Standard 4:Mathematics

Mathematics Cumulative Progress Indicators (CPIs) for the end of the designated grade span

Place a "+" for an expectation that represents a strength & a "-" for a weakness

+ or -

STANDARD 4.1     (NUMBER AND NUMERICAL OPERATIONS)     ALL STUDENTS WILL DEVELOP NUMBER SENSE AND WILL PERFORM STANDARD NUMERICAL OPERATIONS AND ESTIMATIONS ON ALL TYPES OF NUMBERS IN A VARIETY OF WAYS.
 

A.     Number Sense
 

By the end of Grade 2, students will:
 

 1.         Use real-life experiences, physical materials, and technology to construct meanings for numbers (unless otherwise noted, all indicators for grade 2 pertain to these sets of numbers as well).

·        Whole numbers through hundreds

·        Ordinals

·        Proper fractions (denominators of 2, 3, 4, 8, 10)
 

 2.         Demonstrate an understanding of whole number place value concepts.

 

 3.         Understand that numbers have a variety of uses.

 

 4.         Count and perform simple computations with coins.

·        Amounts up to $1.00 (using cents notation)
 

 5.         Compare and order whole numbers.
 

Building upon knowledge and skills gained in preceding grades, by the end of Grade 3, students will:
 

 1.         Use real-life experiences, physical materials, and technology to construct meanings for numbers (unless otherwise noted, all indicators for grade 3 pertain to these sets of numbers as well).

·        Whole numbers through hundred thousands

·        Commonly used fractions (denominators of 2, 3, 4, 5, 6, 8, 10) as part of a whole, as a subset of a set, and as a location on a number line
 

 2.         Demonstrate an understanding of whole number place value concepts.
 

3.         Identify whether any whole number is odd or even.
 

 4.         Explore the extension of the place value system to decimals through hundredths.
 

 5.         Understand the various uses of numbers.

·        Counting, measuring, labeling (e.g., numbers on baseball uniforms)
 

 6.         Compare and order numbers.
 

Building upon knowledge and skills gained in preceding grades, by the end of Grade 4, students will:
 

 1.         Use real-life experiences, physical materials, and technology to construct meanings for numbers (unless otherwise noted, all indicators for grade 4 pertain to these sets of numbers as well).

·        Whole numbers through millions

·        Commonly used fractions (denominators of 2, 3, 4, 5, 6, 8, 10, 12, and 16) as part of a whole, as a subset of a set, and as a location on a number line

·        Decimals through hundredths
 

 2.         Demonstrate an understanding of place value concepts.

 

 3.         Demonstrate a sense of the relative magnitudes of numbers.
 

 4.         Understand the various uses of numbers.

·        Counting, measuring, labeling (e.g., numbers on baseball uniforms), locating (e.g., Room 235 is on the second floor)
 

 5.         Use concrete and pictorial models to relate whole numbers, commonly used fractions, and decimals to each other, and to represent equivalent forms of the same number.
 

 6.         Compare and order numbers.

 

 7.         Explore settings that give rise to negative numbers.

·        Temperatures below 0o, debts

·        Extension of the number line

 

Building upon knowledge and skills gained in preceding grades, by the end of Grade 5, students will:
 

 1.         Use real-life experiences, physical materials, and technology to construct meanings for numbers (unless otherwise noted, all indicators for grade 5 pertain to these sets of numbers as well).

·        All fractions as part of a whole, as subset of a set, as a location on a number line, and as divisions of whole numbers

·        All decimals

 

 2.         Recognize the decimal nature of United States currency and compute with money.
 

 3.         Demonstrate a sense of the relative magnitudes of numbers.
 

 4.         Use whole numbers, fractions, and decimals to represent equivalent forms of the same number.

 

 5.         Develop and apply number theory concepts in problem solving situations.

·        Primes, factors, multiples
 

 6.         Compare and order numbers.

 

Building upon knowledge and skills gained in preceding grades, by the end of Grade 6, students will:
 

 1.         Use real-life experiences, physical materials, and technology to construct meanings for numbers (unless otherwise noted, all indicators for grade 6 pertain to these sets of numbers as well).

·        All integers

·        All fractions as part of a whole, as subset of a set, as a location on a number line, and as divisions of whole numbers

·        All decimals

 

 2.         Recognize the decimal nature of United States currency and compute with money.
 

 3.         Demonstrate a sense of the relative magnitudes of numbers.
 

 4.         Explore the use of ratios and proportions in a variety of situations.
 

 5.         Understand and use whole-number percents between 1 and 100 in a variety of situations.
 

 6.         Use whole numbers, fractions, and decimals to represent equivalent forms of the same number.

 

 7.         Develop and apply number theory concepts in problem solving situations.

·        Primes, factors, multiples

·        Common multiples, common factors
 

 8.         Compare and order numbers.

 

Building upon knowledge and skills gained in preceding grades, by the end of Grade 7, students will:
 

 1.         Extend understanding of the number system by constructing meanings for the following (unless otherwise noted, all indicators for grade 7 pertain to these sets of numbers as well):

·        Rational numbers

·        Percents

·        Whole numbers with exponents

 

 2.         Demonstrate a sense of the relative magnitudes of numbers.
 

 3.         Understand and use ratios, proportions, and percents (including percents greater than 100 and less than 1) in a variety of situations.
 

 4.         Compare and order numbers of all named types.
 

 5.         Use whole numbers, fractions, decimals, and percents to represent equivalent forms of the same number.
 

 6.         Understand that all fractions can be represented as repeating or terminating decimals.
 

Building upon knowledge and skills gained in preceding grades, by the end of Grade 8, students will:
 

 1.         Extend understanding of the number system by constructing meanings for the following (unless otherwise noted, all indicators for grade 8 pertain to these sets of numbers as well):

·        Rational numbers

·        Percents

·        Exponents

·        Roots

·        Absolute values

·        Numbers represented in scientific notation

 

 2.         Demonstrate a sense of the relative magnitudes of numbers.
 

 3.         Understand and use ratios, proportions, and percents (including percents greater than 100 and less than 1) in a variety of situations.
 

 4.         Compare and order numbers of all named types.
 

 5.         Use whole numbers, fractions, decimals, and percents to represent equivalent forms of the same number.
 

 6.         Recognize that repeating decimals correspond to fractions and determine their fractional equivalents.

·        5/7 = 0. 714285714285…  =  0.
 

 7.         Construct meanings for common irrational numbers, such as p (pi) and the square root of 2.
 

Building upon knowledge and skills gained in preceding grades, by the end of Grade 12, students will:
 

 1.         Extend understanding of the number system to all real numbers.

 

 2.         Compare and order rational and irrational numbers.
 

 3.         Develop conjectures and informal proofs of properties of number systems and sets of numbers.
 

B.     Numerical Operations
 

By the end of Grade 2, students will:
 

 1.         Develop the meanings of addition and subtraction by concretely modeling and discussing a large variety of problems.

·        Joining, separating, and comparing

 

 2.         Explore the meanings of multiplication and division by modeling and discussing problems.
 

3.         Develop proficiency with basic addition and subtraction number facts using a variety of fact strategies (such as “counting on” and “near doubles”) and then commit them to memory.
 

 4.         Construct, use, and explain procedures for performing addition and subtraction calculations with:

·        Pencil-and-paper

·        Mental math

·        Calculator
 

 5.         Use efficient and accurate pencil-and-paper procedures for computation with whole numbers.

·        Addition of 2-digit numbers

·        Subtraction of 2-digit numbers
 

 6.         Select pencil-and-paper, mental math, or a calculator as the appropriate computational method in a given situation depending on the context and numbers.
 

 7.         Check the reasonableness of results of computations.
 

 8.         Understand and use the inverse relationship between addition and subtraction.
 

Building upon knowledge and skills gained in preceding grades, by the end of Grade 3, students will:
 

 1.         Develop the meanings of the four basic arithmetic operations by modeling and discussing a large variety of problems.

·        Addition and subtraction:  joining, separating, comparing

·        Multiplication:  repeated addition, area/array

·        Division:  repeated subtraction, sharing
 

 2.         Develop proficiency with basic multiplication and division number facts using a variety of fact strategies (such as “skip counting” and “repeated subtraction”).
 

 3.         Construct, use, and explain procedures for performing whole number calculations with:

·        Pencil-and-paper

·        Mental math

·        Calculator
 

 4.         Use efficient and accurate pencil-and-paper procedures for computation with whole numbers.

·        Addition of 3-digit numbers

·        Subtraction of 3-digit numbers

·        Multiplication of 2-digit numbers by 1-digit numbers
 

5.         Count and perform simple computations with money.

·        Cents notation (˘)
 

 6.         Select pencil-and-paper, mental math, or a calculator as the appropriate computational method in a given situation depending on the context and numbers.
 

 7.         Check the reasonableness of results of computations.
 

Building upon knowledge and skills gained in preceding grades, by the end of Grade 4, students will:
 

  1.         Develop the meanings of the four basic arithmetic operations by modeling and discussing a large variety of problems.

·        Addition and subtraction:  joining, separating, comparing

·        Multiplication:  repeated addition, area/array

·        Division: repeated subtraction, sharing
 

 2.         Develop proficiency with basic multiplication and division number facts using a variety of fact strategies (such as “skip counting” and “repeated subtraction”) and then commit them to memory.

 

 3.         Construct, use, and explain procedures for performing whole number calculations and with:

·        Pencil-and-paper

·        Mental math

·        Calculator
 

4.         Use efficient and accurate pencil-and-paper procedures for computation with whole numbers.

·        Addition of 3-digit numbers

·        Subtraction of 3-digit numbers

·        Multiplication of 2-digit numbers

·        Division of 3-digit numbers by 1-digit numbers
 

 5.         Construct and use procedures for performing decimal addition and subtraction.
 

 6.         Count and perform simple computations with money.

·        Standard dollars and cents notation
 

 7.         Select pencil-and-paper, mental math, or a calculator as the appropriate computational method in a given situation depending on the context and numbers.
 

 8.         Check the reasonableness of results of computations.
 

 9.         Use concrete models to explore addition and subtraction with fractions.

 

 10.     Understand and use the inverse relationships between addition and subtraction and between multiplication and division.
 

Building upon knowledge and skills gained in preceding grades, by the end of Grade 5, students will:
 

 1.         Recognize the appropriate use of each arithmetic operation in problem situations.
 

 2.         Construct, use, and explain procedures for performing addition and subtraction with fractions and decimals with:

·        Pencil-and-paper

·        Mental math

·        Calculator
 

 3.         Use an efficient and accurate pencil-and-paper procedure for division of a 3-digit number by a 2-digit number.
 

 4.         Select pencil-and-paper, mental math, or a calculator as the appropriate computational method in a given situation depending on the context and numbers.
 

 5.         Check the reasonableness of results of computations.
 

 6.         Understand and use the various relationships among operations and properties of operations.
 

Building upon knowledge and skills gained in preceding grades, by the end of Grade 6, students will:
 

 1.         Recognize the appropriate use of each arithmetic operation in problem situations.
 

 2.         Construct, use, and explain procedures for performing calculations with fractions and decimals with:

·        Pencil-and-paper

·        Mental math

·        Calculator
 

3.         Use an efficient and accurate pencil-and-paper procedure for division of a 3-digit number by a 2-digit number.
 

 4.         Select pencil-and-paper, mental math, or a calculator as the appropriate computational method in a given situation depending on the context and numbers.
 

 5.         Find squares and cubes of whole numbers.

 

 6.         Check the reasonableness of results of computations.
 

 7.         Understand and use the various relationships among operations and properties of operations.
 

 8.         Understand and apply the standard algebraic order of operations for the four basic operations, including appropriate use of parentheses.
 

Building upon knowledge and skills gained in preceding grades, by the end of Grade 7, students will:
 

 1.         Use and explain procedures for performing calculations with integers and all number types named above with:

·        Pencil-and-paper

·        Mental math

·        Calculator
 

 2.         Use exponentiation to find whole number powers of numbers.
 

 3.         Understand and apply the standard algebraic order of operations, including appropriate use of parentheses.
 

Building upon knowledge and skills gained in preceding grades, by the end of Grade 8, students will:
 

 1.         Use and explain procedures for performing calculations involving addition, subtraction, multiplication, division, and exponentiation with integers and all number types named above with:

·        Pencil-and-paper

·        Mental math

·        Calculator
 

 2.         Use exponentiation to find whole number powers of numbers.
 

 3.         Find square and cube roots of numbers and understand the inverse nature of powers and roots.
 

 4.         Solve problems involving proportions and percents.
 

 5.         Understand and apply the standard algebraic order of operations, including appropriate use of parentheses.
 

Building upon knowledge and skills gained in preceding grades, by the end of Grade 12, students will:
 

1.         Extend understanding and use of operations to real numbers and algebraic procedures.
 

 2.         Develop, apply, and explain methods for solving problems involving rational and negative exponents.
 

 3.         Perform operations on matrices.

·        Addition and subtraction

·        Scalar multiplication

 

 4.         Understand and apply the laws of exponents to simplify expressions involving numbers raised to power
 

C.     Estimation
 

By the end of Grade 2, students will:
 

 1.         Judge without counting whether a set of objects has less than, more than, or the same number of objects as a reference set.
 

 2.         Determine the reasonableness of an answer by estimating the result of computations (e.g., 15 + 16 is not 211).
 

 3.         Explore a variety of strategies for estimating both quantities (e.g., the number of marbles in a jar) and results of computation.
 

 Building upon knowledge and skills gained in preceding grades, by the end of Grade 3, students will:
 

 1.         Judge without counting whether a set of objects has less than, more than, or the same number of objects as a reference set.
 

 2.         Construct and use a variety of estimation strategies (e.g., rounding and mental math) for estimating both quantities and the result of computations.
 

 3.         Recognize when an estimate is appropriate, and understand the usefulness of an estimate as distinct from an exact answer.
 

 4.         Use estimation to determine whether the result of a computation (either by calculator or by hand) is reasonable.
 

Building upon knowledge and skills gained in preceding grades, by the end of Grade 4, st