Students will learn how Eratosthenes, a Greek mathematician, calculated the circumference of the Earth using measurements from Alexandria and Aswan in Egypt. Working in groups, students will use maps and the Internet to find these cities and measure distances. They will then perform calculations to estimate the circumference, radius, volume, and surface area of the Earth. Finally, students will write a paper evaluating the accuracy of their results and discussing their initial conjectures.

1. Martha Vasquez
Beatriz Sarabia
Alma Vaca
Heidi Miedecke
Geography 300
February 28, 2013
Lesson Plan 5
Lesson Title/Focus: Measuring the Earth
6th Grade
Math GLEs:
1.3.1 Make and test conjectures about two-dimensional and three-dimensional shapes and
their individual attributes and relationships using physical, symbolic and technological
models.
5.2.2. Recognize the mathematical contribution of a person or culture.
NETS-S:
3.b. Students will use the Internet to find map sites. They need to look up Alexandria,
Egypt, and Scene (currently called Aswan) to see where these cities are located. The
students will also locate the cooperating school by the Internet map and evaluate its
distance from their location.
3.d. After collect all values, students use calculators to process data and evaluate problems.
Science GLEs:
1.2.5. Understand that the Solar System is in a galaxy in a universe composed of an
immense number of stars and other celestial bodies.
3.2.1. Analyze how scientific knowledge and technological advances discovered
and developed by individuals and communities in all cultures of the world
contribute to changes in societies.
2. Input: Benjamin Peirce said mathematics is the science that draws necessary
conclusions. To study science, people have to learn math first. This lesson is guiding
students to experience the method that the ancient Greek mathematicians used to estimate
circumference of the Earth by limited tools and math knowledge.
3. Instructional Materials Needed: Calculators, papers, pencils, computers with
Internet, and a meter stick.
4. Lesson Objective (Learning Targets)

2. Teacher: I will ensure that the students can calculate volume, circumference, and surface
area of a sphere.
I will ensure that the students can read maps and discuss planets.
Students: Students will be able to calculate various measurement of the Earth.
Students will be able determine distances on a map.
5. Grouping Students for Instruction: Students will work in small groups of at least
three.
6. LEARNING EXPERIENCES:
Engage
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. This is interesting since the Italians refuted
his claim that the earth was round about 1000 years later. He knew that at noon on the
day of the summer solstice in Alexandrian, Egypt, a vertical post casts a shadow. At
the same time in Scene, a town directly to the south, a vertical post casts no shadow.
Eratosthenes used Euclidean geometry to determine that the angle formed by the
shadow and an imaginary line from the end of the shadow to the top of the post
equals and 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 Scene, and multiplying it by the number of times the angle at the
earth’s center is contained in 360 degrees.
Explore
The students will use the Internet to find map sites. They will look up Alexandria,
Egypt, and Scene (currently called Aswan) to see where these cities are located.
The students should also locate a cooperating school on a map and evaluate its
distance from their location. Students may need to measure the tangent, as this
concept will be used in the calculation. If the students have not yet learned this
concept the instructor will do this portion of the calculation or bring in some pare
calculus students to work with the groups. This experience will show the students of
how math will be applied in upper level classes. Students will need to practice the
activity prior to the actual experiment. The measurements will need to be done at both
schools at noon at the same date. They should measure the height of a stick and the
length of its shadow. Students should be asked prior to the experiment to conjecture
about the circumference, volume, surface area and radius of the earth.
Explain

3. The measure of the angle is found by dividing the length of the shadow by the height
of the object, which would be the tangent of the angle you will be using for this
experiment. The angle measure can then be determined. This is not the central angle.
The angle from the other school must be subtracted from this angle measure and the
absolute value of this difference is the central angle. The circumference of the earth
can then be calculated by setting up a ratio and solving for the circumference.
Elaborate
Once the students have solved the circumference, they can then use the equation 2eπr
to find the radius. The students could then use the radius to find the volume and
surface area of the earth. Students should research these measurements to determine
the accuracy of their calculations.

4. Evaluate
Students would write a short paper about their experiment and the accuracy of their
calculations. This paper would then be graded on the use of proper vocabulary and
the correct use of the equations. Students should be encouraged to include any
conjectures they made during the experiment and whether they were validated or not.
Checking for Understanding/Questions:
What do you think the circumference, volume, surface area, and radius of the earth
are?
How do we calculate each of these measurements?
Why do you suppose this math was lost to later civilizations after
Eratosthenes discovered it?
What other applications are there for these measures?
Closure:
Aristarchus used similar triangles to create a relationship between the radius of the Sun and Earth
and the diameter of the Moon to the distances between these objects. Once Eratosthenes
discovered this method for calculating the circumference of the earth the radius was then
calculated and could be plugged into the equations to determine our distance from the Sun and
Moon. Math and science have evolved this way with conjectures that are later proved and used in
ways the person who discovered the technique had not imagined. Being able to read a map and
determine distances is useful in many situations. You have discovered with this experiment how
this information can be used to calculate volume, surface area, and circumference.