# Progressions for the Common Core State Standards in Mathematics

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1 Progressions for the Common Core State Standards in Mathematics c Common Core Standards Writing Team 8 The Progressions are published under the Creative Commons Attribution (CC BY) license For information about permission to download, reuse, reprint, modify, distribute, or copy their content, see https: //creativecommonsorg/licenses/by// Suggested citation: Common Core Standards Writing Team (8) Progressions for the Common Core State Standards in Mathematics (August draft) Tucson, AZ: Institute for Mathematics and Education, University of Arizona For more information about the Progressions, see For discussion of the Progressions and related topics, see the Mathematical Musings blog: mathematicalmusingsorg Draft, August, 8

2 Number and Operations Fractions, Overview The treatment of fractions in the Standards emphasizes two features: the idea that a fraction is a number and connections with previous learning Fractions in the Standards In the Standards, the word fraction is used to refer to a type of number That number can be expressed in different ways It can be written in the form numerator over denominator (in fraction notation or as a fraction in conventional terminology), or in decimal notation (as a decimal ), or if it is greater than in the form whole number followed by a number less than written as a fraction (as a mixed number ) Thus, in Grades,,, and are all considered fractions, and, in later grades, rational numbers Expectations for computations with fractions appear in the domains of Number and Operations Fractions, Number and Operations in Base Ten, and the Number System To achieve the expectations of the Standards, students need to be able to transform and use numerical and later symbolic expressions, including expressions for numbers For example, in order to get the information they need or to understand correspondences between different approaches to the same problem or different representations for the same situation (MP), students may need to draw on their understanding of different representations for a given number Transforming different expressions for the same number includes the skills traditionally labeled conversion, reduction, and simplification, but these are not treated as separate topics in the Standards Choosing a convenient form for the purpose at hand is an important skill (MP), as is the fundamental understanding of equivalence of forms Thus,,,, and are all considered acceptable expressions for the same number, although their convenience for a given purpose is likely to vary Draft, August, 8

5 NF, numbers and diagrams with measurement scales for length unit measurements (see the Measurement and Data Progression) Both types of scales may be labeled only with whole numbers However, subdivisions between numbers on measurement scales correspond to subdivisions of the length unit, but subdivisions between numbers on count scales may have no referent In Grade, the difference between measurement and count scales becomes more salient because students work with subdivided lengthunits, measuring lengths using rulers marked with halves and fourths of an inch and plotting their data MD Area measurement and area models Students work with area models begins in Grade These diagrams are used in Grade for single-digit multiplication and division strategies (see the Operations and Algebraic Thinking Progression), to represent multi-digit multiplication and division calculations in Grade (see the Number and Operations in Base Ten Progression), and in Grades and to represent multiplication and division of fractions (see this progression and the Number System Progression) The distributive property is central to all of these uses Work with area models builds on previous work with area measurement As with length measurement, area measurement relies on several understandings: MD Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch Show the data by making a line plot, where the horizontal scale is marked off in appropriate units whole numbers, halves, or quarters area is invariant equal areas; Congruent figures enclose regions with area is additive The area of the union of two regions that overlap only at their boundaries is the sum of their areas; area-unit tiling Area is measured by tiling a region with a two-dimensional area-unit (such as a square or rectangle) and parts of the unit, without gaps or overlaps Perceiving a region as tiled by an area-unit relies on spatial structuring For example, second graders learn to see how a rectangular region can be partitioned as an array of squares G Students learn to see an object such as a row in two ways: as a composite of multiple squares and as a single entity, a row (a unit of units) Using rows or columns to cover a rectangular region is, at least implicitly, a composition of units For further discussion, see the K Geometry Progression Addition and subtraction In Grades and, students learn about operations on fractions, extending the meanings of the operations on whole numbers For addition and subtraction, these meanings arise from the Add To, Take From, Put Together/Take Apart, and Compare G Partition a rectangle into rows and columns of same-size squares and count to find the total number of them problem types and are established before Grade For descriptions and examples of these problem types, see the Overview of K in the Operations and Algebraic Thinking Progression In Grade, students compute sums and differences, mainly of fractions and mixed numbers with like denominators In Grade, students use their understanding of equivalent fractions to compute sums and differences of fractions with unlike denominators Draft, August, 8

6 NF, Multiplication The concept of multiplication begins in Grade with an entirely discrete notion of equal groups OA By Grade, students can also interpret a multiplication equation as a statement of comparison involving the notion times as much OA This notion has more affinity to continuous quantities, eg, ˆ might describe how cups of flour are times as much as cup of flour NF,MD By Grade, when students multiply fractions in general, NF products can be larger or smaller than either factor, and multiplication can be seen as an operation that stretches or shrinks by a scale factor NF Grade work with whole-number multiplication and division focuses on two problem types, Equal Groups and Arrays (For descriptions of these problem types and examples that involve discrete attributes, see the Grade section of the Operations and Algebraic Thinking Progression For examples with continuous attributes, see the Geometric Measurement Progression Both illustrate measurement (quotitive) and sharing (partitive) interpretations of division) Initially, problems involve multiplicands that represent discrete attributes (eg, cardinality) Later problems involve continuous attributes (eg, length) For example, problems of the Equal Groups type involve situations such as: There are bags with plums in each bag How many plums are there in all? and, in the domain of measurement: You need lengths of string, each feet long string will you need altogether? How much Both of these problems are about groups of four things each fours in which the group of four can be seen as a whole ( bag or length of string) or as a composite of units ( plums or feet) In the United States, the multiplication expression for groups of four is usually written as ˆ, with the multiplier first (This convention is used in this progression However, as discussed in the Operations and Algebraic Thinking Progression, some students may write ˆ and it is useful to discuss the different interpretations in connection with the commutative property) In Grade, problem types for whole-number multiplication and division expand to include Multiplicative Compare with whole numbers In this grade, Equal Groups and Arrays extend to include problems that involve multiplying a fraction by a whole number For example, problems of the Equal Groups type might be: OA Interpret products of whole numbers, eg, interpret ˆ as the total number of objects in groups of objects each OA Interpret a multiplication equation as a comparison, eg, interpret ˆ as a statement that is times as many as and times as many as Represent verbal statements of multiplicative comparisons as multiplication equations NF Apply and extend previous understandings of multiplication to multiply a fraction by a whole number MD Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale NF Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction a Interpret the product pa{bq ˆ q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a ˆ q b b Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas NF Interpret multiplication as scaling (resizing), by: a Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication b Explaining why multiplying a given number by a fraction greater than results in a product greater than the given number (recognizing multiplication by whole numbers greater than as a familiar case); explaining why multiplying a given number by a fraction less than results in a product smaller than the given number; and relating the principle of fraction equivalence a{b pnˆaq{pnˆbq to the effect of multiplying a{b by You need lengths of string, each string will you need altogether? You need lengths of string, each string will you need altogether? foot long How much feet long How much Draft, August, 8

7 NF, Like the two previous problems, these two problems are about objects that can be seen as wholes ( length of string) or in terms of units However, instead of being composed of units (feet), they are composed of subordinate units ( -feet) In Grade, students connect fractions with division, understanding numerical instances of a b a b for whole numbers a and b, with b not equal to zero (MP8) NF With this understanding, students see, for example, that is one third of, which leads to the meaning of multiplication by a unit fraction: NF Interpret a fraction as division of the numerator by the denominator (a{b a b) Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, eg, by using visual fraction models or equations to represent the problem ˆ This in turn extends to multiplication of any number by a fraction Problem types for multiplication expand to include Multiplicative Compare with unit fraction language, eg, one third as much as," and students solve problems that involve multiplying by a fraction For example, a problem of the Equal Groups type might be: You need of a length of string that is feet long How much string will you need altogether? Measurement conversion At Grades and, expectations for conversion of measurements parallel expectations for multiplication by whole numbers and by fractions In MD, the emphasis is on times as much or times as many, conversions that involve viewing a larger unit as superordinate to a smaller unit and multiplying the number of larger units by a whole number to find the number of smaller units For example, conversion from feet to inches involves viewing a foot as superordinate to an inch, eg, viewing a foot as inches or as times as long as an inch, so a measurement in inches is times what it is in feet In MD, conversions also involve viewing a smaller unit as subordinate to a larger one, eg, an inch is foot, so a measurement in feet is times what it is in inches and conversions require multiplication by a fraction (NF) Division Using their understanding of division of whole numbers and multiplication of fractions, students in Grade solve problems that involve dividing a whole number by a unit fraction or a unit fraction by a whole number In Grade, they extend their work to problems that involve dividing a fraction by a fraction (see the Number System Progression) See the Grade section of the Operations and Algebraic Thinking Progression for discussion of linguistic aspects of as much and related formulations for Multiplicative Compare problems MD Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz; l, ml; hr, min, sec Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit Record measurement equivalents in a two-column table MD Convert among different-sized standard measurement units within a given measurement system (eg, convert cm to m), and use these conversions in solving multi-step, real world problems Draft, August, 8

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