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# National Standards for Grade 8 - Mathematics (page 2)

National Assessment Governing Board
Updated on Mar 14, 2011

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.

 GRADE 8 1) Number sense a) Use place value to model and describe integers and decimals. b) Model or describe rational numbers or numerical relationships using number lines and diagrams. c) Write or rename rational numbers. d) Recognize, translate between, or apply multiple representations of rational numbers (fractions, decimals, and percents) in meaningful contexts. e) Express or interpret numbers using scientific notation from real-life contexts. f) Find or model absolute value or apply to problem situations. g) Order or compare rational numbers (fractions, decimals, percents, or integers) using various models and representations (e.g., number line). h) Order or compare rational numbers including very large and small integers, and decimals and fractions close to zero. 2) Estimation a) Establish or apply benchmarks for rational numbers and common irrational numbers (e.g., π) in contexts. 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. c) Verify solutions or determine the reasonableness of results in a variety of situations including 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 rational numbers. 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. c) Provide a mathematical argument to explain operations with two or more fractions. d) Interpret rational number operations and the relationships between them. e) Solve application problems involving rational numbers and operations using exact answers or estimates as appropriate. 4) Ratios and proportional reasoning a) Use ratios to describe problem situations. b) Use fractions to represent and express ratios and proportions. c) Use proportional reasoning to model and solve problems (including rates and scaling). d) 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) Describe odd and even integers and how they behave under different operations. b) Recognize, find, or use factors, multiples, or prime factorization. c) Recognize or use prime and composite numbers to solve problems. d) Use divisibility or remainders in problem settings. e) Apply basic properties of operations. f) Explain or justify a mathematical concept or relationship (e.g., explain why 17 is prime).