National Standards for Grade 12 - Mathematics (page 2)

— National Assessment Governing Board
Updated on Mar 14, 2011

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. By 12th grade, students should be comfortable dealing with all types of real numbers.


1) Number sense

a) Write, rename, represent, or compare real numbers (e.g., pi , numerical relationships using number lines, models, or diagrams.

b) Represent very large or very small numbers using scientific notation in meaningful contexts.

c) Find or model absolute value or apply to problem situations.

d) Interpret calculator or computer displays of numbers given in scientific notation.

e) Order or compare real numbers, including very large or small real numbers.


2) Estimation

a) Establish or apply benchmarks for real numbers in contexts.

b) Make estimates of very large or very small numbers appropriate to a given situation by:
• identifying when estimation is appropriate or not,
• 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.

c) Verify solutions or determine the reasonableness of results in a variety of situations including scientific notation, calculator, and computer results.

d) Estimate square or cube roots of numbers less than 1,000 between two whole numbers.

3) Number operations

a) Perform computations with real numbers including common irrational numbers or the absolute value of numbers.

b) Describe the effect of multiplying and dividing by numbers including the effect of multiplying or dividing a real number by:
• zero, or
• a number less than zero, or
• a number between zero and one, or
• one, or
• a number greater than one.

c) Solve application problems involving numbers, including rational and common irrationals, using exact answers or estimates as appropriate.

4) Ratios and proportional reasoning

a) Use proportions to model problems.

b) Use proportional reasoning to solve problems (including rates).

c) Solve problems involving percentages (including percent increase and decrease, interest rates, tax, discount, tips, or part/whole relationships).

5) Properties of number and operations

a) Solve problems involving factors, multiples, or prime factorization.

b) Use prime or composite numbers to solve problems.

c) Use divisibility or remainders in problem settings.

d) Apply basic properties of operations.

e) Provide a mathematical argument about a numerical property or relationship.


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. By grade 12, volumes and rates constructed from other attributes, such as speed, are emphasized.

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.


1) Measuring physical attributes

a) Compare objects with respect to length, area, volume, angle measurement, weight, or mass.

b) Estimate the size of an object with respect to a given measurement attribute (e.g., area).

c) Select or use appropriate measurement instrument to determine or create a given length, area, volume, angle, weight, or mass.

d) Solve mathematical or real-world problems involving perimeter or area of plane figures such as or composite figures.

e) Solve problems involving volume or surface area of rectangular solids, cylinders, prisms, or composite shapes.

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.

g) Solve problems involving rates such as speed or population density.

2) System of measurement

a) Select or use appropriate type of unit for the attribute being measured such as length, area, angle, time, or volume.

b) Solve problems involving conversions within the same measurement system such as conversions involving square inches and square feet.

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

d) Determine appropriate size of unit of measurement in problem situation involving such attributes as length, area, or volume.

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.

f) Construct or solve problems (e.g., floor area of a room) involving scale drawings.

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