EW-2000-08-008-JPL

 UNIVERSE?

ld science. We know of early peoples’ interest in celestial

d the sky both day and night — from drawings on cave, Mayan, Egyptian, and Chinese observatories and their

 direct experiences in science, such as chemistry (cooking)

, no one could actually touch the sky. Even climbing to the

 not help humans reach the stars. Unless someone saw a

 a still-warm piece of the meteorite, our ancestors had only

the wonders above. People asked: “How far away are the

ipparchos, a scientist/mathematician of ancient Greece,

catalog of the stars’ brightnesses and positions in the sky.

nd engineers of NASA’s Space Interferometry Mission,ntinuing his work.

HOW WILL WE TAKE THE MEASURE?SIM will measure the positions of thousands ofstars extremely accurately — scientists call thisprocess astrometry. SIM will create a “virtual grid”of reference points on the sky through measuringthe separation between stars. By studying howthe positions change with time, we will be ableto measure distances from Earth to the stars. SIMwill also help us find out whether small planetsare in orbit around stars other than our Sun. 

SIM WILL MEASURE ACCURATEDISTANCES TO STARS THROUGHOUTTHE MILKY WAY GALAXY.

SIM WILL HELP US UNDERSTAND THE LIFE CYCLESOF STARS.

e Measure of the Universe

LP US

UT STARS?

rent positions and

n our own galaxy

scientists will be

ry of these stars

 gain a better

w the Milky Way

n it formed. We

wledge of how

m and will be bet

ir development.

WHAT WILL SIM HELP US UNDERSTANDABOUT PLANETS AROUND OTHER STARS?By observing subtle disturbances in the motionsof stars, we will be able to tell whether these starshave planets orbiting around them. The smallestplanets that we will be able to find are just a littlebigger than Earth. So far, Earth is the only planetin its class that we know of. Other planets that wehave discovered orbiting stars are hundred of timeslarger than Earth. Only when we know of morethan one example will we be able to tell whethera planet like Earth is common or rare.

INTERFEROMETRY AS AMEASURING TOOLSIM’s key measuring tool is a spacecraft to be launched in the secondhalf of this decade. The SIM spacecraft uses interferometry of visiblelight — the same light that our eyessee. But the spacecraft can do something that our eyes cannot. SIM’stelescopes and mirrors collect andcombine light waves coming fromstars and measure the way these light waves interact or “interfere” with eachother. The interferometer data are analyzed to determine the distance to a starand its position as compared with other stars.ABOUT THIS POSTERLike no mission before it, advanced mathematics will be required to understandthe data gathered by SIM. Mathematical analysis and tools are an integral part ofNASA’s space program. This poster and series of activities are designed to makethe 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 abroad and enthusiastic understanding of mathematics is needed not only inspace exploration, but in every part of our society.

NASA ALSO DEVELOPS GROUND-BASEDINTERFEROMETERS SUCH AS THE KECKINTERFEROMETER ON TOP OF MAUNAKEA, THE HIGHEST MOUNTAIN ON THEISLAND OF HAWAII.
EW-2000-08-008-JPL

ad to the class as you do the activity. [Teacher hints are

f objects: STAR, MOON, COMET, METEOR [shooting star],

arth], BLACK HOLE, GALAXY, AIRPLANE. Imagine the

ames of objects you have seen in the night sky with your

tudents may pick any or all of these objects.]

u picked was PLANET, please share with the class what

u decided it was a planet. Did the planet you saw look dif

bright lights in the dark sky? Here are some hints: Did the

winkle, or shine like a flashlight? If you watched several

stay near the same stars every night? [It may have moved

e the color of the planet? [Mars sometimes appears to have

Could you see its shape? [Not without a telescope or bin

aturn has an oblong shape.]

ar system are near Earth. If Earth is the size of a grain of

rice grain about 4 meters (20 feet) away. A nearby star is a

ew York is from Los Angeles. If that “grapefruit” star had its

uld be rice grains as well. Imagine trying to see a rice grain

00 miles) away.

etry Mission (SIM) cannot “see” the planets of nearby stars.

now if other stars besides our own Sun have planets around

asure something that indicates that a planet is present.]

with a friend who suddenly found they had an insect inside

u see the insect right away? [Probably not; it was inside

mall to see from a distance.] Did your friend quietly explain

ct in their clothing? [doubtful] Did your friend yell? [prob

do anything else? [wiggle or jump] Did your friend move

w the class what a person does who has an insect in their

iate, 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 wherestars 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 movearound.] 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 forcecalled? [gravity]

or Planets Without Seeing Them

 Need

paper, pencil

g

• Drinking straws

• Soft clay

ACTIVITYLet’s make a model of a star and planet. [Tie one end of a piece of string arounda drinking straw, allowing the knot to slide back and forth along the straw. Puta golf-ball-size lump of clay around one end of the drinking straw.] What doyou think the clay is supposed to be? [a star] [Put a marble-size lump of clayaround 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.] Observefrom across the room. Can you see the “wobble”? [more difficult]Hold another drinking straw vertically at arm’s length between your eye andour little spinning solar system. [The vertical straw simulates the measuringinstruments on the SIM spacecraft.] Can you see the wobble now? [It shouldbe easier to see.] SIM is like that drinking straw in your hand. It will help ussee the wobble of stars with planets that are very far away.
EW-2000-08-008-JPL

CE MISSION POSSIBLE?

 requires the talents of people of wide-ranging and varied

ccessful. However, almost all the SIM team members see

s an important foundation of their daily work. A good ex

s, SIM Flight System Manager.

M flight system, which includes, among other things, the

nsors, actuators, structure, and software that comprise the

e spacecraft. A major part of Janis’s job involves interacting

tners, Lockheed Martin Missiles and Space and TRW.

SIM team, they will develop the concepts and designs for

 and then build, assemble, and test the system.

M team, Janis was the

ager for one of the sub

i spacecraft, which was

1997 on a mission to

008. Prior to the launch,

uties for Cassini and

ccessful Galileo mission

ns.

of Applied Science degree

ring from the University

ved NASA’s Exceptional

or her leadership of the

stem design for Cassini,

nal Service Medal for her contributions to the development

. Throughout her school career, math and physics were

 subjects. She found the logic and elegance of mathemat

 school, Janis enrolled in advanced math classes and had

rticipate in math contests and team computer games.

ronto, Canada, Janis became interested in working at JPL

e and stimulating nature of the space exploration work per

roughly enjoys the challenge of space missions! Janis and

 in La Cañada–Flintridge, California, with their two sons,

r spare time, Janis is an active soccer and baseball mom

ng in local ballet performances.

SUGGESTED ACTIVITYDiscuss how you think mathematics are used in space missions. You might startby 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 havecomputers, mirrors, sensors, and many other parts, and will have a very largestructure.One part of Janis’s work involves working with the teams that will develop thedesign 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 spacecraftto measure distances to stars? Would you use mathematics in assembling thespacecraft? What kinds of problems might you encounter that can be solved byusing 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 SIMSIM is managed for the National Aeronautics and Space Administration(NASA) by the Jet Propulsion Laboratory (JPL), a division of the CaliforniaInstitute of Technology. For more information about SIM, visit our web site athttp://sim.jpl.nasa.govTo contact the SIM team and send comments and suggestions on this poster,please write via e-mail to sim@jpl.nasa.govSIM is an integral part of NASA’s Origins Program. To learn more about theOrigins Program and about other missions studying the universe, visithttp://origins.jpl.nasa.govFor additional NASA educational materials, visit http://spacelink.nasa.govTeachers — Please take a moment to evaluate this product athttp://ehb2.gsfc.nasa.gov/edcats/educational_wallsheetYour evaluation and suggestions are vital to continually improving NASAeducational materials. Thank you.Permission is given to duplicate this publication for educational purposes.

Math to Go to Space

JANIS CHODAS, SIM FLIGHTSYSTEM MANAGER.
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The Meterstick Master continues on toward the tree, measuring until the treeis reached. Read the measurement and tell the Inspector/Recorder the totaldistance between the Line Sighter’s eye and the base of the tree.The Meterstick Master measures the height of the Math Magician and tellsthe Inspector/Recorder.The Math Magician leads the team in the analysis of the data collected anddesigns a mathematical plan to compute the height of the tree. Suggestions:Make a drawing of the measuring experiment as seen by someone watchingfrom the side. Show the position of each person and draw lines to show whatthe Line Sighter was looking at.What is a ratio? Can it help solve this problem? On your drawing, write downthe data you collected. Circle the object whose height you want to measure.Write out a mathematical plan (equation) and fill in the data numbers. Computethe height of the tree using your plan. Does the answer make sense? (Is theheight too short, or too tall?)If you think you need to try again, that is OK. Ask other teams for help afterseveral attempts, but be sure to take your work with you so they know whatyou were thinking about. Put your team’s agreed-on computed answer for theheight of the tree on the Class Height Data Table. Compare your answer withanswers 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 tomeasure distances in the universe.

a Tree

Distance from Line Sighter’s eye tothe Math Magician’s feet meters

Distance from Line Sighter’s eye to

the base of the tree metersHeight of the Math Magician metersComputed height of the tree meters

ect you can measure with a meterstick? A fence? The

er disagreed with someone about the height of a tree or

 as a team, you can measure the height of any object!

S AND THEIR JOBS

nds between the measuring being done and the object to

ng the end of a string on top of her or his head.

s” using a string to line up the top of the Math Magician’s

t to be measured.

Uses a meterstick to measure many meters and parts of

m together into a total (the total may have decimals); gives

r/Recorder.

 Reads directions, inspects work being done, and records

stick Master. Also gets equipment, and takes a pencil and

le to the field with a clipboard or other item to write on.

EASURING A TREE

er picks a place to measure the tree. It should be a place

r can lie on the ground and have a clear view of the tree.

hooses a place to stand and holds the end of a long string

ead.

s the other end of the string and moves to a place where

n and still see the top of the Math Magician’s head lined

 tree. Use the string pulled tight to help you “sight” these

. You many have to move several times.

r now measures the distance on the ground (not the string)

hter’s eye and the feet of the Math Magician. Without

k, read the measurement and tell the Inspector/Recorder

er that has decimals for the small parts of a meter

move the meterstick!

Height Data Table

56789

 Need

d or other hard surface

write on in the field

• Long piece of string

• Meterstick
EW-2000-08-008-JPL

Earth

34

EARTH

 plan to find out how many pieces of “Earth pie” there are

degrees and a full circle is 360 degrees.

s conclude about the shape of Earth?

567

first to take the measure of his known universe, which

e story as your teacher reads it.

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 distancefrom Syene to Alexandria?How far was that distance?Make a mathematical plan to measure the distance around Earth using unitsof stadia.Now calculate the size of Earth using the information that Eratosthenes had.

TO SUN

ALEXANDRIA

SYENE
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d check each box when the task is completed.

e very hard to imagine. Since few people walk between

to imagine the distance between two widely separated

e a trip from your house to the next town. Can you

e next town with a meterstick, stopping to measure every

? ______ (yes or no) Now try to imagine the distance to the

net, or to a star. Science fiction movies make it seem easy to

hat would it be like to actually try to travel to another star?

S AND THEIR JOBS

 Reads list of materials, finds materials and brings them to

, and returns all materials after the experiment is completed.

alist: Reads the “Directions for Measuring the Galaxy” and

ment; completes the Student Data Table with information

essing Statistician.

atistician: Computes numerical information and gives

Experimental Specialist.

EASURING THE GALAXY

only as big as a grain of rice. Mathematicians call this tech

scale.” Using the Student Data Table, compute the number

Earths”), if placed beside each other in a row, needed to go

n. Hint: To do this, you need to know that Earth is about

er.

Data Table to find the distance from Earth to the Moon.

n, discuss and design a mathematical plan to change the

o the Moon into rice grains laid in a row. Write down your

n with your teacher. Compute the number of rice grains

w the distance from Earth to the Moon. Write this number

o lay this many grains of rice on it, side by side so that each

neighbor. All the rest of the computations will use the scale

e mini-Earth.

Using your mathematical plan, compute the number of rice grains needed tomeasure 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 ricegrains in a row? ______ (yes/no)Measuring length instead of counting: Try changing rows of rice grains intomeasures 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 twodots 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 millimeterfarther away from the last. Using your ruler, draw a straight line through all fivedots. Show your teacher. This represents five rice grains lined up.We need to make a new mathematical plan to change a certain number of ricegrains in a row into millimeters of length. Write down your plan and show it toyour teacher. How many millimeters of rice grains are needed to measure thedistance from Earth to Mars in rice grain (mini-Earth) units? Write this answer inthe Student Data Table and check it with your teacher. Measure a length ofstring this long and stretch it out. This is how many rice grains are needed, linedup, 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 stretchingfrom here to New York City! You can see that distances in the universe are veryhard to imagine. SIM is going to measure them!

he Galaxy

 Need (For each team) 

on of uncooked rice 

with millimeter markings

Distance from In Reality (km) In Rice Grains Length RiceEarth to: Grains OccupyMoon 384,000Mars 78,000,000Sun 150,000,000Pluto 5,900,000,000Sirius 81,000,000,000,000

• Calculators

• 15 meters of string

3

45

Student Data Table
EW-2000-08-008-JPL

 OF THIS ACTIVITY, STUDENTS WILL BE ABLE TO

problem solving to measure inaccessible objects

one variable

mpare similar triangles

ents collect, organize, and represent data sets that have

entify relationships of the data collected. Students will

 of display of data sets.

mbers: Students will demonstrate ways of representing

ips among numbers, and number systems.

em Solving, Algebra, Mathematical Connections, and

munication.

MS

gested to fully involve students of different learning styles.

am members that represent “Kinesthetic” (Meterstick Master),

r), “Quantitative/Mathematical” (Math Magician), and

ector/Recorder). [These suggestions are roughly

rdener’s “Multiple Intelligences.”]

LEMS

 often have trouble sighting. The idea of

o they seem superimposed is concepnts are having trouble with this

o pencils held upright with

ose to the eye and one

 them align the two

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 wherethey see their eyes as well as the reflected pencils.The Meterstick Master must be able to make successive measurements of along distance with one or two metersticks by going end to end and keepingtrack. This student must also be able to measure decimal portions of metersand 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 toencourage students to explore the design of a mathematical plan tosolve the problem (students are often more concerned withgetting the “right answer” and avoiding embarrassment).Have teams discuss their methods before postingdata on the Class Height Data Table. Have adiscussion about interesting mathematicalplans that gave results that did notmake sense. As much praiseshould be shown for analysisof why a method didnot work as forthe answer.

32

 You Will Need

am if possible), or a long tape measure.

er team (enough to stretch from the

op of the Math Magician’s head).

MATHMAGICIAN

LINE

SIGHTER DISTANCE TO MATH MAGICIAN

Time Required• One hour

DISTANCE

TO TREE

TREE
EW-2000-08-008-JPL

 OF THIS ACTIVITY, STUDENTS WILL BE ABLE TO

nd segments of circles to measure Earth

mong numbers

ents collect, organize, and represent data sets.

xplore relationships among numbers and number systems.

ding of Large Numbers: Use benchmarks to understand

erstand and appropriately use various relationships for large

onential, scientific, and calculator notation).

s involving the properties of parallel lines cut by a trans

he properties of circles — Use and measure sides, interior,

 triangles — Classify figures and solve problems — Under

gles and how to measure them.

HE TEACHER

osthenes with the students. Stop the story when you come

ctivity or calculation is suggested. You may wish to make

rency of the illustrations.

ne hour

TOSTHENES

et, and astronomer, Eratosthenes [Err/a/tos/the/neez] lived

s ago in Egypt (276 BC to 194 BC). He is the first person

ated the circumference of Earth.

at there was a deep well in Syene in Egypt. People walked

o get into the well to get water. It was very dark on the steps

alking down. But on one day a year (June 21st), the sunlight

way down to the bottom of the well. He noticed that his

y “short” on that day, only covering his feet but not the

thenes went to another city in Egypt — Alexandria. On

year, sunlight did not reach the bottom of wells. And he

ow was “longer.”

n of the student. She has placed clay on two places on

 in the clay so they are perpendicular to the surface of

Earth

the globe. Notice that one stick has a very small shadow but the other stickhas a longer shadow. What does Earth do to the sticks so that one of them has alonger shadow?Eratosthenes observed the shadow of a tall obelisk (the Washington Monumentin Washington, D.C., is an obelisk). He figured out a way to measure the angleof the shadow from the top of the obelisk. The angle was 7.12 degrees. Hedivided the degrees in a whole circle (360 degrees) by the angle he measuredin Alexandria. He found there were 50 pieces of “pie.” The distance betweenSyene 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 twocities, Syene and Alexandria, measured from the center of Earth. Like piecesof pie, the angle of the obelisk in Alexandria is the angle between that city andEratosthenes’ home, Syene. How many “Eratosthenes pie pieces” are neededto make a whole Earth?Eratosthenes had one last problem to solve: How long was the outer edge of onepiece of “pie” on Earth’s surface? One story says he paid someone to walk fromSyene to Alexandria, measuring with a rod, which is similar to a meterstick butmarked 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 measurethe 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 ofthe hall.) Another story said that Eratosthenes knew that caravans of camels wentfrom 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 numberof those lengths (pieces of “Earth pie” to go once around Earth. He multipliedthe two together (50 x 5,000) and concluded that Earth was 250,000 stadiaaround. 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 prettygood job? Do you think that Columbus knew about the experiment carried outby Eratosthenes?
EW-2000-08-008-JPL

he Galaxy

 OF THIS ACTIVITY, STUDENTS WILL BE ABLE TO

ze of an object using ratio manipulation

 using large numbers

cept of scale (Earth is reduced to rice-grain scale) —

ntations of large numbers — Relationship of large numbers

o a small scale (rice grains) — Exponential and calculator

mbers: Students will demonstrate ways of representing

ips among numbers, and number systems.

umbers — Create benchmarks to understand magnitude —

sentations for large numbers (e.g., exponential, scientific,

on).

IVITY

ggested. Distances in space are very hard to imagine. We

udent sheet aloud, stopping to complete activities and

ccur. Time should also be allowed for group discussion to

atical plans.” Alter the plan if it leads to incorrect solutions

s of the computation.

HEMATICAL PLANS

propriateness, it is hoped that students will be able to think

f converting distances between celestial objects into the

presenting the size of Earth.

The mathematical plan to convert the distance from Earth to the Moon shouldinclude finding the distance to the Moon in kilometers (km) and dividing by thediameter of Earth in km: 384,000 km /13,000 km per Earth diameter = 29.5 Earthdiameters (rice grains).After the students draw the line, it may help to have them go over the line with aglue 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. Thismay 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 grainsmight 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 andmillimeters.The mathematical plan for changing the Earth-to-Mars distance from rice grainsinto 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 proficientin conversion in the metric system, have them change the 6,000 mm into eithercentimeters or meters. Then have them measure a length of string this long andlay it out in the classroom, perhaps crossing their rice grains of the Earth-to-Moondistance. A discussion of relative distances in the solar system may lead to interestin computing distances to other planets, the Sun, or nearby stars.

12345

 You Will Need

mm in diameter

size is unavailable,

bstituted.

eps may be useful

y.

• 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)

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 cmMars 78,000,000 6,000 6 mSun 150,000,000 11,500 11.5 mPluto 5,900,000,000 454,000 454 mSirius 81,000,000,000,000 6,200,000,000 6,200 km

Teacher Data Table