Standard 2 : Number and Operations (Mathematics)

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View this standard as a learning progression
Conceptual Strand
The Number and Operations Standard describes deep and fundamental understanding of, and proficiency with, counting, numbers, and arithmetic, as well as an understanding of number systems and their structures.
Guiding Question
How do students develop number sense that allows them to naturally decompose numbers, use particular numbers as referents, solve problems using the relationships among operations and knowledge about the base-ten system, estimate a reasonable result for a problem, and have a disposition to make sense of numbers, problems, and results?
 
Grade K
Grade Learning Expectations
GLE 0006.2.1  Count objects in a set and use numbers, including written numerals to 25.
GLE 0006.2.2  Create, represent and recognize a set with a given number of objects.
GLE 0006.2.3  Recognize, compare and order sets of numerals by using both cardinal and ordinal meanings.
GLE 0006.2.4  Understand addition as “putting together” and subtraction as “breaking apart.”
GLE 0006.2.5  Model the numbers 1 through 10 as sums or differences of different sets of whole numbers (composing and decomposing numbers).
Checks for Understanding
0006.2.1  Count objects to 25 using one-to-one correspondence and identify the quantity in the counted group.
0006.2.2  Match quantities to 25 with numerals and written words.
0006.2.3  Count backward from 10 to 1.
0006.2.4  Count to 20 by twos.
0006.2.5  Create a set with a given number of objects.
0006.2.6  Quickly recognize the number of objects in a small set.
0006.2.7  Recognize zero (0) as a set with “no objects”.
0006.2.8  Compare sets of ten or fewer objects and identify which are equal to, more than, or less than others.
0006.2.9  Order the numbers through 25 using numerals and words.
0006.2.10  Recognize 6 through 10 as “five and some ones.”
0006.2.11  Recognize and use ordinal numbers (e.g., first, fourth, last).
0006.2.12  Model simple joining and separating situations with objects.
0006.2.13  Add and subtract single-digit numbers whose total or difference is between 0 and 10.
0006.2.14  Understand add as “put together” or “count on” and solve addition problems with sums less than 20.
0006.2.15  Understand subtraction as “break apart” or “take away” and solve subtraction problems using numbers 1 through 10.
0006.2.16  Model, demonstrate, and solve story problems that illustrate addition and subtraction.
0006.2.17  Understand that numbers can be represented by different groupings.
State Performance Indicators
 
Grade 1
Grade Learning Expectations
GLE 0106.2.1  Understand and use number notation and place value to 100.
GLE 0106.2.2  Compare and order whole numbers to 100.
GLE 0106.2.3  Develop strategies for learning basic addition facts and related subtraction facts.
GLE 0106.2.4  Use multiple representations (including groups of ten) to model two-digit addition and subtraction.
Checks for Understanding
0106.2.1  Read and write numerals up to 100.
0106.2.2  Write numbers up to 10 in words.
0106.2.3  Count forward and backward by ones beginning with any number less than 100.
0106.2.4  Skip count by twos, fives, and tens.
0106.2.5  Order and compare (less than, greater than, or equal to) whole numbers to 100.
0106.2.6  Recognize the place value of numbers (tens, ones).
0106.2.7  Develop fluency with addition and subtraction facts of sums through ten.
0106.2.8  Relate “counting on” and “counting back” to addition and subtraction and understand them as inverse operations.
0106.2.9  Add three single-digit numbers.
0106.2.10  Use models (such as discrete objects, connecting cubes, and number lines) to represent “part-whole,” “adding to,” “taking away from,” and “comparing to” situations to develop understanding of the meaning of addition and subtraction.
0106.2.11  Recognize the “part-whole” relationship in representations of basic fractions such as ½ and ¼.
0106.2.12  Use various models to develop strategies for solving arithmetic problems.
0106.2.13  Solve problems that require addition and subtraction of numbers through 100.
0106.2.14  Use composition and decomposition of numbers to identify and discuss patterns.
0106.2.15  Represent whole numbers between 10 and 100 in groups of tens and ones.
0106.2.16  Represent whole numbers up to 100 on a number line.
0106.2.17  Use the number line to create visual representations of sequences (such as even numbers, tens, multiples of five).
State Performance Indicators
 
Grade 2
Grade Learning Expectations
GLE 0206.2.1  Understand and use place value concepts to 1000.
GLE 0206.2.2  Understand and use the base-ten numeration system.
GLE 0206.2.3  Use efficient and accurate strategies to develop fluency with multi-digit addition and subtraction.
GLE 0206.2.4  Develop an initial understanding of multiplication.
Checks for Understanding
0206.2.1  Starting at any number, count by ones, twos, fives, tens, and hundreds up to 1000.
0206.2.2  Read and write numbers up to 1000 using numerals and up to 100 using words.
0206.2.3  Locate and interpret numbers on a number line.
0206.2.4  Recognize that place-value notation represents the sums of multiples of powers of ten (e.g., 853 as 8 hundreds + 5 tens + 3 ones).
0206.2.5  Compare and order multi-digit numbers up to 1000.
0206.2.6  Use various models such as number lines, pictures, and base-ten blocks to illustrate addition and subtraction.
0206.2.7  Develop fluency at recalling basic addition facts and related subtraction facts.
0206.2.8  Use efficient procedures, and understand why they work, to solve problems involving the addition and subtraction of two- and three-digit whole numbers (including those that require regrouping).
0206.2.9  Apply appropriate methods to estimate and mentally calculate sums or differences with ones, tens, and hundreds.
0206.2.10  Add three two-digit numbers.
0206.2.11  Solve addition and subtraction problems in context using various representations.
0206.2.12  Demonstrate skip counting on the number line and relate to repeated addition and multiplication.
0206.2.13  Relate patterns in skip counting to multiplication.
State Performance Indicators
 
Grade 3
Grade Learning Expectations
GLE 0306.2.1  Understand the place value of whole numbers to ten-thousands place including expanded notation for all arithmetic operations.
GLE 0306.2.2  Develop understanding of multiplication and related division facts through multiple strategies and representations.
GLE 0306.2.3  Relate multiplication and division as inverse operations.
GLE 0306.2.4  Solve multiplication and division problems using various representations.
GLE 0306.2.5  Understand the meaning and uses of fractions.
GLE 0306.2.6  Use various strategies and models to compare and order fractions and identify equivalent fractions.
GLE 0306.2.7  Add and subtract fractions with like denominators using various models.
Checks for Understanding
0306.2.1  Represent whole numbers up to 10,000 using various models (such as base-ten blocks, number lines, place-value charts) and in standard form, written form, and expanded form.
0306.2.2  Understand and use the symbols =, < and > to signify order and comparison.
0306.2.3  Use parentheses to indicate grouping.
0306.2.4  Use a variety of methods to perform mental computations and compare the efficiency of those methods.
0306.2.5  Use highest order value (such as tens or hundreds digit) to make simple estimates.
0306.2.6  Solve a variety of addition and subtraction story problems including those with irrelevant information.
0306.2.7  Represent multiplication using various representations such as equal-size groups, arrays, area models, and equal jumps on number lines.
0306.2.8  Represent division using various representations such as successive subtraction, the number of equal jumps, partitioning, and sharing.
0306.2.9  Describe contexts for multiplication and division facts.
0306.2.10  Understand that symbols such as ½, 1/3, and ¼ represent numbers called unit fractions.
0306.2.11  Identify fractions as parts of whole units, as parts of sets, as locations on number lines, and as division of two whole numbers.
0306.2.12  Compare fractions using drawings, concrete objects, and benchmark fractions.
0306.2.13  Understand that when a whole is divided into equal parts to create unit fractions, the sum of all the parts adds up to one.
State Performance Indicators
SPI 0306.2.1  Read and write numbers up to 10,000 in numerals and up to 1,000 in words.
SPI 0306.2.2  Identify the place value of numbers in the ten-thousands, thousands, hundreds, tens, and ones positions.
SPI 0306.2.3  Convert between expanded and standard form with whole numbers to 10,000.
SPI 0306.2.4  Compare and order numbers up to 10,000 using the words less than, greater than, and equal to, and the symbols <, >, =.
SPI 0306.2.5  Identify various representations of multiplication and division.
SPI 0306.2.6  Recall basic multiplication facts through 10 times10 and the related division facts.
SPI 0306.2.7  Compute multiplication problems that involve multiples of ten using basic number facts.
SPI 0306.2.8  Solve problems that involve the inverse relationship between multiplication and division.
SPI 0306.2.9  Solve contextual problems involving the addition (with and without regrouping) and subtraction (with and without regrouping) of two- and three digit whole numbers.
SPI 0306.2.10  Identify equivalent fractions given by various representations.
SPI 0306.2.11  Recognize and use different interpretations of fractions.
SPI 0306.2.12  Name fractions in various contexts that are less than, equal to, or greater than one.
SPI 0306.2.13  Recognize, compare, and order fractions (benchmark fractions, common numerators, or common denominators).
SPI 0306.2.14  Add and subtract fractions with like denominators.
 
Grade 4
Grade Learning Expectations
GLE 0406.2.1  Understand place value of numbers from hundredths to the hundred-thousands place.
GLE 0406.2.2  Develop fluency with multiplication and single-digit division.
GLE 0406.2.3  Identify prime and composite numbers.
GLE 0406.2.4  Understand and use the connections between fractions and decimals.
GLE 0406.2.5  Add and subtract fractions with like and unlike denominators.
GLE 0406.2.6  Solve problems involving whole numbers, fractions, and/or decimals using all four arithmetic operations.
Checks for Understanding
0406.2.1  Compose and decompose quantities according to place value.
0406.2.2  Understand decimal notation as an extension of the base-ten number system.
0406.2.3  Multiply two- and three-digit whole numbers.
0406.2.4  Understand and use a reliable algorithm for multiplying multi-digit numbers and dividing numbers by a single-digit divisor accurately and efficiently.
0406.2.5  Understand that division by zero is undefined.
0406.2.6  Divide three-digit whole numbers by one-digit divisors fluently with pencil and paper.
0406.2.7  Identify factors of whole numbers and model factors and products beyond basic multiplication facts using arrays and area models.
0406.2.8  Generate equivalent forms of whole numbers, decimals, and common fractions (e.g., 1/10, ¼, ½, ¾).
0406.2.9  Compare equivalent forms whole numbers, fractions, and decimals to each other and to benchmark numbers.
0406.2.10  Use models to understand division as the inverse of multiplication, partitioning, and repeated subtraction.
0406.2.11  Use models, benchmarks, and equivalent forms to compare fractions/decimals and locate them on the number line.
0406.2.12  Understand and use decimal numbers up to hundredths and write them as fractions.
0406.2.13  Solve multi-step problems of various types using whole numbers, fractions, and decimals.
0406.2.14  Understand the role of the remainder in division.
State Performance Indicators
SPI 0406.2.1  Read and write numbers from hundredths to hundred-thousands in numerals and in words.
SPI 0406.2.2  Locate and place mixed numbers on the number line.
SPI 0406.2.3  Identify the place value of a specified digit in a number and the quantity it represents.
SPI 0406.2.4  Find factors, common factors, multiples, and common multiples of two numbers.
SPI 0406.2.5  Generate equivalent forms of common fractions and decimals and use them to compare size.
SPI 0406.2.6  Use the symbols < , > and = to compare common fractions and decimals in both increasing and decreasing order.
SPI 0406.2.7  Convert improper fractions into mixed numbers and/or decimals.
SPI 0406.2.8  Add and subtract proper fractions with like and unlike denominators and simplify the answer.
SPI 0406.2.9  Add and subtract decimals through hundredths.
SPi 0406.2.10  Solve contextual problems using whole numbers, fractions, and decimals.
SPI 0406.2.11  Solve problems using whole number multi-digit multiplication.
SPI 0406.2.12  Solve problems using whole number division with one- or two-digit divisors.
 
Grade 5
Grade Learning Expectations
GLE 0506.2.1  Extend the understanding of place value through millions and millionths in various contexts and representations.
GLE 0506.2.2  Write natural numbers (to 50) as a product of prime factors and understand that this is unique (apart from order).
GLE 0506.2.3  Develop fluency with division of whole numbers. Understand the relationship of divisor, dividend, and quotient in terms of multiplication and division.
GLE 0506.2.4  Develop fluency with addition and subtraction of proper and improper fractions and mixed numbers; explain and model the algorithm.
GLE 0506.2.5  Develop fluency in solving multi-step problems using whole numbers, fractions, mixed numbers, and decimals.
Checks for Understanding
0506.2.1  Identify prime numbers up to 50.
0506.2.2  Use the prime factorization of two whole numbers to determine the greatest common factor and the least common multiple.
0506.2.3  Use visual models, benchmarks, and equivalent forms to add and subtract commonly used fractions and decimals.
0506.2.4  Use divisibility rules to factor numbers.
0506.2.5  Make reasonable estimates of fraction and decimal sums and differences.
0506.2.6  Add and subtract mixed numbers.
0506.2.7  Understand the placement of the decimal point in calculations of multiplication and long division, including the placement in the estimation of the answer.
0506.2.8  Understand that division by zero is undefined.
0506.2.9  Explore numbers less than 0 by extending the number line through familiar applications (e.g., temperatures below zero, owing money, measuring elevation below sea level).
0506.2.10  Use exponential notation to represent repeated multiplication of whole numbers.
State Performance Indicators
SPI 0506.2.1  Read and write numbers from millions to millionths in various contexts.
SPI 0506.2.2  Write the prime factorization of numbers through 50 using both exponential and standard notation.
SPI 0506.2.3  Select a reasonable solution to a real-world division problem in which the remainder must be considered.
SPI 0506.2.4  Solve problems involving the division of two- and three-digit whole numbers by one- and two-digit whole numbers.
SPI 0506.2.5  Solve addition and subtraction problems involving both fractions and decimals.
SPI 0506.2.6  Add and subtract proper and improper fractions as well as mixed numbers.
SPI 0506.2.7  Recognize equivalent representations for the same number.
SPI 0506.2.8  Write terminating decimals in the form of fractions or mixed numbers.
SPI 0506.2.9  Compare whole numbers, decimals and fractions using the symbols <, >, and =.
 
Grade 6
Grade Learning Expectations
GLE 0606.2.1  Understand and explain the procedures for multiplication and division of fractions, mixed numbers, and decimals.
GLE 0606.2.2  Solve multi-step mathematical, contextual and verbal problems using fractions, mixed numbers, and decimals.
GLE 0606.2.3  Understand and use ratios, rates and percents.
GLE 0606.2.4  Understand and convert between fraction, decimal, and percent forms of rational numbers.
GLE 0606.2.5  Develop meaning for integers; represent and compare quantities with integers.
Checks for Understanding
0606.2.1  Efficiently compare and order fractions, decimals and percents; determine their approximate locations on a number line.
0606.2.2  Use area models to represent multiplication of fractions.
0606.2.3  Create and solve contextual problems that lead naturally to division of fractions.
0606.2.4  Understand ratio as a fraction used to compare two quantities by division.
0606.2.5  Recognize a:b, a/b, and “a to b” as notations for ratios.
0606.2.6  Recognize common percentages as ratios based on fractions whose denominators are 2, 3, 4, 5, or 10.
0606.2.7  Connect ratio and rate to multiplication and division.
0606.2.8  Recognize that a terminating decimal equals a fraction with a denominator that is a power of ten.
0606.2.9  Recognize that the decimal form of a rational number either terminates or repeats.
0606.2.10  Explore contexts that can be described with negative numbers (such as money, elevation, and temperature).
State Performance Indicators
SPI 0606.2.1  Solve problems involving the multiplication and division of fractions.
SPI 0606.2.2  Solve problems involving the addition, subtraction, multiplication, and division of mixed numbers.
SPI 0606.2.3  Solve problems involving the addition, subtraction, multiplication, and division of decimals.
SPI 0606.2.4  Solve multi-step arithmetic problems using fractions, mixed numbers, and decimals.
SPI 0606.2.5  Transform numbers from one form to another (fractions, decimals, percents, and mixed numbers).
SPI 0606.2.6  Solve problems involving ratios, rates and percents.
SPI 0606.2.7  Locate positive rational numbers on the number line.
SPI 0606.2.8  Locate integers on the number line.
 
Grade 7
Grade Learning Expectations
GLE 0706.2.1  Extend understandings of addition, subtraction, multiplication and division to integers.
GLE 0706.2.2  Understand and work with the properties of and operations on the system of rational numbers.
GLE 0706.2.3  Develop an understanding of and apply proportionality.
GLE 0706.2.4  Use ratios, rates and percents to solve single- and multi-step problems in various contexts.
GLE 0706.2.5  Understand and work with squares, cubes, square roots and cube roots.
GLE 0706.2.6  Introduce the concept of negative exponents.
GLE 0706.2.7  Understand and use scientific notation.
Checks for Understanding
0706.2.1  Understand that the set of rational numbers includes any number that can be written as a ratio of two integers in which the denominator is not zero.
0706.2.2  Develop and analyze algorithms and compute efficiently with integers and rational numbers.
0706.2.3  Recognize that rational numbers satisfy the commutative and associative laws of addition and multiplication and the distributive law.
0706.2.4  Understand that a and –a are additive inverses and are located the same distance from zero on the number line; relate distance from zero to absolute value.
0706.2.5  Understand that –(–a) = a for any number a.
0706.2.6  Use the number line to demonstrate addition and subtraction with integers.
0706.2.7  Write number sentences to solve contextual problems involving ratio and percent.
0706.2.8  Apply ratios, rates, proportions and percents (such as discounts, interest, taxes, tips, distance/rate/time, and percent increase or decrease).
0706.2.9  Efficiently compare and order rational numbers and roots of perfect squares/cubes; determine their approximate locations on a number line.
0706.2.10  Recognize that when a whole number is not a perfect square, then its square root is not rational and cannot be written as the ratio of two integers.
0706.2.11  Estimate square/cube roots and use calculators to find approximations.
0706.2.12  Recognize that the square root of m times n equals the square root of m plus the square root of n and that the square root of m squared equals m.
0706.2.13  Use the meaning of negative exponents to represent small numbers; translate between scientific and standard notation.
0706.2.14  Express numbers in scientific notation and recognize its importance in representing the magnitude of a number.
0706.2.15  Report results of calculations appropriately in a given context (i.e. using rules of rounding, degree of accuracy, and/or significant digits).
State Performance Indicators
SPI 0706.2.1  Simplify numerical expressions involving rational numbers.
SPI 0706.2.2  Compare rational numbers using appropriate inequality symbols.
SPI 0706.2.3  Use rational numbers and roots of perfect squares/cubes to solve contextual problems.
SPI 0706.2.4  Determine the approximate location of square/cube roots on a number line.
SPI 0706.2.5  Solve contextual problems that involve operations with integers.
SPI 0706.2.6  Express the ratio between two quantities as a percent, and a percent as a ratio or fraction.
SPI 0706.2.7  Use ratios and proportions to solve problems.
 
Grade 8
Grade Learning Expectations
GLE 0806.2.1  Extend understanding of the real number system to include irrational numbers.
GLE 0806.2.2  Solve problems involving exponents and scientific notation using technology appropriately.
GLE 0806.2.3  Solve real-world problems using rational and irrational numbers.
GLE 0806.2.4  Understand and use the laws of exponents.
Checks for Understanding
0806.2.1  Recognize and use exponential, scientific, and calculator notation.
0806.2.2  Square numbers and simplify square roots.
0806.2.3  Solve contextual problems involving powers and roots.
0806.2.4  Use a Venn diagram to represent the subsets of the real number system.
0806.2.5  Identify the subset(s) of the real number system to which a number belongs.
0806.2.6  Simplify expressions using the laws of exponents.
0806.2.7  Add, subtract, multiply, and divide numbers expressed scientific notation.
State Performance Indicators
SPI 0806.2.1  Order and compare rational and irrational numbers and locate on the number line.
SPI 0806.2.2  Identify numbers and square roots as rational or irrational.
SPI 0806.2.3  Use scientific notation to compute products and quotients.
SPI 0806.2.4  Solve real-world problems requiring scientific notation.
 
Algebra I
Course Learning Expectations
CLE 3102.2.1  Understand computational results and operations involving real numbers in multiple representations.
CLE 3102.2.2  Understand properties of and relationships between subsets and elements of the real number system.
Checks for Understanding
3102.2.1  Recognize and use like terms to simplify expressions.
3102.2.2  Apply the order of operations to simplify and evaluate algebraic expressions.
3102.2.3  Operate with and simplify radicals (index 2, 3, n) and radical expressions including rational numbers and variables in the radicand.
3102.2.4  Operate efficiently with both rational and irrational numbers.
3102.2.5  Perform operations with numbers in scientific notation (multiply, divide, powers).
3102.2.6  Use appropriate technologies to apply scientific notation to real-world problems.
3102.2.7  Identify the subsets in the real number system and understand their relationships.
3102.2.8  Use multiple strategies to approximate the value of an irrational number including irrational square roots and including location on the real number line.
State Performance Indicators
SPI 3102.2.1  Operate (add, subtract, multiply, divide, simplify, powers) with radicals and radical expressions including radicands involving rational numbers and algebraic expressions.
SPI 3102.2.2  Multiply, divide, and square numbers expressed in scientific notation.
SPI 3102.2.3  Describe and/or order a given set of real numbers including both rational and irrational numbers.
 
Geometry
Course Learning Expectations
CLE 3108.2.1  Establish the relationships between the real numbers and geometry; explore the importance of irrational numbers to geometry.
CLE 3108.2.2  Explore vectors as a numeric system, focusing on graphic representations and the properties of the operation.
CLE 3108.2.3  Establish an ability to estimate, select appropriate units, evaluate accuracy of calculations and approximate error in measurement in geometric settings.
Checks for Understanding
3108.2.1  Analyze properties and aspects of pi (e.g. classical methods of approximating pi, irrational numbers, Buffon’s needle, use of dynamic geometry software).
3108.2.2  Approximate pi from a table of values for the circumference and diameter of circles using various methods (e.g. line of best fit).
3108.2.3  Recognize and apply real number properties to vector operations and geometric proofs (e.g. reflexive, symmetric, transitive, addition, subtraction, multiplication, division, distributive, and substitution properties).
3108.2.4  Add vectors graphically and algebraically.
3108.2.5  Multiply a vector by a scalar graphically and algebraically.
3108.2.6  Analyze precision, accuracy, and approximate error in measurement situations.
State Performance Indicators
SPI 3108.2.1  Analyze, apply, or interpret the relationships between basic number concepts and geometry (e.g. rounding and pattern identification in measurement, the relationship of pi to other rational and irrational numbers)
SPI 3108.2.2  Perform operations on vectors in various representations.
 
Algebra II
Course Learning Expectations
CLE 3103.2.1  Understand the hierarchy of the complex number system and relationships between the elements, properties and operations.
CLE 3103.2.2  Connect numeric, analytic, graphical and verbal representations of both real and complex numbers.
CLE 3103.2.3  Use appropriate technology (including graphing calculators and computer spreadsheets) to solve problems, recognize patterns and collect and analyze data.
CLE 3103.2.4  Understand the capabilities and limitations of technology when performing operations, graphing, and solving equations involving complex numbers.
Checks for Understanding
3103.2.1  Understand that to solve certain problems and equations, the real number system needs to be extended from real numbers to complex numbers.
3103.2.2  Define and give examples of each of the types of numbers in the complex number system.
3103.2.3  Identify and apply properties of complex numbers (including simplification and standard form).
3103.2.4  Add and subtract complex numbers.
3103.2.5  Multiply complex numbers.
3103.2.6  Define and utilize the complex conjugates to write the quotient of two complex numbers in standard form.
3103.2.7  Graph complex numbers in the complex plane and recognize differences and similarities with the graphical representations of real numbers graphed on the number line.
3103.2.8  Solve quadratic equations over the complex number system.
3103.2.11  Understand the capabilities and limitations of technology. Make estimations without a calculator to detect potential errors.
3103.2.12  Select and use appropriate methods to make estimations without technology when solving contextual problems.
3103.2.13  Analyze and evaluate contextual situations involving any type of number from the complex number system.
State Performance Indicators
SPI 3103.2.1  Describe any number in the complex number system.
SPI 3103.2.2  Compute with all real and complex numbers.
SPI 3103.2.3  Use the number system, from real to complex, to solve equations and contextual problems