# Preparation Manual

## Section 3: Overview and Exam Framework

Mathematics 4–8 (115)

### Exam Overview

Exam Name | Mathematics 4–8 |
---|---|

Exam Code | 115 |

Time | 5 hours |

Number of Questions | 100 selected-response questions |

Format | Computer-administered test (CAT) |

The TExES Mathematics 4–8 (115) exam is designed to assess whether an examinee has the requisite knowledge and skills that an entry-level educator in this field in Texas public schools must possess. The 100 selected-response questions are based on the Mathematics 4–8 exam framework and cover grades 4–8. The exam may contain questions that do not count toward the score. Your final scaled score will be based only on scored questions.

### The Standards

Standard I | Number Concepts: The mathematics teacher understands and uses numbers, number systems and their structure, operations and algorithms, quantitative reasoning and technology appropriate to teach the statewide curriculum (Texas Essential Knowledge and Skills [TEKS]) to prepare students to use mathematics. |
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Standard II | Patterns and Algebra: The mathematics teacher understands and uses patterns, relations, functions, algebraic reasoning, analysis and technology appropriate to teach the statewide curriculum (TEKS) to prepare students to use mathematics. |

Standard III | Geometry and Measurement: The mathematics teacher understands and uses geometry, spatial reasoning, measurement concepts and principles and technology appropriate to teach the statewide curriculum (TEKS) to prepare students to use mathematics. |

Standard IV | Probability and Statistics: The mathematics teacher understands and uses probability and statistics, their applications and technology appropriate to teach the statewide curriculum (TEKS) to prepare students to use mathematics. |

Standard V | Mathematical Processes: The mathematics teacher understands and uses mathematical processes to reason mathematically, to solve mathematical problems, to make mathematical connections within and outside of mathematics and to communicate mathematically. |

Standard VI | Mathematical Perspectives: The mathematics teacher understands the historical development of mathematical ideas, the relationship between society and mathematics, the structure of mathematics and the evolving nature of mathematics and mathematical knowledge. |

Standard VII | Mathematical Learning and Instruction: The mathematics teacher understands how children learn and develop mathematical skills, procedures and concepts; knows typical errors students make; and uses this knowledge to plan, organize and implement instruction to meet curriculum goals and to teach all students to understand and use mathematics. |

Standard VIII | Mathematical Assessment: The mathematics teacher understands assessment, and uses a variety of formal and informal assessment techniques appropriate to the learner on an ongoing basis to monitor and guide instruction and to evaluate and report student progress. |

### Domains and Competencies

Domain | Domain Title | Approx. Percentage of Exam | Standards Assessed |
---|---|---|---|

I | Number Concepts | 16% | Mathematics I |

II | Patterns and Algebra | 21% | Mathematics II |

III | Geometry and Measurement | 21% | Mathematics III |

IV | Probability and Statistics | 16% | Mathematics IV |

V | Mathematical Processes and Perspectives | 10% | Mathematics V–VI |

VI | Mathematical Learning, Instruction and Assessment | 16% | Mathematics VII–VIII |

The content covered by this exam is organized into broad areas of content called
**domains**. Each domain covers one or more of the educator standards for this field.
Within each domain, the content is further defined by a set of **competencies**. Each
competency is composed of two major parts:

- The
**competency statement**, which broadly defines what an entry-level educator in this field in Texas public schools should know and be able to do. - The
**descriptive statements**, which describe in greater detail the knowledge and skills eligible for testing.

#### 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 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).

##### Competency 002—The teacher understands number operations and computational algorithms.

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 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 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.

##### Competency 005—The teacher understands and uses linear functions to model and 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 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 of x plus or minus c on the graph of a nonlinear function f of 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 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

##### Competency 008—The teacher understands measurement as a process.

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 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.

##### Competency 010—The teacher analyzes the properties of two- and three-dimensional figures.

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 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 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.

##### Competency 013—The teacher understands the theory of probability.

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 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

##### Competency 015—The teacher understands mathematical reasoning and problem solving.

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 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.
- Explores and applies concepts of financial literacy as it relates to teaching students (e.g., describe the basic purpose of financial institutions, distinguish the difference between gross income and net income, identify various savings options, define different types of taxes, identify the advantages and disadvantages of different methods of payments).
- Applies mathematics to model and solve problems to manage financial resources effectively for lifetime financial security as it relates to teaching students (e.g., distinguish between fixed and variable expenses, calculate profit in a given situation develop a system for keeping and using financial records, describe actions that might be taken to balance a budget when expenses exceed income and balance a simple budget.)

#### Domain VI—Mathematical Learning, Instruction and Assessment

##### Competency 017—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 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 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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