#1824. Measuring the earth (Eratoshenes' method)

Mathematics, level: Senior
Posted Tue Jun 27 17:52:38 PDT 2000 by colbe mazzarella (ccmazz@aol.com).
clark avenue school, chelsea, ma
Materials Required: Math journals, globe, protractor, calculator and safety geometric compass for each student, Professi
Activity Time: five one-hour lessons
Concepts Taught: Geometry, Geography, Ancient History

Lesson Plan
Teacher: Colbe C. Mazzarella Date: June 27, 2000
Class: Sixth Grade Math & History Time: 40 min.
Unit: Greek Geometry Day 1 of 5

Overview: As we study Greece in Ancient Civilizations, we explore Greek math and science.

Essential Questions: How was mathematics developed?
What does Math explain about our world?

Acknowledgments: Material from the following sources has been used in these lesson plans:
http://k12science.stevens-tech.edu/~ihor/sketch/paper_gsp.html
http://k12science.ati.stevens-tech.edu/noonday/noon.html
http://www.ehhs.cmich.edu/~harwoodd/Pi.html
http://ericir.syr.edu/Virtual/Lessons/Mathematics/Geometry/GEO0004.html

Habits of Mind:
Using mathematical skills to interpret information and solve problems, reflecting confidence in ability to do mathematics, exploring the relationship of mathematics to other subject areas, exploring mathematics in the course of daily life, integrating mathematics within a larger network of ideas, appreciating the beauty and fascination of mathematics, viewing mathematics in unconventional ways, generating new ways of thinking.

Massachusetts Frameworks:
Students engage in problem solving, communicating, reasoning, and connecting to represent and solve problems, using geometric models.

Prerequisite knowledge:
LMBAT determine pi by measuring circular objects.
LMBAT demonstrate that equal pairs of angles are formed when a line intersects parallel lines.
(use paper-folding method from http://k12science.stevens-tech.edu/~ihor/sketch/paper_gsp.html)
LMBAT analyze data by removing outliers, computing averages, and finding percentage of error.
LMBAT express measurements as a proportion and solve for an unknown.
LMBAT find a website when given the address, and use a mouse to drag and drop.
LMBAT describe the extent of the volumes in the library at Alexandria.

Obstacles and Solutions:
This is a unique lesson and timing could be hard to predict. (Timing is especially important because the culminating exercise for the five lessons must take place at noon on the equinox.) Students could misuse materials. Must keep pace moving smoothly to minimize gaps. Solution: Do a dry run of lesson to check timing and spot potential slow spots. Have teams picked out in advance with a mix of skill levels.

Student Goal: LWBAT reproduce Eratosthenes' measurement of the Earth.Preparations: Schedule unit to begin four days before equinox.
Contact another school located at some distance from our school, preferably through the Noon Shadow Project: http://k12science.ati.stevens-tech.edu/noonday/noon.html. (Alternatively, use a point on the Equator as a reference location.)
Ask principal about photo releases. Call local newspaper to suggest that they send a photographer when students measure the earth.
Prepare press release including diagram and names of all students as Certified Geometers.
Determine what time sun will be highest on date of equinox (not 12:00 p.m. in all locations)

Materials:
Math journals, globe
The Librarian Who Measured the Earth by Kathryn Lasky (although this is sold by Amazon for ages four through eight, it has been reviewed as suitable for third through fifth grade, and it uses an engaging, aesthetic approach to an adult-level problem in mathematics)

Learning Objective: At the end of Lesson 1, LWBAT describe Eratosthenes' method of measuring the world from a personal and historical perspective.

Time Lesson Procedures
10:00 Introduction: Remind students that we previously learned that the derivation of geometry (geo, earth and metria, measurement) related to its use for surveying land. Explain that we will now use geometry to literally measure the earth, making us geometers.
When was it known that the earth was round? (By 300 BC)
How was that discovered? (Observations such as the circular shadow of the earth in an eclipse, ships apparently sinking below the horizon, etc.)
Can we measure the earth using the methods we have learned for measuring circles? (can't drill a tunnel through earth to measure diameter)
How can the size of the earth be measured? (driving, flying, orbiting around the earth)
10:10 Read aloud to students, The Librarian Who Measured the Earth by Kathryn Lasky, showing them the illustrations.
Locate Alexandria and Syrene on globe.
10:30 Students sketch Eratosthenes' experiment in journals (with equal pairs of angles labeled).
10:40 End

Modifications
Walk around to show pictures and globe to students with limited vision or attention span.

Assignment
Students will write and illustrate a two-page story of Eratosthenes' measurement of the world from the point of view of his slave who paced out the distance from Alexandria to Syrene.

Assessment
Walk around room and review sketches for rough accuracy.
Read stories in journals for accuracy. Lesson Plan
Teacher: Colbe C. Mazzarella Date: June 27, 2000
Class: Sixth Grade Math & History Time: 40 min.
Unit: Greek Geometry Day 2 of 5

Preparations: Rehearse all physical demonstrations

Materials: Math journals, protractor, calculator and safety geometric compass for each student
orange, grapefruit, cantaloupe, honeydew melon, toothpicks, flashlight, marker, knife

Student Goal: LWBAT reproduce Eratosthenes' measurement of the Earth.

Learning Objectives: At the end of Lesson 2, LWBAT conduct a miniature version of Eratos-
thenes' method of measuring the world, and will understand the limitations of the miniature; LWBAT reproduce Eratosthenes' diagram and use it to find a circumference.

Time Lesson Procedures
10:00 Insert two toothpicks into each fruit and use a marker to label them A and S (for Alexandria and Syrene).
Dim classroom lights and shine flashlight onto fruit.
Turn fruit so that light shines directly on toothpick at S and it does not cast a shadow.
Mark length of shadow of toothpick at A. Repeat for each fruit.
Write measurements on board (length of shadow, height of toothpick)
10:10 Have each student diagram all four demonstrations in journal (using protractor and compass) and write in measurements for each fruit.
Write proportion on board: angle : 360 = distance : circumference.
Have each student use proportion to solve for circumference of each fruit.
Have each student use circumference to find radius of each fruit.
10:25 Measure actual radius and circumference of each fruit and post on blackboard.
Slice orange along the "equator" to show how angles meet at the center.
Ask students what sources of error might exist in this experiment (non-spherical fruit, moving shadow, mis-measurement)
10:40 End

Modifications: If students are having difficulty with concept, have them compare fruit to sketch. Redo calculations if necessary.

Assignment
Students recalculate radii and circumferences, underestimating each shadow by one millimeter. Ask them to look for the pattern of error introduced for various-sized fruits. (1mm is a larger percentage of error in the smallest fruit.)
To be graded and redone if calculations are not 100% correct. This will show that students have mastered the basic formula of Eratosthenes' method.

Assessment Check journals for accuracy of sketches and measurements.
Check homework for 100% accuracy. Lesson Plan
Teacher: Colbe C. Mazzarella Date: June 27, 2000
Class: Sixth Grade Math & History Time: 40 min.
Unit: Greek Geometry Day 3 of 5

Preparations: Arrange for video player; locate tape at beginning of segment on Eratosthenes.
One week in advance, and one day in advance, verify that 1) websites are accessible from computer lab,
2) Java is enabled to make graphics movable, and 3) mice are functioning.

Materials:
Cosmos videotape on Eratosthenes (first episode)
Worksheets (see attached)

Student Goal: LWBAT reproduce Eratosthenes' measurement of the Earth.

Learning Objective:
At the end of Lesson 3 LWBAT manipulate computerized geometric models and interpret resultant data showing various approximations of pi and of the earth's circumference.

Time Lesson Procedures
10:00 Watch segment of Cosmos videotape on Eratosthenes.
10:10 File to computer lab
10:15 Pass out worksheet with website addresses and questions (see attached)
Students visit websites and answer questions on worksheet.
Students print out a table of results from schools around the world for use in their homework assignment.
10:30 Give class five-minute warning to be sure they have printed out table for homework.
10:35 File back to class.
Flag each participating school on world map.
10:45 End

Modifications
Supervise students to be sure they can and do locate websites. Tutor students who are unfamiliar with mouse drag-and-drop technique.

Assignment
Estimate the earth's circumference by calculating the average circumference from all schools.
Cross out the outlying circumferences.
Re-estimate the circumference by calculating the average of the remaining circumferences.
Compare your two estimates with the earth's actual circumference of 40,075.16 kilometers.
What is the percentage of error for each of your estimates?
To be turned in and redone if calculations are less than 75% correct.

Assessment
Monitor students for net-literacy
Check homework for substantial accuracy. Lesson Plan
Teacher: Colbe C. Mazzarella Date: June 27, 2000
Class: Sixth Grade Math & History Time: 40 min.
Unit: Greek Geometry Day 4 of 5

Preparations:
Rehearse all physical demonstrations
Divide class into teams of three or four with a mix of skills.

Materials:
Professional plumb bob
Meter stick for each student
Math journals

Student Goal: LWBAT reproduce Eratosthenes' measurement of the Earth.

Learning Objective:
At the end of Lesson 4 LWBAT design and test a mechanism for keeping a meter stick vertical.

Time Lesson Procedures
10:00 Demonstrate the use of a professional plumb bob.
Adjust height of bob by pulling string over finger and point out that it is a simple form of the pulleys shown in our textbook as developed by Archimedes.
Test doorways, blackboards, walls, etc. for a true vertical.
10:05 Read team rosters, break into teams and give one meter stick to each team.
Explain that each team will design and create a mechanism for holding a meter stick vertical, including a method to check for a true vertical.
Teams develop designs and sketch in journals (could use a bookend or soda bottle)
10:20 Each team presents design idea to class
10:30 Teams reassess designs and re-sketch if necessary.
10:40 End

Modifications
Review each design for feasibility. If students cannot visualize design flaws, allow them to implement flawed design but have them make a simple backup mechanism in case of failure.

Assignment
Create mechanism as designed.

Assessment
Team review, class review and teacher review of designs. Lesson Plan
Teacher: Colbe C. Mazzarella Date: June 27, 2000
Class: Sixth Grade Math & History Time: 40 min.
Unit: Greek Geometry Day 5 of 5

Preparations: Rehearse all physical demonstrations

Materials:
Math journals, pencil, protractor, calculator and safety geometric compass for each student
Magnetic compass, masking tape and legal-size paper.
Empty soda bottles and sand for replacement mechanisms.
String and keys for replacement plumb bobs
Table of school results printed for previous assignment
Certificate honoring each student as a Certified Geometer (use card stock; cite Eratosthenes and derivation of word "geometry")

Student Goal: LWBAT reproduce Eratosthenes' measurement of the Earth.

Learning Objective: At the end of Lesson 5 LWBAT measure the earth's circumference.

Time Lesson Procedures
11:40 (time may be adjusted for local sun angle)
File out to schoolyard with mechanisms, pencils, protractors, masking tape and legal paper.
11:45 Teams of 3-4 students tape paper on ground, tape mechanism into position, and measure shadow of meter stick at two-minute intervals.
Check meter sticks for true vertical.
Announce results to all teams. Outlying measurements should be recalculated to give all students sufficient chance to produce usable data.
Between measurements students will make scale drawings of results and use protractor to find angle of sun on drawings.
Use magnetic compass to note which direction the shadow points.
12:15 File back to classroom.
12:20 Analyze pooled data, estimate time of local noon, remove outliers and average results.
(To be sent to Noon Shadow Project for their website.)
12:40 Using table of school results from previous assignment, compute earth's circumference.
12:50 Discuss reasons for variations and select most reliable result.
1:00 Present Certified Geometer certificate to each student.

Modifications
For students with short attention spans, especially in this outdoor environment, keep up momentum between measurements by monitoring students to be sure that they are sketching results accurately in journals.

Assessment
Monitor measurements as they are taken and compare them to results of other teams. Be sure that team locates the source of any errors immediately, to guard against possibility that a team would produce data that the class must discard.Follow-up Acknowledgment Event
Link school webpage to "quilt" at http://www.geocities.com/Athens/8231/noonschs.htm
Check http://www.geocities.com/Athens/8231/nooncal97.htm daily.
When results are published, have students use measurements from other schools to compute circumference of earth in their journals, and write a self-assessment of the results.
Invite other classes to classroom to view mechanisms and journals on display for each team. Bring up http://www.geocities.com/Athens/8231/noonschs.htm on classroom computer, which will show our school on "quilt" of schools, and will play the song "It's a Small World After All."
Websites Related to Eratosthenes' Measurement of the Earth

Visit four of the following websites and answer all questions below.

1.) http://www.geocities.com/Athens/8231/nooncal97.htm

Print out the table of results from schools to use for your homework assignment.

2.) http://k12science.stevens-tech.edu/~ihor/sketch/pi2.html

Use your mouse to adjust the size of a circle, and see how pi is recalculated for the adjusted circle.

What is the highest calculation of pi that you can produce? ______________

What is the lowest calculation of pi that you can produce? ______________

What is the percentage of difference? ______________

3.) http://www.geocities.com/Athens/8231/noongeom.htm
OR
http://mvhs1.mbhs.edu/mvhsproj/suncp.html

What could you add to your diagram to make it clearer? ________________________

_____________________________________________________________________

4.) http://k12science.stevens-tech.edu/noonday/story.html
OR
http://k12science.stevens-tech.edu/~ihor/sketch/paper_gsp.html

Use the mouse to move the two "cities." Notice what Eratosthenes' results would have been if he had used two cities farther apart or closer together.

What is the highest calculation of the earth's circumference that you can produce?

______________

What is the lowest calculation of the earth's circumference that you can produce?

______________

What is the percentage of difference? ______________