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Algebra
Instructional programs from prekindergarten through
grade 12
should enable all students to—
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Understand patterns,
relations, and functions
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Pre-K–2
Expectations:
In
prekindergarten through grade 2 all students should– |
| • |
sort, classify, and order objects by size, number, and other
properties; |
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recognize,
describe, and extend patterns such as sequences of sounds and
shapes or simple numeric patterns and translate from one representation
to another; |
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analyze how both repeating and growing patterns are generated.
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Grades
3–5 Expectations:
In
grades 3–5 all students should– |
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describe,
extend, and make generalizations about geometric and numeric
patterns; |
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represent and analyze patterns and functions, using words, tables,
and graphs.
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Grades
6–8 Expectations:
In
grades 6–8 all students should– |
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represent, analyze, and generalize a variety of patterns with
tables, graphs, words, and, when possible, symbolic rules; |
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relate and compare different forms of representation for a relationship;
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identify functions as linear or nonlinear and contrast their
properties from tables, graphs, or equations.
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Grades
9–12 Expectations:
In
grades 9–12 all students should– |
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generalize
patterns using explicitly defined and recursively defined functions;
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understand relations and functions and select, convert flexibly
among, and use various representations for them; |
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analyze functions of one variable by investigating rates of
change, intercepts, zeros, asymptotes, and local and global
behavior; |
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understand and perform transformations such as arithmetically
combining, composing, and inverting commonly used functions,
using technology to perform such operations on more-complicated
symbolic expressions; |
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understand and compare the properties of classes of functions,
including exponential, polynomial, rational, logarithmic, and
periodic functions; |
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interpret representations of functions of two variables
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Represent and
analyze mathematical situations and structures using algebraic symbols
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Pre-K–2
Expectations:
In
prekindergarten through grade 2 all students should– |
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illustrate
general principles and properties of operations, such as commutativity,
using specific numbers; |
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use
concrete, pictorial, and verbal representations to develop an
understanding of invented and conventional symbolic notations.
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Grades
3–5 Expectations:
In
grades 3–5 all students should– |
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identify such properties as commutativity, associativity, and
distributivity and use them to compute with whole numbers; |
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represent the idea of a variable as an unknown quantity using
a letter or a symbol; |
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express mathematical relationships using equations.
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Grades
6–8 Expectations:
In
grades 6–8 all students should– |
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develop an initial conceptual understanding of different uses
of variables; |
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explore
relationships between symbolic expressions and graphs of lines,
paying particular attention to the meaning of intercept and
slope; |
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use symbolic algebra to represent situations and to solve problems,
especially those that involve linear relationships; |
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recognize and generate equivalent forms for simple algebraic
expressions and solve linear equations
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Grades
9–12 Expectations:
In
grades 9–12 all students should– |
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understand the meaning of equivalent forms of expressions, equations,
inequalities, and relations; |
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write equivalent forms of equations, inequalities, and systems
of equations and solve them with fluency—mentally or with
paper and pencil in simple cases and using technology in all
cases; |
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use
symbolic algebra to represent and explain mathematical relationships;
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use a variety of symbolic representations, including recursive
and parametric equations, for functions and relations; |
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judge the meaning, utility, and reasonableness of the results
of symbol manipulations, including those carried out by technology.
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Use mathematical
models to represent and understand quantitative relationships
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Pre-K–2
Expectations:
In
prekindergarten through grade 2 all students should– |
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model situations that involve the addition and subtraction of
whole numbers, using objects, pictures, and symbols.
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Grades
3–5 Expectations:
In
grades 3–5 all students should– |
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model
problem situations with objects and use representations such
as graphs, tables, and equations to draw conclusions.
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Grades
6–8 Expectations:
In
grades 6–8 all students should– |
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model and solve contextualized problems using various representations,
such as graphs, tables, and equations.
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Grades
9–12 Expectations:
In
grades 9–12 all students should– |
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identify essential quantitative relationships in a situation
and determine the class or classes of functions that might model
the relationships; |
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use symbolic expressions, including iterative and recursive
forms, to represent relationships arising from various contexts;
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draw reasonable conclusions about a situation being modeled.
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Analyze change
in various contexts
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Pre-K–2
Expectations:
In
prekindergarten through grade 2 all students should– |
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describe
qualitative change, such as a student's growing taller; |
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describe quantitative change, such as a student's growing two
inches in one year.
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Grades
3–5 Expectations:
In
grades 3–5 all students should– |
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investigate how a change in one variable relates to a change
in a second variable; |
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identify
and describe situations with constant or varying rates of change
and compare them.
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Grades
6–8 Expectations:
In
grades 6–8 all students should– |
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use graphs to analyze the nature of changes in quantities in
linear relationships.
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Grades
9–12 Expectations:
In
grades 9–12 all students should– |
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approximate and interpret rates of change from graphical and
numerical data. |
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