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 achievementlevel 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 achievementlevel 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 multidigit whole numbers in a variety of forms based on place value
 identify even and odd numbers and understand factors
 add and subtract multidigit whole numbers with singlestep 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 realworld scenarios
 measure or estimate lengths of objects in standard and nonstandard 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 multidigit whole numbers, fractions, and decimals in single and multistep problems
 apply basic properties of operations to solve problems
 divide with whole numbers using onedigit divisors and understand remainders
 solve problems with realworld 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 realworld situations
 solve onestep 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 3digit whole numbers with no remainders
Students performing at the
NAEP Advanced achievement level can likely  select appropriate and reasonable measurements in realworld scenarios
 know vocabulary and units associated with area, perimeter, and volume
 solve onestep and multistep 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 realworld 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 realworld 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 degreesrecognize 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 twostep 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 problemsolving 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 problemsolving 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 yaxis 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 onestep inequalities and multistep 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 yintercept 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 multistep 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 crosssections 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 problemsolving 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 multiplestep 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 singlestep percentages to solve realworld problems
 apply proportional relationships to solve realworld problems
Students performing at the
NAEP Basic achievement level can likely  use the properties of operations to determine equivalent polynomial expressions
 solve singlestep 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 realworld problems using scale or unit rate
 determine how rigid and nonrigid 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 2way tables and verbal descriptions
 determine the characteristics of a welldesigned 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 realworld problems with proportional reasoning
 write numbers in scientific notation and compute with scientific notation
 simplify numeric expressions and perform operations that involve wholenumber exponents
 use common multiples to solve realworld 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 3dimensional shapes by composing/decomposing them and taking cross sections
 determine the type of measurement of a 3dimensional 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 multistep realworld 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 realworld 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 realworld 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 3dimensional shapes to solve problems
 apply the Pythagorean Theorem to find lengths in 3dimensional figures
 apply trigonometric ratios to solve problems involving a single righttriangle
Students performing at the
NAEP Advanced achievement level can likely  determine joint probabilities from 2way 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 2way 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) 
Twelfthgrade students performing at the
NAEP Basic level should demonstrate procedural and conceptual knowledge in solving problems in the five NAEP content areas. Twelfthgrade 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 realworld problems. Twelfthgraders 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) 
Twelfthgrade 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. Twelfthgraders 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 realworld 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) 
Twelfthgrade 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. Twelfthgrade 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)