Pre-Algebra: A practical step-by-step approach
[Pages:380]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: pearblossominc@ Phone: 800-309-3569
FOURTH EDITION
ALL RIGHTS RESERVED. This book (or ebook) contains material protected under International and Federal Copyright Laws and Treaties. Any unauthorized reprint or use of this material is prohibited. No part of this book or ebook (electronic book) may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system without express written permission from the author / publisher. For permission requests, write to the publisher, addressed "Attention: Permissions Coordinator," at the address above. Ordering Information: Quantity sales. Special discounts are available on quantity purchases by corporations, associations, and educational institutions. For details, contact the publisher at the address above. Please contact Pearblossom Academy, Inc.: Tel: (800) 309-3569; Fax: (601) 835-1718
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.
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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 mathematics, 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.
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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.
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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
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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
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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
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