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Problem Solving Strand

Students will build new mathematical knowledge through problem solving

Use a variety of strategies to understand new mathematical content and to develop more efficient methods

Construct appropriate extensions to problem situations

Understand and demonstrate how written symbols represent mathematical ideas

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

Observe patterns and formulate generalizations

Make conjectures from generalizations

Represent problem situations verbally, numerically, algebraically, and graphically

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

Understand that there is no one right way to solve mathematical problems but that different methods have advantages and disadvantages

Understand how to break a complex problem into simpler parts or use a similar problem type to solve a problem

Work backwards from a solution

Use proportionality to model problems

Work in collaboration with others to solve problems

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

Interpret solutions within the given constraints of a problem

Set expectations and limits for possible solutions

Determine information required to solve the problem

Choose methods for obtaining required information

Justify solution methods through logical argument

Evaluate the efficiency of different representations 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 by a variety

Students will make and investigate mathematical conjectures

Use mathematical strategies to reach a conclusion

Evaluate conjectures by distinguishing relevant from irrelevant information to reach a conclusion or make appropriate estimates

Students will develop and evaluate mathematical arguments and proofs

Provide supportive arguments for conjectures

Develop, verify, and explain an argument, using appropriate mathematical ideas and language

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

Support an argument by using a systematic approach to test more than one case

Devise ways to verify results or use counterexamples to refute incorrect statements

Apply inductive reasoning in making and supporting mathematical conjectures

Communication Strand

Students will organize and consolidate their mathematical thinking through communication

Provide a correct, complete, coherent, and clear rationale for thought process used in problem solving

Provide an organized argument which explains 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

Analyze mathematical solutions shared by others

Compare strategies used and solutions found by others in relation to their own work

Formulate mathematical questions that elicit, extend, or challenge strategies, solutions, and/or conjectures of others

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 language, representations, and terminology when describing objects, relationships, mathematical solutions, and rationale

Draw conclusions about mathematical ideas through decoding, comprehension, and interpretation of mathematical visuals, symbols, and technical writing

Connections Strand

Students will recognize and use connections among mathematical ideas

Understand and make connections among multiple representations of the same mathematical idea

Recognize connections between subsets of mathematical ideas

Connect and apply a variety of strategies to solve problems

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

Model situations mathematically, using representations to draw conclusions and formulate new situations

Understand how concepts, procedures, and mathematical results in one area of mathematics can be used to solve problems in other areas of mathematics

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 mathematical ideas 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, areas of interest, and societal issues

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

Recognize, compare, and use an array of representational forms

Explain how different representations express the same relationship

Use standard and non-standard 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

Use representation as a tool for exploring and understanding mathematical ideas

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

Use mathematics to show and understand physical phenomena (e.g., make and interpret scale drawings of figures or scale models of objects)

Use mathematics to show and understand social phenomena (e.g., determine profit from sale of yearbooks)

Use mathematics to show and understand mathematical phenomena (e.g., use tables, graphs, and equations to show a pattern underlying a function)

Number Sense and Operations Strand

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

Distinguish between the various subsets of real numbers (counting/natural numbers, whole numbers, integers, rational numbers, and irrational numbers)

Recognize the difference between rational and irrational numbers (e.g., explore different approximations of pi )

Place rational and irrational numbers (approximations) on a number line and justify the placement of the numbers

Develop the laws of exponents for multiplication and division

Write numbers in scientific notation

Translate numbers from scientific notation into standard form

Compare numbers written in scientific notation

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

Determine multiples and least common multiple of two or more numbers

Determine the prime factorization of a given number and write in exponential form

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

Simplify expressions using order of operations Note: Expressions may include absolute value and/or integral exponents greater than 0

Add, subtract, multiply, and divide integers

Add and subtract two integers (with and without the use of a number line)

Develop a conceptual understanding of negative and zero exponents with a base of ten and relate to fractions and decimals (e.g., 10-2 = .01 = 1/100)

Recognize and state the value of the square root of a perfect square (up to 225)

Determine the square root of non-perfect squares using a calculator

Classify irrational numbers as non-repeating/non-terminating decimals

Students will compute accurately and make reasonable estimates

Identify the two consecutive whole numbers between which the square root of a non-perfect square whole number less than 225 lies (with and without the use of a number line)

Justify the reasonableness of answers using estimation

Algebra Strand

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

Translate two-step verbal expressions into algebraic expressions

Students will perform algebraic procedures accurately

Add and subtract monomials with exponents of one

Identify a polynomial as an algebraic expression containing one or more terms

Solve multi-step equations by combining like terms, using the distributive property, or moving variables to one side of the equation

Solve one-step inequalities (positive coefficients only) (See 7.G.10)

Evaluate formulas for given input values (surface area, rate, and density problems)

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

Draw the graphic representation of a pattern from an equation or from a table of data

Create algebraic patterns using charts/tables, graphs, equations, and expressions

Build a pattern to develop a rule for determining the sum of the interior angles of polygons

Write an equation to represent a function from a table of values

Geometry Strand

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

Calculate the radius or diameter, given the circumference or area of a circle

Calculate the volume of prisms and cylinders, using a given formula and a calculator

Identify the two-dimensional shapes that make up the faces and bases of three-dimensional shapes (prisms, cylinders, cones, and pyramids)

Determine the surface area of prisms and cylinders, using a calculator and a variety of methods

Students will identify and justify geometric relationships, formally and informally

Identify the right angle, hypotenuse, and legs of a right triangle

Explore the relationship between the lengths of the three sides of a right triangle to develop the Pythagorean Theorem

Find a missing angle when given angles of a quadrilateral

Use the Pythagorean Theorem to determine the unknown length of a side of a right triangle

Determine whether a given triangle is a right triangle by applying the Pythagorean Theorem and using a calculator

Students will apply coordinate geometry to analyze problem solving situations

Graph the solution set of an inequality (positive coefficients only) on a number line (See 7.A.5)

Measurement Strand

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

Calculate distance using a map scale

Convert capacities and volumes within a given system

Identify customary and metric units of mass

Convert mass within a given system

Calculate unit price using proportions

Compare unit prices

Convert money between different currencies with the use of an exchange rate table and a calculator

Draw central angles in a given circle using a protractor (circle graphs)

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

Students will develop strategies for estimating measurements

Identify the relationships between relative error and magnitude when dealing with large numbers (e.g., money, population)

Estimate surface area

Determine personal references for customary /metric units of mass

Justify the reasonableness of the mass of an object

Statistics and Probability Strand

Students will collect, organize, display, and analyze data

Identify and collect data using a variety of methods

Display data in a circle graph

Convert raw data into double bar graphs and double line graphs

Calculate the range for a given set of data

Select the appropriate measure of central tendency

Read and interpret data represented graphically (pictograph, bar graph, histogram, line graph, double line/bar graphs or circle graph)

Students will make predictions that are based upon data analysis

Identify and explain misleading statistics and graphs

Students will understand and apply concepts of probability

Interpret data to provide the basis for predictions and to establish experimental probabilities

Determine the validity of sampling methods to predict outcomes

Predict the outcome of an experiment

Design and conduct an experiment to test predictions

Compare actual results to predicted results

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