TEACHER’S
GUIDE: ( download http://www.stick.com.ar/guide.pdf )
Overview
This activity enables students to measure the
circumference of Earth. Groups of students at two
distant schools will take data and then collaborate,
in essentially the same way that Eratosthenes
measured Earth’s circumference millennia ago in
Egypt.
Objectives
• Describe the geometry of how sunlight strikes
Earth at different latitudes
• Describe how the circumference of Earth was first
measured millennia ago
• Describe how to determine local noon
• Measure the angle of the sun at local noon
• Collaborate with another school some distance away
to determine the circumference of Earth.
Logistics
Your class will need measurements of the shortest
shadow length and stick length to be taken at some
other school located a considerable distance away,
either north or south.
You will enter your school's information on the
Stick Project website, <http://stick.com.ar/planilla.html/>,
at the same time as the other school's information.
Once the partner schools have shared the
measurements, the circumference of the earth is
automatically calculated.
National Standards Addressed
This activity addresses the following Benchmarks and
National Science Education Standards (NSES):
Benchmarks, K12: The Mathematical World
Symbolic Relationships
When a relationship is represented in symbols,
numbers can be substituted for all but one of the
symbols and the possible values of the remaining
symbol computed. . . .
Shapes
Distances and angles that are inconvenient to
measure directly can be found from measurable
distances and angles using scale drawings or
formulas.
Models
The basic idea of mathematical modeling is to find a
mathematical relationship that behaves in the same
way as the objects or processes under investigation.
A mathematical model may give insight about how
something really works or may fit observations very
well without any intuitive meaning.
Benchmarks, K12: The Nature of Science
The Scientific Enterprise
The early Egyptian, Greek, Chinese, Hindu, and
Arabic cultures are responsible for many scientific
and mathematical ideas and technological inventions.
NSES, K12: History and Nature of Science
Science as a Human Endeavor
Individuals and teams have contributed and will
continue to contribute to the scientific enterprise.
Historical Perspectives
In history, diverse cultures have contributed
scientific knowledge and technologic inventions.
NSES, K12: Unifying Concepts and Processes
Developing Student Understanding
• Evidence, models, and explanation
• Constancy, change, and measurement
NSES, K12: Science as Inquiry
Abilities necessary to do scientific inquiry
• Design and conduct scientific investigations
• Use technology and mathematics to improve
investigations and communications
• Formulate and revise scientific explanations and
models using logic and evidence
Materials
For each student
• A copy of the
Student Activity Guide
For each group
• stick or dowel, about 60 centimeters (cm) long
• stiff cardboard (to provide a smooth, flat
surface)
• measuring tape
• pieces of blank paper
• pencil
To be shared by several groups
• carpenter’s level
• mason's plumb line
STUDENT’S GUIDE:
THE ERATOSTHENES PROJECT
In this activity, you will work together with
students at another school to measure the
circumference of Earth. You will use the same
methods and principles that Eratosthenes used more
than two thousand years ago.
Eratosthenes was a Greek living in Alexandria,
Egypt, in the third century, BC. He knew that on a
certain day at noon in Syene, a town a considerable
distance to the south, the sun shone straight down a
deep well. This observation meant that the sun was
then directly overhead in Syene, as shown in Figure
1.
Figure 1:
Light rays shining straight down a well in
Syene at noon, when the sun is directly
overhead. No shadow is cast.
Eratosthenes also knew that when the sun was
directly overhead in Syene, it was not directly
overhead in Alexandria, as shown in Figure 2. Notice
that in both drawings, the sun’s rays are shown as
parallel.
Figure 2:
Sunlight shining down a well in Alexandria at
noon, on the same day as the observation shown
in Figure 1. The sun is not directly overhead.
The gray bar at the bottom left shows the shadow
cast by the side of the well. The angle of the
sun’s rays and the size of the shadow are
exaggerated.
In Figure 2, the side of the well casts a shadow on
the bottom.
Eratosthenes used a shadow like this to find the
circumference of Earth. When the sun was directly
overhead in Syene, he measured the shadow of an
object in Alexandria at noon. From the length of the
shadow, the height of the object, and the distance
between Syene and Alexandria, he calculated the
circumference of Earth. His value agreed quite well
with the modern one.
How Eratosthenes Found the
Circumference of Earth
Figure 3:
The sun is directly overhead at noon at Syene
(S). Alexandria is at point “A.”
How did he do it, more than two thousand years ago?
Take a look at Figure 3. Syene is represented by
point “S,” and Alexandria by point “A”. In Figure 3,
the arc length between S and A is d, and the angle
corresponding to the arc SA is ?. The radius of
Earth is R. As suggested above, let’s assume that
the sun’s rays are parallel. Since the ray that
strikes Syene, at point S, is perpendicular to the
surface of Earth, the sun is directly overhead
there.When the sun was directly overhead in Syene,
Eratosthenes measured the shadow of a tower in
Alexandria at noon,1 shown in Figure 4. Since both
the tower at A, which is perpendicular to Earth’s
surface, and the ray of sunlight at point S both
point to the center of Earth, and the rays of
sunlight are parallel, the angle between the
sunlight and the tower is equal to ? (Alternate
interior angles are equal.)
Figure 4:
The geometry of Eratosthenes’ measurement. He
measured the length of the tower and its shadow
at noon at Alexandria. Then he determined the
angle of sunlight with the vertical, which is
the same as the angle subtended by Syene (S) and
Alexandria
(A) at the center of Earth.
The tower and its shadow form two sides of a right
triangle, as shown in Figure 4. Although
trigonometry hadn’t yet been invented, Eratosthenes’
procedure can be expressed in the language of trig
as follows: The length of the shadow, the height of
the tower (which he knew), and the angle ?, given
here in degrees, are related by
Inverting tan? gives the value of ?. Using ratio and
proportion, the arc length d is the same fraction of
Earth’s circumference C as ? is of 360 degrees.
Rearranging for the circumference C,
How You Can Find the Radius of
Earth
Rather than find Earth’s circumference, we
suggest you find its radius, so that you can more
easily compare your measured value with the
accepted value.
Eratosthenes was lucky, because he knew of a place
where the sun was directly overhead at noon on a
certain day. Can you do the experiment even without
that information?
Fortunately you can, as shown in Figures 5 and 6.
You will make the shadow measurements at your school
and share results with a class from a school at
another location.
You can then do a subtraction to find the angle you
need. Be sure to plan ahead—ideally, both
schools make the measurement on the same day.
Figure 5:
The geometry for measuring the radius of Earth
using the data of two collaborating schools
separated by a northsouth distance. Each will
measure the angle of the sun, ?_{A}
at one location and ?_{B}
at the other, at local noon.
We need two points, A and B, separated by a
northsouth distance d, shown on Figure 5. The
experiment will work best if d is as large
as possible.Take a look at Figure 6. Your school and
the collaborating school are represented by the
points A and B, and the angles ?_{A}
and ?_{B}
correspond to points A and B.
Figure 6:
The relationship among the direction of
sunlight, the sticks, and
the two angles ?_{A}
and ?_{B}.
At point A,
and
likewise at point B.
Figures 5 and 6 show that the angle corresponding to
the arc AB is just the difference ?_{B}
 ?_{A}
We can find the radius of Earth in the same way that
Eratosthenes found the circumference in equation.
Rearranging
and simplifying,
Making the Measurement at Local
Noon
On any day, local noon is the instant when the sun
reaches its highest point in the sky. To determine
it, plant the stick in the ground, making sure the
stick is vertical using a plumb bob or a carpenter’s
level. In the late morning, measure the shadow’s
length at regular time intervals. The shadow will
get shorter as noon approaches, and then get longer
again once noon has passed. The shortest length is
what you will substitute into equation above to find
the value of ?_{B}
or ?_{A}
for your location.
Below are three additional activities:
• How Shadows Change During the Day
• Shadows on Earth
• Latitude
How Shadows Change During the Day
If your students have not thought much about
shadows,they might benefit by starting with this
preliminary activity.Give each group a 5cm straw
piece, a sheet of 21cm x 29cm paper, and some tape.
Ask them to tape the straw so it stands in the
center of the paper, as shown in Figure 7, and also
to indicate the direction of north on one of the
long sides. If you provide them with a compass, they
can orient the paper.
Figure 7:
Fivecentimeter straw mounted vertically on a
piece of paper.
Students predict and then measure the shadow of
this straw at different times.
Their
challenge is to imagine that this paper is set on
level ground in sunlight and to predict the location
and length of the shadow of the stick on the hour
during the day. Discuss these predictions to bring
out their thinking. Then have them do the experiment
and compare their predictions and results.
Shadows on Earth
Materials for each group: five 4cm straw pieces,
tape, and a piece of 21cm x 29cm paper.
Explain to students that they will make a model of
shadows at different points on Earth. Have them draw
a straight line across a piece of 21cm x 29cm paper
and tape the five straw pieces, equally spaced,
along this line, so the straws stand straight up.
Ask how the paper and straws could be a model of
sticks placed at different locations on Earth (curve
the paper, with the straws on the convex side).
Explain that to avoid damaging their eyes, they
should never look directly at the sun. In sunlight,
ask students how the paper and straws can model the
Eratosthenes experiment. (Facing the sun, hold the
paper at the ends of the long sides and curve it so
the straws point out. Turn the paper to make the
shadow of one straw disappear. Make this straw point
directly at the sun. The straw without a shadow
models the well at Syene.) Have students describe
what happens to the shadows of the other straws and
relate the shadow of each one to its position. Ask
students to relate these shadows to the shadows
Eratosthenes used to measure Earth’s circumference.
See Figure 8.
Figure
8:
Model of shadows of sticks at noon, at different
latitudes and the same longitude.
Extension Activity: Latitude
Once experimentation is complete and the results
reported, you can have students relate the
measurements they have made to the definition of
latitude.
a. Ask them to define latitude (the length of
arc, or angle from the center of Earth, measured
north or south from the equator).
b. Referring them to Figure 3, ask them to
assume that point S is on the equator. Ask on what
day the sun would be directly overhead at noon at S.
c. If your students made their shadow
measurement on the vernal or autumnal equinox, the
resulting angle would be equal to the latitude, as
shown in Figure 3 (remember point S is on the
equator). If possible, have them try to do this by
measuring the shadow of a stick on or near March 21
or September 21.
d. In the Eratosthenes experiment, the angle
(?_{B}
 ?_{A})
is the same as the difference in latitude of the two
schools, so students could determine this difference
immediately by subtracting the two latitudes of the
collaborating schools. Of course, we want students
to make measurements and compare, rather than look
up the answer in an atlas. If students point this
out, you can remind them that they are reenacting an
historical experiment.
ADDITIONAL ACTIVITIES
Notes on Introduction
In the discussion of Figures 1 and 2, which show
sunlight in wells at two different locations, we
assume the following:
• The rays of sunlight are parallel (see next
section for more detail)
• The sides of the well are vertical.
Notes on How Eratosthenes Found the Circumference
of Earth
Here are Eratosthenes’ assumptions:
Earth is a sphere. In fact, Earth bulges by about
.3% at the equator, but we can safely neglect this
difference.
The sun is very far away, so sunlight can be
represented by parallel rays. The sun is indeed very
far away, but it is not a point source, since its
diameter is about 1/100 of the Earth sun distance.
Figure 9:
Notice the penumbra—the partially illuminated
region—at the
end of the stick’s shadow (drawing not to
scale).
As shown in Figure 9, there is a penumbral region at
the end of the shadow, a region only partially
illuminated by the sun. If the stick is 1 meter (m)
long, the penumbral region will be more than 1
centimeter (cm) wide, which limits the accuracy of
the measurement of the shadow length. The penumbra
size scales up with the length of the stick, so
using a longer stick does not increase the accuracy
of the measurement.
Alexandria is directly north of Syene: This is only
approximately true. Find an atlas and compare the
location of Alexandria and Aswan (built on the site
of Syene).
You might ask students to comment on Eratosthenes’
assumptions, considering that he was working more
tan 2,000 years ago.
Notes on How You Can Find the
Radius of Earth
Ask students to discuss the distance d between the
two schools. Is it better for d to be large or
small? (large) Why? (The larger d, the larger the
value of the angle ?_{B}
 ?_{A}
The larger this angle, the smaller the percentage
error in its measurement). Find the location of the
two schools and ask students how to find the
distance between them (use the scale on the map).
Ask how well the two schools line up northtosouth.
Discuss how an eastwest displacement might affect
the outcome of the experiment. (Convert the
difference in longitudes of the two schools into a
time difference, using ratio and proportion and the
fact that 360° of longitude corresponds to 24 hours.
Then compare the difference with the uncertainty in
identifying local noon.)
Note that near local noon, the shadow length does
not change much with time. For this reason, missing
local noon by a few minutes is not important.
Suggest that students practice measuring the shadow
length at noon in advance. If you need to look up
the time of local noon, see the last reference.
Ask students how to select a suitable date for both
schools to make measurements. Should they check the
weather forecast first?
Try making the measurement of the shadow length
yourself. If possible, drive the stick into the
ground, and check with the level, or a plumb line,
to be sure that the stick is vertical.
Tape copy paper on top of the sheet of cardboard,
and check with the level to make sure that this
surface is horizontal. Be sure your stick is short
enough so the shadow doesn’t extend off the paper.
Ask students to describe what height they will
measure (the distance from the top of the stick to
the surface where they see the shadow).
NOTES ON THE ACTIVITIES
References
The experience is summarized in the following video.
In this video the subtitles in English can be
activated.
https://www.youtube.com/watch?v=4VVx0RTLCEA&feature=emb_logo
Find out the latitude of my city
https://www.latlong.net/
Solar noon in my city
https://www.timeanddate.com/sun/argentina/sanjuan
https://sunsetsunrisetime.com/sun
https://salidaypuestadelsol.com/sun
What Is Solar Noon?
https://www.timeanddate.com/astronomy/solarnoon.html
One of the things we do is "measure with a tape
measure or a ruler" So we are going to remember how
a measurement is read.
https://www.youtube.com/watch?v=WHqns716qiE&feature=youtu.be
Videos with simple explanations
Carl Sagan  Cosmos –
Eratosthenes
https://www.youtube.com/watch?v=G8cbIWMv0rI
ERATOSTHENES https://www.youtube.com/watch?v=wPR3XhIDP9w
ERATÓSTENES Y LA MEDIDA DE LA
TIERRA https://www.youtube.com/watch?v=2tmiWjLSMcA
In this video let's see how the shadow moves. The
shadow will begin to shorten its length until it
reaches a pussycat and then it will begin to be
longer. The shorter shadow is what we need.
https://www.youtube.com/watch?v=pnAlbwWBNSI
PROYECTO STICK (Original)
https://www.youtube.com/watch?v=QeQYYh6zW9I
Project Stick (Subtitled)
https://youtu.be/4VVx0RTLCEA
Web pages to know the weather, (surely you
know a better one)
https://weather.com
https://meteologix.com
https://www.windfinder.com/
https://www.windy.com/
Sign Up
http://form.stick.com.ar
Simulator
http://simu.stick.com.ar
Demonstration
Students may have difficulty understanding how the
method presented here, with two schools along the
same northsouth line (the same meridian) working
together, permits them to measure Earth’s radius on
any day of the year. If you have a globe, mount two
Straw pieces on the same meridian. Place the globe
in the beam of an overhead projector. Ask a student
to rotate the globe so the location of one straw is
at local noon.
Ask what time it is at the other straw (local noon
also).
How can they tell? (The shadows have mínimum
length.) Have a student change the orientation of
the globe axis and repeat.
A different way to visualize the geometry is to
imagine the plane containing the center of Earth and
points A and B. As Earth turns on its axis, this
plane sweeps through all of space. When this plane
is oriented so the sun lies within it, then the
shadow of each stick lies within the plane as well,
so it is local noon at the location of each stick.
Assessment
There are several ways to assess students’
performance in the Stick project.
• Students can choose one or more of the first three
objectives and write or present the description that
is specified in those objectives.
• Students can present a portfolio of student work,
explanations, and drawings to show how they measured
the radius of Earth.
• Students can prepare a written or oral
presentation to a younger student on Eratosthenes’
measurement of the circumference of Earth.

