EW-2000-08-008-JPL
TAKING.THE.MEASURE.OF.THE.UNIVERSE
WHY MEASURE THE UNIVERSE?
Astronomy is a very old science. We know of early peoples’ interest in celestial
objects — they studied the sky both day and night — from drawings on cave
walls and Babylonian, Mayan, Egyptian, and Chinese observatories and their
records. Unlike other direct experiences in science, such as chemistry (cooking)
and physics (hunting), no one could actually touch the sky. Even climbing to the
tops of mountains did not help humans reach the stars. Unless someone saw a
meteor fall and found a still-warm piece of the meteorite, our ancestors had only
their eyes to observe the wonders above. People asked: “How far away are the
stars and the Sun?” Hipparchos, a scientist/mathematician of ancient Greece,
made the first known catalog of the stars’ brightnesses and positions in the sky.
Today, the scientists and engineers of NASA’s Space Interferometry Mission,
known as SIM, are continuing his work.
HOW WILL WE TAKE THE MEASURE?
SIM will measure the positions of thousands of
stars extremely accurately — scientists call this
process astrometry. SIM will create a “virtual grid”
of reference points on the sky through measuring
the separation between stars. By studying how
the positions change with time, we will be able
to measure distances from Earth to the stars. SIM
will also help us find out whether small planets
are in orbit around stars other than our Sun.
SIM’S ASTROMETRIC
REFERENCE GRID.
SIM WILL MEASURE ACCURATE
DISTANCES TO STARS THROUGHOUT
THE MILKY WAY GALAXY.
SIM WILL HELP US UNDERSTAND
THE LIFE CYCLES
OF STARS.
Taking the Measure of the Universe
WHAT WILL SIM HELP US
UNDERSTAND ABOUT STARS?
By measuring the current positions and
motions of the stars in our own galaxy
— the Milky Way — scientists will be
able to trace the history of these stars
back in time. We will gain a better
understanding of how the Milky Way
Galaxy and the stars in it formed. We
will increase our knowledge of how
stars like our Sun form and will be better
able to predict their development.
WHAT WILL SIM HELP US UNDERSTAND
ABOUT PLANETS AROUND OTHER STARS?
By observing subtle disturbances in the motions
of stars, we will be able to tell whether these stars
have planets orbiting around them. The smallest
planets that we will be able to find are just a little
bigger than Earth. So far, Earth is the only planet
in its class that we know of. Other planets that we
have discovered orbiting stars are hundred of times
larger than Earth. Only when we know of more
than one example will we be able to tell whether
a planet like Earth is common or rare.
INTERFEROMETRY AS A
MEASURING TOOL
SIM’s key measuring tool is a spacecraft
to be launched in the second
half of this decade. The SIM spacecraft
uses interferometry of visible
light — the same light that our eyes
see. But the spacecraft can do something
that our eyes cannot. SIM’s
telescopes and mirrors collect and
combine light waves coming from
stars and measure the way these light waves interact or “interfere” with each
other. The interferometer data are analyzed to determine the distance to a star
and its position as compared with other stars.
ABOUT THIS POSTER
Like no mission before it, advanced mathematics will be required to understand
the data gathered by SIM. Mathematical analysis and tools are an integral part of
NASA’s space program. This poster and series of activities are designed to make
the use of measurement and computation in mathematics visible to students.
Every lesson is coordinated with National Council for Teachers of Mathematics
(NCTM) Standards. NASA recognizes that a new generation of people with a
broad and enthusiastic understanding of mathematics is needed not only in
space exploration, but in every part of our society.
NASA ALSO DEVELOPS GROUND-BASED
INTERFEROMETERS SUCH AS THE KECK
INTERFEROMETER ON TOP OF MAUNA
KEA, THE HIGHEST MOUNTAIN ON THE
ISLAND OF HAWAII.
EW-2000-08-008-JPL
TAKING.THE.MEASURE.OF.THE.UNIVERSE
FOR THE TEACHER
This script may be read to the class as you do the activity. [Teacher hints are
in brackets.]
Write down this list of objects: STAR, MOON, COMET, METEOR [shooting star],
PLANET [other than Earth], BLACK HOLE, GALAXY, AIRPLANE. Imagine the
night sky. Circle the names of objects you have seen in the night sky with your
eyes [not pictures]. [Students may pick any or all of these objects.]
If one of the things you picked was PLANET, please share with the class what
you saw and why you decided it was a planet. Did the planet you saw look different
than the other bright lights in the dark sky? Here are some hints: Did the
light from the planet twinkle, or shine like a flashlight? If you watched several
nights, did the planet stay near the same stars every night? [It may have moved
a little.] Could you see the color of the planet? [Mars sometimes appears to have
a slightly red color.] Could you see its shape? [Not without a telescope or binoculars;
with them, Saturn has an oblong shape.]
The planets of our solar system are near Earth. If Earth is the size of a grain of
rice, Mars is another rice grain about 4 meters (20 feet) away. A nearby star is a
grapefruit as far as New York is from Los Angeles. If that “grapefruit” star had its
own planets, they would be rice grains as well. Imagine trying to see a rice grain
that is 6,000 km (3,500 miles) away.
The Space Interferometry Mission (SIM) cannot “see” the planets of nearby stars.
How can it help us know if other stars besides our own Sun have planets around
them? [SIM has to measure something that indicates that a planet is present.]
Have you ever been with a friend who suddenly found they had an insect inside
their clothing? Did you see the insect right away? [Probably not; it was inside
the clothing or too small to see from a distance.] Did your friend quietly explain
to you about the insect in their clothing? [doubtful] Did your friend yell? [probably]
Did your friend do anything else? [wiggle or jump] Did your friend move
around? Can you show the class what a person does who has an insect in their
clothing? [As appropriate, students mime the wiggling.]
Stars that have planets orbiting them are like people with insects in their clothing.
They “jump around” a little. Astronomers call this slight motion “wobble.”
SIM cannot see planets directly, but SIM’s very accurate measurement of where
stars are located can help it see stars that wobble and change their positions.
Why do you think stars with planets wobble, but stars without planets do not?
Here are some more hints: Do planets stay in one place or move? [They move
around.] How do planets move? Does the motion make a shape? [nearly circular]
Is there a force that helps planets move this way? [yes] What is that force
called? [gravity]
Looking for Planets Without Seeing Them
Materials You Will Need
• Students: Notebook paper, pencil
• Strong piece of string
Time Required
• One hour
• Drinking straws
• Soft clay
ACTIVITY
Let’s make a model of a star and planet. [Tie one end of a piece of string around
a drinking straw, allowing the knot to slide back and forth along the straw. Put
a golf-ball-size lump of clay around one end of the drinking straw.] What do
you think the clay is supposed to be? [a star] [Put a marble-size lump of clay
around the other end of the straw.] What does this lump of clay represent?
[a planet] [Hang the model up so it can spin freely. GENTLY turn the model.]
Watch the big lump of clay [the star]. Does it stay in one place or move?
[It moves slightly. Students may be disappointed that the motion is not greater,
but this is close to the slight motion of a star with an orbiting planet.] Observe
from across the room. Can you see the “wobble”? [more difficult]
Hold another drinking straw vertically at arm’s length between your eye and
our little spinning solar system. [The vertical straw simulates the measuring
instruments on the SIM spacecraft.] Can you see the wobble now? [It should
be easier to see.] SIM is like that drinking straw in your hand. It will help us
see the wobble of stars with planets that are very far away.
EW-2000-08-008-JPL
TAKING.THE.MEASURE.OF.THE.UNIVERSE
WHO MAKES A SPACE MISSION POSSIBLE?
A project such as SIM requires the talents of people of wide-ranging and varied
backgrounds to be successful. However, almost all the SIM team members see
mathematical skills as an important foundation of their daily work. A good example
is Janis Chodas, SIM Flight System Manager.
Janis oversees the SIM flight system, which includes, among other things, the
computers, optics, sensors, actuators, structure, and software that comprise the
interferometer and the spacecraft. A major part of Janis’s job involves interacting
with two industry partners, Lockheed Martin Missiles and Space and TRW.
Working with the JPL SIM team, they will develop the concepts and designs for
the SIM flight system, and then build, assemble, and test the system.
Before joining the SIM team, Janis was the
Project Element Manager for one of the subsystems
on the Cassini spacecraft, which was
launched in October 1997 on a mission to
Saturn during 2004–2008. Prior to the launch,
Janis handled other duties for Cassini and
also for the highly successful Galileo mission
to Jupiter and its moons.
Janis holds a Master of Applied Science degree
in Aerospace Engineering from the University
of Toronto. She received NASA’s Exceptional
Achievement Medal for her leadership of the
attitude control subsystem design for Cassini,
and NASA’s Exceptional Service Medal for her contributions to the development
and testing of Galileo. Throughout her school career, math and physics were
among Janis’s favorite subjects. She found the logic and elegance of mathematics
appealing. In high school, Janis enrolled in advanced math classes and had
the opportunity to participate in math contests and team computer games.
Born and raised in Toronto, Canada, Janis became interested in working at JPL
because of the unique and stimulating nature of the space exploration work performed
there. She thoroughly enjoys the challenge of space missions! Janis and
her husband Paul live in La Cañada–Flintridge, California, with their two sons,
Mark and Peter. In her spare time, Janis is an active soccer and baseball mom
and also enjoys dancing in local ballet performances.
SUGGESTED ACTIVITY
Discuss how you think mathematics are used in space missions. You might start
by thinking about what you want a mission to do — for example, the SIM spacecraft
will measure the distances to stars and the positions of the stars in the sky,
and the measurements must be extremely accurate. The spacecraft will have
computers, mirrors, sensors, and many other parts, and will have a very large
structure.
One part of Janis’s work involves working with the teams that will develop the
design for the flight system. Then they will build, assemble, and test the system.
If you were on the team, how would you use mathematics to design a spacecraft
to measure distances to stars? Would you use mathematics in assembling the
spacecraft? What kinds of problems might you encounter that can be solved by
using mathematics?
Math is a powerful tool that anyone can use. Computer programmers use it —
can you think of other jobs needed for space missions that use mathematics?
MORE ABOUT SIM
SIM is managed for the National Aeronautics and Space Administration
(NASA) by the Jet Propulsion Laboratory (JPL), a division of the California
Institute of Technology. For more information about SIM, visit our web site at
http://sim.jpl.nasa.gov
To contact the SIM team and send comments and suggestions on this poster,
please write via e-mail to sim@jpl.nasa.gov
SIM is an integral part of NASA’s Origins Program. To learn more about the
Origins Program and about other missions studying the universe, visit
http://origins.jpl.nasa.gov
For additional NASA educational materials, visit http://spacelink.nasa.gov
Teachers — Please take a moment to evaluate this product at
http://ehb2.gsfc.nasa.gov/edcats/educational_wallsheet
Your evaluation and suggestions are vital to continually improving NASA
educational materials. Thank you.
Permission is given to duplicate this publication for educational purposes.
It Takes Math to Go to Space
JANIS CHODAS, SIM FLIGHT
SYSTEM MANAGER.
EW-2000-08-008-JPL
TAKING.THE.MEASURE.OF.THE.UNIVERSE
The Meterstick Master continues on toward the tree, measuring until the tree
is reached. Read the measurement and tell the Inspector/Recorder the total
distance between the Line Sighter’s eye and the base of the tree.
The Meterstick Master measures the height of the Math Magician and tells
the Inspector/Recorder.
The Math Magician leads the team in the analysis of the data collected and
designs a mathematical plan to compute the height of the tree. Suggestions:
Make a drawing of the measuring experiment as seen by someone watching
from the side. Show the position of each person and draw lines to show what
the Line Sighter was looking at.
What is a ratio? Can it help solve this problem? On your drawing, write down
the data you collected. Circle the object whose height you want to measure.
Write out a mathematical plan (equation) and fill in the data numbers. Compute
the height of the tree using your plan. Does the answer make sense? (Is the
height too short, or too tall?)
If you think you need to try again, that is OK. Ask other teams for help after
several attempts, but be sure to take your work with you so they know what
you were thinking about. Put your team’s agreed-on computed answer for the
height of the tree on the Class Height Data Table. Compare your answer with
answers from the other teams. Do you agree or disagree?
The Space Interferometry Mission will measure stars that are very far away.
The SIM astronomers are using a spacecraft and calculations like these to
measure distances in the universe.
Measure a Tree
Distance from Line Sighter’s eye to
the Math Magician’s feet meters
Distance from Line Sighter’s eye to
the base of the tree meters
Height of the Math Magician meters
Computed height of the tree meters
What is the tallest object you can measure with a meterstick? A fence? The
ceiling? Have you ever disagreed with someone about the height of a tree or
building? If you work as a team, you can measure the height of any object!
THE TEAM MEMBERS AND THEIR JOBS
• Math Magician: Stands between the measuring being done and the object to
be measured, holding the end of a string on top of her or his head.
• Line Sighter: “Sights” using a string to line up the top of the Math Magician’s
head with the object to be measured.
• Meterstick Master: Uses a meterstick to measure many meters and parts of
meters and adds them together into a total (the total may have decimals); gives
data to the Inspector/Recorder.
• Inspector/Recorder: Reads directions, inspects work being done, and records
data from the Meterstick Master. Also gets equipment, and takes a pencil and
the Height Data Table to the field with a clipboard or other item to write on.
DIRECTIONS FOR MEASURING A TREE
The Inspector/Recorder picks a place to measure the tree. It should be a place
where the Line Sighter can lie on the ground and have a clear view of the tree.
The Math Magician chooses a place to stand and holds the end of a long string
on top of his or her head.
The Line Sighter takes the other end of the string and moves to a place where
she or he can lie down and still see the top of the Math Magician’s head lined
up with the top of the tree. Use the string pulled tight to help you “sight” these
two and line them up. You many have to move several times.
The Meterstick Master now measures the distance on the ground (not the string)
between the Line Sighter’s eye and the feet of the Math Magician. Without
moving the meterstick, read the measurement and tell the Inspector/Recorder
(it should be a number that has decimals for the small parts of a meter
measured). Do not remove the meterstick!
Height Data Table
1
2
3
4
5
6
7
8
9
Materials You Will Need
• Pencil and clipboard or other hard surface
(such as a book) to write on in the field
• Long piece of string
• Meterstick
STUDENT.PAGE
EW-2000-08-008-JPL
TAKING.THE.MEASURE.OF.THE.UNIVERSE
Measure Earth
1
2
3
4
HOW TO MEASURE EARTH
Make a mathematical plan to find out how many pieces of “Earth pie” there are
if each piece is 7.12 degrees and a full circle is 360 degrees.
What did Eratosthenes conclude about the shape of Earth?
5
6
7
Eratosthenes was the first to take the measure of his known universe, which
was Earth. Listen to the story as your teacher reads it.
On a separate piece of paper, write the answers to these questions:
How did the angle of the obelisk shadow help Eratosthenes measure Earth?
What are two possible methods that Eratosthenes used to measure the distance
from Syene to Alexandria?
How far was that distance?
Make a mathematical plan to measure the distance around Earth using units
of stadia.
Now calculate the size of Earth using the information that Eratosthenes had.
TO SUN
ALEXANDRIA
SYENE
STUDENT.PAGE
EW-2000-08-008-JPL
TAKING.THE.MEASURE.OF.THE.UNIVERSE
Work as a team, and check each box when the task is completed.
Distances in space are very hard to imagine. Since few people walk between
cities, it is even hard to imagine the distance between two widely separated
places. Try to imagine a trip from your house to the next town. Can you
imagine walking to the next town with a meterstick, stopping to measure every
meter of your journey? ______ (yes or no) Now try to imagine the distance to the
Moon, to another planet, or to a star. Science fiction movies make it seem easy to
travel in space. But what would it be like to actually try to travel to another star?
THE TEAM MEMBERS AND THEIR JOBS
• Materials Engineer: Reads list of materials, finds materials and brings them to
the team, cleans up, and returns all materials after the experiment is completed.
• Experimental Specialist: Reads the “Directions for Measuring the Galaxy” and
performs the experiment; completes the Student Data Table with information
from the Data Processing Statistician.
• Data Processing Statistician: Computes numerical information and gives
information to the Experimental Specialist.
DIRECTIONS FOR MEASURING THE GALAXY
Imagine that Earth is only as big as a grain of rice. Mathematicians call this technique
“changing the scale.” Using the Student Data Table, compute the number
of rice grains (“mini- Earths”), if placed beside each other in a row, needed to go
from here to the Moon. Hint: To do this, you need to know that Earth is about
13,000 km in diameter.
Consult the Student Data Table to find the distance from Earth to the Moon.
Using this information, discuss and design a mathematical plan to change the
distance from Earth to the Moon into rice grains laid in a row. Write down your
plan. Check your plan with your teacher. Compute the number of rice grains
in this scale that show the distance from Earth to the Moon. Write this number
in your table.
Draw a line and try to lay this many grains of rice on it, side by side so that each
rice grain touches its neighbor. All the rest of the computations will use the scale
of one rice grain = one mini-Earth.
Using your mathematical plan, compute the number of rice grains needed to
measure the distance to the other planet. Try laying out the rice grains (miniEarths)
to represent the distance from Earth to Mars. Check this with your teacher.
Write this number in the Student Data Table. Would you like to count this
many rice grains? ______ (yes/no) Would you like to lay out that many rice
grains in a row? ______ (yes/no)
Measuring length instead of counting: Try changing rows of rice grains into
measures of length. (An average coarse rice grain is 1 millimeter in diameter.)
Look at the ruler or meterstick, and find the length called millimeters. Draw two
dots that are 1 millimeter apart. Show your teacher. This is the size of a rice
grain (mini-Earth). Draw four more dots along a line, each dot 1 millimeter
farther away from the last. Using your ruler, draw a straight line through all five
dots. Show your teacher. This represents five rice grains lined up.
We need to make a new mathematical plan to change a certain number of rice
grains in a row into millimeters of length. Write down your plan and show it to
your teacher. How many millimeters of rice grains are needed to measure the
distance from Earth to Mars in rice grain (mini-Earth) units? Write this answer in
the Student Data Table and check it with your teacher. Measure a length of
string this long and stretch it out. This is how many rice grains are needed, lined
up, to measure the distance from Earth to Mars. On this scale (one rice grain =
one mini-Earth), Pluto is over 400 meters (4 soccer fields) away. On this scale,
the distance to Sirius — a nearby bright star — is a line of rice grains stretching
from here to New York City! You can see that distances in the universe are very
hard to imagine. SIM is going to measure them!
Measure the GalaxySTUDENT.PAGE
Materials You Will Need (For each team)
• About one tablespoon of uncooked rice
• Ruler or meterstick with millimeter markings
Distance from In Reality (km) In Rice Grains Length Rice
Earth to: Grains Occupy
Moon 384,000
Mars 78,000,000
Sun 150,000,000
Pluto 5,900,000,000
Sirius 81,000,000,000,000
• Calculators
• 15 meters of string
1
2
3
4
5
Student Data Table
EW-2000-08-008-JPL
TAKING.THE.MEASURE.OF.THE.UNIVERSE
UPON COMPLETION OF THIS ACTIVITY, STUDENTS WILL BE ABLE TO
• Use mathematical problem solving to measure inaccessible objects
• Use data sets with one variable
• Employ ratios to compare similar triangles
NCTM STANDARDS
Grades 6–8
• Data Analysis: Students collect, organize, and represent data sets that have
one variable and identify relationships of the data collected. Students will
know various forms of display of data sets.
• Understanding Numbers: Students will demonstrate ways of representing
numbers, relationships among numbers, and number systems.
Grades 9–12
Mathematics as Problem Solving, Algebra, Mathematical Connections, and
Mathematics as Communication.
COOPERATIVE TEAMS
Teams of four are suggested to fully involve students of different learning styles.
If possible, choose team members that represent “Kinesthetic” (Meterstick Master),
“Spatial” (Line Sighter), “Quantitative/Mathematical” (Math Magician), and
“Verbal/Literal” (Inspector/Recorder). [These suggestions are roughly
based on Howard Gardener’s “Multiple Intelligences.”]
ANTICIPATED PROBLEMS
Line Sighter: Students often have trouble sighting. The idea of
aligning two objects so they seem superimposed is conceptually
foreign. If students are having trouble with this
concept, try using two pencils held upright with
erasers on top, one close to the eye and one
at arm’s length. Have them align the two
Measure a Tree
erasers with a distant object. Or — even better — if a mirror is available,
have them align the pencils by looking at themselves in the mirror where
they see their eyes as well as the reflected pencils.
The Meterstick Master must be able to make successive measurements of a
long distance with one or two metersticks by going end to end and keeping
track. This student must also be able to measure decimal portions of meters
and express the final measurement in one number that includes decimals.
Example: Student measures 5 full metersticks and an additional 40 centimeters.
This is 5.40 meters.
Final display of the data on a Class Height Data Table: The intent is to
encourage students to explore the design of a mathematical plan to
solve the problem (students are often more concerned with
getting the “right answer” and avoiding embarrassment).
Have teams discuss their methods before posting
data on the Class Height Data Table. Have a
discussion about interesting mathematical
plans that gave results that did not
make sense. As much praise
should be shown for analysis
of why a method did
not work as for
the answer.
3
1
2
Notes on Materials You Will Need
• Metersticks (2 per team if possible), or a long tape measure.
• One ball of string per team (enough to stretch from the
Line Sighter to the top of the Math Magician’s head).
MATH
MAGICIAN
LINE
SIGHTER
DISTANCE TO
MATH MAGICIAN
Time Required
• One hour
DISTANCE
TO TREE
TREE
TEACHER.PAGE
EW-2000-08-008-JPL
TAKING.THE.MEASURE.OF.THE.UNIVERSE
UPON COMPLETION OF THIS ACTIVITY, STUDENTS WILL BE ABLE TO
• Use parallel lines and segments of circles to measure Earth
• Use relationships among numbers
NCTM STANDARDS
Grades 6–8
• Data Analysis: Students collect, organize, and represent data sets.
• Identify Variables: Explore relationships among numbers and number systems.
• Develop Understanding of Large Numbers: Use benchmarks to understand
magnitude — Understand and appropriately use various relationships for large
numbers (e.g., exponential, scientific, and calculator notation).
Grades 9–12
Students use theorems involving the properties of parallel lines cut by a transversal
— Investigate the properties of circles — Use and measure sides, interior,
and exterior angles of triangles — Classify figures and solve problems — Understand
the notion of angles and how to measure them.
DIRECTIONS FOR THE TEACHER
Read the story of Eratosthenes with the students. Stop the story when you come
to places where an activity or calculation is suggested. You may wish to make
and project a transparency of the illustrations.
Time Required — One hour
THE STORY OF ERATOSTHENES
A scientific writer, poet, and astronomer, Eratosthenes [Err/a/tos/the/neez] lived
more than 2,000 years ago in Egypt (276 BC to 194 BC). He is the first person
known to have calculated the circumference of Earth.
Eratosthenes knew that there was a deep well in Syene in Egypt. People walked
down circular steps to get into the well to get water. It was very dark on the steps
when people were walking down. But on one day a year (June 21st), the sunlight
at noon shone all the way down to the bottom of the well. He noticed that his
own shadow was very “short” on that day, only covering his feet but not the
ground nearby. Eratosthenes went to another city in Egypt — Alexandria. On
that same day of the year, sunlight did not reach the bottom of wells. And he
noticed that his shadow was “longer.”
Look at the illustration of the student. She has placed clay on two places on
a globe and put sticks in the clay so they are perpendicular to the surface of
Measure Earth
the globe. Notice that one stick has a very small shadow but the other stick
has a longer shadow. What does Earth do to the sticks so that one of them has a
longer shadow?
Eratosthenes observed the shadow of a tall obelisk (the Washington Monument
in Washington, D.C., is an obelisk). He figured out a way to measure the angle
of the shadow from the top of the obelisk. The angle was 7.12 degrees. He
divided the degrees in a whole circle (360 degrees) by the angle he measured
in Alexandria. He found there were 50 pieces of “pie.” The distance between
Syene and Alexandria was about 1/50th of a circle.
Eratosthenes made two observations:
1. The difference in the shadows and the amount of light shining down the
wells is because Earth is round (not flat).
2. The angle of the obelisk is very important. What do you think it told
Eratosthenes?
The diagram shows that this angle is the same as the angle between the two
cities, Syene and Alexandria, measured from the center of Earth. Like pieces
of pie, the angle of the obelisk in Alexandria is the angle between that city and
Eratosthenes’ home, Syene. How many “Eratosthenes pie pieces” are needed
to make a whole Earth?
Eratosthenes had one last problem to solve: How long was the outer edge of one
piece of “pie” on Earth’s surface? One story says he paid someone to walk from
Syene to Alexandria, measuring with a rod, which is similar to a meterstick but
marked in units called stadia. He found that the distance between Syene and Alexandria
was 5,000 stadia, about 800 km. (In a similar fashion, you could measure
the length of your classroom with a meterstick, turning it end over end. Then,
knowing how many classrooms are in your hall, you could calculate the length of
the hall.) Another story said that Eratosthenes knew that caravans of camels went
from Syene to Alexandria in 50 days and traveled at the rate of 100 stadia a day.
Eratosthenes now knew the distance from Alexandria to Syene and the number
of those lengths (pieces of “Earth pie” to go once around Earth. He multiplied
the two together (50 x 5,000) and concluded that Earth was 250,000 stadia
around. In modern measure, that is 40,000 km. Today, using new instruments,
we know that Earth is 41,670 km around. Do you think Eratosthenes did a pretty
good job? Do you think that Columbus knew about the experiment carried out
by Eratosthenes?
TEACHER.PAGE
EW-2000-08-008-JPL
TAKING.THE.MEASURE.OF.THE.UNIVERSE
Measure the Galaxy
UPON COMPLETION OF THIS ACTIVITY, STUDENTS WILL BE ABLE TO
• Convert the scale size of an object using ratio manipulation
• Calculate distances using large numbers
NCTM STANDARDS
Grades 6–8
• Data Analysis: Concept of scale (Earth is reduced to rice-grain scale) —
Concept of representations of large numbers — Relationship of large numbers
(cosmic distances) to a small scale (rice grains) — Exponential and calculator
notation.
• Understanding Numbers: Students will demonstrate ways of representing
numbers, relationships among numbers, and number systems.
Grades 9–12
Analysis using large numbers — Create benchmarks to understand magnitude —
Compute using representations for large numbers (e.g., exponential, scientific,
and calculator notation).
NOTES ON THE ACTIVITY
Teams of three are suggested. Distances in space are very hard to imagine. We
suggest reading the student sheet aloud, stopping to complete activities and
calculations as they occur. Time should also be allowed for group discussion to
create their “mathematical plans.” Alter the plan if it leads to incorrect solutions
and/or analyze results of the computation.
NOTES ON THE MATHEMATICAL PLANS
Depending on age appropriateness, it is hoped that students will be able to think
about the challenge of converting distances between celestial objects into the
scale of rice grains representing the size of Earth.
The mathematical plan to convert the distance from Earth to the Moon should
include finding the distance to the Moon in kilometers (km) and dividing by the
diameter of Earth in km: 384,000 km /13,000 km per Earth diameter = 29.5 Earth
diameters (rice grains).
After the students draw the line, it may help to have them go over the line with a
glue stick. This will keep the rice grains from moving around during placement.
The mathematical plan for calculating the number of rice grains (mini-Earths)
to show the distance to Mars is the same as for the Earth-to-Moon problem. This
may be a time to introduce scientific notation: 78,000,000 km / 13,000 km =
6,000 rice grains. A discussion about the time needed to lay out 6,000 rice grains
might lead to seeing the need for an easier method of measuring scale distances.
This activity begins to make the connection between the size of rice grains and
millimeters.
The mathematical plan for changing the Earth-to-Mars distance from rice grains
into millimeters (mm) of length is: one rice grain = 1 mm of length. Therefore,
6,000 rice grains x 1 mm per rice grain = 6,000 mm. If students are proficient
in conversion in the metric system, have them change the 6,000 mm into either
centimeters or meters. Then have them measure a length of string this long and
lay it out in the classroom, perhaps crossing their rice grains of the Earth-to-Moon
distance. A discussion of relative distances in the solar system may lead to interest
in computing distances to other planets, the Sun, or nearby stars.
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Notes on Materials You Will Need
• Rice grains about 1 mm in diameter
the short way. If this size is unavailable,
tapioca could be substituted.
• Hand lenses or forceps may be useful
but are not obligatory.
Time Required
• One hour
• String, 15 meters long. The string is to
show the distance from Earth to the
Sun using the scale of one Earth =
one grain of rice.
• Glue stick (optional)
TEACHER.PAGE
JPL 400-910 8/00
Distance In Reality (km) In Rice Grains Length Rice
from Earth to: Grains Occupy
Moon 384,000 29.5 2.95 cm
Mars 78,000,000 6,000 6 m
Sun 150,000,000 11,500 11.5 m
Pluto 5,900,000,000 454,000 454 m
Sirius 81,000,000,000,000 6,200,000,000 6,200 km
Teacher Data Table