# Study Guide

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Field 225: Middle Level/Intermediate 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.

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

#### Competency 0001–Apply knowledge of the structure and property of the real number system.

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

- Classify the real number system into various subsets (e.g., rational numbers, irrational numbers, integers).
- Apply properties of the real number system (e.g., distributive, identities, commutative).
- Analyze place value and the properties of real numbers and their representations (e.g., integers, fractions, powers).
- Apply knowledge of radical, exponential, and scientific notation to model and solve problems.
- Evaluate mathematical conjectures, arguments, and informal proofs involving numbers and use counterexamples to evaluate arguments and disprove suppositions.
- Apply basic knowledge of matrices and their operations.

#### Competency 0002–Apply knowledge of number operations and number theory.

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

- Apply operations to real and complex numbers (e.g., integers, fractions, rational exponents) in problem-solving situations.
- Apply ratios, proportional reasoning, and percentages to solve mathematical and real-world problems.
- Apply knowledge of prime numbers, factors, and divisibility to solve problems.
- Apply knowledge of greatest common factors and least common multiples to model and solve mathematical and real-world problems.
- Analyze models of number operations using a variety of representations (e.g., area models, arrays, number lines).

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

#### Competency 0003–Apply 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 using different representations (e.g., tabular, algebraic, graphic).
- Analyze relations and functions and their graphs in terms of domain, range, symmetry, intercepts, maxima, and minima.
- Analyze the effects of transformations such as f of quantity x plus k, f of x plus k, and k times f of x on the graph of the relation or function f of x.
- Evaluate combinations and compositions of functions using functional notation (e.g., f plus or minus g of x, f times g of x, f of g of x.
- Analyze inverse functions (e.g., finding, graphing, applying).
- Analyze and develop algebraic generalizations of different types of patterns (e.g., recursive, exponential, sequences and series).

#### Competency 0004–Apply algebraic techniques.

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

- Convert everyday language into mathematical language, notation, and symbols and vice versa.
- Analyze given mathematical statements, expressions, or definitions in the context of a mathematical or real-world problem.
- Manipulate algebraic expressions and equations (e.g., factoring, simplifying, transforming).
- Justify algebraic techniques using the properties of the real number system and evaluate the mathematical thinking and strategies of others.

#### Competency 0005–Apply the properties of linear functions and relations.

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

- Distinguish between linear and nonlinear data in various contexts (e.g., tables, real-world situations).
- Analyze the relationship between the equation of a line and its graph 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).
- Solve problems involving linear equations and inequalities using algebraic and graphic techniques.
- Solve systems of linear equations and inequalities (maximum of three variables) in mathematical and real-world contexts using a variety of techniques (e.g., substitution, graphing, linear combination, matrices).

#### Competency 0006–Apply the properties of quadratic and higher-order polynomial functions and relations.

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

- Analyze relationships between different representations of quadratic and higher-order polynomial functions (e.g., tabular, algebraic, graphic).
- Model and solve problems involving quadratic and higher-order polynomial functions and inequalities using a variety of techniques (e.g., completing the square, factoring, graphing, quadratic formula).
- Analyze critical points of quadratic and higher-order polynomial equations (e.g., maximums, minimums, turning points, roots).

#### Competency 0007–Analyze advanced functions and the conceptual foundations of calculus.

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

- Manipulate and simplify radical, rational, absolute value, exponential, and logarithmic expressions.
- Analyze characteristics of radical, rational, absolute value, exponential, logarithmic, and piecewise defined functions and their graphs (e.g., domain and range, continuity, asymptotes).
- Convert algebraic representations of radical, rational, absolute value, exponential, logarithmic, and piecewise defined functions into graphic representations and vice versa.
- Model and solve problems involving radical, rational, absolute value, exponential, logarithmic, and piecewise defined equations in mathematical and real-world contexts.
- Analyze concepts of calculus (e.g., limits, rates of change, continuity) for algebraic functions and their graphs.
- Analyze the concepts of the derivative (e.g., instantaneous rate of change, the slope of the line tangent to a curve) and the integral (e.g., area under a curve, cumulative change) and apply these concepts to polynomial functions.

#### subarea roman numeral 3–Measurement and Geometry

#### Competency 0008–Apply principles and procedures related to measurement.

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

- Compare and convert measurements within various measurement systems using appropriate units.
- Apply formulas to find measures (e.g., angles, length, perimeter, area, volume) of a variety of two- and three-dimensional figures.
- Apply proportional and spatial reasoning to solve mathematical and real-world problems, including the effect of scale factors.
- Apply knowledge of the Pythagorean theorem to solve mathematical and real-world problems.
- Solve real-world problems that require converting between different units of measurement.

#### Competency 0009–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

- Apply the properties of lines (e.g., parallel, perpendicular) and angles (e.g., supplementary, vertical) to characterize geometric relationships and solve problems.
- Apply the principles of similarity and congruence to solve problems involving two- and three-dimensional figures.
- Analyze the properties of circles (e.g., intersecting chords and secants) and polygons (e.g., number and length of sides, measure of angles).
- Apply the properties of special right triangles and the sine, cosine, and tangent ratios in right triangles to solve problems.
- Use principles and theorems of geometry to evaluate logical arguments and mathematical conjectures and to construct and evaluate formal and informal proofs.

#### Competency 0010–Apply the principles and properties of coordinate and transformational geometries.

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

- Apply geometric concepts (e.g., distance formula, midpoint formula, slope) to model and solve mathematical and real-world problems.
- Apply the geometric concepts of parallel and perpendicular lines to model and solve problems.
- Use two- and three-dimensional coordinate systems to represent and analyze geometric figures.
- Analyze and apply geometric transformations (e.g., translations, reflections, dilations, rotations).
- Use the techniques of coordinate geometry to construct and evaluate mathematical arguments and conjectures (e.g., formal and informal proofs).

#### subarea roman numeral 4–Probability and Statistics

#### Competency 0011–Apply the principles, properties, and techniques of probability.

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

- Apply knowledge of theoretical and experimental probability events and their characteristics (e.g., conditional, independent, mutually exclusive).
- Solve problems using the techniques of probability (e.g., fundamental counting principle, addition and multiplication rules, sample spaces).
- Interpret graphic representations of probabilities (e.g., tables, Venn diagrams, tree diagrams, frequency graphs).
- Analyze and apply the properties of normal probability distributions to model and solve problems.

#### Competency 0012–Apply the principles, properties, and techniques of statistics.

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

- Evaluate random sampling techniques to collect representative data.
- Interpret data in a variety of graphic formats (e.g., circle graphs, box-and-whisker plots, scatterplots, histograms).
- Apply knowledge of measures of central tendency (mean, median, and mode) and variation (e.g., standard deviation, range).
- Evaluate statistical claims and inferences and make predictions that are based on a set of data.

#### subarea roman numeral 5–Pedagogical Content Knowledge

#### Competency 0013–Analyze a lesson plan for a given content 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.
- Analyze assessment results of student learning 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. Incorporate one of the Standards for Mathematical Practice in your response.
- Describe how this analysis of assessment data or student work can be used to inform future instruction (e.g., differentiation, reteaching) with respect to this content area and the development and reinforcement of sound mathematical practice.