NES Mathematics (326) Practice Tests

Competencies 0001– to 0003 make up about 20% of the test. They ask you to understand real numbers and mathematical problem solving; apply algebraic techniques; understand complex numbers, vectors, and matrices.

• Competency 0001: Understand real numbers and mathematical problem solving. Skills tested: Analyze the group structure of the real numbers (e.g., classifying numbers); Apply properties of the real number system (e.g., distributive, associative, identity, properties of equality); Solve mathematical and real-world problems involving integers, rational numbers, decimals, and percents; Solve mathematical and real-world problems involving ratios, proportions, and average rates of change; Solve mathematical and real-world problems involving radical and exponential expressions (e.g., scientific notation); Solve mathematical and real-world problems using the principles of basic number theory (e.g., prime factorization, greatest common factor, least common multiple).
• Competency 0002: Apply algebraic techniques. Skills tested: Translate informal mathematical language into formal mathematical language, notation, and symbols and vice versa; Manipulate algebraic expressions and equations (e.g., solving, factoring, simplifying, rewriting); Analyze given mathematical statements, expressions, or definitions in the context of a mathematical or real-world problem; Analyze mathematical conjectures, arguments, and informal proofs involving numbers (e.g., use counterexamples to evaluate arguments and disprove claims); Justify algebraic techniques using the properties of the real number system (e.g., analyze and evaluate the mathematical thinking and strategies of others).
• Competency 0003: Understand complex numbers, vectors, and matrices. Skills tested: Apply properties of complex numbers to perform operations and solve problems; Represent complex numbers in rectangular and polar form; Analyze and compare representations of vector quantities (e.g., graphic, verbal, symbolic); Apply vector operations to solve mathematical and real-world problems; Apply properties of matrices to perform operations; Solve mathematical and real-world problems using properties of matrices.

Competencies 0004– to 0007 make up about 30% of the test. They ask you to apply the principles and properties of algebraic functions and relations; understand linear, quadratic, and higher-degree polynomial functions; understand exponential and logarithmic functions, and more.

• Competency 0004: Apply the principles and properties of algebraic functions and relations. Skills tested: Understand properties of functions given different representations (e.g., domain, range, one-to-one correspondence); Analyze properties of functions given different representations (e.g., meaning of intercepts, verbal descriptions); Convert between different representations of relations and functions (e.g., equations, graphs, tables); Solve mathematical and real-world problems using operations on functions, including compositions and inverses; Evaluate sequences and series (e.g., arithmetic, geometric, sums, recursive definitions).
• Competency 0005: Understand linear, quadratic, and higher-degree polynomial functions. Skills tested: Analyze the relationship between a linear, quadratic, or higher-degree polynomial function and its graph; Transform linear, quadratic, and higher-degree polynomial functions and build new functions from existing functions; Solve linear and quadratic equations and inequalities using a variety of methods; Solve systems of linear equations or inequalities using a variety of methods; Solve higher-degree polynomial equations and inequalities in one and two variables; Analyze the characteristics of linear, quadratic, and higher-degree polynomial equations (e.g., relative extrema, end behaviors, average rate of change); Solve real-world problems by modeling them with linear, quadratic, or higher-degree polynomial functions.
• Competency 0006: Understand exponential and logarithmic functions. Skills tested: Apply the properties of exponents and logarithms; Analyze the relationship between exponential and logarithmic functions; Analyze exponential and logarithmic functions and their graphs; Solve real-world problems by modeling them with exponential or logarithmic functions; Transform exponential and logarithmic functions and build new functions from existing functions.
• Competency 0007: Understand rational, radical, absolute value, and piecewise defined functions. Skills tested: Transform rational, radical, and absolute value functions and build new functions from existing functions; Analyze the relationship between a function (e.g., rational, radical, absolute value, piecewise defined) and its graph; Analyze rational, radical, absolute value, and piecewise defined functions in terms of domain, range, and asymptotes; Solve real-world problems by modeling them with rational, radical, absolute value, or piecewise defined functions.

Competencies 0008– to 0010 make up about 20% of the test. They ask you to understand measurement principles and procedures; understand Euclidean geometry in two and three dimensions; understand coordinate and transformational geometry.

• Competency 0008: Understand measurement principles and procedures. Skills tested: Analyze the use of various units and unit conversions within the customary and metric systems (e.g., choosing appropriate units); Solve real-world problems that require converting between different units of measurement; Apply the concepts of similarity, scale factors, and proportional reasoning to solve mathematical and real-world measurement problems; Analyze precision, error, and rounding in measurements and computed quantities; Apply the concepts of perimeter, circumference, area, surface area, and volume to solve problems (e.g., effects of doubling and tripling units on area and/or volume).
• Competency 0009: Understand Euclidean geometry in two and three dimensions. Skills tested: Solve applied and theoretical problems using the characteristics of triangles (e.g., centroid, orthocenter) and of right triangles (e.g., basic trigonometric ratios); Solve mathematical and real-world problems using properties of polygons and circles; Solve mathematical and real-world problems using the Pythagorean theorem and its converse; Analyze formal and informal geometric proofs using geometric concepts (e.g., similarity, congruence, applications of parallel and perpendicular lines); Analyze three-dimensional figures using nets and cross sections; Analyze methods for constructing geometric figures.
• Competency 0010: Understand coordinate and transformational geometry. Skills tested: Analyze two- and three-dimensional figures using coordinate systems; Apply concepts of distance, midpoint, and slope to classify figures and solve problems in the coordinate plane; Apply the concepts of parallel and perpendicular lines to model and solve problems; Analyze the relationship between the equation of a conic section and its graph; Determine the effects of geometric transformations on the graph of a function or relation (e.g., translations, reflections, dilations); Analyze transformations and symmetries of figures in the coordinate plane.

Competencies 0011– to 0012 make up about 15% of the test. They ask you to understand essential concepts in trigonometry; understand essential concepts in calculus.

• Competency 0011: Understand essential concepts in trigonometry. Skills tested: Apply trigonometric functions to solve problems involving distance and angles; Apply trigonometric functions to solve problems involving the unit circle; Rewrite trigonometric expressions to solve equations or prove expressions equivalent by using trigonometric identities; Analyze the relationship between a trigonometric function and its graph; Model periodic relationships using trigonometric functions; Solve mathematical and real-world problems using the law of sines and the law of cosines.
• Competency 0012: Understand essential concepts in calculus. Skills tested: Connect the concepts of limit, derivative, and integration; Analyze limits, rates of change, and continuity for algebraic functions and their graphs; Analyze the concept of the derivative (e.g., instantaneous rate of change, the slope of the line tangent to a curve) and apply this concept to polynomial functions; Analyze the concept of the integral (e.g., area under a curve, cumulative change) and apply this concept to polynomial functions.

Competencies 0013– to 0014 make up about 15% of the test. They ask you to understand principles and techniques of statistics; understand principles and techniques of probability.

• Competency 0013: Understand principles and techniques of statistics. Skills tested: Use appropriate formats for collecting, organizing, and displaying quantitative and qualitative data; Analyze data in a variety of representations (e.g., histograms, two-way tables, scatterplots, box and whisker plots); Analyze and compare measures of central tendency and variability (e.g., mean absolute deviation, interquartile range, standard deviation); Analyze the effects of bias and sampling techniques; Analyze probability distributions (e.g., skewed, uniform, normal); Interpret the correlation coefficient and distinguish between correlation and causation.
• Competency 0014: Understand principles and techniques of probability. Skills tested: Determine probabilities of simple and compound events and conditional probabilities; Calculate probabilities and solve problems using counting principles (e.g., the fundamental counting principle, factorials, permutations, combinations) and set theory; Use a variety of models to identify the size of the sample space for an event and calculate probabilities for an event (e.g., tree diagram, frequency table); Select simulations that model real-world events and predict possible outcomes; Calculate and interpret expected value from a discrete probability distribution.