Head First Algebra by Pilone, Tracey -- Read -- Imperial Library of Trantor

Index

Head First Algebra Dedication Advance Praise for Head First Algebra Authors of Algebra How to use this Book: Intro

Who is this book for?

Who should probably back away from this book?

We know what you’re thinking We know what your brain is thinking Metacognition: thinking about thinking Here’s what WE did

Here’s what YOU can do to bend your brain into submission

Read Me The technical review team Acknowledgments Safari® Books Online

1. What is Algebra?: Solving for unknowns...

It all started with a big gaming sale What does a system really cost?

There’s tax on the system... ... and the extended warranty, too.

Algebra is about solving for unknowns Jo’s got more unknowns X marks the spot unknown Equations are math sentences Now SOLVE for the unknown So which operation do you use when?

And a change for division too...

Jo is ready to accessorize!

Cecking Checking your work... Substitution uses your solution in the original equation

Equation training Jo has an awesome setup! Math Toolbox

2. (More) Complicated Equations: Taking Algebra on the road

Paul loves “Pajama Death”

Can you help Paul?

Always start with what you know There’s a COST for each guy Replace your words with numbers Now solve for g... one step at a time ... but you have to keep the equation equal! If you follow the rules, you’ll ALWAYS get the right answer Whole numbers are usually easier to work with A variable can appear in an equation MORE THAN ONE TIME Checking your work proves your answer What’s a road trip without some girls? We need another variable A term is a chunk of an algebraic equation

“In terms of” is the secret to multiple variables in an equation

Tools for your Algebra Toolbox

3. Rules for Numeric Operations: Follow the rules

Math or No Math

Problem #1

There’s an order for working expressions

Back to Math or No Math... Problem #2

You can re-group your equations It’s an important round...

Problem #3 - The final round

Distributing a value over a grouping doesn’t change a problem’s value A constant stands in for a number Roll the credits... Tools for your Algebra Toolbox

4. Exponent Operations: Podcasts that spread like the plague (that’s a good thing...)

Addie’s got a podcast Let’s mobilize Addie’s listeners Can Addie and Alex get enough hits? Alex is flaking out on his sister There’s always a villain...

You can’t add exponents with different bases

The order of operations says exponents FIRST

We’ve got to work our exponent “backward”

A root is the INVERSE of an exponent Tools for your Algebra Toolbox

5. Graphing: A picture’s worth 1,000 words

Edward’s Lawn Mowing needs help...

Now Ed can know what his cash is at ANY time

Why don’t you just SHOW me the money? Now we can LOOK at Ed’s cash pattern Graphs show an ENTIRE relationship

There’s something else that covers the entire relationship A new situation needs a new equation The Cartesian Plane allows values to go BELOW zero

Let’s graph Ed’s equation on the Cartesian Plane

Ed’s graph has two INTERCEPT points The truth about linear equations...

Ed’s figuring out the SLOPE of lawns Linear equations in point-slope form How does a point and a slope get you a line? Let’s use the point-slope form

Horizontal lines require a different form

Equations also have a standard form The slope-intercept form is EASY to graph

Use the equation form to your advantage

Tools for your Algebra Toolbox

6. Inequalities: Can’t quite get enough?

Kathleen really loves football The cost of all players can’t be more than $1,000,000

You’re really working with a comparison...

Inequalities are COMPARISONS Inequalities involving some negative number operations need special treatment Negative inequalities work BACKWARD

Multiplication and division of negative numbers causes problems for inequalities

FLIP the inequality sign with negative multiplication and division

Use the number line to visualize the relationship

When you’re working with an inequality and negative multiplication or division... You can visualize a solution set on a number line Inequalities can have TWO variables Use a graph to visualize the solutions to an inequality Answers made in the shade

Shading shows potential answers

Are you ready for some football? Tools for your Algebra Toolbox

7. Systems of Equations: Know what you don’t know

A line means infinite solutions You can’t have -1 gallons of liquid! How does the sparkling equation work?

We’ve got another linear equation Now we have TWO linear equations

The INTERSECTION of the lines solves BOTH linear equations Solve multiple unknowns with a SYSTEM of EQUATIONS

More trouble... Zach dropped some glasses

Two kinds of glasses... that’s TWO unknowns

Solve your system of equations using a graph

Let’s solve the glasses problem You can substitute substitution for graphing f is gone with almost no work

Eliminate a variable with the ELIMINATION METHOD The elimination method requires PLANNING

Manipulate your equations for elimination

Which variable?

Zach’s party rocks! Sometimes two equations aren’t two lines Tools for your Algebra Toolbox

8. Expanding Binomials & Factoring: Breaking up is hard to do

Math or No Math semi-regional masters final

Problem #1: Simplify this expression

Who’s right? Binomials are groups of two algebraic terms The distributive property, revisited

Distribute ALL of your first binomial...

Simplify binomials with the distributive property

Problem #2: Simplify this expression - fast! The SQUARE pattern

What about when the signs are the SAME? Sometimes there’s just not a pattern... FOIL ALWAYS works

Problem #3: Another speed round...

Un-distribution is called FACTORING Factoring is un-mulitplying Factor by looking for common terms

Problem #4: Solve this...

Zero times anything is 0

Let’s zero these things out...

Tools for your Algebra Toolbox

9. Quadratic Equations: Getting out of line

Head First U is at war! Jon’s upgrading his technology

Introducing a new type of equation: quadratic

Where does Jon put the catapult?

Uh oh... the president’s relocated

You should always factor with a PLAN Pi Gamma Delta built a wall! 9 feet is not a problem The quadratic formula

What the heck is a discriminant?

Frat Wars, part deux

A graph lets us SEE values...

How can you graph x2? A parabola is the shape of a quadratic equation Graphing a parabola depends upon the vertex

Use and understand the vertex

Work with the parabola, the SMART way The discriminant can help with our graph, too

So here’s our final graph

Tools for your Algebra Toolbox

10. Functions: Everyone has limits

Pajama Death TV

Uh oh... there’s a change in venue

Equations have limits (most of the time)

A function can be expressed as an equation

The input limits are the domain of the function Functions have minimum and maximum outputs

So what’s the MAXIMUM we can make? What’s the MINIMUM? All of the valid outputs are called the range

Algebra is really about relations Relations, equations, and functions all go TOGETHER Function graphs have LIMITS Just before the second episode of Pajama Death TV... Graphing reveals the nature of a relation Functions pass the vertical line test

The FIRST 350 tickets are always free!

But... what about the REST of the tickets?

One function, two parts = real life

Use the function piece you NEED

To evaluate a piecewise function Graph piecewise functions just like “normal” functions!

The numbers are in... and? Pajama Death’s show was a hit! Tools for your Algebra Toolbox

11. Real-World Algebra: Solve the world’s problems

Calculate interest from your interest rate and the principle amount you’re borrowing

Now SUBSTITUTE to figure out Max’s monthly payment

Max doesn’t own that car just yet...

Depreciation is a sad fact of life But the bank still gets their money

You don’t need to GUESS with Algebra

... but remember to keep the context of the problem in mind

Max plans to pay you to be his financial planner

A. Leftovers: The Top Five Things (we didn’t cover)

#1 Negative Exponents Working with negative exponents

Negative exponents also give you flexibility

#2 Table of values for graphing #3 Absolute value equations #4 Calculators #5 More practice, especially for factoring

B. Pre-Algebra Review: Build on a solid foundation

Algebra starts with numbers How do you work with negative numbers?

The number line

Addition and subtraction of integers

Working with mixed integers

Multiplication and division of integers

The rules for integer signs - multiplication and division

Absolute Value What absolute value means Number sets - all together The number sets Three ways to split things up Decimal’s Anatomy

How decimals communicate

Addition and subtraction with decimals Decimal multiplication Decimal division

Let’s do some division!

Decimal division training Special decimals You’re 100% right! Working with percents

First let’s deal with sales tax...

Fractions

Fractions show parts of a whole

Fraction multiplication

Fraction division mixes numerators and denominators

Improper fractions

Divide to make an improper fraction proper

More about improper fractions Invert a fraction to get its reciprocal

Fraction division - option #2

Adding and subtracting fractions

You need a common denominator

Equivalent fractions get you matching denominators Use the lowest common denominator for addition Fraction addition and subtraction training Dividing by one doesn’t change the value Reduce fractions by dividing by 1 Factor trees can eliminate lots of little steps

Pick out the prime factors

Reduce fractions with the factor tree Putting it all together - fractions

Converting from a fraction to a decimal

Converting decimals to fractions

Conversions everywhere

Division by Zero doesn’t work Sometimes multiplication takes forever!

Is there a shorter way?

How quickly things spread... Why does all this matter?

Index About the Authors Copyright

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