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 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 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 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 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 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 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 314: Chandra Studies an Expanding Supernova Shell Using a millimeter ruler and a sequence of images of a gaseous shell between 2000 and 2007, students calculate the speed of the material ejected by Supernova 1987A. [Grade: 6-9 | Topics: Measuring; Metric Units; speed=distance/time]

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 307: The Elementary Particle Masses Students compare the masses and mass differences between elementary particles using units common to physics such as the electron Volt. [Grade: 9-12 | Topics: Scientific Notation; unit conversion]

Problem 306: Unit Conversions; Energy, Power and Flux Students work with more complicated unit conversions involving simple powers of quantities and mixed ratios. [Grade: 8-11 | Topics: unit conversions with compound units]

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 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 300: Earth's Rotation Changes and the Length of the Day? 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 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 297: Atoms - How Sweet They Are! A simple counting activity is based on atoms in a sugar molecule. Students calculate ratios and percentages of various atomic types in the molecule. [Grade: 4-8 | Topics: Counting; Ratios; percentage]

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 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 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 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]

Previous Problems:

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 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 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 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 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 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 282: Exploring the Ares 1-X Launch: The Hard Climb to Orbit Students learn about the energy required to send a payload into orbit by studying the Ares 1-X rocket launch. [Grade: 8-10 | Topics: Algebra II]

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 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 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]

Problem 276: Solid Rocket Boosters and Thrust Students learn how solid rocket boosters work, and calculate the SRB Thrust Curve using a simple geometric model and 'counting squares'.. [Grade: 8-10 | Topics: Geometry, Cylindrical volumes and surface areas, Graphing data]

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 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 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 271: A Simple Model for Atmospheric Carbon Dioxide Students work with the known sources of increasing and decreasing 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]

Problem 269: Parts Per Hundred (pph) Students work with a common unit to describe the number of objects in a population. Other related quantities are the part-per-thousand, part-per-million and part-per-billion. [Grade: 3-5 | Topics: counting, unit conversion]

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 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 the 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 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 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 260: Some Famous Unit Conversion Errors Students examine three famous unit conversion errors that led to catastrophic failures and near-death experiences. [Grade: 6-8 | Topics: unit conversion, metric measure]

Problem 259: Mare Nubium And Las Vegas Students compare 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 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 ]

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 251: Energy at Home Students explore watts and kilowatt-hours as measures of energy and energy consumption. [Grade: 6-8 | Topics: unit conversions; use of kilo and mega]

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 Asteroid 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 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 245: Solid Rocket Boosters Students learn how SRBs actually create thrust, and study the Ares-V booster to estimate its thrust. [Grade: 8-10 | Topics: volume, area, unit conversions]

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 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 242: Counting Atoms in Molecules Students count the number of atoms in a simple molecule and work out some basic fractions, percentages and masses. they also complete the chemical formula for the compound. [Grade: 3-6 | Topics: integers; counting similar things; fractions; percentages ]

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 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 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 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 altitude of the Hubble Space Telescope to estimate its re-entry year. [Grade: 8-10 | Topics: interpreting graphical data; predicting trends]

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

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 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]