTION OF THE VARIABLES L, D, AND H? S = f(L, D, H)

46. AN N-BY-N WINDOW FIND A FORMULA FOR THE TOTAL NUMBER OF WOOD STICKS NEEDED TO BUILD ANY SQUARE WINDOW

47. LOOKING FOR EQUATIONS MAKE IN OUT TABLES OF YOUR GROUPS DATA AND 3 OTHER GROUPS TRY GRAPHING DATA FIND AN EQUATION

48. MORE ABOUT WINDOWS FIND A FORMULA FOR THE TOTAL NUMBER OF WOODEN STICKS NEEDED FOR ANY RECTANGULAR WINDOW M X N WINDOW

49. HOW TO SHRINK IT? LOLA’S : KEEP ANGLES THE SAME AND SUBTRACT 5 FROM EACH SIDE LILY’S : KEEP ALL LENGTHS THE SAME AND DIVIDE THE ANGLES BY 2 LULU’S : KEEP ALL THE ANGLES THE SAME AND DIVIDE THE LENGHTS OF THE SIDES BY 2

50. THE STATUE OF LIBERTY’S NOSE COMPARE YOUR NOSE AND ARM TO THE STATUE OF LIBERTY’S NOSE AND ARM COMPARE YOUR LEG TO THE STATUE OF LIBERTY’S.

51. MAKE IT SIMILAR UNFORTUNATELY WE DO NOT KNOW WHICH SIDE OF THE TRIANGLE HAS THE LENGTH OF 6. TRY EVERY SIDE AS IF IT WERE 6 INCHES.

52. IS THERE A COUNTEREXAMPLE? IF TWO POLYGONS HAVE THEIR CORRESPONDING ANGLES EQUAL, THEN THE POLYGONS ARE SIMILAR. FALSE

53. IS THERE A COUNTEREXAMPLE? IF TWO POLYGONS HAVE THEIR CORRESPONDING SIDES PROPORTIONAL, THEN THE POLYGONS ARE SIMILAR. FALSE

54. IS THERE A COUNTEREXAMPLE? EVERY TRIANGLE WITH TWO EQUAL SIDES ALSO HAS TWO EQUAL ANGELS. TRUE

55. TRIANGULAR COUNTEREXAMPLES IF TWO TRIANGLES HAVE THEIR CORRESPONDING ANGLES EQUAL, THEN THE TRIANGLES ARE SIMILAR. TRUE

56. TRIANGULAR COUNTEREXAMPLES IF TWO TRIANGLES ARE BOTH ISOSCELES, THEN THE TRIANGLES ARE SIMILAR. FALSE

57. TRIANGULAR COUNTEREXAMPLES IF TWO TRIANGLES HAVE THEIR CORRESPONDING SIDES PROPORTIONAL, THEN THE TRIANGLES ARE SIMILAR. TRUE

58. WHY ARE TRIANGLES SPECIAL? PICK FOUR LENGTHS TO FORM A QUADRILATERAL USING THESE SAME STRAWS CAN YOU MAKE A QUADRILATERAL THAT IS NOT SIMILAR TO THE FIRST? NO

59. WHY ARE TRIANGLES SPECIAL? PICK THREE LENGTHS TO FORM A TRIANGLE USING THESE SAME STRAWS CAN YOU MAKE A TRIANGLE THAT IS NOT SIMILAR TO THE FIRST? YES, THEY ARE CONGRUENT

60. SIMILAR PROBLEMS SET UP EQUATIONS FIND THE MISSING LENGTH EXPLAIN HOW YOU FOUND THE SOLUTIONS

61. ARE ANGLES ENOUGH? IF THE LENGTHS OF THE THREE SIDES OF ONE TRIANGLE ARE THE SAME AS THE LENGTHS OF THE THREE SIDES OF A SECOND TRIANGLE, THEN THE TWO TRIANGLES ARE CONGRUENT.

62. ARE ANGLES ENOUGH? EACH PERSON IN THE GROUP MAKE A TRIANGLE WITH THESE THREE ANGLES: 40 60 80 THEY SHOULD BE SIMILAR

63. ARE ANGLES ENOUGH? PICK A DIFFERENT SET OF ANGLES. EACH PERSON IN THE GROUP USE THE SAME ANGLES TO MAKE A TRIANGLE. THEY SHOULD BE SIMILAR

64. ARE ANGLES ENOUGH? THIS TIME MAKE A TRIANGLE USING THE 40, 60, AND 80 DEGREE ANGLES AGAIN PICK A LENGTH TO USE BETWEEN THE 40 AND 60 DEGREE ANGLES THEY SHOULD BE SIMILAR

65. WHAT’S POSSIBLE? CAN ANY THREE ANGLES BE THE ANGLES OF A TRIANGLE? NO, THEY HAVE TO ADD UP TO 180 DEGREES

66. WHAT’S POSSIBLE? CAN ANY THREE NUMBERS BE THE LENTHS OF THE SIDES OF A TRIANGLE? NO, ANY TWO SIDES MUST BE GREATER THAN THE THIRD

67. VERY SPECIAL TRIANGLES LEGS HYPOTENUSE SEGMENT RAY LINE LENGTH

68. INVENTING RULES CROSS MULITPLY

69. WHAT’S THE ANGLE? SUPPLEMENTARY ANGLES COMPLEMENTARY ANGLES STRAIGHT ANLGES

70. MORE ABOUT ANGLES VERTICAL ANGLES CORRESPONDING ANGLES ALTERNATE INTERIOR ANGLES ALTERNATE EXTERIOR ANGLES

71. INSIDE SIMILARITY FOR TRIANGLES TO SIMILAR CORRESPONDING ANGLES MUST BE EQUAL TO ONE ANOTHER. WHEN A TRANSVERSAL CUTS TWO PARALLEL LINES...

72. A PARALLEL PROOF REMEMBER, THE ANGLE SUM PROPERTY OF TRIANGLES REMEMBER, WHEN A TRANSVERSAL CUTS TWO PARALLEL LINES...

73. INS AND OUTS OF PROPORTION FIND AS MANY PAIRS OF EQUAL RATIOS AS YOU CAN.

74. BOUNCING LIGHT FORM AN ANGLE AND MEASURE THE ANGLE OF APPROACHA AND DEPARTURE REPEAT THE EXPERIMENT AND CHANGE THE ANGLE

75. BOUNCING LIGHT PRINCIPLE OF LIGHT RELFECTION: WHEN LIGHT IS REFLECTED OFF A SURFACE, THE ANGLE OF APPROACH IS EQUAL TO THE ANGLE OF DEPARTURE

76. NOW YOU SEE IT, NOW YOU DON’T USE THE PRINCIPLE OF LIGHT REFLECTION

77. MIRROR MAGIC USE PRINCIPLE OF LIGHT REFLECTION USE PROPORTIONS

78. MIRROR MADNESS SISTER IS 48 INCHES MOMMA IS 72 INCHES UNCLE IS 36 INCHES

79. MIRROR MADNESS BABY IS 27 INCHES GRANDDADDY IS 36 INCHES

80. TO MEASURE A TREE REMEMBER, WHEN A TRANSVERSAL CUTS PARALLEL LINES... REMEMBER WHAT MAKES TRIANGLES SIMILAR...

81. A SHADOW OF A DOUBT L = 11 H = 5 D = 12 L = 15H = 5D = 12 L = 15H = 5 D = 60 S = 10 S = 6 S = 30

82. MORE TRIANGLES FOR SHADOWS IS THERE A WAY TO ADD ANOTHER TRIANGLE IN THE DIAGRAM? IS THIS TRIANGLE SIMILAR TO THE OTHER TWO?

83. THE RETURN OF THE TREE IS THERE A WAY TO MAKE ANOTHER TRIANGLE THAT IS SIMILAR TO WOODY’S DIAGRAM? IF TWO TRIANGLES ARE SIMILAR, HOW DO WE FIND A MISSING LENGTH?

84. SIN, COS, AND TAN BUTTONS REVEALED Sin A = opposite/hypotenuse Cos A = adjacent/hypotenuse Tan A = opposite/adjacent

85. THE TREE AND THE PENDULUM Tan 70 = height/12 Height = 32.97 feet

86. SMOKEY AND THE DUDE Sin 6 = 100/hypotenuse Tan 6 = 100/adjacent leg

87. SMOKEY AND THE DUDE Tan 28 = height/50