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Do Bees Build It Best?Do Bees Build It Best?
GeometryGeometry
TrigonometryTrigonometry
BUILDING THE BESTBUILDING THE BEST
NAILING DOWN AREANAILING DOWN AREA
HOW MANY CAN YOU HOW MANY CAN YOU FIND?FIND?
THAT’S ALL THERE IS!THAT’S ALL THERE IS!
HALVING YOUR WAYHALVING YOUR WAY
PARALLELOGRAMS AND PARALLELOGRAMS AND TRAPEZOIDSTRAPEZOIDS
FORMING FORMULASFORMING FORMULAS
GOING INTO THE GALLERYGOING INTO THE GALLERY
A RIGHT-TRIANGLE A RIGHT-TRIANGLE PAINTINGPAINTING
A TRIGONOMETRIC A TRIGONOMETRIC SUMMARYSUMMARY
MORE GALLERY MORE GALLERY MEASUREMENTSMEASUREMENTS
A HOMEMADE TRIG TABLEA HOMEMADE TRIG TABLE
SHADOWS AND SHADOWS AND SAILBOATSSAILBOATS
TRI-SQUARE RUG GAMESTRI-SQUARE RUG GAMES
ANY TWO SIDES WORKANY TWO SIDES WORK
LEG^2 + LEG^2 = HYP^2LEG^2 + LEG^2 = HYP^2
IMPOSSIBLE RUGSIMPOSSIBLE RUGS
MAKE THE LINES COUNTMAKE THE LINES COUNT
PROOF BY RUGSPROOF BY RUGS
THE POWER OF THE POWER OF PYTHAGORASPYTHAGORAS
LESLIE’S FERTILE LESLIE’S FERTILE FLOWERSFLOWERS
FLOWERS FROM FLOWERS FROM DIFFERENT SIDESDIFFERENT SIDES
DON’T FENCE ME INDON’T FENCE ME IN
RECTANGLES ARE RECTANGLES ARE BORING!BORING!
MORE FENCING, BIGGER MORE FENCING, BIGGER CORRALSCORRALS
MORE OPINIONS ABOUT MORE OPINIONS ABOUT CORRALSCORRALS
BUILDING THE BEST BUILDING THE BEST FENCEFENCE
FALLING BRIDGESFALLING BRIDGES
LESLIE’S FLORAL ANGLESLESLIE’S FLORAL ANGLES
FLAT CUBESFLAT CUBES
FLAT BOXESFLAT BOXES
A VOLUMINOUS TASKA VOLUMINOUS TASK
PUT YOUR FIST INTO ITPUT YOUR FIST INTO IT
THE INS AND OUTS OF THE INS AND OUTS OF BOXESBOXES
A SCULPTURE GARDENA SCULPTURE GARDEN
THE WORLD OF PRISMSTHE WORLD OF PRISMS
SHEDDING LIGHT ON SHEDDING LIGHT ON PRISMSPRISMS
PYTHAGORAS AND THE PYTHAGORAS AND THE BOXBOX
BACK ON THE FARMBACK ON THE FARM
WHICH HOLDS MORE?WHICH HOLDS MORE?
CEREAL BOX SIZESCEREAL BOX SIZES
A-TESSELLATING WE GOA-TESSELLATING WE GO
A PORTFOLIO OF A PORTFOLIO OF FORMULASFORMULAS
THAT’S ALL THERE IS!THAT’S ALL THERE IS!
EACH TRIANGLE MUST EACH TRIANGLE MUST HAVE AN AREA OF 2 UNITSHAVE AN AREA OF 2 UNITS
EACH TRIANGLE MUST EACH TRIANGLE MUST HAVE ITS VERTICES ON HAVE ITS VERTICES ON PEGSPEGS
EACH TRIANGLE MUST EACH TRIANGLE MUST HAVE A HORIZONTAL SIDEHAVE A HORIZONTAL SIDE
HALVING YOUR WAYHALVING YOUR WAY
FORMING FORMULASFORMING FORMULAS
LOOK FOR WAYS TO CUT UP LOOK FOR WAYS TO CUT UP THE PARALLELOGRAMS THE PARALLELOGRAMS AND TRAPEZOIDS INTO AND TRAPEZOIDS INTO TRIANGLESTRIANGLES
GOING INTO THE GOING INTO THE GALLERYGALLERY
REMEMBER, THE HEIGHT IS REMEMBER, THE HEIGHT IS ALSO THE ALTITUDE WHICH ALSO THE ALTITUDE WHICH IS DRAWN PERPENDICULAR IS DRAWN PERPENDICULAR TO THE BASETO THE BASE
A RIGHT-TRIANGLE A RIGHT-TRIANGLE PAINTINGPAINTING
DRAW A RIGHT TRIANGLE DRAW A RIGHT TRIANGLE THAT HAS A 55 DEGREE THAT HAS A 55 DEGREE ANGLEANGLE
EXTEND BOTH SIDES SO EXTEND BOTH SIDES SO THAT THEY ARE AT LEAST THAT THEY ARE AT LEAST 10 CM IN LENGTH10 CM IN LENGTH
A TRIGONOMETRIC A TRIGONOMETRIC SUMMARYSUMMARY
SIN A = SIN A = OPPOSITE/HYPOTENUSEOPPOSITE/HYPOTENUSE
COS A = COS A = ADJACENT/HYPOTENUSEADJACENT/HYPOTENUSE
TAN A = TAN A =
OPPOSITE/ADJACENTOPPOSITE/ADJACENT
A HOMEMADE TRIG A HOMEMADE TRIG TABLETABLE
DRAW A RIGHT TRIANGLE DRAW A RIGHT TRIANGLE USING THE ASSIGNED USING THE ASSIGNED ANGLE FOR YOUR GROUPANGLE FOR YOUR GROUP
MEASURE ALL THE SIDESMEASURE ALL THE SIDES COMPUTE THE RATIOS OF COMPUTE THE RATIOS OF
SIN, COS, AND TANSIN, COS, AND TAN
MORE GALLERY MORE GALLERY MESUREMENTSMESUREMENTS
SIN A = SIN A = OPPOSITE/HYPOTENUSEOPPOSITE/HYPOTENUSE
COS A = COS A = ADJACENT/HYPOTENUSEADJACENT/HYPOTENUSE
TAN A = TAN A =
OPPOSITE/ADJACENTOPPOSITE/ADJACENT
SHADOWS AND SHADOWS AND SAILBOATSSAILBOATS
TRI-SQUARE RUG TRI-SQUARE RUG GAMESGAMES
AL WINS WHEN THE AL WINS WHEN THE LARGEST OF THE 3 SQUARES LARGEST OF THE 3 SQUARES HAS THE MOST AREA.HAS THE MOST AREA.
WHAT KIND OF TRIANGLE IS WHAT KIND OF TRIANGLE IS FORMED IN THIS SITUATION?FORMED IN THIS SITUATION?
TRI-SQUARE RUG TRI-SQUARE RUG GAMESGAMES
BETTY WINS IF THE TWO BETTY WINS IF THE TWO SMALLER SQUARES HAVE SMALLER SQUARES HAVE MORE AREA THEN THE MORE AREA THEN THE LARGESTLARGEST
WHAT KIND OF TRIANGLE IS WHAT KIND OF TRIANGLE IS FORMED IN THIS SITUATION?FORMED IN THIS SITUATION?
TRI-SQUARE RUG TRI-SQUARE RUG GAMESGAMES
IF THE TWO SMALLER IF THE TWO SMALLER SQUARES HAVE THE SAME SQUARES HAVE THE SAME AREA AS THE LARGEST, THIS AREA AS THE LARGEST, THIS IS A FAIR GAME.IS A FAIR GAME.
WHAT KIND OF TRIANGLE IS WHAT KIND OF TRIANGLE IS FORMED IN THIS SITUATION?FORMED IN THIS SITUATION?
ANY TWO SIDES WORKANY TWO SIDES WORK
ANY TWO SIDES WORKANY TWO SIDES WORK
ANY TWO SIDES WORKANY TWO SIDES WORK
IMPOSSIBLE RUGSIMPOSSIBLE RUGS
RULE: ANY TWO SIDES OF A RULE: ANY TWO SIDES OF A TRIANGLE MUST BE TRIANGLE MUST BE GREATER THAN THE THIRD GREATER THAN THE THIRD SIDESIDE
MAKE THE LINES MAKE THE LINES COUNTCOUNT
THE POWER OF THE POWER OF PYTHAGORASPYTHAGORAS
THE POWER OF THE POWER OF PYTHAGORASPYTHAGORAS
LESLIE’S FERTILE LESLIE’S FERTILE FLOWERSFLOWERS
FLOWERS FROM FLOWERS FROM DIFFERENT SIDESDIFFERENT SIDES
DON’T FENCE ME INDON’T FENCE ME IN
RANCHER GONZALES CAN ONLY AFFORD 300 FEET OF FECNING
MAKE AN INOUT TABLE OF VARIOUS LENGTHS AND WIDTHS
RECTANGLES ARE RECTANGLES ARE BORING!BORING!
RANCHER GONZALE’S RANCHER GONZALE’S NEPHEW JUAN HAS NEPHEW JUAN HAS SUGGESTED RECTANGLESSUGGESTED RECTANGLES
AGAIN SHE MUST USE 300 AGAIN SHE MUST USE 300 FEET OF FENCINGFEET OF FENCING
MORE FENCING, BIGGER MORE FENCING, BIGGER CORRALSCORRALS
HOW DOES THE AREA OF A HOW DOES THE AREA OF A SQUARE CORRAL MADE 300 SQUARE CORRAL MADE 300 FT OF FENCING COMPARE FT OF FENCING COMPARE TO 600 FT?TO 600 FT?
MORE FENCING, BIGGER MORE FENCING, BIGGER CORRALSCORRALS
HOW DOES THE AREA OF A HOW DOES THE AREA OF A AN EQUILATERAL TRIANGLE AN EQUILATERAL TRIANGLE CORRAL MADE 300 FT OF CORRAL MADE 300 FT OF FENCING COMPARE TO 600 FENCING COMPARE TO 600 FT?FT?
MORE OPINIONS ABOUT MORE OPINIONS ABOUT CORRALSCORRALS
ACES : PENTAGONACES : PENTAGON TWO’S: HEXAGONTWO’S: HEXAGON THREE’S: SEPTAGONTHREE’S: SEPTAGON FOUR’S: OCTAGONFOUR’S: OCTAGON FIVE’S: 9 SIDED REG-POLYFIVE’S: 9 SIDED REG-POLY SIXE’S: 10 SIDED REG-POLYSIXE’S: 10 SIDED REG-POLY
BUILDING THE BEST BUILDING THE BEST FENCEFENCE
AREA=AREA=
(P^2/4n) * tan (90-180/n)(P^2/4n) * tan (90-180/n)
FALLING BRIDGESFALLING BRIDGES
USE OTHER USE OTHER APPROXIMATIONS FOR THE APPROXIMATIONS FOR THE SQUARE ROOT OF 2 SQUARE ROOT OF 2
LESLIE’S FLORAL LESLIE’S FLORAL ANGLESANGLES
REMEMBER, THE INVERSE OF:REMEMBER, THE INVERSE OF:
ADDING?ADDING?
MULTIPLYING?MULTIPLYING?
SOMETHING SQUARED?SOMETHING SQUARED? NOW WE NEED THE INVERSE NOW WE NEED THE INVERSE
OF OUR TRIG FUNCTIONSOF OUR TRIG FUNCTIONS
FLAT BOXESFLAT BOXES
132 CM SQUARED132 CM SQUARED
FLAT BOXESFLAT BOXES
108 CM SQUARED108 CM SQUARED
A VOLUMINOUS TASKA VOLUMINOUS TASK
SURFACE AREASURFACE AREA
VOLUMEVOLUME
#1 #1 #2#2 #3#3 #4#4 #5#5
A VOLUMINOUS TASKA VOLUMINOUS TASK
SURFACE AREASURFACE AREAVOLUMEVOLUME
#7 #7 #8#8 #9#9 #10#10
PUT YOUR FIST INTO ITPUT YOUR FIST INTO IT
THE INS AND OUTS OF THE INS AND OUTS OF BOXESBOXES
A SCULPTURE GARDENA SCULPTURE GARDEN
FIND A WAY TO ARRANGE 8 FIND A WAY TO ARRANGE 8 CUBES THAT USES THE CUBES THAT USES THE LEAST AMOUNT OF PAINTLEAST AMOUNT OF PAINT
THE WORLD OF PRISMSTHE WORLD OF PRISMS
PRISM IS A SPECIAL TYPE OF PRISM IS A SPECIAL TYPE OF SOLID GEOMETRIC FIGURESOLID GEOMETRIC FIGURE– THE INTIAL AND FINAL FACES THE INTIAL AND FINAL FACES
ARE THE BASES ARE THE BASES – USUALLY 1 OR 2 BASESUSUALLY 1 OR 2 BASES– PERPENDICULAR DISTANCE PERPENDICULAR DISTANCE
BETWEEN THE BASES IS THE BETWEEN THE BASES IS THE HEIGHTHEIGHT
THE WORLD OF PRISMSTHE WORLD OF PRISMS
TRIANGULAR PRISMSTRIANGULAR PRISMS HEXAGONAL PRISMSHEXAGONAL PRISMS RECTANGULAR PRISMSRECTANGULAR PRISMS
THE WORLD OF PRISMSTHE WORLD OF PRISMS
RIGHT PRISMSRIGHT PRISMS
OBLIQUE PRISMSOBLIQUE PRISMS
THE WORLD OF PRISMSTHE WORLD OF PRISMS
LATERAL FACESLATERAL FACES LATERAL EDGESLATERAL EDGES BASESBASES
LATERAL SURFACE AREALATERAL SURFACE AREA TOTAL SURFACE AREATOTAL SURFACE AREA
PYTHAGORAS AND THE PYTHAGORAS AND THE BOXBOX
BACK ON THE FARMBACK ON THE FARM