Number Sense

 1NS.1.0 Students understand and use numbers up to 100: 1NS.1.1 Count, read, and write whole numbers to 100. 1NS.1.2 Compare and order whole numbers to 100 by using the symbols for less than, equal to, or greater than (<, =, >). 1NS.1.3 Represent equivalent forms of the same number through the use of physical models, diagrams, and number expressions (to 20) (e.g., 8 may be represented as 4 + 4, 5 + 3, 2 + 2 + 2 + 2, 10 -2, 11 -3). 1NS.1.4 Count and group object in ones and tens (e.g., three groups of 10 and 4 equals 34, or 30 + 4). 1NS.1.5 Identify and know the value of coins and show different combinations of coins that equal the same value. 1NS.2.0 Students demonstrate the meaning of addition and subtraction and use these operations to solve problems: 1NS.2.1 Know the addition facts (sums to 20) and the corresponding subtraction facts and commit them to memory. 1NS.2.2 Use the inverse relationship between addition and subtraction to solve problems. 1NS.2.3 Identify one more than, one less than, 10 more than, and 10 less than a given number. 1NS.2.4 Count by 2s, 5s, and 10s to 100. 1NS.2.5 Show the meaning of addition (putting together, increasing) and subtraction (taking away, comparing, finding the difference). 1NS.2.6 Solve addition and subtraction problems with one-and two-digit numbers (e.g., 5 + 58 = __). 1NS.2.7 Find the sum of three one-digit numbers. 1NS.3.0 Students use estimation strategies in computation and problem solving that involve numbers that use the ones, tens, and hundreds places: 1NS.3.1 Make reasonable estimates when comparing larger or smaller numbers.

The following Number Sense standard is basic:

1NS.1.1 Count, read, and write whole numbers to 100. It is important that students gain a conceptual understanding of numbers and counting, not simply learn to count to 100 by rote. They need to understand, for example, that counting can occur in any order and in any direction, not just in the standard left-to-right counting pattern, as long as each item is tagged once and only once. Students must understand that numbers represent sets of specific quantities of items. Of particular importance is learning and understanding the counting sequence for numbers in the teens and multiples of ten. It should be emphasized that numbers in the teens represent a ten value and a certain number of unit values—12 does not merely represent a set of 12 items; it also represents 1 ten and 2 ones. A related and equally important Number Sense standard is:

1NS.1.2Compare and order whole numbers to 100 by using the symbols for less than, equal to, or greater than (<, =, >). The continuing development of addition and subtraction skills as described in the following standards is basic:

1NS.2.1Know the addition facts (sums to 20) and the corresponding subtraction facts and commit them to memory.

1NS.2.5Show the meaning of addition (putting together, increasing) and subtraction (taking away, comparing, finding the difference). For example, students should understand that the equation 15 - 8 = 7 is the same as 15 = 7 + 8. Particular attention should be paid to the assessment of these competencies because students who fail to learn these topics will have serious difficulties in the later grades. The achievement of these standards will require that students be exposed to and asked to solve simple addition and subtraction problems throughout the school year.

 2NS.1.0 Students understand the relationship between numbers, quantities, and place value in whole numbers up to 1,000: 2NS.1.1 Count, read, and write whole numbers to 1,000 and identify the place value for each digit. 2NS.1.2 Use words, models, and expanded forms (e.g., 45 = 4 tens + 5) to represent numbers (to 1,000). 2NS.1.3 Order and compare whole numbers to 1,000 by using the symbols <, =, >. 2NS.2.0 Students estimate, calculate, and solve problems involving addition and subtraction of two-and three-digit numbers: 2NS.2.1 Understand and use the inverse relationship between addition and subtraction (e.g., an opposite number sentence for 8 + 6 = 14 is 14 - 6 = 8) to solve problems and check solutions. 2NS.2.2 Find the sum or difference of two whole numbers up to three digits long. 2NS.2.3 Use mental arithmetic to find the sum or difference of two two-digit numbers. 2NS.3.0 Students model and solve simple problems involving multiplication and division: 2NS.3.1 Use repeated addition, arrays, and counting by multiples to do multiplication. 2NS.3.2 Use repeated subtraction, equal sharing, and forming equal groups with remainders to do division. 2NS.3.3 Know the multiplication tables of 2s, 5s, and 10s (to "times 10") and commit them to memory. 2NS.4.0 Students understand that fractions and decimals may refer to parts of a set and parts of a whole: 2NS.4.1 Recognize, name, and compare unit fractions from 1/12 to 1/2. 2NS.4.2 Recognize fractions of a whole and parts of a group (e.g., one-fourth of a pie, two-thirds of 15 balls). 2NS.4.3 Know that when all fractional parts are included, such as four-fourths, the result is equal to the whole and to one. 2NS.5.0 Students model and solve problems by representing, adding, and subtracting amounts of money: 2NS.5.1 Solve problems using combinations of coins and bills. 2NS.5.2 Know and use the decimal notation and the dollar and cent symbols for money. 2NS.6.0 Students use estimation strategies in computation and problem solving that involve numbers that use the ones, tens, hundreds, and thousands places: 2NS.6.1 Recognize when an estimate is reasonable in measurements (e.g., closest inch).

Comments: As was the case in grade one, the students’ growing mastery of whole numbers is the basic topic in grade two, although fractions and decimals now appear. These Number Sense standards are particularly important:

2NS.1.1 Count, read, and write whole numbers to 1,000 and identify the place value for each digit. 1.3 Order and compare whole numbers to 1,000 by using the symbols <, =, >. For many of the same reasons, the standards listed below are very important:

2NS.2.1 Understand and use the inverse relationship between addition and subtraction (e.g., an opposite number sentence for 8 + 6 = 14 is 14 – 6 = 8) to solve problems and check solutions.

2NS.2.2 Find the sum or difference of two whole numbers up to three digits long. Standard 2.1 gives students a clear application of the relations between different types of operations (addition and subtraction) and can be used to encourage more flexible methods of thinking about and solving problems; for example, a knowledge of addition can facilitate the solving of subtraction problems and vice versa. The problem 144 - 98 = ? can be solved by realizing that 144 = 100 + 44 = 98 + 2 + 44 = 98 + 46.

Standard 2NS.2.2 asks for the teaching of the addition algorithm for numbers up to three digits. For children at this age, two things should be observed. One is that the teaching should be flexible at the beginning and should not insist on the formalism of that algorithm from the start. For example, one can begin the teaching of 23 + 45 by considering 20 + 3 + 40 + 5 = 20 + 40 + 3 + 5 = 60 + 8 = 68. This process gets children used to the advantage of adding the tens digits and the ones digits separately. A second thing is not to emphasize the special skill of “carrying” at the initial stage. The key idea of this algorithm is the ability to add the numbers column by column, one digit at a time. In other words the important thing is to be able to add digits of the same place (ones digits, tens digits, hundreds digits, etc.) and still obtain the correct answer at the end. Only after this idea has sunk in should the “carrying” skill be taught. The same remark applies to the subtraction algorithm: teachers should emphasize at the beginning the fact that the subtraction of two three-digit numbers can be obtained by performing single-digit subtractions. Thus, 746 – 503 can be computed from three single-digit subtractions: 7 – 5 = 2, 4 – 0 = 4, and 6 – 3 = 3 so that 746 – 503 = 243. Show that this is because 746 – 503 = 700 + 40 + 6 – 500 – 00 – 3. The special skill of “trading” needed for a subtraction of 793 – 568 can be taught only after the idea of the efficacy of single-digit subtractions has taken root.

Formal explanations at this grade level are not necessary; friendly persuasion is the order of the day. The mathematical reasoning behind these algorithms is taken up in grade four. The third Number Sense standard is basic to students’ understanding of arithmetic and the ability to solve multiplication and division problems:

2NS.3.0 Students model and solve simple problems involving multiplication and division. Here, fluency with skip counting is helpful. It is important to remind students that multiplication is a shorthand for repeated addition: the meaning of 5 x 7 is exactly 7 + 7 + 7 + 7 + 7, no more and no less. This is an opportunity to impress on students the fact that every symbol and every concept in mathematics have a precise, unambiguous meaning.

The discussion of fractions and the goals represented in Number Sense Standards 2NS.4.1, 2NS.4.2, and 2NS.4.3 are also essential features of students’ developing arithmetical competencies. Although equivalence of fractions is not explicitly presented in the standards, it is also a good idea to begin the discussion of the topic at this point—students should know, for example, that 2/4 is the same as 1/2, a concept that can (and should) be demonstrated with pictures. Finally, as a practical matter and as a basic application of the topics discussed previously, the material in Number Sense Standards 2NS.5.1 and 2NS.5.2—on modeling and solving problems involving money—is very important. Borrowing money gives a practical context to the concept of subtraction. Special attention should be paid to the need for introducing the symbols \$ and ¢ and to the fact that the order of the symbol for dollars is \$3, not 3\$; but for cents, the order is 31¢, not ¢31.

 3NS.1.0 Students understand the place value of whole numbers: 3NS.1.1 Count, read, and write whole numbers to 10,000. 3NS.1.2 Compare and order whole numbers to 10,000. 3NS.1.3 Identify the place value for each digit in numbers to 10,000. 3NS.1.4 Round off numbers to 10,000 to the nearest ten, hundred, and thousand. 3NS.1.5 Use expanded notation to represent numbers (e.g., 3,206 = 3,000 + 200 + 6). 3NS.2.0 Students calculate and solve problems involving addition, subtraction, multiplication, and division: 3NS.2.1 Find the sum or difference of two whole numbers between 0 and 10,000. 3NS.2.2 Memorize to automaticity the multiplication table for numbers between 1 and 10. 3NS.2.3 Use the inverse relationship of multiplication and division to compute and check results. 3NS.2.4 Solve simple problems involving multiplication of multidigit numbers by one-digit numbers (3,671 x 3 = __). 3NS.2.5 Solve division problems in which a multidigit number is evenly divided by a one-digit number (135 ÷ 5 = __). 3NS.2.6 Understand the special properties of 0 and 1 in multiplication and division. 3NS.2.7 Determine the unit cost when given the total cost and number of units. 3NS.2.8 Solve problems that require two or more of the skills mentioned above. 3NS.3.0 Students understand the relationship between whole numbers, simple fractions, and decimals: 3NS.3.1 Compare fractions represented by drawings or concrete materials to show equivalency and to add and subtract simple fractions in context (e.g., 1/2 of a pizza is the same amount as 2/4 of another pizza that is the same size; show that 3/8 is larger than 1/4). 3NS.3.2 Add and subtract simple fractions (e.g., determine that 1/8 + 3/8 is the same as 1/2). 3NS.3.3 Solve problems involving addition, subtraction, multiplication, and division of money amounts in decimal notation and multiply and divide money amounts in decimal notation by using whole-number multipliers and divisors. 3NS.3.4 Know and understand that fractions and decimals are two different representations of the same concept (e.g., 50 cents is 1/2 of a dollar, 75 cents is 3/4 of a dollar).

Comments: In the Number Sense strand, Standards 1.3 and 1.5 are especially important:

3NS.1.3 Identify the place value for each digit in numbers to 10,000.

3NS.1.5 Use expanded notation to represent numbers (e.g., 3,206 = 3,000 + 200 + 6). For students who show a good conceptual understanding of whole numbers (e.g., place value), the second standard should receive special attention.

Here, Standards 3NS.2.1, 3NS.2.2, 3NS.2.3, and 3NS.2.4 are especially important:

3NS.2.1 Find the sum or difference of two whole numbers between 0 and 10,000.

3NS.2.2 Memorize to automaticity the multiplication table for numbers between 1 and 10.

3NS.2.3 Use the inverse relationship of multiplication and division to compute and check results.

3NS.2.4 Solve simple problems involving multiplication of multidigit numbers by one-digit numbers (3,671 x 3 = __). The foundation that supports standard 3NS.2.1 has been laid in grade two: once students become fluent in adding and subtracting three digit numbers, increasing the number of digits offers no real difficulty. The new wrinkle in grade three is standard 3NS.2.4. Again, the emphasis at the initial stage of teaching the multiplication algorithm should be on the simple cases where “carrying” plays no role. For example, 234 x 2 is the same as doubling 200 + 30 + 4, which is 400 + 60 + 8, which is 468, which is in turn obtained from 234 by multiplying each digit by 2. The same reasoning applies to 123 x 3. Once students perceive the possibility that the answer to a multi-digit multiplication might be assembled from the answers to simple single-digit problems, the the idea of "carrying" can be taught, but in assembling the answer to a such a problem as 234 x 6 = 200 x 6 + 30 x 6 + 4 x 6, the fact that the answer can be assembled from the single-digit multiplications 2x6, 3x6, and 4x6 only should be emphasized; it is this fact that makes learning the multiplication table so important. The relationship between division and multiplication (standard 2.3) should be emphasized from the beginning. In other words, 39 divided by 3 = 13 is the same statement as 39 = 13 x 3. Constant reminder of this fact for children in grade three would seem to be necessary.

Two topics in the third standard also deserve special attention:

3NS.3.2 Add and subtract simple fractions (e.g., determine that 1/8 + 3/8 is the same as 1/2).

3NS.3.3 Solve problems involving addition, subtraction, multiplication, and division of money amounts in decimal notation and multiply and divide money amounts in decimal notation by using whole-number multipliers and divisors. These are the early introductory elements of arithmetic with fractions and decimals—topics that will build over several years.

 4NS.1.0 Students understand the place value of whole numbers and decimals to two decimal places and how whole numbers and decimals relate to simple fractions. Students use the concepts of negative numbers: 4NS.1.1 Read and write whole numbers in the millions. 4NS.1.2 Order and compare whole numbers and decimals to two decimal places. 4NS.1.3 Round whole numbers through the millions to the nearest ten, hundred, thousand, ten thousand, or hundred thousand. 4NS.1.4 Decide when a rounded solution is called for and explain why such a solution may be appropriate. 4NS.1.5 Explain different interpretations of fractions, for example, parts of a whole, parts of a set, and division of whole numbers by whole numbers; explain equivalents of fractions (see Standard 4.0). 4NS.1.6 Write tenths and hundredths in decimal and fraction notations and know the fraction and decimal equivalents for halves and fourths (e.g., 1/2 = 0.5 or .50; 7/4 = 1 3/4 = 1.75). 4NS.1.7 Write the fraction represented by a drawing of parts of a figure; represent a given fraction by using drawings; and relate a fraction to a simple decimal on a number line. 4NS.1.8 Use concepts of negative numbers (e.g., on a number line, in counting, in temperature, in "owing"). 4NS.1.9 Identify on a number line the relative position of positive fractions, positive mixed numbers, and positive decimals to two decimal places. 4NS.2.0 Students extend their use and understanding of whole numbers to the addition and subtraction of simple decimals: 4NS.2.1 Estimate and compute the sum or difference of whole numbers and positive decimals to two places. 4NS.2.2 Round two-place decimals to one decimal or the nearest whole number and judge the reasonableness of the rounded answer. 4NS.3.0 Students solve problems involving addition, subtraction, multiplication, and division of whole numbers and understand the relationships among the operations: 4NS.3.1 Demonstrate an understanding of, and the ability to use, standard algorithms for the addition and subtraction of multidigit numbers. 4NS.3.2 Demonstrate an understanding of, and the ability to use, standard algorithms for multiplying a multidigit number by a two-digit number and for dividing a multidigit number by a one-digit number; use relationships between them to simplify computations and to check results. 4NS.3.3 Solve problems involving multiplication of multidigit numbers by two-digit numbers. 4NS.3.4 Solve problems involving division of multidigit numbers by one-digit numbers. 4NS.4.0 Students know how to factor small whole numbers: 4NS.4.1 Understand that many whole numbers break down in different ways (e.g., 12 = 4 x 3 = 2 x 6 = 2 x 2 x 3). 4NS.4.2 Know that numbers such as 2, 3, 5, 7, and 11 do not have any factors except 1 and themselves and that such numbers are called prime numbers.

Comments: The Number Sense strand for the fourth grades extends students’ knowledge of numbers to both bigger numbers (millions) and smaller numbers (two decimal places). Up to this point, students have been asked to learn to round numbers to the nearest tens, hundreds, and thousands, probably without knowing why. It is now finally possible to explain why rounding is much more than a mechanical exercise and is in fact an essential skill in the application of mathematics to understanding the world around us. One can use the population figure of the United States for this purpose. According to latest census (year 2000), there are 281,421,906 people in this country. Explain to students that, in either daily conversation or strategic planning, it would be more sensible to use the round-off figure of 280 million rather than the precise figure of 281,421,906 (due to the built-in errors of a project of this size, the impossibility of correctly counting all the people in transit, the impossibility of reaching all homeless people, the difficulty of obtaining total participation, etc.). Therefore, rounding to the nearest ten million in this case becomes a matter of necessity in discarding unreliable and nonessential information. Standard 1.5 brings out two facts about fractions that are fundamental for students’ understanding of this topic: different interpretations of a fraction and the equivalence of fractions. We will discuss them one at a time. The fact that a fraction such as 3/5 is not only 3 parts of a whole when the whole (the unit) is divided into 5 equal parts but also one part of 3 when 3 is divided into 5 equal parts is so basic that often one uses it without being aware of it. For example, if we are asked in a daily conversation how long one of the pieces of a 3-foot rod is when it is cut into 5 pieces of equal length, we would say without thinking that it is 3/5 of a foot. In so doing we are using the second (division) interpretation of 3/5. On the other hand, it is important to remember that, according to the part-whole definition of a fraction, 3/5 of a foot is the length of 3 of the pieces when a 1-foot rod is divided into 5 pieces of equal length. Students need an explanation of why these two lengths are equal. One way to explain is to divide each foot of the 3-foot rod

into five equal sections,

Each section is the result of dividing 1 foot into 5 equal parts, and so by the part-whole definition of a fraction, the length of three such sections joined together is 3/5 of a foot. But we can clearly group the 15 (=3 x 5) sections of the 3-foot rod to divide the rod into five equal lengths,

and we see that 3/5 of a foot is identical to the length of one of the pieces when a 3-foot rod is divided into 5 equal lengths.

Therefore the part-whole and division definitions of a fraction coincide. This explanation continues to be valid when the fraction 3/5 is replaced by any other fraction a/b. The concept of the equivalence of fractions lies at the core of almost every mathematical consideration related to fractions. Students should be given every opportunity to understand why 2/5 = 14/35, why 5/4 = 40/32, or why a/b=na/nb for any whole number a, b, n (it will always be understood that b ¹0 and n¹ 0). One can explain 2/5 = 14/35 by use of a picture, provided the context of the picture is carefully laid out. Let then the unit 1 be fixed as the area of the unit square:

The fraction 2/5 is then 2 parts of the unit square when it is divided into 5 parts of equal area. We do the equi-division vertically:

Since each vertical strip represents 1/5, the shaded region represents 2/5. The fraction 14/35 is, on the other hand, 14 parts of the unit square when it is divided into 35 parts of equal area. We can achieve the desired equi-division into 35 parts by adding 7 equally spaced horizontal divisions of the unit square to the preceding vertical division:

Now the unit square is divided into 35 small rectangles of the same size, so each small rectangle is 1/35. Since there are 14 of these small rectangles in the shaded region, the shaded region therefore represents not only 2/5, but also 14/35. The preceding reasoning is perfectly general, but for fourth graders, generality should be soft-pedaled. Mentioning a/b=na/nb in passing may be enough. What needs special emphasis, however, is the immediate consequence of the equivalence of a/b and na/nb , namely, the fact that any two fractions can be written as two fractions with the same denominator. Thus if a/b and c/d are two given fractions, they can be rewritten as ad/bd and bc/bd, which have the same denominator bd. This fact has enormous implications when we come to the addition of fractions. The consideration of why a fraction has a division interpretation, as given above, also sheds light on the teaching of Standard 4NS.1.7. To represent the fraction 3/5 as a decimal, for example, we divide the given unit into 10 equal parts. This is best represented on the number line as 9 equi-distant markings of the line segment from 0 to 1. By taking the 2nd, 4th, 6th, and 8th markings, we obtain a division of the unit into 5 equal parts. Since the fraction 3/5 is 3 of these parts, it is the 6th marking. But the 10 markings represent 0.1, 0.2, . . . 0.9, and therefore the 6th marking is 0.6. This shows 3/5 is 0.6. The next standards are basic and new standards:

4NS.1.8 Use concepts of negative numbers (e.g., on a number line, in counting, in temperature, in “owing”).

4NS.1.9 Identify on the number line the relative position of positive fractions, positive mixed numbers, and positive decimals to two decimal places. These standards can be difficult for students to learn if the required background material—ordering of whole numbers and comparison of fractions and decimals—is not presented carefully. The importance of these standards requires that close attention be paid to assessment. The second standard is about “simple” decimals, that is, decimals up to two decimal places. We have not discussed decimals up to this point, but it is time to note that, for decimals up to two decimal places, their addition and subtraction can be completely modeled by money and can therefore be done informally. Looking ahead, though, to when the arithmetic operations of (finite or terminating) decimals of any number of decimal digits are taken up in grade five, we see it is imperative to inform students that, formally, a finite decimal is a fraction whose denominator is a power of 10. This awareness is important in the teaching of decimals in grade four. (To develop this awareness, it is helpful to describe decimals such as 1.03 verbally as one and three-hundredths, not as one point oh three). The third topic in the Number Sense strand is also especially important. This and its four substandards all involve the use of the standard algorithms for addition, subtraction, and multiplication of multidigit numbers as well as the standard algorithm for division of a multidigit number by a one-digit number. As with simple arithmetic, mastery of these skills will require extensive practice over several grade levels, as described in Chapter 4, “Instructional Strategies.”

The emphasis on Standard 4NS.3.1 is, however, on a formal (mathematical) understanding of the addition and subtraction algorithms for whole numbers. It is important for students to see the prominent role played by the commutative law and especially the associative law of addition in the explanation of these algorithms. We should note also that students’ prior familiarity with the skill component of these algorithms is essential for their understanding here, for the following reason. If they are shaky in the mechanics of these algorithms, their minds would be preoccupied with the mechanics and would not be free to appreciate the reasoning behind the mechanics. Standard 4NS.3.2 is about the reasoning that supports the multiplication and division algorithms at least in simple situations (two-digit multipliers and one-digit divisors). It is a bit awkward here because the key fact is the distributive law, which is not mentioned until grade five (Algebra and Functions, Standard AF5.1.3). However, if care and patience are conjoined, students can learn the distributive law. For the division algorithm, there is a new element, namely, division-with-remainder: if a and b are whole numbers, then there are always whole numbers q and r so that a = qb + r, where r is a whole number strictly smaller than the divisor b. The division algorithm can then be explained as an iterated application of this division-with-remainder. The fourth topic, “students know how to factor small whole numbers,” is needed for the discussion of the equivalence of fractions. It also includes the requirement that students understand what a prime number is. The concept of primality is important yet often difficult for students to understand fully. Students should also know the prime numbers up to 50. For these reasons the preparation for the discussion of prime numbers should begin no later than the third grade. Students who understand prime numbers will find it easier to understand the equivalence of fractions and to multiply and divide fractions in grades five, six, and seven.

 5NS.1.0 Students compute with very large and very small numbers, positive integers, decimals, and fractions and understand the relationship between decimals, fractions, and percents. They understand the relative magnitudes of numbers: 5NS.1.1 Estimate, round, and manipulate very large (e.g., millions) and very small (e.g., thousandths) numbers. 5NS.1.2 Interpret percents as a part of a hundred; find decimal and percent equivalents for common fractions and explain why they represent the same value; compute a given percent of a whole number. 5NS.1.3 Understand and compute positive integer powers of nonnegative integers; compute examples as repeated multiplication. 5NS.1.4 Determine the prime factors of all numbers through 50 and write the numbers as the product of their prime factors by using exponents to show multiples of a factor (e.g., 24 = 2 x 2 x 2 x 3 = 2^3 x 3). 5NS.1.5 Identify and represent on a number line decimals, fractions, mixed numbers, and positive and negative integers. 5NS.2.0 Students perform calculations and solve problems involving addition, subtraction, and simple multiplication and division of fractions and decimals: 5NS.2.1 Add, subtract, multiply, and divide with decimals; add with negative integers; subtract positive integers from negative integers; and verify the reasonableness of the results. 5NS.2.2 Demonstrate proficiency with division, including division with positive decimals and long division with multidigit divisors. 5NS.2.3 Solve simple problems, including ones arising in concrete situations, involving the addition and subtraction of fractions and mixed numbers (like and unlike denominators of 20 or less), and express answers in the simplest form. 5NS.2.4 Understand the concept of multiplication and division of fractions. 5NS.2.5 Compute and perform simple multiplication and division of fractions and apply these procedures to solving problems.

Comments: 5NS.1.2 Interpret percents as a part of a hundred; find decimal and percent equivalents for common fractions and explain why they represent the same value; compute a given percent of a whole number. The fact that a fraction c/d is both “c parts of a whole consisting of d equal parts” and “the quotient of the number c divided by the number d” was first mentioned in Number Sense Standard 4NS.1.5 of grade four. As discussed earlier in grade four, this fact must be carefully explained rather than decreed by fiat, as is the practice in most school textbooks. The importance of providing logical explanations for all aspects of the teaching of fractions cannot be overstated because the students’ fear of fractions and the mistakes related to them appear to underlie the failure of mathematics education. Once c/d is clearly understood to be the division of c by d, then the conversion of fractions to decimals can be explained logically. Students will also continue to learn about the relative positions of numbers on the number line, above all, those of negative whole numbers. Negative whole numbers are especially important because, for the first time, they play a major part in core number-sense expectations. Standard 5NS.1.5 is important in this regard.

5NS.1.5 Identify and represent on a number line decimals, fractions, mixed numbers, and positive and negative integers. The correct placement of positive fractions on the number line implies that students will need to order and compare fractions. Identifying numbers as points on the real line is an important step in relating students’ concepts of arithmetic to geometry. This fusion of arithmetic and geometry, which is ubiquitous in mathematics, adds a new dimension to students’ understanding of numbers. Inasmuch as mixed numbers is one of the things that terrorize elementary students, one must approach Standard 5NS.1.5 carefully. First, students should not be made to think of “proper” and “improper” fractions as distinct objects; they should be given to understand that these are nothing more than different examples of the same concept—namely, a fraction. Identifying fractions as points on the number line (so that one point is no different from any other point) would go a long way toward eliminating most of this misconception. With that understood, the teacher can now mention that for improper fractions, there is an alternate representation.

For example, on the number line 5/4 is beyond 1 by the amount of 1/4, so 1 1/4 is a reasonable alternate notation; 11/3 is 2/3 beyond 3 on the number line, so 3 2/3 is also a reasonable alternate notation. When a fraction such as 5/4 or 11/3 is written as 1 1/4 or 3 2/3, it is said to be a mixed number.

In general, fifth graders should be ready for the general explanation of how to write an improper fraction as a mixed number by using division-with remainder: If we suppose a/b is an improper fraction, then we can rewrite it as a mixed number in the following way. The division of the whole number a by the whole number b is expressed as a = qb + r, where q is the quotient and the remainder r is the whole number strictly less than b. Then the fraction a/b is, by definition, written as the mixed number q r/b. Notice that r/b is a proper fraction.  The important point to emphasize is that a mixed number is just a clearly prescribed way of rewriting a fraction, and no fear needs to be associated with it.  But the most important aspect of students’ work with negative numbers is to learn the rules for doing the basic operations of arithmetic with them, as represented in the following standard:

5NS.2.2 Demonstrate proficiency with division, including division with positive decimals and long division with multidigit divisors.
The most essential number-sense skills that students should learn in the fifth grade are the addition and subtraction of fractions (Standards 2.3) and a little bit of multiplying and dividing fractions (Standards 2.4 and 2.5). At this point of students’ mathematics education, it is necessary that they recognize fractions as numbers, which are on the same footing as whole numbers and can therefore be added, multiplied, and so forth. In other words fractions are a special collection of points on the number line that include the whole numbers. To add
a/b+c/d for example, we look to the addition of whole numbers for a model. Since 3 + 8 is just the length of the combined segments when a segment of length 3 is concatenated with a segment of length 8, likewise we define a/b + c/d to be the length of the combined segments when a segment of length a/b is concatenated with a segment of length c/d. The computation of this combined length is complicated by the fact that b may be different than d. But the concept of equivalent fractions shows how any two fractions can be made to have the same denominator, namely, a/b=ad/bd and c/d=bc/bd.

Therefore, if we think of 1/bd as our basic unit, then a/b is ad copies of such a unit and c/d is bc copies of such a unit. Combining them, therefore, shows that a/b+c/d is ad+bc is copies of such a unit 1/bd; that is, a/b+c/d=(ad+bc)/bd. This is a simple way to obtain a formula for adding fractions. But we should note that this formula is not the definition of adding fractions, which is modeled after the addition of whole numbers. The addition of fractions given should be explained in terms of the least common multiple of the denominators.

Once students have mastered these basic skills with fractions, problems involving concrete applications can be used to provide practice, and to promote students’ technical fluency with fractions. Two main skills are involved in reducing fractions: factoring whole numbers in order to put fractions into reduced forms and understanding the basic arithmetic skills involved in this factoring. The two associated standards that should be emphasized are:

5NS.1.4 Determine the prime factors of all numbers through 50 and write the numbers as the product of their prime factors by using exponents to show multiples of a factor (e.g., 24 = 2 × 2 × 2 × 3 = 23 × 3) and

5NS.2.3 Solve simple problems, including ones arising in concrete situations involving the addition and subtraction of fractions and mixed numbers (like and unlike denominators of 20 or less), and express answers in the simplest form.
The instructional profile with fractions, which appears later in this chapter, gives many ideas of how to approach this topic. Students may profit from the use of the Sieve of Eratosthenes (see the glossary) in connection with Standard 5NS.1.4. Standard 5NS.2.4 asks for the introduction to the multiplication and division of fractions. This is a topic that will be taken up in earnest in grade six, but it is important at this point to remind students of the meaning of division among whole numbers as an alternate way of writing multiplication. In other words if 4 × 7 = 28, then, by definition, we write 28 ÷ 7 = 4, or in general, if a × b
= c, then we write c ÷ b = a. One can get students used to this idea of “division as a different expression of multiplication” by drills or manipulatives. Once this idea sinks in, they will be ready for the corresponding situation with fractions; that is, if a, b, and c are fractions, then again by definition, axb=c means the same as c÷b=a.

Using simple fractions, such as b=1/2 or b=1/3 and c=6 or c=24, and by drawing pictures if necessary, one can easily illustrate why 12 x 1/2 = 6 is the same as 6 ÷ 1/2 =12 or why 24 x 1/3= 8 is the same as 8 ÷ 1/3 = 24.

 6NS.1.0 Students compare and order positive and negative fractions, decimals, and mixed numbers. Students solve problems involving fractions, ratios, proportions, and percentages: 6NS.1.1 Compare and order positive and negative fractions, decimals, and mixed numbers and place them on a number line. 6NS.1.2 Interpret and use ratios in different contexts (e.g., batting averages, miles per hour) to show the relative sizes of two quantities, using appropriate notations ( a/b, a to b, a:b ). 6NS.1.3 Use proportions to solve problems (e.g., determine the value of N if 4/7 = N/ 21, find the length of a side of a polygon similar to a known polygon). Use cross-multiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse. 6NS.1.4 Calculate given percentages of quantities and solve problems involving discounts at sales, interest earned, and tips. 6NS.2.0 Students calculate and solve problems involving addition, subtraction, multiplication, and division: 6NS.2.1 Solve problems involving addition, subtraction, multiplication, and division of positive fractions and explain why a particular operation was used for a given situation. 6NS.2.2 Explain the meaning of multiplication and division of positive fractions and perform the calculations (e.g., 5/8 ÷ 15/16 = 5/8 x 16/15 = 2/3). 6NS.2.3 Solve addition, subtraction, multiplication, and division problems, including those arising in concrete situations, that use positive and negative integers and combinations of these operations. 6NS.2.4 Determine the least common multiple and the greatest common divisor of whole numbers; use them to solve problems with fractions (e.g., to find a common denominator to add two fractions or to find the reduced form for a fraction).

Comments: Most of the standards in the Number Sense strand for the sixth grade are very important. These standards can be organized into four groups. The first is the comparison and ordering of positive and negative fractions (i.e., rational numbers), decimals, or mixed numbers and their placement on the number line:

6NS.1.1 Compare and order positive and negative fractions, decimals, and mixed numbers and place them on a number line. The ordering of fractions is best done through the use of the cross-multiplication algorithm, which says a/b = c/d exactly when ad = bc, and a/b < c/d exactly when ad < bc. Students must not only be fluent in the use of this algorithm but also understand why it is true. The reason for the latter goes back to the observation already made in grades four and five that any two fractions can be rewritten as two fractions with the same denominator. Thus a/b and c/d and can be rewritten ad/bd and bc/bd. The cross- multiplication algorithm now becomes obvious. Of particular importance is the students’ understanding of the positions of the negative numbers and the geometric effect on the numbers of the number line when a number is added or subtracted from them. The second group is represented by the next three standards, all of which refer to ratios and percents:

6NS.1.2 Interpret and use ratios in different contexts (e.g., batting averages, miles per hour) to show the relative sizes of two quantities, using appropriate notations (a/b, a to b, a:b).

6NS.1.3 Use proportions to solve problems (e.g., determine the value of N if 4/7 = N/21, find the length of a side of a polygon similar to a known polygon). Use cross-multiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse.

6NS.1.4 Calculate given percentages of quantities and solve problems involving discounts at sales, interest earned, and tips. Notice that although Standards 6NS.1.2 and 6NS.1.3 come before Standard 6NS.2.1, they need to be taught after students know all about Standard 6NS.2.1; that is, after they have learned about the multiplication and division of fractions (for example, Standard 6NS.1.3 explicitly uses the language of “multiplicative inverse”). Once that is done, a ratio can then be defined as the division of one number by another; for example, the ratio of miles traveled to hours traveled (miles per hour), the ratio of the weights of two bags of potatoes, and so forth. In Standard 6NS.1.4 the teacher must be sure to explain why the concept of percent is useful: it standardizes the comparison of magnitudes and, in most situations, facilitates computations. For example, one can imagine the confusion that would arise if the sales tax of one state is 17/200 and that of another state is 4/45. Which state has a higher sales tax? By agreeing to express the tax as a percent, the two states would normalize their taxes to 8.5% and 8.9% (approx.), respectively. Then one can tell at a glance that the second one is higher. Of course, the expression in terms of percent makes the computation of sales tax relatively easy: an 8.5% tax on an article of \$25.50 is 25.50 × 0.085 = \$2.17. The third group includes the remaining Number Sense standards, all of which relate to fractions:

6NS.2.0 Students calculate and solve problems involving addition, subtraction, multiplication, and division.  Because of the slight ambiguity of the language in Standard 6NS.2.0, it should be made explicit that this standard deals with the four arithmetic operations of positive fractions as well as positive and negative integers. The arithmetic operations of the rational numbers, that is, positive and negative fractions, in full generality are left to grade seven.

Since the addition and subtraction of fractions have been taught in grade five (Number Sense Standard 5NS.2.3), the main emphasis of sub-Standards 6NS.2.1 and 6NS.2.2 is on the multiplication and division of positive fractions. A common mistake is to launch immediately into the formula a/b × c/d = ac/bd without first giving meaning to the product of fractions a/b × c/d. One can define a/b × c/d as the area of a rectangle with side lengths a/b and c/d (in which case the whole of which the product measures a part is the area of the unit square), or as the fraction which is a parts of c/d when c/d is divided into b equal parts. Both interpretations are useful in problem solving, and should be clearly explained. From the explanation of Grade 5 standard 5NS.2.4 (Number Sense) in this chapter, the division of fractions is now straightforward: the expression a/b÷c/d=m/n means the same thing as a/b=m/n x c/d. Grade 4 standard 4AF.2.2 (Algebra and Functions), students know that the equation will hold if both sides are multiplied by d/c, and therefore a/b x d/c = m/n x c/d x d/c , and the invert-and-multiply rule for division of fractions is shown to be valid.

Standard 6NS.2.1 calls for solving problems that make use of multiplication and division of fractions. It is important that students know why the invert-and-multiply rule is sufficient for these applications.

It was mentioned in the discussion of grade five in this chapter that the concept of least common multiple plays a role in the teaching of fractions. The following standard makes this point explicit:

6NS.2.4 Determine the least common multiple and the greatest common divisor of whole numbers; use them to solve problems with fractions (e.g., to find a common denominator to add two fractions or to find the reduced form for a fraction). The use of the lcm (least common multiple) in fractions should be carefully qualified. On the one hand, a knowledge of lcm does lead to simplifications in some situations, e.g.,

where we have made use of the lcm of 16 and 24 being 48. This is obviously simpler than using the denominator 16 × 24. On the other hand, finding the lcm of the denominators can be computationally intensive. For example, is it faster, when adding 2/57 + 3/95, to determine the lcm of the denominators (which is 285), or simply use their product as a common denominator 2/57 + 3/95 as

.

Reducing 361/5415 to 1/15 may be more difficult than finding the lcm first, and then reducing 19/285 to the same, and so the decision on whether to use the lcm should be based on an estimate of the more straightforward method, and whether there is a need to generate a reduced form of the sum. The fourth group stands alone because it consists of only one standard:

6NS.2.3 Solve addition, subtraction, multiplication, and division problems, including those arising in concrete situations that use positive and negative integers and combinations of these operations.

For the first time, students are asked to be completely fluent with the arithmetic of negative integers. Students find this difficult because the reasons for some of the more basic rules seem obscure to them. The addition of positive integers may not be an issue, but if one of a and b is negative in a + b , then how should a student evaluate this sum? The most important thing to remember is that for any integer a , - a is the number satisfying a + (- a ) = 0. We now see how to add two negative numbers, (-3) + (-5) = -(3 + 5), because the number [(-3) + (-5)] satisfies [(-3) + (-5)] + {3+5} = (-3) + 3 + (-5) + 5 = 0 + 0 = 0 (where the associative and commutative laws were employed), so that [(-3) + (-5)] + [3 +5] = 0, which means [(-3) + (-5)] = - (3+5). In general, if a and b are positive integers, then ( - a) + ( - b ) = -( a + b). This is because [(-a) + (-b)] + (a + b) = (-a) + a + (-b) + b = 0 + 0 = 0 (where again the associative and commutative laws were used), so that [(- a ) + (- b )] + ( a + b ) = 0, which then implies (- a ) + (- b ) = -( a + b ). If a and b are positive integers and a < b , then a + (- b ) can be computed in the following way: let c be a positive integer so that a + c = b, then a + (- b) = - c. Here is why. We have just seen that - b = - (a + c) = (- a) + ( - c) and so a + ( - b) = a + ( - a) + (- c) = 0 + ( - c) = - c , as claimed. In like manner, we can show that if a + c = b for positive integers a, b, c , then (- a) + b = c, because (- a) + b = ( - a) + a + c = c . We have just showed how to add any two integers.

Now for the multiplication of integers, we first observe that, say, (-3) × 5 = -(3 × 5). It is sufficient to show, by the usual reasoning, that [(-3) × 5] + [3 × 5] = 0. This is so because we make use of the distributive law and obtain, [(-3) × 5] + [3 × 5] = [(-3) + 3] × 5 = 0 × 5 = 0. More generally, and by the same reasoning, if a and b are any two integers, then ( - a) × b = -( a × b). It similarly follows that

It remains to be shown that (-1) × (-1) = 1. From Grade 4 standard 2.1 (Algebra and Functions), it follows that if this equation is true, then

but by the distributive law,

which is then exactly what is to be proved. To sum up, we now know

7NS.1.0 Students know the properties of, and compute with, rational numbers expressed in a variety of forms:

7NS. 1.1 Read, write, and compare rational numbers in scientific notation (positive and negative powers of 10) with approximate numbers using scientific notation.
7NS. 1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to whole-number powers.
7NS. 1.3 Convert fractions to decimals and percents and use these representations in estimations, computations, and applications.
7NS. 1.4 Differentiate between rational and irrational numbers.
7NS. 1.5 Know that every rational number is either a terminating or repeating decimal and be able to convert terminating decimals into reduced fractions.
7NS. 1.6 Calculate the percentage of increases and decreases of a quantity.
7NS. 1.7 Solve problems that involve discounts, markups, commissions, and profit and compute simple and compound interest.

7NS.2.0 Students use exponents, powers, and roots and use exponents in working with fractions:
7NS. 2.1 Understand negative whole-number exponents. Multiply and divide expressions involving exponents with a common base.
7NS. 2.2 Add and subtract fractions by using factoring to find common denominators.
7NS. 2.3 Multiply, divide, and simplify rational numbers by using exponent rules.
7NS. 2.4 Use the inverse relationship between raising to a power and extracting the root of a perfect square integer; for an integer that is not square, determine without a calculator the two integers between which its square root lies and explain why.
7NS. 2.5 Understand the meaning of the absolute value of a number; interpret the absolute value as the distance of the number from zero on a number line; and determine the absolute value of real numbers.