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.

- Recognize 20 as two groups of 10 and as 10 groups of two.

Example: The fourth ladybug is about to fly.

- Relate the "teen" number words to groups of objects ("ten" + some "ones").

Example: 13 can be called "one ten and three ones," with "thirteen" being a kind of nickname.

- Compare by matching and by counting.
- Use picture graphs (pictographs) to illustrate quantities being compared.

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.

- Write expressions such as 5 + 2 or 7 - 3 to represent situations involving sums or differences of numbers less than 10.

- Understand the meaning of addition problems phrased in different ways to reflect how people actually speak.
- Use fingers and objects to add.
- Attach correct names to objects being added.
Note: This is especially important when the objects are dissimilar. For example, the sum of 3 apples and 4 oranges is 7 fruits.

- Understand the meaning of addition problems phrased in different ways to reflect how people actually speak.
Example: 7 – 3 equals the number of buttons left after 3 buttons are removed from a box that contains 7 buttons.

- Recognize subtraction situations involving missing addends and comparison.
- Use fingers, objects, and addition facts to solve subtraction problems.

- Translate such stories and drawings into numerical expressions such as 7 + 2 or 10 – 8.
- Model, demonstrate (act out), and solve stories that illustrate addition and subtraction.

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.

- Recognize 6 through 10 as "five and some ones."
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.

- Group objects by tens and ones and relate written numerals to counts of the groups by ones, and to counts of the groups by tens.

- Understand and use numbers up to 100 expressed orally.
- Write numbers up to 10 in words.

- Recognize the use of
*digit*to refer to the numerals 0 through 9. - Arrange objects into groups of tens and ones and match the number of groups to corresponding digits in the number that represents the total count of objects.

- Understand that on the number line, bigger numbers appear to the right of smaller numbers.
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. - Use the number line to create visual representations of sequences.
Examples: Even numbers, tens, multiples of five.

- Understand and use relational words such as
*equal, bigger, greater, greatest, smaller*, and*smallest*, and phrases*equal to, greater than, more than, less than*, and*fewer than*.

- Use matching to establish a one-to-one correspondence and count the remainder to determine the size of the difference.
- Connect the meanings of relational terms (bigger, etc.) to the order of numbers, to the measurement of quantities (length, volume, weight, time), and to the operations of adding and subtracting.
Example: If you add something bigger, the result is bigger, but if you take away something bigger, the result will be smaller.

- Know addition and subtraction facts for numbers up to 12.
- Add and subtract efficiently, both mentally and with pencil and paper.
Note: Avoid sums or differences that require numbers greater than 100 or less than 0.

- Be able to explain why the method used produces the correct answer.
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; - Add three single-digit numbers.
Examples: 3 + 4 + 1 = ?; 7 + 5 + 3 = ?

- Understand and solve oral problems with a variety of phrasing, including
*how many more*or*how many fewer*. - Know how to use a calculator to check answers.

- Identify and discuss patterns arising from decompositions.
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. - Represent decomposition situations using terms such as
*put together, add to, take from, break apart*, or*compare*.

- Using objects or drawings, add the tens, add the ones, and regroup if needed.
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.

Examples:

(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.

- Create and discuss problems using drawings, stories, picture graphs, diagrams, symbols, and open equations (e.g., 4 + ? = 17).
- Use the (discrete) number line to illustrate the meaning of addition and subtraction.
- Express answers in a form (verbal or numerical) that is appropriate to the original problem.
- Always check that answers are intuitively reasonable.

- Count accurately for at least 25 terms.
Example: Count by tens from 10 to 200; count by 2s from 2 to 50.

- Begin counts with numbers other than 1.
Example: Count by tens from 200 to 300; count by 5s from 50 to 100.

- Up to 1,000, read and write numerals and understand and speak words; write words up to 100.

- Understand the role of zero in place value notation.
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.

- Recognize that the hundreds place represents numbers that are 10 times as large as those in the tens place and that the units place represents numbers that are 10 times smaller than those in the tens place.
Note: Understanding these relative values provides the foundation for understanding rounding, estimation, accuracy, and significant digits.

- Use meter sticks and related metric objects to understand how the metric system mimics the "power of 10" scaling pattern that is inherent in the place value system.
Example: Write lengths, as appropriate, in centimeters, decimeters, meters, and kilometers.

- Know how to locate zero on the number line.
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.

- Read foot and inch rulers with uneven hash marks to the nearest half inch.

- Add and subtract mentally with ones, tens, and hundreds.
- Use different ways to regroup or ungroup (decompose) to efficiently carry out addition or subtraction both mentally and with pencil and paper.
Example: 389 + 492 = (389 – 8) + (8 + 492) = 381 + 500 = 881.

- Perform calculations in writing and be able to explain reasoning to classmates and teachers.
- Add three two-digit numbers in a single calculation.
- Before calculating, estimate answers based on the left-most digits; after calculating, use a calculator to check the answer.

Note: The expression "related facts" refers to all variations of addition and subtraction facts associated with a particular example.

- Solve addition equations with unknowns in various positions.
Example: 348 + 486 = ?, 348 + ? = 834, ? + 486 = 834, 834 – 486 = ?,

834 – ? = 348, and ? – 486 = 348. - Demonstrate how carrying (in addition) and borrowing (in subtraction) relate to composing and decomposing (or grouping and ungrouping).
- Connect the rollover cases of carrying in addition to the remote borrowing cases in subtraction.
Example: 309 + 296 = 605; 605 – 296 = 309.

- Understand situations described by phrases such as
*put together*or*add to*(for addition), and*take from, break apart*or*compare*(for subtraction). - Recognize and create problems using a variety of settings and language.
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.

- Solve problems that include irrelevant information and recognize when problems do not include sufficient information to be solved.
- Represent problems using appropriate graphical and symbolic expressions.
- Express answers in verbal, graphical, or numerical form, using appropriate units.
- Check results by estimation for reasonableness and by calculator for accuracy.

- Skip count by steps of 2, 3, 4, 5, and 10 and relate patterns in these counts to multiplication.
Example: 3 × 4 is the 3rd number in the sequence 4, 8, 12, 16, 20, . . . .

- Relate multiplication by 10 to the place value system.

- Use objects to represent division of small numbers.
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.

- Use multiplication facts within the 5 × 5 table to solve related division problems.
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.

- Arrange groups of objects into rectangular arrays to illustrate repeated addition and subtraction.
- Rearrange arrays to illustrate that multiplication is commutative.
- Demonstrate skip counting on the number line and then relate this representation of repeated addition to multiplication.

- Identify the thousands, hundreds, tens, and ones positions, and state what quantity each digit represents.
Example: 9,725 – 9,325 = 400 because 7 – 3 = 4 in the hundreds position.

- Understand and use the symbols <, ≤, >, ≥ to signify order and comparison.
- Note especially the distinction between < and ≤ and between > and ≥.
Example: There are 6 numbers that could satisfy 97 < ? ≤ 103, but only five that could satisfy 97 < ? < 103.

- Recognize and use the terms
*sum*and*difference*. - Use parentheses to signify grouping and ungrouping.
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,

- Use a variety of methods appropriate to the problem, including adding or subtracting the smaller number by mental (or finger counting); regrouping to create tens; adding or subtracting an easier number and then compensating; creating mental pictures of manual calculation and others.
- Check answers with a different mental method and compare the efficiency of different methods in relation to different types of problems.

- Use highest order place value (e.g., tens or hundreds digit) to make simple estimates.

- Story problems posed both orally and in writing.
- Problems requiring two or three separate calculations.
- Problems that include irrelevant information.

- Recognize and use the terms
*product*and*quotient*. - Express a multiplication statement in terms of division and vice versa.
Example: 3 x 8 = 24 means that 24 ÷ 3 = 8 and that 24 ÷ 8 = 3.

- Understand multiplication as repeated addition, as area, and as the number of objects in a rectangular array.
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.

- Understand division as repeated subtraction that inverts or "undoes" multiplication.
- Understand division as representing the number of rows or columns in a rectangular array, as the number of groups resulting when a collection is partitioned into equal groups and as the size of each such group.
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.

- Knowing the multiplication table means being able to quickly find missing values in open multiplication or division statements such as 56 ÷ 8 = [ ], 7 x [ ] = 42, or 12÷[ ] = 4.
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.

- Describe contexts for multiplication and division facts.
- Complete sequences of multiples found in the rows and columns of multiplication tables up to 15 by 15.

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.

- Recognize negative numbers as part of the scale of temperature.
- Use negative numbers to count backwards below zero.
- Observe the mirror symmetry in relation to zero of positive and negative numbers.
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.

- Understand that a unit fraction represents the length of a segment that results when the
*unit interval*from 0 to 1 is divided into pieces of equal length.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 .

- Recognize, name, and compare unit fractions with denominators up to 10.
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.

- Understand that is the point to the right of 0 that demarcates the first segment created when the unit interval is divided into n equal segments. Points marking the endpoints of the other segments are labeled in succession with the numbers , . . . . These points represent the numbers that are called
*fractions*. - Understand that a fractional number such as can be interpreted either as the point that lies one-third of the way from 0 to 1 on the number line or as the length of the interval between 0 and this point.
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.

- Understand and use the terms
*numerator*and*denominator*. - Understand that the fraction is a number representing the total length of n segments created when the unit interval from 0 to 1 is divided into d equal parts.
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.

- Recognize that when n = d, the fraction = 1; when n < d, < 1; and when n > d, > 1.
Examples: .

- Recognize the associated vocabulary of
*mixed number*,*proper fraction*, and*improper fraction*.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."

- Understand how to interpret mixed numbers with halves and quarters (e.g., 3½ or 1¼ ) and know how to place them on the number line.
Note: Measurement to the nearest half or quarter inch provides a concrete model.

- Use number lines and rulers to relate fractions to whole numbers.
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.

- When used alone, a fraction such as ½ is a number or a length, but when such as "½ of an apple" the fraction represents a part of a whole.
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.

- When the unit interval is divided into equal parts to create unit fractions, the sum of all the parts adds up to the whole interval, or 1. In other words, the total of
*n*copies of the unit fraction equals 1. Since division is defined as the inverse of multiplication, this is the equivalent of saying that 1 divided by*n*equals .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.

Example: .

- Addition and subtraction of fractions with equal denominators work
subtraction of whole numbers and therefore build on the addition and subtraction of whole numbers.
Note: There is no need to simplify answers to lowest terms.

- Understand that each digit represents a quantity 10 times as great as the digit to its right.

- Use and develop skills such as creating tens and adding columns first down then up to ensure accuracy.
Example: The most common example is a list of prices (e.g., a grocery bill or a shopping list).

- Check answers with a calculator.

*Rounding off*is something done to an overly exact number (e.g., a city's population given as 235,461).*Estimation*and*approximation*are actions taken instead of, or as a check on, an exact calculation. Estimates and approximations are almost always given as round numbers.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.

- Use estimation to check the reasonableness of answers.
- Pay attention to the way answers will be used to determine how much accuracy is important.
Note: There are no formal rules that work in all cases. This expectation is about judgment.

- Understand and use the terms
*factor*and*divisor*. - Apply these definitions to identify prime and composite numbers under 50.
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.

- Find a common factor and a common multiple of two numbers.
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.

- Multiply any multi-digit number by a one-digit number.
- Multiply a three-digit number by a two-digit number.
- Explain why the algorithm works.
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.

- Explain why the algorithm works.
- Understand division as fair shares and as successive subtraction, and explain how the division algorithm yields a result that conforms with these understandings.
- Check results both by multiplying and by using a calculator.

Examples: Common errors are displayed below.

- For whole numbers
*a, b*, and*c*with*b*≠ 0,when

*a*is a multiple of*b*, the statement*a ÷ b = c*is merely a different way of writing*a = c × b*; - when
*a*is not a multiple of*b*, the division*a ÷ b*is expressed as*a = c × b + r*, where the "remainder"*r*is a whole number less than*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 .

- Illustrate equivalent fractions using small numbers with both length and area.
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.

- Understand that equivalent fractions represent the same point on the number line.
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.

- Use length or area drawings to illustrate these equivalences.
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.

- Use the symbols < and > to make comparisons in both increasing and decreasing order.
- Emphasize fractions with denominators of 10 or less.
Example: The fractions 5/6 and 3/8 can be compared using the equivalent fractions and .

- Solve addition and subtraction problems with fractions that are less than 1 and whose denominators are either (a) less than 10 or (b) multiples of 2 and 10 or (c) multiples of each other.
- Add and subtract lengths given as simple fractions (e.g., + ½ inches).
- Find the unknowns in equations such as: .
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).

- Know how to interpret mixed numbers as an addition.
- Locate mixed numbers on the number line.
Example: because on the number line to the right of 5.

- A
*terminating decimal*is place value notation for a special class of fractions with powers of 10 in the denominators. - Understand the values of the digits in a decimal and express them in alternative notations.
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, .

- The arithmetic of decimals becomes arithmetic of whole numbers once they are rewritten as fractions with the same denominator:
- Add and subtract two-decimal numbers, notably currency values, in vertical form.

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.

- Know why parentheses are needed, when and how to use them, and how to evaluate expressions containing them.

Example: Recognize that 185 ÷ 5 = 39 is wrong because 39 × 5 = 195.

- Use multiplication and addition to check the result of a division calculation that produces a non-zero remainder.

- Write composite numbers up to 100 as a product of prime factors.
Note: Prime and composite numbers were introduced in grade 4. Here the goal is to investigate more examples to develop experience with larger numbers.

- Recognize that every decomposition into prime factors involves the same factors apart from order.
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."

Examples:

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.

- Recognize that the division of a whole number
*a*by a whole number*b*(symbolized as*a*÷*b*) is a process to find a*quotient q*and a*remainder r*satisfying*a = q x b + r*, where both*q*and*r*are whole numbers and*r < b*.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. - Understand that the long division algorithm is a repeated application of division-with-remainder.
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.

- Estimate accurately in the steps of the long division algorithm.
Example: To compute 6,512 ÷ 27 requires knowing how many 27s there are in 65, in 111, and in 32.

- Check results by verifying the division equation
*a = q x b + r*, both manually and with a calculator.

- Mentally divide numbers by 10, one hundred, and one thousand.
- Where possible, break apart numbers before dividing to simplify mental calculations.
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.

- Understand and use the general formula .
Note: There is no need to find a

**least**common denominator. The easiest common denominator of is most often**bd**. - When necessary, use calculators to carry out the required multiplications.
Example:

17/19 + 13/14 = [(17 × 14) + (13 × 19)]/(19 × 14)

= (238 + 247)/266 = 485/266.

Example: .

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.

- In general, if
*a, b*, and*c*are whole numbers and*c*≠ 0, then .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 = + + = .

- Understand why the product of two unit fractions is a unit fraction whose denominator is the product of the denominators of the two unit fractions.
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 = .

- Interpret the formula in terms of area.
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).

- Recognize that the formula shows that the multiplication of fractions is commutative.
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.

- In symbols, if a, b, c, and d are all > 0 and < 1, then x <
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.

- Among whole numbers, the answer to
*a ÷ b*is a quotient and a remainder (which may be zero). Among fractions, the answer to*a ÷ b*is the fraction*a/b*.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.")

- As with whole numbers, division of fractions is just a different way to write multiplication. If
*A, B*, and*C*are fractions with*B*≠ 0, then*A/B = C*means*A = C · B*.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 ÷.

- Divide a fraction
*a/b*(where*b*≠ 0) by a non-zero whole number*c*: because = x*c*, this division follows the rules = .Example: = because = x 4. In the partitive interpretation of division, is one part in a division of into 4 equal parts.

- Divide a whole number
*a*by a unit fraction 1/*b*(*b*≠0): because*a = ab*× , this division follows the rule*a*/() =*ab*.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.

- When a division problem
*a ÷ b*is resolved into a quotient*q*and a remainder*r*, then*a = q × b + r*. It follows that*a/b*equals the fraction*(qb + r)/b*, which in turn equals the mixed number =*q*., 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.

- For any numbers
*a*and*b*with*b*≠ 0,*(a × b) ÷ b = a*and*(a ÷ b) × b = a*. In words, if a number (fraction)*a*is first multiplied by*b*and then divided by*b*, the result is the original number*a*and the same is true if we first divide and then multiply.

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.

- When a number is multiplied by a power of 10, the place value of the digits in the number are increased according to the power of 10; the reverse happens when a number is divided by a power of 10.
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.

- If
*p*is a positive whole number, then*a*means^{a}*a x a x a x . . . x a*(*p*times).Note: Emphasize two special cases: powers of 2 and powers of 10.

- Understand and use the language of exponents and powers.
Note: In the expression 10

^{3}, 3 is an exponent and 10^{3}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.

- Provide units in answers.
- Use estimation to judge reasonableness of answers.
- Use calculators to check computations.
- Round off answers as needed to a reasonable number of decimal places.

- If
*a*is a positive number, –*a*is a number that satisfies*a*+ (–*a*) = 0. - On the number line, –
*a*is the mirror image of*a*with respect to 0; it lies as far to the left of 0 as*a*lies to the right.Note: In elementary school, a negative number -a is sometimes called the "opposite" of a, but this terminology is not used in later grades.

- Negative numbers may be either whole numbers or fractions.
- The positive whole numbers together with their negative counterparts and zero are called
*integers*.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.

- Use parentheses, as in –(–
*a*), to distinguish the subtraction operation (minus) from the negative symbol.

- If
*b > a*, the point*c*on the number line that lies at distance*b – a*to the left of zero satisfies the relation*a – b = c*. Thus*a – b = – (b – a)*. - Subtracting a smaller from a larger number is the same as adding the negative of the smaller number to the larger. That is, if
*a > b*then*a - b = a + (-b)*.Note: Formally, a + (–b) = (a − b) + b + (-b) = (a – b) + 0 = a – b.

- Recognize that
*a + (-b) = a – b*(even when*b > a*).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.

- Just as
*A/B = C*means that*A = C × B*for whole numbers, so*(a/b) / (c/d) = M/N*means that*(a/b) = (M/N) x (c/d)*.

- Invert and multiply means:
*(a/b)/(c/d) = (a/b)x(d/c) = (ad)/(bc)*.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.

- Solve problems of the form a ÷ [ ] = b, a × [ ] = b, [ ] ÷ a = b.
Examples: ÷ [ ] = 1; ÷ [] = ; = 1 × [ ]; 2 ÷ 1 = [ ]; 2 ÷ [ ] = 1 .

- Recognize that division by a unit fraction 1/
*n*is the same as multiplying by its denominator*n*.

- Recognize a:b and a/b as alternative notations for ratios.
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.

- Understand that quantities
*a*and*b*can be compared using either subtraction (*a - b*) or division (*a/b*). - Recognize that the terms
*numerator*and*denominator*apply to ratios just as they do to fractions.

- Recognize common percentages and ratios based on fractions whose denominators are 2, 3, 4, 5, or 10.
Examples: 20%, 25%, 33%, 40%, 50%, 66%, 90%, and 100% and their ratio, fraction, and decimal equivalents.

- Express the ratio between two quantities as a percent and a percent as a ratio or fraction.

- Write number sentences and contextual problems involving ratio and percentage.

- Just as
*b × n*(with*b*and*n*positive) can be understood as*b*added to itself*n*times, so*b*can be understood as^{n}*b*multiplied by itself*n*times.Note: The law of exponents for positive exponents is just a restatement of this definition, since both b

^{n}x b^{m}and b^{n+m}mean b multiplied by itself n + m times.Note: If b > 0, b

^{1}= b, b^{0}= 1. The same is true of b < 0. If b = 0, 0^{1}= 0, but 0^{0}is not defined. - If
*n*> 0,*b*means 1/^{-n}*b*(that is, 1 divided by^{n}*b n*times).Note: This definition of b

^{-n}is designed to make the law of exponents work for all integers (positive or negative): b^{n}x b^{-n}means b multiplied by itself n times, then divided by b n times, yielding 1. Thus b^{n}x b^{n}= b^{n+(-n)}= b^{0}= 1.Example: 3

^{-2}= ()^{2}= ; 3^{3}= 27; 3^{3}x 3^{-2}= 3^{(3-2)}= 3^{1}= 3 = 27 x = 27/9.

- Recognize the importance of scientific notation to express very large and very small numbers.
- Locate very large and very small numbers on the number line.
- Understand the concept of
*significant digit*and the role of scientific notatiion in expressing both magnitude and degree of accuracy.

- Calculate with and solve problems involving negative numbers, percentages, ratios, exponents, and scientific notation.
Note: As usual, keep calculations simple in order to focus on the new concepts.

- Compare numbers expressed in different ways and locate them on the number line.
- Write number sentences involving negative numbers, percentages, ratios, exponents, and scientific notation.

- Represent problems mathematically using diagrams, numbers, and symbolic expressions.
- Express answers clearly in verbal, numerical, symbolic, or graphical form.
- Use estimation to check answers for reasonableness and calculators to check for accuracy.
- Describe real situations that require understanding of and calculation with negative numbers, percentages, ratios, exponents, and scientific notation.