# Mathematics 12A Algebra Level I

## About the Author

Rebecca Alano knew at the age of 10 that she wanted to be
a teacher. It

wasn’t until she took algebra in high school that she realized what subject she

wanted to teach. “When I was introduced to the logical , beautiful language of

mathematics through algebra, I was hooked.” Mrs. Alano thoroughly enjoys

studying mathematics and sharing her love of the subject through teaching others

about it. From 1994 to 2001, she taught nine different Independent Study Program

math courses—all nine of them at once for a short period of time. “It was a
great

way for me to stay at home with my children and still be actively teaching math

to students all over the world.” Once her children started school, she because a

full-time copy editor for the instructional development unit of the School of

Continuing Studies, where she helps create the learning guides that accompany

independent study courses. “Writing the learning guide for an independent study

course forces me to develop creative ways of explaining concepts to students.
I’m

still teaching—just not in as direct a fashion.” When not working as a copy
editor,

Mrs. Alano enjoys reading mysteries, cross-stitching, working in her church,

attending plays and concerts with her husband, and reading Harry Potter books

with her two children.

## Table of Contents: Mathematics 12A

Important Information . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . i

Study Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . iii

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . v

Study Materials Order Form

Lessons

1 Slopes and Equations of Lines . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 1

2 Graphing Lines and Linear Inequalities; Exponents . . . . . . . . . . . . 25

3 Power Rules and Graphing Quadratic Equations . . . . . . . . . . . . . . . 51

4 Tips for the First Examination . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 81

Application for the First Examination

5 Solving Quadratic Equations; Square Roots ; Polynomials . . . . . . . 91

6 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 133

7 Tips for the Second Examination . . . . . . . . . . . . . . . . . . . . . . .
. . . . 167

Application for the Second Examination

8 Factoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 175

9 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 213

10 Tips for the Final Examination . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 241

Application for the Final Examination

11 Cumulative Project Evaluation . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 249

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Appendix A: Answers to the Practice Problems and

Practice Exams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 251

Appendix B: Skill Enhancement Exercises

The Slope of a Line . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 325

Products and Powers of Powers . . . . . . . . . . . . . . . . . . . . . .
325

Negative Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 326

Quotients of Powers . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 326

Powers of Products and Quotients . . . . . . . . . . . . . . . . . . . .
327

Using the Quadratic Formula . . . . . . . . . . . . . . . . . . . . . . .
. 327

Multiplying Polynomials . . . . . . . . . . . . . . . . . . . . . . . . .
. . 328

Solving Systems of Equations . . . . . . . . . . . . . . . . . . . . . . .
329

Factoring x ^2 + bx + c . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 330

Factoring ax^2 + bx + c . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 331

Appendix C: Answers to Skill Enhancement Exercises . . . .
. . . . . 333

Appendix D: Instructions for Using GCalc . . . . . . . . . . . . . . . . . . .
341

## Introduction

Welcome to the second semester of algebra• You’re
continuing your study of

a very exciting topic. The word algebra comes from the Arabic al-jabr, which

means “the reunion of broken parts.” I like to view algebra as a puzzle having

many pieces that can be put together to form a bigger, complete picture. When

you work a jigsaw puzzle, the pieces ultimately fit together only one way. Yet

there’s more than one way to arrive at the final picture. Your technique will

probably not be identical to that of a friend who puts the same puzzle together,

but the outcome is the same. Likewise in algebra, while there’s one final
correct

answer, the way that you get that answer may not be identical to someone else’s.

But as long as you each follow the rules of algebra, your answers will be

equivalent.

**Prerequisites**

You should have completed one semester of Algebra I before
taking this course.

If you did not take Math 11A through the Independent Study Program or you

used a different textbook in your first semester of Algebra at your school, you

may discover as you progress through the course that we covered some skills that

you didn’t get to in your course. If so, locate the appropriate topic from
chapters

1–6 of the textbook and do some of the problems for those sections. Check your

answers in the back of the textbook.

**Course Objectives and Content**

Your first year of algebra is the foundation for geometry,
higher-level algebra

courses, trigonometry, and calculus. By the completion of Math 12A, you’ll

be able to graph and use linear equations and inequalities; explore, graph, and

interpret nonlinear equations; perform operations with polynomials; graph and

solve systems of linear equations and inequalities; and demonstrate a basic

understanding of relations and functions. You’ll also see how these topics apply

to science, technology, sports, careers, consumer information, and other fields.

The course is comprised of written assignments, projects,
and examinations.

There are seven written assignments, which you’ll turn in to be graded by your

instructor. You’ll also choose one project from a list to submit with each
written

assignment after the first one. Three examinations, each emphasizing two to two

and a half chapters of the textbook, round out the course.

The following is a breakdown of each lesson’s content:

**Lesson 1—Slopes and Equations of Lines**

1A— Rate of Change

1B—The Slope of a Line

1C—Properties of Slope

1D—Slope-Intercept Equations for Lines

1E—Equations for Lines with a Given Point and Slope

1F— Equations for Lines Through Two Points

**Lesson 2—Graphing Lines and Linear Inequalities;
Exponents**

2A—Fitting a Line to Data

2B—Equations for All Lines

2C—Graphing Linear Inequalities

2D—Compound Interest

2E—Exponential Growth

2F—Comparing Constant Increase and Exponential Growth

2G—Exponential Decay

2H—Chapter 7 Project

**Lesson 3—Power Rules and Graphing Quadratic Equations**

3A—Products and Powers of Powers

3B—Negative Exponents

3C—Quotients of Powers

3D—Powers of Products and Quotients

3E—Remembering Properties of Exponents and Powers

3F—Graphing y = ax^2

3G—Graphing y = ax^2 + bx + c

3H—Graphing Parabolas with an Automatic Grapher

3I—Quadratic Equations and Projectiles

3J—Chapter 8 Project

**Lesson 4—First Exam **(emphasizing lessons 1–3)

**Lesson 5—Solving Quadratic Equations; Square Roots;
Polynomials**

5A—The Quadratic Formula

5B—Analyzing Solutions to Quadratic Equations

5C—Square Roots and Products

5D—Absolute Value, Distance, and Square Roots

5E—Distances in the Plane

5F—What Are Polynomials?

5G—Investments and Polynomials

5H—Multiplying a Polynomial by a Monomial

5I—Multiplying Polynomials

5J—Multiplying Binomials

5K—Special Binomial Products

5L—Chapters 9 and 10 Project

**Lesson 6—Linear Systems**

6A—An Introduction to Systems

6B—Solving Systems Using Substitution

6C—More Uses of Substitution

6D—Solving Systems by Addition

6E—Solving Systems by Multiplication

6F—Systems and Parallel Lines

6G—Situations Which Always or Never Happen

6H—Systems of Inequalities

6I—Chapter 11 Project

**Lesson 7—Second Exam **(emphasizing lessons 5–6)

**Lesson 8—Factoring**

8A—Factoring Integers into Primes

8B—Common Monomial Factoring

8C—Factoring x^2 + bx + c

8D—Solving Some Quadratic Equations by Factoring

8E—Factoring ax^2 + bx + c

8F—How Was the Quadratic Formula Found ?

8G—Rational Numbers and Irrational Numbers

8H—Which Quadratic Expressions Are Factorable?

8I—Chapter 12 Project

**Lesson 9—Functions**

9A—What Is a Function?

9B—Function Notation

9C—Absolute Value Functions

9D—Domain and Range

9E—Probability Functions

9F—Polynomial Functions

9G—The Tangent Function

9H—Functions on Calculators and Computers

9I—Chapter 13 Project

**Lesson 10—Final Exam** (emphasizing lessons 8 and 9)

**Lesson 11—Cumulative Project Evaluation**

**Course Textbook**

The course textbook is The University of Chicago School
Mathematics Project:

Algebra, by John W. McConnell et al. This very readable textbook emphasizes the

application of algebra to everyday situations and shows the connection between

algebra and other topics in mathematics. We’ll study chapters 7–13 in this
course.

(Chapters 1–6 were covered in Math 11A.)

The textbook is part of the University of Chicago School
Mathematics Project

(UCSMP), which at this writing is celebrating its twentieth year of successful

teaching of pre-college math called. The project is supported by corporate and

foundation funding with the goal of having U.S. students gain mathematical

competence on a par with other countries, which have traditionally done better

in math than U.S. students. Students using this textbook and others in the
series

score well on traditional standardized tests while doing better than other
students

in problem solving and fields related to algebra such as statistics. The
textbook’s

applications approach helps students make sense of algebraic ideas. The

Independent Study Program currently offers six courses covering pre-algebra,

algebra, and geometry based on these textbooks.

**Other Course Materials**

For this course you’ll need a notebook or loose-leaf
binder of 8½-by-11-inch lined

paper, a ruler, graph paper, a scientific calculator, and an automatic grapher.
An

automatic grapher called GCalc is included on a disk found in the pocket
envelope

in the back of this learning guide. You’ll use this or another automatic grapher
of

your choice.

**Design of Lessons**

Each nonexam lesson (except lesson 11) begins with a list
of objectives, which

describe the skills you’ll learn in the lesson. Each nonexam lesson is then
divided

into several study sessions, which contain the following parts:

• **Reading Assignment**—The reading assignment tells
you which pages to

read in the course textbook. Go slowly and carefully through the reading

assignment before you read each study session’s discussion. Work out the

textbook examples in your notebook, making sure you understand the point

of each example.

• **Discussion**—The discussion clarifies the material
in the reading assignment

and includes additional examples to help illustrate the topics covered in the

reading assignment. You’ll know that you’ve reached the end of an example

when you see this mark: .

•** Practice Problems**—These problems help you check
your understanding of

the study session’s material. Do the practice problems before you do the study

session’s written assignment. Answers to these problems are located in the

answer section in the back of the textbook and in learning guide appendix A.

Do the practice problems on lined paper and keep them together in a notebook

or loose-leaf binder. Students who take the time to do the practice problems

tend to receive higher scores for the course than those who don’t.

• Written Assignment—Each study session ends with a
written assignment

that you must complete and submit to your instructor for grading. Do the

written assignment problems on 8½-by-11-inch lined paper. Please write on

only one side of the paper and leave space between your answers. Be as neat

as possible since your instructor cannot grade material that can’t be read.

Include all work you do to answer each problem. Circle or box your final

answers to help your instructor locate them. If a problem requires graphing,

you must use graph paper and label the units and axes clearly on your graphs.

Some written assignments provide opportunities for extra credit.

The final study session in each nonexam lesson (except
lessons 1 and 11) is a

chapter project in which you apply the concepts you learned in the lesson. From

a list of three projects, you must choose one project to complete and submit
along

with the lesson’s written assignments. By the end of the course, you’ll have

completed six chapter projects. The five projects receiving the highest scores

will count toward your final course grade. Lesson 11, the cumulative project

evaluation, is where your final project grade will be recorded. We provide you

with the form that shows how your instructor will compute your final project

grade (as well as your overall course grade), but there is nothing for you to
submit.

You will have already completed this lesson before you get to it. Your
instructor

will include your overall project grade on the final examination cover sheet
(with

your final examination grade and your overall course grade).

A written assignment checklist ends each nonexam lesson.
Use this checklist to

double-check that you’ve done all of the lesson’s assigned problems before you

submit them for grading.

**The Exams**

Each of the course’s three exams emphasizes material from
two or three textbook

chapters. Each exam consists largely of open-answer questions with some true/

false, multiple choice, or matching questions. All of the exams’ questions are

similar to the types of questions you’ll do in the written assignments. Lessons
4,

7, and 10 provide further details about the exams, as well as practice exams.

**Grading Standards**

The course’s written assignments are worth 40% of your
final course grade (seven

written assignments worth each), your five
best chapter projects are worth

15% of your course grade (3% each), and the three exams are worth 45% of your

course grade (15% each).

The grading scale for the course is as follows:

90 – 100% A

80 – 89% B

70 – 79% C

60 – 69% D

0 – 59% F

Note: To be in compliance with the Independent Study
Program’s academic

policies, your exam grades must average at least a D– in order for you to pass
the

course. Even if your written assignment grades are excellent, you will not pass
the

course unless you fulfill this requirement.

**Plagiarism**

As an educational institution, IU puts learning first. We
want you to learn, and we

think you value learning as well. We also value honesty and trust. You have
every

right to expect fair exams, fair assignments, and fair grades. By the same
token,

your instructor expects the work you hand in to be your own. You are welcome to

discuss this course with other students and teachers, but when it comes to
writing

your assignments, all the words should come straight from you, unless you are

supporting your assertions with a properly cited quote.

Passing off someone else’s work as your own is plagiarism.
As stated in Indiana

University’s Code of Student Rights, Responsibilities, and Conduct (Art. III, §

A.3), “A student must not adopt or reproduce ideas, words, or statements of

another person without an appropriate acknowledgment. A student must give due

credit to the originality of others and acknowledge an indebtedness whenever he

or she does any of the following:

a. quotes another person’s actual words, either oral or
written;

b. paraphrases another person’s words, either oral or written;

c. uses another person’s idea, opinion, or theory; or

d. borrows facts, statistics, or other illustrative material, unless the
information

is common knowledge.”

We take plagiarism very seriously. If you are caught
plagiarizing, you could

receive an F for the whole course.

So how can you avoid plagiarizing? When is it appropriate
to cite your sources,

and how should you cite them? The answer’ s simple . Ask your instructor. If

you’re unsure whether you’ve cited your sources appropriately, call or e-mail

your instructor before you submit your assignment. Not only will you get answers

to your questions, you’ll reap the fruit of honesty: trust.

**Contacting Your Instructor**

With each lesson you are required to submit an assignment
cover sheet. Every

assignment cover sheet has a space for your questions and comments; you are

strongly encouraged to use this space. If problems arise between assignments,
you

can write to your instructor at the Independent Study Program. Many instructors

can be contacted via e-mail or reached by telephone during established office

hours. To learn your instructor’s e-mail address and/or office hours, please
refer

to the contact information on the back cover of this learning guide.

**Some Tips before We Begin**

Studying mathematics is different from studying other
subject areas. All those

jumbles of numbers can look confusing. Don’t give up• It’s important that you
pay

attention to those jumbles in the examples in the textbook and learning guide.
Try

to figure out what’s different from one step in the example to the next, and
then

ask yourself how or why what was done was done. Look at other examples for

similar steps.

If you begin to get frustrated, don’t throw the textbook
or break your pencil. (My

favorite stress reliever in high school was to bite my pencil. Sometimes I
didn’t

realize how stressed I was until I noticed that I had nearly bitten my pencil in
half•

Then I knew I needed to take a break.) Get away from the problem for a few hours

or even a day. When you come back, things will probably be clearer. If you’re
still

confused after you return to the problem, ask someone for help. You may also

contact your instructor, who’ll be happy to explain the material to you. Just be

sure when you contact your instructor to have the problem in front of you,

including the scratch work you did to try to figure out the problem. This
process

of trying to understand the material, as well as the jumbles of numbers and
steps,

are part of putting together the puzzle of algebra.

While there are a lot of numbers on the pages, the words
are important too.

Learning algebra is like learning any language. You must learn the vocabulary of

algebra to succeed in algebra. If you can’t understand what you’re being asked
to

do, how can you do it? Reread material that doesn’t make sense the first time.

Look at the textbook and learning guide examples to see how the words explain

what’s being done with the numbers. Again, if you still can’t figure it out, ask

someone for help.

Choose to work in an environment that enhances learning
for you. I prefer to study

mathematics with music in the background, but you might require absolute silence

when studying math. (I require a quiet environment when I’m writing papers or

trying to comprehend most science courses.) You may like working in a cool

room to help you think, or you might prefer (as I do) a slightly warmer room
(but

not too warm; you might fall asleep). Maybe you like a very bright room. Perhaps

you need an orderly, organized, almost sterile desk with no materials other than

your math book and papers on it. If you’re a morning person, try working on your

math as early as you can each day; if you’re a night owl, work in the evenings.

The point is that you need to find out what helps promote learning for you and
try

to achieve that environment to the best of your ability each time you study. If
you

don’t already know what you prefer, try some of the environments and times

mentioned above until you find the combination that seems to work best for you.

And now, let’s work on putting together the puzzle of algebra•

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