Transcript: Challenge #3 | Mar 26, 1999

Mister C stands in the studio by a table. He’s in his mid-forties, with a dark beard and wavy black hair. He’s wearing glasses, a white and black T-shirt with Einstein’s face, patterned suspenders and gray trousers.

Three large test tubes sit on the table. One contains a blue liquid. Mister C holds one test tube with a red liquid in it.

He says YOU KNOW, ONE OF THE MOST
IMPORTANT PLACES WHERE
MATHEMATICS IS USED IS IN
THE AREA OF MEASUREMENT
AND IN PARTICULAR,
IN THE AREA OF RATIO.
THIS PARTICULAR
BLUE LIQUID;
THERE IS 800ml OF IT AND
IF WE WERE COMPARING IT
TO THIS PARTICULAR RED
LIQUID, WHICH IS 400ml,
WE WOULD SAY THAT THE
RATIO OF THE VOLUME
IS 800:400 WHICH IS 2:1.
INTERESTINGLY ENOUGH, IF
WE LOOK AT THESE MEASURING
VESSELS, THIS GRADUATED
CYLINDER HAS A TOTAL
CAPACITY OF 1000ml,
THIS ONE HAS 500ml
AND THAT RATIO
IS ALSO 2:1.
MOVING DOWN, WE HAVE TWO
MORE GRADUATED CYLINDERS.
THIS ONE IS 500ml AND
THIS ONE IS 250ml
AND, ONCE AGAIN, THE RATIO
JUST HAPPENS TO BE 2:1.
COINCIDENCE?
OR BY DESIGN?
I SUSPECT IT'S
BY DESIGN.
NOW I'VE GOT A COUPLE
OTHER THINGS THAT
I'D LIKE TO LOOK AT THAT
HAVE LOTS TO DO WITH RATIO.

He takes out the test tubes and continues FOR INSTANCE,
THESE RECTANGLES.
PUTTING THEM OUT HERE,
THEY'RE ALL NICELY NUMBERED.
NOW I'VE GOT A LITTLE
QUESTION TO ASK OF YOU.
ONE, TWO, THREE,
FOUR, FIVE, SIX.
NOW I'M GOING TO STEP BACK
HERE FOR A MOMENT AND I'M
GOING TO ASK YOU TO LOOK
AT ALL OF THOSE RECTANGLES
AND PICK OUT THE ONE
YOU THINK IS THE MOST
PLEASING TO YOUR EYE, THE
ONE YOU LIKE THE BEST.
AND I WOULD LIKE YOU
TO PUT THE NUMBER
OF THAT PARTICULAR
RECTANGLE DOWN
ON YOUR
NOTEBOOK SOMEWHERE.

The papers show different sized rectangles.

He continues I'LL GIVE YOU A SECOND OR
TWO TO THINK ABOUT THAT.
NOW, WITH A LITTLE BIT OF
HELP FROM YOUR TEACHER,
SOMEBODY THAT'S IN THE ROOM
PERHAPS AT THE BLACKBOARD,
I'D LIKE TO RECORD
A BIT OF A VOTE.

He points to rectangle number one and says HOW MANY OF YOU
CHOSE THIS RECTANGLE?
PUT UP YOUR HANDS.
HOW MANY CHOSE
RECTANGLE NUMBER TWO?
PUT UP YOUR HANDS.
HOW MANY CHOSE
RECTANGLE NUMBER THREE?
PUT UP YOUR HANDS.
NUMBER FOUR?
PUT UP YOUR HANDS.
NUMBER FIVE?
HOPE I'M NOT
GOING TOO FAST.
PUT UP YOUR HANDS.
AND NUMBER SIX?
PUT UP YOUR HANDS.
NOW THAT YOU'VE
RECORDED THAT,
INTERESTINGLY ENOUGH,
PSYCHOLOGISTS HAVE DONE
THIS TEST MANY, MANY, MANY
TIMES AND THEY'VE LOOKED
AT THESE RECTANGLES AND
THEY WERE COMPARING
THE RATIO OF THE WIDTH
AND LENGTH AND THEY GAVE
THIS TEST TO MANY, MANY
THOUSANDS OF PEOPLE
AND ONE OF THESE
RECTANGLES IN PARTICULAR
IS PICKED MORE COMMONLY
THAN ALL THE REST.
AS IT TURNS OUT, IT'S
THIS ONE RIGHT HERE
AND THE QUESTION
IS WHY?
He points to rectangle number two and continues WELL, I'M GOING TO SHOW
YOU A COUPLE OF THINGS.
LIKE, FOR INSTANCE,
THIS CREDIT CARD.
IF YOU LOOK AT THE RATIO
OF THE LENGTH TO THE WIDTH
OF THE CREDIT CARD,
IT IS THE SAME
AS THAT RECTANGLE.
INTERESTING.
IS THAT BY DESIGN?
I THINK SO.
LOOK AT THIS
GRAPHING CALCULATOR
IN THE SCREEN
AT THE TOP.
IF WE WERE TO MEASURE THE
LENGTH AND WIDTH OF THE SCREEN,
IT IS ALSO THE SAME AS
THIS PARTICULAR RECTANGLE.
WHAT'S EVEN MORE
AMAZING...
He places a black and white picture over the rectangle and continues IS THAT THIS BUILDING FROM
ANTIQUITY, THE PARTHENON,
WAS ALSO DESIGNED USING
THAT SAME RATIO AND
I'M GOING TO DEMONSTRATE
THAT A LITTLE BIT LATER
OVER AT THE DESK TO
SHOW YOU THAT, IN FACT,
THAT THAT RATIO
APPEARS AGAIN.
I'LL TELL YOU OVER THERE
EXACTLY WHAT THE RATIO
IS BUT A VERY SPECIAL
ONE INDEED.
SPEAKING OF
SPECIAL RATIOS,
I HAVE ANOTHER LITTLE
DEMONSTRATION HERE OF
A RATIO THAT WE'RE PROBABLY
ALL PRETTY FAMILIAR WITH.
LET ME SEE, I'VE
GOT MY CAN HERE.
AND WHAT WE'RE INTERESTED
IN ACTUALLY IS...
THIS IS CALLED THE
CIRCUMFERENCE AND
WE CAN MEASURE THAT.
THIS IS CALLED
THE DIAMETER,
GOING ACROSS THE TOP.
THE WAY I MEASURED IS, I
JUST SIMPLY GOT A PIECE
OF RIBBON, WRAPPED IT AROUND
AND CUT IT AND THAT'S
THIS PIECE OF
RIBBON UP HERE.

He points to a dark blue cardboard with a yellow ribbon stuck to it.

He continues AND THEN I TOOK THE RIBBON
AGAIN AND I PULLED IT
ACROSS THE TOP AND I
CUT IT AND PUT IT HERE.
ONCE I DID THAT...
I ACTUALLY TOOK MY
RULER AND I MEASURED
THE LENGTHS OF EACH.
NOW, I WOULD BE VERY
SURPRISED IF YOU DIDN'T
KNOW WHAT THE RATIO WAS.
THAT IN FACT, THE RATIO
IS SOME MAGICAL NUMBER
WE CALL PI.
AND SOMETIMES YOU LEARN
IT AS 22 OVER 7, 22:7,
THAT'S A RATIO; THE RATIO
OF THE CIRCUMFERENCE
TO THE DIAMETER.
SOMETIMES YOU HEAR THE
DECIMAL EQUIVALENT,
APPROXIMATELY 3.14.
INTERESTINGLY ENOUGH, WHEN
YOU MEASURE THIS YOU'LL
NOTE THAT IT'S ABOUT 31cm
AND THIS IS ABOUT 10cm
AND 31 OVER 10 IS 3.1;
A DEMONSTRATION OF
THE RATIO PI.
WELL, SPEAKING OF
CIRCLES, HERE'S A CIRCLE.
LET'S SEE RIGHT UP HERE.

He places a large wheel on the table and continues
THIS IS A CIRCLE I'M SURE
YOU'RE FAMILIAR WITH.
IT COMES FROM THE BACK
OF AN 18-SPEED BIKE.
18 SPEEDS, NOW WHERE
DO I GET THAT IDEA?
WELL, IT HAS TO DO WITH
THE NUMBER OF SPROCKETS
ON THE REAR AND THE
NUMBER OF SPROCKETS
THAT YOU FIND AT
THE FRONT PEDAL.
AT THE FRONT PEDAL THERE
WOULD BE THREE, AND AS IT
TURNS OUT, THERE'S ONE,
TWO, THREE, FOUR, FIVE,
SIX REAR SPROCKETS.
SO THIS IS AN
18-SPEED BIKE.
THE SPROCKETS ARE
DIFFERENT SIZES,
YOU CAN SEE THEM GETTING
BIGGER AS WE GO DOWN.

A close-up shot shows the sprockets set.

He continues THE EXERCISE THAT I'M GOING
TO HAVE YOU DO THIS WEEK
HAS EVERYTHING IN
THE WORLD TO DO WITH
AN 18-SPEED BIKE.
WHEN YOU FIND OUT A GEAR
RATIO IT IS THE COMPARISON
OF NUMBER OF TEETH ON
THE REAR SPROCKET
TO THE NUMBER OF TEETH
ON THE FRONT SPROCKET.
SO WHAT I DID, OBVIOUSLY,
WITH EACH ONE OF
THESE SPROCKETS FOR THE
EXERCISE YOU'RE DOING,
I COUNTED THE
NUMBER OF TEETH.
FOR INSTANCE, ON THIS ONE
THE SMALL ONE AT THE TOP;
ONE, TWO, THREE, FOUR,
FIVE, SIX, SEVEN, EIGHT,
NIGHT, TEN, ELEVEN,
TWELVE, THIRTEEN, FOURTEEN.
NOW I DID THAT FOR
EVERY SINGLE SPROCKET,
THEN I'VE ASKED YOU
A COUPLE OF RATHER
INTERESTING PROBLEMS
TO DO WITH IT.
WELL, IT'S JUST ABOUT THAT
TIME FOR ME TO GO OVER
TO THE DESK AND LOOK
INTO A COUPLE OF THINGS
THAT I SAID I WOULD.
FOR INSTANCE,
THE PARTHENON.
AS YOU CAN SEE,
THERE'S THE PARTHENON
AND I WANT TO TAKE A
LOOK AT HOW IT COMPARED
TO THAT RECTANGLE BUT
TO DO THAT I THINK
I'D BETTER DRAW A
RECTANGLE AROUND IT.
SO HERE'S THE BOTTOM
OF THE PARTHENON,
THE TOP OF THE PARTHENON
UP AT THE PEAK HERE.

Using a ruler, he draws the lines with a pink marker.

He continues I'LL JUST DRAW A
LINE ACROSS THERE.
THEN I'LL DRAW TO
THE OUTSIDE HERE.
AND I'LL COMPLETE THE
RECTANGLE OVER ON THIS SIDE.
RIGHT ABOUT THERE,
THAT'S REASONABLY CLOSE.
AND WHAT I'M GOING TO DO,
IS I'M GOING TO ACTUALLY
MEASURE THE LENGTH AND THE
WIDTH AND SEE WHAT WE GET.
LET'S START
WITH THE LENGTH.
NOW THIS IS, REMEMBER,
IS AN APPROXIMATION.
THAT'S ABOUT 21cm.
AND THE HEIGHT
OR THE WIDTH,
DEPENDING ON HOW YOU WANT
TO CALL IT, IS ABOUT 13.5.
SO THE LENGTH TO THE
WIDTH IN THE PARTHENON
IS APPROXIMATELY
EQUAL TO 21:13.5.
NOW, I WILL SUGGEST
TO YOU THAT THAT'S
REASONABLY CLOSE TO 1.6.

He writes the equation on a new piece of paper that reads “PHI equals empty is approximately equal to 1.614.”

He continues THIS NUMBER THAT I'VE PUT
UP HERE AT THE TOP
IS THE ACTUAL RATIO THAT WE
CALL THE GOLDEN RATIO.
PHI, WHICH HAS THIS SYMBOL,
IS APPROXIMATELY EQUAL
TO 1.614.
AS IT TURNS OUT,
THERE ARE MANY,
MANY PLACES WHERE
THIS RATIO IS USED.
IT IS FOUND THAT MOST
PEOPLE FIND THAT THIS
RECTANGLE, THE ONE THAT
WE'RE LOOKING AT HERE,
OF THE PARTHENON,
IS PARTICULARLY
PLEASING TO THE EYE.
SO, YOU SEE IT IN ART AND
ARCHITECTURE, THAT RATIO.
YOU ALSO SEE IT
IN PACKAGING,
YOU SEE IT WITH,
FOR INSTANCE,
MY CREDIT CARD
AND SO ON.
INTERESTINGLY ENOUGH,
IT'S NOT JUST ONE GOLDEN
RECTANGLE THAT THEY
USED IN THE PARTHENON.
THEY USED A
WHOLE BUNCH.
IF YOU WERE TO TAKE TWO
OF THE COLUMNS AND DRAW
A RECTANGLE AROUND THAT,
YOU'LL FIND THAT
THE LENGTH TO THE WIDTH
IS ALSO VERY CLOSE TO PHI.
IF YOU WERE TO TAKE ONE
LITTLE SEGMENT UP HERE
OF THIS BLOCK,
RIGHT IN THERE.
YOU'LL FIND THAT IS ALSO
EXTREMELY CLOSE
TO THE SAME RATIO, PHI.
NOW THAT I'VE SORT OF GONE
THROUGH THE WHOLE IDEA
THAT RATIOS ARE
IMPORTANT, THEY ARE USED,
I WANT TO GIVE YOU A COUPLE
OF FORMAL DEFINITIONS
JUST SO THAT YOU CAN NOTE
THEM DOWN IN YOUR BOOK
AND YOU CAN USE
THESE IDEAS WHEN YOU
ARE COMPLETING SOME
OF YOUR EXERCISES.

A blue piece of paper appears. It reads “Ratio. A number or quantity compared with another. 7 colon 3 or 5 to 6.”

He continues VERY SIMPLE
IDEA, ISN'T IT?
LIKE FOR INSTANCE, IT
COULD BE WRITTEN AS 7:3
OR IT CAN BE WRITTEN
AS A FRACTION, 5:6.
THERE ARE A FEW CONCEPTS
OR IDEAS WHICH ARE,
IN FACT, SOMEWHAT RELATED
AND SOMETIMES RATIOS
ARE MIXED UP WITH
THESE OTHER IDEAS
SO WHAT I'VE DONE,
IS I'VE ACTUALLY
PROVIDED ANOTHER DEFINITION
OF ANOTHER CONCEPT.
AND THAT'S THE
IDEA OF RATE.

He shows another piece of paper that reads “One quantity measured in relation to another. Km per hour.”

He continues THIS IS ALSO WHEN -
FOR INSTANCE,
KILOMETERS PER HOUR.
WELL, THAT'S
ANOTHER ONE.

He appears in a small window on the top left corner of the screen.

He continues BUT WHAT'S THE DIFFERENCE
BETWEEN RATIO AND RATE?
WELL, HERE IT IS.

A blue sheet of paper reads “Ratio: The units of the quantities compared must be the same. Rate: The units of the quantities compared must be different.”

He continues IF I GO BACK TO THIS ONE,
THE RATE IS KILOMETRES,
WHICH IS A LENGTH IDEA
COMPARED TO A TIME IDEA,
HENCE, IT'S A RATE.
THE LAST IDEA, WHICH I'LL
ADDRESS FOR A MOMENT OR
TWO IS THE IDEA
OF PROPORTION.
THIS IS WHEN YOU
COMPARE RATIOS.
THE FORMAL DEFINITION
GOES LIKE THIS:

He reads “Two pairs of numbers are in proportion if the ratio formed by the first pair equals the ratio formed by the second pair.”

He continues HMM, WHY WOULD I EVER
WANT TO DO THAT?
I'LL GIVE YOU A
SIMPLE EXAMPLE.
IF I WERE TRYING TO
GET SOME SENSE OF WHAT
MY HEIGHT WAS IN COMPARISON
TO THE CN TOWER,
IT'S PRETTY HARD FOR ME TO
DO IT BECAUSE WHEN I STAND
UNDERNEATH THE CN TOWER,
I CAN'T STAND BACK
FAR ENOUGH TO ACTUALLY
SEE THE HEIGHT OF BOTH
AND GET A REAL SENSE
OF HOW MUCH BIGGER
THE CN TOWER IS THAN ME.
SO WHAT YOU DO IS YOU
SET UP A PROPORTION.
IN THIS CASE, WE
COULD DO IT THIS WAY.
IF MY HEIGHT WAS THE
SAME AS THE CN TOWER,
WHAT WOULD I LOOK LIKE
AT THE BASE OF IT?
WELL, I'VE DONE SOME
CALCULATIONS AND I FOUND
THAT IN FACT, IF YOU TAKE
A LOOK AT THE HEIGHT
OF THIS LEGO BLOCK, WHICH IS
ABOUT HALF A CENTIMETRE.
SEE?
I'M GOING TO LOOK
FROM OVER TOP.

He places a yellow Lego block on a paper and continues
A LEGO BLOCK; YOU KNOW THAT
THESE AREN'T VERY THICK,
IT'S ABOUT
HALF A CENTIMETRE.
IF I WERE TO STAND UP ON
THIS DESK RIGHT BESIDE
THAT LEGO BLOCK.
STAND UP RIGHT THERE THAT
WOULD BE ME AND ME,
I, WOULD BE THE CN TOWER.
SO THAT'S THE IDEA
OF PROPORTION.
IT GIVES ME A REAL SENSE,
IN FACT, HOW MUCH BIGGER
THE CN TOWER IS THAN ME
USING SOME MEASUREMENTS
THAT I, AT LEAST, CAN
SEE AND UNDERSTAND.
NOW, THE LAST THING I'M
GOING TO SAY BEFORE
I LET YOU GO IS THAT YOU'LL
BE WORKING ON EXERCISE
OR CHALLENGE NUMBER 3.
IT INVOLVES RATIO,
MOSTLY RATIO,
NOT THE OTHER
TWO SO MUCH.
IT INVOLVES BICYCLES.
I WISH YOU BEST OF LUCK
WITH THE EXERCISE
AND WE'LL TAKE IT UP WHEN
WE'RE LIVE, NEXT PROGRAM.

Watch: Challenge #3