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Table of Contents
Introduction
Applications
Number Systems: What are They and Why are They Important?
Constants, Coefficients, Variables, and the Invisible Demons
Algebra Here We Come
Light Switch Property
Short-Cuts to Solving Linear
First Order Equations
The Bridge-Toll Method
Working with Unequals, Better Known as Inequalities
Algebra, algebra, it doesn't make sense,
Numbers and variables combined in strange ways,
With symbols and signs, parentheses too,
My footing is lost, I feel stuck in glue,
Help, please help teacher, I'm depending on you!
Then there's those lines, formulas strange,
Point-slope, slope-intercept, that many the days,
I sat there and wondered, why do this to me?
I'm clueless, I'm fried, I'm ready to flee,
Help, please help teacher, I.m going to scream.
Absolute value, never negative we're told,
But insert a variable and I've lost my hold.
It's x, no it's negative x, c'mon what's the deal?
I must be dreaming, this can't be real!
Help, please help teacher, help me to feel.
Onto the galleries, where radicals repose,
Irrational demons whose nature confirms,
The essence of difficulty, why try to fight?
We struggle, we labor, their cause to make right,
Help, please help teacher, I'm losing my might.
Then came the quadratic, with roots complex and real,
"B squared - 4ac" the discriminant we learned,
It baffled the ablest, the roots to extract,
Others sat paralyzed, and this is a fact!
Help, please help teacher, keep us intact!
Polynomials, those creatures, they lurk in the dark,
Harassing and scaring, no walk in the park.
Add them and multiply, exponents confuse,
Like terms, unlike terms, how dare you refuse?
Help, please help teacher, give me some clues.
Then all of those monsters, we find in some words,
Whose meaning we labor to extract from their core,
We scream, we yell, we can't take any more,
These word problems kill us, they rankle our brain,
Help, please help teacher, spare us this pain.
Taken from the poem Help, Please Help Teacher! found in the collection
Poems for the Mathematically Insecure
Introduction
As the above poem illustrates, algebra can be a real nightmare for some. Not content to shake down only the weak, this subject will wreak havoc on even the ablest. And God forbid you have a bad teacher! You will be wishing that you could be a Roman gladiator duking it out with sword and shield.
The right teacher or the right approach could be the difference between a lot of pain and suffering, or a load of fun and enlightenment. The purpose of Algebracadabra is to give you the latter. In this book, I will be taking a different approach than one which you probably might expect to find in a math book. Rather than getting a lot of math in the form of problems and expressions, you will be getting a lot of words in the form of explanation and clarification. These words will help decipher the mysterious algebra code and cut through the pain and suffering that this subject generally inflicts on the high school student. For once you understand the language and symbols of this discipline, the examples and procedures become less intimidating and more amenable to solution.
As I used to teach my students, once you understand a specific technique, then this method will allow you to solve infinitely many problems within that specific genre. In other words, once you've seen one, you've seen them all. That is, once you know the method to solve any particular class of equation, then you can solve all of them. The only thing that changes are the players: the variables and coefficients, and maybe one or two other minor actors. Everything else is the same. Once you have this idea in mind, you will approach this subject with a confidence that will rip through just about any problem at hand. Sure. Practicing many problems does help, but having the confidence to know that you can do every problem in a class because you recognize it for what it is, is the true secret and sought-after shortcut to algebra mastery.
So what is this thing called algebra and why is it important in the real world? For most teenagers, algebra is nothing more than a pain in the neck during their freshman and sophomore high school years. When I taught algebra in high school, I sometimes thought I was teaching how to read hieroglyphics or how to decipher some arcane code. For some reason or another, algebra was simply baffling to many students. And although I was often lauded for my ability to make the math easy and for my teaching skills, I still could not grasp why some just didn.t get it. If you.ve read any of my articles in the Why Study Math series, then you know that you wouldn.t be able to do many things without math, including what you are doing right now: reading this ebook online. It is the power of math that has enabled man to reckon the stars and even galaxies, to discern the properties of planets light years away, and to create the computer.
Algebra, as one of three main branches of this discipline we call mathematics, is indispensable to all higher branches of this subject, and indeed figures in most of the applications of math to the real world. In a nutshell, algebra is concerned with the solution of specific equations. An equation is nothing more than the setting equal of two expressions involving variables, and trying to find the number values that make the equation true. That's pretty much it.
Of course, many interesting things happen in search of those specific solutions and consequently algebra takes on a multicolored aspect, so to speak. I doubtless wonder whether the Persian mathematician Muhammad al-Khwarizmi, who is credited with the creation of algebra, and from whose book we derive the name algebra, knew that this field of mathematics would lay the cornerstone for such future areas as analysis and the calculus. In fact, I would often tell my algebra students who inquired about the mysterious calculus, that this subject was nothing more than glorified algebra; that's how intricately linked the two are. For without a firm grasp of algebra, you could never even contemplate the study of the calculus.
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