# National Aeronautics and Space Administration

## Stars

Problem 669: Exploring Two Nearby Stars to the Sun. Students explore two nearby stars Ross 128 and Gliese 445 and determine when they will be the nearest stars to our sun by working with quaddratic equations that model their distances. [Grade: 9-12 | Topics: Working with quadratic equations; intersection points of quadratic functions] (PDF)

Problem 665: Kepler - Kepler's Latest Count on Goldilocks Planets Students examine the statistics of the latest candidate planets beyond our solar system, work with poercentages and a bar graph, and estimate the number of earth-like planets in our Milky Way. [Grade: 6-8 | Topics: percentages, bar graphs, estimation] (PDF)

Problem 664: HST - The Sun's Nearest Companions?At least for now! Students study a graph that models the distances from the sun of seven nearby stars over a 100,000 year time span. They determine the minimum distances and a timeline of which star will be the suns new closest neighbor in space in the next 80,000 years. [Grade: 6-8 | Topics: Graphical data; finding minimum from a plotted curve] (PDF)

Problem 608: Constellations in 3D
Students create a 3-d model of the constellation Orion and explore how stars are located in space and how this perspective changes from different vantage points. [Grade: 6-8 | Topics: geometry; scale] (PDF)

Problem 564:Exploring the Stars in Orion - Light Year Madness
Students explore the light year and its relationship to light travel time for observing events in different parts of space.When would colonists at different locations observe the star Betelgeuse become a supernova? [Grade: 6-8 | Topics: time lines; time intervalcalculations; time = distance/speed ] (PDF)

Problem 561:Exploring the Evaporating Exoplanet HD189733b
Students estimate how quickly this planet will lose its atmosphere and evaporate at its present loss rate of 6 million tons/second [Grade: 6-8 | Topics:mass=densityx volume; rates; volume of a sphere ] (PDF)

Problem 517: A Distant Supernova Remnant Discovered
Students work with proportions and scaling to discover the size of the supernova remnant compared to the distance from the Sun to the nearest star Alpha Centauri. The also work with time and speed calculations to estimate the speed of the supernova compared to the International Space Station. [Grade: 6-8 | Topics: proportions; speed=distance/time] (PDF)

Problem 494: The Close Encounter to the Sun of Barnards Star
Students use parametric equations and calculus to determine the linear equation for the path of Barnards Star, and then determine when the minimum distance to the sun occurs [Grade: 12 | Topics: Derivitives and minimization] (PDF)

Problem 490: LL Pegasi - A Perfect Spiral in Space
The star LL Persei is ejecting gas like a sprinkler on a lawn. Every 800 years the gas makes one complete orbit, and over time forms a spiral patteri in space. Students explore the timing of this pattern and estimate the size and age of this gas. [Grade: 6-8 | Topics: Distance = speed x time; unit conversions; evaluating formulas ] (PDF)

Problem 483: The Radioactive Dating of a Star in the Milky Way!
Students explore Cayrel's Star, whose age has been dated to 12 billion years using a radioisotope dating technique involving the decay of uranium-238. [Grade: 9-12 | Topics: half-life; exponential functions; scientific notation] (PDF)

Problem 482: Exploring Density, Mass and Volume Across the Universe
Students calculate the density of various astronomical objects and convert them into hydrogen atoms per cubic meter in order to compare how astronomical objects differ enormously in their densities. [Grade: 9-12 | Topics: Density=mass/volume; scientific notation; unit conversion; metric math ] (PDF)

Problem 480: The Expanding Gas Shell of U Camelopardalis
Students explore the expanding U Camelopardalis gas shell imaged by the Hubble Space Telescope, to determine its age and the density of its gas. [Grade: 6-8 | Topics: Scientific Notation; distance = speed x time; density=mass/volume ] (PDF)

Problem 439: Chandra Sees a Distant Planet Evaporating
The planet CoRot2b is losing mass at a rate of 5 million tons per second. Students estimate how long it will take for the planet to lose its atmosphere [Grade: 6-8 | Topics: Scientific Notation; RAte = Amount/Time] (PDF)

Problem 416: Kepler probes the interior of red giant stars Students use the properties of circular arcs to explore sound waves inside stars. [Grade: 8-10 | Topics: geometry of circles and arcs; distance=speed x time] (PDF)

Problem 398: The Crab Nebula - Exploring a pulsar up close! Students work with a photograph to determine its scale and the time taken by light and matter to reach a specified distance from the pulsar. [Grade: 6-8 | Topics: Scale drawings; unit conversion; distance = speed x time] (PDF)

Problem 329: WISE and Hubble: Power Functions: A question of magnitude Students explore the function F(x) = 10^-ax and learn about the stellar magnitude scale used by astronomers to rank the brightness of stars. [Grade: 10-12 | Topics: base-10, evaluating power functions ]

Problem 328: WISE: F(x)G(x): A Tale of Two Functions Students use WISE satellite data to study a practical application of the product of two finctions by graphing them individually, and their product. A calculus-level problem is included for advanced students. [Grade: 10-12 | Topics: Power-law functions; domain and range; graphing; areas under curves; integration]

Problem 327: WISE: Exploring Power-law Functions Using WISE Data Based on a recent press release of the 'First Light' image taken with NASA's new WISE satellite, students explore a practical application of a power law function to count the number of stars in the sky. An additional calculus-level problem is included for advanced students. [Grade: 10-12 | Topics: areas; functions; histograms; unit conversion; power-laws; integration]

Problem 322: Rotation Velocity of a Galaxy Students examine a simple model of the rotation of a galaxy to investigate how fast stars orbit the centers of galaxies in systems such as the Milky Way and Messier-101. [Grade: 10-12 | Topics: Algebra, limiting form of functions; derivitives]

Problem 320: Star Light...Star Bright A simple polynomial function is used to determine the temperature of a star from its brightness at two different visible wavelengths. [Grade: 10-12 | Topics: Algebra II; Polynomials; maxima and minima]

Problem 319: How Many Stars Are In the Sky? A simple polynomial is used to determine how many stars are in the sky. [Grade: 10-12 | Topics: Log Functions; Polynomials]

Problem 318: The Internal Density and Mass of the Sun Students use a simple, spherically symmetric, density profile to determine the mass of the sun using integral calculus. [Grade: 11-12 | Topics: Algebra II; Polynomials; integral calculus]

Problem 314: Chandra Studies an Expanding Supernova Shell Using a millimeter ruler and a sequence of images of a gaseous shell between 2000 and 2005, students calculate the speed of the material ejected by Supernova 1987A. [Grade: 6-9 | Topics: Measuring; Metric Units; speed=distance/time]

Problem 295: Details from an Exploding Star Students work with an image from the Hubble Space Telescope of the Crab Nebula to calculate scales and sizes of various features. [Grade: 6-9 | Topics: Scale; measurement; metric units]

Problem 294: Star Cluster math A simple counting exercise involving star classes lets students work with percentages and ratios. [Grade: 4-6 | Topics: Counting; percentage; scaling]

Problem 284: Calculating the Thickness of a Neutron Star Atmosphere Students determine the thickness of the carbon atmosphere of the neutron star Cas-A using Earth's atmosphere and a set of scaling relationships. [Grade: 9-12 | Topics: Algebra I; Exponential functions; graphing; Scientific notation]

Problem 283: Chandra Observatory Sees the Atmosphere of a Neutron Star Students determine the mass of the carbon atmosphere of the neutron star Cas-A. [Grade: 8-10 | Topics: Volume of spherical shell; mass = density x volume]

Problem 278: Spitzer Studies the Distant Planet Osiris Students learn about the density of the planet HD209458b, also called Osiris, and compare it to that of Jupiter. [Grade: 8-10 | Topics: Spherical volumes; density; Scientific Notation;]

Problem 240: The Eagle Nebula Close-up Students measure a Hubble image of the famous Eagle Nebula 'Pillars of Creation' to determine the sizes of arious features compared to our solar system [Grade: 8-10 | Topics: scale, proportion, angle measure]

Problem 234: The Hand of Chandra Students use an image from the Chandra Observatory to measure a pulsar ejecting a cloud of gas. [Grade: 6-8 | Topics: Scientific Notation; proportions; angle measure]

Problem 232: Star Circles- Students use a photograph of star trails around the North Star Polaris to determine the duration of the timed exposure based on star arc lengths. [Grade: 8-9 | Topics: Lengths of arcs of circles; angular measure.]

Problem 231: Star Magnitudes and Decimals- Students work with the stellar magnitude scale to determine the brightness differences between stars. [Grade: 5-8 | Topics: Multiplying decimals.]

Problem 224: Perimeters; Which constellation is the longest?- Students use tabulated data for the angular distances between stars in the Big Dipper and Orion to determine which constellation has the longest perimeter, and the average star separations. [Grade: 3-5 | Topics: perimeter of a curve; basic fractions; mixed numbers.]

Problem 221: Pulsars and Simple Equations- Students work with linear equations describing the rotation period of a pulsar, and evaluate the equations for various conditions. Students use the equations to predict intersection points in time. [Grade: 6-8 | Topics: Evaluating simple one-variable equations]

Problem 213: Kepler: The hunt for Earth-like planets- Students compare the area of a star with the area of a planet to determine how the star's light is dimmed when the planet passes across the star as viewed from Earth. This is the basis for the 'transit' method used by NASA's Kepler satellite to detect new planets. [Grade: 6-8 | Topics: Area of circle; ratios; percents.]

Problem 212: Finding Mass in the Cosmos- Students derive a simple formula, then use it to determine the masses of objects in the universe from the orbit periods and distances of their satellites. [Grade: 9-12| Topics: Scientific Notation; Algebra II; parametric equations]

Problem 211: Where Did All the Stars Go?- Students learn why NASA photos often don't show stars because of the way that cameras take pictures of bright and faint objects. [Grade: 6-8| Topics: multiplication; division; decimal numbers.]

Problem 209: How to make faint things stand out in a bright world!- Students learn that adding images together often enhances faint things not seen in only one image; the power of averaging data. [Grade: 6-8| Topics: multiplication; division; decimal numbers.]

Problem 197: Hubble Sees a Distant Planet- Students study an image of the dust disk around the star Fomalhaut and determine the orbit period and distance of a newly-discoveblack planet orbiting this young star. [Grade: 6-10| Topics: Calculating image scales; Circle circumferences; Unit conversions; distance-speed-time]

Problem 191: Why are hot things red? - Students explore the Planck Function using graphing skills, and calculus for experts, to determine the relationship between temperature and peak wavelength. [Grade: 10-12| Topics: Algebra, graphing, differential calculus]

Problem 190: Modeling a Planetary Nebula - Students use calculus to create a mathematical model of a planetary nebula [Grade: 10-12| Topics: Algebra, Integral calculus]

Problem 189: Stellar Temperature, Size and Power- Students work with a basic equation to explore the relationship between temperature, surface area and power for a selection of stars. [Grade: 8-10| Topics: Algebra]

Problem 186: Collapsing Gas Clouds and Stability- Students use the derivative to find an extremum of an equation governing the pressure balance of an interstellar cloud. [Grade: 11-12| Topics: differentiation; finding extrema; partial derivitives]

Problem 182: Our Neighborhood in the Milky Way- Students create a scale model of the local Milky Way and estimate distances and travel times for a series of voyages. [Grade: 6-8| Topics: scale models; speed-distance-time]

Problem 172: The Stellar Magnitude Scale- Students learn about positive and negative numbers using a popular brightness scale used by astronomers. [Grade: 3-6| Topics: number relationships; decimals; negative and positive numbers]

Problem 170: Measuring Star Temperatures- Students use a simple formula to determine the temperatures of stars, and to use a template curve to analyze data for a specific star to estimate its temperature. [Grade: 6-8 | Topics: algebra, graph analysis]

Problem 160: The Relative Sizes of the Sun and Stars- Students work through a series of comparisons of the relative sizes of the sun compablack to other stars, to create a scale model of stellar sizes using simple fractional relationships. ( e.g if Star A is 6 times larger than Star B, and Star C is 1/2 the size of Star B, how big is Star C in terms of Star A?) [Grade: 4-6 | Topics: working with fractions; scale models]

Problem 158: The Solar Neighborhod within 17 Light Years - Students create a scale model of the local solar neighborhood and determine the shortest travel distances to several stars. [Grade: 6-8 | Topics: Plotting polar coordinates using a ruler and compass; decimal math]

Problem 156: Spectral Classification of Stars- Students use actual star spectra to classify them into specific spectral types according to a standard ruberic. [Grade: 5-8 | Topics: Working with patterns in data; simple sorting logic

Problem 148 Exploring a Dying Star Students use data from the Spitzer satellite to calculate the mass of a planetary nebula from a dying star. [Grade: 9 - 11 | Topics:Scientific Notation; unit conversions; volume of a sphere ]

Problem 147 Black hole - fade out Students calculate how long it takes light to fade away as an object falls into a black hole. [Grade: 9 - 11 | Topics: Scientific Notation; exponential functions]

Problem 146 Black Hole Power Students calculate how much power is produced as matter falls into a rotating and a non-rotating black hole including solar and supermassive black holes. [Grade: 9 - 11 | Topics:Scientific Notation; Spherical shells; density; power]

Problem 145 Black Holes - What's Inside? Students work with the Pythagorean Theorem for black holes and investigate what happens to space and time on the other side of an Event Horizon. [Grade:9 - 11 | Topics: Scientific Notation; distance; time calculations; algebra]

Problem 144 Exploring Angular Size Students examine the concept of angular size and how it relates to the physical size of an object and its distance. A Chandra Satellite x-ray image of the star cluster NGC-6266 is used, along with its distance, to determine how far apart the stars are based on their angular separations. [Grade: 7 - 10 | Topics:Scientific Notation; degree measurement; physical size=distance x angular size.]

Problem 142 Black Holes---Part VIII Matter that falls into a black hole heats up in an accretion disk, which can emit x-rays and even gamma rays visible from Earth. In this problem, students use a simple algebraic formula to calculate the temperature at various places in an accretion disk. [Grade: 7 - 10 | Topics:Scientific Notation; Working with equations in one variable to first and second power.]

Problem 141 Exploring a Dusty Young Star Students use Spitzer satellite data to learn about how dust emits infrared light and calculate the mass of dust grains from a young star in the nebula NGC-7129. [Grade: 4 - 7 | Topics: Algebra I; multiplication, division; scientific notation]

Problem 140 Black Holes---Part VII If you fell into a black hole, how fast would you be traveling? Students use a simple equation to calculate the free-fall speed as they pass through the event horizon. [Grade: 7 - 10 | Topics:Scientific Notation; Working with equations in one variable to first and second power.]

Problem 138 Black Holes---Part VI Tidal forces are an important gravity phenomenon, but they can be lethal to humans in the vicinity of black holes. This exercise lets students calculate the tidal acceleration between your head and feet while standing on the surface of Earth...and falling into a black hole. [Grade: 7 - 10 | Topics:Scientific Notation; Working with equations in one variable to first and second power.]

Problem 137 Black Holes---Part V Students explore how Kepler's Third Law can be used to determine the mass of a black hole, or the mass of the North Star: Polaris. [Grade: 7 - 10 | Topics:Scientific Notation; Working with equations in one variable to first and second power.]

Problem 136 Black Holes---Part IV Students explore how much energy is generated by stars and gas falling into black holes. The event horizon radius is calculated from a simple equation, R = 2.95 M, and energy is estimated from E = mc^2. [Grade: 7 - 10 | Topics:Scientific Notation; Working with equations in one variable to first and second power.]

Problem 132 Black Holes - III Students learn about how gravity distorts time near a black hole and other massive bodies. [Grade: 8 - 12 | Topics:Simple linear equations; scientific notation]

Problem 131 How Big is It? - Las Vegas up close. Students work with an image taken by the QuickBird imaging satellite of downtown Las Vegas, Nevada. They determine the image scale, and calculate the sizes of streets, cars and buildings from the image. [Grade: 4 - 7 | Topics:image scaling; multiply, divide, work with millimeter ruler]

Problem 130 Black Holes - II Students learn about how gravity distorts time and causes problems even for the Global Positioning System satellites and their timing signals. [Grade: 8 - 12 | Topics:Simple linear equations; scientific notation]

Problem 128 Black Holes - I Students learn about the most basic component to a black hole - the event horizon. Using a simple formula, and scientific notation, they examine the sizes of various kinds of black holes. [Grade: 8 - 12 | Topics:Simple linear equations; scientific notation]

Problem 115 A Mathematical Model of the Sun Students will use the formula for a sphere and a shell to calculate the mass of the sun for various choices of its density. The goal is to reproduce the measured mass and radius of the sun by a careful selection of its density in a core region and a shell region. Students will manipulate the values for density and shell size to achieve the correct total mass. This can be done by hand, or by programming an Excel spreadsheet. [Grade: 8-10 | Topics: scientific notation; volume of a sphere and a spherical shell; density, mass and volume.]

Problem 62 Star light...Star bright - A question of magnitude! - Since the time of the ancient Greek astronomer Hipparchus, astronomers have measured and cataloged the brightness of stars according to the 'apparent magnitude scale'. This activity lets students experience this peculiar numbering system where bright stars have small numbers (even negative: our sun is a -26 magnitude!) and faint stars have large numbers (faintest stars are +29 magnitudes). Students will calculate the brightness differences between stars using multiplication and division. Working with the number line will be a big help and math review! [Grade level: 4-6 | Topics: Positive and negative numbers; decimal math]

Problem 61 Drake's Equation and the Search for Life...sort of! - Way back in the 1960's Astronomer Frank Drake invented an equation that helps us estimate how much life, especially the intelligent kind, might exist in our Milky Way. It has been a lively topic of discussion in thousands of college astronomy courses for the last 30 years. In this simplified version, your students will get to review what we now know about the planetary universe, and come up with their own estimates. The real fun is in doing the research to track down plausible values (or their ranges) for the factors that enter into the equation, and then write a defense for the values that they choose. Lots of opportunity to summarize basic astronomical knowledge towards the end of an astronomy course, or chapter. [Grade level: 6-8 | Topics: decimal math; evaluating functions for given values of variables]

Problem 58 How many stars are there? - For thousands of years, astronomers have counted the stars to determine just how vast the heavens are. Since the 19th century, 'star gauging' has been an important tool for astronomers to assess how the various populations of stars are distributed within the Milky Way. In fact, this was such an important aspect of astronomy between 1800-1920 that many cartoons often show a frazzled astronomer looking through a telescope, with a long ledger at his knee - literally counting the stars through the eyepiece! In this activity, students will get their first taste of star counting by using a star atlas reproduction and bar-graph the numbers of stars in each magnitude interval. They will then calculate the number of similar stars in the sky by scaling up their counts to the full sky area. [Grade level: 6-8 | Topics: Positive and negative numbers; histogramming; extrapolating data]

Problem 53 Astronomy: A Moving Experience! - Objects in space move. To figure out how fast they move, astronomers use many different techniques depending on what they are investigating. In this activity, you will measure the speed of astronomical phenomena using the scaling clues and the time intervals between photographs of three phenomena: A supernova explosion, a coronal mass ejection, and a solar flare shock wave. [Grade level: 6-8 | Topics: Finding the scale of an image; metric measurement; distance = speed x time; scientific notation]

Problem 52 Measuring the size of a Star Cluster - Astronomers often use a photograph to determine the size of astronomical objects. The Pleiades is a famous cluster of hundreds of bright stars. In this activity, students will determine the photographic scale, and use this to estimate the projected (2-D) distances between the stars in this cluster. They will also use internet and library resources to learn more about this cluster. [Grade level: 4-6 | Topics: Online research; Finding the scale of an image; metric measurement; decimal math]

Problem 47 Discovering the Milky Way by Counting Stars. - It is common to say that there are about 8,000 stars visible to the naked eye in both hemispheres of the sky, although from a typical urban setting, fewer than 500 stars are actually visible. Students will use data from a deep-integration image of a region of the sky in Hercules, observed by the 2MASS sky survey project to estimate the number of stars in the sky. This number is a lower-limit to the roughly 250 to 500 billion stars that may actually exist in the Milky Way. [Grade level: 4-6 | Topics: Tallying data; decimal math]

Problem 44 Interstellar Distances with the Pythagorean Theorem - If you select any two stars in the sky and calculate how far apart they are, you may discover that even stars that appear to be far apart are actually close neighbors in space. This activity lets students use the Pythagorean distance formula in 3-dimensions to explore stellar distances for a collection of bright stars, first as seen from Earth and then as seen from a planet orbiting the star Polaris. Requires a calculator and some familiarity with algebra and square-roots. [Grade level: 9-11 | Topics: Decimal math; Pythagorean Theorem; square root]

## Universe

Problem 663: HST - The Hubble Search for the Farthest Galaxy in the Universe Students learn about the recent discovery of z8_GND_5296 what may be the farthest known galaxy in our visible universe whose light left the galaxy when the universe was only 700 million years old. They use a simple linear equation to estimate the galaxys look-back time, and learn about the cosmological redshift. [Grade: 6-8 | Topics: working with simple equations; solving for X] (PDF)

Problem 594: A Number Puzzle about the Origin of Our Universe
Students learn about the Big Bang by solving a number puzzle for missing words using solutions to a variety of problems taken from Algebra 1 topics. [Grade: 6-8 | Topics: distance between two points; slopes; linear equations; dilations; scientific notation] (PDF)

Problem 553:Colliding Galaxies - The future of our Milky Way
Students explore the collision of two galaxies and estimate from their present speed, separation and acceleration how long it will be before they have collided. [Grade: 9-12 | Topics: unit conversions; scientific notation; ballistic equation; solvimg quadratic equations] (PDF)

Problem 513: The Remarkable Gamma Ray Burst GRB 130427A
Students work with the surface area of a sphere, metric conversions and scientific notation to calculate the total power of this distant supernova event. [Grade: 8-10 | Topics: surface area of sphere; scientific notation] (PDF)

Problem 511: Giant Gas Cloud in System NGC 6240
Students use scientific notation and volume of sphere to estimate the density of the gas cloud, and the number of hydrogen atoms per cubic meter. [Grade: 8-10 | Topics:Volume of a sphere; scientific notation; unit conversion ] (PDF)

Problem 510: Planck Mission Sees the Ancient Universe Clearly
Students work with an image of the universe when it was 370,000 years old and determine from simple scaling and proportions the sizes of the features seen in the image compared to the Milky Way. [Grade: 6-8 | Topics: scale and proportion; angular measure] (PDF)

Problem 501: Exploring the Most Distant Galaxies with Hubble
Students use recent Hubble Extreme Deep Field data and a polynomial to determine the light travel time between distant galaxies and Earth. [Grade: 11-12 | Topics: polynomials; linearization] (PDF)

Problem 487: The Hubble eXtreme Deep Field
Students use the Hubble XDF to estimate the number of galaxies in the visible universe. [Grade: 6-8 | Topics: Counting, areas, proportions ] (PDF)

Problem 462: Using a Gravity Lens to Weigh a Cluster of Galaxies
Students explore how the geometry of a gravity lens can be used to measure the mass of the object producing the gravity. [Grade: 9-12 | Topics: algebra; Scientific Notation] (PDF)

Problem 460: Fermi Explores the High-Energy Universe
Students work with percentages to explore the identities of the 1873 gamma-ray sources detected by NASAs Fermi Observatory [Grade: 6-8 | Topics: percentages; pie graphs] (PDF)

Problem 418: Supercomputers: Modeling colliding neutron stars! Students use a series of time-lapse images calculated using a supercomputer to determine the speed of collision of two neutron stars, and whether they will form a black hole afterwards. [Grade: 8-10 | Topics: distance=speed x time; scale model; triangle and circle geometry; circumference] (PDF)

Problem 417: Estimating the Size and Mass of a Black Hole Students use a simple formula to estimate the size of a black hole located 3.8 billion light years from Earth, recently studied by NASA's Chandra and Swift satellites. [Grade: 8-10 | Topics: distance=speed x time] (PDF)

Problem 400: The Most Distant Objects in the Universe Students use a table of the most distant known events and objects in the universe to create a timeline of the universe soon after the Big Bang. [Grade: 6-8 | Topics: Working with millions and billions; elapsed time] (PDF)

Problem 399: A Galactic City in the Far Reaches of the Universe Students work with an image of a distant cluster of galaxies to determine its scale compared to nearby galaxies. [Grade: 6-8 | Topics: Scale; proportion; metric measurement; unit conversion] (PDF)

Problem 388: Hubble Detects More Dark Matter Students learn about how astronomers estimate the amount of invisible dark matter in a cluster of galaxies by comparing its visible mass against the speeds of the galaxies to 'weigh' the cluster' [Grade: 8-10 | Topics: evaluating functions; Scientific notation]

Problem 330: Fermi Detects Gamma-rays from the Galaxy Messier-82 Based on a recent press release, students work with a log-log plot to show that straight lines on this plot represent power-law functions. They use this fact to determine, by interpolation, the strength of the gamma-rays from this galaxy. [Grade: 10-12 | Topics: power-laws; log-log graphing; linear regression]

Problem 323: How Many Quasars are There? Students use a piecewise function that estimates how many quasars are found in a given area of the sky. The function is integrated to determine the estimated total number of quasars across the entire sky. [Grade: 11-12 | Topics: Piecewise functions; integral calculus]

Problem 322: Rotation Velocity of a Galaxy Students examine a simple model of the rotation of a galaxy to investigate how fast stars orbit the centers of galaxies in systems such as the Milky Way and Messier-101. [Grade: 10-12 | Topics: Algebra, limiting form of functions; derivitives]

Problem 313: Exploring the Big Bang with the LHC Two simple equations allow students to compute the temperature and energy of matter soon after the Big Bang, and compare these with energies available at the LHC. [Grade: 9-12| Topics: ALgebra; Scientific Notation; Unit conversions]

Problem 312: Exploring the Large Hadron Collider The Large Hadron Collider collides protons at very high energy to create new forms of matter. Students explore unit conversions related to energy and mass. [Grade: 9-12 | Topics: Scientific Notation]

Problem 311: The Volume of a Hypersphere This problem extends student understanding of volume to include higher-dimensional spheres and their unusual properties. A simple recursion relation is used to calculate the volume formulas for spheres in dimensions 4 through 10. [Grade: 9-12 | Topics: Algebra II; Geometry; recursion relations]

Problem 310: Energy and Mass - Same things but different! Students use unit conversions to explore the relationship between mass and energy. [Grade: 8-10 | Topics: Unit COnversions; Scientific Notation]

Problem 309: The Energy of Empty Space Students explore the energy of 'empty space' and its relationship to the mass of the Higgs Boson using a simple quartic polynomial. [Grade: 10-12 | Topics: Properties of functions; polynomials; Critical points]

Problem 308: The Higgs Boson and the Mystery of Mass The search for the Higgs Boson is underway at the Large Hadron Collider (LHC). In this problem, students explore how the mass of this particle is believed to depend on the energies used to form it by studying a simple quartic polynomial. [Grade: 10-12 | Topics: Properties of functions; polynomials; Critical points]

Problem 291: Calculating Black Hole Power Students use a simple formula to calculate how much power is produced by black holes of various sizes as they absorb matter from nearby stars and gas clouds. [Grade: 9-12 | Topics: Scientific Notation; evaluating simple formulas; unit conversion]

Problem 289: Chandra Spies the Longest Sound Wave in the Universe Students use an image of sound waves produced by a massive black hole to determine wavelength, and comparisons with musical scale to find how many octaves this sound wave is below the wavelength of middle-C. [Grade: 6-8 | Topics: metric measurement; scaling; Scientific Notation; exponents]

Problem 288: Fermi Observatory Measures the Lumps in Space Students use timing data obtained by the Fermi Observatory of a powerful gamma-ray burst 10 billion light years away, to determine how lumpy space is based on travel time delays between the lowest and highest-energy gamma-rays. [Grade: 9-12 | Topics: Scientific Notation; Evaluating an equation with multiple factors]

Problem 285: Chandra Sees the Most Distant Cluster in the Universe Students work with kinetic energy and escape velocity to determine the mass of a distant cluster of galaxies by using information about its x-ray light emissions. [Grade: 9-12 | Topics: Algebra I; Solving for X; Scientific notation]

Problem 239: Counting Galaxies with the Hubble Space Telescope Students use an image of a small area of the sky to estimate the total number of galaxies in the universe visible from Earth. [Grade: 8-10 | Topics: area, angular measure]

Problem 233: The Milky Way: A mere cloud in the cosmos- Students compare the average density of the Milky Way with the density of the universe. [Grade: 8-10 | Topics: Volume of disk, density, scientific notation]

Problem 230: Galaxy Distances and Mixed Fractions- Students use the relative distances to nearby galaxies expressed in mixed numbers to determine distances between selected galaxies. [Grade: 3-5 | Topics: Basic fraction math.]

Problem 219: Variables and Expressions from Around the Cosmos- Students evaluate linear equations describing a variety of astronomical situations. [Grade: 6-8 | Topics: Evaluating simple one-variable equations.]

Problem 213: Kepler: The hunt for Earth-like planets- Students compare the area of a star with the area of a planet to determine how the star's light is dimmed when the planet passes across the star as viewed from Earth. This is the basis for the 'transit' method used by NASA's Kepler satellite to detect new planets. [Grade: 6-8 | Topics: Area of circle; ratios; percents.]

Problem 212: Finding Mass in the Cosmos- Students derive a simple formula, then use it to determine the masses of objects in the universe from the orbit periods and distances of their satellites. [Grade: 9-12| Topics: Scientific Notation; Algebra II; parametric equations]

Problem 196: Angular Size and velocity- Students study a spectacular photo of the ISS passing across the face of the sun, and work out the angular sizes and speeds of the transit to figure out how long the event took in order to photograph it. [Grade: 8-10| Topics: Geometry; Angle measurement]

Problem 192: The Big Bang - Cosmic Expansion - Students explore the expansion of the universe predicted by Big Bang cosmology [Grade: 10-12| Topics: Algebra, Integral Calculus]

Problem 186: Collapsing Gas Clouds and Stability- Students use the derivative to find an extremum of an equation governing the pressure balance of an interstellar cloud. [Grade: 11-12| Topics: differentiation; finding extrema; partial derivitives]

Problem 170: Measuring Star Temperatures- Students use a simple formula to determine the temperatures of stars, and to use a template curve to analyze data for a specific star to estimate its temperature. [Grade: 6-8 | Topics: algebra, graph analysis]

Problem 160: The Relative Sizes of the Sun and Stars- Students work through a series of comparisons of the relative sizes of the sun compablack to other stars, to create a scale model of stellar sizes using simple fractional relationships. ( e.g if Star A is 6 times larger than Star B, and Star C is 1/2 the size of Star B, how big is Star C in terms of Star A?) [Grade: 4-6 | Topics: working with fractions; scale models]

Problem 159: Galaxies to Scale - Students explore the relative sizes of the Milky Way compablack to other galaxies to create a scale model of galaxies, similar to the methods in Problem 161. [Grade: 4-6 | Topics: working with fractions; scale models]

Problem 152: The Hubble Law - Students plot the speed and distance to 7 galaxies and by deriving the slop of the linear model for the data points, obtain an estimate for the expansion rate of the universe known as Hubble's Constant. [Grade: 6-8 | Topics: Plotting data; determining the slope of the data;]

Problem 150: Cosmic Bar Graphs - Students interpret simple bar graphs taken from astronomical data. [Grade: 3-5 | Topics: finding maxima and minima; fractions; extrapolating data]

Problem 146 Black Hole Power Students calculate how much power is produced as matter falls into a rotating and a non-rotating black hole including solar and supermassive black holes. [Grade: 9 - 11 | Topics:Scientific Notation; Spherical shells; density; power]

Problem 144 Exploring Angular Size Students examine the concept of angular size and how it relates to the physical size of an object and its distance. A Chandra Satellite x-ray image of the star cluster NGC-6266 is used, along with its distance, to determine how far apart the stars are based on their angular separations. [Grade: 7 - 10 | Topics:Scientific Notation; degree measurement; physical size=distance x angular size.]

Problem 136 Black Holes---Part IV Students explore how much energy is generated by stars and gas falling into black holes. The event horizon radius is calculated from a simple equation, R = 2.95 M, and energy is estimated from E = mc^2. [Grade: 7 - 10 | Topics:Scientific Notation; Working with equations in one variable to first and second power.]

Problem 128 Black Holes - I Students learn about the most basic component to a black hole - the event horizon. Using a simple formula, and scientific notation, they examine the sizes of various kinds of black holes. [Grade: 8 - 12 | Topics:Simple linear equations; scientific notation]

Problem 123 A Trillion Here...A Trillion There Students learn to work with large numbers, which are the heart and soul of astronomical dimensions of size and scale. This activity explores the number 'one trillion' using examples drawn from the economics of the United States and the World. Surprisingly, there are not many astronomical numbers commonly in use that are as big as a trillion. [Grade: 5-9 | Topics:add, subtract, multiply, divide.]

Problem 111 Scientific Notation III In this continuation of the review of Scientific Notation, students will perform simple multiplication and division problems with an astronomy and space science focus. [Grade: 5-9 | Topics:Scientific notation - multiplication and division]

Problem 110 Scientific Notation II In this continuation of the review of Scientific Notation, students will perform simple addition and subtraction problems. [Grade: 5-9 | Topics:Scientific notation - addition and subtraction]

Problem 109 Scientific Notation I Scientists use scientific notation to represent very big and very small numbers. In this exercise, students will convert some 'astronomical' numbers into SN form. [Grade: 5-9 | Topics:Scientific notation - conversion from decimal to SN]

Problem 58 How many stars are there? - For thousands of years, astronomers have counted the stars to determine just how vast the heavens are. Since the 19th century, 'star gauging' has been an important tool for astronomers to assess how the various populations of stars are distributed within the Milky Way. In fact, this was such an important aspect of astronomy between 1800-1920 that many cartoons often show a frazzled astronomer looking through a telescope, with a long ledger at his knee - literally counting the stars through the eyepiece! In this activity, students will get their first taste of star counting by using a star atlas reproduction and bar-graph the numbers of stars in each magnitude interval. They will then calculate the number of similar stars in the sky by scaling up their counts to the full sky area. [Grade level: 6-8 | Topics: Positive and negative numbers; histogramming; extrapolating data]

Problem 56 The Sombrero Galaxy Close-up - The Sombrero Galaxy in Virgo is a dazzling galaxy through the telescope, and has been observed in detail by both the Hubble Space Telescope and the Spitzer Infrared Observatory. This exercise lets students explore the dimensions of this galaxy as well as its finest details, using simple image scaling calculations. [Grade level: 9-11 | Topics: Finding the scale of an image; measurement; decimal math]

Problem 54 Exploring Distant Galaxies - Astronomers determine the redshifts of distant galaxies by using spectra and measuring the wavelength shifts for familiar atomic lines. The larger the redshift, denoted by the letter Z, the more distant the galaxy. In this activity, students will use an actual image of a distant corner of the universe, with the redshifts of galaxies identified. After histogramming the redshift distribution, they will use an on-line cosmology calculator to determine the 'look-back' times for the galaxies and find the one that is the most ancient galaxy in the field. Can students find a galaxy formed only 500 million years after the Big Bang? [Grade level: 6-8 | Topics: Decimal math; using an online calculator; Histogramming data]

Problem 50 Measuring the Speed of a Galaxy. - Astronomers can measure the speed of a galaxy by using the Doppler Shift. By studying the spectrum of the light from a distant galaxy, the shift in the wavelength of certain spectral lines from elements such as hydrogen, can be decoded to give the speed of the galaxy either towards the Milky Way or away from it. In this activity, students will use the formula for the Doppler Shift to analyze the spectrum of the Seyfert galaxy Q2125-431 and determine its speed. [Grade level: 6-8 | Topics: Interpolating data in a graph; decimal math]

Problem 49 A Spiral Galaxy Up Close. - Astronomers can learn a lot from studying photographs of galaxies. In this activity, students will compute the image scale (light years per millimeter) in a photograph of a nearby spiral galaxy, and explore the sizes of the features found in the image. They will also use the internet or other resources to fill-in the missing background information about this galaxy. [Grade level: 6-8 | Topics: Online research; Finding the scale of an image; metric measurement; decimal math]

## The Search For Extraterrestrial Life

Problem 395: Death Stars Some stars create super-flares that are capable of eliminating life on planets that orbit close to the star. Students learn about these flares on common red-dwarf stars and compare them to flares on our own sun [Grade: 6-9 | Topics: Scientific Notation; percentages; rates of change] (PDF)

Problem 392: Exploring the DNA of an organism based upon arsenic. Students estimate the increase in the mass of the DNA from an arsenic-loving bacterium in which phosphorus atoms have been replaced with arsenic. [Grade: 8-10 | Topics: integer math; percentages] (PDF)

Problem 61: Drake's Equation and the Search for Life...sort of! Way back in the 1960's Astronomer Frank Drake invented an equation that helps us estimate how much life, especially the intelligent kind, might exist in our Milky Way. It has been a lively topic of discussion in thousands of college astronomy courses for the last 30 years. In this simplified version, your students will get to review what we now know about the planetary universe, and come up with their own estimates. The real fun is in doing the research to track down plausible values (or their ranges) for the factors that enter into the equation, and then write a defense for the values that they choose. Lots of opportunity to summarize basic astronomical knowledge towards the end of an astronomy course, or chapter. [Grade level: 6-8 | Topics: decimal math; evaluating functions for given values of variables]

## The Search for Earth-like Planets

Problem 492: Alpha Centauri Bb - a nearby extrasolar planet?
Students plot data for the orbiting planet and determine its orbit period. They use this in a simple formula to determine its distance, then they estimate its surface temperature at this distance. [Grade: 9-12 | Topics: graphing periodic data; finding periods; evaluating simple formulae ] (PDF)

Problem 465: Comparing Planets Orbiting other Stars
Students use simple fraction arithmetic to determine the relative sizes of several new planets recently discovered by the Kepler mission, and compare these sizes to that of Jupiter and Earth. [Grade: 3-5 | Topics: scale models; proportions; fractions] (PDF)

Problem 458: Playing Baseball on the Earth-like Planet Kepler-22b!
The recently-confirmed Earth-like planet Kepler-22b by the Kepler Observatory is a massive planet orbiting its star in the temperature zone suitable for liquid water. This problem explores the gravity and mass of this planet, and some implications for playing baseball on its surface! [Grade: 8-10 | Topics: scale models; proportions; scientific notation; metric math; Evaluating equations] (PDF)

Problem 441: Exploring the new planet Kepler 16b called 'Tatooine'
Using the tangent function, students estimate the angular diameter and separation of the two stars in the Kepler 16 binary system as viewed from the planet's surface...if it had one!! [Grade: 8-10 | Topics: angle measure; tangent] (PDF)

Problem 405: Discovering Earth-like Worlds by their Color Students use recent measurements of the reflected light from solar system bodies to graph their colors and to use this in classifying new planets as Earth-like, moon-like or Jupiter-liike [Grade: 6-8 | Topics: graphing tabular data; interpreting graphical data] (PDF)

Problem 402: Kepler- Earth-like planets by the score! II Students use recent Kepler satellite data summarized in tabular form to estimate the number of planets in the Milky Way galaxy that are about the same size as our Earth, and located in their Habitable Zones were liquid water may exist. [Grade: 6-8 | Topics: Percentage; re-scaling sample sizes] (PDF)

Problem 401: Kepler - Earth-like planets by the score! I Students use recent Kepler satellite data to estimate the number of Earth-like planets in the Milky Way galaxy. [Grade: 6-8 | Topics: Percentage; histograms; Re-scaling sample sizes] (PDF)

Problem 396: Kepler 10b - A matter of gravity Students use the measured properties of the Earth-like planet Kepler 10b to estimate the weight of a human on its surface. [Grade: 8-10 | Topics: Evaluating formulas; mass = density x volume; volume of a sphere; scientific notation] (PDF) Problem 376: The Earth-like Planet Gliese 518g
Students use data for the Gliese 581 planetary system to draw a scaled model of the locations and sizes of the discovered planets. They also identify the location and span of the Habitable Zone for this planetary system. [Grade: 3-5 | Topics: scale models; measurement] (PDF)

Problem 360: Kepler's First Look at 700 Transiting Planets
A statistical study of the 700 transits seen during the first 43 days of the mission. [Grade: 6-8 | Topics: Percentages; area of circle] (PDF)

Problem 333: Hubble: Seeing a Dwarf Planet Clearly Based on a recent press release, students use the published photos to determine the sizes of the smallest discernible features and compare them to the sizes of the 48-states in the USA. They also estimate the density of Pluto and compare this to densities of familiar substances to create a 'model' of Pluto's composition. A supplementary Inquiry Problem asks students to model the interior in terms of two components and estimate what fraction of Pluto is composed of rock or ice. [Grade: 8-12 | Topics: scales and ratios; volume of sphere; density=mass/volume]

Problem 331: Webb Space Telescope: Detecting dwarf planets The 'JWST' will be launched some time in 2014. One of its research goals will be to detect new dwarf planets beyond the orbit of Pluto. In this problem, students use three functions to predict how far from the sun a body such as Pluto could be detected, by calculating its temperature and the amount of infrared light it emits. [Grade: 9-12 | Topics: Evaluating square-roots and base-e exponentials]

Problem 325: Kepler Spies Five New Planets Students count squares on a Bizarro Star to study the transit of a planet, and determine the diameter of the planet. This demonstrates the basic principle used by NASA's Kepler satellite to search for Earth-sized planets orbiting distant stars. [Grade: 4-6 | Topics: Counting; graphing; area of a square]

Problem 213: Kepler: The hunt for Earth-like planets Students compare the area of a star with the area of a planet to determine how the star's light is dimmed when the planet passes across the star as viewed from Earth. This is the basis for the 'transit' method used by NASA's Kepler satellite to detect new planets. [Grade: 6-8 | Topics: Area of circle; ratios; percents.]

Problem 197: Hubble Sees a Distant Planet Students study an image of the dust disk around the star Fomalhaut and determine the orbit period and distance of a newly-discoveblack planet orbiting this young star. [Grade: 6-10| Topics: Calculating image scales; Circle circumferences; Unit conversions; distance-speed-time]

Problem 168: Fitting Periodic Functions - Distant Planets Students work with data from a newly-discovered extra-solar planet to determine its orbit period and other parameters of a mathematical model. [Grade: 9-12 | Topics: trigonometry; functions; algebra]

Problem 160: The Relative Sizes of the Sun and Stars Students work through a series of comparisons of the relative sizes of the sun compablack to other stars, to create a scale model of stellar sizes using simple fractional relationships. ( e.g if Star A is 6 times larger than Star B, and Star C is 1/2 the size of Star B, how big is Star C in terms of Star A?) [Grade: 4-6 | Topics: working with fractions; scale models]

Problem 156: Spectral Classification of Stars Students use actual star spectra to classify them into specific spectral types according to a standard ruberic. [Grade: 5-8 | Topics: Working with patterns in data; simple sorting logic

Problem 155: Tidal Forces: Let 'er rip! Students explore tidal forces and how satellites are destroyed by coming too close to their planet. [Grade: 7-10| Topics: Algebra; number substitution]

Problem 141: Exploring a Dusty Young Star Students use Spitzer satellite data to learn about how dust emits infrared light and calculate the mass of dust grains from a young star in the nebula NGC-7129. [Grade: 4 - 7 | Topics: Algebra I; multiplication, division; scientific notation]

## Black Holes

Problem 613: Measuring the Speed of Gas Near a Black Hole
Students use a graph of intensity and time to estimate thhe orbit period of matter around a black hole. [Grade: 6-8 | Topics: time; graph analysis] (PDF)

Problem 475: Exploring Tidal Forces: Black holes and Saturns rings
Students use the equation for tidal disruption to explore the stability of a star encountering a black hole, and a satellite of Saturn. Why are there no large satellites of Saturn inside the ring system? [Grade: 9-12 | Topics: Evaluating equations; scientific notation] (PDF)

Problem 427: A Black Hole - Up Close
Students explore how the color of a light bulb changes as it gets close to a black hole, demonstrating the principle of the gravitational 'red shift'. [Grade: 9-12 | Topics: Evaluating an equation with one variable; square roots; metric units; nanometers] (PDF)

Problem 426: Black Holes - Hot Stuff!
Students explore the temperature of matter falling into a black hole using a simple equation to calculate the gas temperature at different distances. [Grade: 9-12 | Topics: Evaluating an equation with one variable; fractional exponents] (PDF)

Problem 425: Exploring a Full-sized Black Hole
Students explore how the speed of an orbiting satellite changes if it were near a black hole with 5 times the mass of our Earth. [Grade: 6-8 | Topics: Evaluating an equation with one variable; square roots; speed = distance/time; circumference of a circle] (PDF)

Problem 424: Exploring Black Holes
Students compare the sizes of the planets in our solar system if they were actually black holes. They use a compass and metric ruler to create circles that are the actual sizes of the 'black hole' planets. [Grade: 3-5 | Topics: working with a compass and metric ruler] (PDF)

Problem 423: The Moon as a Black Hole
Students draw a life-sized model of the Earth and Moon as two black holes to explore the actual sizes of these exotic astronomical bodies. [Grade: 3-5 | Topics: Working with a compass; metric ruler] (PDF)

Problem 421: The Lense-Thirring Effect Near the Sun and a Neutron Star Students work with a formula for the Lense-Thirring Effect and estimate how large it will be in orbit around our sun, and in the intense gravitational field of a dense neutron star. [Grade: 9-12 | Topics: algebra; scientific notation, angular measure] (PDF)

Problem 420: Gravity Probe B: Testing Einstein again! Students learn about the Lense-Thirring Effect, and calculate its magnitude near Earth's orbit using an algebraic equation with integer and fractional exponents. [Grade: 9-12 | Topics: algebra; scientific notation, angular measure] (PDF)

Problem 418: Supercomputers: Modeling colliding neutron stars! Students use a series of time-lapse images calculated using a supercomputer to determine the speed of collision of two neutron stars, and whether they will form a black hole afterwards. [Grade: 8-10 | Topics: distance=speed x time; scale model; triangle and circle geometry; circumference] (PDF)

Problem 417: Estimating the Size and Mass of a Black Hole Students use a simple formula to estimate the size of a black hole located 3.8 billion light years from Earth, recently studied by NASA's Chandra and Swift satellites. [Grade: 8-10 | Topics: distance=speed x time] (PDF)

Problem 389: Estimating the diameter of the SN1979c black hole Students use simple equations to learn about the various definitions for the sizes of black holes in terms of their event horizons, last photon orbit, and last stable particle orbit radii, and apply this to the recently discovered 'baby' black hole in the galaxy M-100 [Grade: 6-8 | Topics: evaluating linear functions; integer math; metric units] (PDF)

Problem 128 : Event Horizons Students learn about the most basic component to a black hole - the event horizon. Using a simple formula, and scientific notation, they examine the sizes of various kinds of black holes. [Grade: 8 - 12 | Topics:Simple linear equations; scientific notation]

Problem 145: What's Inside a Black Hole? Students work with the Pythagorean Theorem for black holes and investigate what happens to space and time on the other side of an Event Horizon. [Grade:9 - 11 | Topics: Scientific Notation; distance; time calculations; algebra]

Problem 147: Light Fade-out Students calculate how long it takes light to fade away as an object falls into a black hole. [Grade: 9 - 11 | Topics: Scientific Notation; exponential functions]

Problem 140: Falling Into a Black Hole If you fell into a black hole, how fast would you be traveling? Students use a simple equation to calculate the free-fall speed as they pass through the event horizon. [Grade: 7 - 10 | Topics:Scientific Notation; Working with equations in one variable to first and second power.]

Problem 138: The Intense Gravity of a Black Hole Tidal forces are an important gravity phenomenon, but they can be lethal to humans in the vicinity of black holes. This exercise lets students calculate the tidal acceleration between your head and feet while standing on the surface of Earth...and falling into a black hole. [Grade: 7 - 10 | Topics:Scientific Notation; Working with equations in one variable to first and second power.]

,b>Problem 132: Black Holes and Time Distortion Students learn about how gravity distorts time near a black hole and other massive bodies. [Grade: 8 - 12 | Topics:Simple linear equations; scientific notation]

Problem 130: Gravity and Time Distortion Near Earth Students learn about how gravity distorts time and causes problems even for the Global Positioning System satellites and their timing signals. [Grade: 8 - 12 | Topics:Simple linear equations; scientific notation]

Problem 390: X-rays from hot gases near the black hole SN1979c Students use two functions to estimate the size of a black hole from the gas emitting x-rays which is flowing into it. [Grade: 8-10 | Topics: Functions; substitution; evaluation] (PDF)

Problem 146: Black Hole Power - I Students calculate how much power is produced as matter falls into a rotating and a non-rotating black hole including solar and supermassive black holes. [Grade: 9 - 11 | Topics:Scientific Notation; Spherical shells; density; power]

Problem 291: Black Hole Power - II Students use a simple formula to calculate how much power is produced by black holes of various sizes as they absorb matter from nearby stars and gas clouds. [Grade: 9-12 | Topics: Scientific Notation; evaluating simple formulas; unit conversion]

Problem 136: Black Hole Power - III Students explore how much energy is generated by stars and gas falling into black holes. The event horizon radius is calculated from a simple equation, R = 2.95 M, and energy is estimated from E = mc^2. [Grade: 7 - 10 | Topics:Scientific Notation; Working with equations in one variable to first and second power.]

Problem 142: Accretion Disks Matter that falls into a black hole heats up in an accretion disk, which can emit x-rays and even gamma rays visible from Earth. In this problem, students use a simple algebraic formula to calculate the temperature at various places in an accretion disk. [Grade: 7 - 10 | Topics:Scientific Notation; Working with equations in one variable to first and second power.]

Problem 137: Black Hole Mass Students explore how Kepler's Third Law can be used to determine the mass of a black hole, or the mass of the North Star: Polaris. [Grade: 7 - 10 | Topics:Scientific Notation; Working with equations in one variable to first and second power.]

## ESS1B: The Earth and the Solar System

Problem 353: Dwarf Planets and Kepler's Third Law
Students plot the distance versus period relationship for planets and minor bodies in the solar system and fit it to two functions to determine Kepler's Third Law. [Grade: 9-12 | Topics: Fitting functions to data; Evaluating a polynomial] (PDF)

Problem 84: Beyond the Blue Horizon How far is it to the horizon? Students use geometry, and the Pythagorean Theorem, to determine the formula for the distance to the horizon on any planet with a radius, R, from a height, h, above its surface. Additional problems added that involve calculus to determine the rate-of-change of the horizon distance as you change your height. [Grade level: 9-11 | Topics: Algebra, Pythagorean Theorem; Experts: DIfferential calculus) ]

Problem 612: Exploring Power-laws: Meteor impacts
Students work with logarithmic functions, power-laws and explore the mass functiuon of meteors. [Grade: 9-12 | Topics: logarithmic functions; power-functions; logarithmic graphs] (PDF)

Problem 558:How Quickly are NEOs Being Discovered?
Students work with data presented in bar graphs to estimate how many more hazardous Near Earth Objects (NEOS) remein to be found. [Grade: 6-8 | Topics:bar graphs ] (PDF)

Problem 557:The 10000th Near Earth Asteroid 2013 MZ5
Students graph tabulated data to determine when this asteroid is closest to Earth and its speed at that time. [Grade: 6-8 | Topics: grapohing tabuklar data; solving a linear equation] (PDF)

Problem 556:IRIS Explores the Solar Transition Region
Students use an image from IRIS to examine the sizes and equivalent energy of bright regions in the solar transition region. [Grade: 6-8 | Topics: percentage; proportion; scale; scientific notation; volume of a cylinder] (PDF)

Problem 546: The Relative Sizes of Planets and other Objects
Students use proportional information to determine the relative scales of planets and large moons across the solar system. [Grade: 3-5 | Topics:scale; proportion] (PDF)

Problem 545:Measuring Atmospheric Trace Gases Using Parts Per Million
Students convert from percentage units to parts per million and compare trace gases in the atmospheres of various planets. [Grade: 6-8 | Topics: percentages; unit conversions ] (PDF)

Problem 544:The Composition of Planetary Atmospheres
Students study the composition of planetary atmospheres and compare the amounts of certain compounds in them [Grade: 6-8 | Topics: Pie graphs; percentages; scientific notation] (PDF)

Problem 543:Timeline for Planet Formation
Students calculate time intervals in millions and billions of years from a timeline of events [Grade: 3-5 | Topics: time calculations; integers] (PDF)

Problem 542:The Late Heavy Bombardment Era
Students estimate the average arrival time of large asteroids that impacted the moon. They work with the formula for the volume of a sphere to estimate how much additional mass was added to the moon and Earth during this era. [Grade: 6-8 | Topics: volume of spheres; proportions] (PDF)

Problem 541:How to Build a Planet
Students study planet growth by using a clay model of planetessimals combining to form a planet by investigating volume addition with spheres. [Grade: 3-5 | Topics: graphing; counting] (PDF)

## Moon

Problem 607: The Launch of LADEE to the Moon
Students plot the altitude, range and speed of the LADEE rocket launch and investigate rates of change including acceleration by graphing the tabular data and determining the slope of the graph using the definition of the slope of a line between two points. [Grade: 6-8 | Topics: Graphing tabular data; determining the slope of a line; rates of change] (PDF)

Problem 509:Gail Spacecraft Creates a New Crater on the Moon
Students work with images of the Grail impact sites to estimate the diameter of the crater created after the spacecraft impacted the moon. [Grade: 6-8 | Topics: scale and proportion; volume of cylinder; mass=DensityxVolume] (PDF) Problem 504: Grail Satellites Create a Gravity Map of the Moon
Students explore the gravity field of the moon, and the behavior of simple pendulum clocks in places on the moon where the local gravity is slightly different. [Grade: 9-12 | Topics: square-roots; evaluating equations] (PDF)

Problem 495: The Volume of a Lunar Impact Crater
Students use calculus to determine the volume of a crater whose depth is defined by a fourth-order polynomial [Grade: 12 | Topics: Integration involving vollumes of rotation] (PDF)

Problem 478: The Grail and LRO Encounter in Lunar Orbit
Students explore the May 31, 2012 encounter between NASA's Grail and LRO spacecraft in orbit around the moon. Will the Grail/Ebb spacecraft be able to photograph the LRO spacecraft as it passes-by? [Grade: 9-12 | Topics: formula for an ellipse; semi-major and minor axis] (PDF)

Problem 445: LRO - The relative ages of lunar surfaces
Students examine two Apollo landing areas using images from the LRO spacecraft to estimate the relative ages of the two regions using crater counting. [Grade: 6-8 | Topics: scale; histogramming] (PDF)

Problem 440: LRO explores the Apollo 12 landing area on the moon
Students use a recent image obtained by the LRO spacecraft to estimate how far astronauts walked to get to various points in the landing area. They also estimate how many craters are in this area and the average impact time between crater events. [Grade: 6-8 | Topics: image scale; metric measurement] (PDF)

Problem 435: Apollo-17 Launch from Lunar Surface
Students use a sequence of images to determine the speed of ascent of the Apollo-17 capsule from the lunar surface. [Grade: 6-8 | Topics: scale, metric measurement, speed=distance/time] (PDF)

Problem 394: Probing the lunar core using seismology Students learn about the geometry needed to determine the diameter of the lunar core using a simplified model. [Grade: 9-10 | Topics: Geometry; Properties of Inscribed Arcs] (PDF)

Problem 378: LRO Makes a Temperature Map of the Lunar South Pole
Students use the published LRO temperature map to study the scale of the south polar region, the sizes of its craters, and estimate the volume of water-ice that may be present in the Shackleton Crater. [Grade: 7-9 | Topics: Volume of a circular disk; scale models] (PDF)

Problem 372: LRO Determines Lunar Cratering History
Students count large craters on an LRO coded image of the lunar surface to estimate whether the impacting asteroids that produced the largest craters were from the same population of asteroids during the two different epocs of impacts. [Grade: 6-8 | Topics: Scaled images; probability; percentages] (PDF)

Problem 321: Lunar Crater Frequency Distributions Students use an image from the LRO satellite of the Apollo-11 landing area, along with a power-law model of cratering, to determine what fraction of the landin garea was safe to land upon. [Grade: 11-12 | Topics: Integral calculus]

Problem 296: Getting an Angle on the Sun and Moon Students explore angular size and scale by comparing two images of the sun and moon which have identical angular size, but vastly different scales. [Grade: 8-10 | Topics: Geometry; angle measure; scale; proportion]

Problem 290: The Apollo-11 Landing Area at High Resolution Students use recent images made by the LRO satellite to estimate distances, crater sizes, and how many tons of TNT were needed to create some of the craters by meteor impact. [Grade: 9-12 | Topics: metric measurement; scaling; A = B/C]

Problem 287: LCROSS Sees Water on the Moon Students use information about the plume created by the LCROSS impactor to estimate the (lower-limit) concentration of water in the lunar regolith in a shadowed crater. [Grade: 9-12 | Topics: Geometry; volumes; mass=density x volume]

Problem 275: Water on the Moon! Students estimate the amount of water on the moon using data from Deep Impact/EPOXI and NASA's Moon Minerology Mapper experiment on the Chandrayaan-1 spacecraft. [Grade: 8-10 | Topics: Geometry, Spherical volumes and surface areas, Scientific notation]

Problem 267: Identifying Materials by their Reflectivity The reflectivity of a material can be used to identify it. This is important when surveying the lunar surface for minerals, and also in creating 'green' living environments on Earth. [Grade: 6-8 | Topics: percentage, interpreting tabular data, area ]

Problem 262: LRO Explores Lunar Surface Cratering Students count the number of craters in various size ranges from a high-resolution image of the lunar surface. [Grade: 6-8 | Topics: scale, proportion, ratio, area, density]

Problem 261: LRO - Searching for Lunar Boulders Students use a recent image of the Apollo-11 landing area to search for large lunar boulders. [Grade: 6-8 | Topics: scale, ratio, proportion]

Problem 259: Mare Nubium And Las Vegas Students comare two satellite images taken at the same resolution to appreciate how large lunar features ae compared to more familiar objects. [Grade: 8-10 | Topics: scale, proportion, ratio]

Problem 258: LRO's First Image of Mare Nubium Students examine the first image of this lunar region using the high-resolution camera image provided by the Lunar Reconnaissance Orbiter. [Grade: 6-8 | Topics: scale, ratio, proportion]

Problem 257: LRO and the Apollo-11 Landing Site Students examine a map of the Apollo-11 landing area and determine how well various features will be visible to the Lunar Reconnaissance Orbiter high-resolution camera. [Grade: 6-8 | Topics: scale, proportion, ratios]

Problem 256: A High-resolution Satellite Photo Students examine a satelite photo of the Tennessee Court House from the GEO-1 satellite and determine the sizes of familiar features in the image. [Grade: 6-8 | Topics: scale, ratios, proportions' angle measure, triangle geometry]

Problem 250: The Most Important Equation in Astronomy Students learn about how an instrument's ability to see details depends on its size and its operating wavelength - the key to designing any telescope or camera. [Grade: 8-10 | Topics: geometry, angle measure, scientific notation]

Problem 241: Angular Size and Similar Triangles A critical concept in astronomy is angular size, measured in degrees, minutes or arc-seconds. This is a review of the basic properties of similar triangles for a fixed angle. [Grade: 8-10 | Topics: geometry, similar triangles, proportions]

Problem 236: LRO Sees Apollo-11 on the Moon! Students use the latest image from the Lunar Reconnaissance Orbiter of the Apollo-11 landing site to explore lunar features at 1-meter resolution, and determine the solar elevation angle. [Grade: 6-8 | Topics: scale; ratios; angle measure; right triangles]

Problem 235: Scientific Data: The gift that keeps on giving! Students learn about gigabytes and terrabytes of data and the rates of data generation by NASA missions and how to store it. [Grade: 6-8 | Topics: metric units; rates; money]

Problem 223: Volcanos are a Blast: Working with simple equations- Students examine the famous Krakatoa explosion, asteroid impacts on the moon, and geysers on Enceladus using three equations that describe the height of the plume and initial velocity, to answer questions about the speed of the debris and terminal height. [Grade: 9-11 | Topics: Algebra I; significant figures.]

Problem 218: Craters are a Blast! - Students measure crater diameters in a photo of the moon, and determine the energy required to create them using a simple quadratic equation. [Grade: 8-10 | Topics: Scientific notation; evaluating simple power equations.]

Problem 212: Finding Mass in the Cosmos- Students derive a simple formula, then use it to determine the masses of objects in the universe from the orbit periods and distances of their satellites. [Grade: 9-12| Topics: Scientific Notation; Algebra II; parametric equations]

Problem 211: Where Did All the Stars Go?- Students learn why NASA photos often don't show stars because of the way that cameras take pictures of bright and faint objects. [Grade: 6-8| Topics: multiplication; division; decimal numbers.]

Problem 201: Fly Me To the Moon!- Students learn some basic principles and terminology about how spacecraft change their orbits to get to the moon. [Grade: 6-8| Topics: speed = distance/time; Pythagorean Theorem]

Problem 200: The Moon's Density - What's Inside?- Students develop a simple mathematical model of the moon's interior using two nested spheres with different densities. [Grade: 9-12| Topics: Volume of a sphere; mass = density x volume.]

Problem 181: Extracting Oxygen from Moon Rocks- Students use a chemical equation to estimate how much oxygen can be liberated from a sample of lunar soil. [Grade: 9-11| Topics: ratios; scientific notation; unit conversions]

Problem 179: Is There a Lunar Meteorite Impact Hazard? - Students work with areas, probability and impact rates to estimate whether lunar colonists are in danger of meteorite hazards. [Grade: 5-7| Topics: Area; unit conversions; rates]

Problem 178: The Mass of the Moon - Students use the period and altitude of a NASA lunar spacecraft to determine the mass of the moon. [Grade: 8-11| Topics: Algebra]

Problem 177: Lunar Cratering: Probability and Odds- Students work with crater counting to estimate the area covered by craters and how to convert this into impact probabilities. [Grade: 4-7| Topics: Area; probability]

Problem 155: Tidal Forces: Let 'er rip! - Students explore tidal forces and how satelites are destroyed by coming too close to their planet. [Grade: 7-10| Topics: Algebra; number substitution]

Problem 134 The Last Total Solar Eclipse--Ever! Students explore the geometry required for a total solar eclipse, and estimate how many years into the future the last total solar eclipse will occur as the moon slowly recedes from Earth by 3 centimeters/year. [Grade: 7 - 10 | Topics:Simple linear equations]

Problem 129 How Big is It? - The Moon up close. Students work with an image taken by the Lunar Orbiter III spacecraft to determine image scale, and search for the smallest things seen in a photograph. [Grade: 4 - 7 | Topics:image scaling; multiply, divide, work with millimeter ruler]

Problem 127 How Big is It? - The Moon up close. Students work with an image taken by the Lunar Orbiter IV spacecraft to determine image scale, and search for the smallest things seen in a photograph. [Grade: 4 - 7 | Topics:image scaling; multiply, divide, work with millimeter ruler]

Problem 124 The Moon's Atmosphere! Students learn about the moon's very thin atmosphere by calculating its total mass in kilograms using the volume of a spherical shell and the measured density. [Grade: 8-10 | Topics:volume of sphere, shell; density-mass-volume; unit conversions]

Problem 92 A Lunar Transit of the Sun from Space - One of the STEREO satellites observed the disk of the moon pass across the sun. Students will use simple geometry to determine how far the satellite was from the moon and Earth at the time the photograph was taken. [Grade level: 9-11 | Topics: Geometry; parallax; arithmetic]

Problem 84 Beyond the Blue Horizon - How far is it to the horizon? Students use geometry, and the Pythagorean Theorem, to determine the formula for the distance to the horizon on any planet with a radius, R, from a height, h, above its surface. Additional problems added that involve calculus to determine the rate-of-change of the horizon distance as you change your height. [Grade level: 9-11 | Topics: Algebra, Pythagorean Theorem; Experts: DIfferential calculus) ]

Problem 83 Luner Meteorite Impact Risks - In 2006, scientists identified 12 flashes of light on the moon that were probably meteorite impacts. They estimated that these meteorites were probably about the size of a grapefruit. How long would lunar colonists have to wait before seeing such a flash within their horizon? Students will use an area and probability calculation to discover the average waiting time. [Grade level: 8-10 | Topics: arithmetic; unit conversions; surface area of a sphere) ]

Problem 537:A Solar Storm Number Puzzle
Students solve 10 problems using positive and negative numbers, addition, subtraction and multiplication to find the missing words in a short essay about solar storms. [Grade:3-5 | Topics: integer arithmetic; positive and negative numbers] (PDF) Problem 516: Hinode Observes a Solar Eclipse from Space
Students work with simple proportions to estimate the diameter of the sun using the Moon and the Moon's distance. [Grade: 3-5 | Topics: Proportions; time intervals; calendar arithmetic] (PDF)

Problem 514: Solar Flares and the Stormy Sun
Students work with simple averaging and explore the latest sunspott cycle to find their averages for 2012 and 2013. [Grade: 3-5 | Topics: Averaging integers; rounding] (PDF)

Problem 505: SDO Sees Coronal Rain - Estimating Plasma Speeds
Students estimate the speed of plasma streamers near the solar surface using images from a Solar Dynamics Observatory. [Grade: 6-8 | Topics: scale models; speed=distance/time; proportions] (PDF)

Problem 468: How Common are X-Class Solar Flares?
Students use simple statistics to determine the solar flare frequency during the last 11-year sunspot cycle to estimate the time between X-class flares during the current sunspot cycle [Grade: 5-7 | Topics: mean, median, percentage] (PDF)

Problem 467: Estimating Magnetic Field Speeds on the Sun
Students use two images from the Solar Dynamics Observatory to estimate the speed of the X-class solar flare on March 6, 2012. [Grade: 6-8 | Topics: speed=distance/time; scale model; metric measurement] (PDF)

Problem 466: Exploring the Solar Wind and Coronal Mass Ejections
Students use two images of the solar storm during March 2012 to estimate the speed of the solar wind and a coronal mass ejection. They also estimate arrival times for the CME at Earth and Neptune. [Grade: 3-5 | Topics: scale models; proportions; fractions] (PDF)

Problem 455: The Night Launch of STEREO in 2006
An example of old news seen in a different way! Students use a spectacular time-lapse photo of the launch of the STEREO mission obtained by photographer Dominic Agostini in 2006 to study parabolic curves. [Grade: 8-10 | Topics: time=distance/speed; scale models; metric math; equation of a parabola; curve fitting] (PDF)

Problem 404: STEREO Spacecraft give 360-degree Solar View Students use STEREO satellite images to determine which features can be seen from Earth and which cannot. They learn about the locations and changing positions of the satellites with respect to Earth's orbit. [Grade: 6-8 | Topics: angular measure, extrapolation; distance = speed x time] (PDF)

Problem 395: Death Stars Some stars create super-flares that are capable of eliminating life on planets that orbit close to the star. Students learn about these flares on common red-dwarf stars and compare them to flares on our own sun [Grade: 6-9 | Topics: Scientific Notation; percentages; rates of change] (PDF)

Problem 373: Solar Probe Plus - Having a hot time near the sun!
Students use a simple equation to estimate the temperature reached by the Solar Probe Plus spacecraft as it gets close to the sun. [Grade: 8-10 | Topics: Evaluating a function; square roots and forth roots] (PDF)

Problem 366: Solar Probe Plus - Working with angular diameter
Students use the tangent formula to determine the angular diameter of the sun as seen by the Solar Probe Plus spacecraft as it approaches the sun. [Grade: 8-10 | Topics: angular measure; tangent formula; angular size] (PDF)

Problem 337: SDO Reveals Details on the Surface of the Sun Students use a spectacular colored image of the Sun to calculate the scale of the image in kilometers per millimeter, and then search for the smallest features relative to the size of Earth. [Grade: 6-8 | Topics: image scales; proportions]

Problem 336: SDO: Measuring the Speed of an Eruptive Prominence Students use recent First Light images of the Sun from SDO to calculate the speed of a prominence using a sequence of scaled images and computing position shift over the time interval of the images. [Grade: 6-8 | Topics: image scales; speed=distane/time ]

Problem 334: Solar Dynamics Observatory: Working with Giga, Tera, Peta and Exabytes The recent launch of SDO will bring 'high definition TV' to the study of the sun's surface details. This also means a HUGE amount of data will have to be processed every day to handle the torrent of information. This activity works with the prefixes giga, tera ,peta and exa to familiarize students with how to interpret these quantities in a practical settion. Students already know about 'gigabytes', but the SDO data stream represents terabytes per day, and petabytes per year in data storage demands. [Grade: 8-12 | Topics: powers of ten; time conversion: seconds, minutes, days, years]

Problem 318: The Internal Density and Mass of the Sun Students use a simple, spherically symmetric, density profile to determine the mass of the sun using integral calculus. [Grade: 11-12 | Topics: Algebra II; Polynomials; integral calculus]

Problem 299: Changing Perspectives on the Sun's Diameter Students compare two images of the sun taken by the SHOH satellite to measure the apparent diameter change from different earth obit locations in the winter and summer. [Grade: 6-8 | Topics: measurement; parallax; metric units; percentage change]

Problem 298: Seeing Solar Storms in STEREO - II Students explore the geometry of stereo viewing by studying a solar storm viewed from two satellites. [Grade: 10-12 | Topics: Geometry; Trigonometry]

Problem 296: Getting an Angle on the Sun and Moon Students explore angular size and scale by comparing two images of the sun and moon which have identical angular size, but vastly different scales. [Grade: 8-10 | Topics: Geometry; angle measure; scale; proportion]

Problem 286: STEREO Watches the Sun Kick Up a Storm Students use images from the STEREO observation of a 'solar tsunami' to estimate its speed and kinetic energy. [Grade: 9-12 | Topics: metric measurement; scaling; Scientific Notation; unit conversion; evaluating a simple 2-variable formula for kinetic energy ]

Problem 274: IBEX Uses Fast-moving Particles to Map the Sky! Students learn about Kinetic Energy and how particle energies and speeds are related to each other in a simple formula, which they use to derive the speed of the particles detected by the IBEX satellite. [Grade: 8-10 | Topics: Algebra I, Scientific notation]

Problem 273: IBEX Creates an Unusual Image of the Sky! Students create an image of the sky by using a Bingo-like technique of tallying particles in various sky directions using a simple 5x5 grid. [Grade: 6-8 | Topics: Counting, tallying]

Problem 263: Ice or Water? Whether a planetary surface contains ice or liquid water depends on how much heat is available. Students explore the concepts of Specific heat and Latent Heat of Fusion to better understand the and quantify the energy required for liquid water to exist under various conditions. [Grade: 8-10 | Topics: unit conversion, scientific notation]

Problem 254: Solar Insolation Changes and the Sunspot Cycle Students compare changes in the amount of solar energy reaching earth with the 11-year sunspot cycle to predict the impact on designing a photovoltaic system for a home. [Grade: 8-10 | Topics: graph analysis, correlations, kilowatt, kilowatt-hours]

Problem 248: Seeing Solar Storms in STEREO - I Students work out the details of stereoscopic vision using elementary properties of triangles and the Law of Cosines to determine the distance from earth of a solar storm cloud. [Grade: 8-10 | Topics: geometry, Law of Cosines, V = D/T]

Problem 244: Solar Storms - Fractions and Percentages Students create a Venn Diagram to summarize data on a series of solar storms, and determine how often solar flares occur when a solar plasma eruption happens. [Grade: 4-7 | Topics: precentages; Venn Diagramming]

Problem 212: Finding Mass in the Cosmos- Students derive a simple formula, then use it to determine the masses of objects in the universe from the orbit periods and distances of their satellites. [Grade: 9-12| Topics: Scientific Notation; Algebra II; parametric equations]

Problem 199: Solar Storm Energy and Pie Graphs- Students study two Pie graphs describing solar flares and draw conclusions about percentages and their various forms of energy. [Grade: 6-8| Topics: Interpreting Pie Graphs.]

Problem 198: Solar Storm Timeline- Students read a narrative about the events involved in a solar storm, create a chronology for the sequence of events, and answer some simple time-related questions. [Grade: 6-8| Topics: Time calculations.]

Problem 194: A Magnetic Case for 'What Came First?' - Students create a timeline for events based on several data plots from the THEMIS program, and use their timeline to answer questions about the causes of magnetic storms. [Grade: 6-8| Topics: Time calculations]

Problem 176: Solar Storms: Sequences and Probabilities I - Students continue their study of a stormy week on the sun by working out the probabilities for joint events. [Grade: 4-7| Topics: probability; numerating possible outcomes]

Problem 175: Solar Storms: Sequences and Probabilities II - Students work out the probabilities for various combinations of solar storms during a given week. [Grade: 4-7| Topics: probability; numerating possible outcomes]

Problem 163: Time Intervals - Students calculate time intervals between a number of astronomical events, from milliseconds to years. [Grade: 3-5 | Topics: Time calculations; unit conversions; decimal math]

Problem 160: The Relative Sizes of the Sun and Stars- Students work through a series of comparisons of the relative sizes of the sun compablack to other stars, to create a scale model of stellar sizes using simple fractional relationships. ( e.g if Star A is 6 times larger than Star B, and Star C is 1/2 the size of Star B, how big is Star C in terms of Star A?) [Grade: 4-6 | Topics: working with fractions; scale models]

Problem 134 The Last Total Solar Eclipse--Ever! Students explore the geometry required for a total solar eclipse, and estimate how many years into the future the last total solar eclipse will occur as the moon slowly recedes from Earth by 3 centimeters/year. [Grade: 7 - 10 | Topics:Simple linear equations]

Problem 118 An Application of the Parallax Effect The STEREO mission views the sun from two different locations in space. By combining this data, the parallax effect can be used to determine how far above the solar surface various active regions are located. Students use the Pythagorean Theorem, a bit of geometry, and some actual STEREO data to estimate the height of Active Region AR-978. [Grade: 8-10 | Topics:Pythagorean Theorem; square-root; solving for variables]

Problem 117 CME Kinetic Energy and Mass Coronal Mass Ejections (CMEs) are giant clouds of plasma released by the sun at millions of kilometers per hour. In this activity, students calculate the kinetic energy and mass of several CMEs to determine typical mass ranges and speeds. Students will use the formula for kinetic energy to fill-in the missing entries in a table. They will then use the completed table to answer some basic questions about CMEs. [Grade: 8-10 | Topics:time calculation; Evaluating a simple equation; solving for variables]

Problem 116 The Comet Encke Tail Disruption Event On April 20, 2007 NASA's STEREO satellite captured a rare impact between a comet and the fast-moving gas in a solar coronal mass ejection. In this problem, students analyze a STEREO satellite image to determine the speed of the tail disruption event. [Grade: 8-10 | Topics:time calculation; finding image scale; calculating speed from distance and time]

Problem 115 A Mathematical Model of the Sun Students will use the formula for a sphere and a shell to calculate the mass of the sun for various choices of its density. The goal is to reproduce the measured mass and radius of the sun by a careful selection of its density in a core region and a shell region. Students will manipulate the values for density and shell size to achieve the correct total mass. This can be done by hand, or by programming an Excel spreadsheet. [Grade: 8-10 | Topics: scientific notation; volume of a sphere and a spherical shell; density, mass and volume.]

Problem 114 The Heliopause...a question of balance Students will learn about the concept of pressure equilibrium by studying a simple mathematical model for the sun's heliopause located beyond the orbit of Pluto. They will calculate the distance to the heliopause by solving for 'R' and then using an Excel spreadsheet to examine how changes in solar wind density, speed and interstellar gas density relate to the values for R. [Grade: 8-10 | Topics: Formulas with two variables; scientific notation; spreadsheet programming]

Problem 112 How fast does the sun spin? Students will use two x-ray images of the sun taken by the Hinode satellite to determine how fast the sun rotates. [Grade: 5-9 | Topics:calculating map scales; time calculations; unit conversion]

Problem 111 Scientific Notation III In this continuation of the review of Scientific Notation, students will perform simple multiplication and division problems with an astronomy and space science focus. [Grade: 5-9 | Topics:Scientific notation - multiplication and division]

Problem 108 A Problem in Satellite Synchrony The THEMIS program uses five satellites in five different orbits to study Earth's magnetic field and its changes during a storm. This problem asks students to use the periods of the five satellites to figure out when all 5 satellites will be lined-up as seen from Earth. They will do this by finding the Greatest Common Multiple of the five orbit periods, first for the case of 2 or 3 satellites, which can be easily diagrammed with concentric circles, then the case for all five satellites together. [Grade: 5-9 | Topics:multiplication; Greatest Common Multiple]

Problem 107 Monster Sunspots! Some sunspots are so big that they can be seen from Earth without a telescope. In this problem, students will use images of three super-spots and calculate their sizes from the image scaling information. They will then order the images from the smallest super-spot to the largest super-spot. [Grade: 5-9 | Topics:multiplication; calculating length from image scale]

Problem 104 Loopy Sunspots! Students will analyze data from the Hinode satellite to determine the volume and mass of a magnetic loop above a sunspot. From the calculated volume, based on the formula for the volume of a cylinder, they will use the density of the plasma determined by the Hinode satellite to determine the mass in tons of the magnetically trapped material. [Grade: 9-11 | Topics:image scales; cylinder volume calculation; scientific notation; unit conversions]

Problem 103 The Mysterious Solar Micro-Flares! Students will analyze an image taken by the Hinode solar satellite to determine the scale of the image in kilometers per millimeter, then use this to determine the sizes of solar micro-flares. From the number of micro-flares that they count in the image, the area of the image in square kilometers, and the surface area of a spherical sun, they will calculate the total number of micro-flares on the solar surface. [Grade: 6-9 | Topics:image scales; area calculation; unit conversions]

Problem 101 Super-Fast Solar Flares!! - Students will analyze consecutive images taken of an erupting solar flare, and use the information provided to calculate the speed of the flare. [Grade level: 6-9 | Topics:image scales; time calculations; speed calculations]

Problem 100 The Sunspot Cycle - endings and beginnings - Students will examine a plot of the sunspot cycle and extract information from the plotted data about the previous sunspot cycle, and make predictions about the next one about to start in 2007. [Grade level: 6-9 | Topics:graph reading; extrapolation; time calculations]

Problem 99 The Hinode Satellite Views the Sun - Students will use a full-sun image from the new Hinode satellite to sketch the locations of magnetic fields on the sun's surface using information in the introductory article as a guide. [Grade level: 6-8 | Topics:image interpretation; eye-hand coordination; reading to be informed]

Problem 98 Solar Flare Reconstruction - Students will use data from a solar flare to reconstruct its maximum emission using graphical estimation (pre-algebra), power-law function fitting (Algebra 2), and will determine the area under the profile (Calculus). [Grade level: 9-11 | Topics:plotting tabular date; fitting functions; integration]

Problem 97 Hinode - Closeup of a Sunspot - Students will determine the sizes of sunspots and solar granulation cells from a recent image taken by the Hinode solar observatory. [Grade level: 6-8 | Topics:image scales, metric units, unit conversion]

Problem 96 Hinode Satellite Power - Students will study the design of the Hinode solar satellite and calculate how much power it can generate from its solar panels. [Grade level: 6-8 | Topics:area of rectangle,area of cylinder, unit conversion]

Problem 94 Solar Storms: Odds, Fractions and Percentages - Students will use actual data on solar storms to learn about the different kinds of storms and how common they are. This is a basic science activity that professionals do in order to look for relationships between different kinds of events that might lead to clues about what causes them. Can your students come up with something new that noone has thought about before? The Venn Diagramming activity is a key element of the activity and is reasonably challenging! [Grade level: 6-8 | Topics: Averaging; fractions; percentages; odds; Arithmetic Operations; Venn Diagrams]

Problem 92 A Lunar Transit of the Sun from Space - One of the STEREO satellites observed the disk of the moon pass across the sun. Students will use simple geometry to determine how far the satellite was from the moon and Earth at the time the photograph was taken. [Grade level: 9-11 | Topics: Geometry; parallax; arithmetic]

Problem 89 Atmospheric Shielding from Radiation- III - This is Part III of a 3-part problem on atmospheric shielding. Students use exponential functions to model the density of a planetary atmosphere, then evaluate a definite integral to calculate the total radiation shielding in the zenith (straight overhead) direction for Earth and Mars. [Grade level: 11-12 | Topics: Evaluating an integral, working with exponential functions]

Problem 88 Atmospheric Shielding from Radiation- II - This is the second of a three-part problem dealing with atmospheric shielding. Students use the formula they derived in Part I, to calculate the radiation dosage for radiation arriving from straight overhead, and from the horizon. Students also calculate the 'zenith' shielding from the surface of Mars. [Grade level: 9-11 | Topics: Algebra I; evaluating a function for specific values]

Problem 87 Atmospheric Shielding from Radiation- I - This is the first part of a three-part problem series that has students calculate how much radiation shielding Earth's atmosphere provides. In this problem, students have to use the relevant geometry in the diagram to determine the algebraic formula for the path length through the atmosphere from a given location and altitude above Earth's surface. [Grade level: 9-11 | Topics: Algebra II, trigonometry]

Problem 86 Do Fast CMEs Produce SPEs? - Recent data on solar proton storms (SPEs) and coronal mass ejections (CMEs) are compa black using Venn Diagrams to see if the speed of a CME makes solar proton storms more likely or not. [Grade level: 5-8 | Topics: Venn Diagrams; counting; calculating percentages and odds]

Problem 85 The Solar Tsunami! - Recent data from the Hinode satellite is used to measure the speed of a solar explosion on the surface of the sun using a series of images taken by the satellite at three different times. Students calculate the speed of the blast between the first pair and last pair of images, and determine if the blast wave was accelerating or decellerating in time. [Grade level: 5-8 | Topics: Finding image scale; calculating time differences; calculating speed from distance and time]

Problem 84 Beyond the Blue Horizon - How far is it to the horizon? Students use geometry, and the Pythagorean Theorem, to determine the formula for the distance to the horizon on any planet with a radius, R, from a height, h, above its surface. Additional problems added that involve calculus to determine the rate-of-change of the horizon distance as you change your height. [Grade level: 9-11 | Topics: Algebra, Pythagorean Theorem; Experts: DIfferential calculus) ]

Problem 81 The Pressure of a Solar Storm - Students will examine three mathematical models for determining how much pressure a solar storm produces as it affects Earth's magnetic field. They will learn that magnetism produces pressure, and that this accounts for many of the details seen in solar storms. [Grade level: 9-11 | Topics: Substituting numbers into equations; filling out missing table entries; data interpretation; mathematical models ]

Problem 78 Moving Magnetic Filaments Near Sunspots - Students will use two images from the new, Hinode (Solar-B) solar observatory to calculate the speed of magnetic filaments near a sunspot. The images show the locations of magnetic features at two different times. Students calculate the image scales in kilometers/mm and determine the time difference to estimate the speeds of the selected features. [Grade level: 6-8 | Topics: scaling, estimation, speed calculations, time arithmetic ]

Problem 74 A Hot Time on Mars - The NASA Mars Radiation Environment (MARIE) experiment has created a map of the surface of mars, and measu black the ground-level radiation background that astronauts would be exposed to. This math problem lets students examine the total radiation dosage that these explorers would receive on a series of 1000 km journeys across the martian surface. The students will compare this dosage to typical background conditions on earth and in the International Space Station to get a sense of perspective [Grade level: 6-8 | Topics: decimals, unit conversion, graphing and analysis ]

Problem 73 Monster Functions in Space Science I. - This problem has students employ a pair of complicated algebraic equations to evaluate the strength of the sun's magnetic field near Earth's orbit. The equations are a model of the sun's magnetic field in space based on actual research by a solar physicist. This introduces students to a real-world application of mathematical modeling, and extracting p blackictions from theoretical models that can be tested. Students are provided the values for the relevant variables, and through substitution, calculate the numerical values for two 'vector' components of the sun's magnetic field near Earth's orbit. [Grade level: 9-11 | Topics: decimals, scientific notation, significant figures ]

Problem 70 Calculating Total Radiation Dosages at Mars - This problem uses data from the Mars Radiation Environment Experiment (MARIE) which is orbiting Mars, and measures the daily radiation dosage that an astronaut would experience in orbit around Mars. Students will use actual plotted data to calculate the total dosage by adding up the areas under the data curve. This requires knowledge of the area of a rectangle, and an appreciation of the fact that the product of a rate (rems per day) times the time duration (days) gives a total dose (Rems), much like the product of speed times time gives distance. Both represent the areas under their appropriate curves. Students will calculate the dosages for cosmic radiation and solar proton flares, and decide which component produces the most severe radiation problem. [Grade level: 6-8 | Topics: decimals, area of rectangle, graph analysis]

Problem 64 Solar Activity and Satellite Mathematics - When solar storms cause satellite problems, they can also cause satellites to lose money. The biggest source of revenue from communications satellites comes from transponders that relay television programs, ATM transactions and many other vital forms of information. They are rented to many different customers and can cost nearly \$2 million a year for each transponder. This activity examines what happens to a single satellite when space weather turns bad! [Grade level: 4-6 | Topics: Decimals; money; percents]

Problem 63 Solar Activity and Tree Rings - What's the connection? - Trees require sunlight to grow, and we know that solar activity varies with the sunspot cycle. Can an average tree sense solar activity cycles and change the way it grows from year to year? This activity uses a single tree to compare its growth rings to the sunspot cycle. This is also an interesting suggestion for science fair projects! Here is the accompanying Excell Spreadsheet Data File. [Grade level: 4-6 | Topics: Spreadsheets and technology; decimal math]

Problem 53 Astronomy: A Moving Experience! - Objects in space move. To figure out how fast they move, astronomers use many different techniques depending on what they are investigating. In this activity, you will measure the speed of astronomical phenomena using the scaling clues and the time intervals between photographs of three phenomena: A supernova explosion, a coronal mass ejection, and a solar flare shock wave. [Grade level: 6-8 | Topics: Finding the scale of an image; metric measurement; distance = speed x time; scientific notation]

Problem 51 Sunspots Close-up and Personal - Students will analyze a picture of a sunspot to learn more about its size, and examine the sizes of various other features on the surface of the sun that astronomers study. [Grade level: 6-8 | Topics: Finding the scale of an image; metric measurement; decimal math]

Problem 43 An Interplanetary Shock Wave On November 8, 2000 the sun released a coronal mass ejection that traveled to Earth, and its effects were detected on Jupiter and Saturn several weeks later. In this problem, students will use data from this storm to track its speed and acceleration as it traveled across the solar system. [Grade level: 6-10 | Topics: Time calculations; distance = speed x time ]

Problem 42 Solar Storms in the News - Students will use a newspaper archive to explore how reporters have described the causes of aurora since the 1850's. They will see how some explanations were popular for a time, then faded into oblivion, as better scientific explanations were created. [Grade level: 6-10 | Topics: Online research; tallying data]

Problem 41 Solar Energy in Space Students will calculate the area of a satellite's surface being used for solar cells from an actual photo of the IMAGE satellite. They will calculate the electrical power provided by this one panel. Students will have to calculate the area of an irregular region using nested rectangles. [Grade level: 7-10 | Topics: Area of an irregular polygon; decimal math]

Problem 39 Solar Storm Timeline How long does a solar storm last? How fast does it travel? Students will examine an event timeline for a space weather event and use time addition and subtraction skills to calculate storm durations and speeds. [Grade level: 7-9 | Topics: time math; decimal math; speed = distance/time]

Problem 38 Solar Eclipses and Satellite Power From the ground we see total solar eclipses where the New Moon passes directly between Earth and Sun. Satellites use solar cells to generate electricity, but this is only possible when the Earth is not 'eclipsing' the sun. Students will create a scaled drawing of the orbits of three satellites around Earth, and calculate how long each satellite will be in the shadow of Earth. They will be asked to figure out how to keep the satellites operating even without sunlight to power their solar panels. [Grade: 5 - 8 | Topics: Geometry; decimal math]

Problem 32 Solar Proton Events and Satellite Damage Students will examine the statistics for Solar Proton Events since 1996 and estimate their damage to satellite solar power systems. [Grade: 7 - 9 | Topics: Interpreting tabular data; histogramming]

Problem 27 Satellite Failures and the Sunspot Cycle There are over 1500 working satellites orbiting Earth, representing an investment of 160 billion dollars. Every year, between 10 and 30 of these re-enter the atmosphere. In this problem, students compare the sunspot cycle with the record of satellites re-entering the atmosphere and determine if there is a correlation. They also investigate how pervasive satellite technology has become in their daily lives. [Grade: 6 - 8 | Topics: Graphing tabular data; decimal math]

Problem 26 Super-sized Sunspots and the Solar Cycle. Students compare the dates of the largest sunspots since 1900 with the year of the peak sunspot cycle. They check to see if superspots are more common after sunspot maximum or before. They also compare superspot sizes with the area of earth. [Grade: 6 - 8 | Topics: Interpreting tabular data; decimal math]

Problem 23 Solar Flares and Sunspot Sizes Students compare sunspot sizes to the frequency of solar flares and discover that there is no hard and fast rule that relates sunspot size to its ability to produce very large flares. [Grade: 6 - 8 | Topics: Interpreting tabular data; percentages; decimal math ]

Problem 7 Solar Flares, CME's and Aurora Some articles about the Northern Lights imply that solar flares cause them. Students will use data to construct a simple Venn Diagram, and answer an important question about whether solar flares cause CME's and Aurora. [Grade: 5 - 7 | Topics: Venn Diagramming]

Problem 6 Observing the Sun's rotation Students use a Sunspotter to track sunspots during the week of November 7, 2004, and calculate the rotation period of the sun. [Grade: 6 - 8 | Topics: Lab exercise using a 'Sunspotter' to measure sun's rotation]

Problem 5 The November 8, 2004 solar storm Students calculate the speed of a CME, and describe their aurora observations through writing and drawing. [Grade: 6 - 8 | Topics: Time calculations; distance = speed x time]

## Mercury

Problem 474: MESSENGER Explores the Interior of Mercury
Students work with a simple spherical core and shell model to determine the interior structure of mercury and the size of its dense iron core. [Grade: 9-12 | Topics: working with volumes of speheres; mass = density x volume] (PDF)

Problem 473: MESSENGER Explores the Mass of Mercury
Students use the orbit of NASA's MESSENGER spacecraft to determine the mass of Mercury. [Grade: 9-12 | Topics: working with equations with integer powers and solving for specified values; scientific notation] (PDF)

Problem 415: Mercury and the Moon - Similar but different Students explore the mass and volume of mercury compared to the moon by using the formula for a sphere and scale changes. [Grade: 8-10 | Topics: scale; volume of a sphere; mass = density x volume] (PDF)

Problem 121: Ice on Mercury? Since the 1990's, radio astronomers have mapped Mercury. An outstanding curiosity is that in the polar regions, some craters appear to have 'anomalous reflectivity' in the shadowed areas of these craters. One interpretation is that this is caused by sub-surface ice. The MESSENGER spacecraft hopes to explore this issue in the next few years. In this activity, students will measure the surface areas of these potential ice deposits an calculate the volume of water that they imply. [Grade: 8-10 | Topics:Area of a circle; volume, density, unit conversion] (PDF)

Problem 105: The Transit of Mercury As seen from Earth, the planet Mercury occasionally passes across the face of the sun; an event that astronomers call a transit. From images taken by the Hinode satellite, students will create a model of the solar disk to the same scale as the image, and calculate the distance to the sun. [Grade: 9-11 | Topics:image scales; angular measure; degrees, minutes and seconds] (PDF)

## Mars

Problem 615: Radiation Levels on the Surface of Mars
Students explore radiation dosages on mars and in interplanetary space [Grade: 6-8 | Topics: unit conversions; graph analysis; rates ] (PDF)

Problem 570: Curiosity Heads for Mt Sharp
Tabular data is used to estimate how long it will take the Curiosity rover to reach the base of Mt Sharp using data from its porevious week travels. [Grade: 3-5 | Topics: averaging numbers in a table; time = distance/speed] (PDF)

Problem 536:Exploring a Possible InSight Landing Area on Mars
Students work with latitude and longitude and scaled images of mars to locate the InSight proposed landing area, and describe the terrain of the landing area. [Grade: 6-8 | Topics: degree measure; latitude and longitude; working with scaled images; metric measure] (PDF)

Problem 535:Comparing the InSight Landing Area to a City Block!
Students use scaled images of a proposed InSIght landing area and a scaled image of an urban neighborhood on Earth to compare the sizes of familiar things with the unfamiliar martian landscape. [Grade: 6-8 | Topics: scale; proportion; metric measurement] (PDF)

Problem 534:Exploring Marsquake Energy with the Moment Magnitude Scale
Students are introduced to the Moment Magnitude marsquake scale which gives a logarithmic index for marsquakes of differing energies. They calculate two examples of marsquakes and meteor impacts and compare their Moment Magnitude. [Grade: 8-10 | Topics: logarithms; scientific notation; algebra ] (PDF)

Problem 533:Exploring Logarithms and the Richter Magnitude Scale
Students work with a logarithmic scale to estimate how much ground movement occurs for earthquakes of different strengths. [Grade: 8-10 | Topics: logarithms; base-ten exponents] (PDF)

Problem 532:The Distance to the Martian Horizon
Students devive a basic equation for the distance to the horizon on a spherical body using the Pythagorean Theorem and a bit of algebra. The estimate the number of cell towers needed to cover Mars. [Grade: 8-10 | Topics: Pythagorean Theorem, Algebra; scientific notation; areas of spheres and circles ] (PDF)

Problem 531:Exploring the Interior of Mars with Spheres and Shells
Students use the volume properties of spheres and shells along with the relationship mass=densityxvolume to create a model of the interior of mars. [Grade: 8-10 | Topics: formula for volume of spheres and spherical shells; mass=densityxvolume; scientific notation ] (PDF)

Problem 530:Exploring the Mass of Mars
Students calculate the mass of mars by using satellite data and Keplers Third Law. [Grade: 8-10 | Topics: Algebra; scientific notation ] (PDF)

Problem 529:Exploring Impacts and Quakes on Mars
Students work with logarithmic scales to explore the relationship between the energy of an marsquake and its logarithmic index, which is similar to the Richter Scale used for earthquakes. [Grade: 8-10 | Topics: Logarithmic scales; scientific notation ] (PDF)

Problem 528:Comparing the Heat Output of Mars and Earth
Students learn about the heat flow formula and use it to explore the properties of Earth and Mars in terms of their crust composition. [Grade: 8-10 | Topics: Algebra; temperature gradients] (PDF)

Problem 527:Exploring Heat Flow and Insulation
Students explore how insulation works to reduce heat flow. They convert a verbal description of a formula expressed in proportions, and use it to calculate why aluminum pots heat faster than steel pots, and how we can determine the properties of martian sooil from heat flow and temperature changes. [Grade: 8-10 | Topics: algebra; rates of change ] (PDF)

Problem 526:Exploring Temperature Change in Earth?s Outer Crust
Students explore the rate of temperature change in the crust of Earth and Mars and learn about units expressed as degrees C/km. They calculate how hot the ground will be at various depths, and how gold miners must deal with extreme heat. [Grade: 6-8 | Topics: fahrenheit and celsius degrees; rates of change] (PDF)

Problem 525:Exploring the InSight Lander Telemetry Data Flow
Students explore how long it takes to transmit digital data using examples from downloading songs from their computer to their ipod. [Grade: 6-8 | Topics: working with kilo, mega and rates of data transfer in bytes/sec. ] (PDF)

Problem 524:Seeing the Martian Surface with IDC
Students learn about the IDC camera and calculate resolution and how many images are needed to map the InSight landing area. [Grade: 6-8 | Topics: ANgular measurfe, degrees and seconds; image scal; tiling an area with overlap. ] (PDF)

Problem 523:Telling Time on Mars - Earth Days and Mars Sols
Students work with two clocks on Earth and Mars and learn about earth and mars time given that a day on Mars is 40 minutes longer than an Earth day. [Grade: 6-8 | Topics: time calculations, hours, minutes, seconds; length of day ] (PDF)

Problem 522:Radio Communications with Earth ? The Earth-Sun Angle
The earth-sun angle is given in tabular form in degrees. Students graph the data and find the dates when transmissions to Earth cannot occur. [Grade: 8-10 | Topics: Interpreting tabular data; rates and slopes ] (PDF)

Problem 521:Estimating the Mass of a Martian Dust Devil!
Students estimate the mass of a martian dust devil using the approximation that it is a cylinder with a fixed density of dust. [Grade: 8-10 | Topics: Volume of a cylinder; mass = density x volume ] (PDF)

Problem 520:The Work Area In Front of the Lander
Students estimate the area in front of the InSight lander where experiments will be conducted and instruments moved with a single robotic arm. [Grade: 6-8 | Topics: Area of a circle segment; Area common to two intersecting circles] (PDF)

Problem 519:Scheduling Events in Time for Launch
Students learn about scheduling many events along a timeline (breakfast, packing, driving, etc ) by planning a family trip where the family members have to arrive at the airport for a flight that leaves at a specific date and time. [Grade: 5-7 | Topics: working with time units; creating a timeline] (PDF)

Problem 518:The InSight Seismographic Station Solar Power System
Students explore the properties of decagons to determine the area of the solar panels used on the InSight lander. [Grade: 7-9 | Topics: area of regular polygons; estimating areas of non-square shapes] (PDF) Problem 508: The InSight Seismographic Station - Wave arrival times
Students work with the circumference of Mars and the speed of shock waves in the martian crust to estimate the arrival times of the waves at the InSight Lander. [Grade: 6-8 | Topics: speed=distance/time; Time calculations; circumference of a circle] (PDF)

Problem 500: Curiosity Uses X-Ray DIffraction to Identify Minerals on Mars
Students learn about diffraction geometry and then estimate the distance between crystal planes in a mars rock sample. [Grade: 10-12 | Topics: geometry; trigonometry] (PDF)

Problem 491: The Curiosity Rover on the Move.
Students plot the position of the Curiosity Rover on a cartesian grid covering the satellite image of the landing area. They use the 2-point distance formula to determine how far the rover traveled between stops, and determine it speed. [Grade: 6-8 | Topics: Cartseian graphs; ordered pairs and coordinates; distance = speed x time; metric measure ] (PDF)

Problem 485: Curiosity Discovers Ancient Mars River
Students estimate the speed of an ancient mars river using images from the CUriosity rover. [Grade: 9-12 | Topics: Algebra; trigonometry; evaluating functions ] (PDF)

Problem 479: Exploring Gale Crater with the Curiosity Rover
Students explore the Gale Crater landing area and calculate rover distances to various way stations to determine the round trip distance and travel time. [Grade: 9-12 | Topics: Pythagorean Distance Formula; Coordinate geometry ] (PDF)

Problem 457: The Interplanetary Voyage of MSL
Students use the properties of ellipses to determine the formula for the Hohmann Transfer Orbit taking the Mars Science Laboratory to Mars in 2012 [Grade: 10-11 | Topics: time=distance/speed; scale models; metric math; properties of ellipses] (PDF)

Problem 456: The Launch of the Mars Science Laboratory (MSL) in 2011
Students use a sequence of launch images to determine the Atlas V's launch speed and acceleration. By determining the scale of each image, they estimate average speeds during the first 4 seconds after lift-off. [Grade: 8-10 | Topics: time=distance/speed; scale models; metric math] (PDF)

Problem 393: Taking a stroll around a martian crater! Students use a recent photograph of a crater on Mars to estimate its circumference and the time it will take NASAs Opportunity Rover to travel once around its edge. [Grade: 6-8 | Topics: scale model; distance = speedxtime; metric measure] (PDF)

Problem 237: The Martian Dust Devils Students determine the speed and acceleration of a martian dust devil from time laps images and information about the scale of the image. [Grade: 6-8 | Topics: scales; Determining speed from sequential images; V = D/T (PDF)

Problem 139: How Big Is It? - Mars Students use an image of a crater wall on mars to investigate ancient water gullies discovered in 2008 by the Mars Orbiter. [Grade: 4 - 7 | Topics:image scales; metric measurement; division and multiplication; decimals] (PDF)

Problem 133: How Big is It? - The Mars Rover. Students work with an image taken by the Mars Orbiter satellite of the Spirit landing site. They determine the image scale, and calculate the sizes of various surface features from the image. [Grade: 4 - 7 | Topics:image scaling; multiply, divide, work with millimeter ruler] (PDF)

Problem 126 : How Big is It? - A Martian Avalanche! Students work with a Mars reconnissance Orbiter image to determine image scale, and search for the smallest things seen in a photograph.This avalanche was caught as it occurred on February 19, 2008! [Grade: 4 - 7 | Topics:image scaling; multiply, divide, work with millimeter ruler] (PDF)

Problem 74: A Hot Time on Mars - The NASA Mars Radiation Environment (MARIE) experiment has created a map of the surface of mars, and measu black the ground-level radiation background that astronauts would be exposed to. This math problem lets students examine the total radiation dosage that these explorers would receive on a series of 1000 km journeys across the martian surface. The students will compare this dosage to typical background conditions on earth and in the International Space Station to get a sense of perspective [Grade level: 6-8 | Topics: decimals, unit conversion, graphing and analysis ] (PDF)

Problem 70: Calculating Total Radiation Dosages at Mars - This problem uses data from the Mars Radiation Environment Experiment (MARIE) which is orbiting Mars, and measures the daily radiation dosage that an astronaut would experience in orbit around Mars. Students will use actual plotted data to calculate the total dosage by adding up the areas under the data curve. This requires knowledge of the area of a rectangle, and an appreciation of the fact that the product of a rate (rems per day) times the time duration (days) gives a total dose (Rems), much like the product of speed times time gives distance. Both represent the areas under their appropriate curves. Students will calculate the dosages for cosmic radiation and solar proton flares, and decide which component produces the most severe radiation problem. [Grade level: 6-8 | Topics: decimals, area of rectangle, graph analysis] (PDF)

## Jupiter

Problem 568: Ios Volcanoes and Resurfacing
Students examine how volcanic activity on Jupiters satellite Io can lead to fresurfacing the entire moon in less than a million years covering all new craters. [Grade: 6-8 | Topics: Surface aea of a sphere; rates; scientific notation] (PDF)

Problem 472: Investigating Juno's Elliptical Transfer Orbit
Students use the Standard Formula for an ellipse to study the elliptical orbit of the Juno spacecraft, and relate specific properties of the ellipse to features of the spacecrafts trajectory such as aphelion, perihelion, and ellipticity. [Grade: 9-12 | Topics: formula for an ellipse; semi-major and minor axis] (PDF)

Problem 471: Investigating the Launch of the Juno Spacecraft
Students use a series of images from a launch video to determine the scale of each image and determine the speed of the rocket as it leaves the gantry. [Grade: 6-8 | Topics: scale models; speed = distance/times] (PDF)

Problem 470: The Launch of the Juno Spacecraft - Ascent to orbit
Students use tabulated altitude and range data following the launch of the Juno mission, to determine the speed of the rocket as it travels to arth orbit. [Grade: 6-8 | Topics: scale models; speed = distance/time] (PDF)

Problem 469: Solar Energy and the Distance of Juno from the Sun
Students use the formula for an ellipse, along with the inverse-square law to create a mathemartical model that predicts the declining solar power produced by Junos solar panels as the spacecraft travels from Earth to Jupiter. [Grade: 9-12 | Topics: algebra; trigonometry; distance formula] (PDF)

Problem 135 How Big is It? - Io and Jupiter. Students work with an image taken by the Cassini spacecraft of Jupiter and its satellite Io. They determine the image scale, and calculate the sizes of various features in the image. [Grade: 4 - 7 | Topics:image scaling; multiply, divide, work with millimeter ruler]

## Saturn

Problem 572: How Saturns Moon Mimas Created the Cassini Division
Students calculate the acceleration of gravity in Cassinis Division and estimate the number of years to eject these particles. [Grade: 9-12 | Topics: scientific notation; evaluating a formula for gravity; unit conversions] (PDF)

Problem 569: Orbit Speeds and Times for Saturns Rings
Students learn about the orbit speeds of ring particles and how orbit periods in the Cassini Division relate to the orbit of the moon Mimas. [Grade: 6-8 | Topics: square root formulae; circumference of circle; speed = distance/time ] (PDF)

Problem 550:Comparing the Rings of the Outer Planets
Students compare the dimensions of the rings of Jupiter, Saturn, Uranus and Neptune to the radius of each planet, and the location of the break up Tidal Limit to test an idea of how the rings may have formed. [Grade: 6-8 | Topics: scale model; proportions; number line ] (PDF)

Problem 549:Saturns Rings- Shadows from Moons and Ringlets
Students use an image of a ring of Saturn to investigate its thickness using shadows cast by ringlet material kicked up by a passing moon. [Grade: 6-8| Topics: scales; proportions; triangle geometry; angle measurement] (PDF)

Problem 548:Saturns Rings - A close-up study Students use a Cassini image of Saturns rings to calculate the sizes of the smallest rings, and how their thicknesses change with distance from Saturn. [Grade: 3-5 | Topics: measurement; scales; proportions; metric measure; bar graphs] (PDF)

Problem 547:The Rings of Saturn Students explore the volume and mass of the rings of saturn to estimate the number of ring particles and their separations, and the radius of the equivalent spherical body. [Grade: 9-12 | Topics: volume of a ring and a sphere; scientific notation] (PDF)

Problem 461: Cassini Delivers Holiday Treats from Saturn Students explore proportions and angular size using images of Saturn's moons Titan and Dione [Grade: 7-9 | Topics: scale models; proportions] (PDF)

Problem 335: Methane Lakes on Titan Students use a recent Cassini radar image of the surface of Titan to estimate how much methane is present in the lakes that fill the image, and compare the volume to that of the fresh water lake, Lake Tahoe. [Grade: 6-8 | Topics: estimating irregular areas; calculating volume from area x height; scaled images ]

Problem 315: The Mysterious Hexagon on Saturn A curious hexagon formed by the Saturn polar jet stream, and photographed by the Cassini spacecraft, is used to determine wind speed and acceleration. [Grade: 6-9 | Topics: Measuring; Metric Units; speed=distance/time]

Problem 272: Spitzer Telescope Discovers New Ring of Saturn! Students calculate the volume of the ring and compare it to the volume of Earth to check a news release figure that claims 1 billion Earths could fit inside the new ring. [Grade: 8-9 | Topics: Geometry, Algebra, volumn, scientific notation]

Problem 205: The Io Plasma Torus- Students approximate the Io radiation belts as a cylinder to determine its volume ,and the mass of the particles within it. [Grade: 9-12| Topics: Algebra I - volume of cylinders; calculus - Integrals of volumes.]

Problem 174: A Matter of Timing - Students study satellites of Saturn to work out graphically how often they will line up. [Grade: 3-6| Topics: scale model; time calculation; fractions; working with rulers and compasses]

Problem 154: Pan's Highway and Saturn's Rings - Students use an image from the Cassini spacecraft to determine how large the satellite Pan is, and the scale of Saturn's rings using a millimeter ruler. [Grade: 4-6 | Topics:Finding the scale of an image; measurement; unit conversion]

## General Solar System Problems

Problem 555:Exploring Your Weight Across the Solar System
Students estimate their weight on different planets, moons and asteroids. [Grade: 6-8 | Topics: proportions] (PDF)

Problem 554:Exploring Volcanoes and Geysers Across the Solar System
Students determine the ejection speed and heights of gasses vented by geysers and volcanoes. [Grade: 6-8 | Topics: solving square root equations; ] (PDF)

Problem 546: The Relative Sizes of Planets and other Objects
Students use proportional information to determine the relative scales of planets and large moons across the solar system. [Grade: 3-5 | Topics:scale; proportion] (PDF)

Problem 545:Measuring Atmospheric Trace Gases Using Parts Per Million
Students convert from percentage units to parts per million and compare trace gases in the atmospheres of various planets. [Grade: 6-8 | Topics: percentages; unit conversions ] (PDF)

Problem 544:The Composition of Planetary Atmospheres
Students study the composition of planetary atmospheres and compare the amounts of certain compounds in them [Grade: 6-8 | Topics: Pie graphs; percentages; scientific notation] (PDF)

Problem 543:Timeline for Planet Formation
Students calculate time intervals in millions and billions of years from a timeline of events [Grade: 3-5 | Topics: time calculations; integers] (PDF)

Problem 542:The Late Heavy Bombardment Era
Students estimate the average arrival time of large asteroids that impacted the moon. They work with the formula for the volume of a sphere to estimate how much additional mass was added to the moon and Earth durung this era. [Grade: 6-8 | Topics: volume of spheres; proportions] (PDF)

Problem 541:How to Build a Planet
Students study planet growth by using a clay model of planetessimals combining to form a planet by investigating volume addition with spheres. [Grade: 3-5 | Topics: graphing; counting] (PDF)

Problem 540:Travel Times by Spacecraft Around the Solar System
Students explore how long it takes our fastest rockets to reach each of the planets. [Grade: 6-8 | Topics: time=distance/speed; metric conversion] (PDF)

Problem 539:Visiting the Planets at the Speed of Light
Students learn about the light travel times to the 8 planets by converting the distances in Astronomical Units to travel times at the speed of light. [Grade: 6-8 | Topics: Proportions; unit conversions; time = distance/speed; metric units] (PDF)

Problem 538:How Big is Our Solar System?
Students work with proportions to convert solar system distances into Astronomical Units for the 8 planets. [Grade:6-8 | Topics: Proportions; unit conversions] (PDF)

Problem 496: How to Grow a Planet or a Rain Drop
Students use calculus to slove for the growth in mass of a body, and solve the equation for the case of a raindrop and a planet like Earth. [Grade: 12 | Topics: Solving a simple differential equation.] (PDF)

Problem 464: Big Moons and Small Planets
Students work with a scaled drawing of 26 large moons in the solar system, and together with an exercise in using simple fractions, explore the relative sizes of the moons compared to Earth. [Grade: 3-5 | Topics: scale models; proportions; fractions] (PDF)

Problem 305: From Asteroids to Planetoids Students explore how long it takes to form a small planet from a collection of asteroids in a planet-forming disk of matter orbiting a star. [Grade: 11-12 | Topics: Integral calculus]

Problem 304: From Dust Balls to Asteroids Students calculate how long it takes to form an asteroid-sized body using a simple differential equation. [Grade: 11-12 | Topics: Integral Calculus]

Problem 303: From Dust Grains to Dust Balls Students create a model of how dust grains grow to centimeter-sized dust balls as part of forming a planet. [Grade: 11-12 | Topics: Integral Calculus]

Problem 302: How to Build a Planet from the Inside Out Students model a planet using a spherical core and shell with different densities. The goal is to create a planet of the right size, and with the correct mass using common planet building materials. [Grade: 9-11 | Topics: Geometry; volume; scientific notation; mass=density x volume]

Problem 301: Planetary Alignments Students combine a geometric model with number series to calculate when planets will 'line up' in a simple solar system. [Grade: 4-8 | Topics: Number series; geometry; Least Common Multiple]

Problem 299: Changing Perspectives on the Sun's Diameter Students compare two images of the sun taken by the SHOH satellite to measure the apparent diameter change from different earth obit locations in the winter and summer. [Grade: 6-8 | Topics: measurement; parallax; metric units; percentage change]

Problem 296: Getting an Angle on the Sun and Moon Students explore angular size and scale by comparing two images of the sun and moon which have identical angular size, but vastly different scales. [Grade: 8-10 | Topics: Geometry; angle measure; scale; proportion]

Problem 292: How Hot is That Planet? Students use a simple function to estimate the temperature of a recently discovered planet called CoRot-7b. [Grade: 8-10 | Topics: Algebra II; Evaluating Power functions]

Problem 278: Spitzer Studies the Distant Planet Osiris Students learn about the density of the planet HD209458b, also called Osiris, and compare it to that of Jupiter. [Grade: 8-10 | Topics: Spherical volumes; density; Scientific Notation;]

Problem 268: Planetary Conjunctions Students study a simple solar system with three planets and work out how often planets will 'line up'. [Grade: 3-5 | Topics: geometry, time, patterns]

Problem 267: Identifying Materials by their Reflectivity The reflectivity of a material can be used to identify it. This is important when surveying the lunar surface for minerals, and also in creating 'green' living environments on Earth. [Grade: 6-8 | Topics: percentage, interpreting tabular data, area ]

Problem 264: Water on Planetary Surfaces Students work with watts and Joules to study melting ice. [Grade: 8-10 | Topics: unit conversion, rates]

Problem 263: Ice or Water? Whether a planetary surface contains ice or liquid water depends on how much heat is available. Students explore the concepts of Specific heat and Latent Heat of Fusion to better understand the and quantify the energy required for liquid water to exist under various conditions. [Grade: 8-10 | Topics: unit conversion, scientific notation]

Problem 250: The Most Important Equation in Astronomy Students learn about how an instrument's ability to see details depends on its size and its operating wavelength - the key to designing any telescope or camera. [Grade: 8-10 | Topics: geometry, angle measure, scientific notation]

Problem 249: Spotting an Approaching Asteriod or Comet Students work with a fundamental equation for determing the brightness of an asteroid from its size and distance from Earth. [Grade: 10-12 | Topics: Algebra 1, logarithms, area, scientific notation]

Problem 247: Space Mobile Puzzle Students calculate the missing masses and lengths in a mobile using the basic balance equation m1 x r1 = m2 x r2 for a solar system mobile. [Grade: 8-10 | Topics: metric measur, algebra 1, geometry]

Problem 241: Angular Size and Similar Triangles A critical concept in astronomy is angular size, measured in degrees, minutes or arc-seconds. This is a review of the basic properties of similar triangles for a fixed angle. [Grade: 8-10 | Topics: geometry, similar triangles, proportions]

Problem 212: Finding Mass in the Cosmos- Students derive a simple formula, then use it to determine the masses of objects in the universe from the orbit periods and distances of their satellites. [Grade: 9-12| Topics: Scientific Notation; Algebra II; parametric equations]

Problem 203: Light Travel Times- Students determine the time it takes light to reach various objects in space. [Grade: 6-8| Topics: Scientific Notation; Multiplication; time = distance/speed.]

Problem 180: Planets, Fractions and Scales- Students work with relative planet comparisons to determine the actual sizes of the planets given the diameter of Earth. [Grade: 4-6| Topics: scale models; decimals; fractions]

Problem 114 The Heliopause...a question of balance Students will learn about the concept of pressure equilibrium by studying a simple mathematical model for the sun's heliopause located beyond the orbit of Pluto. They will calculate the distance to the heliopause by solving for 'R' and then using an Excel spreadsheet to examine how changes in solar wind density, speed and interstellar gas density relate to the values for R. [Grade: 8-10 | Topics: Formulas with two variables; scientific notation; spreadsheet programming]

Problem 89 Atmospheric Shielding from Radiation- III - This is Part III of a 3-part problem on atmospheric shielding. Students use exponential functions to model the density of a planetary atmosphere, then evaluate a definite integral to calculate the total radiation shielding in the zenith (straight overhead) direction for Earth and Mars. [Grade level: 11-12 | Topics: Evaluating an integral, working with exponential functions]

Problem 88 Atmospheric Shielding from Radiation- II - This is the second of a three-part problem dealing with atmospheric shielding. Students use the formula they derived in Part I, to calculate the radiation dosage for radiation arriving from straight overhead, and from the horizon. Students also calculate the 'zenith' shielding from the surface of Mars. [Grade level: 9-11 | Topics: Algebra I; evaluating a function for specific values]

Problem 87 Atmospheric Shielding from Radiation- I - This is the first part of a three-part problem series that has students calculate how much radiation shielding Earth's atmosphere provides. In this problem, students have to use the relevant geometry in the diagram to determine the algebraic formula for the path length through the atmosphere from a given location and altitude above Earth's surface. [Grade level: 9-11 | Topics: Algebra II, trigonometry]

Problem 84 Beyond the Blue Horizon - How far is it to the horizon? Students use geometry, and the Pythagorean Theorem, to determine the formula for the distance to the horizon on any planet with a radius, R, from a height, h, above its surface. Additional problems added that involve calculus to determine the rate-of-change of the horizon distance as you change your height. [Grade level: 9-11 | Topics: Algebra, Pythagorean Theorem; Experts: DIfferential calculus) ]

Problem 60 When is a planet not a planet? - In 2003, Dr. Michael Brown and his colleagues at CalTech discovered an object nearly 30% larger than Pluto, which is designated as 2003UB313. It is also known unofficially as Xenia, after the famous Tv Warrior Princess! Is 2003UB313 really a planet? In this activity, students will examine this topic by surveying various internet resources that attempt to define the astronomical term 'planet'. How do astronomers actually assign names to astronomical objects? Does 2003UB313 deserve to be called a planet, or should it be classified as something else? What would the new classification mean for asteroids such as Ceres, or objects such as Sedna, Quaoar and Varuna? [Grade level: 6-8 | Topics: Non-mathematical essay; reading to be informed]

Problem 59 Getting A Round in the Solar System! - How big does a body have to be before it becomes round? In this activity, students examine images of asteroids and planetary moons to determine the critical size for an object to become round under the action of its own gravitational field. Thanks to many Internet image archives this activity can be expanded to include dozens of small bodies in the solar system to enlarge the research data for this problem. Only a few example images are provided, but these are enough for the student to get a rough answer! [Grade level: 6-8 | Topics: Data analysis; decimals; ratios; graphing]

Problem 43 An Interplanetary Shock Wave On November 8, 2000 the sun released a coronal mass ejection that traveled to Earth, and its effects were detected on Jupiter and Saturn several weeks later. In this problem, students will use data from this storm to track its speed and acceleration as it traveled across the solar system. [Grade level: 6-10 | Topics: Time calculations; distance = speed x time ]

## Comet ISON

Problem 584: Comparing Comets Up Close with NASA Spacecraft
Students compare five comets and determine size ranges and percentages. [Grade: 3-5 | Topics: percentages; volume of a cube] (PDF)

Problem 563:Comet ISON and its Close Encounter with Mars
Students use tabular data to determine the date and time of closest approach to Mars [Grade: 3-5 | Topics: graphing tabular data] (PDF)

Problem 562:Exploring the Orbit of Comet ISON
Students use tabulated data to estimate when this comet will make its closest approach to the sun in 2013. [Grade: 6-8 | Topics: graphing tabular data; scale; measurement; distance between points] (PDF)

Problem 560:The Orbit of Comet ISON
Students explore how close Comet ISON will get to Mercury, Venus, Earth and Mars during its 2013 passage. [Grade: 3-5 | Topics: Interpreting tabular data; graphing ] (PDF)

Problem 559:Comet ISON Losing Mass as it Approaches the Sun.
Students estimage how much mass the comet will loose at its present rate. [Grade: 6-8 | Topics: volume of a sphere; rates; mass=density x volume] (PDF)

## Other Comets and Minor Bodies

Problem 587: Comet Encounters after Discovery
Students examine how often newly discovered comets approach Earth and become a hazard, and how soon after discovery these close passes can occur. [Grade: 3-5 | Topics: Averaging, percentages] (PDF)

Problem 586: Searching for Comets
Students use tabular data on the detection of new comets since 1999 to explore detection rates over time. [Grade: 3-5 | Topics: Percentages] (PDF)

Problem 585: Exploring Comet Orbits
Students explore the elliptical orbit of Halleys Comet and determine its period and the speed of the comet. [Grade: 6-8 | Topics: speed=distance/time] (PDF)

Problem 558:How Quickly are Near Earth Objects Being Discovered?
Students work with data presented in bar graphs to estimate how many more hazardous Near Earth Objects (NEOS) remein to be found. [Grade: 6-8 | Topics:bar graphs ] (PDF)

Problem 557:The 10000th Near Earth Asteroid 2013 MZ5
Students graph tabulated data to determine when this asteroid is closest to Earth and its speed at that time. [Grade: 6-8 | Topics: grapohing tabuklar data; solving a linear equation] (PDF)

Problem 481: Pluto's Fifth Moon
Students explore Kepler's Third Law and estimate the orbit period of a hypothetical sixth moon using the distance:period law. They also determine the mass of Pluto using the orbit data, including the recently discovered fifth moon (P5) of Pluto by the Hubble Space Telescope. [Grade: 9-12 | Topics: Power functions; integer exponents; Scientific Notation; tabular data] (PDF)

Problem 454: The Closest Approach of Asteroid 2005YU55 - III
Students work with the equation of a circle and line to find the orbit intersection points, midpoint, and closest distance to earth. [Grade: 8-10 | Topics: time=distance/speed; scale models; metric math] (PDF)

Problem 453: The Closest Approach of Asteroid 2005YU55 - II
Students work with the properties of circles and angular measure to see where the moon will be at the start of the asteroid encounter. [Grade: 6-8 | Topics: angular measure; time=distance/speed; scale models; metric math] (PDF)

Problem 452: The Closest Approach of Asteroid 2005YU55 - I
Students work with a scaled drawing of the orbit of the moon and the asteroid trajectory to predict where the asteroid will be relative to earth and the orbit of the moon. [Grade: 6-8 | Topics: time=distance/speed; scale models; metric math] (PDF)

Problem 434: Dawn Spacecraft Sees Asteroid Vesta Up-Close!
Students use an image of the asteroid to determine the diameters of craters and mountains using a millimeter ruler and the scale of the image in meters per millimeter. [Grade: 6-8 | Topics: scale, metric measurement] (PDF)

Problem 387: A Mathematical Model of Water Loss from Comet Tempel-1 Students use data from the Deep Impact spacecraft to create a simple empirical model for predicting the rate of water loss from a comet based on actual data. [Grade: 8-10 | Topics: graphing; fitting a parabola to data; evaluating functions] (PDF)

Problem 383: Estimating the mass of Comet Hartley 2 using calculus.
Students use a recent image of the nucleus of Comet Hartley 2 taken by the Deep Impact/EPOXI camera and a shape function described by a fourth-order polynomial to calculate the volume of the comet's head using integral calculus. to estimate the volume of the comets nucleus, and its total mass, [Grade: 12 | Topics: Volume integral using disk method; scale model; scientific notation; unit conversion] (PDF)

Problem 382: Estimating the mass and volume of Comet Hartley 2.
Students use a recent image of the nucleus of Comet Hartley 2 taken by the Deep Impact/EPOXI camera and a simple geometric 'dumbell' model based on a cylinder and two spheres, to estimate the volume of the comets nucleus, and its total mass. [Grade: 8-10 | Topics: volume of a sphere and cylinder; scale model; scientific notation; unit conversion] (PDF)

Problem 377: Deep Impact: Approaching Comet Hartley-2
Students use data for the brightness of Comet Hartley-2 measured by the Deep Impact spacecraft to create a linear equation for its approach distance, and use the inverse-square law to estimate its brightness on October 13, 2010. [Grade: 8-10 | Topics: linear modeling from data; inverse-square law] (PDF)

Problem 374: Deep Impact - Closing In on Comet 103P/Hartley 2
Students use the Tangent formula to figure out the angular size of the comet at closest approach, and the scale of the HRI camera image. [Grade: 8-10 | Topics: Scaled images; trigonometry; angle measure] (PDF)

Problem 371: Close Encounters of the Asteroid Kind!
On September 8, 2010 two small asteroids came within 80,000 km of Earth. Their small size of only 15 meters made them very hard to see without telescopes pointed in exactly the right direction at the right time. In this problem, based on a NASA press release, students use a simple formula to calculate the brightness of these asteroids from their distance and size. [Grade: 8-10 | Topics: Evaluating a base-10 power function; graphing; astronomical brightness scale] (PDF)

Problem 365: Meteorite Compositions: A matter of density
Astronomers collect meteorites to study the formation of the solar system 4.5 billion years ago. In this problem, students study the composition of a meteorite in terms of its density and mass, and the percentage of iron and olivine to determine the volumes occupied by each ingredient. [Grade: 8-10 | Topics: density; mass = density x volume; percentages] (PDF)

Problem 612: Exploring Power-laws: Meteor impacts
Students work with logarithmic functions, power-laws and explore the mass functiuon of meteors. [Grade: 9-12 | Topics: logarithmic functions; power-functions; logarithmic graphs] (PDF)

Problem 340: Computing the Orbit of a Comet Students use data from the orbit of Halley's Comet to determine the equation for its elliptical orbit. [Grade: 10-12 | Topics: ellipses; solving quadratic systems; fitting ellipse to data points ] (PDF)

Problem 338: Asteroids and Ice Students calculate how much ice may be present on the asteroid 24-Themis based on recent discoveries by NASA [Grade: 9-12 | Topics: mass=densityxvolume; volume of a spherical shell] (PDF)

Problem 333: Hubble: Seeing a Dwarf Planet Clearly Based on a recent press release, students use the published photos to determine the sizes of the smallest discernible features and compare them to the sizes of the 48-states in the USA. They also estimate the density of Pluto and compare this to densities of familiar substances to create a 'model' of Pluto's composition. A supplementary Inquiry Problem asks students to model the interior in terms of two components and estimate what fraction of Pluto is composed of rock or ice. [Grade: 8-12 | Topics: scales and ratios; volume of sphere; density=mass/volume] (PDF)

Problem 332: Hubble: The Changing Atmosphere of Pluto Based on a recent press release, students determine the aphelion and perihelion of Pluto's elliptical orbit using the properties of ellipses, then calculate the temperature of Pluto at these distances to estimate the thickness of Pluto's atmosphere and its changes during its orbit around the sun. [Grade: 10-12 | Topics: properties of ellipses; evaluating an algebraic function ] (PDF)

Problem 331: Webb Space Telescope: Detecting dwarf planets The 'JWST' will be launched some time in 2014. One of its research goals will be to detect new dwarf planets beyond the orbit of Pluto. In this problem, students use three functions to predict how far from the sun a body such as Pluto could be detected, by calculating its temperature and the amount of infrared light it emits. [Grade: 9-12 | Topics: Evaluating square-roots and base-e exponentials] (PDF)

Problem 326: Hubble Spies Colliding Asteroids Based on a recent press release, students calculate how often asteroids collide in the Asteroid belt using a simple formula. Students estimate belt volume, and asteroid speeds to determine the number of years between collisions. They also investigate how the collision time depends on the particular assumptions they made. An 'extra' integration problem is also provided for calculus students. [Grade: 8-12 | Topics: Volume of a thin disk; Algebra 1; Evaluating a definite integral; power-law] (PDF)

Problem 324: Deep Impact Comet Flyby The Deep Impact spacecraft flew by the Comet Tempel-1 in 2005. Students determine the form of a function that predicts the changing apparent size of the comet as viewed from the spacecraft along its trajectory. [Grade: 9-12 | Topics: Algebra, geometry, differential calculus] (PDF)

Problem 277: Deep Impact Comet Encounter Students learn about the Deep Impact experiment involving Comet Tempel-1, and how the path of an asteroid can be changed by using the Law of Conservation of Momentum. [Grade: 10-12 | Topics: Algebra; Scientific Notation; distance = speedxtime] (PDF)

Problem 255: Tempel-1 - Closeup of a Comet Students examine an image of the Comet Tempel-1 taken by the Deep Impact spacecraft to determine feature sizes and other details. [Grade: 6-8 | Topics: scales, proportions ] (PDF)

Problem 143: So..How big is it? - Asteroid Eros surface Students calculate the scale of an image of the surface of the asteroid Eros from the NEAR mission, and determine how big rocks and boulders are on its surface. [Grade: 4 - 7 | Topics: Scaling; multiplication, division; metric measure] (PDF)

Problem 116: The Comet Encke Tail Disruption Event On April 20, 2007 NASA's STEREO satellite captured a rare impact between a comet and the fast-moving gas in a solar coronal mass ejection. In this problem, students analyze a STEREO satellite image to determine the speed of the tail disruption event. [Grade: 8-10 | Topics:time calculation; finding image scale; calculating speed from distance and time] (PDF)

Problem 57: Asteroids and comets and meteors - Oh My! - Astronomers have determined the orbits for over 30,000 minor planets in the solar system, with hundreds of new ones discovered every year. Working from a map of the locations of these bodies within the orbit of Mars, students will calculate the scale of the map, and answer questions about the distances between these objects, and the number that cross earth's orbit. A great, hands-on introduction to asteroids in the inner solar system! Links to online data bases for further inquiry are also provided. [Grade level: 4-6 | Topics: Scale model; Decimal math; Interpreting 2-D graph] (PDF)

Problem 542:The Late Heavy Bombardment Era
Students estimate the average arrival time of large asteroids that impacted the moon. They work with the formula for the volume of a sphere to estimate how much additional mass was added to the moon and Earth durung this era. [Grade: 6-8 | Topics: volume of spheres; proportions] (PDF)

Problem 541:How to Build a Planet
Students study planet growth by using a clay model of planetessimals combining to form a planet by investigating volume addition with spheres. [Grade: 3-5 | Topics: graphing; counting] (PDF)

Problem 496: How to Grow a Planet or a Rain Drop
Students use calculus to slove for the growth in mass of a body, and solve the equation for the case of a raindrop and a planet like Earth. [Grade: 12 | Topics: Solving a simple differential equation.] (PDF)

Problem 464: Big Moons and Small Planets
Students work with a scaled drawing of 26 large moons in the solar system, and together with an exercise in using simple fractions, explore the relative sizes of the moons compared to Earth. [Grade: 3-5 | Topics: scale models; proportions; fractions] (PDF)

Problem 305: From Asteroids to Planetoids Students explore how long it takes to form a small planet from a collection of asteroids in a planet-forming disk of matter orbiting a star. [Grade: 11-12 | Topics: Integral calculus] (PDF)

Problem 304: From Dust Balls to Asteroids Students calculate how long it takes to form an asteroid-sized body using a simple differential equation. [Grade: 11-12 | Topics: Integral Calculus] (PDF)

Problem 303: From Dust Grains to Dust Balls Students create a model of how dust grains grow to centimeter-sized dust balls as part of forming a planet. [Grade: 11-12 | Topics: Integral Calculus] (PDF)

Problem 302: How to Build a Planet from the Inside Out Students model a planet using a spherical core and shell with different densities. The goal is to create a planet of the right size, and with the correct mass using common planet building materials. [Grade: 9-11 | Topics: Geometry; volume; scientific notation; mass=density x volume] (PDF)

Problem 249: Spotting an Approaching Asteriod or Comet Students work with a fundamental equation for determing the brightness of an asteroid from its size and distance from Earth. [Grade: 10-12 | Topics: Algebra 1, logarithms, area, scientific notation] (PDF)

Problem 59: Getting A Round in the Solar System! - How big does a body have to be before it becomes round? In this activity, students examine images of asteroids and planetary moons to determine the critical size for an object to become round under the action of its own gravitational field. Thanks to many Internet image archives this activity can be expanded to include dozens of small bodies in the solar system to enlarge the research data for this problem. Only a few example images are provided, but these are enough for the student to get a rough answer! [Grade level: 6-8 | Topics: Data analysis; decimals; ratios; graphing] (PDF)

## Planetary Atmospheres and Composition

Problem 545:Measuring Atmospheric Trace Gases Using Parts Per Million
Students convert from percentage units to parts per million and compare trace gases in the atmospheres of various planets. [Grade: 6-8 | Topics: percentages; unit conversions ] (PDF)

Problem 544:The Composition of Planetary Atmospheres
Students study the composition of planetary atmospheres and compare the amounts of certain compounds in them [Grade: 6-8 | Topics: Pie graphs; percentages; scientific notation] (PDF)

Problem 391: Investigating the atmosphere of Super-Earth GJ-1214b Students investigate a simple model for the interior of an exoplanet to estimate the thickness of its atmosphere given the mass size and density of the planet. [Grade: 6-8 | Topics: graphing functions; evaluating functions for given values; volume of a sphere; mass = densityxvolume] (PDF)

Problem 352: Exponential Functions and Atmospheric 'Scale heights'
A study of the way a planet's atmosphere changes as its temperature is changed using exponential functions. [Grade: 9-12 | Topics: Scientific Notation; evaluating exponential functions; Optional calculus] (PDF)

Problem 335: Methane Lakes on Titan Students use a recent Cassini radar image of the surface of Titan to estimate how much methane is present in the lakes that fill the image, and compare the volume to that of the fresh water lake, Lake Tahoe. [Grade: 6-8 | Topics: estimating irregular areas; calculating volume from area x height; scaled images ]

Problem 332: Hubble: The Changing Atmosphere of Pluto Based on a recent press release, students determine the aphelion and perihelion of Pluto's elliptical orbit using the properties of ellipses, then calculate the temperature of Pluto at these distances to estimate the thickness of Pluto's atmosphere and its changes during its orbit around the sun. [Grade: 10-12 | Topics: properties of ellipses; evaluating an algebraic function ]

Problem 278: Spitzer Studies the Distant Planet Osiris Students learn about the density of the planet HD209458b, also called Osiris, and compare it to that of Jupiter. [Grade: 8-10 | Topics: Spherical volumes; density; Scientific Notation;]

Problem 181: Extracting Oxygen from Moon Rocks Students use a chemical equation to estimate how much oxygen can be liberated from a sample of lunar soil. [Grade: 9-11| Topics: ratios; scientific notation; unit conversions]

Problem 124: The Moon's Atmosphere! Students learn about the moon's very thin atmosphere by calculating its total mass in kilograms using the volume of a spherical shell and the measured density. [Grade: 8-10 | Topics:volume of sphere, shell; density-mass-volume; unit conversions]

## Water and Habitable Zones

Problem 403: The Goldilocks Planets - Not too hot or cold Students use a table of the planets discovered by the Kepler satellite, and estimate the number of planets in our Milky Way galaxy that are about the same size as Earth and located in their Habitable Zones. They estimate the average temperature of the planets, and study their tabulated properties using histograms. [Grade: 6-8 | Topics: Averaging; histogramming] (PDF)

Problem 350: Estimating the Temperatures of Exoplanets
Students review the basic properties of ellipses by exploring the orbits of newly-discovered planets orbiting other stars. They also use a simple formula to determine the temperatures of the planets from their orbits.[Grade: 9-12 | Topics: Equation of ellipse; evaluating functions] (PDF)

Problem 349: Exoplanet Orbits and the Properties of Ellipses
Given the formula for the orbits of newly-discovered planets, students determine the basic properties of the elliptical orbits for the planets. [Grade: 9-12 | Topics: Properties of ellipses] (PDF)

Problem 338: Asteroids and Ice Students calculate how much ice may be present on the asteroid 24-Themis based on recent discoveries by NASA [Grade: 9-12 | Topics: mass=densityxvolume; volume of a spherical shell]

Problem 292: How Hot is That Planet? Students use a simple function to estimate the temperature of a recently discovered planet called CoRot-7b. [Grade: 8-10 | Topics: Algebra II; Evaluating Power functions]

Problem 287: LCROSS Sees Water on the Moon Students use information about the plume created by the LCROSS impactor to estimate the (lower-limit) concentration of water in the lunar regolith in a shadowed crater. [Grade: 9-12 | Topics: Geometry; volumes; mass=density x volume]

Problem 275: Water on the Moon! Students estimate the amount of water on the moon using data from Deep Impact/EPOXI and NASA's Moon Minerology Mapper experiment on the Chandrayaan-1 spacecraft. [Grade: 8-10 | Topics: Geometry, Spherical volumes and surface areas, Scientific notation]

Problem 267: Identifying Materials by their Reflectivity The reflectivity of a material can be used to identify it. This is important when surveying the lunar surface for minerals, and also in creating 'green' living environments on Earth. [Grade: 6-8 | Topics: percentage, interpreting tabular data, area ]

Problem 264: Water on Planetary Surfaces Students work with watts and Joules to study melting ice. [Grade: 8-10 | Topics: unit conversion, rates]

Problem 263: Ice or Water? Whether a planetary surface contains ice or liquid water depends on how much heat is available. Students explore the concepts of Specific heat and Latent Heat of Fusion to better understand the and quantify the energy required for liquid water to exist under various conditions. [Grade: 8-10 | Topics: unit conversion, scientific notation]

Problem 189: Stellar Temperature, Size and Power Students work with a basic equation to explore the relationship between temperature, surface area and power for a selection of stars. [Grade: 8-10| Topics: Algebra]

Problem 170: Measuring Star Temperatures Students use a simple formula to determine the temperatures of stars, and to use a template curve to analyze data for a specific star to estimate its temperature. [Grade: 6-8 | Topics: algebra, graph analysis]

Problem 121: Ice on Mercury? Since the 1990's, radio astronomers have mapped Mercury. An outstanding curiosity is that in the polar regions, some craters appear to have 'anomalous reflectivity' in the shadowed areas of these craters. One interpretation is that this is caused by sub-surface ice. The MESSENGER spacecraft hopes to explore this issue in the next few years. In this activity, students will measure the surface areas of these potential ice deposits an calculate the volume of water that they imply. [Grade: 8-10 | Topics:Area of a circle; volume, density, unit conversion]

## ESS1C: The History of Planet Earth

Problem 300: Does Anybody Really Know What Time It Is? Students use tabulated data for the number of days in a year from 900 million years ago to the present, to estimate the rate at which an Earth day has changed using a linear model. [Grade: 4-8 | Topics: Graphing; Finding Slopes; forecasting]

Problem 543:Timeline for Planet Formation
Students calculate time intervals in millions and billions of years from a timeline of events [Grade: 3-5 | Topics: time calculations; integers] (PDF)

Problem 542:The Late Heavy Bombardment Era
Students estimate the average arrival time of large asteroids that impacted the moon. They work with the formula for the volume of a sphere to estimate how much additional mass was added to the moon and Earth durung this era. [Grade: 6-8 | Topics: volume of spheres; proportions] (PDF)

Problem 541:How to Build a Planet
Students study planet growth by using a clay model of planetessimals combining to form a planet by investigating volume addition with spheres. [Grade: 3-5 | Topics: graphing; counting] (PDF)

Problem 305: From Asteroids to Planets Students explore how long it takes to form a small planet from a collection of asteroids in a planet-forming disk of matter orbiting a star based on a very simple physical model. [Grade: 11-12 | Topics: Integral calculus]

Problem 304: From Dust Balls to Asteroids Students calculate how long it takes to form an asteroid-sized body using a simple differential equation based on a very simple physical model. [Grade: 11-12 | Topics: Integral Calculus]

Problem 303: From Dust Grains to Dust Balls Students create a model of how dust grains grow to centimeter-sized dust balls as part of forming a planet based on a very simple physical model. [Grade: 11-12 | Topics: Integral Calculus]

Problem 302: How to Build a Planet from the Inside Out Students model a planet using a spherical core and shell with different densities. The goal is to create a planet of the right size, and with the correct mass using common planet building materials. [Grade: 9-11 | Topics: Geometry; volume; scientific notation; mass=density x volume]

Problem 60: When is a planet not a planet? In 2003, Dr. Michael Brown and his colleagues at CalTech discovered an object nearly 30% larger than Pluto, which is designated as 2003UB313. It is also known unofficially as Xenia, after the famous Tv Warrior Princess! Is 2003UB313 really a planet? In this activity, students will examine this topic by surveying various internet resources that attempt to define the astronomical term 'planet'. How do astronomers actually assign names to astronomical objects? Does 2003UB313 deserve to be called a planet, or should it be classified as something else? What would the new classification mean for asteroids such as Ceres, or objects such as Sedna, Quaoar and Varuna? [Grade level: 6-8 | Topics: Non-mathematical essay; reading to be informed]

Problem 59: Getting A Round in the Solar System! How big does a body have to be before it becomes round? In this activity, students examine images of asteroids and planetary moons to determine the critical size for an object to become round under the action of its own gravitational field. Thanks to many Internet image archives this activity can be expanded to include dozens of small bodies in the solar system to enlarge the research data for this problem. Only a few example images are provided, but these are enough for the student to get a rough answer! [Grade level: 6-8 | Topics: Data analysis; decimals; ratios; graphing]

Problem 8: Making a Model Planet Students use the formula for a sphere, and the concept of density, to make a mathematical model of a planet based on its mass, radius and the density of several possible materials (ice, silicate rock, iron, basalt). [Grade: 7 - 9 | Topics: Volume of sphere; mass = density x volume; decimal math; scientific notation]

## ESS2A: Earth Materials and Systems

Problem 673:VAB - An Improved Model for Van Allen Belt Radiation Dose Students use a detailed model of the path of a satellite and the radiation dose rate along the path to calculate the total radiation dose to the spacecraft. [Grade: 9-11 | Topics: Polynomial equations; trigonometric equations; composite finctions f(f(x)); estimating areas under curves] (PDF)

Problem 672:VAB - Modeling the Radiation Dose of the Van Allen Probes Students create a simple mathematical model of the radiation exposure to the VABP satellites as they travel through the Van Allen belts. [Grade: 11-12 | Topics: Parametric equations;composite functions f(g(x)); integral calculus ] (PDF)

Problem 671:VAB - The Van Allen Probes and Radiation Dose Students study radiation dose units and estimate the exposures for a human living on the gropund; an astronaut in the ISS, and the Van Allen belt environment. [Grade: 8-10 | Topics: Unit conversion; rates] (PDF)

Problem 669: VAB - Exploring the Third Belt with the Van Allen Probes Students use the elliptical equation for the orbit of NASAs Van Allen Probes spaccecraft, and a circle representing the location of the new Van Allen belt, to find where they intersect along the orbit of the spacecraft. [Grade: 9-12 | Topics: Intersection points of circles and ellipses; graphical and algebraic solutions] (PDF)

Problem 668: Meteor Impacts ? How Much Stuff? Students integrate a logarithmic function to estimate the number of tons of meteoritic debris that Earth collects every year. [Grade: 12 | Topics: Integral calculus] (PDF)

Problem 667: Exploring Power-laws: Meteor impacts Students estimate a function for logarithmic data that describes the number of meteor impacts on Earth every year. [Grade: 9-12 | Topics: logarithmic graphs; power laws; linear equations] (PDF)

Problem 666: SAGE - The Ground Track of the International Space Station Students determine how many sunrises and sunset the ISS observes every day. [Grade: 6-8 | Topics: Working with proportions; time calculations] (PDF)

Problem 662: SAGE- Measuring Aerosol Concentration in Parts per Million
Students learn about parts-per-million units by working with percentage and counting squares in different types of grids. [Grade: 6-8 | Topics: Unit conversion; integer counting] (PDF)

Problem 661: SAGE- Measuring Stratospheric Ozone with SAGE-III
Students use a data graph to identify the ozone layer from its concentration of ozone, and use parts-per-million to compare ozone concentration to the atmosphere density. [Grade: 6-8 | Topics: Unit conversion; reading a data graph ] (PDF)

Problem 660: SAGE- Some Basic Properties of the SAGE-III Instrument
Students examine the mass, data, pointing accuracy and power of the SAGE-III instrument and use unit conversions to translate the units into pounds, watts and degrees. [Grade: 6-8 | Topics: Unit conversion; proportions ] (PDF)

Problem 659: VAP- Exploring the Outer Atmosphere ? Gas Density
Students estimate examine the density of gas in the Van Allen belts and use it to estimate how many atoms the Van Allen Probes will encounter. [Grade: 6-8 | Topics: scientific notation; scale model; number = density x volume; volume = area x length; length=speed x time. ] (PDF)

Problem 657: VAP- Exploring the Density of Gas in the Atmosphere
Students examine different ways to represent the density of Earths atmosphere. [Grade: 6-8 | Topics: Scientific notation; density ] (PDF)

Problem 656: VAP- Measuring Earths Magnetic Field in Space
Students work with satellite data to explore Earths magnetic field through graphing data and comparing it with inverse-square and inverse-cube laws. [Grade: 9-12 | Topics: minimum and maximum; graphing data; comparing with models of the form 1/r2 and 1/r3 ] (PDF)

Problem 655: VAP- Estimating the Total Mass of the Van Allen Belts
Students estimate the total mass of the van Allen Belts and compare it to the mass of a donut using the formula for a torus. [Grade: 9-12 | Topics:Volume of torus; scientific notation; mass = density x volume ] (PDF)

Problem 654: VAP- Exploring the Donut-shaped Van Allen Belts
Students estimate the volume of the van Allen Belts in terms of the volume of Earth using a formula for the volume of a torus. [Grade: 9-12 | Topics: Scientific notation; volume of spheres and toriods] (PDF)

Problem 653: VAP-How to Use the RBSP Spacecraft to Measure the Mass of Earth!
Students use a formula to estimate the mass of Earth from data about the orbit of the Van Allen Probes spacecraft. [Grade: 9-12 | Topics: scientific notation; solving formula with integer exponents ] (PDF)

Problem 649: VAP- Electricity from Sunlight: The RBSP Spacecraft Solar Panels
Students work with the area of rectangles to calculate the electrical power produced by solar panels. [Grade: 3-5 | Topics: area of a rectangle; decimal math; unit conversion] (PDF)

Problem 648: SAGE- Using Opacity to Find Aerosol Density
Students examine a mathematical model based on the SAGE geometry and see how it leads to solving a system of linear equations to determine aerosol concentrations at different altitudes. [Grade: 6-8 | Topics: solving a system of linear equations; scientific notation] (PDF)

Problem 647: SAGE- Investigating Opacity and Extinction
Students work with the properties of filters to prove that the product of exponentials leads to the sum of their exponents. [Grade: 9-12 | Topics: exponential functions; exponent math] (PDF)

Problem 646: SAGE- Air Quality Index and Aerosol Density
Students see how the Air Quality Index is related to the number of arosols per cubic meter. [Grade: 6-8 | Topics: density; scientific notation; volume of a sphere; density] (PDF)

Problem 645: SAGE- Exploring the Mass and Density of Aerosol Particles
Students explore the physical sizes of aerosol particles. With unit conversions they convert concentration units of micrograms/m3 to particles/m3. [Grade: 6-8 | Topics: Unit conversions; scientific notation; volume of a sphere; density] (PDF)

Problem 644: SAGE- A Scale Model of Aerosol Sizes
Students work with proportions and scale to create a scale model of aerosol particles. [Grade: 6-8 | Topics: unit conversion; metric units nano and micro] (PDF)

Problem 641: SAGE- A Study of Aerosol Extinction in the Stratosphere
Students work with a table of atmospheric extinction at different altitudes and latitudes to graph selected data and draw a straight line thrlough the graphed data to estimate the slope. They create a linear equation from the graph and use it to predict the extinction at a different altitude. [Grade: 6-8 | Topics: slope of a line; linear equations; forecasting] (PDF)

Problem 640: SAGE- Atmospheric Aerosols by Percentage
Students examine a table that lists the percentages of different aerosol types according to the location on Earth where they are produced. [Grade: 3-5 | Topics: percentages; interpreting tabular data] (PDF)

Problem 639: SAGE- Aerosol Sources in the Stratosphere
Students examine the sources for aerosols in the atmosphere and determine their percentage contributions based upon their individual rates given in megatons/year. [Grade: 6-8 | Topics: Rates; percentage; pie graphs] (PDF)

Problem 638: SAGE- Sunset and Sunrise Geometry
Students explore the tangent geometry used by the SAGE-III instrument, and work with chords to determine their lengths using the Pyhtagorean formula. [Grade: 9-12 | Topics: Pythagorean Theorem; chord lengths] (PDF)

Problem 635: SAGE- Exploring Aerosols
Students compare aerosol sizes to a human hair, calculate volumes and masses from density. [Grade: 6-8 | Topics: density; volume; scale ] (PDF)

Problem 552:Cassini Sees Earth From Space - How Bright is it?
Students explore the logarithmic magnitude scale and estimate how bright Earth appears from Saturn as viewed in a recent Cassini image [Grade: 9-12 | Topics: logarithms; power laws ] (PDF)

Problem 551:Giving Particles a Boost in the van Allen Belts
Students examine a ball bouncing down a flight of stairs and compare this to how van Allen particles gain their energy from numerous small boosts. [Grade: 6-8 | Topics: equations; scientific notation] (PDF)

Problem 512: New NASA Satellite Takes Pictures of Salton Sea
Students work with image of agricultural area to estimate the percentage of area cultivated and the total rainfall in gallons per year. [Grade: 6-8 | Topics: area of square and rectangle; metric units; unit conversion] (PDF)

Problem 506: A New Belt for the Van Allen Belts
Students use a model of the orbit of the van Allen Belts Probes and simulated data to draw the locations of the three van Allen Belts in space. [Grade: 3-5 | Topics: Interpreting Tabular Data] (PDF)

Problem 502:The Frequency of Large Meteor Impacts
Students examine how often a large meteor should be visible like the one that exploded over Russia on February 14, 2013. This asteroid had a mass of 10,000 tons and injured over 1000 people. [Grade: 6-8 | Topics: percercentages, areas] (PDF)

Problem 486: RBSP Hears Dawn Chorus
Students explore the method of triangulation and how it might be used by the RBSP spacecraft to find the origin of the Chorus signals. [Grade: 6-8 | Topics: Graphing on the Cartesian plane; distances between points. ] (PDF)

Problem 484: Exploring Water Use in Kansas
Students use Landsat imagery from 1972 and 2011 to determine how much additional water is being used for irrigation in a small region of Kansas. [Grade: 6-8 | Topics: Area of a circle; unit conversions ] (PDF)

Problem 454: The Closest Approach of Asteroid 2005YU55 - III
Students work with the equation of a circle and line to find the orbit intersection points, midpoint, and closest distance to earth. [Grade: 8-10 | Topics: time=distance/speed; scale models; metric math] (PDF)

Problem 453: The Closest Approach of Asteroid 2005YU55 - II
Students work with the properties of circles and angular measure to see where the moon will be at the start of the asteroid encounter. [Grade: 6-8 | Topics: angular measure; time=distance/speed; scale models; metric math] (PDF)

Problem 452: The Closest Approach of Asteroid 2005YU55 - I
Students work with a scaled drawing of the orbit of the moon and the asteroid trajectory to predict where the asteroid will be relative to earth and the orbit of the moon. [Grade: 6-8 | Topics: time=distance/speed; scale models; metric math] (PDF)

Problem 449: A simple model for the origin of Earth's ocean water
Students create a simple model of the arrival of water to Earth using three sizes of cometary bodies and their arrival rates. [Grade: 8-10 | Topics: volume of a sphere; rates of change] (PDF)

Problem 448: The Declining Arctic Ice Cap During September
Students graph the change in Arctic ice surface area, and perform linear and quadratic regressions to model and forecast trends. [Grade: 9-11 | Topics: Statistics; Regression; graphing tabular data] (PDF)

Problem 447: The Arctic's Vanishing Ozone Layer
Students use ozone data for the Arctic region between 1979 and 2011 to graph the tabulated data, perform simple regression analysis, and forecast trends into the future. How much will there be in the year 2030? [Grade: 9-11 | Topics: Regression; graphing tabular data] (PDF)

Problem 446: Arctic Ozone Hole Continues to Grow in 2011
Students estimate the area of the Arctic ozone hole, and work with the concept of parts-per-million to estimate total ozone volume lost. [Grade: 6-8 | Topics: Area of rectangle; volume; percentage] (PDF)

Problem 409: The 2011 Japan Earthquake Rocks the Earth Using a simple physical model, students explore the principle by which the Japan Earthquake of 2011 caused Earth's rotation to spin up by 1.8 microseconds. [Grade: 9-12 | Topics: Algebra; evaluating an equation] (PDF)

Problem 408: Estimating the Speed of a Tsunami Students use the tsunami arrival times and earthquake start time for the devastating 2011 Japan Earthquake to estimate the speed of a tsunami as it crosses the Pacific Ocean to make landfall in Hawaii and California. [Grade: 6-8 | Topics: Time arithmetic; time zones; speed = distance/time] (PDF)

Problem 406: Growing Grapes in the Middle of the Desert Students use a dramatic Earth Observatory-1 satellite image of agriculture in Namibia to estimate the total cultivated area and water needs of grape growing under desert conditions [Grade: 6-8 | Topics: areas of irregular regions; unit conversion] (PDF)

Problem 375: Terra Satellite Measures Dangerous Dust
Students determine the number of dust particles inhaled by using a satellite map of the dust concentration and a calculation of the mass of a typical dust grain. [Grade: 8-10 | Topics: unit conversion; scientific notation; mass=densityxvolume] (PDF)

Problem 264: Water on Planetary Surfaces Students work with watts and Joules to study melting ice. [Grade: 8-10 | Topics: unit conversion, rates]

Problem 263: Ice or Water? Whether a planetary surface contains ice or liquid water depends on how much heat is available. Students explore the concepts of Specific heat and Latent Heat of Fusion to better understand the and quantify the energy required for liquid water to exist under various conditions. [Grade: 8-10 | Topics: unit conversion, scientific notation]

Problem 254: Solar Insolation Changes and the Sunspot Cycle Students compare changes in the amount of solar energy reaching earth with the 11-year sunspot cycle to predict the impact on designing a photovoltaic system for a home. [Grade: 8-10 | Topics: graph analysis, correlations, kilowatt, kilowatt-hours]

Problem 253: NASA 'Sees' Carbon Dioxide A satellite image of atmospheric carbon dioxide is used to estimate the geographic differences and identify human activity. [Grade: 6-8 | Topics: interpreting a data image, unit conversion, gigatons ]

Problem 252: Carbon Dioxide Increases Students study the Keeling Curve to determine the rates of increase of carbon dioxide in the atmosphere. [Grade: 6-8 | Topics: graph analysis, slope, rates, unit conversion, parts-per-million, gigatons]

Problem 223: Volcanos are a Blast: Working with simple equations- Students examine the famous Krakatoa explosion, asteroid impacts on the moon, and geysers on Enceladus using three equations that describe the height of the plume and initial velocity, to answer questions about the speed of the debris and terminal height. [Grade: 9-11 | Topics: Algebra I; significant figures.]

Problem 204: The Mass of the Van Allen Radiation Belts- Students graph some magnetic field lines in polar coordinates, then estimate the volume and mass of the Belts using the formula for a torus. [Grade: 9-12| Topics: Algebra II.]

Problem 201: Fly Me To the Moon!- Students learn some basic principles and terminology about how spacecraft change their orbits to get to the moon. [Grade: 6-8| Topics: speed = distance/time; Pythagorean Theorem]

Problem 161: Earth and Moon to Scale- Students create a scale model of the Earth-Moon system and compare with artistic renditions and actual NASA spacecraft images. [Grade: 4-6| Topics: Decimals; scaling and similarity]

Problem 151: Time Zone Math - Students learn about time zones and perform simple clock calculations using common United States and European time zones. [Grade: 3-5 | Topics: time units; addition, subtraction]

Problem 131 How Big is It? - Las Vegas up close. Students work with an image taken by the QuickBird imaging satellite of downtown Las Vegas, Nevada. They determine the image scale, and calculate the sizes of streets, cars and buildings from the image. [Grade: 4 - 7 | Topics:image scaling; multiply, divide, work with millimeter ruler]

Problem 123 A Trillion Here...A Trillion There Students learn to work with large numbers, which are the heart and soul of astronomical dimensions of size and scale. This activity explores the number 'one trillion' using examples drawn from the economics of the United States and the World. Surprisingly, there are not many astronomical numbers commonly in use that are as big as a trillion. [Grade: 5-9 | Topics:add, subtract, multiply, divide.]

Problem 71 Are the Van Allen Belts Really Deadly? - This problem explores the radiation dosages that astronauts would receive as they travel through the van Allen Belts enroute to the Moon. Students will use data to calculate the duration of the trip through the belts, and the total received dosage, and compare this to a lethal dosage to confront a misconception that Apollo astronauts would have instantly died on their trip to the Moon. [Grade level: 8-10 | Topics: decimals, area of rectangle, graph analysis]

Problem 68 An Introduction to Space Radiation - Read about your natural background radiation dosages, learn about Rems and Rads, and the difference between low-level dosages and high-level dosages. Students use basic math operations to calculate total dosages from dosage rates, and calculating cancer risks. [Grade level: 6-8 | Topics: Reading to be Informed; decimals, fractions, square-roots]

Problem 46 A Matter of Perspective. - Why can't we see aurora at lower latitudes on Earth? This problem will have students examine the geometry of perspective, and how the altitude of an aurora or other object, determines how far away you will be able to see it before it is below the local horizon. [Grade level: 9-11 | Topics: Geometric proofs]

Problem 37 Time Zone Mathematics. Students will learn about the time zones around the world, and why it is important to keep track of where you are when you see an astronomical phenomenon. A series of simple time calculations teaches students about converting from one time zone to another. [Grade: 5 - 7 | Topics: Time zone math]

Problem 36 The Space Station Orbit Decay and Space Weather Students will learn about the continued decay of the orbit of the International Space Station by studying a graph of the Station's altitude versus time. They will calculate the orbit decay rates, and investigate why this might be happening. [Grade: 5 - 8 | Topics: Interpreting graphical data; decimal math]

Problem 35 Exploring the Plasmasphere Students use an image of the plasmasphere obtained by the IMAGE satellite to calculate how fast it orbits the Earth. They will use this to determine whether gravity or Earth's magnetic field provides the forces responsible for its movement through space. [Grade: 7 - 9 | Topics: Geometry; ratios; decimal math; time arithmetic]

Problem 31 Airline Travel and Space Weather Students will read an excerpt from the space weather book 'The 23rd Cycle' by Dr. Sten Odenwald, and answer questions about airline travel during solar storms. They will learn about the natural background radiation they are exposed to every day, and compare this to radiation dosages during jet travel. [Grade: 6 - 8 | Topics: Reading to be informed; decimal math]

Problem 30 Exploring Earth's Magnetosphere [DOC] Students will examine a NASA website that discusses Earth's magnetosphere, and identify the definitions for key phenomena and parts to this physical system. They will write a short essay that describes, in their own words, how aurora are produced based on what they have read at the NASA site. [Grade: 6 - 8 | Topics: non-mathematical; reading to be informed; online research]

Problem 29 The Wandering Magnetic North Pole Mapmakers have known for centuries that Earth's magnetic North Pole does not stay put. This activity will have students read a map and calculate the speed of the 'polar wander' from 300 AD to 2000 AD. They will use the map scale and a string to measure the distance traveled by the pole in a set period of time and calculate the wander speed in km/year. They will answer questions about this changing speed. [Grade: 6 - 8 | Topics: Interpreting graphical data; speed = distance/time]

Problem 25 The Distance to Earth's Magnetopause Students use an algebraic formula and some real data, to calculate the distance from Earth to the magnetopause, where solar wind and magnetosphere pressure are in balance. [Grade: 8 - 10 | Topics: Evaluating a function with two variables; completing tabular entries]

Problem 22 The Auroral Oval Students learn that the aurora are observed as two 'halos' of light encircling the North and South Poles. Students use measurements made from two satellite images of the 'auroral ovals' to determine the diameter of the rings, and their approximate geographic centers - which are not at the geographic poles! [Grade: 5 - 7 | Topics: Finding the scale of an image; measurement; decimal math]

Problem 21 Exploring the Plasmasphere Students learn that the Pythagorean Theorem is more than a geometric concept. Scientists use a photograph taklen by the IMAGE satellite to measure the size of Earth's plasmasphere region using a ruler and protractor. [Grade: 7 - 9 | Topics: Finding the scale of an image; decimal math; measurement]

Problem 15 Radio Plasma Imaging with IMAGE Students use the Distance=VelocityxTime relationship to determine the distances to plasma clouds seen by the IMAGE satellite. [Grade: 6 - 8 | Topics: Polar graphs; time = distance x speed; decimal math]

Problem 13 Plasma Clouds Students use a simple 'square-root' relationship to learn how scientists with the IMAGE satellite measure the density of clouds of plasma in space. [Grade: 7 - 9 | Topics: Square-root; solving for X; evaluating a function]

Problem 12 The Ring Current Students use the formula for a disk to calculate the mass of the ring current surrounding Earth. [Grade: 7 - 9 | Topics: Volume of a disk; scientific notation; mass = density x volume]

Problem 11 How high is an aurora Students use the properties of a triangle to determine how high up aurora are. They also learn about the parallax method for finding distances to remote objects. [Grade: 5 - 8 | Topics: Geometery; angle measure]

Problem 10 The Life Cycle of an Aurora Students examine two eye-witness descriptions of an aurora and identify the common elements so that they can extract a common pattern of changes. [Grade: 4 - 6 | Topics: Creating a timeline from narrative; ordering events by date and time]

Problem 9 Aurora Power! Students use data to estimate the power of an aurora, and compare it to common things such as the electrical consumption of a house, a city and a country. [Grade: 5 - 7 | Topics: Interpreting tabular data]

Problem 5 The November 8, 2004 solar storm Students calculate the speed of a CME, and describe their aurora observations through writing and drawing. [Grade: 6 - 8 | Topics: Time calculations; distance = speed x time]

Problem 4 Sketching the Northern Lights Students read an account of an aurora seen by an observer, and create a drawing or painting based on the description. [Grade: 5 - 7 | Topics: non-mathematical art problem]

Problem 3 Magnetic Storms II Students learn about the Kp index using a bar graph. They use the graph to answer simple questions about maxima and time. [Grade: 6 - 8 | Topics: Interpreting bar graphs; time calculations]

Problem 1 Magnetic Storms I Students learn about magnetic storms using real data in the form of a line graph. They answer simple questions about data range, maximum, and minimum. [Grade: 7 - 9 | Topics: Interpreting a graph; time calculations]

## ESS2D: Weather and Climate

Problem 634: History of Winter - What is a Snowballs Chance on Mars?
Students explore the phase diagrams for water and carbon dioxide and discover whether astronauts would be able to create snowballs on mars made from carbon dioxide ice. [Grade: 9-12 | Topics: Graph analysis] (PDF)

Problem 630: History of Winter - Snow Density and Volume
Students learn how snow density is measured in the field using cylindrical instruments in a snow pit trench. [Grade: 6-8 | Topics: Density=mass/volume; metric units; decimal math] (PDF)

Problem 629: History of Winter - Snow to Water Ratios
Students learn how to convert between snow volume and equivalent volumes of liquid water. [Grade: 6-8 | Topics: Working with tables; decimal math; proportions] (PDF)

Problem 628: History of Winter - Snowflake Growth Rates and Surface Area
Students study change of scale and dilation by investigating showflake growth. [Grade: 6-8 | Topics: tabular data; rates of change; decimal math] (PDF)

Problem 627: History of Winter - The Surface Area of a Snowflake
Students estimate the area of a single snow flake using the areas of triangles and rectangles. [Grade: 6-8 | Topics: geometry; areas of triangles and rectangles; decimal math] (PDF)

Problem 626: History of Winter - Graphing a Showflake using Symmetry
Students use a simple plotting exercise and reflection symmetry to create a snowflake. [Grade: 6-8 | Topics: geometry; symmetry; plotting points on a Cartesian graph] (PDF)

Problem 625: SCOOL-Cloud Droplets and Rain Drops
Students [Grade: 6-8 | Topics: Volume of a sphere; scientific notation ] (PDF)

Problem 624: SCOOL-Cloud Cover, Albedo, Transmission and Opacity
Students explore the concepts of albedo, transmission and opacity for clouds. [Grade: 9-12 | Topics: logarithmic functions; percentage] (PDF)

Problem 623: SCOOL-Cloud Cover and Solar Radiation
Students examine the relationship between percentage cloud cover and the amount of sunlight that reaches the ground. [Grade: 6-8 | Topics: Graph analysis; evaluating functions] (PDF)

Problem 622: SCOOL-How Clouds Form - Working with Dew Points and Rates of Change
Students learn about the dew point and how clouds form from humid, cooling air. [Grade: 6-8 | Topics: Percentage; rates of change ] (PDF)

Problem 621: SCOOL-Working with Rainfall Rates and Water Volume
Students learn about rain fall rates and how to convert them into the volume of water that falls. [Grade: 9-12 | Topics: scientific notation; rates of change ] (PDF)

Problem 620: SCOOL-Estimating the Mass of a Cloud
Students use the relationship between volume and density to estimate the mass of a common cumulus cloud. [Grade: 6-8 | Topics: VOlume of a sphere; scientific notation; mass = density x volume] (PDF)

Problem 619: SCOOL-Using Proportions to Estimate the Height of a Cloud
Students use the method of triangulation to determine the height of a cloud. [Grade: 6-8 | Topics: geometry of right triangles; proportions] (PDF)

## ESS3B: Natural Hazards

Problem 631: History of Winter - Snow Density, Mass and Roof Failure
Students [Grade: 6-8 | Topics: Density = mass/volume; rates of change; proportions] (PDF)

Problem 621: SCOOL-Working with Rainfall Rates and Water Volume
Students learn about rain fall rates and how to convert them into the volume of water that falls. [Grade: 9-12 | Topics: scientific notation; rates of change ] (PDF)

Problem 316: Counting Craters on the Hubble Space Telescope Students count craters on a piece of the Wide Field Planetary Camera recovered from the Hubble Space Telescope in 2009. They determine the cratering rate and use this to predict how many impacts the solar panels on the International Space Station experiences each day. [Grade: 6-9 | Topics: Counting; Area; density]

Problem 243: ISS - Orbit Altitude Changes Students read an essay describing the increases and decreases in the International Space Station orbit, and calculate the final orbit altitude after all the changes are applied. [Grade: 8-10 | Topics: combining positive and negative mixed numbers; fractions]

Problem 238: Satellite Drag and the Hubble Space Telescope Satellite experience drag with the atmosphere, which eventually causes them to burn up in the atmosphere. Students study various forecasts of the althtiude of the Hubble Space Telescope to estimate its re-entry year. [Grade: 8-10 | Topics: interpreting graphical data; predicting trends]

Problem 179: Is There a Lunar Meteorite Impact Hazard? - Students work with areas, probability and impact rates to estimate whether lunar colonists are in danger of meteorite hazards. [Grade: 5-7| Topics: Area; unit conversions; rates]

Problem 95 A Study on Astronaut Radiation Dosages in SPace - Students will examine a graph of the astronaut radiation dosages for Space Shuttle flights, and estimate the total dosages for astronauts working on the International Space Station. [Grade level: 9-11 | Topics:Graph analysis, interpolation, unit conversion]

Problem 93 An Introduction to Radiation Shielding - Students calculate how much shielding a new satellite needs to replace the ISO research satellite. Students use a graph of the wall thickness versus dosage, and determine how thick the walls of a hollow cubical satellite have to be to blackuce the radiation exposure of its electronics. Students calculate the mass of the satellite and the cost savings by using different shielding. [Grade level: 9-11 | Topics: Algebra; Volume of a hollow cube; unit conversion]

Problem 89 Atmospheric Shielding from Radiation- III - This is Part III of a 3-part problem on atmospheric shielding. Students use exponential functions to model the density of a planetary atmosphere, then evaluate a definite integral to calculate the total radiation shielding in the zenith (straight overhead) direction for Earth and Mars. [Grade level: 11-12 | Topics: Evaluating an integral, working with exponential functions]

Problem 88 Atmospheric Shielding from Radiation- II - This is the second of a three-part problem dealing with atmospheric shielding. Students use the formula they derived in Part I, to calculate the radiation dosage for radiation arriving from straight overhead, and from the horizon. Students also calculate the 'zenith' shielding from the surface of Mars. [Grade level: 9-11 | Topics: Algebra I; evaluating a function for specific values]

Problem 87 Atmospheric Shielding from Radiation- I - This is the first part of a three-part problem series that has students calculate how much radiation shielding Earth's atmosphere provides. In this problem, students have to use the relevant geometry in the diagram to determine the algebraic formula for the path length through the atmosphere from a given location and altitude above Earth's surface. [Grade level: 9-11 | Topics: Algebra II, trigonometry]

Problem 83 Luner Meteorite Impact Risks - In 2006, scientists identified 12 flashes of light on the moon that were probably meteorite impacts. They estimated that these meteorites were probably about the size of a grapefruit. How long would lunar colonists have to wait before seeing such a flash within their horizon? Students will use an area and probability calculation to discover the average waiting time. [Grade level: 8-10 | Topics: arithmetic; unit conversions; surface area of a sphere) ]

Problem 76 Radon Gas in the Basement - This problem introduces students to a common radiation problem in our homes. From a map of the United States provided by the US EPA, students convert radon gas risks into annual dosages. [Grade level: 6-8 | Topics: Unit conversion, arithmetic operations]

Problem 673:VAB - An Improved Model for Van Allen Belt Radiation Dose Students use a detailed model of the path of a satellite and the radiation dose rate along the path to calculate the total radiation dose to the spacecraft. [Grade: 9-11 | Topics: Polynomial equations; trigonometric equations; composite finctions f(f(x)); estimating areas under curves] (PDF)

Problem 672:VAB - Modeling the Radiation Dose of the Van Allen Probes Students create a simple mathematical model of the radiation exposure to the VABP satellites as they travel through the Van Allen belts. [Grade: 11-12 | Topics: Parametric equations;composite functions f(g(x)); integral calculus ] (PDF)

Problem 671:VAB - The Van Allen Probes and Radiation Dose Students study radiation dose units and estimate the exposures for a human living on the gropund; an astronaut in the ISS, and the Van Allen belt environment. [Grade: 8-10 | Topics: Unit conversion; rates] (PDF)

Problem 558:How Quickly are NEOs Being Discovered?
Students work with data presented in bar graphs to estimate how many more hazardous Near Earth Objects (NEOS) remein to be found. [Grade: 6-8 | Topics:bar graphs ] (PDF)

Problem 557:The 10000th Near Earth Asteroid 2013 MZ5
Students graph tabulated data to determine when this asteroid is closest to Earth and its speed at that time. [Grade: 6-8 | Topics: grapohing tabuklar data; solving a linear equation] (PDF)

Problem 502:The Frequency of Large Meteor Impacts
Students examine how often a large meteor should be visible like the one that exploded over Russia on February 14, 2013. This asteroid had a mass of 10,000 tons and injured over 1000 people. [Grade: 6-8 | Topics: percercentages, areas] (PDF)

Problem 454: The Closest Approach of Asteroid 2005YU55 - III
Students work with the equation of a circle and line to find the orbit intersection points, midpoint, and closest distance to earth. [Grade: 8-10 | Topics: time=distance/speed; scale models; metric math] (PDF)

Problem 453: The Closest Approach of Asteroid 2005YU55 - II
Students work with the properties of circles and angular measure to see where the moon will be at the start of the asteroid encounter. [Grade: 6-8 | Topics: angular measure; time=distance/speed; scale models; metric math] (PDF)

Problem 452: The Closest Approach of Asteroid 2005YU55 - I
Students work with a scaled drawing of the orbit of the moon and the asteroid trajectory to predict where the asteroid will be relative to earth and the orbit of the moon. [Grade: 6-8 | Topics: time=distance/speed; scale models; metric math] (PDF)

Problem 414: Radiation Dose and Distance Students explore the dosimetry from the Japan 2011 Earthquake and graph the decline of the radiation dose rates with distance from the nuclear reactors. [Grade: 9-12 | Topics: unit conversions; amount=rate x time; graphing Log-Log data] (PDF)

Problem 413: Exploring Nuclear Decay and Radiation Dose Students compare the dose rates measured from the same location in Japan on two different days, then determine the half-life of the radioisotope causing the radiation exposure by comparing the derived half-life with those of Cesium-137 and Iodine-131. [Grade: 9-12 | Topics: unit conversions; amount=rate x time; Solving exponential equations in base-e] (PDF)

Problem 412: Radiation Dose and Dose Rate Students use radiation measurements across Japan to calculate the total absorbed doses from the 2011 nuclear reactor failures. They also calculate the total dose for passenger trips on jets and the Concorde. [Grade: 6-8 | Topics: unit conversions; amount=rate x time] (PDF)

Problem 411: Lifestyles and Radiation Dose Students see how the kind of lifestyle you lead determines most of your annual absorbed radiation dose. Some factors are under your control, and some are not. [Grade: 6-8 | Topics: unit conversions; amount=rate x time] (PDF)

Problem 410:Exploring Radiation in your Life Students use a pie grap to calculate the total dose and dose rate for various factors that determine your annual radiation exposure while living on Earth. [Grade: 6-8 | Topics: unit conversions; amount=rate x time] (PDF)

Problem 409: The 2011 Japan Earthquake Rocks the Earth Using a simple physical model, students explore the principle by which the Japan Earthquake of 2011 caused Earth's rotation to spin up by 1.8 microseconds. [Grade: 9-12 | Topics: Algebra; evaluating an equation] (PDF)

Problem 408: Estimating the Speed of a Tsunami Students use the tsunami arrival times and earthquake start time for the devastating 2011 Japan Earthquake to estimate the speed of a tsunami as it crosses the Pacific Ocean to make landfall in Hawaii and California. [Grade: 6-8 | Topics: Time arithmetic; time zones; speed = distance/time] (PDF)

Problem 375: Terra Satellite Measures Dangerous Dust
Students determine the number of dust particles inhaled by using a satellite map of the dust concentration and a calculation of the mass of a typical dust grain. [Grade: 8-10 | Topics: unit conversion; scientific notation; mass=densityxvolume] (PDF)

Problem 223: Volcanos are a Blast: Working with simple equations- Students examine the famous Krakatoa explosion, asteroid impacts on the moon, and geysers on Enceladus using three equations that describe the height of the plume and initial velocity, to answer questions about the speed of the debris and terminal height. [Grade: 9-11 | Topics: Algebra I; significant figures.]

## ESS3C: Human Impacts on Earth Systems

Problem 342: The Rate of Oil Leakage in the Gulf Oil Spill of 2010 Students use still images from a video of the oil emitted by the leaking British Petrolium oil well in the Gulf of Mexico to estimate the rate of oil leakage in gallons per day. [Grade: 6-8 | Topics: unit conversions; rates; image scale]

Problem 341: Recent Events: A Perspective on Carbon Dioxide Students compare the carbon dioxide generated by the 2010 Icelandic volcano and the Gulf Oil Spill to see the relative contributions to the atmosphere of a natural and man-made catastrophe. [Grade: 6-8 | Topics: unit conversions; rates ]

Problem 339: Terra Satellite Spies the Great Gulf Oil Catastrophe of 2010 Students use a Terra satellite image of the oil slick in the Gulf of Mexico to calculate its area, mass and thickness. [Grade: 6-8 | Topics: image scales; area of a circle; metric conversions ]

## ESS3D: Global Climate Change

Problem 397: The Changing Pace of Global Warming Students work with a table of global temperatures to forecast the temperature change by 2050 using a linear extrapolation. [Grade: 8-10 | Topics: Graphing tabular data; linear extrapolation; equation of a line y=mx+b] (PDF)

Problem 317: The Global Warming Debate and the Arctic Ice Cap Students use graphical data showing the area of the Arctic Polar Cap in September, and compare this to surveys of what people believe about global warming. Simple linear models are used to extrapolate when we will lose half of the Arctic polar cap, and when the belief in climate change will reach zero. [Grade: 9-11 | Topics: Modeling data with linear equations; forecasting]

Problem 293: Scientists Track the Rising Tide A graph of sea level rise since 1900 provides data for students to fit linear functions and perform simple forecasting for the year 2050 and beyond. [Grade: 8-10 | Topics: Linear equations and modeling data; forecasting]

Problem 271: A Simple Model for Atmospheric Carbon Dioxide Students work with the known sources of increasing and decreasiong carbon dioxide to create a simple model of the rate of change of atmospheric carbon dioxide. [Grade: 10-12 | Topics: Algebra I, rates of change, differential calculus]

Problem 270: Modeling the Keeling Curve with Excel Students create a mathematical model of the growth curve of atmospheric carbon dioxide using an Excel Spreadsheet, and create a future forecast for 2050. [Grade: 11-12 | Topics: Algebra II, properties of functions, Excel Spreadsheet]