Table Of ContentTABLE OF CONTENTS
Introduction: The Subject of Algebra 1
Part I: The Real Numbers 3
Introduction to Part I: What are the Real Numbers?
Chapter 1: The Integers 7
1.1 Definition; Placement on the Real Number Line; Ordering
1.2 Opposites; Adding with Opposites
Chapter 2: Fractions 13
2.1 Basic Definitions and Representations
2.2 Fractions in the Real World: Cutting, Pouring and Constructing
2.3 Fractions in the Real World: Measuring
2.4 Equal or Close to 0 or 1; Complements
2.5 The Number -'-
2
2.6 Obvious Comparisons
2.7 Equivalent Fractions
2.8 Simplifying Fractions
2.9 Anti-simplifying Fractions
2.10 Comparing Fractions
2.11 Locating Fractions on the Real Number Line
2.12 Probability: A Cool Excursion
Chapter 3: Mixed Numbers and Fractions Greater Than One 56
3.1 The Meaning of Mixed Numbers; Representations
3.2 Mixed Numbers in the Real World: Cutting, Pouring and Constructing
3.3 Mixed Numbers in the Real World: Measuring
3.4 Close to Integers; Complements; Halfway Between Integers; Simplifying
3.5 Comparing Mixed Numbers
3.6 Fractions Greater Than One
3.7 Locating Mixed Numbers and Fractions Greater Than One on the Real Number Line.
3.8 Ratios: An Important Application of Fractions
Chapter 4: Decimals and the Nature of 9{ 82
4.1 Definition; Notation; Representations
4.2 Decimals in the Real World
4.3 Close to Integers; Complements; Halfway Between Integers
4.4 Comparing Decimals; Approximating Decimals
4.5 Converting Fractions to Decimals
4.6 Rational Real Numbers; Irrational Real Numbers
4.7 The Nature of 91
Part II: The Operations on the Real Numbers 97
Introduction to Part II: The Two Main Operations
Chapter 5: Addition 101
5.1 The Basics; The Main Principle of Addition
5.2 Adding Fractions, Mixed Numbers, Decimals and Roots
5.3 Applications of Addition
5.4 Subtraction
5.5 Applications of Subtraction
Chapter 6: Multiplication 112
6.1 The Basics; The Main Principle of Multiplication
6.2 Multiplying Fractions, Mixed Numbers, Decimals and Roots
603 Exponents
6.4 Nth Roots
6.5 The Main Application of Multiplication: Rate
6.6 A Special Application of Multiplication: Area
6.7 Fractions of Numbers
6.8 "Out of'
6.9 Division
6.10 Applications of Division
Chapter 7: Combining Operations 137
7.1 Order of Operations
7.2 Applications Involving One Operation
7.3 Applications Involving Multiple Operations
Part III: The Fundamentals of Algebra 145
Introduction to Part III: The Essence of Algebra
Chapter 8: Basic Ideas and Definitions 148
8.1 Looking for Algebraic Facts
8.2 Algebraic Expressions: The Basics
803 Getting the Big Picture: Equal Algebraic Expressions
Chapter 9: Operations on Algebraic Expressions 155
9.1 Addition (and Subtraction) Viewed Algebraically
9.2 Multiplication (and Division) Viewed Algebraically
9.3 Factoring and Simplifying Fractions Viewed Algebraically
9.4 Simplifying Algebraic Expressions
Chapter 10: Equations and Problem-Solving 176
10.1 Solving Equations: The Basics
10.2 Solving Formulas for a Given Variable
1003 Translating from English to Math
lOA Numerical Word Problems
10.5 Real World Word Problems
10.6 Solving Equations: Factoring Zero
10.7 Solving Inequalities: The Basics
Chapter 11: Equations of Two Variables and Graphs 198
11.1 Graphs of Equations
11.2 Graphing Linear Equations
1103 The Slope of a Line
1104 Systems of Equations
11.5 Word Problems With Two Unknowns
11.6 Graphing Quadratic Equations
Appendix A: The Whole Numbers and Their Operations 233
A.1 Reading and Writing Whole Numbers
A.2 Approximating Whole Numbers
Ao3 Adding and Subtracting Whole Numbers; Applications
AA Multiplying and Dividing Whole Numbers; Applications
A.5 Factoring and Divisibility
INTRODUCTION: THE SUBJECT OF ALGEBRA
Algebra is one of the two most fundamentally important subjects in all ofK-12 education - second only,
of course, to reading and writing. As the foundation upon which most of Mathematics rests, and as one of
the most powerful problem-solving tools ever created, Algebra stands as the gateway to most fields of
study and to countless careers in today's society; from computer science to business to medicine to
cngineering and on and on and on. Even more importantly, as a subject in its own right, Algebra, like all
of Mathematics is an amazing power source for the human mind. It teaches you to think-to think clearly
and in very powerful ways; logically, analytically, abstractly. It develops your communication skills and
builds strong work habits in terms of being careful and accurate in your work. It fosters an appreciation
for precision. And it provides nourishment for other cool things like your intellectual curiosity, intuition
and creativity. Algebra is a lot like weight-lifting. Football players lift weights regularly, although never
in an actual game. They do it to prepare for the game. In much the same way, Algebra is the ultimate
exercise for the mind. And whether or not you actually ever use it day to day, in your career or in some
other way, it prepares you for the game. And in this case, we're talking about the game of life.
So, what exactly is Algebra? Well, there's actually a pretty simple answer to that.
Definition: Algebra is the study of the operations on the Real Numbers.
So, learning Algebra really just comes down to understanding two things:
I) The Real Numbers: What are they, what do they look like, and where do they live?
2) The Op"erations on the Real Numbers: What are they and how do they work?
Of course, learning Algebra is just part of what you want to do. You also want to learn how to use it.
Having a car is nice, but it's even nicer if you know how to drive it, and know of a bunch of cool places to
go to. Learning to use Algebra involves learning about some of the most important things in all of
Mathematics, like equations and graphs. And, it definitely involves learning something about problem
solving and how to translate real world problems described in English (or some other language), into the
language of Mathematics. All of these things will be presented here.
1
2
PART I
THE REAL NUMBERS
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