Noon Project Revisited

SCORE Mathematics

Standards Connections

The Noon Project Revisited

By Tom Akashi
Introduction: 2200 years ago, Eratosthenes calculated the circumference of the Earth from the measurements of two shadows (in a well and from a pole). The Noon Project is an annual Internet event for schools to team up and make their own measurements. Using the reported 1995 shadow measurements, you can perform the calculations to determine the circumference of the Earth.

Prior Knowledge: You should have a knowledge of geometry and trig functions.

Grade Level: 4 and 8-12

Task: To determine the circumference of the Earth from the shadow measurements reported from the 1995 Noon Project.

Resources: Noon Project Revisited Worksheet and calculators with trig functions

Process:

Anticipatory Set- Introduce the lesson by stating the objective and/or giving the history on this lesson. For more information on Eratosthenes, go to http://yn.la.ca.us/eratosthenes/welcome.html

Instruction - Model calculating the Angle, the Distance, and the Circumference of the Earth using the data for AUS and CA, from the Noon Project Revisited Worksheet.

d = distance between the two locations, using the formula, d = ( L2 - L1 )k
A = 360 degrees, assumption of round earth
a = the difference of the solar angles, a = angle1 - angle2 .
D = circumference of the Earth.

Guided Practice - Allow for guided practice, using the data for AUS and WI. Check for understanding. Then have students continue to try to complete the rest of the worksheet.

Closure and Independent Practice - Remind students that just like Eratosthenes, they are trying to use math to calculate immeasurable objects. Students are to finish the worksheet at home.

Learning Advice: Teachers need to try out the calculations before having your class do them. Be sure to model each step in a clear and organized manner. Students should take notes which can be included with their own calculations and drawings. Students will need calculators with trig functions. An overhead transparency of the worksheet would be helpful.

Evaluation: Along with accurate calculations, students should show each step in a clear and organized manner. Drawings should be used, where they aid communication. Students should be encouraged to include comments.

Extensions:

You could have students present their calculations and drawings, to the whole class. Create a diagram showing two shadows in the same hemisphere, and explain the reasoning behind subtracting the solar angles, to obtain the central angle. Try participating in the Noon Project, held each year during March or April. http://www.ed.uiuc.edu/coe/projects/noon-project/

Two schools with email accounts can exchange their noon shadow measurements and latitudes, for any one day, and can still get a close approximation of the circumference of the Earth.
Conclusion: This project is a real world application of geometry and trigonometry. Hopefully, by doing such activities, students will begin to see the value and power of mathematics.

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California Mathematics Academic Standards:

Grade 4:
Algebra and Functions
1.0 Students use and interpret variables, mathematical symbols, and properties to write and simplify expressions and sentences.

1.1 use letters, boxes, or other symbols to stand for any number in simple expressions or equations (e.g., demonstrate an understanding and the use of a the concept of a variable)

Grade 8-12:
Geometry
7.0 Students prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles.
12.0 Students find and use measures of sides and of interior and exterior angles of triangles and polygons to classify figures and solve problems.
13.0 Students prove relationships between angles in polygons by using properties of complementary, supplementary, vertical, and exterior angles.
18.0 Students know the definitions of the basic trigonometric functions defined by the angles of a right triangle. They also know and are able to use elementary relationships between them. For example, tan(x) = sin(x)/cos(x), (sin (x))² + (cos (x))² = 1).
19.0 Students use trigonometric functions to solve for an unknown length of a side of a right triangle, given an angle and a length of a side.
20.0 Students know and are able to use angle and side relationships in problems with special right triangles, such as 30^o, 60^o, and 90^o triangles and 45^o, 45^o, and 90^o triangles.
21.0 Students prove and solve problems regarding relationships among chords, secants, tangents, inscribed angles, and inscribed and circumscribed polygons of circles.

Trigonometry
1.0 Students understand the notion of angle and how to measure it, in both degrees and radians. They can convert between degrees and radians.
5.0 Students know the definition of the tangent and cotangent functions and can graph them.
10.0 Students demonstrate an understanding of the addition formulas for sines and cosines and their proofs and can use those formulas to prove and/or simplify other trigonometric identities.
12.0 Students use trigonometry to determine unknown sides or angles in right triangles.
13.0 Students know the laws of sines and the law of cosines and apply those laws to solve problems.
14.0 Students determine the area of a triangle, given one angle and the two adjacent sides.
19.0 Students are adept at using trigonometry in a variety of applications and word problems.

NCTM 9-12:

STANDARD 1: MATHEMATICS AS PROBLEM SOLVING
STANDARD 2: MATHEMATICS AS COMMUNICATION
STANDARD 3: MATHEMATICS AS REASONING
STANDARD 4: MATHEMATICAL CONNECTIONS
STANDARD 5: ALGEBRA
STANDARD 12: DISCRETE MATHEMATICS