Class began with everyone giving a short introduction to the class. The introductions included majors, what lead to us Uteach, plans after graduation, and interesting facts.
After this, we
watched the first five minutes of video clip in which Carl Sagan discussed the ancient
academic Eratosthenes.
Dr. Sagan
related the story of how Eratosthenes deduced that the Earth was curved, and
was able to calculate the circumference of the earth based on the observations
of shadows made by upright sticks in two different African cities. Before
he explained how this was accomplished over two millennia ago, the clip was
paused so that the class could try to solve the problem: Given the observations
that the distance between Alexandria and Syene was approximately 800
kilometers, and that an upright stick in Syene will cast no shadow and a stick
in Alexandria casts a shadow of about 0.1219 meters, how would one calculate
the circumference of the Earth?
The class
broke up into eight groups of three, and each group was given one copy of the
problem to solve (I note this because Tara pointed out that giving a group only
one copy of a problem encourages collaboration, otherwise each member would
simply read their own copy independently). After working on
the problem in groups, we had a class discussion about how our groups came to a
solution.
One group
presented their solution to the class and illustrated it on the board; one
group member explained that since the Sun is so large in relation to Earth and
is so far away, rays of light that reaches the Earth from the Sun travels
parallel to each other. Also, if two parallel lines are cut by a
transversal, the corresponding angles are congruent. The two sticks
(the one at Alexandria and the one at Syene) both point to the center of the
Erath, where they form and angle (Angle A). Based on this information,
one can know the value of Angle A based on the value of its congruent angle,
namely the angle formed by the hypotenuse of the triangle formed by the stick
at Alexandria and its shadow (Angle B). One can get the value of Angle B
by solving for the inverse tangent of the ratio of the stick length over the
shadow length (which is about seven degrees). Since there are 360 degrees
in a circle, it stands to reason the ratio of 7/360 is the same as the ratio of
800 and the value of the Earth’s circumference.
Most of the
groups solved the problem in the same way. When asked if anyone got stumped
anywhere the groups who had not reached a solution said that they just needed
more time (that other groups had gotten to the solution before they had).
Some groups said that when they were setting up the problem, they drew a
diagram with a small sun shown next to a large Earth. They said this was
misleading because, even though many of them knew that the rays of light from
the Sun strike the Earth parallel to each other, their picture made it seem
that they would strike at different angles.
In solving
this problem, it was necessary to know certain factual information (which was
in fact tested in the online Diagnostic Test from our homework).
Conceptual information and problem-solving skills were also needed. These
three prerequisites are also called facts, concepts, and transfer, but these
were not discussed in any detail.
Each day in PBI a different student takes
responsibility for blogging about what goes on in class. Today’s blog is brought to you by Joan.
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