Objectives of this unit: The students will learn: - what it was like to solve geometric problems in the third century BC
- how to use proportional strategies to solve problems
- the relationship between the circumference and diameter (and radius)
- the relationship between the circumference and the measure of the central angle
- (or review) how to measure angles using a protractor
Opening idea: When did people first realize that the
earth
might be round rather than flat? By the third century BC the scholars
of
that time believed that the earth was a sphere. One empirical piece of
evidence is observing ships as they approach the horizon. The sails
appear
to dip into the ocean. But no one knew for sure how big this sphere
was.
It took a curious & ingenious librarian named Eratosthenes to
discover
a remarkably simple method for measuring the circumference of the
earth.
This unit is the story of how Erastosthenes measured the earth.
Show the students the book "The Librarian Who Measured the Earth" by Kathryn Lasky (Little, Brown and Co., 1994). (You can find out more about the book at Amazon.com) As you tell them the story, show them the illustrations in the book. You can also coordinate your talk with the pages on this website. The suggested pages will be indicated below.
As the librarian of the world renown library in Alexandria, Eratosthenes had access to the latest scientific knowledge of his day. He believed the earth was round. But how "round" was it? That was the question that stumped him. He may have imagined that the cross-section of the earth was like an orange sliced in half. You could see the sections of the orange. Also, if you walked along the edge of the orange your path would be circular. We know how to measure distances between two places on a flat surface. But how do you measure something round? Hold up a globe and ask the students how they might measure this globe. The suggestion may be to use a string, wrap it around the globe, and measure the wrap distance. Take a piece of string or rope, wrap it around the earth, mark the string, unwrap it, and measure the rope with a ruler.Have the students try it with the globe. Does it matter where the string is wrapped? (Yes, it should be at the widest possible place.) After the student comes up with an answer, ask the class if Eratosthenes might have done it this way. Walking around the world would be very difficult. Certainly, if he didn't get tired of walking he would not have managed the oceans. Sailing ships were not able to cross the oceans back then. So what other way might he have done this? How about option 2? What if Eratosthenes could drill a tunnel to the other side of the earth and measure that distance could he then figure out the circumference of the earth? The answer is yes. Eratosthenes knew this because he had studied some geometry. Now have the class do an experiment to help them discover the relationship between circumference and diameter. Here are two activities that the students can do. Hold up a circle and a diameter (i.e. a circular hoop and a stick with the same length as the diameter of the hoop.) Show them that the stick fits exactly across the widest part of the hoop.
In this Javasketchpad sketch notice that the diameter is in pixels. If you multiple the diameter by the "times number" it will equal the circumference. On the Javasketchpad, drag the center of the circle to make it bigger (or smaller). Notice that the times number doesn't change much. Why not? What's another name for the "times number"? (pi) The important thing for students to understand is that the cirumference is always a little more than 3 times (pi) the diameter and it doesn't matter what the size of the circle is! Another way to say this is that the ratio of the circumference to the diameter stays the same as you change the length of the diameter. The proportionality of C / D holds! Since straight lines are usually easier to measure than round ones, it is easier to measure the diameter than the circumference. If Eratosthenes knew the earth's diameter all he would have to do is multiply the diameter by 3 1/7 or 3.14 (approximations for pi) to determine the circumference. |