The NAEP Mathematics Achievement Levels by Grade

Grade 4
Grade 8
Grade 12
Grade 12 (1990—2003)

Specific definitions of the NAEP Basic, NAEP Proficient, and NAEP Advanced achievement levels for grades 4, 8, and 12 are presented in the tables that follow. Because of changes made to the NAEP mathematics framework in 2005, the NAEP achievement-level descriptions and cut points indicated below for grade 12 have been updated. To maintain trend, results for grades 4 and 8 are reported on a 0–500 scale while results for grade 12, beginning with the 2005 assessment, are reported on a 0–300 scale. View the grade 12 NAEP achievement-level descriptions and cut scores used prior to 2005.

The NAEP achievement levels are cumulative; therefore, students performing at the NAEP Proficient level also display the competencies associated with the NAEP Basic level, and students at the NAEP Advanced level also demonstrate the skills and knowledge associated with both the NAEP Basic and the NAEP Proficient levels. The cut score indicating the lower end of the score range for each level is noted in parentheses.

Find general information about NAEP achievement levels and the detailed descriptions for other NAEP subjects.

Grade 4

NAEP Basic (214)	Students performing at the NAEP Basic achievement level can likely determine place value of whole numbers up to hundred thousands locate whole numbers on a number line read, write, compose, and decompose multi-digit whole numbers in a variety of forms based on place value identify even and odd numbers and understand factors add and subtract multi-digit whole numbers with single-step and/or regrouping add and subtract decimals to the hundredths place understand inverse operations and their properties and apply concepts of multiplication Students performing at the NAEP Basic achievement level can likely identify appropriate measurement tools in real-world scenarios measure or estimate lengths of objects in standard and non-standard units find the perimeter of polygons given a visual aid Students performing at the NAEP Basic achievement level can likely identify lines of symmetry identify attributes of polygons as well as 3D shapes compare these attributes with the support of visual aids Students performing at the NAEP Basic achievement level can likely correlate information between tables and data displays read and interpret tables and scaled graphs Students performing at the NAEP Basic achievement level can likely identify the rule for a pattern and extend, complete, or determine missing numbers in patterns complete input/output tables locate points on a map/grid system with whole number or letter coordinates
NAEP Proficient (249)	Students performing at the NAEP Proficient achievement level can likely demonstrate an understanding of the relationships between the four operations (addition, subtraction, multiplication, and division) add and subtract multi-digit whole numbers, fractions, and decimals in single and multi-step problems apply basic properties of operations to solve problems divide with whole numbers using one-digit divisors and understand remainders solve problems with real-world contexts involving fractions with like denominators identify and explain factors and multiples compare and order whole numbers identify, understand, and sort even and odd numbers identify and compare decimals, fractions, and whole numbers on a number line identify reasonable estimates Students performing at the NAEP Proficient achievement level can likely identify appropriate units or tools of measurement within the same system convert measurements within the same system measure lengths of objects to the nearest whole or ½ unit solve or estimate problems involving area Students performing at the NAEP Proficient achievement level can likely demonstrate knowledge of 2D shapes by identifying, comparing, contrasting, and analyzing their attributes and describe attributes of 3D shapes with support of visual aids select the final image of translations and reflections with no dashed lines and create parallel lines Students performing at the NAEP Proficient achievement level can likely interpret and analyze data with scales of 2 or greater to solve problems identify possible outcomes in probability events determine the probability of events using terms of likelihood Students performing at the NAEP Proficient achievement level can likely select expressions and equations to represent real-world situations solve one-step equations with whole numbers determine and/or apply rules to write, identify, or extend values in input/output tables locate and name points on a map/grid system with whole number or letter coordinates
NAEP Advanced (282)	Students performing at the NAEP Advanced achievement level can likely compare and order whole numbers, fractions, and decimals to hundredths apply understanding of factors and multiples and the structure of all operations with whole numbers understand and use inverse operations and use simple ratios multiply and divide 2- and 3-digit whole numbers with no remainders Students performing at the NAEP Advanced achievement level can likely select appropriate and reasonable measurements in real-world scenarios know vocabulary and units associated with area, perimeter, and volume solve one-step and multi-step problems involving area and/or perimeter Students performing at the NAEP Advanced achievement level can likely identify and apply attributes of 2D and 3D shapes in more complex contexts compose and decompose 2D shapes to create more complex shapes identify, draw and/or describe parallel lines apply and draw lines of symmetry generalize and reason with attributes of symmetrical figures identify a series of rotations Students performing at the NAEP Advanced achievement level can likely identify or describe events based on general probability categories interpret and analyze data from single or multiple line, bar, and circle graphs determine and interpret probability of an event with more than one condition create a visual representation of equivalent fractions in relation to probability Students performing at the NAEP Advanced achievement level can likely select expressions and equations to represent real-world situations with unknowns in all positions solve for unknowns in all positions with division of whole numbers determine, apply, and/or write a rule for a given pattern or input/output table as well as extend patterns and input/output tables locate and name points (x, y) on a coordinate grid with whole number or letter coordinates
Note: The content descriptions represented within the reporting ALD statements are intended to reflect the content defined within the framework.

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Grade 8

NAEP Basic (262)	Students performing at the NAEP Basic achievement level can likely simplify expressions involving integers use operations to solve real-world problems involving integers or fractions use proportional relationships to find equivalent ratios and create fractions and fractional relationships, with or without models demonstrate understanding of scientific notation Students performing at the NAEP Basic achievement level can likely reason and determine measurements, including length, area, and volume, with descriptions, labeled diagrams, and units provided apply proportional reasoning to solve problems in context using scale factor, distance, unit conversion and quantities apply simple scale factor value to find unknown lengths of triangles and rectangles without setting up a proportion Students performing at the NAEP Basic achievement level can likely find a missing angle in a triangle given two angles and understand that angles of a triangle add to 180 degrees recognize quadrilaterals given a description of their shared attributes Students performing at the NAEP Basic achievement level can likely interpret, create, and/or compare different data set representations to determine a specific set of values for mean, mode, and range while identifying errors and appropriateness Students performing at the NAEP Basic achievement level can likely use a coordinate plane to identify and plot coordinate points precisely find the distance between points recognize and extend patterns within an arithmetic or geometric sequence of numbers in a list or table (arithmetic, geometric) to solve problems in context identify, solve, and/or evaluate one- and two-step equations, and apply slope, given linear relationships
NAEP Proficient (299)	Students performing at the NAEP Proficient achievement level can likely demonstrate an understanding of using and creating ratios to solve problems mathematically or in context calculate GCF and LCM perform basic operations with rational numbers to solve problems in context while applying proper units and converting between fractions, decimals, and percent compare and order rational numbers with rational or common irrational numbers with or without a number line apply problem-solving strategies to solve square roots and ratio and proportions Students performing at the NAEP Proficient achievement level can likely demonstrate an understanding of solving problems that relate to comparing measures of two or three dimensions of space determining possible dimensions given area and volume as well as selecting appropriate units of measure and applying scale factor to area reason abstractly using addition and subtraction in contextual situations solve problems involving capacity, area, and weight classify angle measurements using diagrams and protractors Students performing at the NAEP Proficient achievement level can likely understand concepts of parallel and perpendicular lines use angle relationships and/or measurements formed when parallel lines are cut by a transversal. apply concepts of corresponding parts between similar and congruent figures with some containing composite shapes in context apply problem-solving strategies to solve Pythagorean Theorem problems solve problems in context by creating a figure in the coordinate plane that satisfies area and perimeter criteria reflect a shape on the coordinate plane over the x- and y-axis and plot some of the corresponding points determine unknown side lengths by decomposing a polygon using given constraints determine coordinates of missing endpoints of vertical or horizontal line segments on a coordinate plane Students performing at the NAEP Proficient achievement level can likely use problem solving skills to make calculations based on multiple representations of data sets in context to determine measures of central tendencies, theoretical probability, and basic probability concepts estimate values along the line of best fit identify sources of bias in a sample design calculate the mean from tables of data in multiple sets of values Students performing at the NAEP Proficient achievement level can likely create, model, identify, and solve one-step inequalities and multi-step equations with or without context and with or without constraints evaluate and extend sequential and recursive patterns using tables, models, multiple steps or from translating a written description graph and identify key features of linear and nonlinear functions recognize the effects on a graph when the slope and y-intercept are changed
NAEP Advanced (333)	Students performing at the NAEP Advanced achievement level can likely solve mathematical problems and problems in context with rational numbers including absolute values and variables by interpreting, creating and using diagrams engage with abstract situations and apply properties such as even and/or odd numbers, divisibility rules, and prime and composite numbers in mathematic situations Students performing at the NAEP Advanced achievement level can likely create a proportion to represent scale in context and analyze number lines with fractional intervals use ratios and proportion to find and/or explain measurements in multi-step situations, including unit rate and speed/distance solving problems involving area including composing and decomposing complex figures reason abstractly using multiple steps of addition and subtraction in context estimate length and measure using tools appropriately in context Students performing at the NAEP Advanced achievement level can likely understand angle relationships formed when parallel lines are cut by a transversal(s) to solve complex problems determine relative positions of points using the geometric ideas of midpoint involving directionality on the coordinate plane classify geometric solids by their properties and recognize cross-sections of plane and solid figures predicting results of subdividing complex plane figures in a variety of ways perform a sequence of transformations in the coordinate plane and identify corresponding parts and points understand and apply relationships of circles and central angles evaluate proportionality and similarity to find unknown values justify the sum of interior angles of polygons demonstrate knowledge of the faces of geometric solids determine unknown sides of triangles using Pythagorean Theorem or similar triangles Students performing at the NAEP Advanced achievement level can likely analyze and compare problems using problem-solving skills to make calculations and connections based on multiple representations of data sets to determine measures of central tendencies and their effect, theoretical and experimental probability, and basic probability concepts use a line of best fit or line graph to make predictions and interpretations analyze and critique graphical displays to justify appropriateness and solve problems Students performing at the NAEP Advanced achievement level can likely interpret, analyze, apply, and justify mathematical or contextual linear and nonlinear relationships and their key features represented through equations, tables, and graphs of a line, including with scales other than 1 evaluate, interpret, justify, solve, and write multiple-step equations and/or expressions and inequalities with fractions and/or multiple variables in contextual situations with or without constraints
Note: The content descriptions represented within the reporting ALD statements are intended to reflect the content defined within the framework.

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Grade 12

NAEP Basic (141)	Students performing at the NAEP Basic achievement level can likely use operations with rational numbers apply single-step percentages to solve real-world problems apply proportional relationships to solve real-world problems Students performing at the NAEP Basic achievement level can likely use the properties of operations to determine equivalent polynomial expressions solve single-step radical equations perform a single reflection of a parent function graphically and/or algebraically identify a type of function given a verbal description, table of values, and graph analyze graphs of linear functions to compare rates of change or slope Students performing at the NAEP Basic achievement level can likely apply proportional reasoning to solve real-world problems using scale or unit rate determine how rigid and non-rigid transformations affect an object and its measurements, including area calculate vertical and horizontal distances given the coordinates of two points compare areas of simple figures with or without a grid Students performing at the NAEP Basic achievement level can likely determine probabilities of simple events from 2-way tables and verbal descriptions determine the characteristics of a well-designed survey including valid sampling methods identify the mean from the graph of a normal distribution describe the effect changing the value of a data point has on the mean or median of the data set analyze a scatterplot to identify a correct trend line
NAEP Proficient (176)	Students performing at the NAEP Proficient achievement level can likely demonstrate an understanding of real numbers and operations with real numbers analyze information to solve real-world problems with proportional reasoning write numbers in scientific notation and compute with scientific notation simplify numeric expressions and perform operations that involve whole-number exponents use common multiples to solve real-world problems Students performing at the NAEP Proficient achievement level can likely use the properties of operations and exponents to determine equivalent polynomial expressions in a single step perform a rigid transformation (reflection or translation) of a function graphically and/or algebraically identify a type of function based on equations, verbal descriptions, tables, OR graphs translate between different representations of functions analyze key features of functions (including slope, intercept, domain, and range) given coordinates of points, a table of values, a verbal description, an equation, or a graph evaluate algebraic expressions or functions (including piecewise, linear, radical, quadratic, exponential, step) in the form of an equation or a graph for a given value apply exponential relationships to solve problems or write expressions determine the number of solutions for equations or inequalities write inequalities from a verbal description apply understanding of sequences to solve problems Students performing at the NAEP Proficient achievement level can likely apply proportional relationships to solve problems about similar figures or represent scale relationships using diagrams perform dilations, centered at the origin, with coordinates compare areas of complex figures on a grid reason about properties (angles, line segments, etc.) of diagrams based on perpendicular or parallel lines use properties of plane figures to solve problems demonstrate an understanding of 3-dimensional shapes by composing/decomposing them and taking cross sections determine the type of measurement of a 3-dimensional figure for a given unit complete an expression for a trigonometric ratio from a right triangle Students performing at the NAEP Proficient achievement level can likely distinguish between, find, and compare experimental and theoretical probabilities analyze the characteristics of experimental and survey designs and what can likely be inferred by each describe the impact of increasing a data value on the mean or median analyze data sets to determine or compare measures of center or spread analyze trends in scatterplots to make predictions or determine when predictions are appropriate analyze both appropriate and misleading data displays to draw conclusions
NAEP Advanced (216)	Students performing at the NAEP Advanced achievement level can likely reason with and apply the properties and operations of real numbers solve multi-step real-world problems using percentages compute and/or estimate the values of numeric expressions involving square roots and cube roots Students performing at the NAEP Advanced achievement level can likely use the properties of operations and exponents to determine equivalent polynomial expressions in multiple steps perform a series of rigid transformations (including translations and reflections) of a function graphically and/or algebraically determine an appropriate family of functions to model a real-world problem given a diagram or a table of values analyze multiple key features (including slope, intercept, domain, range, and vertex) of functions (including linear, quadratic, and logarithmic) given coordinates of points, a graph, or an equation analyze real-world problems to determine the meaning of values within an equation solve and/or graph compound, absolute value, and quadratic inequalities determine and apply recursive rules for sequences and functions Students performing at the NAEP Advanced achievement level can likely apply proportional reasoning to solve problems relating to area, similar figures, and converting between measurement systems describe a series of transformations, when two or more types of transformations are used, for a figure on the coordinate plane use coordinate geometry to find the midpoint of a segment reason about the relationship between lines in parallel planes and lines that intersect parallel planes apply properties and measurements of figures, including angles, perimeter, and area to solve problems decompose 3-dimensional shapes to solve problems apply the Pythagorean Theorem to find lengths in 3-dimensional figures apply trigonometric ratios to solve problems involving a single right-triangle Students performing at the NAEP Advanced achievement level can likely determine joint probabilities from 2-way tables or data sets generalize results of an experiment to a population describe or find the effects on summary statistics when the data set is changed create a data set with a given median, mode, and range compare correlation coefficients from scatterplots understand that correlation does not imply causation create data displays, including 2-way tables and boxplots, given data or summary statistics
Note: The content descriptions represented within the reporting ALD statements are intended to reflect the content defined within the framework.

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NAEP Mathematics Achievement Levels for Grade 12, 1990–2003

NAEP Basic (288)	*Twelfth-grade students performing at the NAEP Basic* level should demonstrate procedural and conceptual knowledge in solving problems in the five NAEP content areas.** Twelfth-grade students performing at the NAEP Basic level should be able to use estimation to verify solutions and determine the reasonableness of results as applied to real-world problems. Twelfth-graders performing at the NAEP Basic level should recognize relationships presented in verbal, algebraic, tabular, and graphical forms; and demonstrate knowledge of geometric relationships and corresponding measurement skills. They should be able to apply statistical reasoning in the organization and display of data and in reading tables and graphs. They also should be able to generalize from patterns and examples in the areas of algebra, geometry, and statistics. At this level, they should use correct mathematical language and symbols to communicate mathematical relationships and reasoning processes, and use calculators appropriately to solve problems.
NAEP Proficient (336)	Twelfth-grade students performing at the NAEP Proficient level should consistently integrate mathematical concepts and procedures to the solution of more complex problems in the five NAEP content areas. Twelfth-graders performing at the NAEP Proficient level should demonstrate an understanding of algebraic, statistical, and geometric and spatial reasoning. They should be able to perform algebraic operations involving polynomials, justify geometric relationships, and judge and defend the reasonableness of answers as applied to real-world situations. These students should be able to analyze and interpret data in tabular and graphical form; understand and use elements of the function concept in symbolic, graphical, and tabular form; and make conjectures, defend ideas, and give supporting examples.
NAEP Advanced (367)	Twelfth-grade students performing at the NAEP Advanced level should consistently demonstrate the integration of procedural and conceptual knowledge and the synthesis of ideas in the five NAEP content areas. Twelfth-grade students performing at the NAEP Advanced level should understand the function concept, and be able to compare and apply the numeric, algebraic, and graphical properties of functions. They should apply their knowledge of algebra, geometry, and statistics to solve problems in more advanced areas of continuous and discrete mathematics. They should be able to formulate generalizations and create models through probing examples and counterexamples. They should be able to communicate their mathematical reasoning through the clear, concise, and correct use of mathematical symbolism and logical thinking.

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Last updated 18 October 2022 (FW)