• Mathematical Practices K-12

This link takes you to a parent guide about the mathematical practices. These practices are just as important as any content in all grades K-12. They help the students “think” like mathematicians and connect their learning to mathematics in the real world. These are a shift in thinking  about math instruction and have become a vital part of math learning since the implementation of CommonCore Standards.

Kindergarten Overview

In Kindergarten, instructional time should focus on two areas: (1) developing a sound sense of numbers by representing and comparing numbers, initially using sets of objects; (2) recognizing and describing shapes and using spatial relations. More learning time in Kindergarten should be devoted to number than to any other topic. Please note that while every standard/topic in the grade level has not been included in this overview, all standards should be included in instruction.

1. Through their learning in the Counting and Cardinality and Operations and Algebraic Thinking domains, students:
• develop a more formal sense of numbers;
• use numbers, including written numerals, to represent quantities and to solve quantitative problems, such as counting objects in a set; counting out a given number of objects; comparing sets or numerals; and modeling simple joining and separating situations with sets of objects, or eventually with equations such as 5 + 2 = 7 and 7 – 2 = 5. Note: Kindergarten students should see addition and subtraction equations, and student writing of equations in kindergarten is encouraged, but it is not required; and
• choose, combine, and apply effective strategies for answering quantitative questions, including quickly recognizing the cardinalities of small sets of objects, counting and producing sets of given sizes, counting the number of objects in combined sets, or counting the number of objects that remain in a set after some are taken away.
1. Through their learning in the Geometry and Measurement and Data domains, students:
• describe their physical world using geometric ideas (e.g., shape, orientation, spatial relations) and appropriate vocabulary;
• identify, name, and describe basic two-dimensional shapes, such as squares, triangles, circles, rectangles, and hexagons, presented in a variety of ways (e.g., with different sizes and orientations), as well as three-dimensional shapes such as cubes, cones, cylinders, and spheres;
• use basic shapes and spatial reasoning to model objects in their everyday environment to create and compose more complex shapes; and
• explore* coins and begin identifying of pennies and dimes.

*Note: Explore indicates that the topic is an important concept that builds the foundation for progression toward mastery in later grades. Repeated experiences with these concepts, with immersion in the concrete, are vital.

NYS Next Generation Mathematics Learning Standards - Kindergarten

In Grade 1, instructional time should focus on three areas: (1) developing understanding of addition, subtraction, and strategies for addition and subtraction within 20; (2) developing understanding of whole number relationships and place value, including grouping in tens and ones; and (3) developing understanding of linear measurement and measuring lengths as iterating length units. Please note that while every standard/topic in the grade level has not been included in this overview, all standards should be included in instruction.

1. Through their learning in the Operations and Algebraic Thinking domain, students:
• develop strategies for adding and subtracting whole numbers based on their prior work with small numbers;
• use a variety of models, including discrete objects and length-based models (e.g., cubes connected to form lengths), to model add-to, take-from, put-together, take-apart, and compare situations to develop meaning for the operations of addition and subtraction, and to develop strategies to solve arithmetic problems with these operations;
• understand connections between counting and addition and subtraction (e.g., adding two is the same as counting on two);
• use properties of addition to add whole numbers and to create and use increasingly sophisticated strategies based on these properties (e.g., “making tens”) to solve addition and subtraction problems within 20; and
• build their understanding of the relationship between addition and subtraction by comparing a variety of solution strategies.
1. Through their learning in the Number and Operations in Base Ten domain, students:
• develop, discuss, and use efficient, accurate, and generalizable methods to add within 100 and subtract multiples of 10;
• compare whole numbers (at least to 100) to develop understanding of and solve problems involving their relative sizes;
• think of whole numbers between 10 and 100 in terms of tens and ones (especially recognizing the numbers 11 to 19 as composed of a ten and some ones); and
• understand the order of the counting numbers and their relative magnitudes through activities that build number sense.
1. Through their learning in the Measurement and Data domain, students:
• develop an understanding of the meaning and processes of measurement, including underlying concepts such as iterating (the mental activity of building up the

length of an object with equal-sized units) and the transitivity principle for indirect measurement.*

*Note: Students should apply the transitivity principle of indirect measurement to make comparisons, but they need not use this technical term

In Grade 2, instructional time should focus on four areas: (1) extending understanding of base-ten notation; (2) building fluency with addition and subtraction; (3) using standard units of measure; and (4) analyzing and classifying two dimensional shapes as polygons or non-polygons. Please note that while every standard/topic in the grade level has not been included in this overview, all standards should be included in instruction.

1. Through their learning in the Number and Operations in Base Ten domain, students:
• extend their understanding of the base-ten system. This includes ideas of counting in fives, tens, and multiples of hundreds, tens, and ones, as well as number relationships involving these units, including comparing; and
• understand multi-digit numbers (up to 1000) written in base-ten notation, recognizing that the digits in each place represent amounts of thousands, hundreds, tens, or ones (e.g., 853 is 8 hundreds + 5 tens + 3 ones).
1. Through their learning in the Operations and Algebraic Thinking and Numbers and Operations in Base Ten domains, students:
• use their understanding of addition to develop fluency with addition and subtraction within 100;
• solve problems within 1000 by applying their understanding of models for addition and subtraction, and they develop, discuss, and use efficient, accurate, and generalizable methods to compute sums and differences of whole numbers in base-ten notation, using their understanding of place value and the properties of operations; and select and accurately apply methods that are appropriate for the context and the numbers involved to mentally calculate sums and differences for numbers with only tens or only hundreds.
1. Through their learning in the Measurement and Data domain, students:
• recognize the need for standard units of measure (centimeter and inch) and use rulers and other measurement tools with the understanding that linear measure involves an iteration of units; and
• recognize that the smaller the unit, the more iterations needed to cover a given length.
1. Through their learning in the Geometry domain, students:
• describe and classify shapes as polygons or non-polygons;
• investigate, describe, and reason about decomposing and combining shapes to make other shapes; and
• draw, partition, and analyze two-dimensional shapes to develop a foundation for understanding area, congruence, similarity, and fractions in later grades.

NYS Next Generation Mathematics Learning Standards - Grade 2

In Grade 3, instructional time should focus on four areas: (1) developing understanding of multiplication and division and strategies for multiplication and division within 100; (2) developing understanding of fractions, especially unit fractions (fractions with numerator 1); (3) developing understanding of the structure of rectangular arrays and of area; and (4) describing and analyzing polygons based on the number of sides and vertices. Please note that while every standard/topic in the grade level has not been included in this overview, all standards should be included in instruction.

1. Through their learning in the Operations and Algebraic Thinking domain, students:
• develop an understanding of the meanings of multiplication and division of whole numbers through activities and problems involving equal-sized groups, arrays, and area models; multiplication is finding an unknown product, and division is finding an unknown factor in these situations. For equal-sized group situations, division can require finding the unknown number of groups or the unknown group size;
• use properties of operations to calculate products of whole numbers, using increasingly sophisticated strategies based on these properties to solve multiplication and division problems involving single-digit factors; and
• compare a variety of solution strategies to learn the relationship between multiplication and division.
1. Through their learning in the Number Sense and Operations—Fractions domain, students:
• develop an understanding of fractions, beginning with unit fractions;
• view fractions in general as being built out of unit fractions, and use fractions along with visual fraction models to represent parts of a whole;
• understand that the size of a fractional part is relative to the size of the whole. Use fractions to represent numbers equal to, less than, and greater than one; and
• solve problems that involve comparing fractions by using visual fraction models and strategies based on noticing equal numerators or denominators.
1. Through their learning in the Measurement and Data domain, students:
• recognize area as an attribute of two-dimensional regions;
• measure the area of a shape by finding the total number of same-size units of area required to cover the shape without gaps or overlaps, a square with sides of unit length being the standard unit for measuring area; and
• understand that rectangular arrays can be decomposed into identical rows or into identical columns. By decomposing rectangles into rectangular arrays of squares, students connect area to multiplication, and justify using multiplication to determine the area of a rectangle.
1. Through their learning in the Geometry domain, students:
• classify polygons by examining their sides and vertices; and
• relate their fraction work to geometry by expressing the area of part of a shape as a unit fraction of the whole.

NYS Next Generation Mathematics Learning Standards - Grade 3

In Grade 4, instructional time should focus on three areas: (1) developing understanding and fluency with multi-digit multiplication, and developing understanding of dividing to find quotients involving multi-digit dividends; (2) developing an understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers; and (3) understanding that geometric figures can be analyzed and classified based on their properties, such as having parallel sides, perpendicular sides, particular angle measures, and symmetry. Please note that while every standard/topic in the grade level has not been included in this overview, all standards should be included in instruction.

Through their learning in the Number and Operations in Base Ten domain, students:

• generalize their understanding of place value to 1,000,000, understanding the relative sizes of numbers in each place;
• apply their understanding of models for multiplication (equal-sized groups, arrays, area models), place value, and properties of operations as they develop, discuss, and use efficient, accurate, and generalizable methods to compute products of multi-digit whole numbers;
• select and accurately apply appropriate methods to estimate or mentally calculate products, depending on the numbers and the context;
• develop fluency with efficient procedures for multiplying whole numbers; understand and explain why the procedures work based on place value and properties of operations; and use them to solve problems;
• apply their understanding of models for division, place value, properties of operations, and the relationship of division to multiplication as they develop, discuss, and use efficient, accurate, and generalizable procedures to find quotients involving multi-digit dividends; and
• select and accurately apply appropriate methods to estimate and mentally calculate quotients, and interpret remainders based upon the context.

Through their learning in the Numbers and Operations—Fractions domain, students:

• develop understanding of fraction equivalence and operations with fractions;
• recognize that two different fractions can be equal (e.g., 15/9 = 5/3), and develop methods for generating and recognizing equivalent fractions; and
• extend previous understandings about how fractions are built from unit fractions, composing fractions from unit fractions, decomposing fractions into unit fractions, and using the meaning of fractions and the meaning of multiplication to multiply a fraction by a whole number. Through their learning in the Geometry domain, students:
• deepen their understanding of properties of two-dimensional shapes (e.g., angles, parallelism, and symmetry).

NYS Next Generation Mathematics Learning Standards - Grade 4

In Grade 5, instructional time should focus on three areas: (1) developing fluency with addition and subtraction of fractions, and developing understanding of the multiplication of fractions and of division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions); (2) extending division to 2-digit divisors, integrating decimals into the place value system and developing understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal operations; and (3) developing understanding of volume. Please note that while every standard/topic in the grade level has not been included in this overview, all standards should be included in instruction.

1. Through their learning in the Number and Operations – Fractions and Operations and Algebraic Thinking domains, students:
• apply their understanding of fractions and fraction models to represent the addition and subtraction of fractions with unlike denominators as equivalent calculations with like denominators;
• develop fluency in calculating sums and differences of fractions, and make reasonable estimates of them; and
• use the meaning of fractions, of multiplication and division, and the relationship between multiplication and division to understand and explain why the procedures for multiplying and dividing fractions make sense. (note: this is limited to the case of dividing unit fractions by whole numbers and whole numbers by unit fractions.)
1. Through their learning in the Operations and Algebraic Thinking and Number and Operations in Base Ten domains, students:
• develop understanding of why division procedures work based on the meaning of base-ten numerals and properties of operations;
• apply understandings of models for decimals, decimal notation, and properties of operations to add and subtract decimals to hundredths;
• develop fluency with decimal computations to hundredths, and make reasonable estimates of their results; and
• use the relationship between decimals and fractions, as well as the relationship between finite decimals and whole numbers (i.e., a finite decimal multiplied by an appropriate power of 10 is a whole number), to understand and explain why the procedures for multiplying and dividing finite decimals make sense.
1. Through their learning in the Measurement and Data and Geometry domains, students:
• recognize volume as an attribute of three-dimensional space;
• understand that volume can be measured by finding the total number of same-size units of volume required to fill the space without gaps or overlaps;
• understand that a 1-unit by 1-unit by 1-unit cube is the standard unit for measuring volume;
• select appropriate units, strategies, and tools for solving problems that involve estimating and measuring volume;
• decompose three-dimensional shapes and find volumes of right rectangular prisms by viewing them as decomposed into layers of arrays of cubes; and
• measure necessary attributes of shapes in order to determine volumes to solve real world and mathematical problems.

NYS Next Generation Mathematics Learning Standards - Grade 5