Circumference of the Earth
Lesson Plan

• From National Council of Teachers of Mathematics¹
  • Reasoning and Proof: Instructional programs from pre-kindergarten through grade 12 should enable all students to:
    • Make and investigate mathematical conjectures
  • Communication: Instructional programs from pre-kindergarten through grade 12 should enable all students to:
    • Communicate their mathematical thinking coherently and clearly to peers, teachers, and others
    • Use the language of mathematics to express mathematical ideas precisely.
  • Connections: Instructional programs from pre-kindergarten through grade 12 should enable all students to:
    • Recognize and use connections among mathematical ideas
    • Recognize and apply mathematics in contexts outside of mathematics
  • Representation: Instructional programs from pre-kindergarten through grade 12 should enable all students to:
    • Create and use representations to organize, record, and communicate mathematical ideas
    • Use representations to model and interpret physical, social, and mathematical phenomena
• From Utah Educator Network:
  • Process Objective
    • Reasoning and proof:
      • investigate mathematical conjectures
      • formulate counter examples
      • realize that observing a pattern does not constitute proof
    • Communication:
      • Class and group discussions using precise language
      • Journal
    • Connections:
      • establish connections among mathematical expressions and physical models
      • use real-world applications
    • Representation:
      • use a variety of visual representations and tools (protractor, compass, straight edge, manipulatives)
      • represent patterns verbally, numerically, geometrically and algebraically

At the end of the lesson the student will be able to:

• Describe how at least one method of calculating the circumference of the earth.
• Demonstrate mastery of the principle used by Eratosthenes to calculate the circumference of the by completing a worksheet.

• This lesson requires the use of a straight edge and drawing surface for the teacher. This can be a white board, an overhead transparency, or other.
• The author recommends a large version of the definition rubric to display in the classroom for reference.
• Each student will need a straight edge and a drawing surface such as paper or personal whiteboard to draw figures.
• Each student will need a worksheet and a definition rubric.
• The teacher should have graphics illustrating how Eratosthenes calculated the circumference of the earth.
• The teacher should have made prior arrangements with a teacher at a school at least 100 miles away on a nearly north-south line to also gather data.
• Each student group will need:
  • A worksheet
  • A staff
  • A ruler or tape measure
• The class as a whole will need an accurate clock so that the two sets of measurements are taken at the same time.
• The class as a whole will need at least one plumb bob to make sure the staffs are straight up and down.

• Circumference - the measure of the length of the outside edge of a circle.
• Corresponding angles - the angles that are the same measure when a line crosses two parallel lines
• An entire circle is 360 degrees.

None

1. Review
   1. Review the circumference of a circle.
   2. Review that angles adding up to full circle make 360 degrees.
   3. Review the postulate for corresponding angles (Euclid's Postulate 5).
2. State the problem and objective
   1. State the problem: "How could we calculate the circumference of the earth?" Allow the students to explore ways. Some possible ways include:
      1. Measure the width of the earth from space/the moon.
      2. Travel around the earth at a constant speed.
      3. Eratosthenes method
      Do not reject invalid methods at this time. The students will have the opportunity to evaluate each method later in the lesson.
   2. State and write on the board that at the end of the lesson, students will describe in their math journals at least one method of measuring the circumference of the earth, including samples; and complete a worksheet using Eratosthenes method of calculating the circumference of the earth.
   3. Evaluate the proposed methods.
      1. Form several groups, at least one for each method that will be explored.
      2. Assign each group a single proposed method. Explain the ground rules:
         1. Each group is to determine if this method is valid (if it will give reasonable results). The group is to explain why it is valid or invalid.
         2. Each group will be responsible for explaining the math involved, including a sample.
         3. If the person proposing the idea disagrees with the group they are allowed to give a dissenting opinion after the group presentation.
         4. After the presentation, all students are allowed to ask clarifying questions first, and then offer additional opinions.
         5. The teacher will then assist the students in developing the idea, filling in any missing parts.
      3. Give the groups sufficient time to work, perhaps 30 minutes.
      4. Group presentations
   4. Lecture on Eratosthenes and his method of calculating the circumference of the earth.
      1. Differentiate between negative sign and minus sign. Point to negative sign and ask, "Is this a minus sign?" Lead discussion to conclusion that since backward is not being subtracted from anything it can not be a minus sign. State that it is a negative sign. Have students form groups and make a group statement of the difference between a minus sign and a negative sign.
3. Develop the experiment.
   1. Tell the class that they will duplicate Eratosthenes experiment. Ask them what we need to have and do to duplicate the experiment. As they develop the ideas, introduce or reinforce the arrangements you have made. The arrangements include:
      1. A staff of know length
      2. A tape measure or ruler
      3. A clock, synchronized with the other class
      4. A plumb bob to make sure the staffs are upright.
      5. A worksheet to record the data.
   2. Complete a phone call to the cooperating class to synchronize clocks.
4. Perform the experiment.
   1. Assist the students in setting up and performing the experiment at the appointed time and recording the experiment.
   2. Average the experimental data for the entire class.
   3. Complete a call to the cooperating class and exchange data.
   4. Have the students assist each other in completing the worksheet.
5. Allow the students sufficient time to describe a method of calculating the circumference of the earth in their math journals.

• Blind students will need assistance from sighted students.
• Students with cognitive difficulties will need additional assistance bridging from the concrete to the abstract.
• Cognitively advanced students may be asked to prove that Eratosthenes method is valid.
• Some students may want to orally describe a method of calculating the circumference of the earth, rather than writing.

• An informal assessment will be performed while assisting students to complete their tasks.
• An informal assessment will be performed by listening to student's presentations
• The worksheet can be used as a formal assessment.
• The description in the math journal will be used as a formal assessment.

¹National Council of Teachers of Mathematics, Principles and Standards for School Mathematics, 2000

1. About.com, Eratosthenes, The Father of Geography
2. Livio C. Stecchini, Ancient Measurements of the Circumference of the Earth
3. J J O'Connor and E F Robertson, Eratosthenes of Cyrene

Download Worksheet for Calculating the Circumference of the Earth
Download Rubric for description of a method to calculate the circumference of the earth