A CIESE Collaborative Project

# Project Lessons

## Week 1: Setting the Stage

In addition to completing your letters of introduction and in order to prepare your students for the measurements that will take place later in the month, you might start off by making them more aware of shadows and the "stories they tell."

Have them describe what a shadow does over the course of a day (from sunrise to sunset). Make a graph to tell the shadow's story.

See Making a Shadow Plot in the Northern Hemisphere for details.

### Related Questions

• When is the sun directly overhead?
• What happens to the length of the shadow at that time?
• Is there any time when the sun would be directly overhead and NOT cast a shadow?
• Do you think ancient man thought the earth was round or flat?
• Let's say you belonged to the "Flat Earth Society." What arguments would you make for the earth being flat?
• What arguments would you give if you were a member of the "Earth is Round" Club?

Interesting sidebar: One of the myths that James Loewen debunked in his book "Lies my Teacher Told Me - Everything Your American History Textbook got Wrong" (p. 45-48) is that most people at the time of Columbus' voyages thought the earth was flat. Columbus was not so daring, nor did his crew want to mutiny because they feared they would drop off the edge of the earth. The truth is that most people around 1492 believed the earth was round. Historians give credit to Washington Irving's biography of Columbus in 1828 for starting the myth. He had an account of how Columbus had to convince his investors of the spherical nature of the earth. Pure fiction.

### Introducing the project to your students

Imagine that you are living over 2000 years ago and you are convinced that the earth is round. How would you go about measuring it? One way would be to start walking in one direction and keep track of how far you go. If the earth is round as you believe, eventually you should return to where you started. Do you think anyone actually contemplated trying this back then? Why or why not? What would be some of the obstacles?

Another method would be to drill a hole to the "other side" of the earth and measure the distance. Once you knew that distance, could you determine how round the earth is? Is there a relationship between the distance around a circle (circumference) and the distance through the center of the circle (diameter)? (You might try one of the activities at http://arcytech.org/java/pi/ to see if you can discover a relationship between the circumference of a circle and its diameter.*) Unfortunately, digging a hole through the center of the earth is just as out of the question as circumnavigating the globe by foot.

This brings me to someone named Eratosthenes who was born 2000 years ago in Cyrene, a town in northern Africa. He actually found out how round the earth is by doing an experiment. His idea was to think of the earth as an orange cut in half. The cross section is divided into wedges just like pizza slices.

## Week 2 & 3: Doing the Measurements

• You should try to get your measurement done no later than by the end of week 3. Our target is to do the measurements as close to the Equinox as possible but anytime during this two week span will be fine. (The margin of error is very small if done in this time period.)
• Since you need a sunny day for shadow measuring, we suggest you check your weather forecast and pick a day when you will most likely have sun.
• Do the measurement when the sun is highest in the sky (at your local noon time.)
• To do this experiment you will need some materials to measure shadows accurately. Your "gnomon" will be a meter stick that is perpendicular to the ground. For your measurements to be accurate, it is critical that the meter stick be vertical. (Note the devices used below. Wind can be a major factor.
• It is helpful to have a piece of paper to note where the the end of the shadow is. Also a compass will come in handy to determine in which direction the shadow falls. Since the edge of the shadow is "fuzzy" and the shadow is moving from west to east (northern hemisphere), you want to be careful in deciding where to place your mark.
• Have your class work in groups of 3 or 4.
• Set up your measurement station. Place paper under the station so you can mark where the shadow ends (see photos above.). Since the edge of the shadow is "fuzzy" and the shadow is moving from west to east (northern hemisphere), you will want the students to be careful in deciding where to place their mark. The students may find it interesting that the shadow points towards the north. But does it point to true or magnetic north? A compass will come in handy to determine this.
• Take measurements every 2 minutes beginning at least 10 minutes before local noon which is the time that the sun is highest in the sky. (This will most likely NOT be 12 noon as indicated on your time measuring device (sometimes called a watch). Students should note that when the sun is highest in the sky the shadow length is the shortest.
• After some discussion, each group reports this result to the entire class. The teacher writes each group's best value on the board. Assuming there will be different values, students will need to determine their "best" shadow length and decide which will be the class's best estimate of the shadow length at local noon time.
• Make a scale drawing of your stick and shadow. Complete the triangle and measure the sun's angle with a protractor.

## Week 4: Analyzing Data, Determining Circumference and Submitting Data

### Example Circumference Calculation

If you can measure the sun's angles at two different positions on the earth at the same time, you can figure out the central angle (Angle ABC below). Then you need the distance along the surface from A to C. This gives you the dimension of one slice of the cross section of the Earth. How big is the whole circumference?

Let's look at an example to see how this information leads to determining an empirical value for the circumference of the earth.Our two sites will be at Manasquan, New Jersey and San Juan, Puerto Rico. At local noon on the same day, the experimenters measured the shadow cast by a meter stick. In Manasquan, the shadow length equaled 80.5 cm. In San Juan the shadow length was 35.3.

The next step is to figure out the sun angles at Manasquan and Puerto Rico. A simple method you can use is to create a scale model of your triangles and then directly measure the angle. In this case we could create a triangle for Puerto Rico with sides that are 3.53 cm and 10.0 cm. We could also use 3.53 inches and 10.0 inches. As long as the proportions are the same the angle should be the same. A protractor can be used to measure the angle.

The central angle equals the difference between these angles. Using these measurements, the central angle is 38.85 - 18.59 or 20.26 degrees.

Now that we know the central angle, we can determine how many such angles would make up the full circle. That is, how many "slices" can we make where the angle is 20.26 degrees? Since the total is 360 degrees, then you would have a little less than 18 equal "slices" (or 17.77 to be more precise.) Next, each slice's edge represents the distance between Manasquan and Puerto Rico, or more accurately, the North-South distance between the two sites.

Manasquan, New Jersey: Location: 40:07:34 (40.13) N 74:02:59 (74.05) W

San Juan, Puerto: Location: 18:28:00 (18.47) N 66:07:00 (66.12) W

The distance between these two places along a north-south line is approximately 21.7 degrees or 2404 km (each 1 degree of latitude is about 111 km apart). So if the length of the "slice" is about 2400 km and there are about 18 slices then the projected circumference is 2400 x 18 = 43,200 km which is about an 8% error since the average circumference is 40,008 kms. (More precisely, it is 2404 km times 17.7 "slices".)

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## Week 5: Reflections and Final Report

Other Notes

• What makes this project so special is that this is a collaboration. I am eager to make this project work for you as best as possible and to learn from our collective experiences so we can keep on improving it.
• More about "pi" can be found here.