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Problem Solving Strand

Students will build new mathematical knowledge through problem solving

Explore, examine, and make observations about a social problem or mathematical situation

Understand that some ways of representing a problem are more helpful than others

Interpret information correctly, identify the problem, and generate possible solutions

Students will solve problems that arise in mathematics and in other contexts

Act out or model with manipulatives activities involving mathematical content from literature and/or story telling

Formulate problems and solutions from everyday situations

Translate from a picture/diagram to a numeric expression

Represent problem situations in oral, written, concrete, pictorial, and graphical forms

Select an appropriate representation of a problem

Students will apply and adapt a variety of appropriate strategies to solve problems

Use trial and error to solve problems

Use process of elimination to solve problems

Make pictures/diagrams of problems

Use physical objects to model problems

Work in collaboration with others to solve problems

Make organized lists to solve numerical problems

Make charts to solve numerical problems

Analyze problems by identifying relationships

Analyze problems by identifying relevant versus irrelevant information

Analyze problems by observing patterns

State a problem in their own words

Students will monitor and reflect on the process of mathematical problem solving

Determine what information is needed to solve a problem

Discuss with peers to understand a problem situation

Discuss the efficiency of different representations of a problem

Verify results of a problem

Recognize invalid approaches

Determine whether a solution is reasonable in the context of the original problem

Reasoning and Proof Strand

Students will recognize reasoning and proof as fundamental aspects of mathematics

Use representations to support mathematical ideas

Determine whether a mathematical statement is true or false and explain why

Students will make and investigate mathematical conjectures

Investigate the use of knowledgeable guessing by generalizing mathematical ideas

Make conjectures from a variety of representations

Students will develop and evaluate mathematical arguments and proofs

Justify general claims or conjectures, using manipulatives, models, and expressions

Develop and explain an argument using oral, written, concrete, pictorial, and/or graphical forms

Discuss, listen, and make comments that support or reject claims made by other students

Students will select and use various types of reasoning and methods of proof

Support an argument by trying many cases

Communication Strand

Students will organize and consolidate their mathematical thinking through communication

Understand and explain how to organize their thought process

Verbally explain their rationale for strategy selection

Provide reasoning both in written and verbal form

Students will communicate their mathematical thinking coherently and clearly to peers, teachers, and others

Organize and accurately label work

Share organized mathematical ideas through the manipulation of objects, drawings, pictures, charts, graphs, tables, diagrams, models, symbols, and expressions in written and verbal form

Answer clarifying questions from others

Students will analyze and evaluate the mathematical thinking and strategies of others

Listen for understanding of mathematical solutions shared by other students

Consider strategies used and solutions found in relation to their own work

Students will use the language of mathematics to express mathematical ideas precisely

Increase their use of mathematical vocabulary and language when communicating with others

Describe objects, relationships, solutions and rationale using appropriate vocabulary

Decode and comprehend mathematical visuals and symbols to construct meaning

Connections Strand

Students will recognize and use connections among mathematical ideas

Recognize, understand, and make connections in their everyday experiences to mathematical ideas

Compare and contrast mathematical ideas

Connect and apply mathematical information to solve problems

Students will understand how mathematical ideas interconnect and build on one another to produce a coherent whole

Understand multiple representations and how they are related

Model situations with objects and representations and be able to make observations

Students will recognize and apply mathematics in contexts outside of mathematics

Recognize the presence of mathematics in their daily lives

Apply mathematics to solve problems that develop outside of mathematics

Recognize and apply mathematics to other disciplines

Representation Strand

Students will create and use representations to organize, record, and communicate mathematical ideas

Use verbal and written language, physical models, drawing charts, graphs, tables, symbols, and equations as representations

Share mental images of mathematical ideas and understandings

Recognize and use external mathematical representations

Use standard and nonstandard representations with accuracy and detail

Students will select, apply, and translate among mathematical representations to solve problems

Understand similarities and differences in representations

Connect mathematical representations with problem solving

Construct effective representations to solve problems

Students will use representations to model and interpret physical, social, and mathematical phenomena

Use mathematics to show and understand physical phenomena (e.g., estimate and represent the number of apples in a tree)

Use mathematics to show and understand social phenomena (e.g., count and represent sharing cookies between friends)

Use mathematics to show and understand mathematical phenomena (e.g., use a multiplication grid to solve odd and even number problems)

Number Sense and Operations Strand

Students will understand numbers, multiple ways of representing numbers, relationships among numbers, and number systems

Skip count by 25s, 50s, 100s to 1,000

Read and write whole numbers to 1,000

Compare and order numbers to 1,000

Understand the place value structure of the base ten number system:10 ones = 1 ten; 10 tens = 1 hundred; 10 hundreds = 1 thousand

Use a variety of strategies to compose and decompose three-digit numbers

Use and explain the commutative property of addition and multiplication

Use 1 as the identity element for multiplication

Use the zero property of multiplication

Understand and use the associative property of addition

Develop an understanding of fractions as part of a whole unit and as parts of a collection

Use manipulatives, visual models, and illustrations to name and represent unit fractions (1/2, 1/3, 1/5, 1/6, and 1/10) as part of a whole or a set of objects

Understand and recognize the meaning of numerator and denominator in the symbolic form of a fraction

Recognize fractional numbers as equal parts of a whole

Explore equivalent fractions (1/2, 1/3, 1/4)

Compare and order unit fractions (1/2, 1/3, 1/4) and find their approximate locations on a number line

Identify odd and even numbers

Develop an understanding of the properties of odd/even numbers as a result of addition or subtraction

Students will understand meanings of operations and procedures, and how they relate to one another

Use a variety of strategies to add and subtract 3-digit numbers (with and without regrouping)

Develop fluency with single-digit multiplication facts

Use a variety of strategies to solve multiplication problems with factors up to 12 x 12

Use the area model, tables, patterns, arrays, and doubling to provide meaning for multiplication

Demonstrate fluency and apply single-digit division facts

Use tables, patterns, halving, and manipulatives to provide meaning for division

Develop strategies for selecting the appropriate computational and operational method in problem solving situations

Students will compute accurately and make reasonable estimates

Estimate numbers up to 500

Recognize real world situations in which an estimate (rounding) is more appropriate

Check reasonableness of an answer by using estimation

Algebra Strand

Students will perform algebraic procedures accurately

Use the symbols <, >, = (with and without the use of a number line) to compare whole numbers up to 100 (1/2, 1/3, 1,4, 1/5, 1/6, and 1/10)

Students will recognize, use, and represent algebraically patterns, relations, and functions

Describe and extend numeric (+, -) and geometric patterns

Geometry Strand

Students will use visualization and spatial reasoning to analyze characteristics and properties of geometric shapes

Define and use correct terminology when referring to shapes (circle, triangle, square, rectangle, rhombus, trapezoid, and hexagon)

Identify congruent and similar figures

Name, describe, compare, and sort three-dimensional shapes: cube, cylinder, sphere, prism, and cone

Identify the faces on a three-dimensional shape as two-dimensional

Students will apply transformations and symmetry to analyze problem solving situations

Identify and construct lines of symmetry

Measurement Strand

Students will determine what can be measured and how, using appropriate methods and formulas

Select tools and units (customary) appropriate for the length measured

Use a ruler/yardstick to measure to the nearest standard unit (whole and 1/2 inches, whole feet, and whole yards)

Measure objects, using ounces and pounds

Recognize capacity as an attribute that can be measured

Compare capacities (e.g., Which contains more? Which contains less?)

Measure capacity, using cups, pints, quarts, and gallons

Students will use units to give meaning to measurements

Count and represent combined coins and dollars, using currency symbols ($0.00)

Relate unit fractions to the face of the clock: Whole = 60 minutes; 1/2 = 30 minutes; 1/4 = 15 minutes

Students will develop strategies for estimating measurements

Tell time to the minute, using digital and analog clocks

Select and use standard (customary) and non-standard units to estimate measurements

Statistics and Probability Strand

Students will collect, organize, display, and analyze data

Formulate questions about themselves and their surroundings

Collect data using observation and surveys, and record appropriately

Construct a frequency table to represent a collection of data

Identify the parts of pictographs and bar graphs

Display data in pictographs and bar graphs

State the relationships between pictographs and bar graphs

Read and interpret data in bar graphs and pictographs

Students will make predictions that are based upon data analysis

Formulate conclusions and make predictions from graphs

Information Source: NYSED.GOV CS&IT (standards pdf)