# 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) |
---|---|

Number of Questions | 80 multiple-choice questions |

Time* | 2 hours and 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 | 15% |

II. | Patterns, Algebra, and Functions | 0003–0004 | 20% |

III. | Measurement and Geometry | 0005–0007 | 20% |

IV. | Trigonometry and Advanced Algebraic Functions | 0008–0010 | 25% |

V. | Statistics and Probability | 0011–0012 | 20% |

#### 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.
- Solve applied and theoretical problems using the principles of basic number theory (e.g., prime factorization, divisibility, greatest common factor, least common multiple).
- Solve applied and theoretical problems involving integers, fractions, decimals, and percents.
- Solve applied and theoretical problems involving ratios, proportions, and average rates of change.
- Solve applied and theoretical problems involving irrational numbers and exponents.

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

For example:

- Apply properties of complex numbers to perform operations.
- Translate between representations of vector quantities (e.g., graphic, verbal, symbolic).
- Solve applied and theoretical problems using vector operations.
- Apply properties of matrices to perform operations.
- Solve applied and theoretical problems using properties of matrices.

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

##### Competency 0003–Understand relations and functions.

For example:

- Analyze properties of functions and relations (e.g., domain, range, one-to-one correspondence).
- Translate between different representations of functions and relations.
- Solve applied and theoretical problems using operations with functions, including compositions and inverses.
- Interpret different representations (e.g., equations, graphs, tables) of functions.
- Evaluate sequences and series (e.g., computing term, sums, recursive definitions, limits).

##### 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 and quadratic 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 functions.
- Model and solve applied problems involving linear, quadratic, and higher-order polynomial functions.

#### Domain III—Measurement and Geometry

##### Competency 0005–Understand measurement principles and procedures.

For example:

- Solve applied and theoretical problems using units and unit conversions.
- Solve applied and theoretical problems using the concepts of similarity, scale factors, and proportional reasoning.
- Analyze precision, error, and rounding in measurements and computed quantities.
- Solve applied and theoretical problems using the concepts of perimeter, circumference, area, surface area, and volume.

##### Competency 0006–Understand Euclidean geometry in two and three dimensions.

For example:

- Solve applied and theoretical problems using the characteristics of triangles (e.g., centroid, orthocenter) and of right triangles (e.g., basic trigonometric ratios).
- Solve applied and theoretical problems using the properties of polygons and circles.
- Solve applied and theoretical problems using the Pythagorean theorem and its converse.
- Analyze formal and informal proofs using geometric concepts (e.g., similarity, congruence, properties of angles, applications of parallel and perpendicular lines).
- Analyze 3D figures using 2D diagrams and representations (e.g., nets, cross sections, vertex-edge graphs).
- Analyze methods for constructing geometric figures.

##### Competency 0007–Understand coordinate and transformational geometry.

For example:

- Analyze 2D and 3D figures using coordinate systems.
- Solve applied and theoretical problems in the coordinate plane using the concepts of distance, midpoint, and slope to classify figures.
- 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.
- Analyze transformations to determine similarity and congruence of figures.

#### Domain IV—Trigonometry and Advanced Algebraic Functions

##### Competency 0008–Understand trigonometric functions.

For example:

- Solve applied and theoretical problems involving distance and angles using trigonometric functions.
- Solve problems involving the unit circle by applying trigonometric functions.
- Manipulate trigonometric expressions and solve equations using trigonometric identities.
- Analyze the relationship between a trigonometric function and its graph.
- Model periodic relationships using trigonometric functions and use them to solve applied and theoretical problems.
- Solve applied and theoretical problems using the law of sines and the law of cosines.

##### Competency 0009–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 applied and theoretical mathematical problems involving exponential and logarithmic functions.

##### Competency 0010–Understand rational, radical, absolute value, and piecewise defined functions.

For example:

- Simplify rational, radical, and absolute value expressions.
- Perform operations on rational and radical expressions.
- 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 applied and theoretical problems involving rational, radical, and absolute value equations and inequalities.
- Model and solve applied and theoretical problems involving piecewise defined functions.

#### Domain V—Statistics and Probability

##### Competency 0011–Understand principles and techniques of statistics.

For example:

- Organize and display data using appropriate formats.
- Analyze and interpret data in a variety of representations (e.g., box plots, histograms, dot plots, scatter plots, two-way tables).
- Calculate and analyze measures of central tendency and variability.
- Calculate and interpret a Z-score when given a mean and a standard deviation.
- Analyze the effects of bias and sampling techniques.
- Analyze probability distributions (e.g., uniform, binomial, normal) and determine whether a data set is approximately normally distributed.
- Analyze the correlation coefficient and distinguish between correlation and causation.

##### Competency 0012–Understand principles and techniques of probability.

For example:

- Apply the rules of probability to compute probabilities of simple and compound events and conditional probabilities.
- Calculate probabilities and solve problems using counting principles (e.g., the fundamental counting principle, factorials, permutations, combinations) and set theory.
- Calculate probabilities using a variety of graphical representations.
- Select simulations that model data collection scenarios.
- Calculate and interpret expected value from a discrete probability distribution.