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 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]

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 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 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 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 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 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 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 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 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 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 281: Exploring the Ares 1-X Launch: Energy Changes Students learn about kinetic and potential energy while studying the Ares 1-X rocket launch. [Grade: 8-10 | Topics: Algebra II]

Problem 280: Exploring the Ares 1-X Launch: Parametrics Students learn about parametric equations to determine the path of the Ares 1-X rocket. [Grade: 8-10 | Topics: Algebra II; Parametric Equations]

Problem 279: Exploring the Ares 1-X Launch: Downrange Distance Students learn about the path of the Ares 1-X test launch and calculate its downrange landing distance in the Atlantic Ocean. [Grade: 8-10 | Topics: Algebra; Significant Figures; Metric to English Conversion]

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 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]

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 246: Evaluating Secondary Physical Constants Students evaluate complicated algebraic quantities that define important constants in physics with both integer and fractional exponents. [Grade: 10-12 | Topics: Algebra; significant figures, scientific notation]

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 210: The Mathematics of Ion Rocket Engines- Students learn about the basic physics of ion engines, calculating speeds. [Grade: 9-12| Topics: Scientific Notation; Algebra II; evaluating formulae.]

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 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 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 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 188: Cross Sections and Collision Times - Students explore the relationship between density, speed and size in determining how quickly particles collide in a gas. [Grade: 9-11| Topics: Algebra; Area]

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 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 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 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 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 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 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 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 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 106 Oscillating Spheres Many astronomical bodies have a natural period of oscillation. In this problem, students will use a simple mathematical model to calculate the period of oscillation of a star, a planet, and a neutron star from the estimated densities of these bodies. [Grade: 9-11 | Topics:Algebra; calculating with a formula]

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 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 91 Compound Interest - Students use the 'compound interest' formula to examine rates of growth for space mission costs, and the salaries of astronomers, with allowance for inflation. [Grade level: 8-10 | Topics: Algebra II]

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 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 75 Parametric Functions and Substitution - The relationship between the strength of a solar storm and the resulting magnetic disturbance on Earth is given as a series of equations. Students are asked to create new formulae based on these parametric these equations using the method of substitution. [Grade level: 11-12 | Topics: Algebraic manipulation, integer exponents, scientific notation, significant figures and rounding ]

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 72 Systems of Equations in Space Science - This problem has students solve two problems involving three equations in three unknowns to learn about solar flares, and communication satellite operating power. [Grade level: 8-10 | Topics: decimals, solving systems of equations, matrix math, algebraic substitution ]

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 33 Magnetic Energy From B to V Students will use formulas for the volume of a sphere and cylinder, and magnetic energy, to calculate the total magnetic energy of two important 'batteries' for space weather phenomena- solar prominences and the Earth's magnetotail. This requires scientific notation, a calculator, and experience with algebraic equations with integer powers of 2 and 3. [Grade: 8 - 10 | Topics: Algebra I; volumes; decimal math; scientific notation]

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 18 Magnetic Forces and Particle Motion Students learn about the spiral-shaped trajectories of charged particles moving in magnetic fields, and calculate some basic properties of this 'cyclotron' motion. [Grade: 9 - 11 | Topics: Algebra I; evaluating a function; scientific notation]

Problem 17 Magnetic Forces and Kinetic Energy Students use the formula for the Kinetic Energy of a charged particle to calculate particle speeds for different voltages, and answer simple questions about lightning, aurora and Earth's radiation belts. [Grade: 6 - 8 | Topics: Square root; time=speed x distance; decimal math; significant figures]

Problem 14 Kinetic Energy and Particle Motion Students learn about kinetic energy and how this concept applies to charged particles. They calculate the speed of a particle for various particle energies. [Grade: 7 - 9 | Topics: Evaluating a function; square-roots; scientific notation]

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]

Geometry, Areas, Volumes

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

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 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 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 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 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 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]

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 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 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: 8-10 | Topics: Geometry; parallax; arithmetic]

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 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 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 65 A Perspective on Radiation Dosages - Depending on the kind of career you chose, you will experience different lifetime radiation dosages. This problem compares the cumulative dosages for someone living on Earth, an astronaut career involving travel to the Space Station, and the lifetime dosage of someone traveling to Mars and back. [Grade level: 6-8 | Topics: decimals, unit conversions, graphing a timeline, finding areas under curves using rectangles]

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]

Problem 19 An Application of the Pythagorean Theorem Students learn that the Pythagorean Theorem is more than a geometric concept. Scientists use it all the time when calculating lengths, speeds or other quantities. This problem is an introduction to magnetism, which is a '3-dimensional vector', and how to calculate magnetic strengths using the Pythagorean Theorem. [Grade: 8 - 10 | Topics: Squares and square-roots; Pythagorean Theorem in 3-D]

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]

Calculus, Limits

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 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]

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]

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 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 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 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 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 266: The Ares-V Cargo Rocket Students work with the equations for thrust and fuel loss to determine the acceleration curve of the Ares-v during launch. [Grade: 11-12 | Topics: Algebra II, properties of functions, differential calculus, Excel Spreadsheet]

Problem 265: Estimating Maximum Cell Sizes Students estimate tyhe maximum size of spherical cells based on the rates with which they create waste and remove it through their cell walls. [Grade: 11-12 | Topics: differential calculus, unit conversion]

Problem 234: Calculating Arc Lengths of Simple Functions- Students work with the differential form of the Pythagorean Theorem to determine the basic integral formula for arc length, then evaluate it for a parabola, logrithmic spiral and normal spiral. They evaluate the length of the spiral track on a CDrom. [Grade: 11-12 | Topics: Calculus; differential; integral, U-substitutions; significant figures.]

Problem 225: Areas Under Curves; An astronomical perspective- Students work with a bar graph of the number of planet discoveries since 1995 to evaluate the total discoveries, as areas under the graph, for various combinations of time periods. [Grade: 6-8 | Topics: Adding areas in bar graphs.]

Problem 208: Optimization- Students determine the optimal dimensions of an hexagonal satellite to maximize its surface area given its desiblack volume. [Grade: 9-12| Topics: Calculus; differentiation.]

Problem 202: The Dawn Mission - Ion Rockets and Spiral Orbits- Students determine the shape of the trajectory taken by a spacecraft using a constant-thrust ion motor using differential and integral calculus for arc lengths. [Grade: 9-12| Topics: Calculus - Arc lengths.]

Problem 193: Fluid Level in a Spherical Tank - Students explore the relationship between volume, and the height of fluid in a spherical tank as fluid is being drained at a constant rate. [Grade: 10-12| Topics: Algebra, differential calculus, related rates]

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

Problem 191: Why are hot things black? - 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 187: Differentiation- Students explore partial derivatives by calculating rates of change in simple equations taken from astrophysics. [Grade: 11-12| Topics: differentiation; 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 184: The Ant and the Turntable: Frames of reference - Students pblackict the motion of an ant crawling from the center of a spinning CDrom to the edge. They also use calculus to estimate the length of the spiral path seen by a stationary observer. [Grade: 11-12| Topics: integration; parametric equations; polar coordinates]

Problem 183: Calculating Arclengths of Simple Functions- Students determine the basic equation for arclength and its integral, and evaluate it for simple polar functions. [Grade: 11-12| Topics: calculus; integration; parametric equations]

Problem 169: The Limiting Behavior of Functions- Students work with two complex formulae to determine their limiting behavior as the independent variables approach infinity and zero. [Grade: 9-12 | Topics: Algebra II, pre-calculus]

Problem 157: Space Shuttle Launch Trajectory - I - Students use the parametric equation for the altitude and range for an actual Shuttle launch to determine the speed and acceleration of the Shuttle during launch and orbit insertionh [Grade: 11-12 | Topics: Algebra; Calculus; Parametric Equations; Differentiation

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 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) ]

Measurement

Problem 119 A Star Sheds a Comet Tail! The GALEX satellite captured a spectacular image of the star Mira shedding a tail of gas and dust nearly 13 light years long. Students use the GALEX image to determine the speed of the star, and to translate the tail structures into a timeline extending to 30,000 years ago. [Grade: 8-10 | Topics:Image scaling; Unit conversion; Calculating speed from distance and time]

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 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]

Data Analysis and Probability

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 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 120 Benford's Law Students will explore a relationship called Benford's Law, which describes the frequency of the integers 1-9 in various data. This law is used by the IRS to catch fradulent tax returns, but also applies to astronomical data and other surprising situations. [Grade: 8-10 | Topics:Calculating frequency tables; Histogramming; Statistics]

Problem 102 How fast does the sun rotate? Students will analyze consecutive images taken by the Hinode satellite to determine the sun's speed of rotation, and the approximate length of its 'day'. [Grade: 6-9 | Topics:image scales; time calculations; speed calculations, unit conversions]

Problem 90 A Career in Astronomy - This problem looks at some of the statistics of working in a field like astronomy. Students will read graphs and answer questions about the number of astronomers in this job area, and the rate of increase in the population size and number of advanced degrees. [Grade level: 6-8 | Topics: graph reading; percentages; interpolation]

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 69 Single Event Upsets in Aircraft Avionics - Radiation is problem for high-altitude commercial and research aircraft. Showers of high-energy neutrons cause glitches in computer electronics and other aircraft systems. This problem investigates the neutron background radiation at 30,000 to 100,000 feet based on actual flight data, and has students calculate how many computer memory glitches will happen over a set amount of flight time. [Grade level: 8-10 | Topics: decimals, unit conversions, graph analysis]

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 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 2 Satellite Surface Area Students calculate the surface area of an octagonal cylinder and calculate the power it would yield from solar cells covering its surface. [Grade: 7 - 9 | Topics: surface areas; hexagone; decimal math]

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]

Problem Solving

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 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 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 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 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 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 122 XZ Tauri's Super CME! Ordinarily, the SOHO satellite and NASA's STEREO mission spot coronal mass ejections (CMEs) but the Hubble Space Telescope has also spotted a few of its own...on distant stars! Students will examine a sequence of images of the young star XZ Tauri, and measure the average speed and density of this star's CME event between 1955 and 2000. [Grade: 8-10 | Topics:Calculate image scale; speed from distance and time; mass:volume:density]

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 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 66 Background Radiation and Lifestyles - Living on Earth, you will be subjected to many different radiation environments. This problem follows one person through four different possible futures, and compares the cumulative lifetime dosages. [Grade level: 6-8 | Topics: fractions, decimals, unit conversions]

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]