# Preparation Manual

## Section 3: Overview and Exam Framework TX PACT: Mathematics: Grades 7–12 (735)

### Exam Overview

Table outlining the test format, number of questions, time, and passing score.
Exam Name TX PACT: Mathematics: Grades 7–12
Exam Code 735
Time 5 hours total appointment time
• 15 minutes for CAT tutorial and compliance agreement
• 4 hours and 45 minutes testing time
Number of Questions 125 selected-response questions

The TX PACT: Mathematics: Grades 7–12 (735) exam is designed to assess whether a test taker has demonstrated the requisite knowledge and skills for admission to an educator preparation program. The 125 selected-response questions are based on the Mathematics: Grades 7–12 exam framework. Questions on this exam range from grades 7–12. Your final scaled score will be based only on scored questions.

### Domains and Competencies

Table outlining test content and subject weighting by sub area and objective.
Domain Domain Title Approx. Percentage of Exam
I Mathematical Processes and Number Sense 19%
II Patterns, Algebra, and Functions 24%
III Measurement and Geometry 19%
IV Trigonometry and Calculus 19%
V Statistics, Probability, and Discrete Mathematics 19%

The content covered by this exam is organized into broad areas of content called domains. Each domain covers one or more of the 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 individual should know and be able to do in order to perform effectively in a Texas-approved educator preparation program.
• The descriptive statements, which describe in greater detail the knowledge and skills eligible for testing.

#### Domain I—Mathematical Processes and Number Sense

##### Competency 001—Understand mathematical problem solving.

For example:

1. Identify an appropriate problem-solving strategy for a particular problem.
2. Analyze the use of estimation in a variety of situations (e.g., rounding, area, plausibility).
3. Solve mathematical and real-world problems involving integers, fractions, decimals, and percents.
4. Solve mathematical and real-world problems involving ratios, proportions, and average rates of change.
##### Competency 002—Understand mathematical communication, connections, and reasoning.

For example:

1. Translate between representations (e.g., graphic, verbal, symbolic).
2. Recognize connections between mathematical concepts.
3. Analyze inductive and deductive reasoning.
4. Apply principles of logic to solve problems.
5. Demonstrate knowledge of the historical development of major mathematical concepts, including contributions from diverse cultures.
##### Competency 003—Understand number theory.

For example:

1. Analyze the group structure of the real numbers.
2. Use complex numbers and their operations.
3. Analyze the properties of numbers and operations.
4. Apply the principles of basic number theory (e.g., prime factorization, greatest common factor, least common multiple).

#### Domain II—Patterns, Algebra, and Functions

##### Competency 004—Understand relations and functions.

For example:

1. Demonstrate knowledge of relations and functions and their applications.
2. Perform operations with functions, including compositions and inverses.
3. Analyze characteristics of functions.
4. Interpret different representations of functions.
##### Competency 005—Understand linear, quadratic, and higher-order polynomial functions.

For example:

1. Analyze the relationship between a linear, quadratic, or higher-order polynomial function and its graph.
2. Solve linear and quadratic equations and inequalities using a variety of methods.
3. Solve systems of linear equations or inequalities using a variety of methods.
4. Solve higher-order polynomial equations and inequalities in one and two variables.
5. Analyze the characteristics of linear, quadratic, and higher-order polynomial equations.
6. Analyze real-world problems involving linear, quadratic, and higher-order polynomial functions.
##### Competency 006—Understand exponential and logarithmic functions.

For example:

1. Apply the laws of exponents and logarithms.
2. Analyze the relationship between exponential and logarithmic functions.
3. Analyze exponential and logarithmic functions and their graphs.
4. Analyze real-world problems involving exponential and logarithmic functions.
##### Competency 007—Understand rational, radical, absolute value, and piece-wise defined functions.

For example:

1. Manipulate rational, radical, and absolute value expressions, equations, and inequalities.
2. Analyze the relationship between a rational, radical, absolute value, or piece-wise defined function and its graph.
3. Analyze rational, radical, absolute value, and piece-wise defined functions in terms of domain, range, and asymptotes.
4. Analyze real-world problems involving rational, radical, absolute value, and piece-wise defined functions.

#### Domain III—Measurement and Geometry

##### Competency 008—Understand measurement principles and procedures.

For example:

1. Analyze the use of various units and unit conversions within the customary and metric systems.
2. Apply the concepts of similarity, scale factors, and proportional reasoning to solve measurement problems.
3. Analyze precision, error, and rounding in measurements and computed quantities.
4. Apply the concepts of perimeter, circumference, area, surface area, and volume to solve real-world problems.
##### Competency 009—Understand Euclidean geometry in two and three dimensions.

For example:

1. Demonstrate knowledge of axiomatic systems and of the axioms of non-Euclidean geometries.
2. Use the properties of polygons and circles to solve problems.
3. Apply the Pythagorean theorem and its converse.
4. Analyze formal and informal geometric proofs, including the use of similarity and congruence.
5. Use nets and cross sections to analyze three-dimensional figures.
##### Competency 010—Understand coordinate and transformational geometry.

For example:

1. Analyze two- and three-dimensional figures using coordinate systems.
2. Apply concepts of distance, midpoint, and slope to classify figures and solve problems in the coordinate plane.
3. Analyze conic sections.
4. Determine the effects of geometric transformations on the graph of a function or relation.
5. Analyze transformations and symmetries of figures in the coordinate plane.

#### Domain IV—Trigonometry and Calculus

##### Competency 011—Understand trigonometric functions.

For example:

1. Apply trigonometric functions to solve problems involving distance and angles.
2. Apply trigonometric functions to solve problems involving the unit circle.
3. Manipulate trigonometric expressions and equations using techniques such as trigonometric identities.
4. Analyze the relationship between a trigonometric function and its graph.
5. Use trigonometric functions to model periodic relationships.
##### Competency 012—Understand differential calculus.

For example:

1. Evaluate limits.
2. Demonstrate knowledge of continuity.
3. Analyze the derivative as the slope of a tangent line and as the limit of the difference quotient.
4. Calculate the derivatives of functions (e.g., polynomial, exponential, logarithmic).
5. Apply differentiation to analyze the graphs of functions.
6. Apply differentiation to solve real-world problems involving rates of change and optimization.
##### Competency 013—Understand integral calculus.

For example:

1. Analyze the integral as the area under a curve and as the limit of the Riemann sum.
2. Calculate the integrals of functions (e.g., polynomial, exponential, logarithmic).
3. Apply integration to analyze the graphs of functions.
4. Apply integration to solve real-world problems.

#### Domain V—Statistics, Probability, and Discrete Mathematics

##### Competency 014—Understand principles and techniques of statistics.

For example:

1. Use appropriate formats for organizing and displaying data.
2. Analyze data in a variety of representations.
3. Analyze the use of measures of central tendency and variability.
4. Analyze the effects of bias and sampling techniques.
##### Competency 015—Understand principles and techniques of probability.

For example:

1. Determine probabilities of simple and compound events and conditional probabilities.
2. Use counting principles to calculate probabilities.
3. Use a variety of graphical representations to calculate probabilities.
4. Select simulations that model real-world events.
5. Analyze uniform, binomial, and normal probability distributions.
##### Competency 016—Understand principles of discrete mathematics.

For example:

1. Apply concepts of permutations and combinations to solve problems.
2. Analyze sequences and series including limits and recursive definitions.
3. Perform operations on matrices and vectors.
4. Apply set theory to solve problems.