## Elementary Mathematics and Arithmetic

**Quick Navigation** - click on the links below to jump to related math problems in this archive.

Integers | | Fractions and Mixed Numbers | | Working with Decimals | | Positive and Negative Numbers | | |

Simple Equations | | Basic Geometry | | Measurement | | Data Analysis and Probability | Problem Solving | |

## Integers

**Problem 537:A Solar Storm Number Puzzle**

Students solve 10 problems using positive and negative numbers, addition, subtraction and multiplication
to find the missing words in a short essay about solar storms.
[Grade:3-5 | Topics: integer arithmetic; positive and negative numbers]
(PDF)

**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]
(PDF)

**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 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 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: 3-5 | Topics: combining positive and negative mixed numbers; fractions]

**Problem 228: Nuclear Arithmetic ** Students use the equation N = A - Z to solve for A, Z or N given values for the other two variables.
[Grade: 4-6 | Topics: Evaluating a simple equation.]

**Problem 173: Groups, Clusters and Individuals ** Students determine the number of individual objects given the number of groups and the number of individuals in an average group for clusters of stars and galaxies.
[Grade: 3-5 | Topics: multiplication]

**Problem 20: A Space Science Crossword Puzzle** Students work with positive and negative numbers to
solve a crossword puzzle. The theme is 'Scientists use math to explore
Nature'. Good exercise for pre-algebra review of adding and subtracting
positive and negative numbers.
[Grade: 4 - 6 | Topics: Integer arithmetic; associative and distributive laws]

## Fractions and Mixed Numbers

**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]
(PDF)

**Problem 493: Fun with Gears and Fractions **

Students learn about how simple fractions are used to describe gears and gear trains that reduce or increase speed.
[Grade: 4-7 | Topics: multiplying simple fractions]
(PDF)

**Problem 465: Comparing Planets Orbiting other Stars **

Students use simple fraction arithmetic to determine the relative sizes of several new planets recently discovered by the Kepler mission,
and compare these sizes to that of Jupiter and Earth.
[Grade: 3-5 | Topics: scale models; proportions; fractions]
(PDF)

**Problem 464: Big Moons and Small Planets **

Students work with a scaled drawing of 26 large moons in the solar system, and together with an exercise in using simple
fractions, explore the relative sizes of the moons compared to Earth.
[Grade: 3-5 | Topics: scale models; proportions; fractions]
(PDF)

**Problem 347: More Molecular Madness!**

Students count the number of atoms in a molecule of ciprofloaxcin to determine its chemical formula and mass. [Grade: 3-5 | Topics: Counting; multiplication]
(PDF)

**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 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 230: Galaxy Distances and Mixed Fractions-** Students use the relative distances to nearby galaxies expressed in mixed numbers to determine distances between selected galaxies.
[Grade: 3-5 | Topics: Basic fraction math.]

**Problem 229: Atomic Numbers and Multiplying Fractions-** Students use a piece of the Periodic Table of the Elements to figure out the identities of atoms based on numerical clues expressed as mixed numbers.
[Grade: 3-5 | Topics: Basic fraction math; mixed numbers.]

**Problem 217: Fractions and Chemistry-** Students study simple chemical equations by using simple proportions and mixed numbers.
[Grade: 3-6 | Topics: Basic fraction math; ratios.]

**Problem 216: Atomic Fractions-** Students study the energy ladders of an atom and work out, using differences between mixed numbers, the energy gained
or lost by an electron as it moves up and down the ladder.
[Grade: 3-6 | Topics: Basic fraction math]

**Problem 215: More Atomic Fractions-** Students study the energy ladders of an atom and work out, using differences between mixed numbers, the energy gained
or lost by an electron as it moves up and down the ladder.
[Grade: 3-6 | Topics: Basic fraction math.]

**Problem 214: Atomic Fractions III-** Students study the energy ladders of an atom and work out, using differences between mixed numbers, the energy gained
or lost by an electron as it moves up and down the ladder.
[Grade: 3-6 | Topics: Basic fraction math.]

**Problem 180: Planets, Fractions and Scales-** Students work with relative planet comparisons to determine the actual sizes of the planets given the diameter of Earth.
[Grade: 4-6| Topics: scale models; decimals; fractions]

**Problem 165: Fractions in Space -** Students explore the many ways that simple fractions come up in the study of planetary motion.
[Grade: 3-5 | Topics: working with fractions; time calculations]

**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)]

## Time

**Problem 174: A Matter of Timing -** Students study satellites of Saturn to work out graphically how often they will line up.
[Grade: 3-6| Topics: scale model; time calculation; fractions; working with rulers and compasses]

**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]

** 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]

**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]
Jump to Top of Page

## Positive and Negative Numbers

**Problem 231: Star Magnitudes and Decimals-** Students work with the stellar magnitude scale to determine the brightness differences between stars.
[Grade: 5-8 | Topics: Multiplying decimals.]

**Problem 172: The Stellar Magnitude Scale-** Students learn about positive and negative numbers using a popular brightness scale used by astronomers.
[Grade: 3-6| Topics: number relationships; decimals; negative and positive numbers]

**Problem 62: Star light...Star bright - A question of magnitude! -**
Since the time of the ancient Greek astronomer Hipparchus, astronomers have measured and cataloged the brightness of stars
according to the 'apparent magnitude scale'. This activity lets students experience this peculiar numbering system
where bright stars have small numbers (even negative: our sun is a -26 magnitude!) and faint stars have large numbers
(faintest stars are +29 magnitudes). Students will calculate the
brightness differences between stars using multiplication and division. Working with the number line will be a big
help and math review!
[Grade level: 4-6 | Topics: Positive and negative numbers; decimal math]

## 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]
Jump to Top of Page

## Geometry

**Problem 584: Comparing Comets Up Close with NASA Spacecraft**

Students compare five comets and determine size ranges and percentages. [Grade: 3-5 | Topics: percentages; volume of a cube] (PDF)

**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 232: Star Circles-** Students use a photograph of star trails around the North Star Polaris to determine the duration of the timed exposure based on star arc lengths.
[Grade: 8-9 | Topics: Lengths of arcs of circles; angular measure.]

**Problem 59: Getting A Round in the Solar System! - **
How big does a body have to be before it becomes round? In this
activity, students examine images of asteroids and planetary moons to determine the critical
size for an object to become round under the action of its own gravitational field. Thanks to many Internet image archives
this activity can be expanded to include dozens of small bodies in the solar system to enlarge the
research data for this problem. Only a few example images are provided, but these are enough for
the student to get a rough answer!
[Grade level: 6-8 | Topics: Data analysis; decimals; ratios; graphing]

**Problem 51: Sunspots Close-up and Personal - **
Students will analyze a picture of a sunspot to learn more about its size, and examine the
sizes of various other features on the surface of the sun that astronomers study.
[Grade level: 6-8 | Topics: Finding the scale of an image; metric measurement; decimal math]

**Problem 38: Solar Eclipses and Satellite Power** From the ground we see total solar eclipses where
the New Moon passes directly between Earth and Sun. Satellites use solar
cells to generate electricity, but this is only possible when the Earth is
not 'eclipsing' the sun. Students will create a scaled drawing of the
orbits of three satellites around Earth, and calculate how long each
satellite will be in the shadow of Earth. They will be asked to figure out
how to keep the satellites operating even without sunlight to power their
solar panels. [Grade: 5 - 8 | Topics: Geometry; decimal math]

**Problem 16: Solar Power and Satellite Design** Students perform simple surface area calculations to
determine how much solar power a satellite can generate, compared to the
satellite's needs. [Grade: 5 - 8 | Topics: Area of irregular polygons]

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

**Problem 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]
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## Measurement

**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]
(PDF)

**Problem 464: Big Moons and Small Planets **

Students work with a scaled drawing of 26 large moons in the solar system, and together with an exercise in using simple
fractions, explore the relative sizes of the moons compared to Earth.
[Grade: 3-5 | Topics: scale models; proportions; fractions]
(PDF)

**Problem 423: The Moon as a Black Hole **

Students draw a life-sized model of the Earth and Moon as two black holes to
explore the actual sizes of these exotic astronomical bodies.
[Grade: 3-5 | Topics: Working with a compass; metric ruler]
(PDF)

**Problem 376: The Earth-like Planet Gliese 518g **

Students use data for the Gliese 581 planetary system to draw a scaled model of the locations and sizes of the discovered planets. They also
identify the location and span of the Habitable Zone for this planetary system.
[Grade: 3-5 | Topics: scale models; measurement]
(PDF)

**Problem 232: Star Circles-** Students use a photograph of star trails around the North Star Polaris to determine the duration of the timed exposure based on star arc lengths.
[Grade: 8-9 | Topics: Lengths of arcs of circles; angular measure.]

**Problem 161: Earth and Moon to Scale-** Students create a scale model of trhe Earth-Moon system and compare with artistic renditions and actual NASA spacecraft images.
[Grade: 4-6| Topics: Decimals; scaling and similarity]

**Problem 160: The Relative Sizes of the Sun and Stars-** Students work through a series of comparisons of the relative sizes of the sun compablack to other stars, to create a scale model of stellar sizes using simple fractional relationships. ( e.g if Star A is 6 times larger than Star B, and Star C is 1/2 the size of Star B, how big is Star C in terms of Star A?)
[Grade: 4-6 | Topics: working with fractions; scale models]

**Problem 159: Galaxies to Scale -** Students explore the relative sizes of the Milky Way compablack to other galaxies to create a scale model of galaxies, similar to the methods in Problem 161.
[Grade: 4-6 | Topics: working with fractions; scale models]

**Problem 154: Pan's Highway and Saturn's Rings -** Students use an image from the Cassini spacecraft to
determine how large the satellite Pan is, and the
scale of Saturn's rings using a millimeter ruler.
[Grade: 4-6 | Topics:Finding the scale of an image; measurement; unit conversion]

**Problem 143: So..How big is it? - Asteroid Eros surface **
Students calculate the scale of an image of the surface of the asteroid Eros from the NEAR mission, and
determine how big rocks and boulders are on its surface.
[Grade: 4 - 7 | Topics: Scaling; multiplication, division; metric measure]

** Problem 139 How Big Is It? - Mars **
Students use an image of a crater wall on mars to investigate ancient
water gullies discovered in 2008 by the Mars Orbiter.
[Grade: 4 - 7 | Topics:image scales; metric measurement; division and multiplication; decimals]

**Problem 135 How Big is It? - Io and Jupiter.**
Students work with an image taken by the Cassini spacecraft of Jupiter and its satellite Io.
They determine the image scale, and calculate the sizes of various features in the image.
[Grade: 4 - 7 | Topics:image scaling; multiply, divide, work with millimeter ruler]

**Problem 133 How Big is It? - The Mars Rover.**
Students work with an image taken by the Mars Orbiter
satellite of the Spirit landing site. They
determine the image scale, and calculate the sizes of various surface features from the image.
[Grade: 4 - 7 | Topics:image scaling; multiply, divide, work with millimeter ruler]

** 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]

**Problem 129 How Big is It? - The Moon up close. **
Students work with an image taken by the Lunar Orbiter III spacecraft
to determine image scale, and search for the smallest things seen in a photograph.
[Grade: 4 - 7 | Topics:image scaling; multiply, divide, work with millimeter ruler]

**Problem 127 How Big is It? - The Moon up close. **
Students work with an image taken by the Lunar Orbiter IV spacecraft
to determine image scale, and search for the smallest things seen in a photograph.
[Grade: 4 - 7 | Topics:image scaling; multiply, divide, work with millimeter ruler]

**Problem 126 How Big is It? - A Martian Avalanche! **
Students work with a Mars reconnissance Orbiter image to determine image scale, and search for
the smallest things seen in a photograph.This avalanche was caught as it occurred on February 19, 2008!
[Grade: 4 - 7 | Topics:image scaling; multiply, divide, work with millimeter ruler]

**Problem 125 How Big is It? - Washington DC up close. **
Students work with an image taken by ISS astronauts to determine image scale, and search for the smallest things seen in a photograph.
[Grade: 4 - 7 | Topics:image scaling; multiply, divide, work with millimeter ruler]

**Problem 97 Hinode - Closeup of a Sunspot - **
Students will determine the sizes of sunspots and solar granulation cells from a recent image
taken by the Hinode solar observatory.
[Grade level: 6-8 | Topics:image scales, metric units, unit conversion]

**Problem 52 Measuring the size of a Star Cluster - **
Astronomers often use a photograph to determine the size of astronomical objects. The
Pleiades is a famous cluster of hundreds of bright stars. In this activity, students will determine the photographic scale,
and use
this to estimate the projected (2-D) distances between the stars in this cluster. They will also use internet and library resources to learn more about this
cluster.
[Grade level: 4-6 | Topics: Online research; Finding the scale of an image; metric measurement; decimal math]

**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]
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## Data Analysis and Probability

**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 150: Cosmic Bar Graphs -** Students interpret simple bar graphs taken from astronomical data.
[Grade: 3-5 | Topics: finding maxima and minima; fractions; extrapolating data]

**Problem 100 The Sunspot Cycle - endings and beginnings - **
Students will examine a plot of the sunspot cycle and extract information from the plotted data about
the previous sunspot cycle, and make predictions about the next one about to start in 2007.
[Grade level: 6-9 | Topics:graph reading; extrapolation; time calculations]

**Problem 36 The Space Station Orbit Decay and Space **
Weather Students will learn about
the continued decay of the orbit of the International Space Station by
studying a graph of the Station's altitude versus time. They will
calculate the orbit decay rates, and investigate why this might be
happening. [Grade: 5 - 8 | Topics: Interpreting graphical data; decimal math]

**Problem 7 Solar Flares, CME's and Aurora** Some articles about the Northern Lights imply that
solar flares cause them. Students will use data to construct a simple Venn
Diagram, and answer an important question about whether solar flares cause
CME's and Aurora. [Grade: 5 - 7 | Topics: Venn Diagramming]

## Problem Solving

**Problem 587: Comet Encounters after Discovery **

Students examine how often newly discovered comets approach Earth and become a hazard, and how soon after discovery these close passes can occur.
[Grade: 3-5 | Topics: Averaging, percentages]
(PDF)

**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 582: How do Telescopes Magnify? **

Students use a simple ratio formula to calculatethe magnification of a telescope.
[Grade: 3-5 | Topics: division of two decimal numbers; evaluating simple ratios. ]
(PDF)

**Problem 581: How Telescopes Work **

Students compare how much light a telescope can gather compared to the human eye.
[Grade: 3-5 | Topics: Area of a circle ]
(PDF)

**Problem 570: Curiosity Heads for Mt Sharp **

Tabular data is used to estimate how long it will take the Curiosity rover to reach the base of Mt Sharp using data from its porevious week travels.
[Grade: 3-5 | Topics: averaging numbers in a table; time = distance/speed]
(PDF)

**Problem 563:Comet ISON and its Close Encounter with Mars**

Students use tabular data to determine the date and time of closest approach to Mars
[Grade: 3-5 | Topics: graphing tabular data]
(PDF)

**Problem 560:The Orbit of Comet ISON**

Students explore how close Comet ISON will get to Mercury, Venus, Earth and Mars during its 2013 passage.
[Grade: 3-5 | Topics: Interpreting tabular data; graphing ]
(PDF)

**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]
(PDF)

**Problem 198: Solar Storm Timeline-** Students read a narrative about the events involved in a solar storm, creates a chronology for the sequence of events, and answer some simple time-related questions.
[Grade: 6-8| Topics: Time calculations.]

**Problem 100 The Sunspot Cycle - endings and beginnings - **
Students will examine a plot of the sunspot cycle and extract information from the plotted data about
the previous sunspot cycle, and make predictions about the next one about to start in 2007.
[Grade level: 6-9 | Topics:graph reading; extrapolation; time calculations]