Grade: K
Number, Number Sense and Operations Standard
Computation and Estimation
Standard Number: 13
Recognize the number or quantity of sets up to 5 without counting; e.g., recognize without counting the dot arrangement on a domino as 5
Meaning of Operations
Standard Number: 10
Model and represent addition as combining sets and counting on, and subtraction as take-away and comparison
a Combine and separate small sets of objects in contextual situations; e.g., add or subtract one, two, or another small amount
b Count on (forward) and count back (backward) on a number line between 0 and 10
Standard Number: 11
Demonstrate joining multiple groups of objects, each containing the same number of objects; e.g., combining 3 bags of candy, each containing 2 pieces
Standard Number: 12
Partition or share a small set of objects into groups of equal size; e.g., sharing 6 stickers equally among 3 children
Number and Number Systems
Standard Number: 1
Compare and order whole numbers up to 10
Standard Number: 2
Explain rules of counting, such as each object should be counted once and that order does not change the number.
Standard Number: 3
Count to twenty; e.g., in play situations or while reading number books.
Standard Number: 4
Determine “how many” in sets (groups) of 10 or fewer objects
Standard Number: 5
Relate, read and write numerals for single-digit numbers (0 to 9).
Standard Number: 6
Construct multiple sets of objects each containing the same number of objects.
Standard Number: 7
Compare the number of objects in two or more sets when one set has one or two more, or one or two fewer objects
Standard Number: 8
Represent and use whole numbers in flexible ways, including relating, composing and decomposing numbers; e.g., 5 marbles can be 2 red and 3 green or 1 red and 4 green
Standard Number: 9
Identify and state the value of a penny, nickel and dime
Grade: Gr. 1
Number, Number Sense and Operations Standard
Computation and Estimation
Standard Number: 16
Develop strategies for basic addition facts such as:
a counting all
b counting on
c one more, two more
d doubles
e doubles plus or minus one
f make ten
g using tens frames
h identity property (adding zero).
Standard Number: 17
Develop strategies for basic subtraction facts
a relating to addition
b one less, two less
c all but one
d using tens frames
e missing addends
Meaning of Operations
Standard Number: 11
Model, represent and explain subtraction as take-away and comparison
a Model and explain subtraction using physical materials in contextual situations
b Draw pictures to model subtraction
c Write number sentences to represent subtraction
d Explain that subtraction of whole numbers yields an answer smaller than the original number
Standard Number: 12
Use conventional symbols to represent the operations of addition and subtraction
Standard Number: 13
Model and represent multiplication as repeated addition and rectangular arrays in contextual situations; e.g., four people will be at my party and if I want to give 3 balloons to each person, how many balloons will I need to buy?
Standard Number: 14
Model and represent division as sharing equally in contextual situations; e.g., sharing cookies.
Standard Number: 15
Demonstrate that equal means “the same as” using visual representations
Number and Number Systems
Standard Number: 1
Use ordinal numbers to order objects; e.g., first, second, third
Standard Number: 2
Recognize and generate equivalent forms for the same number using physical models, words and number expressions; e.g., concept of ten is described by “10 blocks,” full tens frame, numeral 10, 5 + 5, 15 - 5, one less than 11, my brother’s age
Standard Number: 3
Read and write the numerals for numbers to 100
Standard Number: 4
Count forward to 100, count backwards from 100, and count or backward starting at any number between 1 and 100
Standard Number: 5
Use place value concepts to represent whole numbers using numerals, words, expanded notation and physical models with ones and tens
a Develop a system to group and count by twos, fives and tens
b Identify patterns and groupings in a 100’s chart and relate to place value concepts
c Recognize the first digit of a two-digit number as the most important to indicate size of a number and the nearness to 10 or 100
Standard Number: 6
Identify and state the value of a penny, nickel, dime, quarter and dollar
Standard Number: 7
Determine the value of a small collection of coins (with a total value up to one dollar) using 1 or 2 different type coins, including pennies, nickels, dimes and quarters
Standard Number: 8
Show different combinations of coins that have the same value
Standard Number: 9
Represent commonly used fractions using words and physical models for halves, thirds and fourths, recognizing fractions are represented by equal size parts of a whole and of a set of objects
Standard Number: 10
Model, represent and explain addition as combining sets (part + part = whole) and counting on
a Model and explain addition using physical materials in contextual situations
b Draw pictures to model addition
c Write number sentences to represent addition
d Explain that adding two whole numbers yields a larger whole number
Grade: Gr. 2
Number, Number Sense and Operations Standard
Computation and Estimation
Standard Number: 9.0
Model and use the commutative property for addition
Standard Number: 10.0
Demonstrate fluency in addition facts with addends through 9 and corresponding subtractions; e.g., 9 + 9 = 18, 18 – 9 = 9.
Standard Number: 11
Add and subtract multiples of 10
Standard Number: 12
Demonstrate multiple strategies for adding and subtracting 2- or 3-digit whole numbers
a compatible numbers
b compensatory numbers
c informal use of commutative and associative properties of addition
Standard Number: 13.0
Estimate the results of whole number addition and subtraction problems using front-end estimation, and judge the reasonableness of the answers
Meaning of Operations
Standard Number: 6.0
Model, represent and explain subtraction as comparison, take-away and part-to-whole; e.g., solve missing addend problems by counting up or subtracting, such as “I had six baseball cards, my sister gave me more, and I now have ten. How many did she give me?” can be represented as 6 + ? = 10 or 10 - 6 = ?.
Standard Number: 7.0
Model, represent and explain multiplication as repeated addition, rectangular arrays and skip counting
Standard Number: 8.0
Model, represent and explain division as sharing equally and repeated subtraction
Number and Number Systems
Standard Number: 1.0
Use place value concepts to represent, compare and order whole numbers using physical models, numerals and words, with ones, tens and hundreds
a Recognize 10 can mean “10 ones” or a single entity (1 ten) through physical models and trading games
b Read and write 3-digit numerals (e.g., 243 as two hundred forty three, 24 tens and 3 ones, or 2 hundreds and 43 ones, etc.) and construct models to represent each
Standard Number: 2.0
Recognize and classify numbers as even or odd
Standard Number: 3.0
Count money and make change using coins and a dollar bill.
Standard Number: 4.0
Represent and write the value of money using the ¢ sign and in decimal form when using the $ sign.
Standard Number: 5.0
Represent fractions (halves, thirds, fourths, sixths and eighths), using words, numerals and physical models.
a Recognize that a fractional part can mean different amounts depending on the original quantity
b Recognize that a fractional part of a rectangle does not have to be shaded with contiguous parts
c Identify and illustrate parts of a whole and parts of sets of objects
d Compare and order physical models of halves, thirds and fourths in relations to 0 and 1
Grade: Gr. 3
Number, Number Sense and Operations Standard
Computation and Estimation
Standard Number: 12.0
Add and subtract whole numbers with and without regrouping
Standard Number: 13.0
Demonstrate fluency in multiplication facts through 10 and corresponding division facts.
Standard Number: 14.0
Divide 2- and 3-digit numbers by a single-digit number, without remainders for division
Standard Number: 15.0
Evaluate the reasonableness of computations based upon operations and the numbers involved; e.g., considering relative size, place value, and estimates.
Meaning of Operations
Standard Number: 8.0
Model, represent and explain multiplication; e.g., repeated addition, skip counting, rectangular arrays and area model
a Use conventional mathematical symbols to write equations for word problems involving multiplication.
b Understand that, unlike addition and subtraction, the factors in multiplication and division may have different units; e.g., 3 boxes of 5 cookies each
Standard Number: 9.0
Model, represent and explain division; e.g., sharing equally, repeated subtraction, rectangular arrays and area model
a Translate contextual situations involving division into conventional mathematical symbols
b Explain how a remainder may impact an answer in a real-world situation
Standard Number: 10.0
Explain and use relationships between operations
a relate addition and subtraction as inverse operations
b relate multiplication and division as inverse operations
c relate addition to multiplication (repeated addition)
d relate subtraction to division (repeated subtraction)
Standard Number: 11.0
Model and use the commutative and associative properties for addition and multiplication
Number and Number Systems
Standard Number: 1.0
Identify and generate equivalent forms of whole numbers; e.g., 36, 30 + 6, 9 x 4, 46 - 10, number of inches in a yard
Standard Number: 2.0
Use place value concepts to represent whole numbers and decimals using numerals, words, expanded notation and physical models
a Recognize 100 means “10 tens” as well as a single entity (1 hundred) through physical models and trading games.
b Describe the multiplicative nature of the number system
c Model the size of 1000 in multiple ways
Standard Number: 3.0
Use mathematical language and symbols to compare and order; e.g., less than, greater than, at most, at least, <, >, =, =, =.
Standard Number: 4.0
Count money and make change using coins and paper bills to ten dollars.
Standard Number: 5.0
Represent fractions and mixed numbers using words, numerals and physical models
Standard Number: 6.0
Compare and order commonly used fractions and mixed numbers using number lines, models (such as fraction circles or bars), points of reference (such as more or less than ), and equivalent forms found using physical or visual models.
Standard Number: 7.0
Recognize and use decimal and fraction concepts and notations as related ways of representing parts of a whole or a set; e.g., 3 of 10 marbles are red can also be described as and 3 tenths are red
Grade: Gr. 4
Number, Number Sense and Operations Standard
Computation and Estimation
Standard Number: 8.0
Solve problems involving counting money and making change, using both coins and paper bills
Standard Number: 9.0
Estimate the results of computations involving whole numbers, fractions and decimals, using a variety of strategies
Standard Number: 10.0
Use physical models, visual representations, and paper and pencil to add and subtract decimals and commonly used fractions with like denominators
Standard Number: 11.0
Develop and explain strategies for performing computations mentally
Standard Number: 12.0
Analyze and solve multi-step problems involving addition, subtraction, multiplication and division using an organized approach, and verify and interpret results with respect to the original problem
Standard Number: 13.0
Use a variety of methods and appropriate tools for computing with whole numbers; e.g., mental math, paper and pencil, and calculator
Standard Number: 14.0
Demonstrate fluency in adding and subtracting whole numbers and in multiplying and dividing whole numbers by 1- and 2-digit numbers and multiples of ten
Meaning of Operations
Standard Number: 6.0
Use associative and distributive properties to simplify and perform computations; e.g., use left to right multiplication and the distributive property to find an exact answer without paper and pencil, such as 5 x 47 = 5 x 40 + 5 x 7 = 200 + 35 = 235
Standard Number: 7.0
Recognize that division may be used to solve different types of problem situations and interpret the meaning of remainders; e.g., situations involving measurement, money
Number and Number Systems
Standard Number: 1.0
Identify and generate equivalent forms of fractions and decimals
a Connect physical, verbal and symbolic representations of fractions, decimals and whole numbers; e.g. “five tenths,” 0.5, shaded rectangles with half, and five tenths
b Understand and explain that ten tenths is the same as one whole in both fraction and decimal form
Standard Number: 2.0
Use place value structure of the base-ten number system to read, write, represent and compare whole numbers through millions and decimals through thousandths
Standard Number: 3.0
Round whole numbers to a given place value
Standard Number: 4.0
Identify and represent factors and multiples of whole numbers through 100, and classify numbers as prime or composite
Standard Number: 5.0
Use models and points of reference to compare commonly used fractions
Grade: Gr. 5
Number, Number Sense and Operations Standard
Computation and Estimation
Standard Number: 12
Use physical models, points of reference, and equivalent forms to add and subtract commonly used fractions with like and unlike denominators and decimals
Standard Number: 13
Estimate the results of computations involving whole numbers, fractions and decimals, using a variety of strategies
Meaning of Operations
Standard Number: 6.0
Represent and compare numbers less than 0 by extending the number line and using familiar applications; e.g., temperature, owing money
Standard Number: 7.0
Use commutative, associative, distributive, identity and inverse properties to simplify and perform computations
Standard Number: 8.0
Identify and use relationships between operations to solve problems
Standard Number: 9.0
Use order of operations, including use of parentheses, to simplify numerical expressions
Standard Number: 10.0
Justify why fractions need common denominators to be added or subtracted
Standard Number: 11.0
Explain how place value is related to addition and subtraction of decimals; e.g., 0.2 + 0.14; the two tenths is added to the one tenth because they are both tenths
Number and Number Systems
Standard Number: 1.0
Use models and visual representation to develop the concept of ratio as part-to-part and part-to-whole, and the concept of percent as part-to-whole
Standard Number: 2.0
Use various forms of “one” to demonstrate the equivalence of fractions
Standard Number: 3.0
Identify and generate equivalent forms of fractions, decimals and percents
Standard Number: 4.0
Round decimals to a given place value and round fractions (including mixed numbers) to the nearest half
Standard Number: 5.0
Recognize and identify perfect squares and their roots
Grade: Gr. 6
Number, Number Sense and Operations Standard
Computation and Estimation
Standard Number: 1
Decompose and recompose whole numbers using factors and exponents (e.g., 32 = 2 x 2 x 2 x 2 x 2 = 25 ), and explain why “squared” means “second power” and “cubed” means “third power
Standard Number: 11
Perform fraction and decimal computations and justify their solutions; e.g., using manipulatives, diagrams, mathematical reasoning
Standard Number: 12
Develop and analyze algorithms for computing with fractions and decimals, and demonstrate fluency in their use
Standard Number: 13
Estimate reasonable solutions to problem situations involving fractions and decimals
Standard Number: 14
Use proportional reasoning, ratios and percents to represent problem situations and determine the reasonableness of solutions
Standard Number: 15
Determine the percent of a number and solve related problems; e.g., find the percent markdown if the original price was $140, and the sale price is $100
Meaning of Operations
Standard Number: 6
Use the order of operations, including the use of exponents, decimals and rational numbers, to simplify numerical expressions
Standard Number: 7
Use simple expressions involving integers to represent and solve problems; e.g., if a running back loses 15 yards on the first carry but gains 8 yards on the second carry, what is the net gain/loss?
Standard Number: 8
Represent multiplication and division situations involving fractions and decimals with models and visual representations
Standard Number: 9
Give examples of how ratios are used to represent comparisons; e.g., part-to-part, part-to-whole, whole-to-part
Standard Number: 10
Recognize that a quotient may be larger than the dividend when the divisor is a fraction
Number and Number Systems
Standard Number: 2
Find and use the prime factorization of composite numbers
a Use the prime factorization to recognize the greatest common factor (GCF)
b Use the prime factorization to recognize the least common multiple (LCM
c Apply the prime factorization to solve problems and explain solutions
Standard Number: 3
Explain why a number is referred to as being “rational,” and recognize that the expression can mean a parts of size each, a divided by b, or the ratio of a to b
Standard Number: 4
Describe what it means to find a specific percent of a number, using real-life examples
Standard Number: 5
Use models and pictures to relate concepts of ratio, proportion and percent, including percents less than 1 and greater than 100
Grade: Gr. 7
Number, Number Sense and Operations Standard
Computation and Estimation
Standard Number: 6
Explain the meaning and effect of adding, subtracting, multiplying and dividing integers; e.g., how adding two integers can result in a lesser value
Standard Number: 7
Solve problems using the appropriate form of a rational number (fraction, decimal or percent).
Standard Number: 8
Develop and analyze algorithms for computing with percents and integers, and demonstrate fluency in their use
Standard Number: 9
Represent and solve problem situations that can be modeled by and solved using concepts of absolute value, exponents and square roots (for perfect squares).
Meaning of Operations
Standard Number: 4
Use order of operations and properties to simplify numerical expressions involving integers, fractions and decimals
Standard Number: 5
Explain the meaning and effect of adding, subtracting, multiplying and dividing integers; e.g., how adding two integers can result in a lesser value
Number and Number Systems
Standard Number: 1
Demonstrate an understanding of place value using powers of 10 and write large numbers in scientific notation
Standard Number: 2
Explain the meaning of exponents that are negative or 0
Standard Number: 3
Describe differences between rational and irrational numbers; e.g., use technology to show that some numbers (rational) can be expressed as terminating or repeating decimals and others (irrational) as non-terminating and non-repeating decimals
Grade: Gr. 8
Number, Number Sense and Operations Standard
Computation and Estimation
Standard Number: 5
Determine when an estimate is sufficient and when an exact answer is needed in problem situations, and evaluate estimates in relation to actual answers; e.g., very close, less than, greater than
Standard Number: 6
Estimate, compute and solve problems involving rational numbers, including ratio, proportion and percent, and judge the reasonableness of solutions
Standard Number: 7
Find the square root of perfect squares, and approximate the square root of non-perfect squares as consecutive integers between which the root lies
Standard Number: 8
Add, subtract, multiply, divide and compare numbers written in scientific notation
Meaning of Operations
Standard Number: 3
Apply order of operations to simplify expressions and perform computations involving integer exponents and radicals
Standard Number: 4
Explain and use the inverse and identity properties and use inverse relationships (addition/subtraction, multiplication/division, squaring/square roots) in problem solving situations
Number and Number Systems
Standard Number: 1
Use scientific notation to express large numbers and small numbers between 0 and 1
Standard Number: 2
Recognize that natural numbers, whole numbers, integers, rational numbers and irrational numbers are subsets of the real number system
Grade: Gr. 9
Number, Number Sense and Operations Standard
Computation and Estimation
Standard Number: 4
Demonstrate fluency in computations using real numbers
Standard Number: 5
Estimate the solutions for problem situations involving square and cube roots
Meaning of Operations
Standard Number: 3
Explain the effects of operations such as multiplication or division, and of computing powers and roots on the magnitude of quantities
Number and Number Systems
Standard Number: 1
Identify and justify whether properties (closure, identity, inverse, commutative and associative) hold for a given set and operations; e.g., even integers and multiplication
Standard Number: 2
Compare, order and determine equivalent forms for rational and irrational numbers.
Grade: Gr. 10
Number, Number Sense and Operations Standard
Computation and Estimation
Standard Number: 3
Use factorial notation and computations to represent and solve problem situations involving arrangements
Standard Number: 4
Approximate the nth root of a given number greater than zero between consecutive integers when n is an integer; e.g., the 4th root of 50 is between 2 and 3.
Meaning of Operations
Standard Number: 2
Explain the meaning of the nth root
Number and Number Systems
Standard Number: 1
Connect physical, verbal and symbolic representations of irrational numbers
Grade: Gr. 11
Number, Number Sense and Operations Standard
Computation and Estimation
Standard Number: 6
Compute sums, differences and products of matrices using paper and pencil calculations for simple cases, and technology for more complicated cases
Standard Number: 7
Compute sums, differences, products and quotients of complex numbers
Standard Number: 8
Use fractional and negative exponents as optional ways of representing and finding solutions for problem situations
Standard Number: 9
Use vector addition and scalar multiplication to solve problems
Meaning of Operations
Standard Number: 4
Use matrices to represent given information in a problem situation
Standard Number: 5
Model using the coordinate plane, vector addition and scalar multiplication
Number and Number Systems
Standard Number: 1
Determine what properties hold for matrix addition and matrix multiplication; e.g., use examples to show addition is commutative and when multiplication is not commutative
Standard Number: 2
Determine what properties hold for vector addition and multiplication, and for scalar multiplication
Standard Number: 3
Represent complex numbers on the complex plane
Grade: Gr. 12
Number, Number Sense and Operations Standard
Computation and Estimation
Standard Number: 2
Apply combinations as a method to create coefficients for the Binomial Theorem, and make connections to everyday and workplace problem situations
Number and Number Systems
Standard Number: 1
Determine what properties (closure, identity, inverse, commutative and associative) hold for operations with complex numbers
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