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Unformatted text preview: The Navigations series seeks to guide readers through the ﬁve .
strands of Principles and Srandardrﬁar School Mathematics in order to help
them translate the Standards and Principles into action and to illustrate
the growth and connectedness of content ideas from prekindergarten
through grade 12. The Navigations through the algebra curriculum
reﬂect NCTM’s vision of how algebraic concepts should be introduced,
how they grow, what to expect of students during and at the end of each
grade band, how'to assess what students know, and how selected
instructional activities can contribute to learning. Fundamental Components
of Algebraic Thinking The Algebra Standard emphasizes relationships among quantities
and the ways in which quantities change relative to one another. To '
think algebraically, one must be able to underStand patterns, relations,
and functions; represent and analyze mathematical situations and struc—
tures using algebraic symbols; use mathematical models to represent:
and understand quantitative relationships; and analyze change in vari
ous contexts. Each of these basic components evolves as students grow
and mature. Understanding patterns, relatiOns, and functions Young children begin to explore patterns in the world around them
through experiences with such things as color, size, shape, design,
words, syllables, musical tones, rhythms, movements, and physical objects. They observe, describe, repeat, extend, compare, and create patterns; they sort, classify, and order objects according to various char—_ acteristics; they predict what comes next and identify missing elements
in patterns, they learn to distinguish types of patterns, such as growing
or repeating patterns .  In the higher elementary grades, children learn to represent patterns
numerically, graphically, and symbolically,a as well as verbally. They
begin to look for relationships in numerical and geometric patterns and 
analyze how patterns grow or change. By using tables, charts, physical
objects, and symbols, students make and explain generalizations about
patterns and use relationships in patterns to make predictions. Students in the middle grades explore patterns expressed in tables,
graphs, words, or symbols, with an emphasis on patterns that exhibit
linear relationships (constant rate of change). They learn to relate sym
bolic and graphical representations and develop an understanding of
the signiﬁcance of slope and y—intercept. They also explore “What if?”
questions to investigate how patterns change, and they distinguish lin
ear frorn nonlinear patterns. In high school, students create and use tables, symbols, graphs, and
verbal representations to generalize and analyze patterns, relations, and
functions with increasing sophistication, and they convert ﬂexibly
among various representations. They compare and contrast situations
modeled by different types of functions, and they develop an under— Navigating throogh Algebrain PrekindergartennGrade 2 standing of classes of functions, both linear and nonlinear, and their
properties. Their understanding expands to include functions of more
than one variable, and they learn to perform transformations such as
composing and inverting commonly used functions.  Representing and analyzing mathematical situations
and structures using algebraic symbols Young children can illustrate mathematical properties (e.g., the com—
mutativity of addition) with objects or speciﬁc numbers. They use
objects, pictures, words, or symbols to represent mathematical ideas
and relationships, including the relationship of equality, and to solve
problems. When children are encouraged to describe and represent
quantities in different ways, they learn to recognize equivalent repre—
sentations and expand their ability to use symbols to communicate their
ideas. Later in the elementary grades, children investigate, represent,
describe, and explain mathematical properties, and they begin to gener—
alize relationships and to use them in computing with whole numbers.
They develop notions of the idea and usefulness of variables, which
they may express with a box, letter, or other symbol to signify the idea
of a variable as a placeholder. They also learn to use variables to
describe a rule that relates two quantities or to express relationships
using equations. During the middle grades, students encounter additional uses of vari—
ables as changing quantities in generalized patterns, formulas, identi—
ties, expressions of mathematical properties, equations, and inequalities.
They explore notions of dependence and independence as variables
change in relation to one another, and they develop facility in recogniz—
ing the equivalence of mathematical representations, which they can
use to transform expressions; to solve problems; and to relate graphical,
tabular, and symbolic representations. They also acquire greater facility
with linear equations and demonstrate how the values of slope and y—
intercept affect the line. ' High school students continue to develop fluency with mathematical
symbols and become proﬁcient in operating on algebraic expressions in
solving problems. Their facility with representation expands to include
equations, inequalities, systems of equations, graphs, matrices, and func—
tions, and they recognize and describe the advantages and'disadvantages
of various representations for a particular situation.'Such facility with
symbols and alternative representations enables them to analyze a math—
ematical situation, choose an appropriate model, select an appropriate
solution method, and evaluate the plausibility of their solution. Using mathematical. models to represent and
understand quantitative relationships Very young children learn to use objects or pictures, and, eventually,
symbols to enact stories or model situations that involve the addition or subtraction of whole numbersrAs they progress into the upper elemen—
tary grades, children begin to realize that mathematics can be used to
model numerical and geometric patterns, scientiﬁc experiments, and introduction other physical situations, and they discover that mathematical models
have the power to predict as well as to describe. As they employ graphs,
tables, and equations to represent relationships and use their models to
draw conclusions, students compare various models and investigate
whether different models of a particular situation yield the same results. Contextualized problems that can be modeled andsolved using vari
ous representations, such as graphs, tables, and equations, engage
middle—grades students. With the aid of technology, they learn to use
functions to model patterns of change, including situations in which
they generate and represent real data. Although the emphasis is on con—
texts that are modeled by linear relationships, students also explore
examples of nonlinear relationships, and they use their models to
develop and test conjectures. High school students develoP skill in identifying essential quantita—
tive relationships in a situation and in determining the type of function
with which to model the relationship. They use symbolic expressions to
represent relationships arising from various contexts, including situa—
tions in which they generate and use data. Using their models, students
conjecture about relationships, formulate and test hypotheses, and draw
conclusions about the situations being modeled. Analyzing change in various contexts From a very early age, children recognize examples of change in
their environment and describe change in qualitative terms, such as get—
ting taller, colder,.darker, or heavier. By measuring and comparing
quantities, children also learn to describe change quantitatively, such as
in keeping track of variations in temperature or the growth of a class—
room plant or pet. They learn that some changes are predictable but
others are not and that often change can be described mathematically.
Later in the elementary grades, children represent change in numerical,
tabular, or graphical form, and they observe that patterns of change
often involve more than one quantity, such as that the length of a spring
increases as additional weights are hung from it. Students in the upper _
grades also begin to study differences in patterns of change and to C0m~
pare changes that occur at a constant rate, such as the cost of buying
various numbers of pencils at $0.20 each, with changes whose rates
increase or decrease, such as the growth of a seedling. Middle—grades students explore many examples of quantities that
change and the graphs that represent those changes; they answer ques—
tions about the relationShipsrepresented in the graphs and learn to dis—
tinguish rate of change from accumulation (total amount of change). By
varying parameters such as the rate of change, students can observe the
corresponding changes in the graphs, equations, or tables of values of
the relationships. High school students deepen this understanding of
how mathematical quantities change and, in particular, of the concept
of rate of change. They investigate numerous mathematical situations
and real—world phenomena to analyze and make sense of changing rela—
tionships; interpret change and rates of change ﬁom graphical and
numerical data, and use algebraic techniques and appropriate technol—
ogy to develop and evaluate models of dynamic situations. Navigating through Algebra in PrekindergartenGrade 2 ...
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 Fall '11
 McAnelly
 Algebra

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