Mathematics 6–12

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Competencies and Skills and Blueprint

The test design below describes general testing information. The blueprints that follow provide a detailed outline that explains the competencies and skills that this test measures.

Test Design

table describing the format of the test, including test time, number of questions, and required passing score
Format Computer-based test (CBT)
Number of Questions Approximately 50 multiple-choice questions
Time 2 hours and 30 minutes
Passing Score A scaled score of at least 200

 

Competencies, Skills, and Approximate Percentages of Questions

graph of percentages of each competency's weight toward overall test score, described in table below

table describing the competencies, skills, and approximate percentage of each competency's weight toward overall test score
Competency Approximate Percentage of Total Test Questions
1 Knowledge of number sense, operations, and proportionality 8%
2 Knowledge of algebra 19%
3 Knowledge of geometry and trigonometry 28%
4 Knowledge of data analysis, statistics, and probability 19%
5 Knowledge of precalculus and calculus 16%
6 Knowledge of student reasoning and instructional practice 10%

Competencies and Skills

Competency 1—Knowledge of number sense, operations, and proportionality

  1. Compare and convert between rational numbers represented in various ways (i.e., fractions, terminating and repeating decimals, percentages, number line).

  2. Solve problems by performing operations with rational numbers, using estimates and algorithms, and evaluate multi-step expressions using order of operations (e.g., expressions with rational exponents, multiple levels of grouping symbols, radicals, absolute value).

  3. Estimate irrational numbers, including square roots, and compare them to rational numbers.

  4. Represent real number approximations with scientific notation.

  5. Solve problems involving ratios and proportions (e.g., mixtures, comparisons, rates, measurement conversions, graphs, percent growth, taxes, depreciation).

Competency 2—Knowledge of algebra

  1. Generate equivalent expressions (e.g., polynomials, radical expressions, rational expressions) to determine a quantity of interest within a context or solve for an unknown.

  2. Determine key features of 2-variable relationships presented in various forms (e.g., tables, graphs, equations, written descriptions, function notation), and evaluate whether a relationship is a function.

  3. Identify function types (i.e., linear, quadratic, cubic, exponential growth and decay, absolute value, square root, cube root, reciprocals of linear functions, step, piecewise linear), and determine whether a function is even or odd.

  4. Solve equivalent algebraic expressions (e.g., polynomials, radical expressions, rational expressions) in mathematical and real-world problems using operations (e.g., associative, commutative and distributive laws, order of operations, laws of exponents) and properties of equality.

  5. Interpret slopes and intercepts of a linear function and determine the equation of a line (i.e., passes through two given points, through one given point, perpendicular to a given line, parallel to a given line, has a given slope).

  6. Solve absolute value and quadratic inequalities with one or two variables, within real-world or mathematical contexts representing solutions algebraically or graphically.

  7. Evaluate systems of linear equations or linear inequalities that describe quantities or relationships in mathematical and real-world contexts.

  8. Interpret the x-intercepts, y-intercept, vertex, line of symmetry, and concavity of a quadratic function representing real-world and mathematical situations.

  9. Solve quadratic equations over the real and complex number systems.

  10. Select graphical representations of and determine exponential functions representing real-world problems of exponential growth and decay.

  11. Determine the impacts of shifting and scaling transformations on various representations (e.g., tables, graphs, formulas) presented in mathematical and real-world contexts.

  12. Determine the inverse of a given function with description of its domain and range, the composition of 2 functions, and interpret inverses and compositions in mathematical and real-world contexts.

Competency 3—Knowledge of geometry and trigonometry

  1. Classify triangles, quadrilaterals, and solids based on their defining attributes.

  2. Solve mathematical and real-world problems involving properties of angles, using the Triangle Sum Theorem, parallel lines cut by a transversal, relationships between angles of triangles, and the formula for the sum of interior angles of polygons.

  3. Prove and apply relationships about lines, angles, triangles, and quadrilaterals in mathematical and real-world contexts.

  4. Apply Side-Side-Side, Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, Angle-Angle and Hypotenuse-Leg criteria to prove pairs of triangles, overlapping triangles, and figures composed of triangles are congruent or similar, including the concept of congruence and similarity of triangles to solve mathematical and real-world problems.

  5. Apply translations, rotations, reflections and scaling transformations and sequences, based on the relationship between a 2D geometric figure and its pre-image, to demonstrate congruence and similarity.

  6. Apply coordinate geometry to classify and justify properties of lines, triangles, quadrilaterals, and circles, and solve mathematical and real-world problems (e.g., using the distance formula to determine distances between points and between parallel lines).

  7. Identify the shapes of 2D cross sections of 3D figures and of 3D figures generated by rotations of 2D figures.

  8. Solve mathematical and real-world problems involving proportional relationships between similar 2D and 3D figures, including problems involving scale factors of measurements of similar figures (e.g., perimeter, volume).

  9. Solve mathematical and real-world problems involving formulas for the perimeter, circumference and area of 2D figures and the surface area and volume of 3D figures.

  10. Determine and justify geometric constructions (e.g., copy or bisect a segment or an angle).

  11. Solve mathematical and real-world problems involving the lengths and intersection angles of secants, chords, and tangent segments, the measures of central and inscribed angles in circles and semicircles, and the areas of sectors of circles.

  12. Select equations and graphs representing a conic section (e.g., circle, parabola) on a coordinate plane, and apply conic sections to model real-world situations.

  13. Apply basic right triangle trigonometry and the Pythagorean Theorem to determine unknown sides of right triangles.

  14. Solve mathematical and real-world problems using trigonometric ratios and the laws of sines and of cosines.

  15. Determine the values for and identify the graphs of the six trigonometric functions, including their key features.

  16. Select graphical representations of and solve periodic mathematical and real-world problems that are modeled with trigonometric functions.

  17. Convert between rectangular and polar coordinates, and select graphical representations of polar equations in the polar plane.

  18. Select graphical representations of curves in a plane represented parametrically, and apply parametric equations to model problems involving motion in the plane.

  19. Interpret the language of logic, including the converse, inverse, and contrapositive of an “if … then” statement, and evaluate the validity of informal and formal geometric arguments.

Competency 4—Knowledge of data analysis, statistics, and probability

  1. Identify and determine appropriate measures of central tendency and measures of variation of a numerical data set.

  2. Apply the mean and standard deviation of a numerical data set to fit it to a normal distribution and estimate population percentages.

  3. Interpret information and patterns from a numerical data display and the shape of its distribution (i.e., symmetry, gaps, clusters, outliers, mode, range).

  4. Identify displays for univariate numerical and categorical data (e.g., histograms, box plots, bar charts, frequency tables).

  5. Determine population total, mean, and percentage using data from a survey, considering margin of error.

  6. Identify appropriate processes used to design statistical experiments, including the use of randomized trials and proper sampling methods to reduce possible sources of bias.

  7. Determine the properties of correlations in bivariate data displayed in scatter plots and 2-way frequency tables, representing real-world situations.

  8. Select linear functions that fit to real-world bivariate numerical data that suggest a possible linear association, and interpret the x- and y-intercepts.

  9. Use a variety of methods (e.g., permutations, combinations, tables, tree diagrams, addition and multiplication rules) to determine the probabilities and conditional probabilities of events in probability experiments (e.g., complements of events and unions, intersections of independent and dependent events).

  10. Analyze 2-way frequency tables and relative frequency tables summarizing real-world bivariate categorical data, using the data or conditional, marginal, or joint relative frequencies from the data, and interpret the quantities in such a table within a context.

  11. Determine an appropriate sample space for a simple experiment (e.g., rolling a die, drawing an object randomly from a bag, spinning a spinner) or a repeated simple experiment (e.g., repeatedly tossing a coin, rolling several dice, drawing an object repeatedly from a bag with replacement).

  12. Compare experimental and theoretical probabilities to make predictions and draw conclusions about real-world situations.

  13. Calculate the expected value and standard deviation of a random variable to solve real-world problems (e.g., compare the expected payoffs and risks of different decisions).

Competency 5—Knowledge of precalculus and calculus

  1. Solve mathematical and real-world problems involving quantities that can be represented by vectors (e.g., displacement, velocity, acceleration, force, magnetic fields), using addition, subtraction, and scalar multiplication of vectors represented both algebraically and on the coordinate plane.

  2. Solve mathematical and real-world problems using exponential and logarithmic functions and equations (e.g., exponential functions with base e and natural logarithms) and apply the properties of logarithmic functions.

  3. Apply the Fundamental Theorem of Algebra and the Factor Theorem to analyze polynomials (e.g., using the graph of a polynomial to help determine its algebraic formula).

  4. Select rational functions representing mathematical and real-world problems in written and graphical form, and interpret key features of the graphs.

  5. Determine and apply arithmetic and geometric sequences, defined explicitly or recursively, in mathematical and real-world problems.

  6. Determine limits using theorems concerning sums, products, and quotients of functions.

  7. Determine derivatives of algebraic, exponential, logarithmic, and trigonometric functions and combinations of functions formed using arithmetic operations and compositions.

  8. Apply differentiation to determine features of functions representing mathematical and real-world problems (e.g., intervals, concavity, local maxima and minima, tangent lines of graphs).

  9. Apply properties of integration and substitution to determine anti-derivatives of algebraic, exponential, logarithmic, and trigonometric functions and combinations formed by using arithmetic operations.

  10. Apply definite integrals to solve mathematical and real-world problems involving distance, area, and volume.

  11. Calculate binomial expansions to a positive integral power or determine a specified term in the expansion.

Competency 6—Knowledge of student reasoning and instructional practices

  1. Identify appropriate methods to facilitate instruction in using mathematical strategies, concepts, and procedures with fluency to solve problems in various real-world or mathematical contexts.

  2. Identify opportunities for students to evaluate the reasonableness of their results, and assess the validity of students’ mathematical arguments.

  3. Identify patterns to make connections between mathematical and real-world problems across subject areas, and analyze a sequence of concepts for mathematical continuity.

  4. Select appropriate mathematical representations (e.g., verbal statements, tables, algebraic expressions) and instructional tools for teaching mathematical concepts to all students.

  5. Analyze learning progressions to demonstrate how students’ mathematical knowledge and skills develop over time among concrete, representational, and abstract modes of understanding.

  6. Analyze and interpret individual student mathematics assessment data using a variety of assessment formats to guide instructional decisions and differentiate instruction.

  7. Analyze students' mathematical misconceptions, errors, and gaps in knowledge and choose instructional approaches to promote student achievement.