NAEP Mathematics Achievement Level Descriptions
Basic
Eighth-grade students performing at the Basic level should exhibit evidence of conceptual and procedural understanding in the five NAEP content areas. This level of performance signifies an understanding of arithmetic operations—including estimation—on whole numbers, decimals, fractions, and percents.
Eighth graders performing at the Basic level should complete problems correctly with the help of structural prompts such as diagrams, charts, and graphs. They should be able to solve problems in all NAEP content areas through the appropriate selection and use of strategies and technological tools, including calculators, computers, and geometric shapes. Students at this level also should be able to use fundamental algebraic and informal geometric concepts in problem solving. As they approach the Proficient level, students at the Basic level should be able to determine which of the available data are necessary and sufficient for correct solutions and use them in problem solving. However, these eighth graders show limited skill in communicating mathematically.
Proficient
Eighth-grade students performing at the Proficient level should apply mathematical concepts and procedures consistently to complex problems in the five NAEP content areas.
Eighth graders performing at the Proficient level should be able to conjecture, defend their ideas, and give supporting examples. They should understand the connections among fractions, percents, decimals, and other mathematical topics such as algebra and functions. Students at this level are expected to have a thorough understanding of Basic level arithmetic operations—an understanding sufficient for problem solving in practical situations. Quantity and spatial relationships in problem solving and reasoning should be familiar to them, and they should be able to convey underlying reasoning skills beyond the level of arithmetic. They should be able to compare and contrast mathematical ideas and generate their own examples. These students should make inferences from data and graphs, apply properties of informal geometry, and accurately use the tools of technology. Students at this level should understand the process of gathering and organizing data and be able to calculate, evaluate, and communicate results within the domain of statistics and probability.
Advanced
Eighth-grade students performing at the Advanced level should be able to reach beyond the recognition, identification, and application of mathematical rules in order to generalize and synthesize concepts and principles in the five NAEP content areas.
Eighth graders performing at the Advanced level should be able to probe examples and counterexamples in order to shape generalizations from which they can develop models. Eighth graders performing at the Advanced level should use number sense and geometric awareness to consider the reasonableness of an answer. They are expected to use abstract thinking to create unique problem-solving techniques and explain the reasoning processes underlying their conclusions.
NAEP Mathematics Objectives – Mathematical Content Areas
Number Properties and Operations
Number sense is a major expectation of the 2007 NAEP. At fourth grade, students are expected to have a solid grasp of whole numbers, as represented by the decimal system, and to have the beginnings of understanding fractions. By eighth grade, they should be comfortable with rational numbers, represented either as decimal fractions (including percents) or as common fractions. They should be able to use them to solve problems involving proportionality and rates. Also in middle school, number should begin to coalesce with geometry via the idea of the number line. This should be connected with ideas of approximation and the use of scientific notation. Eighth graders should also have some acquaintance with naturally occurring irrational numbers, such as square roots and pi.
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GRADE 8
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1) Number sense
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a) Use place value to model and describe integers and decimals.
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b) Model or describe rational numbers or numerical relationships using number lines and diagrams.
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c) Write or rename rational numbers.
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d) Recognize, translate between, or apply multiple representations of rational numbers (fractions, decimals, and percents) in meaningful contexts.
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e) Express or interpret numbers using scientific notation from real-life contexts.
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f) Find or model absolute value or apply to problem situations.
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g) Order or compare rational numbers (fractions, decimals, percents, or integers) using various models and representations (e.g., number line).
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h) Order or compare rational numbers including very large and small integers, and decimals and fractions close to zero.
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2) Estimation
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a) Establish or apply benchmarks for rational numbers and common irrational numbers (e.g., π) in contexts.
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b) Make estimates appropriate to a given situation by:
• identifying when estimation is appropriate,
• determining the level of accuracy needed,
• selecting the appropriate method of estimation, or
• analyzing the effect of an estimation method on the accuracy of results.
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c) Verify solutions or determine the reasonableness of results in a variety of situations including calculator and computer results.
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d) Estimate square or cube roots of numbers less than 1,000 between two whole numbers.
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3) Number operations
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a) Perform computations with rational numbers.
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b) Describe the effect of multiplying and dividing by numbers including the effect of multiplying or dividing a rational number by:
• zero, or
• a number less than zero, or
• a number between zero and one,
• one, or
• a number greater than one.
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c) Provide a mathematical argument to explain operations with two or more fractions.
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d) Interpret rational number operations and the relationships between them.
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e) Solve application problems involving rational numbers and operations using exact answers or estimates as appropriate.
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4) Ratios and proportional reasoning
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a) Use ratios to describe problem situations.
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b) Use fractions to represent and express ratios and proportions.
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c) Use proportional reasoning to model and solve problems (including rates and scaling).
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d) Solve problems involving percentages (including percent increase and decrease, interest rates, tax, discount, tips, or part/whole relationships).
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5) Properties of number and operations
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a) Describe odd and even integers and how they behave under different operations.
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b) Recognize, find, or use factors, multiples, or prime factorization.
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c) Recognize or use prime and composite numbers to solve problems.
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d) Use divisibility or remainders in problem settings.
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e) Apply basic properties of operations.
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f) Explain or justify a mathematical concept or relationship (e.g., explain why 17 is prime).
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Measurement
In this NAEP Mathematics Framework, attributes such as capacity, weight/mass, time, and temperature are included, as well as the geometric attributes of length, area, and volume. Although many of these attributes are included in the grade 4 framework, the emphasis is on length, including perimeter, distance, and height. More emphasis is placed on area and angle in grade 8.
Units involved in items on the NAEP assessment include non-standard, customary, and metric units. At grade 4, common customary units such as inch, quart, pound, and hour and the common metric units such as centimeter, liter, and gram are emphasized. Grades 8 and 12 include the use of both square and cubic units for measuring area, surface area, and volume; degrees for measuring angles; and constructed units such as miles per hour. Converting from one unit in a system to another (such as from minutes to hours) is an important aspect of measurement included in problem situations. Understanding and using the many conversions available is an important skill. There are a limited number of common, everyday equivalencies that students are expected to know.
Items classified in this content area depend on some knowledge of measurement. For example, an item that asks the difference between a 3-inch and a 1¾-inch line segment is a number item, while an item comparing a 2-foot segment with an 8-inch line segment is a measurement item. In many secondary schools, measurement becomes an integral part of geometry; this is reflected in the proportion of items recommended for these two areas.
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GRADE 8
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1) Measuring physical attributes
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a) Compare objects with respect to length, area, volume, angle measurement, weight, or mass.
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b) Estimate the size of an object with respect to a given measurement attribute (e.g., area).
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c) Select or use appropriate measurement instrument to determine or create a given length, area, volume, angle, weight, or mass.
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d) Solve mathematical or real-world problems involving perimeter or area of plane figures such as or composite figures.
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e) Solve problems involving volume or surface area of rectangular solids, cylinders, prisms, or composite shapes.
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f) Solve problems involving indirect measurement such as finding the height of a building by comparing its shadow with the height and shadow of a known object.
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g) Solve problems involving rates such as speed or population density.
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2) System of measurement
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a) Select or use appropriate type of unit for the attribute being measured such as length, area, angle, time, or volume.
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b) Solve problems involving conversions within the same measurement system such as conversions involving square inches and square feet.
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c) Estimate the measure of an object in one system given the measure of that object in another system and the approximate conversion factor. For example:
• Distance conversion: 1 kilometer is approximately e of a mile.
• Money conversion: U.S. dollar is approximately 1.5 Canadian dollars.
• Temperature conversion: Fahrenheit to Celsius
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d) Determine appropriate size of unit of measurement in problem situation involving such attributes as length, area, or volume.
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e) Determine appropriate accuracy of measurement in problem situations (e.g., the accuracy of each of several lengths needed to obtain a specified accuracy of a total length) and find the measure to that degree of accuracy.
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f) Construct or solve problems (e.g., floor area of a room) involving scale drawings.
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Geometry
By grade 4, students are expected to be familiar with a library of simple figures and their attributes, both in the plane (lines, circles, triangles, rectangles, and squares) and in space (cubes, spheres, and cylinders). In middle school, understanding of these shapes deepens, with the study of cross-sections of solids and the beginnings of an analytical understanding of properties of plane figures, especially parallelism, perpendicularity, and angle relations in polygons. Right angles and the Pythagorean theorem are introduced, and geometry becomes more and more mixed with measurement. The basis for analytic geometry is laid by study of the number line.
Symmetry is an increasingly important component of geometry. Elementary students are expected to be familiar with the basic types of symmetry transformations of plane figures, including flips (reflection across lines), turns (rotations around points), and slides (translations). In middle school, this knowledge becomes more systematic and analytical, with each type of transformation being distinguished from other types by their qualitative effects. For example, a rigid motion of the plane that leaves at least two points fixed (but not all points) must be a reflection in a line.
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GRADE 8
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1) Dimension and shape
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a) Draw or describe a path of shortest length between points to solve problems in context.
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b) Identify a geometric object given a written description of its properties.
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c) Identify, define, or describe geometric shapes in the plane and in three-dimensional space given a visual representation.
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d) Draw or sketch from a written description polygons, circles, or semicircles.
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e) Represent or describe a three-dimensional situation in a two-dimensional drawing from different views.
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f) Demonstrate an understanding about the two- and three-dimensional shapes in our world through identifying, drawing, modeling, building, or taking apart.
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2) Transformation of shapes and preservation of properties
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a) Identify lines of symmetry in plane figures or recognize and classify types of symmetries of plane figures.
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b) Recognize or informally describe the effect of a transformation on two-dimensional geometric shapes (reflections across lines of symmetry, rotations, translations, magnifications, and contractions).
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c) Predict results of combining, subdividing, and changing shapes of plane figures and solids (e.g., paper folding, tiling, and cutting up and rearranging pieces).
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d) Justify relationships of congruence and similarity, and apply these relationships using scaling and proportional reasoning.
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e) For similar figures, identify and use the relationships of conservation of angle and of proportionality of side length and perimeter.
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3) Relationships between geometric figures
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a) Apply geometric properties and relationships in solving simple problems in two and three dimensions.
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b) Represent problem situations with simple geometric models to solve mathematical or real-world problems.
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c) Use the Pythagorean theorem to solve problems.
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d) Describe or analyze simple properties of, or relationships between, triangles, quadrilaterals, and other polygonal plane figures.
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e) Describe or analyze properties and relationships of parallel or intersecting lines.
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4) Position and direction
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a) Describe relative positions of points and lines using the geometric ideas of midpoint, points on common line through a common point, parallelism, or perpendicularity.
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b) Describe the intersection of two or more geometric figures in the plane (e.g., intersection of a circle and a line).
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c) Visualize or describe the cross section of a solid.
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d) Represent geometric figures using rectangular coordinates on a plane.
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5) Mathematical reasoning
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a) Make and test a geometric conjecture about regular polygons.
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Data Analysis and Probability
By grade 4, students should be expected to apply their understanding of number and quantity to pose questions that can be answered by collecting appropriate data. They should be expected to organize data in a table or a plot and summarize the essential features of center, spread, and shape both verbally and with simple summary statistics. Simple comparisons can be made between two related data sets, but more formal inference based on randomness should come later. The basic concept of chance and statistical reasoning can be built into meaningful contexts, though, such as, “If I draw two names from among those of the students in the room, am I likely to get two girls?” Such problems can be addressed through simulation.
Building on the same definition of data analysis and the same principles of describing distributions of data through center, spread, and shape, grade 8 students will be expected to use a wider variety of organizing and summarizing techniques. They can also begin to analyze statistical claims through designed surveys and experiments that involve randomization, with simulation being the main tool for making simple statistical inferences. They will begin to use more formal terminology related to probability and data analysis.
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GRADE 8
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1) Data representation
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Histograms, line graphs, scatter plots, box plots, circle graphs, stem and leaf plots, frequency distributions, tables, and bar graphs.
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a) Read or interpret data, including interpolating or extrapolating from data.
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b) For a given set of data, complete a graph and then solve a problem using the data in the graph (histograms, line graphs, scatter plots, circle graphs, and bar graphs).
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c) Solve problems by estimating and computing with data from a single set or across sets of data.
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d) Given a graph or a set of data, determine whether information is represented effectively and appropriately (histograms, line graphs, scatter plots, circle graphs, and bar graphs).
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e) Compare and contrast the effectiveness of different representations of the same data.
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2) Characteristics of data sets
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a) Calculate, use, or interpret mean, median, mode, or range.
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b) Describe how mean, median, mode, range, or interquartile ranges relate to the shape of the distribution.
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c) Identify outliers and determine their effect on mean, median, mode, or range.
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d) Using appropriate statistical measures, compare two or more data sets describing the same characteristic for two different populations or subsets of the same population.
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e) Visually choose the line that best fits given a scatter plot and informally explain the meaning of the line. Use the line to make predictions.
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3) Experiments and samples
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a) Given a sample, identify possible sources of bias in sampling.
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b) Distinguish between a random and non-random sample.
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c) Evaluate the design of an experiment.
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4) Probability
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a) Analyze a situation that involves probability of an independent event.
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b) Determine the theoretical probability of simple and compound events in familiar contexts.
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c) Estimate the probability of simple and compound events through experimentation or simulation.
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d) Use theoretical probability to evaluate or predict experimental outcomes.
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e) Determine the sample space for a given situation.
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f) Use a sample space to determine the probability of the possible outcomes of an event.
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g) Represent probability of a given outcome using fractions, decimals, and percents.
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h) Determine the probability of independent and dependent events. (Dependent events should be limited to linear functions with a small sample size.)
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i) Interpret probabilities within a given context.
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Algebra
By grade 4, students are expected to be able to recognize and extend simple numeric patterns as one foundation for a later understanding of function. They can begin to understand the meaning of equality and some of its properties, as well as the idea of an unknown quantity as a precursor to the concept of variable.
As students move into middle school, the ideas of function and variable become more important. Representation of functions as patterns, via tables, verbal descriptions, symbolic descriptions, and graphs, can combine to promote a flexible grasp of the idea of function. Linear functions receive special attention. They connect to the ideas of proportionality and rate, forming a bridge that will eventually link arithmetic to calculus. Symbolic manipulation in the relatively simple context of linear equations is reinforced by other means of finding solutions, including graphing by hand or with calculators.
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GRADE 8
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1) Patterns, relations, and functions
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a) Recognize, describe, or extend numerical and geometric patterns using tables, graphs, words, or symbols.
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b) Generalize a pattern appearing in a numerical sequence or table or graph using words or symbols.
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c) Analyze or create patterns, sequences, or linear functions given a rule.
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d) Identify functions as linear or non-linear or contrast distinguishing properties of functions from tables, graphs, or equations.
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e) Interpret the meaning of slope or intercepts in linear functions.
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2) Algebraic representations
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a) Translate between different representations of linear expressions using symbols, graphs, tables, diagrams, or written descriptions.
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b) Analyze or interpret linear relationships expressed in symbols, graphs, tables, diagrams, or written descriptions.
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c) Graph or interpret points that are represented by ordered pairs of numbers on a rectangular coordinate system.
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d) Solve problems involving coordinate pairs on the rectangular coordinate system.
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e) Make, validate, and justify conclusions and generalizations about linear relationships.
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f) Identify or represent functional relationships in meaningful contexts including proportional, linear, and common non-linear (e.g., compound interest, bacterial growth) in tables, graphs, words, or symbols.
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3) Variables, expressions, and operations
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a) Write algebraic expressions, equations, or inequalities to represent a situation.
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b) Perform basic operations, using appropriate tools, on linear algebraic expressions (including grouping and order of multiple operations involving basic operations, exponents, roots, simplifying, and expanding).
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4) Equations and inequalities
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a) Solve linear equations or inequalities (e.g., ax + b = c or ax + b = cx + d or ax + b > c).
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b) Interpret "=" as an equivalence between two expressions and use this interpretation to solve problems.
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c) Analyze situations or solve problems using linear equations and inequalities with rational coefficients symbolically or graphically (e.g., ax + b = c or ax + b = cx + d).
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d) Interpret relationships between symbolic linear expressions and graphs of lines by identifying and computing slope and intercepts (e.g., know in y = ax + b, that a is the rate of change and b is the vertical intercept of the graph).
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e) Use and evaluate common formulas [e.g., relationship between a circle’s circumference and diameter (C = ï°d), distance and time under constant speed].
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Reprinted with the permission of the National Assessment Governing Board.
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