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Problem Solving Strand

Students will build new mathematical knowledge through problem solving

Know the difference between relevant and irrelevant information when solving problems

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

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

Formulate problems and solutions from everyday situations

Translate from a picture/diagram to a numeric expression

Represent problem situations verbally, numerically, algebraically, and/or graphically

Select an appropriate representation of a problem

Understand the basic language of logic in mathematical situations (and, or, not)

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

Work in collaboration with others to solve problems

Translate from a picture/diagram to a number or symbolic expression

Use trial and error and the process of elimination to solve problems

Model problems with pictures/diagrams or physical objects

Analyze problems by observing patterns

Make organized lists or charts to solve numerical problems

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

Discuss with peers to understand a problem situation

Determine what information is needed to solve problem

Determine the efficiency of different representations of a problem

Differentiate between valid and invalid approaches

Understand valid counterexamples

Explain the methods and reasoning behind the problem solving strategies used

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

Verify results of a problem

Reasoning and Proof Strand

Students will recognize reasoning and proof as fundamental aspects of mathematics

Recognize that mathematical ideas can be supported using a variety of strategies

Understand that mathematical statements can be supported, using models, facts, and relationships to explain their thinking

Students will make and investigate mathematical conjectures

Investigate conjectures, using arguments and appropriate mathematical terms

Make and evaluate conjectures, using a variety of strategies

Students will develop and evaluate mathematical arguments and proofs

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

Develop and explain an argument verbally, numerically, and/or graphically

Verify claims other students make, using examples and counterexamples when appropriate

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

Support an argument through examples/counterexamples and special cases

Communication Strand

Students will organize and consolidate their mathematical thinking through communication

Provide an organized thought process that is correct, complete, coherent, and clear

Explain a rationale for strategy selection

Organize and accurately label work

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

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

Answer clarifying questions from others

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

Understand mathematical solutions shared by other students

Raise questions that elicit, extend, or challenge others' thinking

Consider strategies used and solutions found by others 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

Use appropriate vocabulary when describing objects, relationships, mathematical solutions, and rationale

Decode and comprehend mathematical visuals and symbols to construct meaning

Connections Strand

Students will recognize and use connections among mathematical ideas

Understand and make connections and conjectures in their everyday experiences to mathematical ideas

Explore and explain the relationship between 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 draw conclusions

Students will recognize and apply mathematics in contexts outside of mathematics

Recognize and provide examples of the presence of mathematics in their daily lives

Apply mathematics to problem situations that develop outside of mathematics

Investigate the presence of mathematics in careers and areas of interest

Recognize and apply mathematics to other disciplines and areas of interest

Representation Strand

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

Use physical objects, drawings, charts, tables, graphs, symbols, equations, or objects created using technology as representations

Explain, describe, and defend mathematical ideas using representations

Read, interpret, and extend external models

Use standard and nonstandard representations with accuracy and detail

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

Use representations to explore problem situations

Investigate relationships between different representations and their impact on a given problem

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

Use mathematics to show and understand physical phenomena (e.g., determine the perimeter of a bulletin board)

Use mathematics to show and understand social phenomena (e.g., construct tables to organize data showing book sales)

Use mathematics to show and understand mathematical phenomena (e.g., find the missing value that makes the equation true: (3 + 4) + 5 = 3 + (4 + ___ )

Number Sense and Operations Strand

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

Read and write whole numbers to millions

Compare and order numbers to millions

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

Create equivalent fractions, given a fraction

Compare and order fractions including unlike denominators (with and without the use of a number line) Note: Commonly used fractions such as those that might be indicated on ruler, measuring cup, etc.

Understand the concept of ratio

Express ratios in different forms

Read, write, and order decimals to thousandths

Compare fractions using <, >, or =

Compare decimals using <, >, or =

Understand that percent means part of 100, and write percents as fractions and decimals

Recognize that some numbers are only divisible by one and themselves (prime) and others have multiple divisors (composite)

Calculate multiples of a whole number and the least common multiple of two numbers

Identify the factors of a given number

Find the common factors and the greatest common factor of two numbers

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

Use a variety of strategies to multiply three-digit by three-digit numbers Note: Multiplication by anything greater than a three-digit multiplier/ multiplicand should be done using technology

Use a variety of strategies to divide three-digit numbers by one-and two-digit numbers Note: Division by anything greater than a two-digit divisor should be done using technology

Evaluate an arithmetic expression using order of operations including multiplication, division, addition, subtraction and parentheses

Simplify fractions to lowest terms

Convert improper fractions to mixed numbers, and mixed numbers to improper fractions

Use a variety of strategies to add and subtract fractions with like denominators

Add and subtract mixed numbers with like denominators

Use a variety of strategies to add, subtract, multiply, and divide decimals to thousandths

Students will compute accurately and make reasonable estimates

Round numbers to the nearest hundredth and up to 10,000

Estimate sums and differences of fractions with like denominators

Estimate sums, differences, products, and quotients of decimals

Justify the reasonableness of answers using estimation

Algebra Strand

Students will represent and analyze algebraically a wide variety of problem solving situations

Define and use appropriate terminology when referring to constants, variables, and algebraic expressions

Translate simple verbal expressions into algebraic expressions

Students will perform algebraic procedures accurately

Substitute assigned values into variable expressions and evaluate using order of operations

Solve simple one-step equations using basic whole-number facts

Solve and explain simple one-step equations using inverse operations involving whole numbers

Evaluate the perimeter formula for given input values

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

Create and explain patterns and algebraic relationships (e.g.,2,4,6,8...) algebraically: 2n (doubling)

Create algebraic or geometric patterns using concrete objects or visual drawings (e.g., rotate and shade geometric shapes)

Geometry Strand

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

Calculate the perimeter of regular and irregular polygons

Students will identify and justify geometric relationships, formally and informally

Identify pairs of similar triangles

Identify the ratio of corresponding sides of similar triangles

Classify quadrilaterals by properties of their angles and sides

Know that the sum of the interior angles of a quadrilateral is 360 degrees

Classify triangles by properties of their angles and sides

Know that the sum of the interior angles of a triangle is 180 degrees

Find a missing angle when given two angles of a triangle

Identify pairs of congruent triangles

Identify corresponding parts of congruent triangles

Students will apply transformations and symmetry to analyze problem solving situations

Identify and draw lines of symmetry of basic geometric shapes

Students will apply coordinate geometry to analyze problem solving situations

Identify and plot points in the first quadrant

Plot points to form basic geometric shapes (identify and classify)

Calculate perimeter of basic geometric shapes drawn on a coordinate plane (rectangles and shapes composed of rectangles having sides with integer lengths and parallel to the axes)

Measurement Strand

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

Use a ruler to measure to the nearest inch, 1/2, 1/4, and 1/8 inch

Identify customary equivalent units of length

Measure to the nearest centimeter

Identify equivalent metric units of length

Convert measurement within a given system

Determine the tool and technique to measure with an appropriate level of precision: lengths and angles

Students will use units to give meaning to measurements

Calculate elapsed time in hours and minutes

Measure and draw angles using a protractor

Students will develop strategies for estimating measurements

Determine personal references for customary units of length (e.g., your pace is approximately 3 feet, your height is approximately 5 feet, etc.)

Determine personal references for metric units of length

Justify the reasonableness of estimates

Statistics and Probability Strand

Students will collect, organize, display, and analyze data

Collect and record data from a variety of sources (e.g., newspapers, magazines, polls, charts, and surveys)

Display data in a line graph to show an increase or decrease over time

Calculate the mean for a given set of data and use to describe a set of data

Students will make predictions that are based upon data analysis

Formulate conclusions and make predictions from graphs

Students will understand and apply concepts of probability

List the possible outcomes for a single-event experiment

Record experiment results using fractions/ratios

Create a sample space and determine the probability of a single event, given a simple experiment (e.g., rolling a number cube)

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