Kind: captions
Language: en
in 1982 there was one sat question that
every single student got
wrong here it is in the figure above the
radius of circle a is 1/3 the radius of
Circle B starting from the position
shown in the figure Circle A rolls
around Circle B at the end of how many
Revolutions of circle a will the center
of the circle first reach its starting
point is it a three Hales B B 3 C 6 D 9
Hales or E
9 sat questions are designed to be quick
this exam gave students 30 minutes to
solve 25 problems so about a minute each
so feel free to pause the video here and
try to solve
it what is your
answer I'll tell you right now that
option b or three is not
correct when I first saw this problem my
intuitive answer was B because the
circumference of a circle is just 2 pi r
and since the radius of Circle B is 3
times the radius of circle a the
circumference of circle B must also be
three times the circumference of circle
a so logically it should take three full
rotations of circle a to roll around
Circle B So my answer was
three this is
wrong but so our answers a c d and
e the reason no one got question 17
correct is that the test writers
themselves got it wrong they also
thought the answer was three so the
actual correct answer was not listed as
an option on the
test mistakes like this aren't supposed
to happen on the SAT for decades it was
the one exam every student had to take
to go to college in the US it had a
reputation for determining ing people's
entire Futures as a newspaper from the
time stated if you mess up on your sat
tests you can forget it your life as a
productive citizen is over hang it up
son of 300,000 test takers just three
students wrote about the error to the
College Board the company that
administers the SAT Shivan cartha Bruce
to and Doug young
rice I did a lot of math problems when I
was young for the competitions I
probably did thousands of mouth problems
I read it and I was amazed how badly
it's worded I just put three down I
figured that's what they wanted the
three students were confident none of
the listed answers were correct and
their letters showed it as a director at
the testing service recalled they didn't
say they had come up with possible
alternative answers or that maybe we
were wrong they said flat out you're
wrong and they proved
it I discussed it with some other people
and said I think there was a mistake and
they mostly said no one
cares I wrote a letter to the
educational testing service it was a
little while later they called us and
said I was
correct here is their
argument the simplest version of this
problem is with two identical coins
these have the exact same circumference
so by our initial logic this coin should
rotate exactly once as it rolls around
the other so let's try it
[Music]
okay but wait we can see it's already
right side up at the halfway point so if
we finish rolling it around the other
coin it'll have rotated not once but
twice even though the coins are the
exact same size there are no tricks here
you can try it for yourself and I'll do
it again
slowly it's
one
[Music]
two this is known as the coin rotation
Paradox this Paradox also applies to
question 17 I've made a two-scale model
of the problem one useful tip for
standardized tests even though they say
their images are not to scale they
almost always
are so when we roll Circle A around
Circle B we can see that it rot R Ates
once
twice three
times and four times in total so the
correct answer to this question is
actually Four once again the circle
rotates one more time than we expected
to understand this let's wrap this
larger Circle in some
[Music]
ribbon
and I'll make it the same length as the
circumference and then I will stick it
down to the table as a straight line I'm
adding some paper here uh so there's
something for this to roll
[Music]
on and now it rolls one
two three times what's happening when we
turn this straight path into a circular
one is that Circle A is now rolling the
length of the circumference and it's
going around a circle the shape of the
circular path itself makes Circle A do
an additional rotation to return to its
starting point so this is the general
solution to the problem find the ratio
between the circumferences of Circle B
and circle a and then add one rotation
to account for the circular path
traveled but there is a way to correctly
get three let's count the rotations of
circle a from the perspective of Circle
B looking out at a we can see Circle A
rotates
one
2 three
times and it doesn't matter which Circle
you are looking from to Circle A it also
rotates three times to come back to its
starting position around Circle
B similarly from the perspective of the
coins we can see that the outer coin
only rotates once as it rolls around the
inner
coin using the perspective of a circle
is just like turning the circle's
circumference into a straight
line it's only as external observers
that we actually see the Outer Circle
travel a circular path back to its
starting point giving us the one extra
rotation but there's even another
answer if you look closely at question
17 it asks how how many revolutions
Circle A makes as it rolls around Circle
B back to its starting point now in
astronomy the definition of a revolution
is precise it's a complete orbit around
another
body the Earth revolves around the sun
which is different from it rotating
about its axis so by the astronomical
definition of a revolution circle a only
revolves around Circle B once it goes
around one
time now other definitions of Revolution
do include the notion of an object
rotating about its own axis so one isn't
a definitive answer but the wording of
this question is extremely ambiguous if
you can justify at least three different
solutions after reviewing the letters
from the students the College Board
publicly admitted their mistake a few
weeks later and nullified the question
for all test
takers they said they were discounting
the problem and they were calling us
because they were going to tell the news
and they thought that that we should be
warned that the news might contact us I
did a bunch of phone interviews and NBC
news they came to my school they said
they were they said I was right and they
were discounting and so that was
great but there's more to the
explanation it's easy to get an
intuitive reason but it's really hard to
formally prove that the answer is for I
could give you some proofs if you want
well that would be wonderful I think
that would be um we'd appreciate that
for sure I have a whiteboard because I'm
a mathematician so I just happen to have
a whiteboard here hold
on can you see that yep it turns out
that the amount the small circle rotates
is always the same as the distance the
center travels all right so why is this
true suppose you had a camera and the
camera was always pointed at the center
so in your movie it looks like the
center doesn't
move in the real world the center is
going around the circle
let's say it's going at some speed V
what's the velocity of this point it's
zero and that's because it's rolling
without slipping if it had any component
in that direction that's what slipping
would be I mean this is something I
think they should have spelled out in
the problem but when you change your
frame of reference the relative
velocities don't change in the movie The
Center always has velocity zero so this
point would have to have velocity NE V
so that means the speed that this is
turning is the same as the speed the
center is moving so if they always have
the same speed they have to go the same
total
distance the total distance this turns
has to be the same as the total distance
the center
moves in this problem the center of the
small circle goes around a circle of
radius 4 so the total distance that the
center moves is 8
Pi What's the total amount that the
small circle rotates it rotates four
times
and its circumference is 2 pi it's the
same number if it rolls without slipping
the total distance the center travels is
the same as the total amount of
turns and this is always true take a
circle rolling without slipping on any
surface from a polygon to a blob on the
outside or the inside the distance
travel by the center of the circle is
equal to the amount the circle has
rotated so just find this distance and
divide it by the circle's circumference
to get how many rotations it's made this
is an even more General solution than
our answer to the coin Paradox where we
just took our expected answer which
we'll call n and added one and it
reveals where this shortcut comes from
if a circle is rolling continuously
around a shape the Circle Center goes
around the outside increasing its
distance traveled by exactly one
circumference of the circle so the
distance traveled by the circle's Center
is just the perimeter of the shape plus
the circle circle's circumference when
we ultimately divide this by the circle
circumference to get the total number of
rotations we get n + 1 if a circle is
rolling continuously within a shape the
distance traveled by the Circle Center
decreases by one circumference of the
circle making the total number of
rotations n minus one if the circle is
rolling along a flat line the distance
traveled by the Circle Center is equal
to the length of the line which divided
by the circle circumference is just
n this General principle extends far
beyond a mathematical fun fact in fact
it's essential in astronomy for accurate
timekeeping when we count 365 days going
by in a year
365.24 to be precise we say we're just
counting how many rotations the earth
makes in one orbit around the Sun but
it's not that simple all this counting
is done from the perspective of you on
Earth to an external Observer they'll
see the Earth do one extra rotation to
account for for its circular path around
the Sun so while we count
365.24 days in a year they count
36.2 4 days in a year this is called a
siderial year siderial meaning with
respect to the Stars where an external
Observer would be but what happens to
that one extra day a normal solar day is
the time it takes the sun to be directly
above you again on Earth but the Earth
isn't just rotating it's orbiting the
Sun at the same time so in a 24-hour
solar day Earth actually has to rotate
more than 360° in order to bring the sun
directly overhead again but Earth's
orbit is negligible to distant stars to
see a star directly overhead again Earth
just needs to rotate exactly
360°. so while it takes the sun exactly
24 hours to be directly above you again
a star at night takes only 23 hours 56
minutes and 4 seconds to be above you
again that's a siderial
day this explains where the extra day
goes in the siderial
year if we start a solar day and a
siderial day at the same time we'd see
them slowly diverge throughout the year
after 6 months the siderial day would be
12 hours ahead of the solar Day meaning
that noon would be midnight and it would
keep moving up until it's finally one
full day ahead of the solar day at which
point a new year and orbit
begins
365.24 days that are each 24 hours long
are equal to
366.50 6 minutes and 4 seconds
long so it makes no sense to use
siderial time on Earth because 6 months
down the line day and night would be
completely swapped but equally it's
useless to use solar time while tracking
objects in space because the region
you're observing would shift between say
10: p.m. one night and 10 p.m. the next
night so instead astronomers use
siderial time for their telescopes to
ensure that they're looking at the same
region of space each night and all
geostationary satellites like those used
for communication or navigation they use
siderial time to keep their orbits
locked with the Earth
rotation so the coin Paradox actually
explains the difference between how we
track time on Earth and how we track
time in the
universe the rescoring of the 1982 sat
wasn't all good news with question 17
scrapped students scores were scaled
without it moving their final result up
or down by 10 points out of 800 now
while that doesn't seem like much some
universities and Scholar scholarships
use strict minimum test score cutoffs
and as one admissions expert put it
there are instances even if we do not
consider them justified in which 10
points can have an impact on a person's
educational opportunities it might not
keep someone out of law school but it
might affect which one he could go
to this mistake didn't only cost points
off the exam according to the testing
service rescoring would cost them over
$100,000 money that came out of the
pockets of test
takers
the question 17 Circle problem was far
from the last error on the
SAT but errors are likely the least of
their concerns these days I mean the SAT
is slowly becoming a thing of the Past
after covid-19 nearly 80% of
undergraduate colleges in the US no
longer require any standardized testing
and that 1982 exam well it didn't turn
out too badly for some how did you uh
how did you do on your math sat if I can
ask I got an 800
even before that it was clear I was
going to go into math I would I did math
competitions I did I really liked math
do you end up writing any math questions
these days a while back I wrote problems
for a math competition and were you
careful with how you wrote them the
wording and I hope
so I
tried today's Deep dive on one sat
question proves there's no substitute
for Hands-On exploration to understand
and appreciate the everyday phenomena of
our world but you don't have to observe
Earth from space or make cardboard
cutouts to get hands- on with math
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