# 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 July Suggested citation: Common Core Standards Writing Team. (, July ). Progressions for the Common Core State Standards in Mathematics (draft). Grades 6 8, The Number System; High School, Number. Tucson, AZ: Institute for Mathematics and Education, University of Arizona. For updates and more information about the Progressions, see For discussion of the Progressions and related topics, see the Tools for the Common Core blog: http: //commoncoretools.me. Draft, 9 July, comment at commoncoretools.wordpress.com.

2 The Number System, 6 8 Overview In Grades 6 8, students build on two important conceptions which have developed throughout K, in order to understand the rational numbers as a number system. The first is the representation of whole numbers and fractions as points on the number line, and the second is a firm understanding of the properties of operations on whole numbers and fractions. Representing numbers on the number line In early grades, students see whole numbers as counting numbers, Later, they also understand whole numbers as corresponding to points on the number line. Just as the 6 on a ruler measures 6 inches from the mark, so the number 6 on the number line measures 6 units from the origin. Interpreting numbers as points on the number line brings fractions into the family as well; fractions are seen as measurements with new units, creating by partitioning the whole number unit into equal pieces. Just as on a ruler we might measure in tenths of an inch, on the number line we have halves, thirds, fifths, sevenths; the number line is a sort of ruler with every denominator. The denominators,, etc. play a special role, partioning the number line into tenths, hundredths, etc., just as a metric ruler is partioned into centimeters and millimeters. Starting in Grade students see addition as concatenation of lengths on the number line..md.6 By Grade they are using the same model to represent the sum of fractions with the same denominator: is seen as putting together a length that is units of one fifth long with a length that is units of one fifth long, making units of one fifths in all. Since there are five fifths in (that s what it means to be a fifth), and is fives, we get. Two fractions with different denominators are added by representing them in terms of a common unit. Representing sums as concatenated lengths on the number line is important because it gives students a way to think about addition that makes sense independently of how numbers are represented symbolically. Although addition calculations may look different for numbers represented in base ten and as fractions, addition is the.md.6 Represent whole numbers as lengths from on a number line diagram with equally spaced points corresponding to the numbers,,,..., and represent whole-number sums and differences within on a number line diagram. Representing and on the number line place a length of next to a length of Draft, 9 July, comment at commoncoretools.wordpress.com.

4 NS, 6 8 and that is indeed half of five, as we have understood in Grade..NF. When students extend their conception of multiplication to include negative rational numbers, the properties of operations become crucial. The rule that the product of negative numbers is positive, often seen as mysterious, is the result of extending the properties of operations (particularly the distributive property) to rational numbers..nf. Interpret a fraction as division of the numerator by the denominator ( ). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. Draft, 9 July, comment at commoncoretools.wordpress.com.

7 NS, 6 8 not as wholly different types of numbers but as as part of the same number system, represented by the number line. In many traditional treatments of fractions greatest common factors occur in reducing a fraction to lowest terms, and least common multiples occur in adding fractions. As explained in the Fractions Progression, neither of these activities is treated as a separate topic in the standards. Indeed, insisting that finding a least common multiple is an essential part of adding fractions can get in the way of understanding the operation, and the excursion into prime factorization and factor trees that is often entailed in these topics can be time-consuming and distract from the focus of K. In Grade 6, however, students experience a modest introduction to the concepts 6.NS. and put the idea of greatest common factor to use in a rehearsal for algebra, where they will need to see, for example, that 6. Apply and extend previous understandings of numbers to the system of rational numbers In Grade 6 the number line is extended to include negative numbers. Students initially encounter negative numbers in contexts where it is natural to describe both the magnitude of the quantity, e.g. vertical distance from sea level in meters, and the direction of the quantity (above or below sea level). 6.NS. In some cases has an essential meaning, for example that you are at sea level; in other cases the choice of is merely a convention, for example the temperature designated as in Farenheit or Celsius. Although negative integers might be commonly used as initial examples of negative numbers, the Standards do not introduce the integers separately from the entire system of rational numbers, and examples of negative fractions or decimals can be included from the beginning. Directed measurement scales for temperature and elevation provide a basis for understanding positive and negative numbers as having a positive or negative direction on the number line. 6.NS.6a Previous understanding of numbers on the number line related the position of the number to measurement: the number is located at the endpoint of an line segment units long whose other endpoint is at. Now the line segments acquire direction; starting at they can go in either the positive or the negative direction. These directed numbers can be represented by putting arrows at the endpoints of the line segments. Students come to see as the opposite of, located an equal distance from in the opposite direction. In order to avoid the common misconception later in algebra that any symbol with a negative sign in front of it should be a negative number, it is useful for students to see examples where is a positive number, for example if then. Students come to see the operation of putting a negative sign in front of a number as flipping the direction of the number from positive to negative or negative to 6.NS. Find the greatest common factor of two whole numbers less than or equal to and the least common multiple of two whole numbers less than or equal to. Use the distributive property to express a sum of two whole numbers with a common factor as a multiple of a sum of two whole numbers with no common factor. 6.NS. Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in realworld contexts, explaining the meaning of in each situation. 6.NS.6 Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. a Recognize opposite signs of numbers as indicating locations on opposite sides of on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g.,, and that is its own opposite. Representation of rational numbers on the number line represented by an interval represented by an arrow represented by an arrow Showing on the number line Draft, 9 July, comment at commoncoretools.wordpress.com.

8 NS, positive. Students generalize this understanding of the meaning of the negative sign to the coordinate plane, and can use it in locating numbers on the number line and ordered pairs in the coordinate plane. 6.NS.6bc b Understand signs of numbers in ordered pairs as indicating With the introduction of negative numbers, students gain a new locations in quadrants of the coordinate plane; recog- nize that when two ordered pairs differ only by signs, the sense of ordering on the number line. Whereas statements like locations of the points are related by reflections across could be understood in terms of having more of or less of a one or both axes. certain quantity I have apples and you have, so I have fewer c Find and position integers and other rational numbers on than you comparing negative numbers requires closer attention a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a to the relative positions of the numbers on the number line rather than their magnitudes. 6.MS.a Comparisons such as can coordinate plane. initially be confusing to students, because is further away from than, and is therefore larger in magnitude. Referring back to 6.NS. Understand ordering and absolute value of rational numbers. contexts in which negative numbers were introduced can be helpful: meters below sea level is lower than meters below sea level, a Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. and F is colder than F. Students are used to thinking of colder temperatures as lower than hotter temperatures, and so the mathematically correct statement also makes sense in terms of the context. 6.NS.b b Write, interpret, and explain statements of order for rational numbers in real-world contexts. At the same time, the prior notion of distance from as a measure of size is still present in the notion of absolute value. To avoid confusion it can help to present students with contexts where it makes sense both to compare the order of two rational numbers and to compare their absolute value, and where these two comparisons run in different directions. For example, someone with a balance of \$ in their bank account has more money than someone with a balance of \$, because. But the second person s debt is much larger than the first person s credit. 6.NS.cd c Understand the absolute value of a rational number as This understanding is reinforced by extension to the coordinate plane. 6.NS.8 its distance from on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. d Distinguish comparisons of absolute value from statements about order. 6.NS.8 Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate. Draft, 9 July, comment at commoncoretools.wordpress.com.

11 NS, 6 8 choice we make, not something we can justify by appeals to real world situations. Why do we make the choice of saying that a negative times a negative is positive? Because we want to extend the operation of multiplication to rational number is such a way that all of the properties of operations are satisfied..ns.a In particular, the property that really makes a difference here is the distributive property. If you want to be able to say that.ns.a Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. a Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. you have to say that 8, because the number on the left is and the first addend on the right is. This leads to the rules positive negative negative and negative positive negative If you want to be able to say that then you have to say that 8, since now we know that the number on the left is and the first addend on the right is. This leads to the rule negative negative positive Why is it important to maintain the distributive property? Because when students get to algebra, they use it all the time. They must be able to say 6 without worrying about whether and are positive or negative. The rules about moving negative signs around in a product result from the rules about multiplying negative and positive numbers. Think about the various possibilities for and in If and are both positive, then this just a restatement of the rules above. But it still works if, for example, is negative and is positive. In that case it says negative negative positive positive positive Just as the relationship between addition and subtraction helps students understand subtraction of rational numbers, so the relationship between multiplication and division helps them understand division. To calculate 8, students recall that 8, and so 8. By the same reasoning, 8 8 because 8 8 Draft, 9 July, comment at commoncoretools.wordpress.com.

12 NS, 6 8 This means it makes sense to write 8 as 8 Until this point students have not seen fractions where the numerator or denominator could be a negative integer. But working with the corresponding multiplication equations allows students to make sense of such fractions. In general, they see that.ns.b for any integers and, with. Again, using multiplication as a guide, students can extend division to rational numbers that are not integers..ns.c For example because And again it makes sense to write this division as a fraction:.ns.b Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. b Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If and are integers, then. Interpret quotients of rational numbers by describing real-world contexts. c Apply properties of operations as strategies to multiply and divide rational numbers. because Note that this is an extension of the fraction notation to a situation it was not originally designed for. There is no sense in which we can think of as parts where one part is obtained by dividing the line segment from to into equal parts! But the fact that the properties of operations extend to rational numbers means that calculations with fractions extend to these so-called complex fractions.ns., where and could be rational numbers, not only integers. By the end of Grade, students are solving problems involving complex fractions..ns. Decimals are special fractions, those with denominator,,, etc. But they can also be seen as a special sort of measurement on the number line, namely one that you get by partitioning into equal pieces. In Grade students begin to see this as a possibly infinite process. The number line is marked off into tenths, each of which is marked off into hundreths, each of which is marked off into thousandths, and so on ad infinitum. These finer and finer partitions constitute a sort of address system for numbers on the number line: 6 is, first, in the neighborhood betweeen 6 and, then in part of that neighborhood between 6 and 6, then exactly at 6. The finite decimals are the rational numbers that eventually come to fall exactly on one of the tick marks in this decimal address system. But there are numbers that never come to rest, no Solve real-world and mathematical problems involving the four operations with rational numbers. Zooming in on The finite decimal 6 can eventually be found sitting one of the tick marks at the thousandths level. Draft, 9 July, comment at commoncoretools.wordpress.com.

13 NS, 6 8 Zooming in on matter how far down you go. For example, is always sitting one third of the way along the third subdivision. It is plus one-third of a thousandth, and plus one-third of a ten thousandth, and so on. The decimals,, are successively closer and closer approximations to. We summarize this situation by saying that has an infinite decimal expansion consisting entirely of s where the bar over the indicates that it repeats indefinitely. Although a rigorous treatment of this mysterious infinite expansion is not available in middle school, students in Grade start to develop an intuitive understanding of decimals as a (possibly) infinite address system through simple examples such as this..ns.d For those rational numbers that have finite decimal expansions, students can find those expansions using long division. Saying that a rational number has a finite decimal expansion is the same as saying that it can be expressed as a fraction whose numerator is a Division base-ten unit (,,, etc.). So if is a fraction with a finite expansion, then or or for some whole number. If this is the case, then or or or or So we can find the whole number by dividing successively into,,, and so on until there is no remainder..ns.d The margin illustrates this process for 8, where we find that there is no remainder for the division into, so The fraction is never eventually on one of the tick marks. It is always one third the way along the third subdivision..ns.d Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. d Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in s or eventually repeats. of 8 into times a base-ten unit Notice that it is not really necessary to restart the division for each new base-ten unit, since the steps from the previous calculation carry over to the next one. 8 which means that 8 Draft, 9 July, comment at commoncoretools.wordpress.com.

14 NS, 6 8 Grade 8 Know that there are numbers that are not rational, and approximate them by rational numbers In Grade students encountered infinitely repeating decimals, such as. In Grade 8 they understand why this phenomenon occurs, a good exercise in expressing regularity in repeated reasoning (MP8). 8.NS. Taking the case of, for example, we can try to express it as a finite decimal using the same process we used for 8 in Grade. We successively divide into,,, hoping to find a point at which the remainder is zero. But this never happens; there is always a remainder of. After a few tries it is clear that the long division will always go the same way, because the individual steps always work the same way: the next digit in the quotient is always resulting in a reduction of the dividend from one base-unit to the next smaller one (see margin). Once we have seen this regularity, we see that can never be a whole number of decimal base-ten units, no matter how small they are. A similar investigation with other fractions leads to the insight that there must always eventually be a repeating pattern, because there are only so many ways a step in the algorithm can go. For example, considering the possibility that might be a finite decimal with, we try dividing into,,, etc., hoping to find a point where the remainder is zero. But something happens when we get to dividing into, the left-most division in the margin. We find ourselves with a remainder of. Since we started with a numerator of, the algorithm is going to start repeating the sequence of digits from this point on. So we are never going to get a remainder of. All is not in vain, however. Each calculation gives us a decimal approximation of. For example, the left-most calculation in the margin tells us that and the next two show that Noticing the emergence of the repeating pattern 8 in the digits, we say that 8 The possibility of infinite repeating decimals raises the possibility of infinite decimals that do not ever repeat. From the point of view of the decimal address system, there is no reason why such number should not correspond to a point on the number line. For 8.NS. Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. Division of into,, and Repeated division of into larger and larger base ten units shows the same pattern. Division of into multiples of times larger and larger base-ten units The remainder at each step is always a single digit multiple of a base-ten unit so eventually the algorithm has to cycle back to the same situation as some earlier step. From then on the algorithm produces the same sequence of digits as from the earlier step, ad infinitum. Draft, 9 July, comment at commoncoretools.wordpress.com.

15 NS, 6 8 example, the number π lives between and, and between and, and between and, and so on, with each successive decimal digit narrowing its possible location by a factor of. Numbers like π, which do not have a repeating decimal expansion and therefore are not rational numbers, are called irrational. 8.NS. Although we can calculate the decimal expansion of π to any desired accuracy, we cannot describe the entire expansion because it is infinitely long, and because there is no pattern (as far as we know). However, because of the way in which the decimal address system narrows down the interval in which a number lives, we can use the first few digits of the decimal expansion to come up with good decimal approximations of π, or any other irrational number. For example, the fact that π is between and tells us that π is between 9 and 6; the fact that π is between and tells us that π is between 96 and, and so on. 8.NS. 8.NS. Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. Zooming in on π The number π has an infinite non-repeating decimal expansion which determines each successive sub-interval to zoom in on. 8.NS. Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π ). Draft, 9 July, comment at commoncoretools.wordpress.com.

16 High School, Number* The Real Number System Extend the properties of exponents to rational exponents In Grades 6 8 students began to widen the possible types of number they can conceptualize on the number line. In Grade 8 they glimpse the existence of irrational numbers such as. In high school, they start a systematic study of functions that can take on irrational values, such as exponential, logarithmic, and power functions. The first step in this direction is the understanding of numerical expressions in which the exponent is not a whole number. Functions such as, or more generally polynomial functions, have the property that if the input is a rational number, then so is the output. This is because their output values are computed by basic arithmetic operations on their input values. But a function such as does not necessarily have rational output values for every rational input value. For example, is irrational even though is rational. The study of such functions brings with it a need for an extended understanding of the meaning of an exponent. Exponent notation is a remarkable success story in the expansion of mathematical ideas. It is not obvious at first that a number such as can be represented as a power of. But reflecting that and thinking about the properties of exponents, it is natural to define since if we follow the rule then Similar reasoning leads to a general definition of the meaning of whenever and are rational numbers. N-RN. It should be noted high school mathematics does not develop the mathematical ideas necessary to prove that numbers such as and actually exist; N-RN. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. *This progression concerns Number and Quantity standards related to number. The remaining standards are discussed in the Quantity Progression. Draft, 9 July, comment at commoncoretools.wordpress.com.

17 HS, N in fact all of high school mathematics depends on the fundamental assumption that properties of rational numbers extend to irrational numbers. This is not unreasonable, since the number line is populated densely with rational numbers, and a conception of number as a point on the number line gives reassurance from intuitions about measurement that irrational numbers are not going to behave in a strangely different way from rational numbers. Because rational exponents have been introduced in such a way as to preserve the laws of exponents, students can now use those laws in a wider variety of situations. For example, they can rewrite the formula for the volume of a sphere of radius, V π to express the radius in terms of the volume, N-RN. V π Use properties of rational and irrational numbers An important difference between rational and irrational numbers is that rational numbers form a number system. If you add, subtract, multiply, or divide two rational numbers, you get another rational number (provided the divisor is not in the last case). The same is not true of irrational numbers. For example, if you multiply the irrational number by itself, you get the rational number. Irrational numbers are defined by not being rational, and this definition can be exploited to generate many examples of irrational numbers from just a few. N-RN. For example, because is irrational it follows that and are also irrational. Indeed, if were an irrational number, call it, say, then from we would deduce. This would imply is rational, since it is obtained by subtracting the rational number from the rational number. But it is not rational, so neither is. Although in applications of mathematics the distinction between rational and irrational numbers is irrelevant, since we always deal with finite decimal approximations (and therefore with rational numbers), thinking about the properties of rational and irrational numbers is good practice for mathematical reasoning habits such as constructing viable arguments and attending to precisions (MP., MP.6). N-RN. Rewrite expressions involving radicals and rational exponents using the properties of exponents. N-RN. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Draft, 9 July, comment at commoncoretools.wordpress.com.

19 HS, N 9 Students who continue to study geometric representations of complex numbers in the complex plane use both rectangular and polar coordinates which leads to a useful geometric interpretation of the operations. N-CN., N-CN. The restriction of these geometric interpretations to the real numbers yields and interpretation of these operations on the number line. One of the great theorems of modern mathematics is the Fundamental Theorem of Algebra, which says that every polynomial equation has a solution in the complex numbers. To put this into perspective, recall that we formed the complex numbers by creating a solution,, to just one special polynomial equation,. With the addition of this one solution, it turns out that every polynomial equation, for example, also acquires a solution. Students have already seen this phenomenon for quadratic equations. N-CN.9 Although much of the study of complex numbers goes beyond the college and career ready threshold, as indicated by the () on many of the standards, it is a rewarding area of exploration for advanced students. N-CN. () Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number. N-CN. () Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. N-CN.9 () Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. Draft, 9 July, comment at commoncoretools.wordpress.com.

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