Elementary Mathematics Benchmarks, Grade 4

Number (N)

N.4.1 Read, write, add, and subtract positive whole numbers.

a. Read and write numbers in numerals and in words.

b. Recognize the place values in numbers and understand what quantities each digit represents.

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

c. Compare natural numbers expressed in place value notation.

d. Add columns consisting of several three- and four-digit numbers.

• 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.

N.4.2 Understand why and how to approximate or estimate.

a. Round off numbers to the nearest 5, 10, 25, 100, or 1,000.

• 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.

b. Estimate answers to problems involving addition, subtraction, and multiplication.

c. Judge the accuracy appropriate to given problems or situations.

• 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.

N.4.3 Identify small prime and composite numbers.

a. Understand and use the definitions of prime and composite number.

• 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.

b. List all factors of integers up to 50.

c. Determine if a one-digit number is a factor of a given integer and whether a given integer is a multiple of a given one-digit number.

• 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.

d. Recognize that some integers can be expressed as a product of factors in more than one way.

Example: 12 = 4 × 3 = 2 × 6 = 2 × 2 × 3.

N.4.4 Multiply small multi-digit numbers and divide by single-digit numbers.

a. Understand and use a reliable algorithm for multiplying multi-digit numbers accurately and efficiently.

• 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.

b. Understand and use a reliable algorithm for dividing numbers by a single-digit number accurately and efficiently.

• 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.

c. Recognize, understand, and correct common computational errors.

Examples: Common errors are displayed below.

d. Understand the role and function of remainders in division.

• 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.

N.4.5 Understand and use the concept of equivalent fractions.

a. Understand that two fractions are equivalent if they represent the same number.

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 .

Using length to illustrate equivalent fractions:

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

.

Using area to illustrate equivalent fractions:

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.

b. Place fractions on the number line.

• 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.

c. Understand that any two fractions can be written as equivalent fractions with equal denominators.

• 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.

d. Use equivalent fractions to compare fractions.

• 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 .

N.4.6 Add and subtract simple fractions.

a. Add and subtract fractions by rewriting them as equivalent fractions with a comm denominator.

• 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).

b. Recognize mixed numbers as an alternate notation for fractions greater than 1.

• 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.

N.4.7 Understand and use decimal numbers up to hundredths.

a. Understand decimal digits in the context of place value for terminating decimals with up to two decimal places.

• 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, .

b. Add and subtract decimals with up to two decimal places.

• 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.

c. Write tenths and hundredths in decimal and fraction notation and recognize the fraction and decimal equivalents of halves, fourths, and fifths.

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.

d. Use decimal notation to convert between grams and kilograms, meters and kilometers, and cents and dollars.

N.4.8 Solve multi-step problems using whole numbers, fractions, decimals, and all four arithmetic operations.

a. Solve problems of various types (mathematical tasks, word problems, contextual questions, and real-world settings) that require more than one of the four arithmetic operations.

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.

b. Understand and use parentheses to specify the order of operations.

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

c. Use the inverse relation between multiplication and division to check results when solving problems.

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.

d. Translate a problem's verbal statements or contextual details into diagrams and numerical expressions and express answers in appropriate verbal or numerical form, using units as needed.

e. Use estimation to judge the reasonableness of answers.

f. Create verbal and contextual problems representing a given number sentence and use the four operations to write number sentences for given situations.

Measurement (M)

M.4.1 Understand and use standard measures of length, area, and volume.

a. Know and use common units of measure of length, area, and volume in both metric and English systems.

• Always use units when recording measurements.
• Know both metric and English units: centimeter, square centimeter, cubic centimeter; meter, square meter, cubic meter; inch, square inch, cubic inch; foot, square foot, cubic foot.
• Use abbreviations: m, cm, in, ft, yd; m², cm², in², ft², yd²; sq m, sq cm, sq in, sq ft, sq yd; m³, cm³, in³, ft³ and yd³.

b. Convert measurements of length, weight, area, volume, and time within a single system.

Note: Emphasize conversions that are common in daily life. Common conversions typically involve adjacent units—for example, hours and minutes or minutes and seconds, but not hours and seconds. Know common within-system equivalences.

• Use unit cubes to build solids of given dimensions and find their volumes.
• 1 square foot = 12² square inches; 1 square meter as 100² square centimeters; 1 cubic foot = 12³ cubic inches; 1 cubic meter as 100³ cubic centimeters.

  

c. Visualize, describe, and draw the relative sizes of length, area, and volume units in the different measurement systems.

• Estimate areas of rectangles in square inches and square centimeters.

  Note: Avoid between-system conversions.

  Examples: Centimeter vs. inch, foot and yard vs. meter; square centimeter vs. square inch; square yard vs. square meter; cubic foot vs. cubic meter.

  

d. Recognize that measurements are never exact.

• Both recorded data and answers to calculations should be rounded to a degree of precision that is reasonable in the context of a given problem and the accuracy of the measuring instrument.

  Note: All measurements of continuous phenomena such as length, capacity, or temperature are approximations. Measurements of discrete items such as people or bytes can be either exact (e.g., size of an athletic team) or approximate (e.g., size of a city).

e. Solve problems involving area, perimeter, surface area, or volume of rectangular figures.

• Select appropriate units to make measurements of everyday objects, record measurements to a reasonable degree of accuracy, and use a calculator when appropriate to compute answers.
• Know that answers to measurement problems require appropriate units in order to have any meaning.

  Note: Include figures whose dimensions are given as fractions or mixed numbers.

Probability and Statistics (S)

PS.4.1 Record, arrange, present, and interpret data using tables and various types of graphs.

a. Create and interpret line, bar, and circle graphs and their associated tables of data.

• Create and label appropriate scales for graphs.
• Prepare labels or captions to explain what a table or graph represents.
• Solve problems using data presented in graphs and tables.
• Employ fractions and mixed numbers, as needed, in tables and graphs.

Geometry (G)

G.4.1 Understand and use the definitions of angle, polygon, and circle.

a. An angle in a plane is a region between two rays that have a common starting point.

Note: According to this definition, a right angle (as determined by perpendicular rays) is indeed an angle.

b. If angle A is contained in another angle B, then angle B is said to be bigger than angle A.

• The figures below illustrate how to determine whether an angle is larger than, smaller than, or close to a right angle.

  The angle is not contained in a right angle, so this tells us that it is larger than a right angle.

  

  The angle is contained in a right angle, so this tells us that it is smaller than a right angle.

  

  Note: When two rays come from the same point (see figure below), they divide the plane into two regions, giving two angles. Except where otherwise indicated, the angle determined by the two rays is defined, by convention, as the smaller region.

  

• Understand that shapes such as triangles, squares, and rectangles have angles.

  Note: Technically, polygons do not contain rays, which are required for the definition of angles. Their sides are line segments of finite length. Nonetheless, if we imagine the sides extending indefinitely away from each corner, then each corner becomes an angle.

  Example: Describe the difference between the two figures below:

  

• Identify acute, obtuse and right angles.

c. Know and use the basic properties of squares; rectangles; and isosceles, equilateral and right triangles.

• Identify scalene, acute, and obtuse triangles.
• Know how to mark squares, rectangles, and triangles appropriately:

  

  

d. Know what a polygon is and be able to identify and draw some examples.

• A polygon is a figure that lies in a plane consisting of a finite number of line segments called edges (or sides) with the properties that (a) each edge is joined to exactly two other edges at the end points; edges do not meet each other except at end points; and the edges enclose a single region.

  Example: The figures below are not polygons:

  

e. Know and use the basic properties of a circle.

• A circle is the set of points in a plane that are at a fixed distance from a given point.

  

• Know that a circle is not a polygon.

Algebra (A)

A.4.1 Use properties of arithmetic to solve simple problems.

a. Understand and use the commutative, associative, and distributive properties of numbers.

• Use these terms appropriately in oral descriptions of mathematical reasoning.
• Use parentheses to illustrate and clarify these properties.

b. Find the unknown in simple linear equations.

• Use a mixture of whole numbers, fractions, and mixed numbers as coefficients.

  Examples: 24 + n = n – 2; ¾ + p = 5/4 – p

  Note: "Simple" equations for grade 4 are those that require only addition or subtraction (e.g., 3/4 + [ ] = 7/4) or a single division whose answer is a whole number (e.g., 3 x [ ] = 12).

  Note: There is no need to use term linear since these are the only kinds of equations encountered in grade 4.

A.4.2 Evaluate simple expressions.

a. Find the value of expressions such as na + b, and na – b where a, b and n are whole numbers or fractions and where na ≥ b.

• Make tables and graphs to display the results of evaluating expressions for different values of n such as n = 1, 2, 3, . . . .

  Note: Evaluating an expression involves two distinct steps: substituting specific values for letter variables in the expression, and then carrying out the arithmetic operations implied by the expression. Working with expressions both introduces the processes of algebra and also reinforces skills in arithmetic.

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

b. Evaluate expressions such as , where a, b, c, and n are whole numbers.

c. Evaluate expressions such as where a and b are single-digit whole numbers.

Example: The value of when a = 1, b = 2, c = 3, and n = 4 is .

Note: Addition of fractions is limited to cases included in the grade 4 expectations—namely, unit fractions with denominators under 10 and other fractions where one denominator is a multiple of the other.

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