NAEP Mathematics Achievement Level Descriptions
Basic
Fourth-grade students performing at the Basic level should show some evidence of understanding the mathematical concepts and procedures in the five NAEP content areas.
Fourth graders performing at the Basic level should be able to estimate and use basic facts to perform simple computations with whole numbers; show some understanding of fractions and decimals; and solve some simple real-world problems in all NAEP content areas. Students at this level should be able to use—though not always accurately—four-function calculators, rulers, and geometric shapes. Their written responses are often minimal and presented without supporting information.
Proficient
Fourth-grade students performing at the Proficient level should consistently apply integrated procedural knowledge and conceptual understanding to problem solving in the five NAEP content areas.
Fourth graders performing at the Proficient level should be able to use whole numbers to estimate, compute, and determine whether results are reasonable. They should have a conceptual understanding of fractions and decimals; be able to solve real-world problems in all NAEP content areas; and use four-function calculators, rulers, and geometric shapes appropriately. Students performing at the Proficient level should employ problem-solving strategies such as identifying and using appropriate information. Their written solutions should be organized and presented both with supporting information and explanations of how they were achieved.
Advanced
Fourth-grade students performing at the Advanced level should apply integrated procedural knowledge and conceptual understanding to complex and non-routine real-world problem solving in the five NAEP content areas.
Fourth graders performing at the Advanced level should be able to solve complex non-routine real-world problems in all NAEP content areas. They should display mastery in the use of four-function calculators, rulers, and geometric shapes. These students are expected to draw logical conclusions and justify answers and solution processes by explaining why, as well as how, they were achieved. They should go beyond the obvious in their interpretations and be able to communicate their thoughts clearly and concisely.
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.
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Grade 4
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1) Number sense
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a) Identify the place value and actual value of digits in whole numbers.
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b) Represent numbers using models such as base 10 representations, number lines, and two-dimensional models.
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c) Compose or decompose whole quantities by place value (e.g., write whole numbers in expanded notation using place value: 342 = 300 + 40 + 2).
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d) Write or rename whole numbers (e.g., 10: 5 + 5, 12 – 2, 2 x 5).
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e) Connect model, number word, or number using various models and representations for whole numbers, fractions, and decimals.
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f) Order or compare whole numbers, decimals, or fractions.
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2) Estimation
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a) Use benchmarks (well-known numbers used as meaningful points for comparison) for whole numbers, decimals, or fractions in contexts (e.g., ½ and .5 may be used as benchmarks for fractions and decimals between 0 and 1.00).
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b) Make estimates appropriate to a given situation with whole numbers, fractions, or decimals by:
• knowing when to estimate,
• selecting the appropriate type of estimate, including overestimate, underestimate, and range of estimate, or
• selecting the appropriate method of estimation (e.g., rounding).
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c) Verify solutions or determine the reasonableness of results in meaningful contexts.
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3) Number operations
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a) Add and subtract:
• whole numbers, or
• fractions with like denominators, or
• decimals through hundredths.
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b) Multiply whole numbers:
• no larger than two-digit by two-digit with paper and pencil computation, or
• larger numbers with use of calculator.
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c) Divide whole numbers:
• up to three-digits by one-digit with paper and pencil computation, or
• up to five-digits by two-digits with use of calculator.
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d) Describe the effect of operations on size (whole numbers).
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e) Interpret whole number operations and the relationships between them.
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f) Solve application problems involving numbers and operations.
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4) Ratios and proportional reasoning
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a) Use simple ratios to describe problem situations.
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5) Properties of number and operations
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a) Identify odd and even numbers.
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b) Identify factors of whole numbers.
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c) Apply basic properties of operations.
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d) Explain or justify a mathematical concept or relationship (e.g., explain why 15 is an odd number or why 7–3 is not the same as 3–7).
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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.
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. 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 4
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1) System of measurement
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a) Select or use appropriate type of unit for the attribute being measured such as length, time, or temperature.
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b) Solve problems involving conversions within the same measurement system such as conversions involving inches and feet or hours and minutes.
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c) Determine appropriate size of unit of measurement in problem situation involving such attributes as length, time, capacity, or weight.
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d) Determine situations in which a highly accurate measurement is important.
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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).
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).
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Grade 4
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1) Dimension and shape
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a) Explore properties of paths between points.
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b) Identify or describe (informally) real-world objects using simple plane figures (e.g., triangles, rectangles, squares, and circles) and simple solid figures (e.g., cubes, spheres, and cylinders).
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c) Identify or draw angles and other geometric figures in the plane.
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d) Describe attributes of two- and three-dimensional shapes.
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2) Transformation of shapes and preservation of properties
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a) Identify whether a figure is symmetrical, or draw lines of symmetry.
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b) Identify the images resulting from flips (reflections), slides (translations), or turns (rotations).
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c) Recognize which attributes (such as shape and area) change or don’t change when plane figures are cut up or rearranged.
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d) Match or draw congruent figures in a given collection.
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3) Relationships between geometric figures
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a) Analyze or describe patterns of geometric figures by increasing number of sides, changing size or orientation (e.g., polygons with more and more sides).
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b) Assemble simple plane shapes to construct a given shape.
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c) Recognize two-dimensional faces of three-dimensional shapes.
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d) Describe and compare properties of simple and compound figures composed of triangles, squares, and rectangles.
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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 parallelism or perpendicularity.
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b) Construct geometric figures with vertices at points on a coordinate grid.
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5) Mathematical reasoning
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a) Distinguish which objects in a collection satisfy a given geometric definition and explain choices.
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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.
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Grade 4
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1) Data representation
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Pictographs, bar graphs, circle graphs, line graphs, line plots, tables, and tallies.
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a) Read or interpret a single set of data.
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b) For a given set of data, complete a graph (limits of time make it difficult to construct graphs completely).
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c) Solve problems by estimating and computing within a single set of data.
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2) Characteristics of data sets
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a) Given a set of data or a graph, describe the distribution of the data using median, range, or mode.
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b) Compare two sets of related data.
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3) Probability
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a) Use informal probabilistic thinking to describe chance events (i.e., likely and unlikely, certain and impossible).
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b) Determine a simple probability from a context that includes a picture.
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c) List all possible outcomes of a given situation or event.
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d) Represent the probability of a given outcome using a picture or other graphic.
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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.
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Grade 4
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1) Patterns, relations, and functions
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a) Recognize, describe, or extend numerical patterns.
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b) Given a pattern or sequence, construct or explain a rule that can generate the terms of the pattern or sequence.
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c) Given a description, extend or find a missing term in a pattern or sequence.
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d) Create a different representation of a pattern or sequence given a verbal description.
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e) Recognize or describe a relationship in which quantities change proportionally.
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2) Algebraic representations
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a) Translate between the different forms of representations (symbolic, numerical, verbal, or pictorial) of whole number relationships (such as from a written description to an equation or from a function table to a written description).
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b) Graph or interpret points with whole number or letter coordinates on grids or in the first quadrant of the coordinate plane.
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c) Verify a conclusion using algebraic properties.
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3) Variables, expressions, and operations
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a) Use letters and symbols to represent an unknown quantity in a simple mathematical expression.
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b) Express simple mathematical relationships using number sentences.
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4) Equations and inequalities
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a) Find the value of the unknown in a whole number sentence.
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