## Functions

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

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

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

** Problem 153: Number Sentence Puzzles -** Students are presented with three number sentences such
as 145 + N = 375, and asked to select which
'spacy' word problem they belong to.
[Grade: 3-4 | Topics: Number Topics and problem solving]

**Problem 149: Equations with One Variable -** Students solve formulas of the form 2001 = 1858 + 11x to find 'X'.
[Grade: 3-5 | Topics: addition, subtraction, multiplication, division; solving simple equations]

**Problem 24: Reading Between the Lines** Students solve simple equations for x, (like 2x + 3
= 5) to discover which words complete an essay on the causes of aurora,
and answer questions after reading the completed essay.
[Grade: 5 - 7 | Topics: solving for X; distributive law; associative law]

**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: graphing tabular data; solving a linear equation]
(PDF)

**Problem 580: Measuring Gravity with a Pendulum **

Students design pendulum clocks for mars and the moon, and how pendulums can be used for mining on Earth.
[Grade: 6-8 | Topics: evaluating square-root equations; scientific notation ]
(PDF)

**Problem 554:Exploring Volcanoes and Geysers Across the Solar System**

Students determine the ejection speed and heights of gasses vented by geysers and volcanoes.
[Grade: 6-8 | Topics: solving square root equations; ]
(PDF)

**Problem 489: The Van Allen Probes and the location of Dawn Chorus - III **

The location of the Chorus signal from each of the VAP spacecraft is given by a linear equation that represents the direction along which the signal is detected by each spacecraft. Students solve the two linear
equations for the common intersection point representing the location of the Chorus signal in space. This can be done graphically by plotting each linear equation, or solved algebraically.
[Grade: 6-8 | Topics: Linear equations; solving systems of equations; graphical solutions ]
(PDF)

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

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

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

**Problem 164: Equations with One Variable-** Students work with equations like '4.3 = 3.26D' to solve for D in a number of simple astronomical problems involving distances, speed and temperature conversion.
[Grade: 6-8 | Topics: equations in one variable; multiplication; division; decimals]

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

**Problem 654: VAP- Exploring the Donut-shaped Van Allen Belts **

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

**Problem 653: VAP-How to Use the VAP Spacecraft to Measure the Mass of Earth! **

Students use a formula to estimate the mass of Earth from data about the orbit of the Van Allen Probes spacecraft.
[Grade: 9-12 | Topics: scientific notation; solving formula with integer exponents ]
(PDF)

**Problem 648: SAGE- Using Opacity to Find Aerosol Density**

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

**Problem 647: SAGE- Investigating Opacity and Extinction**

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

**Problem 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]
(PDF)

**Problem 638: SAGE- Sunset and Sunrise Geometry **

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

**Problem 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]
(PDF)

**Problem 572: How Saturns Moon Mimas Created the Cassini Division **

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

**Problem 569: Orbit Speeds and Times for Saturns Rings **

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

**Problem 552:Cassini Sees Earth From Space - How Bright is it?**

Students explore the logarithmic magnitude scale and estimate how bright Earth appears from Saturn as viewed in a recent Cassini image
[Grade: 9-12 | Topics: logarithms; power laws ]
(PDF)

**Problem 547:The Rings of Saturn**

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

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

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

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

**Problem 383: Estimating the mass of Comet Hartley 2 using calculus.**

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

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

**Problem 478: The Grail and LRO Encounter in Lunar Orbit **

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

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

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

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

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

**Problem 501: Exploring the Most Distant Galaxies with Hubble**

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

**Problem 481: Pluto's Fifth Moon **

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

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

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

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

**Problem 472: Investigating Juno's Elliptical Transfer Orbit **

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

**Problem 396: Kepler 10b - A matter of gravity**
Students use the measured properties of the Earth-like planet Kepler 10b to estimate the weight of a human on its surface.
[Grade: 8-10 | Topics: Evaluating formulas; mass = density x volume; volume of a sphere; scientific notation]
(PDF)

**Problem 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 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 373: Solar Probe Plus - Having a hot time near the sun! **

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

**Problem 442: Modeling the Atmospheric Re-entry of UARS **

Students graph the altitude of the UARS satellite in the weeks before re-enrty to explore the accelerating
effects of atmospheric drag. They create a mathematical model that fits the data, and use this to make their own
prediction of the re-entry date. [Grade: 8 - 11 | Topics: graphing data; linear equations; exponential and power functions]
(PDF)

**Problem 381: The Cosmological Redshift - Changing the light from a galaxy.**

Students learn about the redshift unit of measurement in astronomy, and solve a simple linear equation to explore
how the light from very distant galaxies is reddened compared to nearby galaxies.
[Grade: 8-10 | Topics: solving a simple equation for X]
(PDF)

**Problem 380: Seeing the Distant Universe Clearly**

Students calculate the angular sizes and scales of distant objects to study how different sized
telescopes see details with varying degrees of clarity.
[Grade: 7-9 | Topics: solving a simple equation for X; angular measure; Scientific Notation]
(PDF)

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

**Problem 136 : Energy Generation near Black Holes**
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 501: Exploring the Most Distant Galaxies with Hubble**

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

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

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

**Problem 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]