An Explorer lesson in which students use shadows and geometry to measure the earth.
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Author: Jane Rich, Shawnee High School, Shawnee, OK
Grade Level/Subject: 9-12
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 casts 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 equalled 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.
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:
- Work effectively in a small group to take accurate measurements at a specific time.
- Use their knowledge of geometry and trig. to determine the measure of an angle.
- Use significant digits in their reports.
- Calculate percent error.
- Use their research skills to determine accepted values.
- Understand the value of cooperation in achieving a common goal.
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:
- 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.
- 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.
- 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.
- 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 pushing 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
- 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:
- Discuss the sources of error and the fact that your results depend on other people making accurate measurements.
- I like to show the first tape of the "Cosmos," which tells about Eratosthenes.
- The next activity might be to measure the height of our flagpole indirectly.
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