## Math problems about Earth

**Problem 681:A Practical Application of Vector Dot and Cross Products **
Students work with coordinate vectors describing the corners of the roof of a house, calculate the area of the roof using dot products; calculate the normal vector to the roof using cross products; and the amount of sunlight striking the roof using dot products to determine how much solar power could be generated by solar panels on the roof.
[Grade: 10-12 | Topics: vectors; dot and cross product; normal vectors; unit conversions ]
[Click here]

**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 functions f(g(x)); estimating areas under curves]
[Click here]

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

**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 ground; an astronaut in the ISS, and the Van Allen belt environment.
[Grade: 8-10 | Topics: Unit conversion; rates]
[Click here]

**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]
[Click here]

**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]
[Click here]

**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]
[Click here]

**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]
[Click here]

**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]
[Click here]

**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 ]
[Click here]

**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 ]
[Click here]

**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. ]
[Click here]

**Problem 658: VAP- Exploring Gas Density in Space**

Students explore how gas density is related to the average distances between molecules in the air using a simple geometric mode of a cube with 64 cells.
[Grade: 6-8 | Topics: geometry; density=number/volume; scale models and proportions; scientific notation ]
[Click here]

**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 ]
[Click here]

**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 ]
[Click here]

**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 ]
[Click here]

**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]
[Click here]

**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 ]
[Click here]

**Problem 652: VAP- Telemetry Math**

Students work with data rates for the spacecraft and determine how much data needs to be stored.
[Grade: 6-8 | Topics: megabytes; rates in time]
[Click here]

**Problem 651: VAP- The RBSP Satellite: Working with Octagons **

Students work with the area formula for squares, rectangles and triangles to find the surface area of an octagonal satellite.
[Grade: 9-12 | Topics: areas of simple figures; algebraic manipulation]
[Click here]

**Problem 650: VAP- Working with Areas of Rectangles and Circles **

Students use the formulas for simple rectangle and circle areas to determine the areas of the holes in a satellite panel.
[Grade: 3-5 | Topics: area of a rectangle; area of a circle]
[Click here]

**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]
[Click here]

**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]
[Click here]

**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]
[Click here]

**Problem 646: SAGE- Air Quality Index and Aerosol Density**

Students see how the Air Quality Index is related to the number of aerosols per cubic meter.
[Grade: 6-8 | Topics: density; scientific notation; volume of a sphere; density]
[Click here]

**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]
[Click here]

**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]
[Click here]

**Problem 643: SAGE- The Sources and Sinks of Carbonyl Sulfide**

Students explore a molecule important in forming stratospheric aerosols. They calculate total rates of change from
a table of sources and sinks, and estimate the change in the number of molecules per year.
[Grade: 6-8 | Topics: Scientific notation; rates]
[Click here]

**Problem 642: SAGE-Three Mathematical Ways to Describe Light Extinction **

Studens explore the three common ways that scientists record extinction using base-10 and base-e functions.
[Grade: 9-12 | Topics: Base-10 and Base-e functions; exponential equations]
[Click here]

**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 through 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]
[Click here]

**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]
[Click here]

**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]
[Click here]

**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 Pythagorean formula.
[Grade: 9-12 | Topics: Pythagorean Theorem; chord lengths]
[Click here]

**Problem 637: SAGE-Light Attenuation Using Exponential Functions **

Students work with the extinction formula for light and see how light dimming is an exponetial process.
[Grade: 9-12 | Topics: exponential functions; natural logarithm, e]
[Click here]

**Problem 636: SAGE-Aerosols and Light Dimming **

Students explore how light is dimmed as it passes through a series of filters.
[Grade: 6-8 | Topics: percentage; multiplication ]
[Click here]

**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 ]
[Click here]

**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]
[Click here]

**Problem 633: History of Winter - Exploring Temperature and States of Matter **

Students learn how to read a simple phase diagram and how states of matter are related to temperature and pressure.
[Grade: 9-12 | Topics: Rates of Change; Unit conversions; decimal math ]
[Click here]

**Problem 632: History of Winter - Exploring Energy and Temperature **

Students learn about the relationship between temperature and the kinetic energy of particles.
[Grade: 9-12 | Topics: Evaluating equations]
[Click here]

**Problem 631: History of Winter - Snow Density, Mass and Roof Failure **

Students
[Grade: 6-8 | Topics: Density = mass/volume; rates of change; proportions]
[Click here]

**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]
[Click here]

**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]
[Click here]

**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]
[Click here]

**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]
[Click here]

**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]
[Click here]

**Problem 625: SCOOL-Cloud Droplets and Rain Drops **

Students
[Grade: 6-8 | Topics: Volume of a sphere; scientific notation ]
[Click here]

**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]
[Click here]

**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]
[Click here]

**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 ]
[Click here]

**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 ]
[Click here]

**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]
[Click here]

**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]
[Click here]

**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]
[Click here]

**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]
[Click here]

**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 ]
[Click here]

**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]
[Click here]

**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]
[Click here]

**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]
[Click here]

**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]
[Click here]

**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) remain to be found.
[Grade: 6-8 | Topics:bar graphs ]
[Click here]

**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]
[Click here]

**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 ]
[Click here]

**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]
[Click here]

**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]
[Click here]

**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]
[Click here]

**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]
[Click here]

**Problem 499: The Equation of a Magnetic Field Line **

Students use calculus to determine the formula for a magnetic field line.
[Grade: 12 | Topics: differential and integral calculus; slope; parametric equations]
[Click here]

**Problem 498: The Slope of a Magnetic Field Line **

Students graph a magnetic field line in the First Quadrant, then calculate the
segment midpoints using the Midpoint Formula, and then draw tangent lines at each midpoint to determine compass direction.
[Grade: 7-8 | Topics: Graphing in the XY plane; midpoint formula; tangent lines to curves]
[Click here]

**Problem 497: Graphing a Magnetic Field Line **

Students plot points along a magnetic field line in the First Quadrant,
then use reflection symmetry to complete the field line shape in all four quadrants.
[Grade: 6-8 | Topics: graphing in XY plane; reflection symmetry]
[Click here]

**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. ]
[Click here]

**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 ]
[Click here]

**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]
[Click here]

**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]
[Click here]

**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]
[Click here]

**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]
[Click here]

**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]
[Click here]

**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]
[Click here]

**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]
[Click here]

**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]
[Click here]

**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]
[Click here]

**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]
[Click here]

**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]
[Click here]

**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]
[Click here]

**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]
[Click here]

**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]
[Click here]

**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]
[Click here]

**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]
[Click here]

**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]
[Click here]

**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]
[Click here]

**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]
[Click here] [Click here]

**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 ] [Click here] [Click here]

**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 ] [Click here]

**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] [Click here]

**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] [Click here]

**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] [Click here]

**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] [Click here]

**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] [Click here]

**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] [Click here]

**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 ] [Click here]

**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] [Click here]

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

**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] [Click here]

**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] [Click here]

**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 ] [Click here]

**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] [Click here]

**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] [Click here]

**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] [Click here]

**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.] [Click here]

**Problem 222: Kelvin Temperatures and Very Cold Things-**
Students convert from Celsius to Fahrenheit and to Kelvin using three linear equations.
[Grade: 5-8 | Topics: Evaluating simple linear equations for given values.] [Click here]

**Problem 220: The Many Faces of Energy-**
Students convert between several different energy units.
[Grade: 8-10 | Topics: Scientific notation; unit conversions.] [Click here]

**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] [Click here]

**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.] [Click here]

**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] [Click here]

**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] [Click here]

**Problem 166: The Dollars and Cents of Research -**
Students work with dollar amounts, hourly salary rates, percentages to explore various models of the cost of scientific research as seen by the individual scientist.
[Grade: 4-6 | Topics: percentages, decimal math, simple rates (e.g dollars/hour)] [Click here]

**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] [Click here]

**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] [Click here]

**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] [Click here]

**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 three centimeters/year.
[Grade: 7 - 10 | Topics:Simple linear equations] [Click here]

**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] [Click here]

**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.] [Click here]

**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] [Click here]

**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] [Click here]

**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] [Click here]

**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] [Click here]

**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) ] [Click here]

**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] [Click here]

**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] [Click here]

**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] [Click here]

**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] [Click here]

**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] [Click here]

**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] [Click here]

**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] [Click here]

**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] [Click here]

**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] [Click here]

**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] [Click here]

**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] [Click here]

**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] [Click here]

**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] [Click here]

**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] [Click here]

**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] [Click here]

**Problem 21
Exploring the Plasmasphere **
Students learn that the Pythagorean Theorem is more
than a geometric concept. Scientists use a photograph taken 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] [Click here]

**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] [Click here]

**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] [Click here]

**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] [Click here]

**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] [Click here]

**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] [Click here]

**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] [Click here]

**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] [Click here]

**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] [Click here]

**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] [Click here]

**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] [Click here]

**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] [Click here]