SHADOWS

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?
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WHATS POSSIBLE?
17
SIMILAR PROBLEMS
18
VERY SPECIAL TRIANGLES
19
INVENTING RULES
20
WHATS 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 DONT
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...
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YOUR OPPOSITE IS MY ADJACENT
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THE TREE AND THE PENDULUM
37
SMOKEY AND THE DUDE
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THE SUN SHADOW PROBLEM
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THE SUN SHADOW
40
BEGINNING PORTFOLIO SELECTION
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HOW LONG IS A SHADOW?

• THINK OF VARIABLES THAT WOULD AFFECT THE LENGTH
  OF A SHADOW
• MAKE A LIST

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HOW LONG IS A SHADOW?

• IN YOUR GROUP, CHOOSE ONE VARIABLE TO EXPERIMENT
  WITH AT HOME
• KEEP EVERYTHING ELSE CONSTANT

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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

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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?

• LOLAS KEEP ANGLES THE SAME AND SUBTRACT 5 FROM
  EACH SIDE
• LILYS KEEP ALL LENGTHS THE SAME AND DIVIDE THE
  ANGLES BY 2
• LULUS KEEP ALL THE ANGLES THE SAME AND DIVIDE
  THE LENGHTS OF THE SIDES BY 2

50
THE STATUE OF LIBERTYS NOSE

• COMPARE YOUR NOSE AND ARM TO THE STATUE OF
  LIBERTYS NOSE AND ARM
• COMPARE YOUR LEG TO THE STATUE OF LIBERTYS.

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
WHATS POSSIBLE?

• CAN ANY THREE ANGLES BE THE ANGLES OF A TRIANGLE?

NO, THEY HAVE TO ADD UP TO 180 DEGREES
66
WHATS 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
WHATS 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...

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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 DONT

• 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

S 10

• L 15 H 5 D 12

S 6
S 30

• L 15 H 5 D 60

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 WOODYS 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