**
DOMAIN I - NUMBER
CONCEPTS **
Module One

**Competency 001**

**The teacher understands the
structure of number systems, the development of a sense of quantity, and the
relationship between quantity and symbolic representations.**

Analyzes the structure of
numeration systems and the roles of place value and zero in the base ten system.

Understands the relative
magnitude of whole numbers, integers, rational numbers, and real numbers.

Demonstrates an
understanding of a variety of models for representing numbers (e.g., fraction
strips, diagrams, patterns, shaded regions, number lines).

Demonstrates an
understanding of equivalency among different representations of rational
numbers.

Selects appropriate
representations of real numbers (e.g., fractions, decimals, percents, roots,
exponents, scientific notation) for particular situations.

Understands the
characteristics of the set of whole numbers, integers, rational numbers, real
numbers, and complex numbers (e.g., commutativity, order, closure, identity
elements, inverse elements, density).

Demonstrates an
understanding of how some situations that have no solution in one number system
(e.g., whole numbers, integers, rational numbers) have solutions in another
number system (e.g., real numbers, complex numbers).

**
Competency 002**

**The teacher understands number
operations and computational algorithms.**

Works proficiently with
real and complex numbers and their operations.

Analyzes and describes
relationships between number properties, operations, and algorithms for the four
basic operations involving integers, rational numbers, and real numbers.

Uses a variety of concrete
and visual representations to demonstrate the connections between operations and
algorithms.

Justifies procedures used
in algorithms for the four basic operations with integers, rational numbers, and
real numbers, and analyzes error patterns that may occur in their application.

Relates operations and
algorithms involving numbers to algebraic procedures (e.g., adding fractions to
adding rational expressions, division of integers to division of polynomials).

Extends and generalizes
the operations on rationals and integers to include exponents, their properties,
and their applications to the real numbers.

**
Competency 003**

**The teacher understands ideas
of number theory and uses numbers to model and solve problems within and outside
of mathematics.**

Demonstrates an
understanding of ideas from number theory (e.g., prime factorization, greatest
common divisor) as they apply to whole numbers, integers, and rational numbers,
and uses these ideas in problem situations.

Uses integers, rational
numbers, and real numbers to describe and quantify phenomena such as money,
length, area, volume, and density.

Applies knowledge of place
value and other number properties to develop techniques of mental mathematics
and computational estimation.

Applies knowledge of
counting techniques such as permutations and combinations to quantify situations
and solve problems.

Applies properties of the
real numbers to solve a variety of theoretical and applied problems.

**
DOMAIN Il -
PATTERNS AND ALGEBRA**
Module Two

** Competency
004**

**The teacher understands and
uses mathematical reasoning to identify, extend, and analyze patterns and
understands the relationships among variables, expressions, equations,
inequalities, relations, and functions.**

Uses inductive reasoning
to identify, extend, and create patterns using concrete models, figures,
numbers, and algebraic expressions.

Formulates implicit and
explicit rules to describe and construct sequences verbally, numerically,
graphically, and symbolically.

Makes, tests, validates,
and uses conjectures about patterns and relationships in data presented in
tables, sequences, or graphs.

Gives appropriate
justification of the manipulation of algebraic expressions.

Illustrates the concept of
a function using concrete models, tables, graphs, and symbolic and verbal
representations.

Uses transformations to
illustrate properties of functions and relations and to solve problems.

**
Competency 005**

**The teacher understands and
uses linear functions to model and solve problems.**

Demonstrates an
understanding of the concept of linear function using concrete models, tables,
graphs, and symbolic and verbal representations.

Demonstrates an
understanding of the connections among linear functions, proportions, and direct
variation.

Determines the linear
function that best models a set of data.

Analyzes the relationship
between a linear equation and its graph.

Uses linear functions,
inequalities, and systems to model problems.

Uses a variety of
representations and methods (e.g., numerical methods, tables, graphs, algebraic
techniques) to solve systems of linear equations and inequalities.

Demonstrates an
understanding of the characteristics of linear models and the advantages and
disadvantages of using a linear model in a given situation.

**
Competency 006**

**The teacher understands and
uses nonlinear functions and relations to model and solve problems.**

Uses a variety of methods
to investigate the roots (real and complex), vertex, and symmetry of a quadratic
function or relation.

Demonstrates an
understanding of the connections among geometric, graphic, numeric, and symbolic
representations of quadratic functions.

Analyzes data and
represents and solves problems involving exponential growth and decay.

Demonstrates an
understanding of the connections among proportions, inverse variation, and
rational functions.

Understands the effects of
transformations such as f(x
? *
c) *
on the graph of a nonlinear function f(x).

Applies properties,
graphs, and applications of nonlinear functions to analyze, model, and solve
problems.

Uses a variety of
representations and methods (e.g., numerical methods, tables, graphs, algebraic
techniques) to solve systems of quadratic equations and inequalities.

Understands how to use
properties, graphs, and applications of nonlinear relations including
polynomial, rational, radical, absolute value, exponential, logarithmic,
trigonometric, and piecewise functions and relations to analyze, model, and
solve problems.

** **

**Competency 007**

**The teacher uses and
understands the conceptual foundations of calculus related to topics in middle
school mathematics. **

Relates topics in middle
school mathematics to the concept of limit in sequences and series.

Relates the concept of
average rate of change to the slope of the secant line and instantaneous rate of
change to the slope of the tangent line.

Relates topics in middle
school mathematics to the area under a curve.

Demonstrates an
understanding of the use of calculus concepts to answer questions about rates of
change, areas, volumes, and properties of functions and their graphs.

**
DOMAIN Ill -
GEOMETRY AND MEASUREMENT**
Module Three
Module Four

**
Competency 008**

** The teacher
understands measurement as a process. **

Selects and uses
appropriate units of measurement (e.g., temperature, money, mass, weight, area,
capacity, density, percents, speed, acceleration) to quantify, compare, and
communicate information.

Develops, justifies, and
uses conversions within measurement systems.

Applies dimensional
analysis to derive units and formulas in a variety of situations (e.g., rates of
change of one variable with respect to another) and to find and evaluate
solutions to problems.

Describes the precision of
measurement and the effects of error on measurement.

Applies the Pythagorean
theorem, proportional reasoning, and right triangle trigonometry to solve
measurement problems.

**
Competency 009**

**The teacher understands the
geometric relationships and axiomatic structure of Euclidean geometry. **

Understands concepts and
properties of points, lines, planes, angles, lengths, and distances.

Analyzes and applies the
properties of parallel and perpendicular lines.

Uses the properties of
congruent triangles to explore geometric relationships and prove theorems.

Describes and justifies
geometric constructions made using a compass and straight edge and other
appropriate technologies.

Applies knowledge of the
axiomatic structure of Euclidean geometry to justify and prove theorems.

** **

**Competency 010 **

**The teacher analyzes the
properties of two- and three-dimensional figures. **

Uses and understands the
development of formulas to find lengths, perimeters, areas, and volumes of basic
geometric figures.

Applies relationships among
similar figures, scale, and proportion and analyzes how changes in scale affect
area and volume measurements.

Uses a variety of representations
(e.g., numeric, verbal, graphic, symbolic) to analyze and solve problems
involving two- and three-dimensional figures such as circles, triangles,
polygons, cylinders, prisms, and spheres.

Analyzes the relationship
among three-dimensional figures and related two-dimensional representations
(e.g., projections, cross-sections, nets) and uses these representations to
solve problems.

**Competency 011**

**The teacher understands
transformational geometry and relates algebra to geometry and trigonometry using
the Cartesian coordinate system. **

Describes and justifies
geometric constructions made using a reflection device and other appropriate
technologies.

Uses translations,
reflections, glide-reflections, and rotations to demonstrate congruence and to
explore the symmetries of figures.

Uses dilations (expansions
and contractions) to illustrate similar figures and proportionality.

Uses symmetry to describe
tessellations and shows how they can be used to illustrate geometric concepts,
properties, and relationships.

Applies concepts and
properties of slope, midpoint, parallelism, and distance in the coordinate plane
to explore properties of geometric figures and solve problems.

Applies transformations in
the coordinate plane.

Uses the unit circle in
the coordinate plane to explore properties of trigonometric functions.

**
DOMAIN IV -
PROBABILITY AND STATISTICS **
Module Five

**Competency 012**

**The teacher understands how to
use graphical and numerical techniques to explore data, characterize patterns,
and describe departures from patterns. **

Organizes and displays
data in a variety of formats (e.g., tables, frequency distributions,
stem-and-leaf plots, box-and-whisker plots, histograms, pie charts).

Applies concepts of
center, spread, shape, and skewness to describe a data distribution.

Supports arguments, makes
predictions, and draws conclusions using summary statistics and graphs to
analyze and interpret one-variable data.

Demonstrates an
understanding of measures of central tendency (e.g., mean, median, mode) and
dispersion (e.g., range, interquartile range, variance, standard deviation).

Analyzes connections among
concepts of center and spread, data clusters and gaps, data outliers, and
measures of central tendency and dispersion.

Calculates and interprets
percentiles and quartiles.

**Competency 013**

**The teacher understands the
theory of probability. **

Explores concepts of
probability through data collection, experiments, and simulations.

Uses the concepts and
principles of probability to describe the outcome of simple and compound events.

Generates, simulates, and
uses probability models to represent a situation.

Determines probabilities
by constructing sample spaces to model situations.

Solves a variety of
probability problems using combinations, permutations, and geometric probability
(i.e., probability as the ratio of two areas).

Uses the binomial,
geometric, and normal distributions to solve problems.

**Competency 014**

**The teacher understands the
relationship among probability theory, sampling, and statistical inference, and
how statistical inference is used in making and evaluating predictions. **

Applies knowledge of designing,
conducting, analyzing, and interpreting statistical experiments to investigate
real-world problems.

Demonstrates an understanding of
random samples, sample statistics, and the relationship between sample size and
confidence intervals.

Applies knowledge of the use of
probability to make observations and draw conclusions from single variable data
and to describe the level of confidence in the conclusion.

Makes inferences about a
population using binomial, normal, and geometric distributions.

Demonstrates an understanding of
the use of techniques such as scatter plots, regression lines, correlation
coefficients, and residual analysis to explore bivariate data and to make and
evaluate predictions.