Middle Grades Mathematics 5–9

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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

Pie chart of approximate test weighting outlined in the 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 15%
2 Knowledge of algebra 30%
3 Knowledge of geometry 20%
4 Knowledge of data analysis, statistics, and probability 15%
5 Knowledge of student reasoning and instructional practice 20%

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 integer exponents, multiple levels of grouping symbols, and absolute value).

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

  4. Represent and perform operations with real number approximations with scientific notation, giving attention to significant digits.

  5. Apply factors of whole numbers to arithmetic operations (e.g., common factors, LCD, GCM).

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

  7. Apply properties of operations (i.e., associative, commutative, distributive, inverse relationships between operations) in performing multi-step arithmetic operations with rational numbers.

  8. Solve problems by performing operations with numbers involving radicals and with rational numbers with rational exponents, making use of the laws of exponents.

  9. Interpret operations on rational numbers and radicals within mathematical and real-world contexts.

Competency 2—Knowledge of algebra

  1. Identify and apply numerical and algebraic patterns, using tables, graphs, written descriptions, and formulas.

  2. Evaluate a function at a given value of its input to determine whether a relationship presented in various forms (e.g., tables, written descriptions, function notation) represents a function and to determine its type (i.e., linear, quadratic, cubic, exponential growth and decay, absolute value, square root, cube root).

  3. Apply operations with exponents and radicals to generate equivalent expressions (e.g., polynomials, radical expressions, rational expressions).

  4. Solve linear and absolute value equations or inequalities with one or two variables, representing solutions algebraically or graphically, and interpret the key features (vertex, line of symmetry) of an absolute value function within real-world or mathematical contexts.

  5. Identify the slope and intercepts of a line using a graph, table, or equation, 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 and interpret systems of two-variable linear equations and inequalities, algebraically, graphically, and in real-world contexts.

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

  8. Analyze key features of quadratic functions presented in mathematical and real-world contexts, and solve using a variety of methods (e.g., factoring, quadratic formula, completing the square, graphing).

  9. Determine and select graphical representations of exponential functions in the form abx and a(1 + r)x that represent real-world problems of exponential growth and decay (e.g., problems about depreciation, compound interest, population growth).

  10. Determine the impacts of shifting and scaling transformations on the formulas for linear, quadratic, and absolute value functions.

Competency 3—Knowledge of geometry

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

  2. Apply formulas for the area of a triangle and composite figures to find solutions for various shapes (e.g., rectangles, trapezoids, parallelograms, rhombi).

  3. Apply formulas for volume and surface area of solids (i.e., right solids, Cavalieri’s principle, nets for non-right solids).

  4. 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.

  5. Solve mathematical and real-world problems using the coordinate plane.

  6. Solve mathematical and real-world problems involving proportional relationships between similar 2D and 3D figures.

  7. Solve mathematical and real-world problems using the Triangle Inequality Theorem, the Pythagorean Theorem, and the Pythagorean Theorem converse.

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

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

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

  11. Determine the center, the radius, and the equation of a circle, and select graphical representations of a circle on a coordinate plane.

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

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

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

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

  4. Determine estimates for a population total, mean, and percentage using data from a survey.

  5. Identify statistical questions and samples to draw inferences about a population.

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

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

  8. Determine the theoretical probabilities of outcomes (e.g., rolling a 3 on a standard 6-sided die) and events (e.g., drawing two red balls in a row when drawing with replacement from a bag containing a given number of red and green balls) in simple and repeated experiments.

  9. Determine and compare experimental and theoretical probabilities to make predictions and draw conclusions about real-world situations.

Competency 5—Knowledge of student reasoning and instructional practice

  1. Analyze real-world contexts across subject areas to represent them with appropriate mathematical expressions and equations.

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

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

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

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

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

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

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