DoDEA Curriculum
Pacing Guide for Algebra 1
FIRST QUARTER PERIOD
ALGEBRAIC THINKING
Patterns and
Variables
Data
Representations
Proportional
Reasoning
SECOND QUARTER PERIOD
MODELING RATES OF CHANGES
Linear and
Non-linear
Models
Representing
Linear Models
Graphing
Linear
Models
THIRD QUARTER PERIOD
PROCESSES AND SYMBOLS
Evaluating
Expressions
Solving
Equations and
Inequalities
Systems of
Equations and
Inequalities
FOURTH QUARTER PERIOD
ALGEBRAIC STRUCTURES
Functions and
Their Graphs
Exponential
Functions
Families of
Functions
Purpose
The purpose of the Algebra I Curriculum Guide is to provide teachers with the content and
teaching resources that support teaching to the DoDEA Mathematics Standards. The Guide
contains pathways to understanding, alignment with teaching resources, including the adopted
textbook, and suggested teaching and assessment strategies for classroom use.
All DoDEA teachers should follow the prescribed curriculum as outlined in the Guide to
maximize student achievement. Quarterly and End-of-Course examinations are used to assess
achievement of all DoDEA Algebra I students.
The components of the Algebra I Curriculum Guide provide essential information and examples
that allow teachers to create specific lessons to maximize student learning. The Guide is
designed to support teachers in providing all students with the skills, knowledge and experiences
they need to succeed in algebra.
The Guide is divided into four sections, each presenting one quarter of instruction and each
centered around a Big Idea. Each quarter has three units, and each unit contains
•
a table of contents, to identify information and activities in that unit that assist in building
student knowledge and understanding,
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an alignment to the content strands found in the DoDEA Mathematics Standards,
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identification of students’ prior knowledge, which is the foundation for the unit’s
development,
•
identification of the unit understanding to be developed,
•
identification of the vocabulary students encounter in this unit,
•
a section called “Pathways to knowledge and understanding,” which identifies the skills
and concepts essential to the development of the Big Idea,
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an essay that provides an overview of the unit, the rationale for studying the Big Idea, and
identification of major misconceptions based upon research,
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correlations to algebra materials used in the unit, such as:
▫
Cognitive Tutor, Teacher Assisted Package (TAPS), Mathematics Standardized Test
Prep, Algebra End of Course Examination, and the Algebra I: Prentice Hall
This Algebra I curriculum guide is intended to be a living document that is revised and improved
periodically. It is our hope that teachers will initiate the review process as they work with this
guide in their schools and classrooms, making recommendations for change along the way.
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Introduction
Algebra is a prerequisite for virtually all mathematics beyond middle school, and thus is a
gatekeeper for studying mathematics beyond the basics. The type of quantitative reasoning
inherent to the study of algebra is necessary not only for members of today’s and tomorrow’s
technical workforce, but also for well-informed and responsible citizens in our democracy. The
two primary goals for studying algebra are (1) preparation for further study—of mathematics and
other subjects such as science, statistics, and economic—and (2) preparation for the life and work
required in a world of global information.
The New Algebra
Studying the algebra of the past does not sufficiently prepare the students of today for the needs
of the future. All students need to study algebra, and all teachers need to consider how their
teaching of algebra must change. As all students need to study algebra to progress in mathematics
and many other subjects, we must equip all students in order to maximize their options. We
cannot afford to deny anyone the opportunity to be part of the high-skill workforce.
Why must we change how we teach algebra? In the past, many students became technically
competent at solving, simplifying, and other skills of symbol manipulation, but few were able to
use these skills to solve different problems or apply algebra to new situations. In addition, it is
less critical to be able to manipulate symbols with speed and accuracy, since can use computers
and hand-held calculators to do that part of problem solving. What is needed are more people
able to do the thinking part of problem solving—to make sense of problems, to represent
problems in symbols, to analyze solutions for fit with the original situation, to find the best
solution, and to check results using several strategies—since real problems rarely have a solution
in the “back of the book.”
The impact of technology is significant in terms of how mathematics is used. Everyday
problems and tasks require citizens who are able to analyze mathematics-related situations and
to communicate knowledgeably with others. Math is everywhere, and modern life requires the
ability to speak the language of mathematics. Algebra is necessary in common situations such as
in selecting a mobile telephone plan or updating an investment plan with a spreadsheet. We need
to change the way we teach mathematics because the world has changed, the needs of citizens
and society have changed, and—most importantly—technology has changed what is important
for people to know and be able to do. We need students who have strong algebraic thinking, not
just algebraic skills.
How Students Learn Algebra
We know much about what teaching practices lead to successful learning in algebra. For
example, we do know that learning builds on prior knowledge, that the framework for learning
must integrate concepts and skills, and that students must reflect on their own learning. We
also know that when students have opportunities to approach learning through real-life contexts
they are more motivated. Thus learning is accessible to more students and makes more sense.
Traditionally we have taught skills prior to teaching applications. Teachers have often skipped
or set aside the applications for extra credit, which allows most students little chance to even
attempt applying skills or concepts in new situations.
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Everyone needs to be able to use algebra – as a way to model and explain ideas, to use patterns to
predict, to synthesize models of change in data, shapes, and discrete mathematics, to make sense
of experience. All students, regardless of their education level or career goals, need to understand
the fundamental concepts of the subject and to be able to use technology as an algebraic tool.
Today’s Algebra Classroom
In a classroom that prepares students for today’s world, students represent their algebraic
thinking in multiple formats: verbally, in tables, graphically, and with algebraic symbols.
They engage in conversations with peers, communicate and share their ideas orally and in
writing, present solutions and strategies to the class, connect their learning with other parts of
mathematics and other fields of study, solve real life and mathematical problems, and reason with
continually increasing sophistication. They are able to tolerate some ambiguity in solving real
problems, in which they must make assumptions about which facets of a problem are important
and which can be ignored or simplified. Students should be able to explain not only how they
might produce a reasonable answer, but why and when to do so, and what that answer should look
like. Students should be skilled at procedures and also demonstrate conceptual understanding.
The DoDEA Algebra Curriculum
The DoDEA curriculum has four sections, each with one Big Idea, lasting approximately one
quarter. The first section, about algebraic thinking, picks up from where students are likely to be
after a typical middle-school program. Beginning with a unit on patterns and variables, students
start with an idea with which they are likely to be quite familiar: searching for patterns in a
situation. Here they use variables informally, and then more formally as the course continues.
Next, students study the foundational concept of data representation, an idea that may be less
familiar. This introductory section ends with a unit on proportional reasoning. These three
units together serve as the basic toolkit in the study of algebra. Students who are able to search
for patterns in data and situations and represent elements in terms of variables, to reason
proportionally, and to represent data in graphic displays are well equipped to succeed in algebra.
The second section focuses on modeling rates of change. Students first consider changing
quantities in general, examining examples of both linear and nonlinear change. Next they study
representing linear models, looking at this important and omnipresent type of function is depth,
paying particular attention to the idea that linear functions have a constant rate of change.
Though the idea of function underlies this unit, it does not present formal definitions, vocabulary,
symbols, or notation. Next, students represent linear models in multiple forms in a unit about
graphing linear models. Linear change is a good place to start pedagogically—not only because
of the relative simplicity of the algebra, but because it can be applied in numerous situations, with
contexts that are easily accessible to all algebra students.
The third section attends to processes and symbols, once considered the entirety of algebra, and
still important as a tool for students. In this section students learn some mechanics of algebra
as well as the meaning of those mechanics. They begin with evaluating expressions, move next
to solving equations and inequalities, and conclude with systems of equations and inequalities.
These topics should be more meaningful to students, anchored as they now are with earlier units
on patterns, variables, and the informal functions studied in section two. The symbols will not be
meaningless abstractions following arbitrary rules, but will represent quantities and relationships.
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The fourth and final section is about algebraic structures. This includes more formal approaches
to functions and their graphs, since students have the context, skills, and experience to help them
understand functions and appreciate the power of functions to model and predict. This section
also includes exponential functions, another type of function that is common in applications.
Finally, as a culminating unit, students examine families of functions in general, paying special
attention to those studied in depth—linear and exponential—characterizing their rates of change,
typical graphs and equations, and the type of situations that can be modeled with each family.
At the end of the DoDEA Algebra I course, students have a strong conceptual understanding of
algebra and are experienced algebraic thinkers. They have algebraic habits of mind, and are able
to model situations algebraically, abstract and generalize from such situations, represent concepts
in several forms, and analyze the algebraic structure of problems. They understand rates of
change and how these are related to important types of functions. They are able to work forward
and backwards with ideas and algebraic symbols to solve problems, demonstrate equivalence,
or reason about functions. Their understanding of various aspects of algebra is connected, and
they approach problems from multiple perspectives, collaborating and reasoning to determine
effective approaches to messy problems with real data. Students completing the Algebra I course
emerge not only well prepared for future mathematics coursework, but also able to speak the
language of mathematics—the language of tomorrow’s technical workforce and society.
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