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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







DOMAIN II — PATTERNS AND ALGEBRA

1. Uses inductive reasoning to identify, extend and create patterns using concrete models, figures, numbers and algebraic expressions.
   
   
2. Formulates implicit and explicit rules to describe and construct sequences verbally, numerically, graphically and symbolically.
   
   
3. Makes, tests, validates and uses conjectures about patterns and relationships in data presented in tables, sequences or graphs.
   
   
4. Gives appropriate justification of the manipulation of algebraic expressions.
   
   
5. Illustrates the concept of a function using concrete models, tables, graphs and symbolic and verbal representations.
   
   
6. Uses transformations to illustrate properties of functions and relations and to solve problems.

1. Demonstrates an understanding of the concept of linear function using concrete models, tables, graphs and symbolic and verbal representations.
   
   
2. Demonstrates an understanding of the connections among linear functions, proportions and direct variation.
   
   
3. Determines the linear function that best models a set of data.
   
   
4. Analyzes the relationship between a linear equation and its graph.
   
   
5. Uses linear functions, inequalities and systems to model problems.
   
   
6. Uses a variety of representations and methods (e.g., numerical methods, tables, graphs, algebraic techniques) to solve systems of linear equations and inequalities.
   
   
7. Demonstrates an understanding of the characteristics of linear models and the advantages and disadvantages of using a linear model in a given situation.

1. Uses a variety of methods to investigate the roots (real and complex), vertex and symmetry of a quadratic function or relation.
   
   
2. Demonstrates an understanding of the connections among geometric, graphic, numeric and symbolic representations of quadratic functions.
   
   
3. Analyzes data and represents and solves problems involving exponential growth and decay.
   
   
4. Demonstrates an understanding of the connections among proportions, inverse variation and rational functions.
   
   
5. Understands the effects of transformations such as f (x ± c) on the graph of a nonlinear function f (x).
   
   
6. Applies properties, graphs and applications of nonlinear functions to analyze, model and solve problems.
   
   
7. Uses a variety of representations and methods (e.g., numerical methods, tables, graphs, algebraic techniques) to solve systems of quadratic equations and inequalities.
   
   
8. 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.

1. Relates topics in middle school mathematics to the concept of limit in sequences and series.
   
   
2. 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.
   
   
3. Relates topics in middle school mathematics to the area under a curve.
   
   
4. 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

1. 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.
   
   
2. Develops, justifies and uses conversions within measurement systems.
   
   
3. 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.
   
   
4. Describes the precision of measurement and the effects of error on measurement.
   
   
5. Applies the Pythagorean theorem, proportional reasoning and right triangle trigonometry to solve measurement problems.

1. Understands concepts and properties of points, lines, planes, angles, lengths and distances.
   
   
2. Analyzes and applies the properties of parallel and perpendicular lines.
   
   
3. Uses the properties of congruent triangles to explore geometric relationships and prove theorems.
   
   
4. Describes and justifies geometric constructions made using a compass and straight edge and other appropriate technologies.
   
   
5. Applies knowledge of the axiomatic structure of Euclidean geometry to justify and prove theorems.

1. Uses and understands the development of formulas to find lengths, perimeters, areas and volumes of basic geometric figures.
   
   
2. Applies relationships among similar figures, scale and proportion and analyzes how changes in scale affect area and volume measurements.
   
   
3. 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.
   
   
4. 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.

1. Describes and justifies geometric constructions made using a reflection device and other appropriate technologies.
   
   
2. Uses translations, reflections, glide-reflections and rotations to demonstrate congruence and to explore the symmetries of figures.
   
   
3. Uses dilations (expansions and contractions) to illustrate similar figures and proportionality.
   
   
4. Uses symmetry to describe tessellations and shows how they can be used to illustrate geometric concepts, properties and relationships.
   
   
5. Applies concepts and properties of slope, midpoint, parallelism and distance in the coordinate plane to explore properties of geometric figures and solve problems.
   
   
6. Applies transformations in the coordinate plane.
   
   
7. Uses the unit circle in the coordinate plane to explore properties of trigonometric functions.

DOMAIN IV — PROBABILITY AND STATISTICS

1. 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).
   
   
2. Applies concepts of center, spread, shape and skewness to describe a data distribution.
   
   
3. Supports arguments, makes predictions and draws conclusions using summary statistics and graphs to analyze and interpret one-variable data.
   
   
4. Demonstrates an understanding of measures of central tendency (e.g., mean, median, mode) and dispersion (e.g., range, interquartile range, variance, standard deviation).
   
   
5. Analyzes connections among concepts of center and spread, data clusters and gaps, data outliers and measures of central tendency and dispersion.
   
   
6. Calculates and interprets percentiles and quartiles.

1. Explores concepts of probability through data collection, experiments and simulations.
   
   
2. Uses the concepts and principles of probability to describe the outcome of simple and compound events.
   
   
3. Generates, simulates and uses probability models to represent a situation.
   
   
4. Determines probabilities by constructing sample spaces to model situations.
   
   
5. Solves a variety of probability problems using combinations, permutations and geometric probability (i.e., probability as the ratio of two areas).
   
   
6. Uses the binomial, geometric and normal distributions to solve problems.

1. Applies knowledge of designing, conducting, analyzing and interpreting statistical experiments to investigate real-world problems.
   
   
2. Demonstrates an understanding of random samples, sample statistics and the relationship between sample size and confidence intervals.
   
   
3. 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.
   
   
4. Makes inferences about a population using binomial, normal and geometric distributions.
   
   
5. 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

1. Demonstrates an understanding of proof, including indirect proof, in mathematics.
   
   
2. Applies correct mathematical reasoning to derive valid conclusions from a set of premises.
   
   
3. Demonstrates an understanding of the use of inductive reasoning to make conjectures and deductive methods to evaluate the validity of conjectures.
   
   
4. Applies knowledge of the use of formal and informal reasoning to explore, investigate and justify mathematical ideas.
   
   
5. Recognizes that a mathematical problem can be solved in a variety of ways and selects an appropriate strategy for a given problem.
   
   
6. Evaluates the reasonableness of a solution to a given problem.
   
   
7. Applies content knowledge to develop a mathematical model of a real-world situation and analyzes and evaluates how well the model represents the situation.
   
   
8. Demonstrates an understanding of estimation and evaluates its appropriate uses.

1. 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).
   
   
2. Uses mathematics to model and solve problems in other disciplines, such as art, music, science, social science and business.
   
   
3. Expresses mathematical statements using developmentally appropriate language, standard English, mathematical language and symbolic mathematics.
   
   
4. Communicates mathematical ideas using a variety of representations (e.g., numeric, verbal, graphic, pictorial, symbolic, concrete).
   
   
5. Demonstrates an understanding of the use of visual media such as graphs, tables, diagrams and animations to communicate mathematical information.
   
   
6. Uses the language of mathematics as a precise means of expressing mathematical ideas.
   
   
7. Understands the structural properties common to the mathematical disciplines.

DOMAIN VI — MATHEMATICAL LEARNING, INSTRUCTION AND ASSESSMENT

1. Applies theories and principles of learning mathematics to plan appropriate instructional activities for all students.
   
   
2. Understands how students differ in their approaches to learning mathematics with regard to diversity.
   
   
3. Uses students’ prior mathematical knowledge to build conceptual links to new knowledge and plans instruction that builds on students’ strengths and addresses students’ needs.
   
   
4. Understands how learning may be assisted through the use of mathematics manipulatives and technological tools.
   
   
5. 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.
   
   
6. Understands how to provide instruction along a continuum from concrete to abstract.
   
   
7. Recognizes the implications of current trends and research in mathematics and mathematics education.

1. Demonstrates an understanding of a variety of instructional methods, tools and tasks that promote students’ ability to do mathematics described in the TEKS.
   
   
2. Understands planning strategies for developing mathematical instruction as a discipline of interconnected concepts and procedures.
   
   
3. Develops clear learning goals to plan, deliver, assess and reevaluate instruction based on the TEKS.
   
   
4. Understands procedures for developing instruction that establishes transitions between concrete, symbolic and abstract representations of mathematical knowledge.
   
   
5. Applies knowledge of a variety of instructional delivery methods, such as individual, structured small-group and large-group formats.
   
   
6. 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.
   
   
7. Demonstrates an understanding of a variety of questioning strategies to encourage mathematical discourse and to help students analyze and evaluate their mathematical thinking.
   
   
8. Understands how technological tools and manipulatives can be used appropriately to assist students in developing, comprehending and applying mathematical concepts.
   
   
9. Understands how to relate mathematics to students’ lives and a variety of careers and professions.

1. Demonstrates an understanding of the purpose, characteristics and uses of various assessments in mathematics, including formative and summative assessments.
   
   
2. Understands how to select and develop assessments that are consistent with what is taught and how it is taught.
   
   
3. 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.
   
   
4. 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.
   
   
5. 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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