Pre-Algebra: A practical step-by-step approach

380 Pages·2015·8.94 MB·English

Pre-Algebra: A practical step-by-step approach
Gary S. Goldman, PhD
Dedicated to my 5th grade math teacher—Mr. Hivesly

Author: Gary S. Goldman, Ph.D.
Title: Pre-Algebra: A Practical Step-by-Step Approach

Editor: Preston R. Meltzer, BA, Educator (35 years experience), Life-time Teaching Credential

Copyright © 2012, 2013, 2019 Gary S. Goldman
Publisher:
Pearblossom Academy, Inc., 1882 Mill Creek Ln SW, Bogue Chitto, MS 39629
Email: [email protected] Phone: [phone]

FOURTH EDITION

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About the Author
I am 59 years old, but I still recall an event from 1964 that had a significant impact on my
life. I was 10-years-old in 5th grade at Gardenhill Elementary School located in La Mirada,
California. Two individuals in that grade especially influenced me back then: (1) my friend Jill
Hagan, who slapped me in the face when I failed to vote for her as class president; and (2) my
classroom teacher, Mr. Hivesly, who contributed greatly to my comprehension of math concepts.
On the day that I received the slap (which occurred during recess), all students in Mr. Hivesly’s
classroom had voted for class president and had taken a math test that involved problems on
simple division. Several days later, these tests had now all been graded. Students were in
anticipation as the teacher returned each test paper with the letter grade shown prominently as a
large letter at the top of the test page. I missed virtually every problem and could not stop
focusing on the large “F” at the top of my test. I was near to tears, but was too embarrassed to
cry in front of my classmates—so I was unusually quiet for the remainder of class trying to
maintain a normal composure without any tears. For me, time seemed to be dragging on, until
finally the bell rang all students were formally dismissed by the teacher. I purposely was slow in
organizing my homework and to carry home—making certain that all students had exited the
class when I finally approached my teacher. I think at this point, I did have tears in my eyes as I
politely asked, “Mr. Hivesly, I just don’t understand how to divide.”
Mr. Hivesly discerned my tears (though I tried to fight them back by wiping my eyes as they
teared up). He patiently took the time right then to show me an example problem on the
blackboard: 43 divided by 8. I wiped back my tears and tried to focus as he took the chalk in his
hands and kindly asked, “How many times does 8 go into 43?”
I told him, ”I just don’t know—I don’t understand how to divide.” He told me, “Then just
make a guess, any guess.”
I said, “How about 2?” He said, “Well, 2 times 8 is 16, that’s too low, we’re trying to get to
43, without going over that number. So guess again.”
I said, “How about 4?” Mr. Hivesly said, “Well, 4 times 8 is 32—you’re getting closer!”
I said, “Try 5.” He said, “5 times 8 is 40. Now, 43 minus 40 is 3, so the answer is 5 remainder
3.”
I said, “I think I am starting to understand, can we do another problem?”
Mr. Hivesly proceeded to show me two more division problems before I thanked him,
dismissed myself, then walked home (without my “friend” Jill)—arriving just 10 minutes later
than usual. But I learned a valuable lesson (at least in math—not about females) that lasted a
lifetime! From that point on, I not only understood division; but I learned other math concepts
often as quickly as they were presented to me. For the remainder of 5th grade, continuing through
college, where I took Calculus, Vector Analysis, and other math courses, I always earned an “A”.
In fact, in an advanced course called Fourier Series and Laplace Transforms, I actually never
missed a single test problem—math came easy to me. And in this book, I will try to make math
easy for you!
It is not uncommon for a student in a math class to ask such questions as: “Why is this
important? What benefit could this possibly have?” Where possible, in this book, math concepts
will be put to use in real world problems and realistic applications.
iii
In college, I had a double major: engineering and computer science. I received two Bachelor
of Science degrees. Later, I received a Ph.D. (or doctorate) degree in Computer Science. In my
employment, I developed the first microcomputer-based CAD (Computer Aided Drafting)
System (prior to today’s widely known AutoCAD system). I also patented a “Power Wheel”—an
efficient motor in a wheel for electric cars. The CAD applications software allowed a user to
create various engineering drawings. Writing that software required mastery of basic mathe-
matics, algebra, geometry, and trigonometry. Later work involved some statistical methods and
modeling of disease trends within a community. Math, to me, was not just variables and
equations, it was a way to analyze and model real world applications. While some persons are
satisfied recording data and tabulating figures, I have always been interested in understanding the
underlying reasons for any trends or patterns that exist among the numbers.
Over the past 40+ years I have acquired real-world experience as a (1) designer and tester of
electronic modules for Hughes Ground Systems, Fullerton, (2) contractor with a Class B General
Contractor's License, (3) senior systems analyst in the Computer Aided Drafting (CAD)
department at Holmes & Narver, Inc. (in Orange, CA), (4) computer consultant to CSUF faculty,
staff, and students, (5) vice-president of Systems Development for Cascade Graphics
Development (in Irvine, CA) producing the first microcomputer-based CAD system, (6) research
analyst for the varicella surveillance project funded by the CDC, responsible for authoring
medical journal papers on capture-recapture methods, varicella and herpes zoster, and other
epidemiological subjects, (7) inventor of the “Power-Wheel” (U.S. Patent)―a highly efficient
electric motor for vehicular transportation, (8) independent computer consultant working on a
variety of projects ranging from the automation of the design of vertical cylindrical heaters to
earth retainer walls with traffic loads (involving heuristic programming) and (9) substitute
teacher for all subjects and all grades in three local school districts.
This real-world work experience combined with the answering of student math questions
during the past twenty years as founder and director of Pearblossom Private School, Inc. has
contributed to my ability to understand the math difficulties that students frequently encounter
and direct students to a straightforward, practical, step-by-step procedure for solving problems.
It is highly recommended that you, the student, do every single exercise in this book. The
repetition and variety of problems well help to re-enforce the concepts in each section so that you
gain self-confidence and competency. I have always maintained that if a student received a poor
grade in math, this was more a reflection on the qualities of the teacher—rather than the ability
of the student. This course will provide you with algebra fundamentals that will help you to score
well on most college entrance exams (such as S.A.T.) or placement tests. But it may also impact
your ability to think logically and apply math concepts in other endeavors.
It is recommended that you NOT utilize a calculator for most problems in this book. As a
prerequisite for this course, you should already be adept at reciting multiplication tables from
memory and knowing the basic computational skills associated with addition, subtraction,
multiplication, and division, as well as some basics working with fractions and decimals.
Sincerely,

Gary S. Goldman, Ph.D.
iv
About this Book

This pre-algebra textbook (or ebook) is different from other math texts you have previously used
in school where you were taught by a classroom teacher. This book, for the most part, is your
teacher and tutor and as such, it will guide your step-by-step learning. Just as you should pay
strict attention to a teacher’s lecture in a classroom, you will need to carefully read each and
every sentence of the book narrative, Your learning will be enhanced by doing the problems in
the exercise presented at the end of each section.

You may not actually have read any of your previous math texts—perhaps only opening them to
do various even- or odd-numbered problems assigned as homework by the teacher. But more is
required of you as an independent learner studying pre-algebra. First, you must closely follow
the logic or reason behind each and every step that is used to solve various example problems
that are worked in extreme detail. Try to comprehend the specific solution to the problem as it is
explained in the narrative. Then after considering all the example problems, try to grasp the
general approach and method of solution that might be applicable to solve other similar
problems.

Reinforce your mastery of each newly introduced concept by doing all the problems that follow
each section. These problems provide further learning opportunities and reinforce the concepts
that were previously introduced. Oftentimes, concepts previously learned in earlier sections of
the book will need to be recalled and applied to achieve a solution to a current problem. By
having to recall knowledge previously acquired, this serves as a sort of ongoing review. While
some problems will be straightforward and vary little from the example problems discussed in
the narrative, you will find other problems progressively more difficult to solve—perhaps
requiring more innovative or challenging thinking that is often characteristic of the mindset of a
mathematician or scientist.

Answers are provided to each and every question so that you can confirm each of your answers
is correct. If your answer is incorrect, review the step-by-step procedures used to solve the
example problems in the narrative and try to determine at what step you deviated from the
correct procedure or method of solution.

At the conclusion of this course, how well you understand pre-algebra concepts and maintain
pre-algebra skills will directly depend on how closely you have followed the above suggestions.
To develop mathematical insight and gain an understanding of abstract concepts and their
application takes time. If taken seriously, this course can profoundly influence your perception of
things and equip you with analytical skills that may be useful in your future pursuits and
endeavors—especially with respect to taking your next math course in Algebra I. If you master
the concepts presented in this pre-algebra course, you will find Algebra I to be largely a review
with the introduction of only a few new topics.

v
Table of Contents
About the Author ..................................................................................................................... iii
About this Book ..........................................................................................................................v
Chapter 1: Expressions and Integers
1.1 Constants, Variables, Terms, Expressions, and Equations .................................... 1
1.2 Applying Capture-Recapture Methods .................................................................. 7
1.3 Adding and Subtracting Integers ......................................................................... 11
1.4 Multiplying and Dividing Integers ....................................................................... 15
1.5 Exponentiation ..................................................................................................... 18
1.6 Rounding .............................................................................................................. 21
1.7 Order of Operations ............................................................................................. 23
1.8 Word Problems .................................................................................................... 26
Answers to Chapter 1 Exercises .......................................................................... 29

Chapter 2: One-Step Equations and Decimals
2.1 Solving Equations using Addition and Subtraction ............................................. 34
2.2 Solving Equations using Multiplication and Division ......................................... 37
2.3 Adding and Subtracting Decimals ....................................................................... 41
2.4 Multiplying and Dividing Decimals .................................................................... 43
2.5 Properties ..............................................................................................................48
2.6 Mean, Median, Mode, and Range ........................................................................ 51
Answers to Chapter 2 Exercises .......................................................................... 55

Chapter 3: Two-Step Equations and Scientific Notation
3.1 Solving Two-Step Equations ............................................................................... 56
3.2 Solving Equations When the Variable is on Both Sides ...................................... 60
3.3 Torque, Lever Arms, Momentum, and Distance ................................................. 63
3.4 Lever Arms with a Given Weight (Optional) ....................................................... 71
3.5 Scientific Notation ............................................................................................... 74
Answers to Chapter 3 Exercises .......................................................................... 78

Chapter 4: Expressions with Fractions
4.1 Working with Fractions ....................................................................................... 79
4.2 Factors and Prime Factorization .......................................................................... 82
4.3 Greatest Common Factor and Least Common Multiple ...................................... 86
4.4 Addition and Subtraction of Equivalent Fractions ............................................... 90
4.5 Solving Algebra Using Sums and Differences .................................................... 94
4.6 Multiplying Fractions ........................................................................................... 98
4.7 Solving Algebra Using Products of Fractions ................................................... 101
4.8 Multiplying and Dividing Fractions ................................................................... 104
4.9 Solving Algebra Using Quotients of Fractions .................................................. 107
Answers to Chapter 4 Exercises ........................................................................ 109

vi
Chapter 5: Rational Numbers, Expressions, and Equations
5.1 Rational Numbers .............................................................................................. 110
5.2 Writing Rational Numbers ................................................................................. 113
5.3 Solving Rational Equations …………………………………….........................117
5.4 Simplifying Expressions by Combining Like Terms ......................................... 118
5.5 Solving Rational Equations .................................................................................126
5.6 Solving Equations by Combining Like Terms ................................................... 130
5.7 Using Multiplicative Inverses to Solve Equations ............................................. 133
5.8 Application of Pre-Algebra to Electronics (Optional) ....................................... 136
Answers to Chapter 5 Exercises ........................................................................ 141

Chapter 6: Polynomials
6.1 Monomials and Polynomials .............................................................................. 142
6.2 Adding and Subtraction Polynomials ................................................................ 146
6.3 Multiplying Monomials and Polynomials ......................................................... 149
6.4 Multiplying Binomials ....................................................................................... 151
6.5 Dividing Polynomials ........................................................................................ 154
6.6 Factoring Trinomials in the Standard Form ....................................................... 161
6.7 Solving Equations Containing Polynomials ...................................................... 163
Answers to Chapter 6 Exercises ........................................................................ 165

Chapter 7: Linear Equations and Graphing
7.1 The Rectangular Coordinate System and Graphing Points ................................ 166
7.2 Linear Equations ................................................................................................ 169
7.3 Graphing a System of Linear Equations ............................................................ 172
7.4 Slope and y-Intercept ......................................................................................... 176
7.5 Solving a System of Linear Equations by Elimination ...................................... 184
7.6 Solving a System of Linear Equations by Substitution ..................................... 189
7.7 Solving and Graphing Inequalities ..................................................................... 191
7.8 Cramer’s Rule (Optional) ...................................................................................197
Answers to Chapter 7 Exercises ........................................................................ 199

Chapter 8: Geometry
8.1 Polygons and Polyhedrons ................................................................................. 205
8.2 Triangles and Quadrilaterals .............................................................................. 210
8.3 Areas of Parallelograms and Trapezoids ............................................................ 215
8.4 Circumference and Area of a Circle ................................................................... 219
8.5 Volumes of Prisms and Cylinders....................................................................... 223
8.6 Volumes of Pyramids and Cones ........................................................................ 227
8.7 Surface Areas of Prisms and Cylinders .............................................................. 232
8.8 Surface Areas of Pyramids and Cones ................................................................ 235
8.9 Metric Units of Length, Area, Volume, and Weight .......................................... 239
8.10 Volume, Mass, and Density ............................................................................... 244
Answers to Chapter 8 Exercises ........................................................................ 247

vii
Chapter 9: Triangles, Trigonometry, and Transformations

9.1 Similar Figures ................................................................................................... 249
9.2 Similar Right Triangles ...................................................................................... 252
9.3 Sine and Cosine of an Angle ............................................................................... 255
9.4 Tangent of an Angle ........................................................................................... 262
9.5 Square Roots ...................................................................................................... 268
9.6 Calculating Square Roots (Optional) ...................................................................270
9.7 The Pythagorean Theorem ................................................................................. 275
9.8 Finding the Distance Between Two Points ..........................................................278
9.9 Translations ........................................................................................................ 281
9.10 Reflections, Rotations and Symmetry ................................................................ 284
Answers to Chapter 9 Exercises ........................................................................ 290

Chapter 10: Radicals

10.1 Introduction to Radicals ..................................................................................... 292
10.2 Simplifying Radicals .......................................................................................... 295
10.3 Adding and Subtracting Radicals ....................................................................... 298
10.4 Multiplying and Dividing Radicals .................................................................... 302
10.5 Simplifying Radical Expressions with Variables (Optional) ............................. 305
10.6 Solving Equations Containing Radicals...............................................................307
Answers to Chapter 10 Exercises ...................................................................... 310

Chapter 11: Quadratic Equations

11.1 Solving Quadratic Equations Using the Square Root Method ........................... 311
11.2 Solving Quadratic Equations by Completing the Square .................................. 315
11.3 Solving Quadratic Equations Using the Quadratic Formula .............................. 317
11.4 Derivation of the Quadratic Formula (Optional) ............................................... 323
11.5 Graphing Quadratic Equations ........................................................................... 325
Answers to Chapter 11 Exercises ...................................................................... 329

Chapter 12: Statistics and Probability

12.1 Permutations ...................................................................................................... 330
12.2 Combinations ......................................................................................................334
12.3 Probability .......................................................................................................... 337
12.4 Percentages .........................................................................................................343
12.5 Data Bias (Optional) ...........................................................................................349
12.6 Analyzing Data - Graphs ....................................................................................351
12.7 Misleading Statistics ...........................................................................................355
Answers to Chapter 12 Exercises .......................................................................358

viii
Appendix I: Trigonometric Values Corresponding to Angles 0 to 90 Degrees ................360

Appendix II: Square Roots of Numbers 1 through 200......................................................361

Appendix III: List of Prime Numbers from 2 through 9733 .............................................362

Appendix IV: Trigonometric Values for Some Common Angles ......................................363

Index ......................................................................................................................................... 364

More About the Author .......................................................................................................... 367

ix
Chapter 1
Chapter 1: Expressions and Integers

Before you start... did you read the introductory section on page v, About the Book? Please do
not proceed further without going back and first reading that section.
Pre-Algebra Defined
Pre-algebra is a branch of mathematics marked chiefly by the use of symbols to represent
numbers. Pre-algebra is a generalization of arithmetic.
In about the year 830 C.E., Mohammad ibn-Musa al-Khwarizmi of Baghdad (Iraq) wrote a
book in Baghdad with the Arabic title Hisab al-jabr w'al-muqabala, (or Science of the Reunion
and the Opposition). The reunion, or al-jabar, became what we presently call algebra, which
deals not so much with numbers themselves (or arithmetic) but with solving equations that
consist of operations (such as addition and multiplication) and relationships (such as equality).
1.1 Constants, Variables, Terms, Expressions, and Equations
I promised to make pre-algebra easy despite the fact that pre-algebra is somewhat more
abstract than general math. In general math you learned very early that 2 + 2 equals 4. Any
positive or negative whole number is called an integer (pronounced IN’TA-JEAR). So numbers
such as -103, -12, 0, 8, 25 are integers—they are also called constants (but constants can also
include fractions, mixed numbers, and decimals). Perhaps you did not realize it at the time, but in
the process of performing the indicated addition or calculating the answer to two plus two, you
were actually using one of the most basic concepts of pre-algebra—that of solving for a number
or quantity that is unknown. When given the problem:
two plus two equals what number
or written another way, 2 + 2 = ?

we are actually doing pre-algebra since we are trying to obtain an answer which is unknown or at
least not given in the stated problem. The unknown value, in pre-algebra (and later in algebra), is
called a variable and often some letter of the alphabet is used to represent an unknown value. In
the above problem, if we select n as the unknown value or variable, we could write: 2 + 2 = n. To
solve for n you simply perform the addition (2 + 2) and obtain the result of 4. So we could say n
= 4 in the equation 2 + 2 = n.
One or more different variables can actually appear anywhere in an equation. An equation or
equality, by definition, always contains an equal sign. Consider another related pre-algebra
problem: two plus what number equals 4? If we represent the unknown number as the variable x,
then we could write the equation:
2 + x = 4

If you told me that x is 2 in the above equation, you would be correct and at this point you
have solved a typical, basic pre-algebra problem and so you are actually doing pre-algebra. Now
was that difficult? Of course not!
If we had written simply 2 + x (without the equal 4), this is called an algebraic expression.
We cannot know the value of a variable (such as x) that appears in an expression until it is set
1