Elementary Mathematics Benchmarks, Grade 5

Number (N)

N.5.1 Understand that every natural number can be written as a product of prime numbers in only one way (apart from order).

a. Extend knowledge of prime and composite numbers up to 100.

• 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.

b. Decompose composite numbers into products of factors in different ways and identify which of these combinations are products of prime factors.

• 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.

N.5.2 Know how to divide whole numbers.

a. Understand and use a reliable algorithm for division of whole numbers.

• 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.

b. Divide numbers up to 1,000 by numbers up to 100 using long division or some comparable approach.

• 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.

c. Know and use mental methods to calculate or estimate the answers to division problems.

• 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.

N.5.3 Understand how to add and subtract fractions.

a. Add fractions with unequal denominators by rewriting them as equivalent fractions with equal denominators.

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.

b. Add and subtract mixed numbers.

Example: .

c. Find the unknown in simple equations involving fractions and mixed numbers.

Examples: + [ ] = ; [ ] x 14 + 3 = 101.

N.5.4 Understand what it means to multiply fractions and know how to do it.

a. Understand how multiplying a fraction by a whole number can be interpreted as repeated addition of the fraction.

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 = + + = .

  

b. Understand how multiplying two fractions can be interpreted in terms of an area model.

• 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.

c. Understand why " of c" is the same as " x c."

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.

d. Understand that the product of a positive number with a positive fraction less than 1 is smaller than the original number.

• 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.

N.5.5 Understand and use the interpretation of a fraction as division.

a. Understand why the fraction a/b can be considered an answer to the division problem a ÷ b.

• 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.")

b. Understand how to divide a fraction by a fraction and to solve related problems.

• 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.

c. Express division with remainder in the form of mixed numbers.

• 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.

d. Understand division as the inverse of multiplication and vice versa.

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.

N.5.6 Understand how to multiply terminating decimals by whole numbers.

a. Multiplying a terminating decimal by a whole number is equivalent to multiplying a fraction by a whole number.

Example: 7.53 x 5 = (753/100) x 5 = (753 x 5) /100 = 3,765/100 = 37.65.

b. Understand how to place the decimal point in an answer to a multiplication problem both by estimation and by calculation.

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.

c. Demonstrate with examples that multiplication of a number by a decimal or a fraction may result in either a smaller or a larger number.

Note: Decimals, like fractions, can be greater than one.

N.5.7 Understand the notation and calculation of positive whole number powers.

a. Recognize and use the definition and notation for exponents.

• If p is a positive whole number, then a^a means 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 10³, 3 is an exponent and 10³ is a power of 10.

N.5.8 Solve multi-step problems using multi-digit positive numbers, fractions, and decimals.

a. Solve problems of various types—mathematical tasks, word problems, contextual questions, and real-world settings.

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.

b. Translate a problem's verbal statements or contextual details into diagrams, symbols, and numerical expressions.

c. Express answers in appropriate verbal or numerical form.

• 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.

d. Solve problems that require a mixture of arithmetic operations, parentheses, and arithmetic laws (commutative, distributive, associative).

e. Use mental arithmetic with simple multiplication and division of whole numbers, fractions, and decimals.

Measurement (M)

M.5.1 Make, record, display, and interpret measurements of everyday objects.

a. Select appropriate units to make measurements and include units in answers.

b. Recognize and use measures of weight, information, and temperature.

• For information: bytes, kilobytes (K or Kb), megabytes (M), gigabytes (G). 1G = 1,000M, 1M = 1,000K, 1K = 1,000 bytes.

  Note: Literally, the multiplier is 1,024 = 2¹⁰, but for simplicity in calculation, 1,000 is generally used instead.

• For weight: kilogram (kg), gram (g), pound (lb), ounce (oz). 1 kg= 1,000 g, 1 lb = 16 oz.
• For temperature: Centigrade and Fahrenheit degrees. 32°F = 0°C; 212°F = 100°C.

c. Record measurements to a reasonable degree of accuracy, using fractions and decimals as needed to achieve the desired detail.

d. When needed, use a calculator to find answers to questions associated with measurements.

• Understand the role of significant digits in signaling the accuracy of measurements and associated calculations.

  Example: Report a city's population as 210,000, not as 211,513.

e. Create graphs and tables to present and communicate data.

Probability and Statistics (PS)

PS.5.1 Find, interpret, and use the average (mean) of a set of data.

a. Calculate the average of a set of data that includes whole numbers, fractions, and decimals.

Note: Emphasize that data is plural and datum is singular, the name for a single number in a set of data.

• Infer characteristics of a data set given the mean and other incomplete information.

Geometry (G)

G.5.1 Measure angles in degrees and solve related problems.

a. Understand the definition of degree and be able to measure angles in degrees.

• A degree is one part of the circumference of a circle of radius 1 unit (a unit circle) that is divided into 360 equal parts. The measure of an angle in degrees is defined to be the number of degrees of the arc of the unit circle, centered at the vertex of the angle, that is intercepted by the angle.
• The measure of an angle in degrees can also be interpreted as the amount of counter-clockwise turning from one ray to the other.

  Note: Earlier (in grade 4), the angle determined by two rays was defined to be the smaller of the two options. For consistency, therefore, when an angle is measured by the amount of turning necessary to rotate one ray into another, it is important to start with the particular ray that will produce an angle measure no greater than 180°.

• The symbol ° is an abbreviation for "degree" (e.g., 45 degrees = 45°).
• As a shorthand, angles are called equal if the measures of the angles are equal.

b. Know and use the measures of common angles.

• Recognize that angles on a straight line add up to 180° and that angles around a point add up to 360°. An angle of 180° is called a straight angle.
• A right angle is an angle of 90°. An acute angle is an angle of less than 90°, while an obtuse angle is an angle of more than 90°.

  Note: Since a pair of perpendicular lines divides the plane into 4 equal angles, the measure of a right angle is 360°/4 = 90°.

  

c. Interpret and prepare circle graphs (pie charts).

G.5.2 Know how to do basic constructions using a straightedge and compass.

a. Basic constructions include (a) drop a perpendicular from a point to a line, (b) bisect an angle, (c) erect the perpendicular bisector of a line, and (d) construct a hexagon on a circle.

Note: A straightedge is a physical representation of a line, not a ruler that is used for measuring. The role of a straightedge in constructions is to draw lines through two points, just as the role of the compass is to draw a circle based on two points, the center and a point on the circumference.

Note: Students need extended practice with constructions, since constructions embody the elements of geometry—lines and circles—independent of numbers and measurement. Since constructions are so central to Euclidean geometry, they are often called Euclidean constructions.

Note: These constructions are basic in the sense that other important constructions introduced in later grades (e.g., of an equilateral triangle given one side; of a square inscribed in a circle) build on them.

• Use informal arguments such as paper folding to verify the correctness of constructions.

G.5.3 Recognize and work with simple polyhedra.

a. Represent and work with rectangular prisms by means of orthogonal views, projective views, and nets.

• A net is a flat (two-dimensional) pattern of faces that can be folded to form the surface of a solid.

  Note: Because a net represents the surface of a polyhedra spread out in two dimensions, the area of a net equals the surface area of the corresponding solid.

• Orthogonal views are from top, front, and side; picture views are either projective or isometric; and nets are plane figures that can be folded to form the surface of the solid.

  Example: An orthogonal view (a), a projective view (b), and a net (c) of the same rectangular prism:

  

b. Recognize the five regular ("Platonic") solids.

• Count faces, edges, and vertices, and make a table with the results.

G.5.4 Find the area of shapes created out of triangles.

a. Understand, derive, and use the formula A = ½bh for the area of a triangle.

• Arrange two identical right triangles with base b and height h to form a rectangle whose area is bh. Since the area of each right triangle is half that of the rectangle, A = ½bh.

  

• If triangle ABC is not a right triangle, then placing two copies together will form a parallelogram with base b and height h. This parallelogram can be transformed into a rectangle of area bh by moving a right triangle of height h from one side of the parallelogram to the other. So here too, A = ½bh.

  

• Alternatively, divide a general triangle ABC into two right triangles as shown below and combine the areas of the two parts:

  

  Note: As the diagrams show, there are two cases to consider: For an acute triangle (where all angles are smaller than a right angle), the parts are added together. For an obtuse triangle (where one angle is larger than a right angle), one right triangle must be subtracted from the other).

b. Find the area of a convex polygon by decomposing it into triangles.

• A polygon is called convex if a line segment joining any two points on the perimeter of the polygon will lie inside or on the polygon.
• Any convex polygon of n sides can be decomposed into (n – 2) triangles.

c. Find the area of other geometric figures that can be paved by triangles.

G.5.5 Interpret and plot points on the coordinate plane.

a. Associate an ordered pair of numbers with a point in the first (upper right) quadrant and, conversely, any such point with an ordered pair of numbers.

• Positions on the coordinate plane are determined in relation to the coordinate axes, a pair of number lines that are placed perpendicular to each other so that the zero point of each coincides.

  

  Note: The coordinate plane is a two-dimensional extension of the number line and builds on extensive (but separate) prior work with the number line and with perpendicular lines.

• Recognize the similarity between locating points on the coordinate plane and locating positions on a map.
• Recognize and use the terms vertical and horizontal.

b. Identify characteristics of the set of points that define vertical and horizontal line segments.

• Use subtraction of whole numbers, fractions, and decimals to find the length of vertical or horizontal line segments identified by the ordered pairs of its endpoints.

  Example: What is the length of the line segment determined by (3/5, 0) and (1.5, 0)?

Algebra (A)

A.5.1 Find the unknown in simple linear equations.

a. Equations that require only simple calculation should be solved mentally (that is, "by inspection"):

96 + 67 = b + 67

+ - = p

a + =

39 – k = 39 – 40

- + = d + -

+ = b +

78 + b = 57 + 79

53 + 76 = 51 +76 + d

A.5.2 Evaluate and represent simple expressions.

a. Translate between simple expressions, tables of data and graphs in the coordinate plane.

b. Understand and use the conventions for order of operations (including powers).

Example: ax² + bx = (a(x²)) + (bx), not (ax)² + bx.

c. Evaluate expressions such as

• nr where n is a whole number and r is a fraction.
• nab/(na-b) when n, a, and b are whole numbers and where na > b.
• + , where a, b, c, d are positive whole numbers.
• 1/ab where a and b are positive whole numbers.
• a/b where one of a or b is a positive whole number and one is a fraction.

  Note: Avoid expressions that introduce negative numbers since systematic treatment of operations on negative numbers is not introduced until grade 6.

  Note: Working with expressions both introduces the processes of algebra and also reinforces skills in arithmetic.

d. Understand the importance of not dividing by zero.

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