ERATOSTHENES VISITS FIFTH GRADE: ASSESSING THE ABILITY OF
STUDENTS TO WORK WITH MODELS OF THE EARTH
by Judith Powers and Sergio Torres*
Abstract. Fifth grade students, using inexpensive materials and simple geometry, were
able to reproduce the measurement of the size of the Earth done by the Greeks more than
2,000 years ago.
Imagine the excitement of fifth grade students participating in a scientific project with
students in a remote country to measure the size of the Earth, and coming home to tell
their parents “mom, dad I measured the Earth today”! That is precisely what we did in
science class, using meter sticks and measuring their shadows in two distant locations. To
obtain the size of the Earth, students had to understand the connection between these
measurements and a model of the spherical Earth following the method developed by
Eratosthenes. In the process students learned about the history of science, the value of
collaborating with students in an international setting, and appreciated the fact that a
complex and large object as is our planet can be measured using simple geometric
concepts.
Background
The method of Eratosthenes (circa 273 - 194 BC) to estimate the Earth’s girth consists of
measuring the distance separating two distant locations along the north-south line,
together with their angular separation determined with measurements of the shadows cast
by sticks at the two locations (Sarton 1952; Ferguson 1999).
To illustrate Eratosthenes observations we made a demonstration in class by
shining light on models of a flat and spherical Earth with two sticks of the same size
inserted perpendicular to the surface. Whereas the shadows projected on a flat earth are
equal, those on the sphere have different length depending on their location. The ensuing
discussion with students and teachers resulted in the realization that replicating
Eratosthenes measurement would be an exciting hands-on activity that would offer
multiple opportunities to involve students in the scientific process.
Measuring the size of the Earth in fifth grade provides a suitable experience to
begin the development of a precise model of the Earth and the Solar system throughout
the middle school years consistent with the recommendations of the National Science
Education Standards (Agan and Sneider 2004). Of particular interest to middle school
science teachers is to know whether the students have the level of abstract reasoning
required to analyze the data, and to evaluate how effectively students can work with
abstract models (Greene 2008).
*
At the time of the measurements Judith Powers was a Science teacher for the Lower School at HoltonArms in Bethesda, Maryland. Sergio Torres (father of one of the fifth graders) is an astrophysicist.
The Earth is modeled as a sphere and to connect it with the measurements we
used a circle representing the cross section of the Earth that passes through the two
locations where measurements take place. The geometry, as depicted in Figure 3, is
defined by this circle and a wedge shaped section (or “slice”) as shown. The number of
slices in the circle is 360° divided by the angle of the slice (C) and the circumference of
the Earth is the distance (S) multiplied by the number of slices (Webb and Bustin 1988):
Earth’s Circumference = S x (360°/C)
Replicating Eratosthenes Measurements
For this activity students from Holton-Arms in Bethesda (Maryland) teamed with
students of Liceo Benalcazar in Cali (Colombia). The latter was chosen because it meets
the conditions for Eratosthenes’ method: it is located 3,936 Km directly south of
Bethesda. Larger distances results in smaller errors.
We established a project web site (Torres 2008) with reference material to help
teachers prepare for the activity. Pictures and results were added later.
Measuring the Shadows
To determine the angle of the slice (C) we needed the angles of the meter sticks with the
sun’s rays measured simultaneously in the two cities when the sun reached its highest
altitude. Measuring the shadows of the sticks allowed us to compute these angles.
We arranged to take simultaneous measurements on March 11, 2008 at 13:18
(daylight savings) in Bethesda and 12:18 in Cali. At this time the sun reaches highest
altitude (USNO 2008). We distributed meter sticks and meter tapes to groups of
approximately five in a grade of 44 students. Since it was cloudy, we took measurements
starting few minutes prior to the target time and continued taking them through 13:19.
We had the girls check that their meter sticks were perpendicular to the ground. The
most dramatic effect, which was readily appreciated by the girls, was the visible
difference in the length of the shadows (92 cm in Bethesda, 13 cm in Cali) when viewing
the pictures side-by-side (see Figure 1 and Figure 2)
Figure 1. Students in Bethesda taking measurement of the shadow
One of the challenges in this project was keeping the girls focused on the time and
the task. A beautiful spring day has many distractions, but we were able to collect
enough consistent data to begin our calculations.
Figure 2. Shadow cast in the Cali location
Calculating the Circumference of the Earth in Class
We completed the data analysis in class three weeks later. Using an LCD projector, we
showed the girls how measuring the angle of the “slice” we could estimate the
circumference of the circle. We divided the task into three steps: a) determination of the
angle of the slice, b) estimation of the Bethesda-Cali distance and c) computing the
circumference.
Measuring the Angular Separation of the Two Cities
We instructed the students to draw a triangle rectangle with the measured lengths of the
stick and the shadow scaled down by a factor of 10 to make it fit in a sheet of paper. The
triangle is built with the shadow as the base and the stick as the perpendicular side. The
students measured the angle opposite the base with a protractor. This is done for the
observations from the two cities (angles A and B). The angle of the slice is the difference
C = A - B.
Figure 3. Geometry that relates the measurements of shadows with the angle of the slice
The Bethesda-Cali Distance
The challenge was to find a method to estimate the distance between two locations that
did not require prior knowledge of the radius of the Earth. Eratosthenes sent surveyors to
walk the entire distance. We thought that parents would be reluctant to subject their
daughters to a 1,357 Km walk, combined with 2,579 Km of swimming; thus we settled
for the more practical solution of measuring the distance on flat maps. After all, it was
Eratosthenes himself who made the first maps by locating points on a longitude-latitude
grid.
To reduce errors due to projection effects, we assembled a mosaic of three maps
covering the Bethesda-Cali great circle. Students measured distances in cm directly on
the maps, and from the Km scales on the maps they had to figure how many Km
corresponded to 1 cm.
We started the computation with one of the groups. Managing the measurements
from three maps using different scales and making sure that the measurement covered a
continuous arc was the main source of errors. We observed that for most students, this
step required spatial reasoning skills that not all had developmentally acquired. The
students in the first group were more concrete learners, who had many questions about
the individual steps and how they related to our final solution. The second class followed,
and went more smoothly because we knew where questions would arise and because this
group had more analytical students.
We made use of an LCD projector to show the angles that we were measuring,
and then used the diagram to reiterate why we were following the steps in the experiment.
Once the experiment was completed and reviewed, the students were visibly pleased that
they had replicated the Greeks’ method for determining the circumference of the Earth.
Results
We analyzed the report sheets to assess the accuracy of the combined results. For the
angle C the average of all groups came to within 2° from the actual value of 35.5°
(USNO 2008). The average of the distance S was 3,722 Km. Since the actual distance is
3,936 Km the absolute error is 214 Km. The average for the circumference was 40,231
Km for an error of 223 Km (using 40,008 Km for the Earth’s circumference as derived
from the average radius). The fact that the average comes so close to the actual value is
due to the partial cancellation of errors in C and S. Based on the spread of individual
measurements, the overall error of the circumference is 7%.
Lessons Learned
Carrying out the data analysis with the students in class was important because we were
able to identify issues and give them immediate verbal and visual explanations. We knew
that many of the students would be challenged with the spatial reasoning skills required
to use three different maps to compute the distance, and to relate this with the
circumference of the Earth. In future experiments, we would use one map instead of a
combination of maps (at the expense of a larger error due to map projection effects).
The use of an LCD projector was also essential because we could show
illustrations of the concepts and visually reinforce how and why we made the sun’s
shadow measurements. This repetition and reiteration was especially important when
some students became lost in the procedure.
Working together with students and teachers from another country provided a
concrete illustration of how students from different backgrounds and living in different
countries can collaborate to solve scientific problems. We would have liked more
interaction between the students. In the future, we could use Internet connections to see
each other’s measurements in real time and to jointly discuss the analysis and results.
Potential Extensions of the Project
We foresee future applications of the project targeted at middle school students that could
add the following extensions: a) the source of the seasons is the 23.5° inclination of the
Earth’s equator which can be measured following the stick and shadow method during
the solstices; b) the difference between the magnetic and geographic north can be made
explicit by comparing the direction of the compass needle with the direction of the
shadow when the sun is at the maximum elevation; c) errors can be appreciated by
checking how the circumference varies as we use values for C and S, adding and
subtracting one standard deviation; d) demonstrating that the Earth is not a perfect sphere
can be experienced by noticing that even if we make no errors in our measurements, we
would not get the right answer.
Conclusions
With actual measurements in the field followed by analysis in the classroom, students
experienced first hand the nuances of scientific research. Working with a geometric
model was an effective tool to connect measurement with theory. Adapting the geometry
used by the Greeks in a tangible experiment expanded the girls understanding of how
mathematics is an essential component of science. The girls realized that even the most
careful measurements are subject to errors. They were able to appreciate that length
scales tremendously larger than the human scale can be measured with means within their
reach. Judging by the level of engagement of the students and the feedback obtained from
students and parents, we concluded that this experiment made a positive impact by
raising the interest of the students in science and increasing their understanding of the
scientific process.
Acknowledgements
We would like to thank the fifth grade teachers at Holton-Arms, Rachel Zukowski in
particular, for their enthusiastic support. We are indebted to Anne Marie Leroy, academic
director of the Liceo Benalcazar in Cali, for coordinating the activity in Colombia.
References
Agan, L. and Sneider, C. 2004. “Learning About the Earth’s Shape and Gravity: A
Guide for Teachers and Curriculum Developers”. Astronomy Education Review, 2 (2):
90-117.
Ferguson, K. 1999. Measuring the Universe. New York: Walker and Company.
Greene, B. 2008. “Put a Little Science in Your Life”. The New York Times, Op-Ed, June
1.
Sarton, G. 1952. A History of Science - II. Cambridge: Harvard University Press
Torres, S. 2008. The Size of Earth Project web site at http://astroverada.com/earth/
USNO 2008. US Naval Observatory. Available at http://aa.usno.navy.mil/data/
Webb, J., and R. Bustin. 1988. “Eratosthenes Revisited”. The Physics Teacher, 26 (3):
154 – 155.