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.