# Assessment Design and

Framework

## Field 027: Mathematics

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

### Assessment Design

Format | Computer-based test (CBT) and online-proctored test |
---|---|

Number of Questions | 150 multiple-choice questions |

Time* | 255 minutes |

Passing Score | 220 |

*Does not include 15-minute tutorial or optional 15-minute break for online-proctored assessment

### Framework

Pie chart of approximate test weighting outlined in the table below.

Domain | Range of Competencies | Approximate Percentage of Assessment Score | |
---|---|---|---|

I | Mathematical Processes and Number Sense | 0001–0003 | 19% |

II | Patterns, Algebra, and Functions | 0004–0007 | 24% |

III | Measurement and Geometry | 0008–0010 | 19% |

IV | Trigonometry and Calculus | 0011–0013 | 19% |

V | Statistics, Probability, and Discrete Mathematics | 0014–0016 | 19% |

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

**0001 Understand mathematical problem solving.**

Includes:

- Identify an appropriate problem-solving strategy for a particular problem.
- Analyze the use of estimation in a variety of situations (e.g., rounding, area, plausibility).
- Solve mathematical and real-world problems involving integers, rational numbers, decimals, and percents.
- Solve mathematical and real-world problems involving ratios, proportions, and average rates of change.

**0002 Understand mathematical communication, connections, and reasoning.**

Includes:

- Translate between representations (e.g., graphic, verbal, symbolic).
- Recognize connections between mathematical concepts.
- Identify viable arguments based on inductive and deductive reasoning.
- Apply principles of logic to solve problems.
- Demonstrate knowledge of the historical development of major mathematical concepts, including contributions from diverse cultures.

**0003 Understand number theory.**

Includes:

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

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

**0004 Understand relations and functions.**

Includes:

- Demonstrate knowledge of relations and functions and their applications.
- Perform operations with functions, including compositions and inverses.
- Analyze characteristics of functions.
- Interpret different representations of functions.

**0005 Understand linear, quadratic, and higher-order polynomial functions.**

Includes:

- 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.
- Solve real-world problems by modeling them with linear, quadratic, and higher-order polynomial functions.

**0006 Understand exponential and logarithmic functions.**

Includes:

- Apply the laws of exponents and logarithms.
- Analyze the relationship between exponential and logarithmic functions.
- Analyze exponential and logarithmic functions and their graphs.
- Solve real-world problems by modeling them with exponential and logarithmic functions.

**0007 Understand rational, radical, absolute value, and piece-wise defined functions.**

Includes:

- Manipulate rational, radical, and absolute value expressions, equations, and inequalities.
- Analyze the relationship between a rational, radical, absolute value, or piece-wise defined function and its graph.
- Analyze rational, radical, absolute value, and piece-wise defined functions in terms of domain, range, and asymptotes.
- Solve real-world problems by modeling them with rational, radical, absolute value, and piece-wise defined functions.

##### Domain III–Measurement and Geometry

**0008 Understand measurement principles and procedures.**

Includes:

- Analyze the use of various units and unit conversions within the customary and metric systems.
- Apply the concepts of similarity, scale factors, and proportional reasoning to 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-world problems.

**0009 Understand Euclidean geometry in two and three dimensions.**

Includes:

- Demonstrate knowledge of axiomatic systems and of the axioms of Euclidian and non-Euclidean geometries.
- Use the properties of polygons and circles to solve problems.
- Apply the Pythagorean theorem and its converse.
- Analyze formal and informal geometric proofs, including the use of similarity and congruence.
- Use nets and cross sections to analyze three-dimensional figures.

**0010 Understand coordinate and transformational geometry.**

Includes:

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

##### Domain IV–Trigonometry and Calculus

**0011 Understand trigonometric functions.**

Includes:

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

**0012 Understand differential calculus.**

Includes:

- 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-world problems involving rates of change and optimization.

**0013 Understand integral calculus.**

Includes:

- 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-world problems.

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

**0014 Understand principles and techniques of statistics.**

Includes:

- 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.

**0015 Understand principles and techniques of probability.**

Includes:

- Determine probabilities of simple and compound events and conditional probabilities.
- Use counting principles to calculate probabilities.
- Use a variety of graphical representations to calculate probabilities.
- Select simulations that model real-world events.
- Analyze uniform, binomial, and normal probability distributions.

**0016 Understand principles of discrete mathematics.**

Includes:

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