Wednesday, September 28, 2011

Class 2: 08/30/2011-Introduction/ Eratosthenes Problem

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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