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