Mathematics+standards

|| In prekindergarten through grade 2 all students should– ||
 * These standards can be found in greater detail at []. || [[image:http://standards.nctm.org/document/appendix/images/shim.gif width="120" height="1"]] || [[image:http://standards.nctm.org/document/appendix/images/shim.gif width="1" height="1"]] ||
 * [[image:http://standards.nctm.org/document/appendix/images/number_f2.gif width="120" height="28" link="http://standards.nctm.org/document/appendix/numb.htm"]] || [[image:http://standards.nctm.org/document/appendix/images/shim.gif width="1" height="28"]] ||
 * [[image:http://standards.nctm.org/document/appendix/images/algebra.gif width="120" height="17" link="http://standards.nctm.org/document/appendix/alg.htm"]] || [[image:http://standards.nctm.org/document/appendix/images/shim.gif width="1" height="17"]] ||
 * [[image:http://standards.nctm.org/document/appendix/images/geometry.gif width="120" height="17" link="http://standards.nctm.org/document/appendix/geom.htm"]] || [[image:http://standards.nctm.org/document/appendix/images/shim.gif width="1" height="17"]] ||
 * [[image:http://standards.nctm.org/document/appendix/images/measurement.gif width="120" height="17" link="http://standards.nctm.org/document/appendix/meas.htm"]] || [[image:http://standards.nctm.org/document/appendix/images/shim.gif width="1" height="17"]] ||
 * [[image:http://standards.nctm.org/document/appendix/images/data.gif width="120" height="28" link="http://standards.nctm.org/document/appendix/data.htm"]] || [[image:http://standards.nctm.org/document/appendix/images/shim.gif width="1" height="28"]] ||
 * [[image:http://standards.nctm.org/document/appendix/images/process.gif width="120" height="17" link="http://standards.nctm.org/document/appendix/process.htm"]] || [[image:http://standards.nctm.org/document/appendix/images/shim.gif width="1" height="17"]] ||
 * [[image:http://standards.nctm.org/document/appendix/images/problem.gif width="120" height="17" link="http://standards.nctm.org/document/appendix/process.htm"]] || [[image:http://standards.nctm.org/document/appendix/images/shim.gif width="1" height="17"]] ||
 * [[image:http://standards.nctm.org/document/appendix/images/reasoning.gif width="120" height="17" link="http://standards.nctm.org/document/appendix/process.htm#bp2"]] || [[image:http://standards.nctm.org/document/appendix/images/shim.gif width="1" height="17"]] ||
 * [[image:http://standards.nctm.org/document/appendix/images/communication.gif width="120" height="17" link="http://standards.nctm.org/document/appendix/process.htm#bp3"]] || [[image:http://standards.nctm.org/document/appendix/images/shim.gif width="1" height="17"]] ||
 * [[image:http://standards.nctm.org/document/appendix/images/connections.gif width="120" height="17" link="http://standards.nctm.org/document/appendix/process.htm#bp4"]] || [[image:http://standards.nctm.org/document/appendix/images/shim.gif width="1" height="17"]] ||
 * [[image:http://standards.nctm.org/document/appendix/images/representation.gif width="120" height="17" link="http://standards.nctm.org/document/appendix/process.htm#bp5"]] || [[image:http://standards.nctm.org/document/appendix/images/shim.gif width="1" height="17"]] ||
 * [[image:http://standards.nctm.org/images/spacer_black.gif width="120" height="1"]]
 * ||  ||   ||   || ===Number and Operations Standard Instructional programs from prekindergarten through grade 12 should enable all students to— === ||
 * ====**Understand numbers, ways of representing numbers, relationships among numbers, and number systems** ==== ||
 * || **Pre-K–2 Expectations: **

In grades 3–5 all students should– ||
 * • || count with understanding and recognize "how many" in sets of objects; ||
 * • || use multiple models to develop initial understandings of place value and the base-ten number system; ||
 * • || develop understanding of the relative position and magnitude of whole numbers and of ordinal and cardinal numbers and their connections; ||
 * • || develop a sense of whole numbers and represent and use them in flexible ways, including relating, composing, and decomposing numbers; ||
 * • || connect number words and numerals to the quantities they represent, using various physical models and representations; ||
 * • || understand and represent commonly used fractions, such as 1/4, 1/3, and 1/2. ||  ||
 * || **Grades 3–5 Expectations: **

In grades 6–8 all students should– ||
 * • || understand the place-value structure of the base-ten number system and be able to represent and compare whole numbers and decimals; ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">recognize equivalent representations for the same number and generate them by decomposing and composing numbers; ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">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; ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">use models, benchmarks, and equivalent forms to judge the size of fractions; ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">recognize and generate equivalent forms of commonly used fractions, decimals, and percents; ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">explore numbers less than 0 by extending the number line and through familiar applications; ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">describe classes of numbers according to characteristics such as the nature of their factors. ||  ||
 * || <span style="font-family: Geneva,Arial,sans-serif;">**<span style="color: #000000; font-family: Geneva,Arial,Helvetica,sans-serif;">Grades 6–8 Expectations: **

<span style="color: #000000; font-family: Geneva,Arial,Helvetica,sans-serif;">In grades 9–12 all students should– ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">work flexibly with fractions, decimals, and percents to solve problems; ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">compare and order fractions, decimals, and percents efficiently and find their approximate locations on a number line; ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">develop meaning for percents greater than 100 and less than 1; ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">understand and use ratios and proportions to represent quantitative relationships; ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">develop an understanding of large numbers and recognize and appropriately use exponential, scientific, and calculator notation; ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">use factors, multiples, prime factorization, and relatively prime numbers to solve problems; ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">develop meaning for integers and represent and compare quantities with them. ||  ||
 * || <span style="font-family: Geneva,Arial,sans-serif;">**<span style="color: #000000; font-family: Geneva,Arial,Helvetica,sans-serif;">Grades 9–12 Expectations: **

<span style="font-family: Geneva,Arial,Helvetica,sans-serif;">In prekindergarten through grade 2 all students should– ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">develop a deeper understanding of very large and very small numbers and of various representations of them; ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">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; ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">understand vectors and matrices as systems that have some of the properties of the real-number system; ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">use number-theory arguments to justify relationships involving whole numbers. ||  ||   ||
 * [[image:http://standards.nctm.org/document/appendix/images/top.gif width="94" height="22" align="center" link="http://standards.nctm.org/document/appendix/numb.htm#TOP"]] ||  ||   || ====<span style="font-family: Geneva,Arial,Helvetica,sans-serif;">Understand meanings of operations and how they relate to one another ==== ||
 * || <span style="font-family: Geneva,Arial,sans-serif;">**<span style="font-family: Geneva,Arial,Helvetica,sans-serif;">Pre-K–2 Expectations: **

<span style="color: #000000; font-family: Geneva,Arial,Helvetica,sans-serif;">In grades 3–5 all students should– ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">understand various meanings of addition and subtraction of whole numbers and the relationship between the two operations; ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">understand the effects of adding and subtracting whole numbers; ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">understand situations that entail multiplication and division, such as equal groupings of objects and sharing equally. ||  ||
 * || <span style="font-family: Geneva,Arial,sans-serif;">**<span style="color: #000000; font-family: Geneva,Arial,Helvetica,sans-serif;">Grades 3–5 Expectations: **

<span style="color: #000000; font-family: Geneva,Arial,Helvetica,sans-serif;">In grades 6–8 all students should– ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">understand various meanings of multiplication and division; ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">understand the effects of multiplying and dividing whole numbers; ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">identify and use relationships between operations, such as division as the inverse of multiplication, to solve problems; ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">understand and use properties of operations, such as the distributivity of multiplication over addition. ||  ||
 * || <span style="font-family: Geneva,Arial,sans-serif;">**<span style="color: #000000; font-family: Geneva,Arial,Helvetica,sans-serif;">Grades 6–8 Expectations: **

<span style="color: #000000; font-family: Geneva,Arial,Helvetica,sans-serif;">In grades 9–12 all students should– ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">understand the meaning and effects of arithmetic operations with fractions, decimals, and integers; ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">use the associative and commutative properties of addition and multiplication and the distributive property of multiplication over addition to simplify computations with integers, fractions, and decimals; ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">understand and use the inverse relationships of addition and subtraction, multiplication and division, and squaring and finding square roots to simplify computations and solve problems. ||  ||
 * || <span style="font-family: Geneva,Arial,sans-serif;">**<span style="color: #000000; font-family: Geneva,Arial,Helvetica,sans-serif;">Grades 9–12 Expectations: **

<span style="font-family: Geneva,Arial,Helvetica,sans-serif;">In prekindergarten through grade 2 all students should– ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">judge the effects of such operations as multiplication, division, and computing powers and roots on the magnitudes of quantities; ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">develop an understanding of properties of, and representations for, the addition and multiplication of vectors and matrices; ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">develop an understanding of permutations and combinations as counting techniques. ||  ||   ||
 * [[image:http://standards.nctm.org/document/appendix/images/top.gif width="94" height="22" link="http://standards.nctm.org/document/appendix/numb.htm#TOP"]] ||  ||   || ====<span style="font-family: Geneva,Arial,Helvetica,sans-serif;">Compute fluently and make reasonable estimates ==== ||
 * || <span style="font-family: Geneva,Arial,sans-serif;">**<span style="font-family: Geneva,Arial,Helvetica,sans-serif;">Pre-K–2 Expectations: **

<span style="color: #000000; font-family: Geneva,Arial,Helvetica,sans-serif;">In grades 3–5 all students should– ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">develop and use strategies for whole-number computations, with a focus on addition and subtraction; ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">develop fluency with basic number combinations for addition and subtraction; ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">use a variety of methods and tools to compute, including objects, mental computation, estimation, paper and pencil, and calculators. ||  ||
 * || <span style="font-family: Geneva,Arial,sans-serif;">**<span style="color: #000000; font-family: Geneva,Arial,Helvetica,sans-serif;">Grades 3–5 Expectations: **

<span style="color: #000000; font-family: Geneva,Arial,Helvetica,sans-serif;">In grades 6–8 all students should– ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">develop fluency with basic number combinations for multiplication and division and use these combinations to mentally compute related problems, such as 30[[image:http://standards.nctm.org/document/images/inlines/times.gif width="15" height="9" caption="times"]]50; ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">develop fluency in adding, subtracting, multiplying, and dividing whole numbers; ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">develop and use strategies to estimate the results of whole-number computations and to judge the reasonableness of such results; ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">develop and use strategies to estimate computations involving fractions and decimals in situations relevant to students' experience; ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">use visual models, benchmarks, and equivalent forms to add and subtract commonly used fractions and decimals; ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">select appropriate methods and tools for computing with whole numbers from among mental computation, estimation, calculators, and paper and pencil according to the context and nature of the computation and use the selected method or tools. ||  ||
 * || <span style="font-family: Geneva,Arial,sans-serif;">**<span style="color: #000000; font-family: Geneva,Arial,Helvetica,sans-serif;">Grades 6–8 Expectations: **

<span style="color: #000000; font-family: Geneva,Arial,Helvetica,sans-serif;">In grades 9–12 all students should– ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">select appropriate methods and tools for computing with fractions and decimals from among mental computation, estimation, calculators or computers, and paper and pencil, depending on the situation, and apply the selected methods; ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">develop and analyze algorithms for computing with fractions, decimals, and integers and develop fluency in their use; ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">develop and use strategies to estimate the results of rational-number computations and judge the reasonableness of the results; ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios. ||  ||
 * || <span style="font-family: Geneva,Arial,sans-serif;">**<span style="color: #000000; font-family: Geneva,Arial,Helvetica,sans-serif;">Grades 9–12 Expectations: **


 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">develop fluency in operations with real numbers, vectors, and matrices, using mental computation or paper-and-pencil calculations for simple cases and technology for more-complicated cases. ||
 * • || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">judge the reasonableness of numerical computations and their results. ||  ||   ||
 * ||  ||   || <span style="font-family: Geneva,Arial,Helvetica,sans-serif;">[[image:http://standards.nctm.org/document/images/previous.gif width="80" height="22" link="http://standards.nctm.org/document/chapter8/index.htm"]][[image:http://standards.nctm.org/document/images/shim.gif width="1" height="40"]] [[image:http://standards.nctm.org/document/images/top.gif width="70" height="21" caption="Back to top" link="http://standards.nctm.org/document/appendix/numb.htm#TOP"]] [[image:http://standards.nctm.org/document/images/next.gif width="80" height="22" link="http://standards.nctm.org/document/appendix/alg.htm"]] ||   ||