Algebra Textbook Volume 1 Student Edition wc

Now, I Can Understand Algebra
Algebra Text, Volume 1

The purpose of this integrated algebra textbook is to enhance the mathematical literacy
of below grade level to moderately at-risk students. The textbook is laid out in a
straightforward, linear manner designed to develop and deepen conceptual
understanding of topics. Hence, its name “Now, I can Understand Algebra.” The hope is
that every student expresses that sentiment at some point.
This textbook is part of a two-volume series. The first volume covers the first half of a
standard algebra course. Included in the first four chapters are the requisite
foundational skills necessary for algebra success. The second volume covers the second
half of a standard algebra course. It can also be used as an algebra review textbook for
those who need to shore up their algebra in order to successfully master calculus.
Albert Einstein is attributed with saying, “If you can’t explain it simply, you don’t
understand it well enough.” This book subscribes to that principle. The language is
simple and the arithmetic computation basic which allows the general, at-risk and
special education populations the opportunity to learn without the difficulty of
cumbersome arithmetic and the manipulation of large numbers. Essential skill
development is addressed and reinforced throughout the book so that pre−existing
deficiencies and lack of prior knowledge is not problematic. Key foundational skills are
introduced prior to that skill being utilized in a mathematical application.
Additionally, integrated throughout the book are opportunities for all students to delve
deeper and to explore the mathematical concepts taught.
Within each chapter are real−world, interdisciplinary problem−based explorations.
These activities help students internalize the material taught and answer the universal
question, “Why must I learn this?”
Instructional videos are provided on selected topics. These topics were chosen because
they have often caused unnecessary difficulty; however, in these videos they are
presented in a simple, yet thorough manner.
Throughout the textbook are teacher notes, based upon evidenced-based (best
practices) and over 100 years of combined teaching experience among the authors.
These practical notes will help any teacher in the art of questioning, eliciting conceptual
understanding, and assessment; but they are especially helpful to the novice teacher or
one new to teaching algebra.
This textbook adheres to both the Common Core State Standards and the National
Council of Teachers of Mathematics Principles and Standards.

1|Page

Annotated Chapter Contents
Chapter One

Chapter One is divided into three sections:
1. Cancelling and Integer Arithmetic − reviews basic math facts, integer
arithmetic, subtraction and division of positive and negative numbers.
2. Graphing −− introduces basic graphing vocabulary, identification of points
on coordinate axes and graphing linear and circle equations.
3. Basic Operations −− includes the bedrock operations for algebra (exponents,
combining like terms and order of operations).

For many students, this section might be a review of previously covered material, but it
will also strengthen their proficiency and numeric understanding. For others it may be
new, but it will provide the requisite understanding that, up to now, may have been
lacking. The objective is to provide the students with the opportunity to strengthen
these basic skills and to achieve success, often an obstacle in the study of mathematics

Chapter Two
Topics in Chapter Two include solving equations and working with proportions and
percentages. There are four sections to this chapter, each including several basic skill
developments. These skill development exercises are designed to accomplish three
goals: (a) to shore up computational or basic conceptual weak areas that will adversely
affect students continued algebraic development; (b) to provide the necessary practice
needed to promote efficiency and accuracy; and (c) to make future concept areas of
algebra easier to master. Hence each section is comprised of an algebraic and a skill
development component.
There are four sections in Chapter 2.

1. Solving Simple Equations −− introduces solving simple one- and two-step
equations using addition, subtraction, multiplication, and division.
Skill Development – greatest common factors, prime factorization, fractions,
and words and their equivalent mathematical expressions

2. Solving Multi−step Equations −− introduces the four-step method for solving
multi−step equations.
Skill Development: place value and rounding, decimals, percents, absolute
value, and scientific notation

2|Page

3. Solving Absolute Value Equations – builds upon the skill review of the last
section and includes common misconceptions in solving absolute value
equations, such as distributing over the absolute value bars.

4. Ratio and Rates – introduces comparing two numbers with the same and
different units (ratio and rates), determining unit rates, conversions, and
proportions.
Chapter Three

This chapter is an introduction to word problems involving the simplest one equation,
one unknown sort. The purpose of dedicating a complete chapter to word problems is
to promote student success, reduce word problem anxiety, and create a foundation or
platform from which students can solve more complex word problems in the future.
In this chapter, word problems are categorized based upon similar strategies required to
solve them. In this way, students begin to recognize patterns which facilitate
comprehension of the words; they see how each type of word problem can be written
algebraically.
There are two sections in Chapter 3; each section is comprised of an algebraic and a skill
development component.

1. Solving Pre−defined Word Problems −− introduces a category of word
problems which involve defining one variable in terms of another (relational
values). One term is defined directly (pre-defined), and the second is derived
from the first. Two types of these word problems are presented: solving
number equality word problems and consecutive number word problems
(including even and odd).
Skill Development – adding fractions and lowest common multiples;
multiplication and division of fractions; addition and subtraction of fractions
with different denominators and mixed numbers.

2. Solving Not Pre−defined Word Problems −− introduces the second category
of word problems, those in which the student must define the relational
values or quantities of the problem. Four types of such word problems are
presented: solving sums of numbers, perimeter and area, angles, and money
(cost) / purchasing word problems.
Skill Development: Review of all fraction and mixed number problems

Chapters 1-3 cover the major, basic skill development areas required to have success in
Algebra. Therefore, Chapters 4-7 do not contain separate skill development material.

3|Page

Chapter Four
Chapter four begins the second half of the book, where all foundational skills have been
covered.
Chapter 4 introduces inequalities, one-step through multi-step inequalities and
compound inequalities. This chapter builds on the material presented in Chapter Two−−
the solving of equations with the four-step method.
There are two sections in this chapter.

1. Solving Inequalities – explains algebraic expressions and graphing and
inequalities. Simple one−step and multi−step inequalities are solved.

2. Solving Compound Inequalities −− compound inequalities are illustrated
(graphed on a line) and solved.

Chapter Five
Chapter Five introduces relationships of numbers (arithmetic sequences, relations and
functions) and ordered pairs – the prerequisite material for understanding graphing of
equations later on in the text/course. Range and domain, independent/dependent
variables, and correlation graphs are also explored.
There are three sections in this chapter.

1. Relationships – explains arithmetic sequences, relations and functions. The
methods used to identify functions are also covered.

2. Function Notation – explores function equations, including variable
relationships (independent and dependent) and function notation (F(x)).

3. Interpreting Graphs – illustrates the relationship between ordered pairs
(graphing and interpreting correlations).

4|Page

Chapter Six
Chapter Six builds upon Chapter Five emphasizing understanding the relationship
between x− and y− values, including horizontal and vertical lines; graphing elements of a
line (slope, y−intercept); understanding the forms for the equation of a line, and
graphing linear equations. Finally, the chapter covers two aspects of linear relationships
– direct variation and functions.
There are three sections in this chapter.

1. Linear Equation Elements – introduces, interprets, and graphs the elements
of a linear equation (ordered pairs, slope, and y−intercept).

2. Writing and Graphing Linear Equations – interprets and graphs linear
equations expressed in the three major forms: slope-intercept, standard, and
point−slope.

3. Linear Relationships – examines another method for determining whether a
line is a function (via coordinates) and direct variation (y=kx).

Chapter Seven
Chapter Seven encompasses solving and graphing two equations (or inequalities) which
contain two variables (i.e., solving systems of equations). The three most common forms
are reviewed (standard form, slope−intercept form, and point−slope form). Then four
methods of solving these systems are explained (graphing, substitution, elimination, and
Cramer’s Rule (optional)).
There are three sections in this chapter

1. Solution Methods & Graphing – The methods for solving systems of
equations (graphing, substitution, elimination, and Cramer’s Rule) are
introduced individually.

2. Systems of Linear Inequalities and Graphing – solves and graphs linear
inequalities to determine the intersection. Linear programming is introduced
as a real−world application.

3. Review of Solution Methods – allows students to demonstrate mastery of the
solution methods. While Section 1 introduced the methods individually, this
section requires determining which method is most appropriate for solving a
given set of equations.

5|Page

Table of Contents
Introduction of Textbook

Introduction ....................................................................................................... 1.0A
Annotated Chapter Contents............................................................................. 1.0B
Table of Contents............................................................................................... 1.0C
Preface – Textbook Overview ............................................................................ 1.0D
Chapters (See next page for detailed chapter contents)

Chapter One: Integers and Skill Development ........................................ 1.0
Chapter Two: Solving Equations & Skill Development ............................ 2.0
Chapter Three: Word Problems & Skill Development ............................. 3.0
Chapter Four: Inequalities and Graphs.................................................... 4.0
Chapter Five: Functions and Graphs........................................................ 5.0
Chapter Six: Linear Equations and Inequalities ....................................... 6.0
Chapter Seven: Solving Systems of Equations......................................... 7.0
Video Clips and Project−based Learning Exploration.......................................... 8.0
Appendix: Explorations (Chapters 1-7) ............................................................... 9.0
Practice Final and Final Exam............................................................................. 10.0
Annotated Glossary of Key Vocabulary Words and Concepts........................... 11.0

6|Page

Preface – About the Textbook
Purpose of the Common Core Based, Integrated Algebra Textbook.
The purpose of this contextualized algebra modular textbook is to enhance students’
conceptual understanding of algebra taught in the “hands on, minds on” inquiry-based
platform of the sciences and the arts (STEAM) and to increase mathematics literacy.
This algebraic textbook focuses on the language of mathematics (specifically algebraic
concepts) as a method of communication. Thus, the textbook affords opportunities to
use mathematics to communicate (in writing, graphs, and symbols) while solving in real-
world situations.
The textbook is designed to help all students in the algebra classroom, but is especially
helpful for the below grade level and moderately at-risk. All of the topics specified by
the Common Core Standards are taught in a conceptually relevant way that does NOT
confuse the learning with complicated computations. In other words, a lack of well-
developed computational skills (addition, subtraction, multiplication, division, fraction
usage, etc.) would not impede learning. Additionally, the textbook is procedural in
nature, providing step-by-step, scaffolded instruction. Students gain a better conceptual
understanding of the material because the foundations are well laid. The book is
self−contained in that all needed knowledge for a topic is taught. This means that
students who are lacking key background knowledge will not be left
floundering. Finally, to enhance mathematical understanding, throughout the book are
project−based, inquiry−based, interdisciplinary explorations.
The creation of such an algebra textbook contributes to the math education and special
education fields by providing teachers with a practical teaching tool that integrates the
Common Core State Standards. The textbook adheres to the Core’s overarching idea of
promoting mathematically skilled and literate (both verbally and computationally),
analytically sound, and reasoning proficient students.
The entire book is predicated on the concept of the progression from direct to guided to
mental inquiry methods. The bulk of the text may look like simple worksheets, but they
are designed to encourage students to make connections, see relationships, and draw
conclusions. Each chapter in the book guides the teacher in successfully integrating the
mathematics Common Core State Standards (CCSS) to diverse populations.

In order to become a good writer, it is necessary to have a good vocabulary and
knowledge of sentence structure. Likewise, in math, it is necessary to develop a
familiarity with numbers and an ease when working with them. Developing that
familiarity and ease is one of the main objectives of this text. Students will then view
numbers as more than just symbols and have an understanding of how they behave and
interact. This number sense will allow a deeper conceptual understanding of higher level

7|Page

mathematics and other pursuits that involve mathematics.
Value of the modular textbook
Why a Common Core State Standards based textbook: State education chiefs and
governors in 48 states came together to develop the Common Core, a set of clear
college− and career−ready standards for kindergarten through 12th grade in English
language arts/literacy and mathematics. Today, almost every state has adopted and is
working to implement the standards, designed to ensure that students graduating from
high school are prepared to take credit−bearing introductory courses in two− or
four−year college programs or to enter the workforce [Common Core Standards, 2014].
Why the interactive, interdisciplinary pedagogy: This integrated math/science/art
approach to teaching algebra will, for diverse students, lead to a greater appreciation
for the field of mathematics and its vital importance as a tool to help us study and
explain our world. Teachers will more likely emulate this evidence-based, pedagogical
approach to learning (inherent in the Common Core) when they can see and implement
the teacher−friendly, step−by−step, modules. The general, at−risk and special education
students will be especially served by this approach. Success is the best motivator for
these students, and this approach garners success.
While a plethora of literature states the need for our secondary teachers to incorporate
the Common Core, very few give practical suggestions as to how to accomplish it. This
textbook will help to fill this gap (Dodson, 2013; Gallagher, Rosenthal, & Stepien, 1992;
Hiebert et al.,1996; Huntley, Rasmussen, Villarubi, & Fey,2000; Lubienski, 2000;
Mergendoller, Maxwell, & Bellisimo, 2006.; National Mathematics Advisory Panel 2008;
National Research Council, 2011; National Mathematics Advisory Panel, 2008 ; Schoen &
Charles, 2003).
Why algebraic concepts: The importance of algebraic modules should not be
undervalued as algebra is the gatekeeper for successful completion of higher levels of
mathematics (Frankenstein, 1995; Moses & Cobb, 2001; Stintson, 2004). A sound
foundation of algebraic concepts can facilitate student engagement and can lead to
more STEM career choices (Dodson, 2013; National Mathematics Advisory Panel 2008;
National Research Council, 2011).
Textbook format
The textbook consists of seven chapters. Each chapter contains an introductory cloze,
guided modeling practice exercises, progressive practice exercises, quizzes, cloze
responses, practice tests (which can also serve as the pre-test for data driven
instruction), tests, and a companion interdisciplinary exploration. The practice final is a
review for the entire book; a final exam is also provided. The book ends with an
annotated glossary, a handy, quick reference for students.

8|Page

Cloze – The cloze (fill in the blanks) introduces the concepts of each lesson and
serves as a summary of key vocabulary, points, and ideas. As opposed to simply a
lecture, students are involved in the development of ideas as they work through
the cloze with the teacher.

By completing a cloze together, students are involved even in the key
idea review. Pertinent vocabulary is emphasized, explained and defined
through this method. Student note−taking is also enhanced.
Because this text is also designed to be a study aide for students, in the
back of the book answers to each cloze are given. In that way, students
who need to review material or were absent can still complete the work.
They are not solely teacher−dependent. Additionally, it allows those
students who wish to advance to do so.
Guided modeling – This practice helps the students make the conversion from a
written explanation to problem solving. The teacher models and guides the
student in completing the problem and gradually increases student input until
students can do the entire problem by themselves. This is an example of
progression learning, going from direct instruction to mental inquiry. Students
apply the explanations or the problems with decreasing teacher guidance. The
student sets the pace and guides the teacher to the extent to which s/he needs
to provide guidance. The first exercise will go from the simplest arithmetic
problem to the more complex ones. Students are not distracted by
computational complexity. Thus, their focus is on the concept being taught.
Progressive practice exercises – Each section contains several practice exercises.
The purpose of the first exercise is to simply assure that the student is capable of
the mechanics of solving the problem. Subsequent exercises have varying levels
of complexity, but always emphasize an understanding of the procedures used
and an appreciation of where they need to be applied. If a practice quiz is
desired, teachers can use the last practice exercise. The design of the exercises
also supports the way in which students learn (i.e. The dynamics of learning:
practice the basic procedure, apply the procedure with multiple mistakes, try
again and make fewer and new ones, and then reach proficiency.)
The most effective way for a teacher to guide the students is through
questions that are well thought−out. These questions should point the
student in the right direction and should require careful thought on the
part of the students. This key element is essential to the development of
the student into a life−long learner. This is inquiry / problem−solving
reasoning at its best.
There are several practice exercise sets for each concept. The first
problem set includes simple arithmetic to avoid any unnecessary

9|Page

confusion; then, more involved arithmetic computations are introduced.
These are progressive learning exercises. In the first exercise it is
expected that students will make copious mistakes. That is perfectly fine.
Mistakes are learning experiences and opportunities for teachers to
correct misconceptions. In the next exercise set, students will have
corrected the original mistakes, but inevitably make new ones. Once the
teacher has addressed those, the student will tackle the third exercise set
which should show student proficiency of material covered. Because this
is not "drill and kill," but an interactive student−teacher assessment of
understanding, the teacher manages pacing with the student. If no
additional practice is needed, the teacher should progress directly to the
summary assessment. This is data driven assessment at its best. The
reason exercises become drill and kill is that many teachers robotically
assign worksheets. This is not what is happening here. This textbook
encourages and relies on teachers constantly monitoring the proficiency
of the students in dealing with the topic at hand.
Quizzes and Tests – Each quiz and test is modeled after the practice exercises.
The quizzes and end−of−chapter tests cover the main concepts of
each chapter section and skill development. Each major topic is represented by
at least two questions, one basic and one involving more conceptual
understanding. Quizzes should take approximately 15 minutes, and chapter tests
should be completed in one class period; lengths will vary. A review and final
exam for the entire text are also provided.
Note: As good practice dictates, the practice tests can serve as the pre-
tests for data driven instruction).
Interdisciplinary, common core explorations – Complete interdisciplinary
explorations are located in an appendix prior to the practice final and final exam.
However, they should be assigned as specified in the table of contents for each
chapter. These explorations highlight material covered in the companion chapter
and allow teachers to teach key algebra topics in an engaging, problem solving
manner. Students utilize mathematics as a tool to explore the world. These
lessons include plans, activities, assessments, and answer keys. They are easily
adaptable to diverse audiences and include scaffolding and teacher notes.
This textbook provides scaffolding of content knowledge, the integration of topics (what
topics can be taught simultaneously without interference and within topic progression),
and when to teach them. This is especially helpful for new teachers. Topics are taught
with the emphasis on understanding, not the memorization and application of rules,
thus conceptual knowledge of the math is highlighted.

10 | P a g e

The design of the exercises also supports the way in which students learn (i.e. The
dynamics of learning: practice the basic procedure, apply the procedure with multiple
mistakes, try again and make fewer and new ones, and then reach proficiency.)

After a teacher starts using the book for a few weeks, re−reading this introduction is
encouraged. The rationale will become clearer, and implementation will flow more
smoothly.

Our desire is that after implementing this textbook, a change in student’s
comprehension and enthusiasm for mathematics in general and algebra in particular will
become evident. Please feel free to contact us with any suggestions for improvement,
activities, resources, or even lessons you would like shared in the next edition.

The Algebra Textbook Team Dr. Karla Spence, Science & Mathematics
Dr. Vicki−Lynn Holmes, Mathematics Dr. Shelia Ingram, Mathematics Education
Dr. Jane Finn, Special Education

References

Common Core State Standards Initiative (2014), NGA Center/CCSSO2014. Overview. Available at

http://www.corestandards.org/about−the−standards/frequently−asked−questions/
Dodson, A. P. (2013). “STEM Education is Important to Our Future.” Diverse: Issues in Higher Education. Vol. 29,

No. 26: 16.
Frankenstein, M. (1995). Equity in mathematics education: Class in the world outside the class. In W. G. Secada,

E. Fennema, &L. Byrd (Eds.), New directions for equity in mathematics education (pp. 165–190). Cambridge:
Cambridge University Press.
National Research Council. (2011). Successful K−12 STEM Education: Identifying Effective Approaches in Science,
Technology, Engineering, and Mathematics. Committee on Highly Successful Science Programs for K−12
Science Education. Board on Science Education and Board on Testing and Assessment, Division of
Behavioral and Social Sciences and Education. Washington, DC: The National Academies Press.

Gallagher, S. A., Rosenthal, H., & Stepien, W. (1992). The effects of problem−based learning on problem−solving.

Gifted Child Quarterly, 36(4), 195−200.
Hiebert, J., Carpenter, T., Fennema, E., Fuson, K., Human, P., Murray, H., . . . Werne, D. (1996). Problem solving as a

basis for reform in curriculum and instruction: The case of mathematics. Educational Researcher, 24(4),
12−21.

Huntley, M. A., Rasmussen, C., Villarubi, R. S., & Fey, J. T. (2000). Effects of standard−based mathematics

education: A study of the Core−Plus Mathematics Project algebra and functions strand. Journal for Research
in Mathematics Education, 31(3), 328−361.
Lubienski, S. T. (2000). Problem solving as a means toward mathematics for all: An exploratory look through a

class lens. Journal for Research in Mathematics Education, 31(4), 454−482.
Mergendoller, J. R., Maxwell, N. L., & Bellisimo, Y. (2006). The effectiveness of problem−based instruction: A

comparative study of instructional methods and student characteristics. The Interdisciplinary Journal of
Problem−based Learning, 1(2), 49−69.
Moses, R. P., & Cobb, C. E. (2001). Radical equations: Math literacy and civil rights. Boston: Beacon Press.
National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the
NationalMathematics Advisory Panel. Washington, DC: U.S. Department of Education. Available at:
http://www2.ed.gov/about/bdscomm/list/mathpanel/report/finalreport. pdf.
Schoen, H., & Charles, R. (Eds.). (2003). Teaching mathematics through problem solving grades 6−12. Reston: The
National Council of Teachers of Mathematics, Inc.

Stintson, D.W. (2004). Mathematics as “Gate−Keeper”(?): Three Theoretical Perspectives that Aim Toward

Empowering All Children With A Key to the Gate. The Mathematics Educator, 14(1), 8−18.

11 | P a g e