# Study Guide

##
Field 111: Advanced Mathematics

Test Design and Framework

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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

*Does not include 15-minute C B T tutorial

### Test Framework

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

subareas | range of competencies | approximate percentage of test | |
---|---|---|---|

selected-response | |||

roman numeral 1 | number properties and number sense | 0001–0002 | 12 percent |

roman numeral 2 | relations, functions, and algebra | 0003–0009 | 27 percent |

roman numeral 3 | trigonometry and calculus | 0010–0014 | 20 percent |

roman numeral 4 | measurement and geometry | 0015–0017 | 13 percent |

roman numeral 5 | probability, statistics, and discrete mathematics | 0018–0020 | 13 percent |

this cell intentionally left blank. | 85 percent |

subareas | range of competencies | approximate percentage of test | |
---|---|---|---|

constructed-response | |||

roman numeral 6 | pedagogical content knowledge | 0021 | 15 percent |

#### Subarea roman numeral 1–Number Properties and Number Sense

##### Competency 0001–Apply knowledge of the structure, properties, and operations of the real number system.

start italics *The following topics are
examples of content that may be covered under this competency.* end
italics

- Identify the hierarchy of the real number system and its classification into various subsets.
- Analyze the properties of real numbers and their representations (e.g., fractions, radicals, exponents, scientific notation).
- Apply the properties of rational numbers and their operations in problem-solving situations.
- Apply proportional thinking and knowledge of ratios and percentages to represent and solve problems.
- Evaluate mathematical conjectures, arguments, and informal proofs involving numbers, and use counterexamples to evaluate arguments and disprove suppositions.
- Apply the principle of mathematical induction in proving or disproving arguments.

##### Competency 0002–Apply knowledge of number theory and the complex number system.

start italics *The following topics are
examples of content that may be covered under this competency.* end
italics

- Apply the fundamental principles of number theory (e.g., prime numbers, divisibility).
- Apply knowledge of greatest common factors and least common multiples to model and solve problems.
- Analyze algebraic and geometric representations of complex numbers (e.g., polar form, vector form).
- Apply operations on complex numbers (e.g., difference, product, root, geometric interpretation of the sum).

#### Subarea roman numeral 2–Relations, Functions, and Algebra

##### Competency 0003–Analyze the principles and properties of algebraic relations and functions.

start italics *The following topics are
examples of content that may be covered under this competency.* end
italics

- Distinguish between relations and functions.
- Analyze relationships between different representations (e.g., tabular, algebraic, graphic) of relations and functions.
- Analyze relations and functions and their graphs in terms of domain, range, intercepts, maxima, and minima.
- Determine the effects of transformations [e.g.,
*f*(*x*+*k*),*kf*(*x*)] on the graph of a relation or function. - Apply knowledge of technological tools (e.g., spreadsheets, graphing utilities) to analyze functions and solve problems.

##### Competency 0004–Apply algebraic techniques.

*The following topics are
examples of content that may be covered under this competency.* end
italics

- Convert written words and expressions into mathematical notation and symbols, and vice versa.
- Analyze given mathematical statements, expressions, or definitions, and deduce the underlying assumptions.
- Manipulate algebraic expressions and equations (e.g., factoring, simplifying, transforming).
- Justify algebraic techniques using the properties of the real number system, and evaluate mathematical strategies and arguments.

##### Competency 0005–Analyze the principles and properties of linear algebra.

*The following topics are
examples of content that may be covered under this competency.* end
italics

- Apply basic knowledge of the properties of vectors (e.g., components, magnitude) and vector operations to model and solve problems.
- Analyze and apply properties involving matrices (e.g., commutative property of addition, associative property of multiplication).
- Analyze the inverse and determinant of a matrix.
- Use matrices to represent and solve systems of linear equations.
- Analyze the matrix of a linear transformation.

##### Competency 0006–Apply the properties of linear functions and relations.

*The following topics are
examples of content that may be covered under this competency.* end
italics

- Apply the slope and intercept(s) of a linear equation in mathematical and real-world contexts.
- Determine the equation of a line using different types of information (e.g., two points on the line, the slope and one point on the line).
- Model and solve problems involving linear equations and inequalities using algebraic and graphic techniques.
- Solve systems of linear equations and inequalities using a variety of techniques (e.g., substitution, graphing).

##### Competency 0007–Analyze the properties of quadratic and higher-order polynomial relations and functions.

*The following topics are
examples of content that may be covered under this competency.* end
italics

- Analyze relationships between tabular, algebraic, and graphic representations of quadratic and higher-order polynomial functions.
- Model and solve problems involving quadratic and higher-order polynomial equations and inequalities using a variety of techniques (e.g., completing the square, factoring, graphing).
- Analyze the zeros of quadratic and higher-order polynomial functions and apply their characteristics to solve problems.
- Analyze the equations and graphs of conic sections.

##### Competency 0008–Apply the principles and properties of rational, radical, piecewise, and absolute value functions.

*The following topics are
examples of content that may be covered under this competency.* end
italics

- Manipulate and simplify expressions involving rational, radical, piecewise, and absolute value functions.
- Analyze characteristics of rational, radical, piecewise, and absolute value functions and their graphs (e.g., intercepts, asymptotes, domain, range).
- Convert between algebraic and graphic representations of rational, radical, piecewise, and absolute value functions.
- Model and solve problems involving rational, radical, piecewise, and absolute value equations.

##### Competency 0009–Apply the principles and properties of exponential and logarithmic functions.

*The following topics are
examples of content that may be covered under this competency.* end
italics

- Apply the properties of exponents and logarithms to manipulate expressions and solve equations.
- Analyze and apply the inverse relationship between exponential and logarithmic functions.
- Convert between algebraic and graphic representations of exponential and logarithmic functions.
- Model and solve problems involving exponential and logarithmic functions (e.g., compound interest, exponential decay) in mathematical and real-world contexts.

#### Subarea roman numeral 3–Trigonometry and Calculus

##### Competency 0010–Analyze the principles and properties of and relationships involving trigonometric functions and their graphic representations.

*The following topics are
examples of content that may be covered under this competency.* end
italics

- Analyze the relationships between right triangle ratios, trigonometric functions, and the unit circle.
- Analyze graphs of trigonometric functions in terms of frequency, period, amplitude, and shifts (vertical and phase).
- Analyze the effects of transformations on the graph of a trigonometric function
[e.g.,
*f*(*x*) =*a*sin(*bx*+*c*) +*d*]. - Simplify expressions using trigonometric identities.
- Verify trigonometric identities.

##### Competency 0011–Apply the principles and techniques of trigonometry to model and solve problems.

*The following topics are
examples of content that may be covered under this competency.* end
italics

- Solve mathematical and real-world problems using the trigonometry of right triangles.
- Apply trigonometric functions and relationships (e.g., law of sines) to model and solve problems involving angles, length, and area.
- Model and solve problems involving trigonometric equations and inequalities using algebraic and graphic techniques.
- Use trigonometric functions to model periodic phenomena in mathematics and other disciplines.

##### Competency 0012–Apply the principles and properties of limits, continuity, and average rates of change.

*The following topics are
examples of content that may be covered under this competency.* end
italics

- Apply the concept of limits to algebraic functions and their graphs.
- Analyze the characteristics of functions using the concept of limit (e.g., continuity, asymptotes).
- Apply the relationship between the slope of a secant line and the derivative of a function.
- Solve problems involving average rates of change (e.g., average velocity and acceleration).

##### Competency 0013–Apply the principles and techniques of differential calculus.

*The following topics are
examples of content that may be covered under this competency.* end
italics

- Relate the concept of the derivative to instantaneous rate of change and the concept of the slope of the tangent line to a curve.
- Find the derivative of a function.
- Use the concepts of differential calculus to analyze the graph of a function (e.g., extrema, concavity, points of inflection).
- Model and solve real-world problems using differential calculus (e.g., rates of change, optimization, related rates).

##### Competency 0014–Apply the principles and techniques of integral calculus.

*The following topics are
examples of content that may be covered under this competency.* end
italics

- Relate the concept of the integral to the area under a curve.
- Find the definite and indefinite integral of a function.
- Apply integration to solve problems (e.g., area, velocity, volume, distance).
- Model and solve problems involving first-order differential equations (e.g., separation of variables, initial value problems).

#### Subarea roman numeral 4–Measurement and Geometry

##### Competency 0015–Apply principles and procedures related to measurement.

*The following topics are
examples of content that may be covered under this competency.* end
italics

- Apply formulas to find measures (e.g., angles, length, perimeter, area, volume) for a variety of two- and three-dimensional figures.
- Solve problems involving derived units (e.g., density, pressure, rates of change).
- Convert measurements within and between various measurement systems.
- Find angle and arc measures related to circles.

##### Competency 0016–Apply the principles and properties of Euclidean geometry in two and three dimensions.

*The following topics are
examples of content that may be covered under this competency.* end
italics

- Characterize geometric relationships and solve problems using the properties of lines (e.g., parallel, perpendicular) and angles (e.g., supplementary, vertical).
- Apply the principles of similarity and congruence to solve problems involving two- and three-dimensional figures.
- Apply the properties of circles (e.g., intersecting chords and secants) and polygons (e.g., number and length of sides, measure of angles) to analyze and solve problems.
- Use definitions, postulates, and theorems of geometry (e.g., Pythagorean theorem) to evaluate mathematical conjectures and arguments, and construct and analyze proofs.

##### Competency 0017–Apply the principles and properties of coordinate geometry.

*The following topics are
examples of content that may be covered under this competency.* end
italics

- Apply geometric concepts (e.g., distance, midpoint, slope) to model and solve problems.
- Apply the geometric concepts of parallel and perpendicular lines to model and solve problems.
- Analyze and apply geometric transformations (e.g., translations, reflections, dilations, rotations).
- Represent two- and three-dimensional geometric figures in various coordinate systems (e.g., Cartesian, polar).
- Apply knowledge of connections between mathematical concepts in different fields (e.g., algebra, coordinate geometry, probability).

#### Subarea roman numeral 5–Probability, Statistics, and Discrete Mathematics

##### Competency 0018–Apply the principles, properties, and techniques of probability.

*The following topics are
examples of content that may be covered under this competency.* end
italics

- Evaluate descriptions and calculate the probabilities of different kinds of events (e.g., conditional, independent, mutually exclusive).
- Solve problems using the techniques of probability (e.g., addition and multiplication rules).
- Interpret graphic representations of probabilities (e.g., tables, Venn diagrams, tree diagrams, frequency graphs, the normal curve).
- Analyze and apply the properties of probability distributions (e.g., binomial, normal) to model and solve problems.

##### Competency 0019–Apply the principles, properties, and techniques of statistics.

*The following topics are
examples of content that may be covered under this competency.* end
italics

- Apply random sampling techniques to collect representative data.
- Interpret data in a variety of graphic formats (e.g., charts, bar graphs, circle graphs, stem-and-leaf plots, histograms, scatter plots).
- Analyze and apply measures of central tendency (e.g., mean, median) and dispersion (e.g., standard deviation).
- Analyze statistical measures (e.g., correlation coefficients, confidence intervals, linear regression equations) and make valid inferences and predictions that are based on the measures.

##### Competency 0020–Apply the principles of discrete mathematics.

*The following topics are
examples of content that may be covered under this competency.* end
italics

- Apply various counting strategies (e.g., permutations, combinations) to problem-solving situations.
- Analyze and apply recurrence relations (e.g., Fibonacci sequence, triangular numbers) to model and solve problems.
- Analyze and apply sequences and series (e.g., arithmetic, geometric) to model and solve problems.
- Apply the basic elements of discrete mathematics (e.g., graph theory, linear programming, finite difference methods) to model real-world problems.

#### Subarea roman numeral 6–Pedagogical Content Knowledge

##### Competency 0021–Analyze a lesson plan for a given learning standard in the Oklahoma Academic Standards for Mathematics, including examples of student work and/or assessments, and describe subsequent activities that address student needs on the basis of your analysis.

*The following topics are
examples of content that may be covered under this competency.* end
italics

- Apply knowledge of standards-based learning goals for mathematical content and processes.
- Analyze assessment results of student learning and/or samples of student work for a particular lesson in mathematics, citing specific evidence from the exhibits that identifies a significant mathematical strength as well as a significant area of need shown by the student or students.
- Describe an appropriate instructional strategy or intervention that would help the student or students improve in the identified area of need.
- Describe how your analysis of assessment data and/or student work can be used to inform future instruction with respect to this content area and the development and reinforcement of sound mathematical practice.