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.