# Progressions for the Common Core State Standards in Mathematics (draft)

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1 Progressions for the Common Core State Standards in Mathematics (draft) cthe Common Core Standards Writing Team 2 April 202

2 K 5, Number and Operations in Base Ten Overview Students work in the base-ten system is intertwined with their work on counting and cardinality, and with the meanings and properties of addition, subtraction, multiplication, and division. Work in the base-ten system relies on these meanings and properties, but also contributes to deepening students understanding of them. Position The base-ten system is a remarkably efficient and uniform system for systematically representing all numbers. Using only the ten digits 0,, 2, 3, 4, 5, 6, 7, 8, 9, every number can be represented as a string of digits, where each digit represents a value that depends on its place in the string. The relationship between values represented by the places in the base-ten system is the same for whole numbers and decimals: the value represented by each place is always 0 times the value represented by the place to its immediate right. In other words, moving one place to the left, the value of the place is multiplied by 0. In moving one place to the right, the value of the place is divided by 0. Because of this uniformity, standard algorithms for computations within the base-ten system for whole numbers extend to decimals. Base-ten units Each place of a base-ten numeral represents a base-ten unit: ones, tens, tenths, hundreds, hundredths, etc. The digit in the place represents 0 to 9 of those units. Because ten like units make a unit of the next highest value, only ten digits are needed to represent any quantity in base ten. The basic unit is a one (represented by the rightmost place for whole numbers). In learning about whole numbers, children learn that ten ones compose a new kind of unit called a ten. They understand two-digit numbers as composed of tens and ones, and use this understanding in computations, decomposing ten into 0 ones and composing a ten from 0 ones. The power of the base-ten system is in repeated bundling by ten: 0 tens make a unit called a hundred. Repeating this process of 2

4 4 Mathematical practices Both general methods and special strategies are opportunities to develop competencies relevant to the NBT standards. Use and discussion of both types of strategies offer opportunities for developing fluency with place value and properties of operations, and to use these in justifying the correctness of computations (MP.3). Special strategies may be advantageous in situations that require quick computation, but less so when uniformity is useful. Thus, they offer opportunities to raise the topic of using appropriate tools strategically (MP.5). Standard algorithms can be viewed as expressions of regularity in repeated reasoning (MP.8) used in general methods based on place value. Numerical expressions and recordings of computations, whether with strategies or standard algorithms, afford opportunities for students to contextualize, probing into the referents for the symbols involved (MP.2). Representations such as bundled objects or math drawings (e.g., drawings of hundreds, tens, and ones) and diagrams (e.g., simplified renderings of arrays or area models) afford the mathematical practice of explaining correspondences among different representations (MP). Drawings, diagrams, and numerical recordings may raise questions related to precision (MP.6), e.g., does that represent one or ten? This progression gives examples of representations that can be used to connect numerals with quantities and to connect numerical representations with combination, composition, and decomposition of base-ten units as students work towards computational fluency.

5 5 Kindergarten In Kindergarten, teachers help children lay the foundation for understanding the base-ten system by drawing special attention to 0. Children learn to view the whole numbers through 9 as ten ones and some more ones. They decompose 0 into pairs such as 9, 2 8, 3 7 and find the number that makes 0 when added to a given number such as 3 (see the OA Progression for further discussion). Work with numbers from to 9 to gain Figure foundations : for place value K.NBT Children use objects, math drawings, and equations to describe, explore, and explain how the teen numbers, the counting numbers from through 9, are ten ones and some more ones. Children can count out a given teen number of objects, e.g., 2, and group the objects to see the ten ones and the two ones. It is also helpful to structure the ten ones into patterns that can be seen as ten objects, such as two fives (see the OA Progression). 5 strip A difficulty in the English-speaking world is that the words for 7 teen numbers do not make their base-ten meanings evident. For example, eleven and twelve do not sound like ten and one and ten and two. The numbers thirteen, fourteen, fifteen,..., nineteen Figure : 0 7 reverse the order of the ones and tens digits by saying the ones digit first. Also, teen must be interpreted as meaning ten 0 strip and or equations. the prefixes thir and fif do not clearly say three and five. In contrast, the corresponding East Asian number words are ten one, ten two, ten three, and so on, fitting directly with the base-ten layered place value cards structure and drawing attention to the role of ten. Children could layered separated 5 strip learn to say numbers in this East Asian way in addition to learning0 7 the standard English number names. Difficulties with number front: words 07 beyond nineteen are discussed in the Grade section. The numerals need special attention for children to understand them. The first nine numerals 2 39, and back: 0 strip 0 are essentially arbitrary marks. These same marks are used again to represent larger numbers. Children need to learn the differences in the ways these marks are used. For example, initially, a numeral such as 6 looks like one, six, not ten and 6 ones. Layered place value cards can help children see the 0 hiding under the ones place and that the in the tens place really is 0 (ten ones). By working with teen numbers in this way in Kindergarten, students gain a foundation for viewing 0 ones as a new unit called a ten in Grade. K.NBT Compose and decompose numbers from to 9 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (e.g., 8 = 0 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones. Decomposing 7 as 0 and 7 Math drawings are simple drawings that make essential mathematical features and relationships salient while suppressing details that are not relevant to the mathematical ideas. Number Bond Drawing Number-bond diagram and equation Number Bond Drawing = Equation 7 = Decompositions of teen numbers can be recorded with diagrams Figure : Decomposing 7 as 0 and 7 Equation 5 strip 5- and 0-frames 0 strip layered place value cards layered separated Children can place small objects into 0 0-frames 7 to show the ten as two rows of five and the extra ones within the next 0-frame, or work with strips that show ten ones in a column. front: back: layered Decomposing 7 as 0 and 7 Number Bond Drawing 7 layered place value cards Place value cards separated Equation 7 = front: Children can use layered place value cards to see the 0 hiding inside any teen number. Such decompositions can be connected to numbers represented with objects and math drawings.

10 0 ple, they might say, 00 and 400 is 500. And 70 and 30 is another hundred, so 600. Then 8, 9, 0,... and the other 0 is 2. So, 62. Keeping track of what is being added is easier using a written form of such reasoning and makes it easier to discuss. There are two kinds of decompositions in this strategy. Both addends are decomposed into hundreds, tens, and ones, and the first addend is decomposed successively into the part already added and the part still to add. Students should continue to develop proficiency with mental computation. They mentally add 0 or 00 to a given number between 00 and 900, and mentally subtract 0 or 00 from a given number between 00 and NBT.8 2.NBT.8 Mentally add 0 or 00 to a given number , and mentally subtract 0 or 00 from a given number

11 Grade 3 At Grade 3, the major focus is multiplication, so students work with addition and subtraction is limited to maintenance of fluency within 000 for some students and building fluency to within 000 for others. Use place value understanding and properties of operations to perform multi-digit arithmetic Students continue adding and subtracting within NBT.2 They achieve fluency with strategies and algorithms that are based on place value, properties of operations, and/or the relationship between addition and subtraction. Such fluency can serve as preparation for learning standard algorithms in Grade 4, if the computational methods used can be connected with those algorithms. Students use their place value understanding to round numbers to the nearest 0 or NBT They need to understand that when moving to the right across the places in a number (e.g., 456), the digits represent smaller units. When rounding to the nearest 0 or 00, the goal is to approximate the number by the closest number with no ones or no tens and ones (e.g., so 456 to the nearest ten is 460; and to the nearest hundred is 500). Rounding to the unit represented by the leftmost place is typically the sort of estimate that is easiest for students. Rounding to the unit represented by a place in the middle of a number may be more difficult for students (the surrounding digits are sometimes distracting). Rounding two numbers before computing can take as long as just computing their sum or difference. The special role of 0 in the base-ten system is important in understanding multiplication of one-digit numbers with multiples of 0. 3.NBT.3 For example, the product 3 50 can be represented as 3 groups of 5 tens, which is 5 tens, which is 50. This reasoning relies on the associative property of multiplication: It is an example of how to explain an instance of a calculation pattern for these products: calculate the product of the non-zero digits, then shift the product one place to the left to make the result ten times as large. See the progression on Operations and Algebraic Thinking. 3.NBT.2 Fluently add and subtract within 000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. 3.NBT Use place value understanding to round whole numbers to the nearest 0 or NBT.3 Multiply one-digit whole numbers by multiples of 0 in the range 0 90 (e.g., 9 80, 5 60) using strategies based on place value and properties of operations. Grade 3 explanations for 5 tens is 50 Skip-counting by tens is 50, 00, 50. Counting on by 5 tens. 5 tens is 50, 5 more tens is 00, 5 more tens is 50. Decomposing 5 tens. 5 tens is 0 tens and 5 tens. 0 tens is tens is 50. So 5 tens is 00 and 50, or 50. Decomposing All of these explanations are correct. However, skip-counting and counting on become more difficult to use accurately as numbers become larger, e.g., in computing 5 90 or explaining why 45 tens is 450, and needs modification for products such as The first does not indicate any place value understanding.

14 Simplified array/area drawing for = = 8 5 hundreds = 8 40 = 8 4 tens = 8 9 = hundreds 32 tens Three accessible ways to record the standard algorithm: Computation of 8 549: Ways to record general methods One component of understanding general methods for multiplication is understanding how to compute products of one-digit numbers Left to right Right to left Right to left showing the showing the recording the and multiples of 0, 00, and 000. This extends work in Grade 3 on partial products partial products carries below products of one-digit numbers and multiples of 0. We can calculate by calculating 6 7 and then shifting the result to the left 8 thinking: 8 thinking: two places (by placing two zeros at the end to show that these are hundreds hundreds) because 6 groups of 7 hundred is 6 7 hundreds, which is tens tens hundreds 42 hundreds, or Students can use this place value reasoning, which can also be supported with diagrams of arrays or areas, as they develop and practice using the patterns in relationships among The first method proceeds from left to right, and the others from right to left. In the third method, the digits representing new units products such as 6 7, 6 70, 6 700, and Products of 5 are written below the line rather than above 549, thus keeping and even numbers, such as 5 4, 5 40, 5 400, and 4 5, the digits of the products close to each other, e.g., the 7 from 4 50, 4 500, might be discussed and practiced separately is written diagonally to the left of the 2 rather than afterwards because they may seem at first to violate the patterns above the 4 in 549. by having an extra 0 that comes from the one-digit product. Simplified array/area drawing for Computation of connected with an area model Another part of understanding general base-ten methods for multidigit multiplication is understanding the role played by the distributive property. This allows numbers to be decomposed into base-ten units, products of the units to be computed, then combined. By decomposing the factors into like base-ten units and applying the distributive property, multiplication computations are reduced to single-digit multiplications and products of numbers with multiples of 0, of 00, and of 000. Students can connect diagrams of areas or arrays to numerical work to develop understanding of general base-ten multiplication methods. Computing products of two two-digit numbers requires using the distributive property several times when the factors are decomposed into base-ten units. For example, General methods for computing quotients of multi-digit numbers and one-digit numbers rely on the same understandings as for multiplication, but cast in terms of division. 4.NBT.6 One component is quotients of multiples of 0, 00, or 000 and one-digit numbers. For example, 42 6 is related to and Students can draw on their work with multiplication and they can also reason that means partitioning 42 hundreds into 6 equal groups, so there are 7 hundreds in each group. Another component of understanding general methods for multidigit division computation is the idea of decomposing the dividend into like base-ten units and finding the quotient unit by unit, starting with the largest unit and continuing on to smaller units. As with multiplication, this relies on the distributive property. This can be viewed as finding the side length of a rectangle (the divisor is the length of the other side) or as allocating objects (the divisor is the number of groups). See the figures on the next page for examples Simplified array/area drawing for tens 9 tens = 27 hundreds = = 6 9 tens = tens = tens 54 tens = = 24 The products of like base-ten units are shown as parts of a rectangular region. Two accessible, right to left ways to record Two the accessible, standard right algorithm: to left ways to record the standard algorithm: Computation of 36 94: Ways to record general methods Showing the Recording the carries below Showing the Recording the carries below partial products for correct place value placement partial products for correct place value placement = 30 4 = tens 9 tens = 3 tens 4 = 27 hundreds = 2 tens = = 30 4 = 20 3 tens 4 = 2 tens = 20 thinking: thinking: tens 6 9 tens 3 tens 4 3 tens 4 3 tens 9 tens 3 tens 9 tens 6 4 = because we are multiplying by 3 tens in 0 this because row we are multiplying by 3 tens in this row These proceed from right to left, but could go left to right. On the right, digits that represent newly composed tens and hundreds are written below the line instead of above 94. The digits 2 and are surrounded by a blue box. The from is placed correctly in the hundreds place and the digit 2 from is placed correctly in the thousands place. If these digits had been placed above 94, they would be in incorrect places. Note that the 0 (surrounded by a yellow box) in the ones place of the second line of the method on the right is there because the whole line of digits is produced by multiplying by 30 (not 3). 4.NBT.6 Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

15 5 Multi-digit division requires working with remainders. In preparation for working with remainders, students can compute sums of a product and a number, such as In multi-digit division, students will need to find the greatest multiple less than a given number. For example, when dividing by 6, the greatest multiple of 6 less than 50 is Students can think of these greatest multiples in terms of putting objects into groups. For example, when 50 objects are shared among 6 groups, the largest whole number of objects that can be put in each group is 8, and 2 objects are left over. (Or when 50 objects are allocated into groups of 6, the largest whole number of groups that can be made is 8, and 2 objects are left over.) The equation (or ) corresponds with this situation. Cases involving 0 in division may require special attention. Case a 0 in the dividend: 6 ) What to do about the 0? 3 hundreds = 30 tens Cases involving 0 in division Case 2 a 0 in a remainder part way through: 4 2 ) Stop now because of the 0? No, there are still 3 ones left. Case 3 a 0 in the quotient: 3 2 ) Stop now because is less than 2? No, it is tens, so there are still = 4 left =? Thinking: A rectangle has area 966 and one side of length 7. Find the unknown side length. Find hundreds first, then tens, Division asthen finding ones. side length? hundreds +? tens +? ones = ??? 7 ) ) is viewed as finding the unknown side length of a rectangular region with area 966 square units and a side of length 7 units. The amount of hundreds is found, then tens, then ones. This yields a decomposition into three regions of dimensions 7 by 00, 7 by 30, and 7 by 8. It can be connected with the decomposition of 966 as By the distributive property, this is , so the unknown side length is 38. In the recording on the right, amounts of hundreds, tens, and ones are represented by numbers rather than by digits, e.g., 700 instead of 7. 3 groups 3 groups =? 2 hundr. 2 hundr. 2 hundr. 7 hundreds 3 each group gets 2 hundreds; hundred is left. 2 3 )745-6 Unbundle hundred. Now I have 0 tens + 4 tens = 4 tens. 2 3 ) Division as finding group size 3 groups Thinking: Divide 7 hundreds, 4 tens, 5 ones equally among 3 groups, starting with hundreds hundr. + 4 tens 2 hundr. + 4 tens 2 hundr. + 4 tens 4 tens 3 each group gets 4 tens; 2 tens are left ) Unbundle 2 tens. Now I have = 25 left ) groups 3 )745 2 hundr. + 4 tens hundr. + 4 tens hundr. + 4 tens each group gets 8; is left ) Each group got 248 and is left can be viewed as allocating 745 objects bundled in 7 hundreds, 4 tens, and 3 ones equally among 3 groups. In Step, the 2 indicates that each group got 2 hundreds, the 6 is the number of hundreds allocated, and the is the number of hundreds not allocated. After Step, the remaining hundred is decomposed as 0 tens and combined with the 4 tens (in 745) to make 4 tens.

18 8 the decimal point in based on estimation. Students can summarize the results of their reasoning such as those above as specific numerical patterns and then as one general overall pattern such as the number of decimal places in the product is the sum of the number of decimal places in each factor. General methods used for computing quotients of whole numbers extend to decimals with the additional issue of placing the decimal point in the quotient. As with decimal multiplication, students can first examine the cases of dividing by 0 and 00 to see that the quotient becomes 0 times or 00 times as large as the dividend (see also the Number and Operations Fractions Progression). For example, students can view 7 0 as asking how many tenths are in 7. 5.NF.7b Because it takes 0 tenths make, it takes 7 times as many tenths to make 7, so Or students could note that 7 is 70 tenths, so asking how many tenths are in 7 is the same as asking how many tenths are in 70 tenths, which is 70. In other words, 7 0 is the same as 70. So dividing by 0 moves the number 7 one place to the left, the quotient is ten times as big as the dividend. As with decimal multiplication, students can then proceed to more general cases. For example, to calculate 7 02, students can reason that 02 is 2 tenths and 7 is 70 tenths, so asking how many 2 tenths are in 7 is the same as asking how many 2 tenths are in 70 tenths. In other words, 7 02 is the same as 70 2; multiplying both the 7 and the 02 by 0 results in the same quotient. Or students could calculate 7 02 by viewing 02 as 2 0, so they can first divide 7 by 2, which is 35, and then divide that result by 0, which makes 35 ten times as large, namely 35. Dividing by a decimal less than results in a quotient larger than the dividend 5.NF.5 and moves the digits of the dividend one place to the left. Students can summarize the results of their reasoning as specific numerical patterns then as one general overall pattern such as when the decimal point in the divisor is moved to make a whole number, the decimal point in the dividend should be moved the same number of places. 5.NF.7b Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. 3 b Interpret division of a whole number by a unit fraction, and compute such quotients. 5.NF.5 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 to the effect of multiplying by.

19 9 Extending beyond Grade 5 At Grade 6, students extend their fluency with the standard algorithms, using these for all four operations with decimals and to compute quotients of multi-digit numbers. At Grade 6 and beyond, students may occasionally compute with numbers larger than those specified in earlier grades as required for solving problems, but the Standards do not specify mastery with such numbers. In Grade 6, students extend the base-ten system to negative numbers. In Grade 7, they begin to do arithmetic with such numbers. By reasoning about the standard division algorithm, students learn in Grade 7 that every fraction can be represented with a decimal that either terminates or repeats. In Grade 8, students learn informally that every number has a decimal expansion, and that those with a terminating or repeating decimal representation are rational numbers (i.e., can be represented as a quotient of integers). There are numbers that are not rational (irrational numbers), such as the square root of 2. (It is not obvious that the square root of 2 is not rational, but this can be proved.) In fact, surprisingly, it turns out that most numbers are not rational. Irrational numbers can always be approximated by rational numbers. In Grade 8, students build on their work with rounding and exponents when they begin working with scientific notation. This allows them to express approximations of very large and very small numbers compactly by using exponents and generally only approximately by showing only the most significant digits. For example, the Earth s circumference is approximately m. In scientific notation, this is m. The Common Core Standards are designed so that ideas used in base-ten computation, as well as in other domains, can support later learning. For example, use of the distributive property occurs together with the idea of combining like units in the NBT and NF standards. Students use these ideas again when they calculate with polynomials in high school. The distributive property and like units: Multiplication of whole numbers and polynomials decomposing as like units (powers of 0 or powers of ) using the distributive property using the distributive property again combining like units (powers of 0 or powers of )

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