Algebra in Words: A Guide of Hints, Strategies and Simple Explanations

191 Pages·2014·1.2 MB·English

by  Gregory P. Bullock Ph.D.

ALGEBRA
IN
WORDS

A Guide of Hints,
Strategies and
Simple Explanations

Gregory P. Bullock, Ph.D.
Copyright © 2014 Gregory P. Bullock, Ph.D.

All Rights Reserved. This book may not be used or reproduced in part, in whole,
or by any other means whatsoever without written permission.

Bullock, Gregory P.
Algebra in words: a guide of hints, strategies and simple explanations

MATHEMATICS/Algebra/General

STUDY AIDS/Study Guides

First Edition

The United States of America
Table of Contents
INTRODUCTION
What Is This Book?
Why Do You Need Algebra?
REVIEW OF THE BASICS
The Real Order of Operations: GEMA
The Truth about PEMDAS
The Unwritten 1
Property Crises of Zeros, Ones & Negatives
Integers & Whole Numbers
Prime Numbers
Is 51 a Prime Number?
What is a Term?
What is a “Like-Term”?
What is a Factor?
Factoring
The Procedure for Prime Factoring
The Prime Number Multiples Table
The Greatest Common Factor (GCF)
The Least Common Denominator (LCD)
GCF vs. LCD
FRACTIONS
Procedure for Adding & Subtracting Fractions
Multiplying Fractions
Dividing Fractions
OPERATIONS OF BASES WITH EXPONENTS
Multiplying Bases With Exponents
Dividing Bases With Exponents
Exponents of Exponents (a.k.a. Powers of Powers)
SOLVING SIMPLE ALGEBRAIC EQUATIONS
Solving a Simple Algebraic Equation with One Variable (First Degree)
Arrangement: Descending Order
Expressions vs. Equations
LINEAR EQUATIONS
A Diagonal Line:
A Horizontal Line:
A Vertical Line:
What Does “Undefined” Mean?
How to Graph a Linear Equation
The Slope Equation
The 4 Important Equations for Lines
When x =x :
1 2
When y =y :
1 2
Parallel & Perpendicular Lines on a Graph
SOLVING A SYSTEM OF (TWO) LINEAR EQUATIONS
What Does “Solving In Terms Of” Mean?
Graph & Check
The Substitution Method
The Addition/Elimination Method
Examples for Choosing the Method
Interpreting the “Solutions”
One Solution - Consistent
No Solution - Inconsistent, Parallel
Infinite Solutions - Dependent
TRINOMIALS & QUADRATICS
What Are “Solutions” to Quadratic Equations?
Solving Quadratic Equations
Factor & Solve
Trial & Error/Reverse FOIL Method
The ac/Grouping Method
Complete the Square
The Quadratic Formula
The Part Everyone Forgets (The Last Step of the Quadratic Equation)
Graph & Check
Quadratics with Zero
2
When c is 0: ax + bx = 0
2
When Both b & c are 0: ax = 0
2
When b is 0: ax + c = 0
“The Difference of Two Squares”
Conjugate Pair Binomials
Taking the Square Root of Both Sides
The Sum of Two Squares
Special Words for Special Cases
Perfect Square Trinomial
The Difference of Two Squares
Prime vs. No Solution
Clarification: When the Solution is 0
RATIONAL EXPRESSIONS
Procedure for Simplifying Rational Expressions
Procedure for Adding & Subtracting Rational Expressions
Simplifying a Complex Rational Expression
All-LCD Method (detailed version):
Simplify Overall Numerator & Overall Denominator Separately Method
(detailed version)
All-LCD Method (short version)
Simplify Overall Numerator & Overall Denominator Separately Method
(short version)
Annotated Example 1 Using the All-LCD Method
Annotated Example 2 Using the Overall Numerator & Denominator Method
The Wrong Way to Simplify a Rational Expression
Extraneous Solutions
Procedure for Solving Equations with Rational Expressions & Extraneous
Solutions
Cross Multiplication
Cross-Multiplication vs. Cross Cancelling
RADICALS, ROOTS & POWERS
Perfect Squares & Associated Square Roots
List of Perfect Squares & Associated Square Roots
Common Perfect Cubes & Associated Cube Roots
Other Powers & Relationships of 2, 3, 4 & 5
Manipulating & Simplifying Radicals
List of Common Radical Fingerprints
Extraneous Roots in Radical Equations
FMMs (FREQUENTLY MADE MISTAKES)
The Two Meanings of “Cancelling Out”
Checking Your Answers
Miscellaneous Mistakes
Scientific Notation on Your Calculator
What Does “Error” on a Calculator Mean?
CLOSING
INTRODUCTION
What Is This Book?

This book is a guide of common math and algebra topics that are
explained in a non-traditional way. It is not a textbook, nor is it a conventional
study guide. This is a book where basic mathematical and algebraic topics are
explained in laymen’s terms, sometimes even purposefully redundant terms, to
make your understanding easier and your learning curve faster. It’s more of a
guide of supplemental information and perspectives on the math you must learn.
I tutored Calculus for the Math Department in undergraduate school as a
part-time job, then began teaching math at the collegiate level (Basic Math and
Arithmetic through College Algebra/Pre-Calc I) in 2009 to a wide range of
students of various ages and math education backgrounds. During that time, I
began noticing trends among my students and classes. One major trend I noticed
was the divide among people who seemed to “get it,” and those who didn’t “get
it” as easily, as quickly, or in the same way as those who did. Although it’s
pointless to classify students into groups, my job as an instructor is to help
bridge the gap and find mechanisms to help everyone “get it.”
As my teaching style evolved, I noticed that a lot of math (either in the
books or traditional lectures) was taught in a sort of “math language,” meaning
mostly in numbers, variables and lines of equations and simplification steps…
which is all well and good, because that’s what math is. But I found that much of
it was left to interpretation, which some would get and some wouldn’t. So I
started translating the math into worded steps and notes and found that students
responded positively to it. This was the bridge over the gap I was looking for.
Since then, I began giving explicitly worded notes including, but not limited to,
step by step instructions. Through observing common learning patterns among
students, I also was able to predict common questions or areas of confusion, so I
would give notes to preemptively answer questions such as “What do I look
for?” or,
“When should I use this?” or,
“What will it look like?”
and prepare students for frequent mistake areas by also showing what not to do,
along with what to do.
These experiences inspired me to record my notes and make them
available to any student who wishes they had another resource to make learning
math and algebra easier. As I stated, this book is not a textbook, and by that I
mean I don’t give extensive examples and practice questions the way textbooks
do. Math textbooks are generally very good at giving them and contain a wealth
of information. But traditional textbooks also teach in a very rigid and often
bottom-up way. I’ve found that many textbooks teach certain topics to such a
sub-categorical level of detail that students lose sight of how it connects to the
bigger picture. So what I offer are other perspectives to the math from the
textbooks, and I sometimes unveil them in a more top-down way.