Do Bees Build It Best?

1
Do Bees Build It Best?

• Geometry
• Trigonometry

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BUILDING THE BEST
3
NAILING DOWN AREA
4
HOW MANY CAN YOU FIND?
5
THATS ALL THERE IS!
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HALVING YOUR WAY
7
PARALLELOGRAMS AND TRAPEZOIDS
8
FORMING FORMULAS
9
GOING INTO THE GALLERY
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A RIGHT-TRIANGLE PAINTING
11
A TRIGONOMETRIC SUMMARY
12
MORE GALLERY MEASUREMENTS
13
A HOMEMADE TRIG TABLE
14
SHADOWS AND SAILBOATS
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TRI-SQUARE RUG GAMES
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ANY TWO SIDES WORK

• LEG2 LEG2 HYP2

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IMPOSSIBLE RUGS
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MAKE THE LINES COUNT
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PROOF BY RUGS
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THE POWER OF PYTHAGORAS
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LESLIES FERTILE FLOWERS
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FLOWERS FROM DIFFERENT SIDES
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DONT FENCE ME IN
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RECTANGLES ARE BORING!
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MORE FENCING, BIGGER CORRALS
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MORE OPINIONS ABOUT CORRALS
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BUILDING THE BEST FENCE
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FALLING BRIDGES
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LESLIES FLORAL ANGLES
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FLAT CUBES
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FLAT BOXES
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A VOLUMINOUS TASK
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PUT YOUR FIST INTO IT
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THE INS AND OUTS OF BOXES
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A SCULPTURE GARDEN
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THE WORLD OF PRISMS
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SHEDDING LIGHT ON PRISMS
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PYTHAGORAS AND THE BOX
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BACK ON THE FARM
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WHICH HOLDS MORE?
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CEREAL BOX SIZES
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A-TESSELLATING WE GO
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A PORTFOLIO OF FORMULAS
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THATS ALL THERE IS!

• EACH TRIANGLE MUST HAVE AN AREA OF 2 UNITS
• EACH TRIANGLE MUST HAVE ITS VERTICES ON PEGS
• EACH TRIANGLE MUST HAVE A HORIZONTAL SIDE

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HALVING YOUR WAY
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FORMING FORMULAS

• LOOK FOR WAYS TO CUT UP THE PARALLELOGRAMS AND
  TRAPEZOIDS INTO TRIANGLES

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GOING INTO THE GALLERY

• REMEMBER, THE HEIGHT IS ALSO THE ALTITUDE WHICH
  IS DRAWN PERPENDICULAR TO THE BASE

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A RIGHT-TRIANGLE PAINTING

• DRAW A RIGHT TRIANGLE THAT HAS A 55 DEGREE ANGLE
• EXTEND BOTH SIDES SO THAT THEY ARE AT LEAST 10 CM
  IN LENGTH

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A TRIGONOMETRIC SUMMARY

• SIN A OPPOSITE/HYPOTENUSE
• COS A ADJACENT/HYPOTENUSE
• TAN A
• OPPOSITE/ADJACENT

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A HOMEMADE TRIG TABLE

• DRAW A RIGHT TRIANGLE USING THE ASSIGNED ANGLE
  FOR YOUR GROUP
• MEASURE ALL THE SIDES
• COMPUTE THE RATIOS OF SIN, COS, AND TAN

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MORE GALLERY MESUREMENTS

• SIN A OPPOSITE/HYPOTENUSE
• COS A ADJACENT/HYPOTENUSE
• TAN A
• OPPOSITE/ADJACENT

52
SHADOWS AND SAILBOATS
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TRI-SQUARE RUG GAMES

• AL WINS WHEN THE LARGEST OF THE 3 SQUARES HAS THE
  MOST AREA.
• WHAT KIND OF TRIANGLE IS FORMED IN THIS SITUATION?

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TRI-SQUARE RUG GAMES

• BETTY WINS IF THE TWO SMALLER SQUARES HAVE MORE
  AREA THEN THE LARGEST
• WHAT KIND OF TRIANGLE IS FORMED IN THIS SITUATION?

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TRI-SQUARE RUG GAMES

• IF THE TWO SMALLER SQUARES HAVE THE SAME AREA AS
  THE LARGEST, THIS IS A FAIR GAME.
• WHAT KIND OF TRIANGLE IS FORMED IN THIS SITUATION?

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ANY TWO SIDES WORK
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ANY TWO SIDES WORK
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ANY TWO SIDES WORK
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IMPOSSIBLE RUGS

• RULE ANY TWO SIDES OF A TRIANGLE MUST BE GREATER
  THAN THE THIRD SIDE

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MAKE THE LINES COUNT
61
THE POWER OF PYTHAGORAS
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THE POWER OF PYTHAGORAS
63
LESLIES FERTILE FLOWERS
64
FLOWERS FROM DIFFERENT SIDES
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DONT FENCE ME IN

• RANCHER GONZALES CAN ONLY AFFORD 300 FEET OF
  FECNING
• MAKE AN INOUT TABLE OF VARIOUS LENGTHS AND WIDTHS

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RECTANGLES ARE BORING!

• RANCHER GONZALES NEPHEW JUAN HAS SUGGESTED
  RECTANGLES
• AGAIN SHE MUST USE 300 FEET OF FENCING

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MORE FENCING, BIGGER CORRALS

• HOW DOES THE AREA OF A SQUARE CORRAL MADE 300 FT
  OF FENCING COMPARE TO 600 FT?

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MORE FENCING, BIGGER CORRALS

• HOW DOES THE AREA OF A AN EQUILATERAL TRIANGLE
  CORRAL MADE 300 FT OF FENCING COMPARE TO 600 FT?

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MORE OPINIONS ABOUT CORRALS

• ACES PENTAGON
• TWOS HEXAGON
• THREES SEPTAGON
• FOURS OCTAGON
• FIVES 9 SIDED REG-POLY
• SIXES 10 SIDED REG-POLY

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BUILDING THE BEST FENCE

• AREA
• (P2/4n) tan (90-180/n)

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FALLING BRIDGES

• USE OTHER APPROXIMATIONS FOR THE SQUARE ROOT OF 2

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LESLIES FLORAL ANGLES

• REMEMBER, THE INVERSE OF
• ADDING?
• MULTIPLYING?
• SOMETHING SQUARED?
• NOW WE NEED THE INVERSE OF OUR TRIG FUNCTIONS

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FLAT BOXES

• 132 CM SQUARED

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FLAT BOXES

• 108 CM SQUARED

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A VOLUMINOUS TASK

• SURFACE AREA VOLUME
• 1
• 2
• 3
• 4
• 5

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A VOLUMINOUS TASK

• SURFACE AREA VOLUME
• 7
• 8
• 9
• 10

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PUT YOUR FIST INTO IT
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THE INS AND OUTS OF BOXES
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A SCULPTURE GARDEN

• FIND A WAY TO ARRANGE 8 CUBES THAT USES THE LEAST
  AMOUNT OF PAINT

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THE WORLD OF PRISMS

• PRISM IS A SPECIAL TYPE OF SOLID GEOMETRIC FIGURE
• THE INTIAL AND FINAL FACES ARE THE BASES
• USUALLY 1 OR 2 BASES
• PERPENDICULAR DISTANCE BETWEEN THE BASES IS THE
  HEIGHT

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THE WORLD OF PRISMS

• TRIANGULAR PRISMS
• HEXAGONAL PRISMS
• RECTANGULAR PRISMS

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THE WORLD OF PRISMS

• RIGHT PRISMS
• OBLIQUE PRISMS

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THE WORLD OF PRISMS

• LATERAL FACES
• LATERAL EDGES
• BASES
• LATERAL SURFACE AREA
• TOTAL SURFACE AREA

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PYTHAGORAS AND THE BOX
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BACK ON THE FARM