## Expressions & Equations

## Percentage, Scientific Notation, Unit Conversions

**Problem 586: Searching for Comets **

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

**Problem 545:Measuring Atmospheric Trace Gases Using Parts Per Million**

Students convert from percentage units to parts per million and compare trace gases in the atmospheres of various planets.
[Grade: 6-8 | Topics: percentages; unit conversions ]
(PDF)

**Problem 544:The Composition of Planetary Atmospheres**

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

**Problem 539:Visiting the Planets at the Speed of Light**

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

**Problem 538:How Big is Our Solar System?**

Students work with proportions to convert solar system distances into Astronomical Units for the 8 planets.
[Grade:6-8 | Topics: Proportions; unit conversions]
(PDF)

**Problem 446: Arctic Ozone Hole Continues to Grow in 2011 **

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

**Problem 444: Predicting the Transits of the Stars Kepler-16A and 16B from Tatooine - II **

Students determine how often the two stars Kepler 16 A and B will line up with Tatooine on
the same day of the year. [Grade: 6-8 | Topics: comparing two sequences of numbers; patterns, Least Common Multiple]
(PDF)

**Problem 407: Cryo-testing the Webb Space Telecope ISIM**
Students explore scaling by creating an enlarged geometric model of the ISIM to better
appreciate the small changes due to expansion and contraction
[Grade: 6-8 | Topics: scale models; proportions; unit conversion]
(PDF)

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

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

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

**Problem 345: How many stars are there?**

A starfield image taken by the 2MASS survey is analyzed to estimate how many stars are in the sky. [Grade: 6-8 | Topics: Scaling; unit conversion; angular measure]
(PDF)

**Problem 344: Hubble Spies an Asteroid - Yes it does move!**

The track of an asteroid in a Hubble image of a cluster of galaxies is analyzed to determine speed of the asteroid.[Grade: 6-8 | Topics: Scaling; unit conversion; speed=distance/time]
(PDF)

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

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

**Problem 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 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 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 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 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 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 235: Scientific Data: The gift that keeps on giving! ** Students learn about gigabytes and terabytes 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 195: Unit Conversions III-** Students work with more unit conversions and use them to solve a series of practical problems in science and solar energy.
[Grade: 6-10| Topics: unit conversions.]

**Problem 171: Unit Conversions II-** Students work with four unit conversion problems that are a bit tricky!
[Grade: 6-8 | Topics: unit conversions]

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

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

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

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

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

**Problem 82 ****Unit Conversions I - **
Students will use a number of obscure English units measures to convert from metric to English units and back, and answer some
unusual questions!
[Grade level: 6-8 | Topics: arithmetic; unit conversions involving 1 to 5 steps) ]

**Problem 67 ****Unit Conversion Exercises -**
Radiation dosages and exposure calculations allow students to
compare several different ways that scientists use to compare how radiation exposure is delive black
and accumulated over time.Like converting 'centimeters per sec' to 'kilometers per year' ,this
activity reinforces student Topics in converting from one set of units to another.
[Grade level: 6-8 | Topics: fractions, decimals, units]

**Problem 48 ****Scientific Notation - An Astronomical Perspective. - **
Astronomers use scientific notation because the numbers they work with are usually..astronomical in size. This
collection of problems will have students reviewing how to perform multiplication and division with large and small numbers, while
learning about some interesting astronomical applications. They will learn about the planet Osiris, how long it takes to
download all of
NASA's data archive, the time lag for radio signals to Pluto, and many more real-world applications.
[Grade level: 8-10 | Topics: Scientific notation; decimal math]

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

## Simple Equations

**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]

## Grade 6-8: Working with Equations and Formulae

**Problem 585: Exploring Comet Orbits **

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

**Problem 583: Buying a Telescope **

Students compare several telescopes and select the one with the best performance and lowest cost.
[Grade: 6-8 | Topics: simple ratio formula; decimal math]
(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 574: Telescope Light Gathering Ability - Seeing Faint Stars **

Students calculate the light gathering ability of various telescopes compared to the human eye.
[Grade: 6-8 | Topics: Area of a circle ]
(PDF)

**Problem 573: Calculating the Magnification of a Telescope **

Students fill in missing numbers in a table using proportions and evaluating a simple equation for magnification.
[Grade: 6-8 | Topics: proportions]
(PDF)

**Problem 571: Focal Lengths, Apertures and F/numbers **

Students learn about the basic terms that define the performance of a digital camera or a telescope.
[Grade: 6-8 | Topics: fractions; integer division; evaluating simple equations ]
(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 568: Ios Volcanoes and Resurfacing **

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

**Problem 561:Exploring the Evaporating Exoplanet HD189733b**

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

**Problem 559:Comet ISON Losing Mass as it Approaches the Sun.**

Students estimage how much mass the comet will loose at its present rate.
[Grade: 6-8 | Topics: volume of a sphere; rates; mass=density x volume]
(PDF)

**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 551:Giving Particles a Boost in the van Allen Belts**

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

**Problem 540:Travel Times by Spacecraft Around the Solar System**

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

**Problem 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]
(PDF)

**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]
(PDF)

**Problem 489: RBSP and the location of Dawn Chorus - III **

The location of the Chorus signal from each of the RBSP 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 480: The Expanding Gas Shell of U Camelopardalis **

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

**Problem 425: Exploring a Full-sized Black Hole **

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

**Problem 422: Supercomputers: Getting the job done FAST!**
Students use a simple counting problem to explore how much faster a supercomputer is compared to as hand-calculation.
[Grade: 6-8 | Topics: algebra]
(PDF)

**Problem 419: The Space Shuttle: Fly me to the moon?**
Students discuss the popular misconception that the Space Shuttle can travel to the moon by examining the required orbit speed change and the capacity of the Shuttle engines to provide the necessary speed changes.
[Grade: 6-8 | Topics: amount = rate x time ]
(PDF)

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

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

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

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

**Problem 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 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 357: The Fastest Sea Level Rise in the United States**

Global climate change is causing measurable sea level changes. Which part of the United States is sinking the fastest? [Grade: 6-8 | Topics: fitting linear equations to graphical data]
(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 206: Can You Hear me now? -** Students learn about how the transmission of data is affected by how far away a satellite is, for a variety of spacecraft in the solar system
[Grade: 6-8| Topics: multiplication; division; decimal numbers.]

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

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

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