AppendixTable of Standards and ExpectationsPre-K–2ExpectationsGrades 3–5ExpectationsNumber and OperationsInstructional programs from prekindergartenthrough grade 12 should enable all students to—StandardUnderstand numbers, ways of representingnumbers, relationships among numbers,and number systemsUnderstand meanings of operations andhow they relate to one anotherCompute fluently and make reasonable estimatesIn prekindergarten through grade 2 all students should—• count with understanding and recognize “how many” in sets of objects;• use multiple models to develop initial understandings of placevalue and the base-ten number system;• develop understanding of the relative position and magnitude ofwhole numbers and of ordinal and cardinal numbers and their connections;• develop a sense of whole numbers and represent and use them inflexible ways, including relating, composing, and decomposingnumbers;• 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.• understand various meanings of addition and subtraction of wholenumbers and the relationship between the two operations;• understand the effects of adding and subtracting whole numbers;• understand situations that entail multiplication and division, such asequal groupings of objects and sharing equally.• develop and use strategies for whole-number computations, with afocus on addition and subtraction; • develop fluency with basic number combinations for addition andsubtraction;• use a variety of methods and tools to compute, including objects,mental computation, estimation, paper and pencil, and calculators.In grades 3–5 all students should—• understand the place-value structure of the base-ten number system and be able to represent and compare whole numbers anddecimals;• recognize equivalent representations for the same number andgenerate them by decomposing and composing numbers;• develop understanding of fractions as parts of unit wholes, as partsof a collection, as locations on number lines, and as divisions ofwhole numbers;• use models, benchmarks, and equivalent forms to judge the size offractions;• recognize and generate equivalent forms of commonly used fractions, decimals, and percents;• explore numbers less than 0 by extending the number line andthrough familiar applications;• describe classes of numbers according to characteristics such asthe nature of their factors.• understand various meanings of multiplication and division;• understand the effects of multiplying and dividing whole numbers;• identify and use relationships between operations, such as divisionas the inverse of multiplication, to solve problems;• understand and use properties of operations, such as the distributivity of multiplication over addition.• develop fluency with basic number combinations for multiplicationand division and use these combinations to mentally compute related problems, such as 30 × 50;• develop fluency in adding, subtracting, multiplying, and dividingwhole numbers;• develop and use strategies to estimate the results of whole-numbercomputations and to judge the reasonableness of such results;• develop and use strategies to estimate computations involving fractions and decimals in situations relevant to students’ experience;• use visual models, benchmarks, and equivalent forms to add andsubtract commonly used fractions and decimals;• select appropriate methods and tools for computing with wholenumbers from among mental computation, estimation, calculators,and paper and pencil according to the context and nature of thecomputation and use the selected method or tool.392Grades 6–8ExpectationsGrades 9–12ExpectationsNumber and OperationsInstructional programs from prekindergartenthrough grade 12 should enable all students to—StandardUnderstand numbers, ways of representingnumbers, relationships among numbers,and number systemsUnderstand meanings of operations andhow they relate to one anotherCompute fluently and make reasonable estimatesIn grades 6–8 all students should—• work flexibly with fractions, decimals, and percents to solve problems;• compare and order fractions, decimals, and percents efficiently andfind their approximate locations on a number line;• develop meaning for percents greater than 100 and less than 1;• understand and use ratios and proportions to represent quantita-tive relationships;• develop an understanding of large numbers and recognize and appropriately use exponential, scientific, and calculator notation;• use factors, multiples, prime factorization, and relatively prime numbers to solve problems;• develop meaning for integers and represent and compare quantities with them.• understand the meaning and effects of arithmetic operations withfractions, decimals, and integers; • use the associative and commutative properties of addition andmultiplication and the distributive property of multiplication over addition to simplify computations with integers, fractions, and decimals;• understand and use the inverse relationships of addition and subtraction, multiplication and division, and squaring and findingsquare roots to simplify computations and solve problems.• select appropriate methods and tools for computing with fractionsand decimals from among mental computation, estimation, calculators or computers, and paper and pencil, depending on thesituation, and apply the selected methods;• develop and analyze algorithms for computing with fractions, decimals, and integers and develop fluency in their use; • develop and use strategies to estimate the results of rational-number computations and judge the reasonableness of the results;• develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios.In grades 9–12 all students should—• develop a deeper understanding of very large and very small numbers and of various representations of them;• compare and contrast the properties of numbers and numbersystems, including the rational and real numbers, and understandcomplex numbers as solutions to quadratic equations that do nothave real solutions;• understand vectors and matrices as systems that have some of theproperties of the real-number system;• use number-theory arguments to justify relationships