Principles and Standards for School Mathematics

Number and Operations Standard

Number and Operations Standard (pre-K - 2)

count with understanding and recognize "how many" in sets of objects;


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use multiple models to develop initial understandings of place value and the base-ten number system;


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develop understanding of the relative position and magnitude of whole numbers and of ordinal and cardinal numbers and their connections;


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develop a sense of whole numbers and represent and use them in flexible ways, including relating, composing, and decomposing numbers;


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connect number words and numerals to the quantities they represent, using various physical models and representations;


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understand and represent commonly used fractions, such as 1/4, 1/3, and 1/2.


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Number and Operations Standard (3 - 5)

understand the place-value structure of the base-ten number system and be able to represent and compare whole numbers and decimals;


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recognize equivalent representations for the same number and generate them by decomposing and composing numbers;


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develop understanding of fractions as parts of unit wholes, as parts of a collection, as locations on number lines, and as divisions of whole numbers;


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use models, benchmarks, and equivalent forms to judge the size of fractions;


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recognize and generate equivalent forms of commonly used fractions, decimals, and percents;


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explore numbers less than 0 by extending the number line and through familiar applications;


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describe classes of numbers according to characteristics such as the nature of their factors.


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Number and Operations Standard (6 - 8)

work flexibly with fractions, decimals, and percents to solve problems;


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compare and order fractions, decimals, and percents efficiently and find their approximate locations on a number line;


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develop meaning for percents greater than 100 and less than 1;


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understand and use ratios and proportions to represent quantitative relationships;


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develop an understanding of large numbers and recognize and appropriately use exponential, scientific, and calculator notation;


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use factors, multiples, prime factorization, and relatively prime numbers to solve problems;


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develop meaning for integers and represent and compare quantities with them.


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Number and Operations Standard (9 - 12)

develop a deeper understanding of very large and very small numbers and of various representations of them;


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compare and contrast the properties of numbers and number systems, including the rational and real numbers, and understand complex numbers as solutions to quadratic equations that do not have real solutions;


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understand vectors and matrices as systems that have some of the properties of the real-number system;


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use number-theory arguments to justify relationships involving whole numbers.


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