Study Guide
Field 211: 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 | 13 percent |
| roman numeral 2 | relations, functions, and algebra | 0003–0008 | 23 percent |
| roman numeral 3 | trigonometry and calculus | 0009–0013 | 18 percent |
| roman numeral 4 | measurement and geometry | 0014–0016 | 16 percent |
| roman numeral 5 | probability, statistics, and discrete mathematics | 0017–0019 | 15 percent |
| this cell intentionally left blank. | 85 percent | ||
| subareas | range of competencies | approximate percentage of test | |
|---|---|---|---|
| constructed-response | |||
| roman numeral 6 | pedagogical content knowledge | 0020 | 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.
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 represented in various ways (e.g., graphs, tables, sets).
- Analyze the relationships between different representations (e.g., tabular, algebraic, graphic) of relations and functions.
- Analyze graphs of relations and functions in terms of their characteristics (e.g., domain, range, intercepts, extrema).
- Determine the effects of transformations [e.g., f (x + k), kf (x)] on the graph of a relation or function.
- Determine whether a function has an inverse and analyze the inverse of invertible functions.
- Analyze composite functions and apply operations to functions.
Competency 0004–Analyze the principles and properties of linear algebra.
start italics 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 determinant or inverse of a 2 by 2 matrix.
- Apply matrices to represent and solve systems of linear equations.
- Analyze the matrix of a linear transformation.
Competency 0005–Apply the properties of linear expressions, relations, and functions.
start italics The following topics are examples of content that may be covered under this competency. end italics
- Determine and interpret the slope and intercepts of a linear equation for 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, parallel and perpendicular lines).
Competency 0006–Analyze the properties of quadratic and higher-order polynomial expressions, relations, and functions.
start italics The following topics are examples of content that may be covered under this competency. end italics
- Manipulate and simplify polynomial expressions (e.g., factoring, combining like terms).
- Analyze the 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 characteristics of quadratic and higher-order polynomial functions (e.g., real and complex roots, zeros, x-intercepts, factors, extrema, intervals of increase or decrease, end behavior) and apply these characteristics to solve problems.
- Analyze the equations and graphs of conic sections.
Competency 0007–Apply the principles and properties of rational, radical, piecewise, and absolute value expressions and functions.
start italics 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 0008–Apply the principles and properties of exponential and logarithmic expressions and functions.
start italics 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 0009–Analyze properties of trigonometric expressions, trigonometric functions, and their graphic representations.
start italics 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 of X equals A times sine times the quantity B times X plus C times D ].
- Simplify expressions using trigonometric identities.
- Convert between rectangular and polar coordinates.
- Convert between radians and degrees.
Competency 0010–Apply the principles and techniques of trigonometry to model and solve problems.
start italics The following topics are examples of content that may be covered under this competency. end italics
- Solve mathematical and real-world problems using trigonometric ratios (e.g., angles of elevation and depression).
- Apply trigonometric functions and relationships (e.g., law of sines) to model and solve problems involving angles, length, and area.
- Apply trigonometric functions to model periodic phenomena in mathematics and other disciplines.
- Model and solve problems involving trigonometric equations and inequalities using algebraic and graphic techniques.
Competency 0011–Apply the principles and properties of limits, continuity, and average rates of change.
start italics The following topics are examples of content that may be covered under this competency. end italics
- Apply the concept of a limit to algebraic functions and their graphs (e.g., difference quotient).
- Analyze the characteristics of functions using the concept of a limit (e.g., continuity, asymptotes).
- Apply the relationship between the slope of a secant line and the derivative of a function.
- Apply the relationship between the slope of a secant line and the derivative of a function.
Competency 0012–Apply the principles and techniques of differential calculus.
start italics 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.
- Identify the derivative of a polynomial function, an exponential function, a trigonometric function, or the product or quotient of these types of functions.
- Apply 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 0013–Apply the principles and techniques of integral calculus.
start italics 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.
- Identify the definite and indefinite integral of a polynomial function or an exponential 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 0014–Apply principles and procedures related to measurement.
start italics The following topics are examples of content that may be covered under this competency. end italics
- Apply formulas to find measurements (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).
- Solve problems using the concepts of similarity, scale factors, and proportional reasoning.
- Convert measurements within and between various measurement systems.
- Find angle and arc measurements related to circles using both radians and degrees.
Competency 0015–Apply the principles and properties of Euclidean geometry in two and three dimensions.
start italics 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., corresponding, supplementary, vertical).
- Identify the shapes of two-dimensional cross sections of three-dimensional figures and apply mathematical models (e.g., nets, formulas) to solve problems involving the surface area and volume of three-dimensional figures.
- 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, angle measurements) to analyze and solve problems.
- Apply definitions, postulates, and theorems of geometry (e.g., Pythagorean theorem) to evaluate mathematical conjectures and arguments and to construct and analyze proofs.
Competency 0016–Apply the principles and properties of coordinate geometry.
start italics 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).
Subarea roman numeral 5–Probability, Statistics, and Discrete Mathematics
Competency 0017–Apply the principles, properties, and techniques of probability.
start italics 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, complements).
- 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 0018–Apply the principles, properties, and techniques of statistics.
start italics The following topics are examples of content that may be covered under this competency. end italics
- Analyze sampling techniques used in statistical studies, the role of randomization in study design, and how bias impacts the result of a study.
- Interpret data in a variety of graphic formats (e.g., charts, bar graphs, circle graphs, stem-and-leaf plots, histograms, scatter plots, two-way frequency tables).
- Analyze and apply measures of central tendency (e.g., mean, median) and spread (e.g., standard deviation, variance, interquartile range, range).
- Apply the mean and standard deviation of a numerical data set to fit the data into a normal distribution and to estimate population percentages.
- Analyze statistical measures (e.g., confidence intervals, correlation coefficients, linear regression equations) and make valid inferences and predictions that are based on the measures.
Competency 0019–Apply the principles of discrete mathematics.
start italics The following topics are examples of content that may be covered under this competency. end italics
- Apply various counting strategies (e.g., permutations, combinations, binomial expansions) to problem-solving situations.
- Analyze and apply recurrence relations (e.g., Fibonacci sequence, triangular numbers, Pascal's triangle) 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, logic, linear programming, finite difference methods) to model real-world problems.
Subarea roman numeral 6–Pedagogical Content Knowledge
Competency 0020–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.
start italics 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.