NCTM Pre-K – 2
Number and Operations
Count with understanding and recognize "how many" in sets of objects. Use multiple models to develop
initial understandings of place value and the base 10 number system. Develop understanding of the
relative position and magnitude of whole numbers and of ordinal and cardinal numbers and their
connections. Develop a sense of whole numbers and represent and use them in flexible ways, including
relating, composing, and decomposing numbers. Connect number words and numerals to the quantities
they represent, using various physical models and representations. Understand and represent
commonly used fractions, such as 1/4, 1/3, and 1/2. Understand various meanings of addition and
subtraction of whole numbers and the relationship between the two operations. Understand the
effects of adding and subtracting whole numbers. Understand situations that entail multiplication and
division, such as equal groupings of objects and sharing equally. Develop and use strategies for wholenumber computations, with a focus on addition and subtraction. Develop fluency with basic number
combinations for addition and subtraction. Use a variety of methods and tools to compute, including
objects, mental computation, estimation, paper and pencil, and calculators.
NCTM 3-5th Grade Number and Operations
Understand place value, base 10, compare whole number and decimals, equivalent representations,
decomposing and composing numbers, fractions as parts of unit wholes, parts of a collection, locations
on number lines, and as divisions of whole numbers, use models, benchmarks, and equivalent forms to
judge the size of fractions, recognize and generate equivalent forms of commonly used fractions,
decimals and percents, explore numbers less than 0 by extending the number line through familiar
applications, describe classes of numbers according to characteristics such as the nature of their factors.
Understand the meaning of multiplication and division, understand the effects of multiplying and
dividing whole numbers, identify and use relationships between operations, such as division as the
inverse of multiplication to solve problems, understand the properties of operations such as the
distributivity of multiplication over addition. Develop fluency with basic number combinations for
multiplication and division, develop fluency in adding, subtracting, multiplying and dividing numbers,
estimate the results of whole number computations and judge the reasonableness of such results,
estimate computations involving fractions and decimals in situations relevant to students' experience,
use visual models, benchmarks, and equivalent forms to add and subtract commonly used fractions and
decimals, select appropriate methods and tools for computing with whole numbers according to the
context and nature of the computation.
NCTM 6th-8th Grade Number and Operations
Work flexibly with fractions, decimals, and percents to solve problems. Compare and order fractions,
decimals and percents efficiently and find their approximate locations on a number line. Develop
meaning for percents greater than 100 or less than 1. Understand and use ratios and proportions to
represent quantitative relationships. Develop an understanding of large numbers and recognize and
appropriately use exponential, scientific, and calculator notation. Use factors, multiples, prime
factorization, and relatively prime numbers to solve problems. Develop meaning for integers and
represent and compare quantities with them. Understand the meaning and effects of arithmetic
operations with fractions, decimals, and integers. Use the associative and commutative properties of
addition and multiplication and the distributive property of multiplication over addition to simplify
computations with integers, fractions and decimals. Understand and use the inverse relationships of
addition and subtraction, multiplication and division, and squaring and finding square roots to simplify
computations and solve problems. Select appropriate methods and tools for computing with fractions
and decimals from among mental computation, estimation, calculators or computers, and paper and
pencil, depending on the situation, and apply the selected methods. Develop and analyze algorithms for
computing with fractions, decimals, and integers and develop fluency in their use. Develop and use
strategies to estimate the results of rational-number computations and judge the reasonableness of the
results. Develop, analyze and explain methods for solving problems involving proportions, such as
scaling and finding equivalent ratios.
GPS Number and Operations Kindergarten
MKN1. Students will connect numerals to quantities they represent. A. Count a number of objects up to
30. B. Produce models for number words through ten. C. Write numerals through 20 to label sets. D.
Sequence and identify using ordinal numbers (1st -10th). E. Compare two or more sets of objects (1-10)
and identify which set is equal to, more than, or less than the other. F. Estimate quantities using five
and ten as a benchmark. (e.g. 9 is one five and four more. It is closer to 10, which can be represented as
one ten or two fives, than it is to five.) G. Use informal strategies to share objects equally (divide)
between two or three people or sets. H. Identify coins by name and value (penny, nickel, dime, and
quarter). I. Count out pennies to buy items that together cost less than 30 cents. J. Make fair trades
using combinations involving pennies and nickels and pennies and dimes.
MKN2. Students will
use representations to model addition and subtraction. A. Use counting strategies to find out how
many items are in two sets when they are combined, separated, or compared. B. Build number
combinations up to 10 (e.g. 4 and 1, 2 and 3, 3 and 2, 4 and 1 for five) and for doubles to 10 (3 and 3 for
six). C. Use objects, pictures, numbers, or words to create, solve and explain story problems (combining,
separating, or comparing) for two numbers that are each less than 10.
GPS Number and Operations 1st Grade
M1N1. Students will estimate, model, compare, order, and represent whole numbers up to 100. A.
Represent numbers up to 100 using a variety of models, diagrams, and number sentences. Represent
numbers larger than 10 in terms of tens and ones using manipulatives and pictures. B. Correctly count
and represent the number of objects in a set using numerals. C. Compare small sets using the terms
greater than, less than, and equal to. D. Understand the magnitude and order of numbers up to 100 by
making ordered sequences and representing them on a number line. E. Exchange equivalent quantities
of coins by making fair trades involving combinations of pennies, nickels, dimes and quarters up to one
dollar; count out a combination of coins needed to purchase items up to one dollar. F. Identify bills
($1, $5, $10, $20) by name and value and exchange equivalent quantities by making fair trades involving
combinations of bills; count out a combination of bills needed to purchase items that total up to twenty
dollars.
M1N2. Students will understand place value notation for the numbers 1 to 99.
(Discussions may allude to 3-digit number to assist in understanding place value.) A. Determine to
which ten a given number is closest using tools such as a sequential number line or chart. B. Represent
collections of less than 30 objects with 2-digit numbers and understand the meaning of place value. C.
Decompose numbers from 10 to 99 as the appropriate number of tens and ones. M1N3. Students
will add and subtract numbers less than 100, as well as understand and use the inverse relationship
between addition and subtraction. A. Identify one more than, one less than, 10 more than, and 10 less
than a given number. B. Skip-count by 2s, 5s, and 10s, forward and backwards, to and from numbers up
to 100. C. Compose and decompose numbers up to 10 (e. g. 3+5=8, 8=5+1+2). D. Understand a variety
of situations to which subtraction may apply; taking away from a set, comparing two sets, and
determining how many more or how many less. E. Understand addition and subtraction number
combinations using strategies such as counting on, counting back, doubles and making tens. F. Know
the single digit addition facts to 18 using corresponding subtraction facts with understanding and
fluency. (Use strategies such as relating facts already known, applying the commutative property, and
grouping facts into families.) G. Apply addition and subtraction to 2 digit numbers without regrouping
(e.g. 15+4, 80-60, 56+10, 100-30, 52+5). H. Solve and create word problems involving addition and
subtraction to 100 without regrouping. Use words, pictures and concrete models to interpret story
problems and reflect the combining of sets as addition and taking away or comparing elements of sets
as subtraction. M1N4. Students will count collections of up to 100 objects by dividing them into equal
parts and represent the results using words, pictures, or diagrams. A. Use informal strategies to share
objects equally between two to five people. B. Build number patterns, including concepts of even and
odd, using various concrete representations. (Examples of concrete representations include a hundreds
chart, ten grid frame, place value chart, number line, counters, or other objects.) C. Identify label and
relate fractions (halves, fourths) as equal parts of a collection of objects or a whole using pictures or
models. D. Understand halves and fourths as representations of equal parts of a whole.
GPS Number and Operations 2nd Grade
M2N1. Students will use multiple representations of numbers to connect symbols to quantities. A.
Represent numbers using a variety of models, diagrams, and number sentences (e.g. 4703 represented
as 4,000+700+3, and units, 47 hundreds+ 3, or 4,500+203). B. Understand the relative magnitudes of
numbers using 10 as a unit, 100 as a unit, or 1000 as a unit. Represent 2-digit numbers with drawings of
tens and ones and 3-digit numbers with drawings of hundreds, tens, and ones. C. Use money as a
medium of exchange. Make change and use decimal notation and the dollar and cent symbols to
represent the collection of coins and currency. M2N2. Students will build fluency with multi-digit
addition and subtraction. A. Correctly add and subtract two whole numbers up to three digits each with
regrouping. B. Understand and use the inverse relationship between addition and subtraction to solve
problems and check solutions. C. Use mental math strategies such as benchmark numbers to solve
problems. D. Use basic properties of addition (commutative, associative, and identity) to simplify
problems (e.g. 98+17 by taking two from 17 and adding it to the 98 to make 100 and replacing the
original problem by the sum of 100+15). E. Estimate to determine if solutions are reasonable for
addition and subtraction. M2N3. Students will understand multiplication, multiply numbers, and verify
results. A. Understand multiplication as repeated addition. B. Use repeated addition, arrays, and
counting by multiples (skip counting) to correctly multiply 1-digit numbers and construct a multiplication
table. C. Use the multiplication table (grid) to determine the product of two numbers. D. Use repeated
subtraction, equal sharing, and forming equal groups to divide large collections of objects and
determine factors for multiplication. M2N4. Students will understand and compare fractions. A.
Model, identify, label, and compare fractions (thirds, sixths, eighths, tenths) as a representation of equal
parts of a whole or of a set. B. Know that when all fractional parts are included, such as three thirds,
the result is equal to the whole. M2N5. Students will represent and interpret quantities and
relationships using mathematical expressions including equality and inequality signs. A. Include the use
of boxes or _ to represent a missing value. B. Represent problem solving situations where addition,
subtraction or multiplication may be applied using mathematical expressions.
GPS Number and Operations 3rd Grade
M3N1 Develop understanding of whole numbers and ways of representing them. A. Identify place
values from tenths to ten thousands. Understand the relative sizes of digits in place value notation and
ways to represent them. M3N3 Understanding of multiplication of whole numbers and develop the
ability to apply it in problem solving. a. Describe the relationship between multiplication and division. B.
Know the multiplication facts with understanding and fluency to 10 X10. C. Use arrays and area models
to develop understanding of the distributive property and to determine partial products for
multiplication of a 2 or 3 digit numbers by a 1 digit number. M3M4 Understand the meaning of division
and develop the ability to apply it in problem solving. M3M5 Understand the meaning of decimal
fractions and common fractions in simple cases and apply them in problem solving situations.
GPS Number and Operations 4th Grade
M4N1 Understanding of how whole numbers are represented in the base 10 numeration system. A.
Identify place value names and places from the hundredths through one million. B. Equate a number's
word name, its standard form, and its expanded form. M4N2 Understand and apply the concept of
rounding numbers. A. Round to the nearest ten, hundred, or thousand. B. Describe situations in which
rounding numbers would be appropriate and determine whether to round to the nearest ten, hundred
or thousand. C. Determine to which whole number or tenth a given decimal is closest using tools such
as a number line, and/or charts. D. Round a decimal to the nearest whole number or tenth. E.
Represent the results of computation as a rounded number with appropriate and estimate a sum or
difference by rounding numbers. M4N3 Solve problems involving multiplication of 2 and 3 digit
numbers by 1 and 2 digit numbers. M4N4 Understand division of whole numbers and divide in problem
solving situations without calculators. A. Know the division facts with understanding and fluency. B.
Solve problems involving division by 1 or 2 digit numbers (including those that generate a remainder). C.
Understand the relationship between the dividend, divisor, quotient, and remainder. D. Understand
and explain the effect on the quotient of multiplying or dividing both the divisor and the dividend by the
same number. (2050/50 = 205/5). M4N5 Further develop understanding of the meaning of decimals
and use them in computations. A. Understand decimals are part of the base 10 system. B. Understand
the relative size of numbers and order two digit decimals. C. Add and subtract both one and two digit
decimals. D. Model multiplication and division of decimals by whole numbers. E. Multiply and divide
both one and two digit decimals by whole numbers. M4N6 Understand the meaning of common
fractions and use them in computations. a. Understand representations of simple equivalent fractions.
b. Add and subtract fractions and mixed numbers with common denominators. c. Convert and use
mixed numbers and improper fractions interchangeably. M4N7 Explain and use properties of the four
arithmetic operations to solve and check problems. A. Describe situations in which the four operations
may be used and relationships among them. B. Compute using the order of operations, including
parentheses. C. Compute using the commutative, associative, and distributive properties. D. Use
mental math and estimation strategies to compute.
GPS Number and Operations 5th Grade
M5M1 Understand whole numbers. A. Classify the set of counting numbers into subsets with
distinguishing characteristics (odd/even, prime/composite) B. Find multiples and factors. C. Analyze and
use divisibility rules. M5N2 Understand decimals as part of the base-ten number system. A. Understand
place value. B. Analyze the effect on the product when a number is multiplied by 10,100, 1000, 0.1, and
0.01. C. Use greater than, less than, or = to compare decimals and justify the comparison. M5N3
Understand the meaning of multiplication and division with decimal fractions and use them. A. Model
multiplication and division of decimal fractions by another decimal fraction. B. Explain the process of
multiplication, division, including situations in which the multiplier and divisor are both whole numbers
and decimal fractions. C. Multiply and divide with decimal fractions including decimal fractions less than
one and greater than one. D. Understand the relationships and rules for multiplication and division of
whole numbers also apply to decimal fractions. M5N4 Understand meaning of common fractions and
compute with them. A. Division of whole numbers can be represented by a fraction. B. Understand the
value of a fraction is not changed when both its numerator and denominator are multiplied or divided
by the same number because it is the same as multiplying or dividing by 1. C. Find equivalent fractions
and simplify fractions. D. Model the multiplication and division of common fractions. E. Explore finding
common denominators by using concrete, pictoral, and computational models. F. Use greater than, less
than or = to compare fractions and justify the comparison. G. Add and subtract common fractions with
mixed numbers and unlike denominators. H. Use fractions (proper and improper) and decimals
interchangeably. I. Estimate products and quotients. M5N5. Students will understand the meaning of
percentage. A. Explore and model percents using multiple representations. B. Apply percents to circle
graphs.
GPS Number and Operations 6th Grade
M6N1. Students will understand the meaning of the four arithmetic operations as related to positive
rational numbers and will use these concepts to solve problems. A. Apply factors and multiples. B.
Decompose numbers into their prime factorization (Fundamental Theorem of Arithmetic). C. Detemine
the greatest common factor (GCF) and the least common multiple (LCM) for a set of numbers. D. Add
and subtract fractions and mixed numbers with unlike denominators. E. Multiply and divide fractions
and mixed numbers. F. Use fractions, decimals, and percents interchangeably. G. Solve problems
involving fractions, decimals and percents.
GPS Number and Operations 7th Grade
M7N1. Students will understand the meaning of positive and negative rational numbers and use them
in computation. A. Find the absolute value of a number and understand it as the distance from zero on
a number line. B. Compare and order rational numbers, including repeating decimals. C. Add, subtract,
multiply and divide positive and negative rational numbers. D. Solve problems using rational numbers.
GPS Number and Operations 8th Grade
M8N1. Students will understand different representations of numbers including square roots,
exponents, and scientific notation. A. Find square roots of perfect squares. B. Recognize the (positive)
square root of a number as a length of a side of a square with a given area. C. Recognize square roots
as points and as lengths on a number line. D. Understand that the square root of 0 is 0 and that every
positive number has two square roots that are opposite in sign. Recognize and use the radical sign to
denote the positive square root of a positive number.
Common Core Number and Operations Kindergarten
Know number names and count sequence. Count to tell the number of objects. Compare numbers.
Common Core Number and Operations 1st Grade
Represent and solve problems involving addition and subtraction. Add and subtract within 20.
Understand and apply properties of operations and the relationship between addition and subtraction.
Extend the counting sequence. Understand place value. Use place value understanding to add and
subtract.
Common Core Number and Operations 2nd Grade
Represent and solve problems involving addition and subtraction. Add and subtract within 20. Work
with equal groups to gain foundations for multiplication. Understand place value. Use place value
understanding and properties of operations to add and subtract.
Common Core Number and Operations 3rd Grade
Represent and solve problems involving multiplication and division. Understand properties of
multiplication and the relationship between X and division. Multiply and divide within 100. Solve
problems involving the four operations. Use place value and properties of operations to perform multidigit arithmetic. Develop understanding of fractions as numbers.
Common Core Number and Operations 4th Grade
Use place value for multi-digit arithmetic. Use the four operations with whole numbers to solve
problems. Gain familiarity with factors and multiples. Generalize place value understanding for multidigit whole numbers. Use place value understanding and properties of operations to perform multi-digit
arithmetic. Extend understanding of fraction equivalence and ordering. Build fractions from unit
fractions. Understand decimal notation for fractions and compare decimal fractions.
Common Core Number and Operations 5th Grade
Place value operations with multi digit numbers and with decimals to the hundredths. Use equivalent
fractions as a strategy to add and subtract fractions. Multiply and divide fractions. Division of a fraction
by a fraction is not required at this grade.
Common Core Number and Operations 6th Grade
Divide fractions by fractions. Compute fluently with multi-digit numbers and find common factors and
multiples. Understand rational numbers. Understand ratio and use ratio reasoning to solve problems.
Common Core Number and Operations 7th Grade
Analyze proportional relationships and use them to solve real-world and mathematical problems. Use
fractions to add, subtract, multiply and divide rational numbers.
Common Core Number and Operations 8th Grade
Know that there are numbers that are not rational, and approximate them by rational numbers.
Massachusetts Former Standards Number and Operations
Pre K-4th Grade
Pre K - 4 Number Sense and Numeration: Construct number meaning by using manipulatives and other
physical materials to represent concepts of numbers in the real world. Demonstrate an understanding
of our numeration system by relating counting, grouping, and place value concepts. Interpret the
multiple uses of numbers by taking real world situations and then translating them into numerical
statements. Whole Number Computation: Students engage in problem solving, communicating,
reasoning, and connecting to model, explain, and develop proficiency with basic facts and algorithms.
Students use calculators in appropriate computational situations.
5th- 8th Grade
5- 8 Number and Number Relationships: Students engage in problem solving, communicating,
reasoning, and connecting to: Represent and use equivalent forms of numbers, including integers,
fractions, decimals, percents, exponents, and scientific notation. Apply ratios, proportions, and
percents. Investigate and describe the relationships among fractions, decimals and percents. Represent
numerical relationships in one-and two- dimensional graphs. Number Systems and Number Theory:
Students engage in problem solving, communicating, reasoning, and connecting to: Explain the need for
numbers other than whole numbers. Know and use order relations for whole numbers, fractions,
decimals, integers, and rational numbers. Use operation involving fractions, decimals, integers, and
rational numbers. Demonstrate how basic operations are related to one another. Create and apply
number theory concepts, including prime numbers, factors, and multiples. Computation and Estimation:
Students engage in problem solving, communicating, reasoning, and connecting to: Compute with
whole numbers, fractions, decimals, integers, and rational numbers. Develop, analyze and explain
procedures for computing, estimating, and solving problems. Select and use an appropriate method for
computing from among mental arithmetic, paper and pencil, calculator, and computer models. Use
computation, estimation, and proportions to solve problems. Estimate and check the reasonableness of
results of computations and problems involving rational numbers.