Lesson Plan

1. Lesson Title/Focus: Measuring the Earth

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.

Teacher: I will ensure that the students can calculate volume, circumference, and surface area of a sphere.

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.

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.

5. Grouping Students for Instruction: Students will work in small groups of at least three.

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.

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.

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.

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.

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.

What do you think the circumference, volume, surface area, and radius of the earth are?

Why do you suppose this math was lost to later civilizations after Eratosthenes discovered it?

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.

Rubric:

CATEGORY	4	3	2	1
Mathematical Concepts	Explanation shows complete understanding of the mathematical concepts used to solve the problem(s).	Explanation shows substantial understanding of the mathematical concepts used to solve the problem(s).	Explanation shows some understanding of the mathematical concepts needed to solve the problem(s).	Explanation shows very limited understanding of the underlying concepts needed to solve the problem(s) OR is not written.
Mathematical Terminology and Notation	Correct terminology and notation are always used, making it easy to understand what was done.	Correct terminology and notation are usually used, making it fairly easy to understand what was done.	Correct terminology and notation are used, but it is sometimes not easy to understand what was done.	There is little use, or a lot of inappropriate use, of terminology and notation.
Explanation	Explanation is detailed and clear.	Explanation is clear.	Explanation is a little difficult to understand, but includes critical components.	Explanation is difficult to understand and is missing several components OR was not included.
Working with Others	Student was an engaged partner, listening to suggestions of others and working cooperatively throughout lesson.	Student was an engaged partner but had trouble listening to others and/or working cooperatively.	Student cooperated with others, but needed prompting to stay on-task.	Student did not work effectively with others.
Completion	All problems are completed.	All but 1 of the problems are completed.	All but 2 of the problems are completed.	Several of the problems are not completed.

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1. Let’s set Ellensburg High School as the cooperating school. We use an Internet map site to locate our school and Ellensburg High School and then measure the distance between them.

2. Let’s do these both on the same day and at noon, one at our school and another at Ellensburg High School: we put an equal length stick on a blank sheet and measure its shadow at noon.

3. Suppose we only get a shadow at noon at one of the school. We need to measure the height of the stick and the length of its shadow.

4. Suppose the earth was round and the sun’s rays are parallel. We can calculate the angle formed by the shadow and the imaginary line from the end of the shadow to the top of the post by using the formula of tangent. We can input our measurement in a calculator to get the value of the angle. This angle will be equal to the angle at the earth’s center formed by imaginary lines from the two schools.

5. After we figure out the angle, we can use it to calculate the circumference, radius, volume, and surface area of the earth.