## TExES 4-8 Mathematics Competencies

The University of Houston's free online mathematics quizzes are based on the TExES Mathematics 4-8 Domains and Competencies, which are listed below. For information about how each online quiz relates to these competencies, click on the "Overview of Quizzes" link in the navigation pane at the left.

DOMAIN I — NUMBER CONCEPTS

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.

THE TEACHER UNDERSTANDS THE STRUCTURE OF NUMBER SYSTEMS, THE DEVELOPMENT OF A SENSE OF QUANTITY AND THE RELATIONSHIP BETWEEN QUANTITY AND SYMBOLIC REPRESENTATIONS.

The
beginning teacher:

- 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).

The
beginning teacher:

- 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.

THE TEACHER UNDERSTANDS IDEAS OF NUMBER THEORY AND USES NUMBERS TO MODEL AND SOLVE PROBLEMS WITHIN AND OUTSIDE OF MATHEMATICS.

The
beginning teacher:

- 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 II — PATTERNS AND ALGEBRA

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.

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.

The
beginning teacher:

- 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.

The
beginning teacher:

- 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.

THE TEACHER UNDERSTANDS AND USES NONLINEAR FUNCTIONS AND RELATIONS TO MODEL AND SOLVE PROBLEMS.

The
beginning teacher:

- 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.

THE TEACHER USES AND UNDERSTANDS THE CONCEPTUAL FOUNDATIONS OF CALCULUS RELATED TO TOPICS IN MIDDLE SCHOOL MATHEMATICS.

The
beginning teacher:

- 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 III — GEOMETRY AND MEASUREMENT

The
beginning teacher:

- 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.

THE TEACHER UNDERSTANDS THE GEOMETRIC RELATIONSHIPS AND AXIOMATIC STRUCTURE OF EUCLIDEAN GEOMETRY.

The
beginning teacher:

- 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.

The
beginning teacher:

- 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.

THE TEACHER UNDERSTANDS TRANSFORMATIONAL GEOMETRY AND RELATES ALGEBRA TO GEOMETRY AND TRIGONOMETRY USING THE CARTESIAN COORDINATE SYSTEM.

The
beginning teacher:

- 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

COMPETENCY
012

THE TEACHER UNDERSTANDS HOW TO USE GRAPHICAL AND NUMERICAL TECHNIQUES TO EXPLORE DATA, CHARACTERIZE PATTERNS AND DESCRIBE DEPARTURES FROM PATTERNS.

THE TEACHER UNDERSTANDS HOW TO USE GRAPHICAL AND NUMERICAL TECHNIQUES TO EXPLORE DATA, CHARACTERIZE PATTERNS AND DESCRIBE DEPARTURES FROM PATTERNS.

The
beginning teacher:

- 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.

The
beginning teacher:

- 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.

THE TEACHER UNDERSTANDS THE RELATIONSHIP AMONG PROBABILITY THEORY, SAMPLING AND STATISTICAL INFERENCE AND HOW STATISTICAL INFERENCE IS USED IN MAKING AND EVALUATING PREDICTIONS.

The
beginning teacher:

- 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.

DOMAIN V — MATHEMATICAL PROCESSES AND PERSPECTIVES

The
beginning teacher:

- Demonstrates an understanding of proof, including indirect proof,
in mathematics.

- Applies correct mathematical reasoning to derive valid conclusions
from a set of premises.

- Demonstrates an understanding of the use of inductive reasoning to
make conjectures and deductive methods to evaluate the validity of
conjectures.

- Applies knowledge of the use of formal and informal reasoning to
explore, investigate and justify mathematical ideas.

- Recognizes that a mathematical problem can be solved in a variety
of ways and selects an appropriate strategy for a given problem.

- Evaluates the reasonableness of a solution to a given problem.

- Applies content knowledge to develop a mathematical model of a
real-world situation and analyzes and evaluates how well the model
represents the situation.

- Demonstrates an understanding of estimation and evaluates its appropriate uses.

COMPETENCY
016

THE TEACHER UNDERSTANDS MATHEMATICAL CONNECTIONS WITHIN AND OUTSIDE OF MATHEMATICS AND HOW TO COMMUNICATE MATHEMATICAL IDEAS AND CONCEPTS.

THE TEACHER UNDERSTANDS MATHEMATICAL CONNECTIONS WITHIN AND OUTSIDE OF MATHEMATICS AND HOW TO COMMUNICATE MATHEMATICAL IDEAS AND CONCEPTS.

The
beginning teacher:

- Recognizes and uses multiple representations of a mathematical
concept (e.g., a point and its coordinates, the area of circle as a
quadratic function in r, probability as the ratio of two areas).

- Uses mathematics to model and solve problems in other disciplines,
such as art, music, science, social science and business.

- Expresses mathematical statements using developmentally
appropriate language, standard English, mathematical language and
symbolic mathematics.

- Communicates mathematical ideas using a variety of representations
(e.g., numeric, verbal, graphic, pictorial, symbolic, concrete).

- Demonstrates an understanding of the use of visual media such as
graphs, tables, diagrams and animations to communicate mathematical
information.

- Uses the language of mathematics as a precise means of expressing
mathematical ideas.

- Understands the structural properties common to the mathematical disciplines.

DOMAIN VI — MATHEMATICAL LEARNING, INSTRUCTION AND ASSESSMENT

COMPETENCY
017

THE TEACHER UNDERSTANDS HOW CHILDREN LEARN AND DEVELOP MATHEMATICAL SKILLS, PROCEDURES AND CONCEPTS.

THE TEACHER UNDERSTANDS HOW CHILDREN LEARN AND DEVELOP MATHEMATICAL SKILLS, PROCEDURES AND CONCEPTS.

The
beginning teacher:

- Applies theories and principles of learning mathematics to plan
appropriate instructional activities for all students.

- Understands how students differ in their approaches to learning
mathematics with regard to diversity.

- Uses students’ prior mathematical knowledge to build conceptual
links to new knowledge and plans instruction that builds on
students’ strengths and addresses students’ needs.

- Understands how learning may be assisted through the use of
mathematics manipulatives and technological tools.

- Understands how to motivate students and actively engage them in
the learning process by using a variety of interesting, challenging
and worthwhile mathematical tasks in individual, small-group and
large-group settings.

- Understands how to provide instruction along a continuum from
concrete to abstract.

- Recognizes the implications of current trends and research in mathematics and mathematics education.

COMPETENCY
018

THE TEACHER UNDERSTANDS HOW TO PLAN, ORGANIZE AND IMPLEMENT INSTRUCTION USING KNOWLEDGE OF STUDENTS, SUBJECT MATTER AND STATEWIDE CURRICULUM (TEXAS ESSENTIAL KNOWLEDGE AND SKILLS [TEKS]) TO TEACH ALL STUDENTS TO USE MATHEMATICS.

THE TEACHER UNDERSTANDS HOW TO PLAN, ORGANIZE AND IMPLEMENT INSTRUCTION USING KNOWLEDGE OF STUDENTS, SUBJECT MATTER AND STATEWIDE CURRICULUM (TEXAS ESSENTIAL KNOWLEDGE AND SKILLS [TEKS]) TO TEACH ALL STUDENTS TO USE MATHEMATICS.

The
beginning teacher:

- Demonstrates an understanding of a variety of instructional
methods, tools and tasks that promote students’ ability to do
mathematics described in the TEKS.

- Understands planning strategies for developing mathematical
instruction as a discipline of interconnected concepts and
procedures.

- Develops clear learning goals to plan, deliver, assess and
reevaluate instruction based on the TEKS.

- Understands procedures for developing instruction that establishes
transitions between concrete, symbolic and abstract representations
of mathematical knowledge.

- Applies knowledge of a variety of instructional delivery methods,
such as individual, structured small-group and large-group formats.

- Understands how to create a learning environment that provides all
students, including English-language learners, with opportunities to
develop and improve mathematical skills and procedures.

- Demonstrates an understanding of a variety of questioning
strategies to encourage mathematical discourse and to help students
analyze and evaluate their mathematical thinking.

- Understands how technological tools and manipulatives can be used
appropriately to assist students in developing, comprehending and
applying mathematical concepts.

- Understands how to relate mathematics to students’ lives and a variety of careers and professions.

COMPETENCY
019

THE TEACHER UNDERSTANDS ASSESSMENT AND USES A VARIETY OF FORMAL AND INFORMAL ASSESSMENT TECHNIQUES TO MONITOR AND GUIDE MATHEMATICS INSTRUCTION AND TO EVALUATE STUDENT PROGRESS.

THE TEACHER UNDERSTANDS ASSESSMENT AND USES A VARIETY OF FORMAL AND INFORMAL ASSESSMENT TECHNIQUES TO MONITOR AND GUIDE MATHEMATICS INSTRUCTION AND TO EVALUATE STUDENT PROGRESS.

The
beginning teacher:

- Demonstrates an understanding of the purpose, characteristics and
uses of various assessments in mathematics, including formative and
summative assessments.

- Understands how to select and develop assessments that are
consistent with what is taught and how it is taught.

- Demonstrates an understanding of how to develop a variety of
assessments and scoring procedures consisting of worthwhile tasks
that assess mathematical understanding, common misconceptions and
error patterns.

- Understands how to evaluate a variety of assessment methods and
materials for reliability, validity, absence of bias, clarity of
language and appropriateness of mathematical level.

- Understands the relationship between assessment and instruction
and knows how to evaluate assessment results to design, monitor and
modify instruction to improve mathematical learning for all
students, including English-language learners.

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