Study Guide

Mathematics
Test Design and Framework

The test design below describes general assessment information. The framework that follows is a detailed outline that explains the knowledge and skills that this test measures.

Test Design

Format	Computer-based test (CBT) and online-proctored test
Number of Questions	80 multiple-choice questions
Time*	2 hours, 30 minutes
Passing Score	220

*Does not include 15-minute tutorial

Test Framework

 

Content Domain		Range of Competencies	Approximate Percentage of Test Score
I.	Numbers and Quantity	0001–0002	14%
II.	Patterns, Algebra, and Functions	0003–0006	30%
III.	Measurement and Geometry	0007–0009	21%
IV.	Trigonometry and Calculus	0010–0012	21%
V.	Statistics and Probability	0013–0014	14%

Domain I–Numbers and Quantity

Competency 0001–Understand real numbers and mathematical problem solving.

For example:

• Analyze the structure of the real numbers.
• Analyze the properties of numbers and operations.
• Apply the principles of basic number theory (e.g., prime factorization, greatest common factor, least common multiple) to solve real-life and mathematical problems.
• Solve real-life and mathematical problems involving integers, fractions, decimals, and percents.
• Solve real-life and mathematical problems involving ratios, proportions, and average rates of change.
• Solve real-life and mathematical problems involving irrational numbers and exponents.

Competency 0002–Understand complex numbers, vectors, and matrices.

For example:

• Apply complex numbers and their operations to solve problems.
• Translate between representations of vector quantities (e.g., graphic, verbal, symbolic).
• Perform vector operations and solve problems that can be modeled using vectors.
• Demonstrate knowledge of matrix operations.
• Apply properties of matrices to model and solve problems.

Domain II–Patterns, Algebra, and Functions

Competency 0003–Understand relations and functions.

For example:

• Demonstrate knowledge of relations and functions and their applications.
• Translate between different representations of functions.
• Perform operations with functions, including compositions and inverses.
• Analyze characteristics of functions.
• Interpret different representations (e.g., equations, graphs, tables) of functions.
• Analyze sequences and series, including limits and recursive definitions, and use them to solve problems.

Competency 0004–Understand linear, quadratic, and higher-order polynomial functions.

For example:

• Analyze the relationship between a linear, quadratic, or higher-order polynomial function and its graph.
• Solve linear and quadratic equations and inequalities using a variety of methods.
• Solve systems of linear equations or inequalities using a variety of methods.
• Solve higher-order polynomial equations and inequalities in one and two variables.
• Analyze the characteristics of linear, quadratic, and higher-order polynomial equations.
• Model and solve real-life problems involving linear, quadratic, and higher-order polynomial functions.

Competency 0005–Understand exponential and logarithmic functions.

For example:

• Apply the laws of exponents and logarithms.
• Analyze the relationship between exponential and logarithmic functions.
• Analyze exponential and logarithmic functions and their graphs.
• Model and solve real-life and mathematical problems involving exponential and logarithmic functions.

Competency 0006–Understand rational, radical, absolute value, and piecewise defined functions.

For example:

• Manipulate rational, radical, and absolute value expressions, equations, and inequalities.
• Analyze the relationship between a rational, radical, absolute value, or piecewise defined function and its graph.
• Analyze rational, radical, absolute value, and piecewise defined functions in terms of domain, range, and asymptotes.
• Model and solve real-life and mathematical problems involving rational, radical, absolute value, and piecewise defined functions.

Domain III–Measurement and Geometry

Competency 0007–Understand measurement principles and procedures.

For example:

• Reason quantitatively and use units and unit conversions to solve problems.
• Apply the concepts of similarity, scale factors, and proportional reasoning to model and solve measurement problems.
• Analyze precision, error, and rounding in measurements and computed quantities.
• Apply the concepts of perimeter, circumference, area, surface area, and volume to solve real-life and mathematical problems.

Competency 0008–Understand Euclidean geometry in two and three dimensions.

For example:

• Demonstrate knowledge of inductive and deductive reasoning and of axiomatic systems.
• Use the properties of triangles (e.g., centroid, orthocenter) and of right triangles (e.g., basic trigonometric ratios) to model and solve problems.
• Use the properties of polygons and circles to model and solve problems.
• Apply the Pythagorean theorem and its converse to model and solve real-life and mathematical problems.
• Analyze formal and informal proofs, using geometric concepts (e.g., similarity, properties of angles, parallel lines).
• Use two-dimensional diagrams and representations (e.g., nets, cross sections, vertex-edge graphs) to analyze three-dimensional figures.

Competency 0009–Understand coordinate and transformational geometry.

For example:

• Analyze two- and three-dimensional figures using coordinate systems.
• Apply concepts of distance, midpoint, and slope to classify figures and model and solve problems in the coordinate plane.
• Analyze conic sections, their equations, and the relationship between the equation of a conic section and its graph.
• Determine the effects of transformations on the graph of a function or relation.
• Analyze transformations and symmetries of figures and of the graphs of functions in the coordinate plane.

Domain IV–Trigonometry and Calculus

Competency 0010–Understand trigonometric functions.

For example:

• Apply trigonometric functions to solve problems involving distance and angles.
• Apply trigonometric functions to solve problems involving the unit circle.
• Manipulate trigonometric expressions and solve equations using techniques such as trigonometric identities.
• Analyze the relationship between a trigonometric function and its graph.
• Use trigonometric functions to model periodic relationships.

Competency 0011–Understand differential calculus.

For example:

• Evaluate limits.
• Demonstrate knowledge of continuity.
• Analyze the derivative as the slope of a tangent line and as the limit of the difference quotient.
• Calculate the derivatives of functions (e.g., polynomial, exponential, logarithmic).
• Apply differentiation to analyze the graphs of functions.
• Apply differentiation to solve real-life and mathematical problems involving rates of change and optimization.

Competency 0012–Understand integral calculus.

For example:

• Apply the fundamental theorem of calculus to model and solve problems.
• Analyze the integral as the area under a curve and as the limit of the Riemann sum.
• Calculate the integrals of functions (e.g., polynomial, exponential, logarithmic).
• Apply integration to analyze the graphs of functions.
• Apply integration to solve real-life and mathematical problems, including problems involving solids of revolution.

Domain V–Statistics and Probability

Competency 0013–Understand principles and techniques of statistics.

For example:

• Use appropriate formats for organizing and displaying data.
• Analyze data in a variety of representations.
• Analyze the use of measures of central tendency and variability.
• Analyze the effects of bias and sampling techniques.

Competency 0014–Understand principles and techniques of probability.

For example:

• Apply the rules of probability to compute probabilities of simple and compound events and conditional probabilities.
• Use counting principles (e.g., permutations, combinations) and set theory to calculate probabilities and solve problems.
• Use a variety of graphical representations to calculate probabilities.
• Select simulations that model real-life events.
• Analyze uniform, binomial, and normal probability distributions.

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