Note: Accuracy depends on not skipping objects or counting objects twice. Counting objects foreshadows the important mathematical concept of one-to-one correspondence.
Note: Going past 20 is important to move beyond the irregular "teen" pattern into the regular twenty-one, twenty-two, . . . counting routine.
Note: In early grades "number" generally means "natural number" or, more mathematically, "non-negative integer."
Note: The emphasis in kindergarten is on the sequence of numbers as discrete objects. The "number line" that displays continuous connection from one number to the next is introduced in grade 2.
Example: The fourth ladybug is about to fly.
Example: 13 can be called "one ten and three ones," with "thirteen" being a kind of nickname.
Note: Zero is the answer to "how many are left?" when all of a collection of objects has been taken away.
Example: Zero is the number of buttons left after 7 buttons are removed from a box that contains 7 buttons.
Note: This is especially important when the objects are dissimilar. For example, the sum of 3 apples and 4 oranges is 7 fruits.
Example: 7 – 3 equals the number of buttons left after 3 buttons are removed from a box that contains 7 buttons.
Note: Decomposition and composition of single-digit numbers into other single-digit numbers is of fundamental importance to develop meaning for addition and subtraction.
Example: 5 = 4 + 1 = 3 + 2; 10 = 9 + 1 = 8 + 2 = 7 + 3 = 6 + 4 = 5 + 5.
Note: This is an important special case because of its relation to finger counting.
Example: 6 = 5 + 1; 7 = 5 + 2; 8 = 5 + 3; 9 = 5 + 4; 10 = 5 + 5.
Note: The discrete number line is not the continuous number line that will be used extensively in later grades, but a visual device for holding numbers in their proper regularly spaced positions The focus in grades K-2 is on the uniformly spaced natural numbers, not on the line that connects them. However, for simplicity, in these grades the discrete number line is often called the number line.
Examples: Even numbers, tens, multiples of five.
Example: If you add something bigger, the result is bigger, but if you take away something bigger, the result will be smaller.
Note: Avoid sums or differences that require numbers greater than 100 or less than 0.
Note: Any correct method will suffice; there is no reason to insist on a particular algorithm since there are many correct methods. Common methods include "adding on" (often using fingers) and regrouping to make a 10.
Examples: 6 + 8 = 6 + 4 + 4 = 10 + 4 = 14;
or 6 + 8 = 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14;
Examples: 3 + 4 + 1 = ?; 7 + 5 + 3 = ?
Example: 8 = 7 + 1 = 1 + 7 = 6 + 2 = 2 + 6 = 5 + 3 = 3 + 5 = 4 + 4;
9 = 8 + 1 = 1 + 8 = 7 + 2 = 2 + 7 = 3 + 6 = 6 + 3 = 5 + 4 = 4 + 5.
Note: Grouping relies on the commutative and associative properties of addition. Examples in early grades foreshadow more formal treatments later. The vocabulary should await later grades.
(a) 17 + 24 = 17 + 23 + 1 = 17 + 3 + 20 + 1 = 20 + 20 + 1 = 40 + 1 = 41.
(b) 58 + 40 = 50 + 8 + 40 = 50 + 40 + 8 = 98.
(c) 58 + 6 = 50 + 8 + 6 = 50 + 14 = 50 + 10 + 4 = 60 + 4 = 64.
(d) 58 + 26 = (58 + 2) + (26 - 2) = 60 + 24 = 84.
Example: Count by tens from 10 to 200; count by 2s from 2 to 50.
Example: Count by tens from 200 to 300; count by 5s from 50 to 100.
Example: In 508 = 5 hundreds, 0 tens, and 8 ones, the 0 tens cannot be ignored (even though it is equal to zero), because in place value notation, it is needed to separate the hundreds position from the ones position.
Note: Grade 2 begins the process of numerical abstraction—of dealing with numbers beyond concrete experience. Place value, invented in ancient India, provides an efficient notation that makes this abstract process possible and comprehensible.
Note: Understanding these relative values provides the foundation for understanding rounding, estimation, accuracy, and significant digits.
Example: Write lengths, as appropriate, in centimeters, decimeters, meters, and kilometers.
Note: The number line is an important unifying idea in mathematics. It ties together several aspects of number, including size, distance, order, positive, negative, and zero. Later it will serve as the basis for understanding rational and irrational numbers and after that for the limit processes of calculus. In grade 2 the interpretation of the number line advances from discrete natural numbers to a continuous line of indefinite length in both directions. Depending on context, a number N (e.g., 1 or 5) can be thought of either as a single point on the number line, or as the interval connecting the point 0 to the point N, or as the length of that interval.
Note: A meter stick marked in centimeters is a useful model of the number line because it reflects the place value structure of the decimal number system.
Example: 389 + 492 = (389 – 8) + (8 + 492) = 381 + 500 = 881.
Note: The expression "related facts" refers to all variations of addition and subtraction facts associated with a particular example.
Example: 348 + 486 = ?, 348 + ? = 834, ? + 486 = 834, 834 – 486 = ?,
834 – ? = 348, and ? – 486 = 348.
Example: 309 + 296 = 605; 605 – 296 = 309.
Caution: Avoid being misled by (or dependent on) stock phrases such as more or less as signals for adding or subtracting.
Note. Since the challenge here is to deal with multi-step problems, the numbers are limited to those already mastered in the previous grade.
Example: 3 × 4 is the 3rd number in the sequence 4, 8, 12, 16, 20, . . . .
Note: As multiplication is repeated addition, so division is repeated subtraction. Consequently, division reverses the results of multiplication and vice versa.
Note: Since division is defined here as the inverse of multiplication, only certain division problems make sense, namely those that arise from a multiplication problem.
Example: 8 ÷ 4 is 2 since 4 × 2 = 8, but 8 ÷ 3 is not (yet) defined.
Note: Multiplication facts up to 5 x 5 are easy to visualize in terms of objects or pictures, so introducing them in grade 2 lays the foundation for the more complex 10 x 10 multiplication expectation that is central to grade 3.
Example: 9,725 – 9,325 = 400 because 7 – 3 = 4 in the hundreds position.
Example: There are 6 numbers that could satisfy 97 < ? ≤ 103, but only five that could satisfy 97 < ? < 103.
Example: 375 + 726 = (3 + 7) × 100 + (7 + 2) x 10 + (5 + 6)
= 10×100 + 9×10 + 10+1 = 10×100 + 10×10 + 1
=(1 x 1000) + (1 x 100) + 0 x 10 + 1 = 1101,
Example: 3 x 8 = 24 means that 24 ÷ 3 = 8 and that 24 ÷ 8 = 3.
Example: Compare a class with 4 rows of 9 seats, a sheet of paper that is 4 inches wide and 9 inches high, and a picnic with 4 groups of 9 children each. Contrast with a class that has 9 rows of 4 seats, a sheet of paper that is 9 inches wide and 4 inches high, and a picnic that involves 9 groups of 4 children each.
Example: When 12 objects are partitioned into equal groups, 3 can represent either the number of groups (because 12 objects can be divided into three groups of four [4, 4, 4]) or the size of each group (because 12 objects can be divided into four groups of three [3, 3, 3, 3]).
Note: In early grades, use only ÷ as the symbol for division—to avoid confusion when the slash (/) is introduced as the symbol for fractions.
Note: Knowing by instant recall is the goal, but recalling patterns that enable a correct rapid response is an important early stage in achieving this skill.
Note: Although solving problems is implicit in every expectation (and thus often not stated explicitly), this particular standard emphasizes the important skill of employing two different arithmetical operations in a single problem.
Note: Problem selection should be guided by two principles: To avoid excess reliance on verbal skills, use real contexts as prompts as much as possible. And to focus on problem-solving skills, keep numbers simple, typically within the computational expectations one grade earlier.
Caution: In grade 3, negative numbers are introduced only as names for points to the left of zero on the number line. They are not used in arithmetic at this point (e.g., for subtraction). In particular the minus sign (–) prefix on negative numbers should not at this stage be interpreted as subtraction.
Note: A unit fraction is determined not just by the number of parts into which the unit interval is divided, but by the number of equal parts. For example, in the upper diagram that follows, each of the four line segments represents , but in the lower diagram none represents .
Example: The unit fraction is smaller than the unit fraction , since when the unit interval is divided into 6 equal parts, each part is smaller than if it were divided into four equal parts. The same thing is true of cookies or pizzas: One-sixth of something is smaller than one-fourth of that same thing.
Note: When the unit interval is divided into n segments, the point to the right of the last (nth) segment is . This point, the right end-point of the unit interval, is also the number 1.
Note: The standard of meeting this expectation is not that children be able to explain these interpretations but that they are able to use different interpretations appropriately and effectively.
Note: This definition applies even when n > d (i.e., the numerator is greater than the denominator): Just lay n segments of size d end to end. It will produce a segment of length regardless of whether n is less than, equal to, or greater than d. Consequently, there is no need to require that the numerator be smaller than the denominator.
Note: These terms are somewhat archaic and not of great significance. It makes no difference if the numerator of a fraction is larger than the denominator, so there is nothing "improper" about so-called "improper fractions."
Note: Measurement to the nearest half or quarter inch provides a concrete model.
Note: The denominators 2, 4, and 8 appear on inch rulers and are created by repeatedly folding strips of paper; the denominator 10 appears on co understand place value.
Note: A similar distinction also applies to whole numbers: The phrase "I'll take 3 oranges" is not about taking the number 3, but about counting 3 oranges. Similarly, "½ of an orange" is not about the number (or unit fraction) ½, but is a reference to a part of the whole orange.
Note: The vocalization of unit fractions (one-half, one-third, one-fourth) are expressions children will know from prior experience (e.g., one-half cup of sugar, one-quarter of an hour). Mathematical fractions extend this prior knowledge to numbers by dividing an interval of length 1. In this way, the unit fraction ½ can be defined as the number representing one-half of the unit interval.
Example: Since 4 copies of the unit fraction 1/4 combine to make up the unit interval, 4 x (¼) = 1. Equivalently, 1 ÷ 4 = ¼.
Caution: At first glance, the statement "1 ÷ 4 = ¼ might appear to be a tautology. It is anything but. Indeed, understanding why this innocuous equation is expressing something oimportance is an important step in understanding fractions. The fraction 1/4 is the name of a point on the number line, the length of part of the unit interval. The open equation 1 ÷ 4 = ? asks for a number with the property that 4 × ? = 1. By observing that the four parts of the unit interval add up to the whole interval, whose length is 1, we discover that the length of one of these parts is the unknown needed to satisfy the equation: 4 x ¼ = 1. This justifies the assertion that 1 ÷ 4 = ¼.
Note: Common synonyms for equal are common denominators or like denominators or same denominators. The latter appear to emphasize the form of the denominator (e.g., all 4s), whereas "equal" correctly focuses on what matters, namely, the value of denominator.
Note: There is no need to simplify answers to lowest terms.
Example: The most common example is a list of prices (e.g., a grocery bill or a shopping list).
Examples: In estimating the number of students to be served school lunch, round the number to the nearest 10 students. In estimating a town's population, rounding to the nearest 50 or 100 is generally more appropriate.
Note: There are no formal rules that work in all cases. This expectation is about judgment.
Note: A prime number is a natural number that has exactly two positive divisors, 1 and itself. A composite number is a natural number that has more than two divisors. By convention, 1 is neither prime nor composite.
Note: Common factors and multiples provide a foundation for arithmetic of fractions and for the concepts of greatest common factor and least common multiple, which are developed in later grades.
Example: 12 = 4 × 3 = 2 × 6 = 2 × 2 × 3.
Example: Justification of a multiplication algorithm relies on the distributive property applied to place value—an analysis that helps prepare students for algebra. For example, using the distributive property, 2 × 35 can be written as 2(30 + 5) = 60 + 10 = 70. Here's how the analysis applies to a more complex problem: 258 × 35 can be written as (200 + 50 + 8) × 35.This becomes:
200 x 35 + 50 × 35 + 8 × 35 = 200(30 + 5) + 50(30 + 5) + 8(30 + 5).
From this point computations can be done mentally:
6000 + 1000 + 1500 + 250 + 240 + 40 = 9030.
Examples: Common errors are displayed below.
when a is a multiple of b, the statement a ÷ b = c is merely a different way of writing a = c × b;
Note: Because equivalent fractions represent the same number, we often say, more simply, that they are the same, or equal.
Examples: Just as 2 + 2 represents the same number as 4, so represents the same number as . The diagram below shows that .
Example: The examples below demonstrate in two different ways (length and area) how the fact that 3 x 3 = 9 and 3 x 4 = 12 makes equivalent to .
Let the whole be the length of a line segment. Divide it into 4 equal parts:
The length of each part represents ¼ by the definition of the fraction ¼. Therefore is represented by the length of the thickened line segment, because it has 3 of the 4 equal parts.
Divide the length of each equal part of the whole into 3 equal parts:
Here the length of each small line segment represents . Now of the whole takes up 9 of these small line segments:
Therefore the thickened line segment represents . Since the thickened line segment also represents , we see that equals
Let the whole be the area of a square. Divide it into 4 equal parts:
The area of each part represents ¼ by the definition of the fraction ¼. Therefore is represented by the area of the shaded region:
Divide each equal part of the whole into 3 equal parts:
The area of each small rectangle represents . Now of the area of the whole takes up 9 of these small rectangles:
Therefore the shaded area represents . Since the shaded area also represents , we see that equals .
Note: Adults use three symbols interchangeably to represent division: ÷, /, and –. The latter two are also used interchangeably to represent fractions. Indeed, the symbol 2/3 is as often used to represent a fraction as the result of the act of division. In school, however, since fractions and division are introduced in a specific sequence, it is important that these not be used interchangeably until their equivalence has been well established and rehearsed.
Note: As introduced in grade 3, fractions can be interpreted as a point on the number line; as a number that lies between two consecutive whole numbers; as the the length of a segment on the real number line; and as a part of a whole. Two fractions are equivalent in each of these interpretations if they refer to the same point, number, length or part of a whole.
Note: The phrase "like denominator" is often used in this context. However, it is equality, not form or "likeness," that is important.
Example: and are equivalent because both represent one-third of the unit interval. Similarly, and are also equivalent because both represent one-fifth of the unit interval.
Example: is equivalent to , and is equivalent , both of which have the same denominator.
Note: More generally, a/b and c/d are equivalent to the fractions , respectively. This shows a general method for transforming fractions into equivalent fractions with equal (common) denominators.
Note: The calculations that create equivalent fractions require multiplying both the numerator and the denominator separately, by the same number. This is, of course, the same as multiplying the fraction itself by 1—which is why the two fractions are equivalent. However, it is premature at this stage to suggest that students think of as x 1 because multiplication of fractions by whole numbers is not yet addressed.
Example: The fractions 5/6 and 3/8 can be compared using the equivalent fractions and .
Note: The idea of common denominator is a natural extension of common multiples introduced above. Addition and subtraction of fractions with common denominators was introduced in grade 3.
Note: To keep calculations simple, do not use mixed numbers (e.g., 3½) or sums involving more than two different denominators (e.g., + + ). Also, do not stress reduction to a 'simplest' form (because, among many reasons, such forms may not be the simplest to use in subsequent calculations).
Example: because on the number line to the right of 5.
Examples: The terminating decimal 0.59 equals the fraction 59/100. Similarly, the decimal 12.3 is just another way of expressing the fraction 123/10 or the mixed number .
Note: Two-place decimals were introduced in grade 3 to represent currency. The concept of two-place decimals as representing fractions with denominator 100 is equivalent to saying that the same amount of money can be expressed either as dollars ($1.34) or as cents (134¢).
Note: The denominators of fractions associated with decimal numbers, being powers of 10, are multiples of one another. This makes adding such fractions relatively easy. For example, .
Note: "Thirds" are missing from this list since 1/3 cannot be represented by a terminating decimal. This is because no power of 10 is a multiple of three, so the fraction 1/3 does not correspond to any terminating decimal.
Note: Problem-solving is an implied part of all expectations, but also sometimes worth special attention, as here where all four arithmetic operations are available for the first time. As noted earlier, to focus on strategies for solving problems that are cognitively more complex than those previously encountered, computational demands should be kept simple.
Example: Recognize that 185 ÷ 5 = 39 is wrong because 39 × 5 = 195.
Note: Prime and composite numbers were introduced in grade 4. Here the goal is to investigate more examples to develop experience with larger numbers.
Note: It is this uniqueness ("the same factors apart from order") of the prime decomposition of integers that makes this fact important—so much so that this result is often called "the fundamental theorem of arithmetic."
24 = 2 × 12 = 2 x 3 x 4 = 2 x 3 x 2 x 2.
24 = 3 x 8 = 3 x 4 x 2 = 3 x 2 x 2 x 2.
24 = 4 x 6 = 2 x 2 x 2 x 3.
Note: The division algorithm most widely used in the United States is called long division. Although the term itself is often taken to mean division by a two-digit number, the algorithm applies equally well to division of a multi-digit number by a single-digit number.
Example: To divide 85 by 6, write 85 = 80 + 5. Dividing 80 by 6 yields 6 10s with 20 left over. In other words, 85 = 80 + 5 = (10 × 6) + 20 + 5 = (10 × 6) + 25. In the long division algorithm, this is written as 6 in the tens place with a remainder of 25. Next, in long division, we divide the remainder, 25, by 6: 25 = (4 × 6) + 1. Combining both steps yields 85 = (10 × 6) + 25 = (10 × 6) + (4 × 6) +1 = (14 × 6) + 1.
Note: Since long division is a process in which the same steps are repeated until an answer is obtained, the example just given offers sufficient understanding of the general process.
Example: To compute 6,512 ÷ 27 requires knowing how many 27s there are in 65, in 111, and in 32.
Example: Divide 49 by 4 by writing 49 = 48 + 1. Since 48/4 = 12, 49 ÷ 4 = yields the quotient 12 and remainder 1.
Note: In grade 4, addition of fractions was restricted to unit fractions, or to those in which one denominator was a multiple of the other. In both cases, these restrictions simplify the required calculations. Here the goal is to understand and learn to do the most general case.
Note: There is no need to find a least common denominator. The easiest common denominator of is most often bd.
17/19 + 13/14 = [(17 × 14) + (13 × 19)]/(19 × 14)
= (238 + 247)/266 = 485/266.
Examples: + [ ] = ; [ ] x 14 + 3 = 101.
Example: 3 × can be thought of as + + = .
Note: As introduced in grade 3, fractions can be interpreted as a point on the number line; as a number that lies between two consecutive whole numbers; as the length of a segment on the real number line; and as a part of a whole. Defining multiplication of fractions by whole numbers as repeated addition is analogous to how the multiplication of whole numbers is understood and readily conforms to the number and length interpretations of fractions.
Note: In interpreting multiplication of a fraction by a whole number as repeated addition, we introduce a curious asymmetry. 3 × is added to itself three times, but it does not make sense to think of × 3 as 3 being added to itself times. This leaves × 3 undefined under this interpretation. If we were sure that multiplication of fractions is commutative, as is multiplication of whole numbers, then we would be able to say that × 3 = 3 × . But to do this requires the "part of a whole" interpretation of fractions.
Example: The multiplication of a fraction by a whole number can also be interpreted by means of a length or area model. Here's an example of using an area model for 3 × . Taking the whole as the area of a unit square, 3 × 1 would be the area of the tower consisting of three unit squares. Since means dividing the whole into 5 equal parts and taking 2 of them, the sum + + can be represented by the shaded area in the middle figure on the right. The figure on the far right rearranges the small shaded rectangles to show that 3 x = + + = .
Note: Taking the whole to be a unit square, then x is by definition the area of a rectangle with length and width . In symbols, x = .
Example: Let the whole be the area of a unit square. Then x is by definition the area of a rectangle with sides of length x . The shaded rectangle in this drawing of the unit square is such a rectangle. The shaded area is also of the area of the whole. Therefore, x = .
Note: If the area of the whole is a unit square, then a/b × c/d is by definition the area of a rectangle with length a/b and width c/d.
Example: To illustrate the multiplication using area models, let the whole be the area of a unit square. Then × is the area of the shaded rectangle with length of side and width . By definition, and . These are illustrated in the diagrams below. The large rectangle has been made from 5 × 4 copies of the small shaded rectangle shown above. Since and are unit fractions, the area of the shaded rectangle is × = . Therefore, the area of the large rectangle (5 x 4)/(2 x 3).
Note: By validating commutativity of multiplication, the area model provides the crucial feature that is missing from the "repeated addition" model for multiplication of fractions. This shows that x 3 = 3 x .
Note: The formula for multiplying fractions can be used to show that fractions also obey the associative and distributive laws of whole number arithmetic. Experience with examples is sufficient to gain insight into just how this works.
Example: The phrase " of 3" means of a whole that is 3 units (e.g., of 3 pizzas, of 3 cups of sugar). To take of 3 units, take of each unit and add them together: + + = 3 × . Since multiplication of fractions is commutative, 3 × = × 3.
Example: ¾ of the length of a 12-inch ruler is 9 inches, while ¾ of the length of a 100-centimeter ruler is 75 centimeters.
Note: Area is again the easiest model: a/b × c/d can be represented by a rectangle with dimensions a/b and c/d, whereas c/d can be represented by a rectangle of dimensions 1 and c/d. When a/b < 1, the former will fit inside the latter, thus showing that it has a smaller area.
Note: The expression a ÷ b where a and b are whole numbers signifies a process to find a quotient q and a remainder r satisfying a = q × b + r, where both q and r are whole numbers and r < b. If we permit fractions as answers, then q = a/b and r = 0 will always solve the division problem, since a = × b + 0.
Note: This fact justifies using the fraction bar (— or /) to denote division rather than the divsion symbol (÷). Beyond elementary school, this is the common convention, since the limitation of integer answers (quotient and remainder) is much less common.
Example: To illustrate the assertion that a/b = a ÷ b with the fraction 3/4, begin as usual with the whole being a unit square. 3 ÷ 4 is the area of one part when three wholes are divided into 4 equal parts as shown. By moving all three shaded rectangles into the same whole, as shown, they form 3 parts of a whole that has been divided into 4 equal parts. That is the definition of the fraction 3/4. Thus 3 ÷ 4 = 3/4.
Note: Another way to think about the relation between fractions and division is to begin with 4 × 3/4 = 3. This says that 4 equal parts, each of size 3/4, make up 3 wholes. Therefore, 3/4 is one part when 3 is divided into 4 equal parts— which is one interpretation of 3 ÷ 4. (This latter interpretation of division is often called "equal shares" or "partitive.")
Note: The dot (·) is an alternative to the cross (x) as a notation for multiplication. (Computers generally use the asterisk (*) in place of a dot.) In written mathematics, but never on a computer, the dot is often omitted (e.g., ab means a · b). As students move beyond the arithmetic of whole numbers to the arithmetic of fractions and decimals, the symbols · and / tend to replace x and ÷.
Example: = because = x 4. In the partitive interpretation of division, is one part in a division of into 4 equal parts.
Example: Because 5 = (5 × 6)×, 5/() = 5 × 6. In the measurement sense of division, 5() = 5 × 6 is the answer to the question "how many parts of size can 5 be divided into?" Since there are 6 parts of size in one whole, there are 5 × 6 parts of size in 5 wholes.
, which is equal to by definition.
Note: Fractions greater than 1 are often called improper fractions, although there is no justification or need for this label.
Note: Division was defined in grade 2 as an action that reverses the results of multiplication. At that time, using only integers, division was limited to composite numbers and their factors (e.g., 6 ÷ 3, but not 6 ÷ 4). Only now, using fractions as well as whole numbers, can this inverse relationship be fully understood.
Note: Although in previous grades the word "number" meant positive whole number, hereafter it will generally mean positive fraction, which encompasses all whole and mixed numbers.
Example: 7.53 x 5 = (753/100) x 5 = (753 x 5) /100 = 3,765/100 = 37.65.
Example. 5 × 0.79 = 3.95 because . This can easily be estimated because 0.79 is less than 1, so 5 × 0.79 must be less than 5. Therefore, the answer cannot be 395.0 or 39.5. Similarly, since 5 > 1, 5 × 0.79 must be greater than .79, so the answer cannot be .395. Thus it must be 3.95.
Note: As a consequence, when multiplying a whole number by 10, 100, or 1,000, the decimal point shifts to the right by 1, 2, or 3 places. Similarly, when dividing a whole number by 10, 100, or 1,000, the decimal point shifts to the left.
Note: Decimals, like fractions, can be greater than one.
Note: Emphasize two special cases: powers of 2 and powers of 10.
Note: In the expression 103, 3 is an exponent and 103 is a power of 10.
Note: As noted earlier, problem-solving is an implied part of all expectations. To focus on strategies for solving problems that are cognitively more complex than those previously encountered, computational demands should be kept simple.
Note: In elementary school, a negative number -a is sometimes called the "opposite" of a, but this terminology is not used in later grades.
Note: The properties of negative numbers apply equally to integers and to fractions. Thus it is just as effective (and certainly easier) to limit to integers all examples that introduce the behavior of negative numbers.
Note: The positive fractions together with the negative fractions and zero (which include all integers) are called rational numbers. In grade 6, these are all the numbers we have, so they are usually referred to just as "numbers." Later when irrational numbers are introduced, the distinction between rational and irrational will be important--but not now.
Note: Formally, a + (–b) = (a − b) + b + (-b) = (a – b) + 0 = a – b.
Example: 3 – 8 = –5 because 5 + (3 – 8) = 5 + (3 + (–8)) = (5 + 3) + (–8) = 8 + (–8) = 0. Therefore, 3 – 8 satisfies the definition of –5 as being that number which, when added to 5, yields zero.
Note: Demonstrations of these laws are part of Algebra, below. Here recognition and fluent use are the important issues.
Note: To verify that (a/b)/(c/d) = (ad)/(bc), we need to check that (ad)/(bc) satisfies the definition of (a/b)/(c/d), namely, that (a/b) = (ad)/(bc) x (c/d): (ad)/(bc) x (c/d) = (a/b)x(d/c) x (c/d) = (a/b) x 1 = a/b.
Examples: ÷ [ ] = 1; ÷  = ; = 1 × [ ]; 2 ÷ 1 = [ ]; 2 ÷ [ ] = 1 .
Note: A ratio is often thought of as a pair of numbers rather than as a single number. Two such pairs of numbers represent the same ratio if one is a non-zero multiple of the other or, equivalently, if when interpreted as fractions, they are equivalent.
Example: 2:4 is the same ratio as 6:12, 8:16,or 1:2.
Examples: 20%, 25%, 33%, 40%, 50%, 66%, 90%, and 100% and their ratio, fraction, and decimal equivalents.
Note: The law of exponents for positive exponents is just a restatement of this definition, since both bn x bm and bn+m mean b multiplied by itself n + m times.
Note: If b > 0, b1 = b, b0 = 1. The same is true of b < 0. If b = 0, 01 = 0, but 00 is not defined.
Note: This definition of b-n is designed to make the law of exponents work for all integers (positive or negative): bn x b-n means b multiplied by itself n times, then divided by b n times, yielding 1. Thus bn x bn = bn+(-n) = b0 = 1.
Example: 3-2 = ()2 = ; 33 = 27; 33 x 3-2 = 3(3-2) = 31 = 3 = 27 x = 27/9.
Note: As usual, keep calculations simple in order to focus on the new concepts.