Space Math VII - NASA

188 Pages·2012·11.07 MB·English

National Aeronautics and Space Administration
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This collection of activities is based on a weekly series of space
science problems distributed to thousands of teachers during the 2010-
2011 school year. They were intended for students looking for
additional challenges in the math and physical science curriculum in
grades 5 through 12. The problems were created to be authentic
glimpses of modern science and engineering issues, often involving
actual research data.
The problems were designed to be ‘one-pagers’ with a Teacher’s
Guide and Answer Key as a second page. This compact form was
deemed very popular by participating teachers.
For more weekly classroom activities about astronomy and space visit
the NASA website,
http://spacemath.gsfc.nasa.gov

Add your email address to our mailing list by contacting Dr. Sten
Odenwald at
[email protected]
Front and back cover credits: Front) Top - Gravity Probe-B sketch of warped space near
Earth, and the equation from General Relativity that predicts the precession and 'frame-dragging'
effects detected by the satellite. Illustration courtesy NASA/GP-B/Stanford University. Bottom -
Three moments in the supercomputer calculation of colliding neutron stars showing the
rearrangements in the magnetic field 'ribbons' during the collision and merger. Courtesy…. Back)
A combined image of the galaxy NGC-5128 also called Centarus-A; A galaxy containing a
supermaassive black hole at its core, and ejecting jets of matter. The image is a superposition of
X-ray (blue) data from the NASA Chandra Observatory, radio (orange) and visible light data.
Credit: ESO/WFI (visible) ,MPIfR/ESO/APEX/A. Weiss (mictrowave); and NASA/CXC/CfA/R. Kraft
et al (x-ray).

Interior Illustrations: 1) Black hole art (Anon Internet); 2)Black hole art (Anon Internet); 4)
ISS and Sun (John Stetson); 11) Black hole (David Aguilar, CfA); 12) NASA/Artemis 13) Bone
loss (MenaQ7.com); 16) Exoplanet art (G. Bacon ,STScI/NASA); 17) Black hole (Dana Berry,
NASA/SkyWorks Digital); 20) Esquel Pallasite (Anon Internet); 22) Line Spectrum (Anon Internet);
23) Supercomputer (Anon Internet); 32) Gliese 581g Artwork (Lynette Cook/NASA); 38) Dreath
Star (NASA/G. Bacon); 43) Transiting planets (NASA/Tim Pyle); 45) Habitable Zones
(NASA/Kepler); 61) Solar Probe (JHU/APL); 73) Bacterium (NASA/Jodi Blum); 81) Warped space
(Pancho Eekers/GPB); 78) Dosimeter (AP/Kyodo News)
This booklet was created through an education grant NNH06ZDA001N-
EPO from NASA's Science Mission Directorate.
Space Math http://spacemath.gsfc.nasa.gov
Table of Contents
ii

Grade Page
Acknowledgments i
Table of Contents ii
Mathematics Topic Matrix v
How to use this Book ix
Alignment with Standards x
Teacher Comments x

Exploring Black Holes 3-5 1
The Moon as a Black Hole 3-5 2
The Oldest Lunar Rocks 3-5 3
ISS and a Sunspot - Angular Scales 6-8 4
The Fastest Sea Level Rise in the United States 6-8 5
An Atom Counting Exercise 6-8 6
Calculating Molecular Mass 6-8 7
Counting Atoms in a Molecule 6-8 8
How Many Stars Are There? 6-8 9
Yes…It Moves! 6-8 10
Exploring a Full-sized Black Hole 6-8 11
Wacky Spacecraft Orbits - They only seem crazy! 6-8 12
Astronaut Bone Loss In Space 6-8 13
Earth's Polar Wander - The Chandler Wobble 6-8 14
Growing Grapes in the Middle of the Desert 6-8 15
The Cometary Planet - HD209458b 9-11 16
Black Holes - Hot Stuff! 9-11 17
A Black Hole Up Close 9-11 18
The Most Massive Stars Known 9-11 19
Meteorite Compositions - a matter of density 9-11 20
Significant Figures 9-11 21
The Cosmological Redshift - The light from a distant galaxy 9-11 22
Supercomputer Math - Getting the job done fast! 9-11 23

In the News…
A Flyby of Asteroid Lutetia 6-8 24
Hinode Discovers the Origin of White Light Flares 6-8 25
Kepler's First Look at Transiting Planets 6-8 26
Terra Spies a Major Glacier Break-Up 6-8 27
The Hexagonal Tiles in the Webb Telescope Mirror 6-8 28
Super-sizing the Webb Space Telescope Mirror 6-8 29
LRO Determines Lunar Cratering History 6-8 30
The Most Distant Supernova in the Universe 6-8 31
The Earth-like Planet Gliese 581g 6-8 32
Approaching Comet Hartley-2 6-8 33
LRO Makes a Temperature Map of the Lunar South Pole 6-8 34
Space Math http://spacemath.gsfc.nasa.gov
Table of Contents (contd)
iii

Grade Page
Exploring the Cosmos with Supercomputers 6-8 35
Taking a Stroll Around a Martian Crater 6-8 36
Estimating the Diameter of the SN1979c Black Hole 6-8 37
Death Stars 6-8 38
The Crab Nebula - Exploring a Pulsar Up Close 6-8 39
The Changing Pace of Global Warming 6-8 40
A Galactic City in the Far reaches of the Universe 6-8 41
The Most Distant Objects in the Visible Universe 6-8 42
Earth-sized Planets by the Score! 6-8 43
Earth-like Planets and Habitable Zones 6-8 44
Habitable Zones and the Search for Goldilocks Planets 6-8 45
STEREO Spacecraft Give 360-degree Solar View 6-8 46
Cryo-testing the Webb Space Telescope ISIM 6-8 47
Estimating the Speed of a Tsunami 6-8 48
Exploring Radiation in Your Life 6-8 49
Lifestyles and Radiation Dose 6-8 50
Mercury and the Moon - Similar but different 6-8 51
Kepler Probes the Interior of Red Giant Stars 6-8 52
Estimating the Size and Mass of a Black Hole 6-8 53
Supercomputers - Modeling colliding neutron stars 6-8 54
The Space Shuttle - Fly me to the moon? 6-8 55
Dwarf Planets and Kepler's Third Law 6-8 56
Celestial Fireworks near Cluster NGC-3603 9-11 57
The Sky is Falling? - Well not quite! 9-11 58
Close Encounters of the Asteroid Kind 9-11 59
6-Fold Symmetry and the Webb Space Telescope Mirror 9-11 60
Solar Probe Plus - Having a hot time near the sun! 9-11 61
Solar Probe Plus - Working with angular diameter 9-11 62
Deep Impact - Closing in on comet Hartley-2 9-11 63
Terra Satellite Measures Dangerous Dust 9-11 64
Seeing the Distant Universe Clearly 9-11 65
Estimating the Volume and Mass of Comet Hartley-2 9-11 66
A Mathematical Model of Water Loss from Comet Tempel-1 9-11 67
Gamma-ray Bubbles in the Milky Way 9-11 68
Hubble Detects More Dark Matter 9-11 69
X-Rays from the Hot Gas Near the SN1979c Black Hole 9-11 70
Investigating the Atmosphere of Super-Earth GJ 1214b 9-11 71
Probing the Lunar Core with Seismology 9-11 72
An Organism Based on Arsenic not Phosphorus 9-11 73
Planet Kepler 10b - A matter of gravity 9-11 74
Discovering Earth-like Worlds by their Color 9-11 75

Space Math http://spacemath.gsfc.nasa.gov
Table of Contents (contd)
iv

Grade Page
The 2011 Japan Earthquake Rocks Earth 9-11 76
Exploring Nuclear Decay and Radiation Dose 9-11 77
Radiation Dose and Distance 9-11 78
Radiation Dose and Dose Rate 9-11 79
Gravity Probe-B - Testing Einstein Agani! 9-11 80
The Lens-Thirring Effect Near Massive Bodies 9-11 81
Taking a Log-Log Look at the Universe 9-11 82
Exoplanet Orbits and the Properties of Ellipses 9-11 83
Estimating the Temperatures of Exoplanets 9-11 84
Exponential Functions and the Atmosphere (calculus) 12 85
Estimating the Mass of Comet Hartley-2 (calculus) 12 86

Useful Links 87
Author Comments 88

Note to Teachers:

The problems in this math guide use metric units throughout, but
occasional problems (15, 27, 47, 55) use English measurements such as acres,
miles or pounds to be consistent with American engineering practice. Students
will be responsible as adults working in the STEM environment of the aerospace
industry for being intimately familiar with metric conversions to English units.
Since this book is a collection of real-world math problems it is appropriate that
a mixture of math problems in metric and English units appear.

Although we do use metric units in all scientific discussions, we do not
use SI units (meters-kilograms-seconds) exclusively because this particular
metric standard common in K12 instruction is not always convenient for
scientific research. For example, we do not state the diameter of Earth in
meters but use kilometers. We do not determine the scale of a photographic
image by using meters, but use millimeters, etc. Students need to learn that in
the real world of day-to-day science, a variety of metric units are used in each
discipline depending on the particular system being studied. Astronomers for
example do not state the masses of stars in kilograms, but use the 'solar mass'
as 1-unit of stellar mass. This is clearly not an SI unit, but it greatly serves to
improve comprehension, which is always the point of scientific research and
communication. It is not clear to the Author how stating stellar masses in the SI
units of kilograms rather than solar units would improve student comprehension
of astronomical scales.
Space Math http://spacemath.gsfc.nasa.gov
Mathematics Topic Matrix
v
Topic Problem Numbers
1 2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
Inquiry X
Technology, X X X X X X X X X
rulers
Numbers, X X X X X X X X X X X
patterns,
percentages
Averages X
Time, distance, X X
speed
Mass, density, X X
volume
Areas and X X X X X
volumes
Scale X X X X X X X X X
drawings
Geometry X X X X X
Scientific X X X X
Notation
Unit
Conversions
Significant X
Figures
Fractions
Graph or Table X X X X X X
Analysis
Solving for X X X
Evaluating Fns X X X X X X X X
Modeling X
Probability X X
Rates/Slopes X X
Linear X X
Equations
Logarithmic
Fns
Polynomials
Power Fns
Exponential
Fns
Conics
Piecewise Fns
Trigonometry
Integration
Differentiation
Limits
Space Math http://spacemath.gsfc.nasa.gov
Mathematics Topic Matrix (cont'd)
vi
Topic Problem Numbers
3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 6 6 6
2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2
Inquiry
Technology, x x x x x x x
rulers
Numbers, x x x x
patterns,
percentages
Averages x x x
Time, x x x x x x x
distance,speed
Mass, density, x x
volume
Areas and
volumes
Scale x x x x x x x x x x x x
drawings
Geometry x x x x x
Scientific x x x x x x x
Notation
Unit x x x x
Conversions
Significant
Figures
Fractions
Graph or Table x x x x x x x x x x x
Analysis
Solving for X
Evaluating Fns x x x x x x
Modeling
Probability x x x x
Rates/Slopes x x x
Linear x
Equations
Logarithmic x
Fns
Polynomials x
Power Fns x
Exponential x
Fns
Conics
Piecewise Fns
Trigonometry x
Integration
Differentiation
Limits
Space Math http://spacemath.gsfc.nasa.gov
Mathematics Topic Matrix (cont'd)
vii
Topic Problem Numbers
6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8
3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6
Inquiry
Technology,
rulers
Numbers, x x
patterns,
percentages
Averages
Time, x x
distance,speed
Mass, density, x x x x x
volume
Areas and x x x x
volumes
Scale x x x
drawings
Geometry x x x x x x x x
Scientific x x x x x x x x x x x x x
Notation
Unit x x
Conversions
Significant x x
Figures
Fractions
Graph or Table x x x x x x
Analysis
Solving for X x
Evaluating Fns x x x x x x x x x x x x
Modeling x x x
Probability x x
Rates/Slopes x x x
Linear
Equations
Logarithmic x x
Fns
Polynomials x
Power Fns
Exponential x x
Fns
Conics x x
Piecewise Fns
Trigonometry x x
Integration x x
Differentiation
Limits
Space Math http://spacemath.gsfc.nasa.gov
How to use this book
viii
Teachers continue to look for ways to make math meaningful by providing students
with problems and examples demonstrating its applications in everyday life. Space Math
offers math applications through one of the strongest motivators-Space. Technology
makes it possible for students to experience the value of math, instead of just reading
about it. Technology is essential to mathematics and science for such purposes as “access
to outer space and other remote locations, sample collection and treatment, measurement,
data collection and storage, computation, and communication of information.” 3A/M2
authentic assessment tools and examples. The NCTM standards include the statement
that "Similarity also can be related to such real-world contexts as photographs, models,
projections of pictures" which can be an excellent application for all of the Space Math
applications.

This book is designed to be used as a supplement for teaching mathematical
topics. The problems can be used to enhance understanding of the mathematical concept,
or as a good assessment of student mastery.

An integrated classroom technique provides a challenge in math and science
classrooms, through a more intricate method for using Space Math VII. Read the scenario
that follows:

Ms. Green decided to pose a new activity using Space Math for her students. She
challenged each student team with math problems from the Space Math VII book. She
copied each problem for student teams to work on. She decided to have the students
develop a factious space craft. Each team was to develop a set of criteria that included
reasons for the research, timeline and budget. The student teams had to present their
findings and compete for the necessary funding for their space craft. The students were to
use the facts available in the Space Math VII book and images taken from the Space
Weather Media Viewer, http://sunearth.gsfc.nasa.gov/spaceweather/FlexApp/bin-
debug/index.html#

Space Math VII can be used as a classroom challenge activity, assessment tool,
enrichment activity or in a more dynamic method as is explained in the above scenario. It
is completely up to the teacher, their preference and allotted time. What it does provide,
regardless of how it is used in the classroom, is the need to be proficient in math. It is
needed especially in our world of advancing technology and physical science.

Space Math http://spacemath.gsfc.nasa.gov
Alignment with Standards
ix

AAAS: Project:2061 Benchmarks

(3-5) - Quantities and shapes can be used to describe objects and events in the world around
us. 2C/E1 --- Mathematics is the study of quantity and shape and is useful for describing
events and solving practical problems. 2A/E1 (6-8) Mathematicians often represent things with
abstract ideas, such as numbers or perfectly straight lines, and then work with those ideas
alone. The "things" from which they abstract can be ideas themselves; for example, a
proposition about "all equal-sided triangles" or "all odd numbers". 2C/M1 (9-12) -
Mathematical modeling aids in technological design by simulating how a proposed system
might behave. 2B/H1 ---- Mathematics provides a precise language to describe objects and
events and the relationships among them. In addition, mathematics provides tools for solving
problems, analyzing data, and making logical arguments. 2B/H3 ----- Much of the work of
mathematicians involves a modeling cycle, consisting of three steps: (1) using abstractions to
represent things or ideas, (2) manipulating the abstractions according to some logical rules,
and (3) checking how well the results match the original things or ideas. The actual thinking
need not follow this order. 2C/H2

NCTM: Principles and Standards for School Mathematics
Grades 6–8 :
• work flexibly with fractions, decimals, and percents to solve problems;
• understand and use ratios and proportions to represent quantitative relationships;
• develop an understanding of large numbers and recognize and appropriately use
exponential, scientific, and calculator notation; .
• understand the meaning and effects of arithmetic operations with fractions, decimals,
and integers;
• develop, analyze, and explain methods for solving problems involving proportions, such
as scaling and finding equivalent ratios.
• represent, analyze, and generalize a variety of patterns with tables, graphs, words, and,
when possible, symbolic rules;
• model and solve contextualized problems using various representations, such as
graphs, tables, and equations.
• use graphs to analyze the nature of changes in quantities in linear relationships.
• understand both metric and customary systems of measurement;
• understand relationships among units and convert from one unit to another within the
same system.
Grades 9–12 :
• judge the reasonableness of numerical computations and their results.
• generalize patterns using explicitly defined and recursively defined functions;
• analyze functions of one variable by investigating rates of change, intercepts, zeros,
asymptotes, and local and global behavior;
• understand and compare the properties of classes of functions, including exponential,
polynomial, rational, logarithmic, and periodic functions;
• draw reasonable conclusions about a situation being modeled.

Space Math http://spacemath.gsfc.nasa.gov