Elementary Mathematics Benchmarks, Grade 2
Number (N)
N.2.1 Understand and use number notation and place value up to 1,000.
a. Count by ones, twos, fives, tens, and hundreds.
- 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.
b. Read and write numbers up to 1,000 in numerals and in words.
- Up to 1,000, read and write numerals and understand and speak words; write words up to 100.
c. Recognize the place values of numbers (hundreds, tens, ones).
- 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.
d. Understand and utilize the relative values of the different number places.
- 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.
e. Compare numbers up to 1,000.
N.2.2 Locate and interpret numbers on the number line.
a. Recognize the continuous interpretation of the number line where points correspond to distances from the origin (zero).
- 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.
b. Use number line pictures and manipulatives to illustrate addition and subtraction as the adding and subtracting of lengths.
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.
c. Understand the symbol ½ and the word half as signifying lengths and positions on the number line that are midway between two whole numbers.
- Read foot and inch rulers with uneven hash marks to the nearest half inch.
N.2.3 Add, subtract, and use numbers up to 1,000.
a. Add and subtract two- and three-digit numbers with efficiency and understanding.
- 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.
b. Understand "related facts" associated with adding and subtracting.
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.
c. Create stories, make drawings, and solve problems that illustrate addition and subtraction with unknowns of various types.
- 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.
d. Solve problems that require more than one step and that use numbers below 50.
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.
N.2.4 Understand multiplication as repeated addition and division as the inverse of multiplication.
a. Multiply small whole numbers by repeated addition.
- 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.
b. Understand division as the inverse of multiplication.
- 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.
c. Know the multiplication table up to 5 × 5.
- 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.
d. Solve multiplication and division problems involving repeated groups and arrays of small whole numbers.
- 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.
Measurement (M)
M.2.1 Add, subtract, compare, and estimate measurements.
a. Estimate, measure, and calculate length in meters, centimeters, yards, feet, and inches.
- Recognize and use standard abbreviations: m, cm, yd, ft, and in, as well as the symbolic notation 3'6".
- Understand and use units appropriate to particular situations.
Example: Standard U.S .school notepaper is sized in inches, not centimeters.
- Add and subtract mixed metric units (e.g., 8m,10cm + 3m,5cm) but defer calculation with mixed English units (e.g., 3ft,1in + 1ft,8in) until third grade.
Note: Conversion between systems awaits a later grade.
b. Measure the lengths of sides and diagonals of common two-dimensional figures such as triangles, rectangles (including squares), and other polygons.
- Measure to the nearest centimeter or half inch using meter sticks, yardsticks, rulers, and tape measures marked in either metric or English units.
Note: Measure within either system without conversion between systems.
- Create and use hand-made rulers by selecting an unconventional unit length (e.g., a hand-width), marking off unit and half-unit lengths.
- Explore a variety of ways to measure perimeter and circumference.
Examples: Encircle with a tape measure; measure and sum various pieces; wrap with a string and then measure the length of the string. Compare answers obtained by different strategies and explain any differences.
Note: Comparing the result of a direct measurement (encircling) with that of adding component pieces underscores the importance of accuracy and serves as a prelude to understanding the significance of significant digits.
c. Estimate and measure weight and capacity in common English and metric units.
- Recognize, use, and estimate common measures of volume (quarts, liters, cups, gallons) and weight (pound, kilogram).
- Understand and use common expressions such as half a cup or quarter of a pound that represent fractional parts of standard units of measurement.
d. Compare lengths, weights, and capacities of pairs of objects.
- Demonstrate that the combined length of the shorter pieces from two pairs of rods is shorter than the combined lengths of the two longer pieces.
- Recognize that the same applies to combined pairs of weights or volumes.
Note: Even though this relation may seem obvious, it is an important demonstration of the fundamental relation between addition and order, namely, that if a ≤ b and c ≤ d, then a + c ≤ b + d.
Probability and Statistics (PS)
PS.2.1 Tell, estimate, and calculate with time.
a. Tell, write, and use time measurements from analog (round) clock faces and from digital clocks and translate between the two.
- Round off to the nearest five minutes.
- Understand and use different ways to read time, e.g., "nine fifteen" or "quarter past nine," "nine fifty" or "ten to ten."
- Understand a.m. and p.m.
b. Understand the meaning of time as an interval and be able to estimate the passage of time without clock measurement.
c. Understand and use comparative phrases such as "in fifteen minutes," "half an hour from now," "ten minutes late."
PS.2.2 Count, add, and subtract money.
a. Read, write, add, and subtract money up to 10 dollars.
- Handle money accurately and make change for amounts of $10 or less by counting up.
- Use the symbols $ and ¢ properly.
- Recognize and use conventional ("decimal") monetary notation and translate back and forth into $ and ¢ notation.
- Add and subtract monetary amounts in both $ and ¢ and conventional notation.
- Use a calculator to check monetary calculations and also to add lists of three or more amounts.
- Estimate answers to check for reasonableness of hand or calculator methods.
PS.2.3 Represent measurements by means of bar graphs.
a. Collect data and record them in systematic form.
b. Select appropriate scales for a graph and make them explicit in labels.
- Employ both horizontal and vertical configurations.
- Recognize an axis with a scale as a representation of the number line.
- Compare scales on different graphs.
- Use addition and subtraction as appropriate to translate data (gathered or provided) into measurements required to construct a graph.
c. Create and solve problems that require interpretation of bar or picture graphs.
Geometry (G)
G.2.1 Recognize, classify, and transform geometric figures in two and three dimensions.
a. Identify, describe, and compare common geometric shapes in two and three dimensions.
- Define a general triangle and identify isosceles, equilateral, right, and obtuse special cases.
Note: The goal of naming triangles is not the names themselves but to focus on important differences. Triangles (or quadrilaterals) are not all alike, and it is their differences that give them distinctive mathematical features.
- Identify various quadrilaterals (rectangles, trapezoids, parallelograms, squares) as well as pentagons and hexagons.
Note: In this grade parallel is used informally and intuitively; it receives more careful treatment at a later grade.
Note: A square is a special kind of rectangle (since it has four sides and four right angles); a rectangle is a special kind of parallelogram (since it has four sides and two pairs of parallel sides); and a parallelogram is a special kind of trapezoid (since it has four sides and at least one pair of parallel sides); and a trapezoid is a polygon (since it is a figure formed of several straight sides). So for example, contrary to informal usage, in mathematics, a square is a trapezoid.
- Understand the terms perimeter and circumference.
Note: The primary meaning of both terms is the outer boundary of a two-dimensional figure; circumference is used principally in reference to circles. A secondary meaning for both is the length of the outer boundary. Which meaning is intended needs to be determined from context.
- Distinguish circles from ovals; recognize the circumference, diameter, and radius of a circle.
- In three dimensions, identify spheres, cones, cylinders, and triangular, and rectangular prisms.
b. Describe common geometric attributes of familiar plane and solid objects.
- Common geometric attributes include position, shape, size, and roundness, and numbers of corners, edges, and faces.
- Distinguish between geometric attributes and other characteristics such as weight, color, or construction material.
- Distinguish between lines and curves and between flat and curved surfaces.
c. Rotate, flip, and fold shapes to explore the effect of transformations.
- Use paper folding to find lines of symmetry.
- Recognize congruent shapes.
- Identify shapes that have been moved (flipped, slid, rotated), enlarged, or reduced.
G.2.2 Understand and interpret rectangular arrays as a model of multiplication.
a. Create square cells from segments of the discrete number line used as sides of a rectangle.
- Match cells to discrete objects lined up in regular rows of the same length.
b. Understand rectangular arrays as instances of repeated addition.
Algebra (A)
A.2.1 Create, identify, describe, and extend patterns.
a. Fill in tables based on stated rules to reveal patterns.
- Find patterns in both arithmetic and geometric contexts.
b. Record and study patterns in lists of numbers created by repeated addition or subtraction.
- Create patterns mentally (by counting up and down), by hand (with paper and pencil), and by repeated action on a calculator.
Examples: 3, 8, 13, 18, 23, . . .; 50, 46, 42, 38, 34, . . .
A.2.2 Find unknowns in simple arithmetic problems.
a. Solve equations and problems involving addition, subtraction, and multiplication with the unknown in any position.
Note: In the early grades, it is better to signify the unknown with a symbol, such as [ ], ?, or __, that carries the connotation of unknown rather than with an alphabetic letter such as x.
b. Understand and use the facts that addition and multiplication are commutative and associative.
- Use parentheses to clarify groupings and order of operation.
- Recognize terms such as commutative and associative.
Note: It is not necessary for children at this grade to use or write these words, merely to recognize them orally and to know the properties to which they refer.
c. Recognize how multiplication and division are, like addition and subtraction, inverse operations.
A.2.3 Understand basic properties of odd and even numbers.
a. Explain why the sum of two even numbers is even and that the sum of two odd numbers is also even.
- Use diagrams to represent even and odd numbers and to explain their behavior.
Example: The representation at the right shows that 14 is even and 13 is odd.
