TITLE:   Measuring the Earth

AUTHOR:  Jane Rich, Shawnee High School, Shawnee, OK

GRADE LEVEL/SUBJECT:  (9-12)

OVERVIEW:  Many students have little understanding or
appreciation for the process of science. I like to get my
students involved in doing science, taking measurements,
collecting data at the very beginning of the school year.
so, I usually start this activity during the first week and
call the local newspaper to come out and take pictures of
the students.

Eratosthenes, a Greek mathematician, was the first to
measure the circumference of the earth. he based his
measurement of the earth on the assumptions that the earth
was round and the sun's rays are parallel. He knew that at
noon on the day of the summer solstice in Alexandria, Egypt,
a vertical post cats a shadow. At the same time in Syene, a
town directly to the south, a vertical post casts no shadow.
Eratosthenes used Euclidean geometry to determine that the
angle formed by the post and an imaginary line from the end
of the shadow to the top of the post equals an angle at the
earth's center formed by imaginary lines from the two towns.
He calculated the earth's circumference by measuring the
distance between Alexandria and Syene, and multiplying it by
the number of times the angle at the earth's center is
contained in 360 degrees.

PURPOSE:  The purpose of this activity is to get students
interested and involved in doing science, give them a reason
to use some of the math they have learned, and develop a
feeling of cooperation in working with people from other
schools.

OBJECTIVE(s):  As a result of this activity, the students
will:
1.   Work effectively in a small group to take accurate
     measurements at a specific time.
2.   Use their knowledge of geometry and trig. to determine
     the measure of an angle.
3.   Use significant digits in their reports.
4.   Calculate percent error.
5.   Use their research skills to determine accepted values.
6.   Understand the value of cooperation in achieving a
     common goal.

RESOURCES/MATERIALS:  The students will need a meter stick
or measuring tape and a scientific calculator. I like to
video tape the students doing the activity and send it to
the other school. Your geography teacher may have some good
maps to get the distance between schools.

ACTIVITIES AND PROCEDURES:
1.   Contact a class directly north or south of you (in a
     different state if possible) and set a specific date
     and time to take the measurements.
2.   Divide the class into groups and practice at least once
     before the day of the activity. They are to measure the
     height of an object ( a pole is good) and the length of
     its shadow at a specific time. I have them start 15
     minutes before the stated time.
3.   Assign several people to research the circumference of
     the earth and others to find several ways to determine
     the distance from your school to the other group's
     school (maps, auto clubs, etc.). Eratosthenes had a
     slave to pace off the distance between the two cities
     and report back to him.
4.   The measure of the angle is found by dividing the
     length of the shadow by the height of the object on
     your scientific calculator and then push 2nd function
     tangent. However, this is not the central angle. The
     angle from the other school must be subtracted from
     your angle and the absolute value of this difference is
     the central angle. The circumference of the earth can
     them be calculated by setting up a ratio and solving
     for the circumference.

     central angle       =         distance from schools
     ---------------               ----------------------
       360 degrees                     circumference

5.   The students will have to decide how many significant
     digits to use in their results and then calculate the
     percent error from the value they found in their
     research.

TYING IT ALL TOGETHER:
1.   Discuss the sources of error and the fact that your
     results depend on other people making accurate
     measurements.
2.   I like to show the first tape of the "Cosmos" which
     tells about Eratosthenes.
3.   The next activity might be to indirectly measure the
     height of our flagpole.