Elementary Mathematics Benchmarks, Grade 3

Number (N)

N.3.1 Read, write, add, subtract, and comprehend five-digit numbers.

a. Read and write numbers up to 10,000 in numerals and in words.

b. Understand that digits in numbers represent different values depending on their location (place) in the number.

• 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.

c. Compare numbers up to 10,000.

• 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.

d. Understand and use grouping for addition and ungrouping for subtraction.

• 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,

e. Add and subtract two-digit numbers mentally.

• 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.

f. Judge the reasonableness of answers by estimation.

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

g. Solve a variety of addition and subtraction problems.

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

N.3.2 Multiply and divide with numbers up to 10.

a. Understand division as an alternative way of expressing multiplication.

• 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.

b. Recognize different interpretations of multiplication and division and explain why they are equivalent.

• 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.

c. Know the multiplication table up to 10 x 10.

• 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.

d. Count aloud the first 10 multiples of each one-digit natural number.

e. Create, analyze, and solve multiplication and division problems that involve groups and arrays.

• 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.

f. Make comparisons that involve multiplication or division.

N.3.3 Solve contextual, experiential, and verbal problems that require several steps and more than one arithmetic operation.

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.

a. Represent problems mathematically using diagrams, numbers, and symbolic expressions.

b. Express answers clearly in verbal, numerical, or graphical (bar or picture) form, using units whenever appropriate.

c. Use estimation to check answers for reasonableness and calculators to check for accuracy.

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.

N.3.4 Recognize negative numbers and fractions as numbers and know where they lie on the number line.

a. Know that symbols such as –1, –2, –3 represent negative numbers and know where they fall on the number line.

• 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.

b. Understand that symbols such as represent numbers called unit fractions that serve as building blocks for all fractions.

• 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.

c. Understand that each unit fraction generates other fractions of the form , . . . and know how to locate these fractions on the number line.

• 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.

N.3.5 Understand, interpret, and represent fractions.

a. Recognize and utilize different interpretations of fractions, namely, as a point on the number line; as a number that lies between two consecutive (whole) numbers; as the length of a segment of the real number line; and as a part of a whole.

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.

b. Understand how a general fraction is built up from n unit fractions of the form .

• 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."

c. Locate fractions with denominator 2, 4, 8, and 10 on the number line.

• 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.

  

d. Understand and use the language of fractions in different contexts.

• 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.

e. Recognize fractions as numbers that solve division problems.

• 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.

N.3.6 Understand how to add, subtract, and compare fractions with equal denominators.

a. Recognize how adding and subtracting fractions with equal denominators can be thought of as the joining and taking away, respectively, of contiguous segments on the number line.

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.

b. Understand that a fraction is the sum of n unit fractions of the form .

Example: .

c. Compare, add, and subtract fractions with equal denominators.

• 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.

  

Measurement (M)

M.3.1 Recognize why measurements need units and know how to use common units.

a. Understand that all measurements require units and that a quantity accompanied by a unit represents a measurement.

• Know and use the names and approximate magnitudes of common units:

  For length: kilometer, meter, centimeter; mile, yard, foot, inch.

  For capacity: liter, milliliter; gallon, quart, pint, cup.

  For time: year, month, week, day, hour, minute, second.

  For money: pennies, nickels, dimes, quarters, dollars.

  Note: Many of these units have been introduced in prior grades; others will be introduced in later grades. Here some are pulled together for reinforcement and systematic use. Each year in grades 2-6 some new measures should be introduced and previous ones reinforced. Which are done in which grades is of lesser importance.

b. Know common within-system equivalences:

• 1 meter = 100 centimeters, 1 yard = 3 feet, 1 foot = 12 inches.
• 1 liter = 1,000 milliliters, 1 gallon = 4 quarts, 1 quart= two pints.
• 1 year = 12 months, 1 week = 7 days, 1 hour = 60 minutes, 1 minute = 60 seconds.
• 1 dollar = 4 quarters = 10 dimes = 100 pennies, 1 quarter= 5 nickels = 25 pennies, 1 dime = 2 nickels = 10 pennies, 1 nickel = 5 pennies.

c. Choose reasonable units of measure, estimate common measurements, use appropriate tools to make measurements, and record measurements accurately and systematically.

• Make and record measurements that use mixed units within the same system of measurement (e.g., feet and inches, hours and minutes).

  Note: Many situations admit various approaches to measurement. Using different means and comparing results is a valuable activity.

• Understand that errors are an intrinsic part of measurement.
• Understand and use time both as an absolute (12:30 p.m.) and as a duration of a time interval (20 minutes).
• Understand and use idiomatic expressions of time (e.g., "10 minutes past 5," "quarter to 12," "one hour and ten minutes").

d. Use decimal notation to express, add, and subtract amounts of money.

Note: Dealing with money enables students to become accustomed to decimal notation, i.e., $1.49 + $0.25 = $1.74.

e. Solve problems requiring the addition and subtraction of lengths, weights, capacities, times, and money.

• Include use of common abbreviations: m, cm, kg, g, l, ml, hr, min, sec, in, ft, lb, oz, $, ¢.

  Note: Add and subtract only within a single system, using quantities within students' experience. Use real data where possible, but limit the size and complexity of numbers so that problem solving, not computation, is the central challenge of each task.

Geometry (G)

G.3.1 Recognize basic elements of geometric figures and use them to describe shapes.

a. Identify points, rays, line segments, lines, and planes in both mathematical and everyday settings.

• A line is a straight path traced by a moving point having no breadth nor end in either direction.

  Examples: Each figure on the left above represents a line; the arrows indicate that the lines keep going in the indicated directions without end. The number line with both positive and negative numbers is a line.

• A part of a line that starts at one point and ends at another is called a line segment. Line segments are drawn without arrows on either end because line segments end at points.

  Examples: The figure in the center above is a line segment. The edges of a desk or door or piece of paper are everyday examples of line segments.

• Part of a line that starts at one point and goes on forever in one direction is called a ray.

  Examples: The figure above on the right is a ray. The positive number line (to the right of 0) is a ray. On the other hand, none of the four examples below are lines:

  

  Caution: Not all sources distinguish carefully among the terms line, segment, or ray, nor do all sources employ the convention of arrowheads in exactly the manner described above. Often context is the best guide to distinguish among these terms.

• Know that a plane is a flat surface without thickness that extends indefinitely in every direction.

  Examples: Everyday examples that illustrate a part of a plane are the flat surfaces of a floor, desk, windowpane, or book. Examples that are not part of a plane are the curved surfaces of a light bulb, a ball, or a tree.

b. Understand the meaning of parallel and perpendicular and use these terms to describe geometric figures.

• Lines and planes are called parallel if they do not meet no matter how far they are extended.
• Lines and planes are called perpendicular if the corners formed when they meet are equal.
• Identify parallel and perpendicular edges and surfaces in everyday settings (e.g., the classroom).

  Examples: The lines on the page of a notebook are parallel, as are the covers of a closed book. Corners of books, walls, and rectangular desks are perpendicular, as are the top and side edges of a chalk board and a wall and a floor in a classroom.

• The corner where two perpendicular lines meet is called a right angle.

  Note: The general concept of "angle" is developed later; here the term is used merely as the name for this specific and common configuration.

• Understand and use the terms vertical and horizontal.
• Recognize that vertical and horizontal lines or planes are perpendicular, but that perpendicular lines or planes are not necessarily vertical or horizontal.

c. Use terms such as line, plane, ray, line segment, parallel, perpendicular, and right angle correctly to describe everyday and geometric figures.

G.3.2 Identify and draw perpendicular and parallel lines and planes.

a. Draw perpendicular, parallel, and non-parallel line segments using rulers and squares.

• Recognize that lines that are parallel to perpendicular lines will themselves be perpendicular.

  Example: Fold a piece of paper in half from top to bottom, then fold it in half again from left to right. This will give two perpendicular fold lines and four right angles.

  

• Edges of a polygon are called parallel or perpendicular if they lie on parallel or perpendicular lines, respectively.
• Similarly, faces of a three-dimensional solid are called parallel or perpendicular if they lie in parallel or perpendicular planes, respectively.

G.3.3 Explore and identify familiar two- and three-dimensional shapes.

a. Describe and classify plane figures and solid shapes according to the number and shape of faces, edges, and vertices.

• Plane figures include circles, triangles, squares, rectangles, and other polygons; solid shapes include spheres, pyramids, cubes, and rectangular prisms.
• Recognize that the exact meaning of many geometric terms (e.g., rectangle, square, circle and triangle) depends on context: Sometimes they refer to the boundary of a region and sometimes to the region contained within the boundary.

b. Know how to put shapes together and take them apart to form other shapes.

Examples: Two identical right triangles can be arranged to form a rectangle. Two identical cubes can be arranged to form a rectangular prism. See figures below.

c. Identify edges, vertices (corners), perpendicular and parallel edges and right angles in two-dimensional shapes.

Example: A rectangle has four pairs of perpendicular edges, two pairs of parallel edges and four right angles.

d. Identify right angles, edges, vertices, perpendicular and parallel planes in three-dimensional shapes.

G.3.4 Understand how to measure length, area, and volume.

a. Understand that measurements of length, area, and volume are based on standard units.

• Fundamental units are: a unit interval of length 1 unit, a unit square whose sides have length 1 unit, and a unit cube whose sides have length 1 unit.
• The volume of a rectangular prism is the number of unit cubes required to fill it exactly (with no space left over).

  

  Note: The common childhood experience of pouring water or sand offers a direct representation of volume.

• The area of a rectangle is the number of unit squares required to pave the rectangle—that is, to cover it completely without any overlapping.

  Note: Area provides a critical venue for developing the conceptual underpinnings of multiplication.

• The length of a line segment is the number of unit intervals that are required to cover the segment exactly with nothing left over.

b. Know how to calculate the perimeter, area, and volume of shapes made from rectangles and rectangular prisms.

• The perimeter of a rectangle is the number of unit intervals that are required to enclose the rectangle.
• Measure and compare the areas of shapes using non-standard units (e.g., pieces in a set of pattern blocks).
• Recognize that the area of a rectangle is the product of the lengths of its base and height (A = b × h) and that the volume of a rectangular prism is the product of the lengths of its base, width, and height (V = b × w × h).

  Example: Build solids with unit cubes and use the formula for volume (V = bwh) to verify the count of unit cubes; make similar comparisons with rectilinear figures in the plane that are created from unit squares.

• Find the area of a complex figure by adding and subtracting areas.
• Compare rectangles of equal area and different perimeter and also rectangles of equal perimeter and different area.
• Measure surface area of solids by covering each face with copies of a unit square and then counting the total number of units.

Algebra (A)

A.3.1 Explore and understand arithmetic relationships among positive whole numbers .

a. Understand the inverse relationships between addition and subtraction and between multiplication and division, and the commutative laws of multiplication and addition.

• Show that subtraction and division are not commutative.

b. Find the unknown in simple equations that involve one or more of the four arithmetic operations.

Note: To emphasize the process of solving for an unknown, limit coefficients and solutions to small positive whole numbers.

Examples: 3 × ? = 3 + 6; ? ÷ 5 = 5 x 55; 36 = ? × ?.

c. Create, describe, explain and extend patterns based on numbers, operations, geometric objects and relationships.

• Explore both arithmetic (constant difference) and geometric (constant multiple) sequences.

  Examples: 100, 93, 86, 79, 72, . . .; 2, 4, 8, 16, . . .; 3, 9, 27, 81, . . . .

• Understand that patterns do not imply rules; rules imply patterns.

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