Progressions for the Common Core State Standards …

1 Progressions for the Common CoreState Standards in Mathematics (draft)c The Common Core Standards Writing Team21 April 2012 Draft, 4/21/2012, comment 5, Number andOperations in Base TenOverviewStudents work in the base-ten system is intertwined with their workon counting and cardinality, and with the meanings and propertiesof addition, subtraction, multiplication, and division. Work in thebase-ten system relies on these meanings and properties, but alsocontributes to deepening students understanding of base-ten system is a remarkably efficient and uni-form system for systematically representing all numbers.

2 Using onlythe ten digits0,1,2,3,4,5,6,7,8,9, every number can be repre-sented as a string of digits, where each digit represents a value thatdepends on its place in the string. The relationship between valuesrepresented by the places in the base-ten system is the same forwhole numbers and decimals: the value represented by each placeis always10 times the value represented by the place to its imme-diate right. In other words, moving one place to the left, the valueof the place is multiplied by10.

3 In moving one place to the right,the value of the place is divided by10. Because of this uniformity,standard algorithms for computations within the base-ten system forwhole numbers extend to unitsEach place of a base-ten numeral represents abase-ten unit: ones, tens, tenths, hundreds, hundredths, etc. Thedigit in the place represents0to9of those units. Because tenlike units make a unit of the next highest value, only ten digits areneeded to represent any quantity in base ten. The basic unit isaone(represented by the rightmost place for whole numbers).

4 Inlearning about whole numbers, children learn that ten ones com-pose a new kind of unit called aten. They understand two-digitnumbers as composed of tens and ones, and use this understandingin computations, decomposing1ten into10ones and composing aten power of the base-ten system is in repeated bundling byten:10tens make a unit called a hundred. Repeating this process ofDraft, 4/21/2012, comment new units by bundling in groups of ten creates units calledthousand,ten thousand,hundred thousand.

5 In learning aboutdecimals, children partition a one into10equal-sized smaller units,each of which is a tenth. Each base-ten unit can be understood interms of any other base-ten unit. For example, one hundred can beviewed as a tenth of a thousand,10tens,100ones, or1 for operations in base ten draw on such relationshipsamong the base-ten algorithms for base-ten computations withthe four operations rely on decomposing numbers written in base-ten notation into base-ten units. The properties of operations thenallow any multi-digit computation to be reduced to a collection ofsingle-digit computations.

6 These single-digit computations some-times require the composition or decomposition of a base-ten in Kindergarten, the requisite abilities develop grad-ually over the grades. Experience with addition and subtractionwithin20is a Grade 1 fluency is a Grade and subtract within 20, demonstrating fluency for ad-dition and subtraction within 10. Use strategies such as countingon; making ten ( ,8682410414); decom-posing a number leading to a ten ( ,13413311019); using the relationship between addition and subtrac-tion ( , knowing that8412, one knows1284); andcreating equivalent but easier or known sums ( , adding67by creating the known equivalent66112113).

7 Within20that cross 10, such add and subtract within 20 using mental end of Grade 2, know from memory all sums of twoone-digit , are especially relevant to NBT because they afford thedevelopment of the Level 3 make-a-ten strategies for addition andsubtraction described in the OA Progression. From the NBT per-spective, make-a-ten strategies are (implicitly) the first instances ofcomposing or decomposing a base-ten unit. Such strategies are afoundation for understanding in Grade 1 that addition may requirecomposing a in Grade 2 that subtraction may within 100, including adding a two-digit number anda one-digit number, and adding a two-digit number and a multipleof 10, using concrete models or drawings and strategies basedon place value, properties of operations, and/or the relationshipbetween addition and subtraction; relate the strategy to a writ-ten method and explain the reasoning used.

8 Understand thatin adding two-digit numbers, one adds tens and tens, ones andones; and sometimes it is necessary to compose a a and subtract within 1000, using concrete models ordrawings and strategies based on place value, properties of op-erations, and/or the relationship between addition and subtrac-tion; relate the strategy to a written method. Understand thatin adding or subtracting three-digit numbers, one adds or sub-tracts hundreds and hundreds, tens and tens, ones and ones;and sometimes it is necessary to compose or decompose tens and algorithmsThe Standards distinguish strategiesfrom algorithms.

9 For example, students use strategies for addition Computation set of predefined steps applicableto a class of problems that gives the correct result in every casewhen the steps are carried out correctly. See also: manipulations that may bechosen for specific problems, may not have a fixed order, andmay be aimed at converting one problem into another. See also:computation subtraction in Grades K-3, but are expected to fluently addand subtract whole numbers using standard algorithms by the endof Grade 4.

10 Use of the standard algorithms can be viewed as theculmination of a long progression of reasoning about quantities, thebase-ten system, and the properties of progression distinguishes between two types of computa-tional strategies: special strategies and general methods. For ex-ample, a special strategy for computing39817is to decompose17as215, and evaluate398215. Special strategies eithercannot be extended to all numbers represented in the base-ten sys-tem or require considerable modification in order to do so.