Common Topics Covered in Standard Algebra II Textbooks

Algebra II is a course in mathematics offered in the United States public and private school systems taken by approximately 85% of all graduating high school seniors by the age of 17. Two major studies by the U.S Department of Education have shown that Algebra II is considered by many as a gateway course that predicts student graduation from college, and their eventual qualification for high-paying careers. The course is typically taught in Grade 10 as a two-semester series following prerequisite courses in Algebra I and/or Geometry. The course stresses student mastery of the analysis and graphing of polynomials, logarithmic, exponential and trigonometric functions, with some applications to real-world problems in which these modeling techniques can often be seen to apply.

In keeping with the intent to show how Algebra II topics connect with real-world applications, textbooks commonly include several hundred word problems that are generally culled from situations that students may encounter, often involving economics. What appears to be absent from the selection are an adequate number of problems in earth or space science. For example, out of 700 application problems in the textbook Algebra II (McDougal-Littell, 2004) one finds fewer than 30 that connect with physical science or space science. Many of these are fairly generic and do not leverage recent discoveries in earth or space science as a way to stimulate the students interest in these topics and prospective careers.

Since 2004, Space Math@ NASA has developed math problems for grades 3-12 designed to showcase how NASA discoveries in earth and space science are connected to a variety of math topics and skills. By 2010, over 400 of these problems are available online, or can be found in a series of special-topic books (Black Hole Math, Earth Math, etc). Frequently, NASA press releases serve as the hook to provide a suitable topic from which an appropriate mathematical problem is developed. This also allows students to read about a new discovery on the Evening News or CNN.com, and then within a few days they can work through some mathematical issue presented by the news release. For example, the Gulf Oil Spill of 2010 was viewed by the NASA, Terra satellite and students used the satellite image to calculate its total area, mass and density. In other examples, students can read a press release announcing the discovery of a new planet, and calculate from two points on its elliptical orbit, the equation of the orbit, its semi-major axis and the orbit period of the planet.

This book contains over 200 problems spanning over 70 specific topic areas covered in a typical Algebra II course. The content areas have been extracted from the McDougal-Littell Algebra II textbook according to the sequence used therein. A selection of application problems featuring astronomy, earth science and space exploration were then designed to support each specific topic, often with more than one example in a specific category. Each problem is introduced with a brief paragraph about the underlying science, written in a simplified, non-technical jargon where possible. Problems are often presented as a multi-step or multi-part activities. The intent of these problems is not to follow an explicitly inquiry-based approach, but to systematically show students how problems and questions of a specific type are often solved. Once students have mastered a particular approach, there are many opportunities available for students to go beyond each problem and inquire about other connections that may suggest themselves as the student completes each problem, or a collection of problems.

Real Numbers and Operations

1.1.1 -- Unit Conversions I
1.1.2 -- Unit Conversions II

Algebraic Expressions and Operations

1.2.1 -- Calculating Star Distances
1.2.2 -- Black Hole Tidal Forces
1.2.3 -- Moon Crater Explosions
1.2.4 -- Secondary Physical Constants
1.2.5 -- Magnetic Fields
1.2.6 -- Ares IX Rocket Launch
1.2.7 -- Temperature of a Planet I
1.2.8 -- Density of the Solar Interior
1.2.9 -- Temperature of a Planet II

Solving Linear Equations

1.3.1 -- Solving for X in Astronomy

Rewriting Equations and Formulas

1.4.1 -- Lunar Escape Speed
1.4.2 -- Keplers Third Law

Problem Solving using Algebraic Models

1.5.1 -- A Model for the Lunar Interior
1.5.2 -- Modeling Atmospheric CO2
1.5.3 -- Spitzer: New Saturn Ring
1.5.4 -- Spitzer: Weather on a New Planet
1.5.5 -- LRO: Water on the Moon
1.5.6 -- Spitzer: A Model for Planet Osiris

Solving Linear Inequalities

1.6.1 -- Brown Dwarf Stars
1.6.2 -- Asteroid Math
1.6.3 -- The Higgs Boson Mass Limits
1.6.4 -- Neutron Star Mass Limits

Solving Absolute Value Equations and Inequalities

1.7.1 -- Variable Stars

Functions and their Graphs

2.1.1 -- Telescope Resolving Power
2.1.2 -- Sunspot Numbers

Slope and Rate of Change

2.2.1 -- Slope in Astronomy
2.2.2 -- Mixed Units

Quick Graphs of Linear Equations

2.3.1 -- The Moon�s Orbit
2.3.2 -- Temperature in Deep Mines
2.3.3 -- Solar Power

Writing Equations of Lines

2.4.1 -- Recent Sea Level Rise
2.4.2 -- Loss of Arctic Ice

Correlation and Best-fitting Lines

2.5.1 -- Gamma Ray Bursts

Linear Equalities In Two Variables

2.6.1 -- Star Cluster Study
2.6.2 -- WIMPS and Dark Matter
2.6.3 -- Exoplanet Habitable Zones

Piecewise Functions

2.7.1 -- The Expanding Universe

Absolute Value Functions

2.8.1 -- Reflecting Light Rays

Solving Linear Systems By Graphing

3.1.1 -- Cratering on the Moon

Solving Linear Systems Algebraically

3.2.1 -- Calculating Molecular Structure

Graphing and Solving Systems of Linear Equalities

3.3.1 -- Graphing the Cosmos I
3.3.2 -- Graphing the Cosmos II
3.3.3 -- Graphing the Cosmos III
3.3.4 -- Graphing the Cosmos IV

Linear Programming

3.4 No Current Examples

Graphing Linear Equations in Three Variables

3.5 No Current Examples

Solving Systems of Equations in Three Variables

3.6.1 -- Solving Molecular Structure
3.6.2 -- Solving Molecular Structure

Matrix Operations

4.1.1 -- The Sunspot Cycle
4.1.2 -- Star Brightness and Distance
4.1.3 -- Analyzing Astronomical Photos
4.1.4 -- Star Brightnesses

Multiplying Matrices

4.2.1 -- Rotation Matrices
4.2.2 -- Mass and Weight

Determinants and Cramers Rule

4.3 No Current Examples

Identity and Inverse Matrices

4.4.1 -- Coordinate Transformations
4.4.2 -- Astronomical Image Processing

Solving Systems Using Inverse Matrices

4.5.1 -- Studying Solar Storms with Matrices
4.5.2 -- Communication Satellies and Matrices
4.5.3 -- Rotation Matrix Math
4.5.3 -- Solving Molecular Structure

Solving Systems Using Augmented Matrices

4.6 No Current Examples

Graphing Quadratic Functions

5.1.1 -- Vertical Motion under Gravity

Solving Quadratic Equations by Factoring

5.2.1 -- LRO Creates a Water Fountain

Solving Quadratic Equations by Finding Square Roots

5.3.1 -- The Speed of Sound
5.3.2 -- Gravitational Collapse
5.3.3 -- Comet Impact

Complex Numbers

5.4.1 -- Interstellar Extinction

Completing the Square

5.5 No Current Examples

The Quadratic Formula and the Discriminant

5.6.1 -- Supernova Explosion
5.6.2 -- Detecting Exoplanets

Graphing and Solving Quadratic Functions

5.7 No Current Examples

Modeling with Quadratic Functions

5.8.1 -- Atmospheric Carbon Dioxide
5.8.2 -- The Power of a Supernova
5.8.3 -- Water Emission by Comets
5.8.4 -- The Pace of Exoplanet Discovery

Using Properties of Exponents

6.1.1 -- Scientific Notation I
6.1.2 -- Scientific Notation II
6.1.3 -- Scientific Notation III

Evaluating and Graphing Polynomial Functions

6.2.1 -- White Dwarf Fadeout
6.2.2 -- The Higgs Boson Mass
6.2.3 -- The Energy of Empty Space
6.2.4 -- The Interior of the Sun

Adding, Subtracting and Multiplying Polynomial Functions

6.3.1 -- The Ares-1X Acceleration Curve

Factoring and Solving Polynomial Equations

6.4 No Current Examples

The Remainder and Factor Theorems

6.5 No Current Examples

Finding Rational Zeros

6.6 No Current Examples

Using the Fundamental Theorem of Algebra

6.7 No Current Examples

Analyzing Graphs of Polynomial Functions

6.8.1 -- The Energy of the Vacuum

Modeling with Polynomials

6.9.1 -- The Rotation of the Sun
6.9.2 -- Ejection of Water from Comets
6.9.3 -- An Erupting Solar Prominence

Nth Root and Rational Exponents

7.1.1 -- Passing Time near a Black Hole
7.1.2 -- The Size of a Nebula
7.1.3 -- The Earth�s Bow Shock

Properties of Rational Exponents

7.2.1 -- Accreting Gas near a Black Hole
7.2.2 -- Temperature of a Planet
7.2.3 -- The Temperature of the Big Bang
7.2.4 -- The Planet Gliese-581c

Power Functions and Operations

7.3.1 -- No current example

Inverse Functions

7.4.1 -- Expanding Interstellar Nebula
7.4.2 -- Time Distortion Near a Black Hole
7.4.3 -- The Longest Sound Wave
7.4.4 -- Collapsing Star Clouds

Graphing Square and Cube-root Functions

7.5.1 -- Gravity and Time Distortion

Solving Radical Equations

7.6.1 -- The Shape of a Galaxy
7.6.2 -- The Growth of Cosmic Structure

Statistics and Statistical Graphs

7.7.1 -- The Average Speeds of Galaxies
7.7.2 -- Sunspot Cycles
7.7.3 -- Analyzing Astronomical Images

Exponential Growth

8.1.1 -- Compound Interest

Exponential Decay

8.3.1 -- Carbon-- 14 Dating
8.3.2 -- Supernova Fadeout

Logarithmic Functions

8.4.1 -- Star Counting
8.4.2 -- The LogLog Universe I
9.4.3 -- The LogLog Universe II

Properties of Logarithms

8.5.1 -- The Star Magnitude Scale

Solving Exponential and Logarithmic Equations

8.6.1 -- The Distances to Stars
8.6.2 -- The Brightness of Stars and Magnitudes

Modeling with Exponential and Power Functions

8.7.1 -- Keplers Third Law
8.7.2 -- Satellite Orbit Decay
8.7.3 -- Atmospheric Attenuation
8.7.4 -- The Thickness of the Atmosphere
8.7.5 -- Gamma Ray Bursts

Logistic Growth Functions

8.8.1 -- Planet Formation and Growth

Inverse and Joint Variation

9.1.1 -- Some Astronomical Examples
9.1.2 -- Sea Level Rise

Graphing Simple Rational Functions

9.2.1 -- The Distance to Galaxies
9.2.2 -- The Doppler Shift

Graphing General Rational Functions

9.3.1 -- Inside a Neutron Star
9.3.2 -- The Sun�s Corona

Multiplying and Dividing Rational Expressions

9.4 No Current Examples

Adding and Subtracting Complex Fractions

9.5.1 -- Simple Atomic Energy Diagrams
9.5.2 -- Chemistry Made Simple
9.5.3 -- The Periodic Table of the Elements
9.5.4 -- Relative Distances Between Galaxies

Solving Rational Equations

9.6 No Current Examples

The Distance and Midpoint Formulas

10.1.1 -- Astronomical Distances
10.1.2 -- Distances to Globular Star Clusters
10.1.3 -- Calculating Horizon Distances

Parabolas

10.2.1 -- Comet Orbits
10.2.2 -- Solving for a Comet Orbit

Circles

10.3.1 -- The Transit of Venus 2012

Ellipses

10.4.1 -- The Orbit of an Exoplanet
10.4.2 -- The Orbit of a Comet
10.4.3 -- The Comet Wild-2

Hyperbolas

10.5.1 -- The Butterfly Nebula
10.5.2 -- The Unusual Comet Lulin

Graphing and Classifying Conics

10.6 No Current Examples

Solving Quadratic Systems

10.7.1 -- The Large Hadron Collider
10.7.2 -- Determining a Comets Orbit
10.7.3 -- Halley�s Comet Orbit

An Introduction to Sequences and Series

11.1.1 -- The Lyman Series of Hydrogen
11.1.2 -- The Titius-Bode Law of Planets

Arithmetic Sequences and Series

11.2.1 -- Areas Under Curves as Series
11.2.2 -- The Speed of an Ion Spacecraft

Geometric Sequences and Series

11.3.1 -- Fading Starlight Through a Cloud

Infinite Geometric Series

11.4.1 -- The Brightness of a Star Field
11.4.2 -- The Maximum Speed of a Rocket

Recursive Rules for Sequences

11.5.1 -- The Volume of a Hypersphere

The Fundamental Counting Principle and Permutations

12.1.1 -- Solar Storms I
12.1.2 -- Solar Storms II
12.1.3 -- Craters on the Moon

Combinations and the Binomial Theorem

12.2.1 -- Solar X-Ray Flares
12.2.2 -- Severe Space Weather

Introduction to Probability

12.3 No Current Examples

Probability of Compound Events

12.4.1 -- Solar Flares and Storms
12.4.2 -- Meteor Impacts on Earth
12.4.3 -- Craters on the Moon
12.4.4 -- Solar Storms and Compound Events

Probability of Dependent and Independent Events

12.5.1 -- Reliable Decisions

Binomial Distribuitons

12.6.1 -- Space Weather and Stormy Days
12.6.2 -- Stormy Space Weather
12.6.3 -- The Apollo Moon Landings

Normal DIstributions

12.7.1 -- Astronomical Applications in Imaging
12.7.2 -- The Speed of Gas Particles
12.7.3 -- Measurement Error: Up Close

Right Triangle Trigonometry

13.1.1 -- STEREO views of the Sun
13.1.2 -- Basic Similar Triangle Geometry

General Angle and Radian Measure

13.2.1 -- Radians and Degrees in Astronomy
13.2.2 -- Angular and Linear Size
13.2.3 -- Degrees, Minutes, Seconds of Arc
13.2.4 -- Angular Resolution and Moon Details
13.2.5 -- Transits and Eclipses

Trigonometric Functions of Any Angle

13.3.1 -- Rotation of Images

Inverse Trig Functions

13.4 No Current Examples

Law of Sines

13.5.1 -- Location of the Planets During the Transit of Venus in 2012

Law of Cosines

13.6.1 -- Angular Distance Between Planets
13.6.2 -- The STEREO View of the Sun

Parametric Equations and Projectile Motion

13.7.1 -- The Ares 1X Trajectory

Graphing Sin, Cosine and Tangent Functions

14.1.1 -- Solar Power
14.1.2 -- Delta Cephi

Translations and Reflections of Trig Graphs

14.2 No Current Examples

Verifying Trig Identities

14.3 No Current Examples

Solving Trigonometric Equations

14.4.1 -- Temperature on Mars
14.4.2 -- Spinning Satellites
14.4.3 -- The Distances to Stars and Nebula
14.4.4 -- The Surveyor�s Challenge

Modeling with Trigonometric Functions

14.5.1 -- Electric Power Usage
14.5.2 -- Carbon Dioxide in the Atmosphere
14.5.3 -- The Liquid Mirror Telescope

Using Sum and Difference Formulas

14.6.1 -- The Rising and Setting Formula

Using Half and Double-angle Formulas

14.7.1 -- The Acceleration of Gravity on Earth