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Problem Solving Strand

Students will build new mathematical knowledge through problem solving

Use a variety of problem solving strategies to understand new mathematical content

Recognize and understand equivalent representations of a problem situation or a mathematical concept

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

Observe and explain patterns to formulate generalizations and conjectures

Use multiple representations to represent and explain problem situations (e.g., verbally, numerically, algebraically, graphically)

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

Choose an effective approach to solve a problem from a variety of strategies (numeric, graphic, algebraic)

Use a variety of strategies to extend solution methods to other problems

Work in collaboration with others to propose, critique, evaluate, and value alternative approaches to problem solving

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

Determine information required to solve the problem, choose methods for obtaining the information, and define parameters for acceptable solutions

Interpret solutions within the given constraints of a problem

Evaluate the relative efficiency of different representations and solution methods of a problem

Reasoning and Proof Strand

Students will recognize reasoning and proof as fundamental aspects of mathematics

Support mathematical ideas using a variety of strategies

Students will make and investigate mathematical conjectures

Investigate and evaluate conjectures in mathematical terms, using mathematical strategies to reach a conclusion

Evaluate conjectures and recognize when an estimate or approximation is more appropriate than an exact answer

Recognize when an approximation is more appropriate than an exact answer

Students will develop and evaluate mathematical arguments and proofs

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

Construct logical arguments that verify claims or counterexamples that refute claims

Present correct mathematical arguments in a variety of forms

Evaluate written arguments for validity

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, using counterexamples and informal indirect proof

Extend specific results to more general cases

Apply inductive reasoning in making and supporting mathematical conjectures

Communication Strand

Students will organize and consolidate their mathematical thinking through communication

Communicate verbally and in writing a correct, complete, coherent, and clear design (outline) and explanation for the steps used in solving a problem

Use mathematical representations to communicate with appropriate accuracy, including numerical tables, formulas, functions, equations, charts, graphs, and diagrams

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

Present organized mathematical ideas with the use of appropriate standard notations, including the use of symbols and other representations when sharing an idea in verbal and written form

Explain relationships among different representations of a problem

Communicate logical arguments clearly, showing why a result makes sense and why the reasoning is valid

Support or reject arguments or questions raised by others about the correctness of mathematical work

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

Read and listen for logical understanding of mathematical thinking shared by other students

Reflect on strategies of others in relation to one's own strategy

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

Use correct mathematical language in developing mathematical questions that elicit, extend, or challenge other students' conjectures

Represent word problems using standard mathematical notation

Understand and 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

Understand the corresponding procedures for similar problems or mathematical concepts

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

Understand how quantitative models connect to various physical models and representations

Students will recognize and apply mathematics in contexts outside of mathematics

Recognize and apply mathematics to situations in the outside world

Recognize and apply mathematical ideas to problem situations that develop outside of mathematics

Develop an appreciation for the historical development of mathematics

Representation Strand

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

Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts

Recognize, compare, and use an array of representational forms

Use representation as a tool for exploring and understanding mathematical ideas

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

Select appropriate representations to solve problem situations

Investigate relationships among 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., investigate sound waves using the sine and cosine functions)

Use mathematics to show and understand social phenomena (e.g., interpret the results of an opinion poll)

Use mathematics to show and understand mathematical phenomena (e.g., use random number generator to simulate a coin toss)

Number Sense and Operations Strand

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

Evaluate numerical expressions with negative and/or fractional exponents, without the aid of a calculator (when the answers are rational numbers)

Perform arithmetic operations (addition, subtraction, multiplication, division) with expressions containing irrational numbers in radical form

Perform arithmetic operations with polynomial expressions containing rational coefficients

Perform arithmetic operations on irrational expressions

Rationalize a denominator containing a radical expression

Write square roots of negative numbers in terms of i

Simplify powers of i

Determine the conjugate of a complex number

Perform arithmetic operations on complex numbers and write the answer in the form a + bi Note: This includes simplifying expressions with complex denominators

Know and apply sigma notation

Algebra Strand

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

Solve absolute value equations and inequalities involving linear expressions in one variable

Use the discriminant to determine the nature of the roots of a quadratic equation

Solve systems of equations involving one linear equation and one quadratic equation algebraically Note: This includes rational equations that result in linear equations with extraneous roots

Solve quadratic inequalities in one and two variables, algebraically and graphically

Use direct and inverse variation to solve for unknown values

Solve an application which results in an exponential function

Students will perform algebraic procedures accurately

Factor polynomial expressions completely, using any combination of the following techniques: common factor extraction, difference of two perfect squares, quadratic trinomials

Apply the rules of exponents to simplify expressions involving negative and/or fractional exponents

Rewrite algebraic expressions that contain negative exponents using only positive exponents

Rewrite algebraic expressions with fractional exponents as radical expressions

Rewrite algebraic expressions in radical form as expressions with fractional exponents

Evaluate exponential expressions, including those with base e

Simplify radical expressions

Perform addition, subtraction, multiplication, and division of radical expressions

Rationalize denominators involving algebraic radical expressions

Perform arithmetic operations with rational expressions and rename to lowest terms

Simplify complex fractional expressions

Evaluate logarithmic expressions in any base

Apply the properties of logarithms to rewrite logarithmic expressions in equivalent forms

Determine the sum and product of the roots of a quadratic equation by examining its coefficients

Determine the quadratic equation, given the sum and product of its roots

Solve radical equations

Solve rational equations and inequalities

Know and apply the technique of completing the square

Solve quadratic equations, using the quadratic formula

Find the solution to polynomial equations of higher degree that can be solved using factoring and/or the quadratic formula

Solve exponential equations with and without common bases

Solve a logarithmic equation by rewriting as an exponential equation

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

Identify an arithmetic or geometric sequence and find the formula for its nth term

Determine the common difference in an arithmetic sequence

Determine the common ratio in a geometric sequence

Determine a specified term of an arithmetic or geometric sequence

Specify terms of a sequence, given its recursive definition

Represent the sum of a series, using sigma notation

Determine the sum of the first n terms of an arithmetic or geometric series

Apply the binomial theorem to expand a binomial and determine a specific term of a binomial expansion

Define a relation and function

Determine when a relation is a function

Determine the domain and range of a function from its equation

Write functions in functional notation

Use functional notation to evaluate functions for given values in the domain

Find the composition of functions

Determine if a function is one-to-one, onto, or both

Define the inverse of a function

Determine the inverse of a function and use composition to justify the result

Perform transformations with functions and relations: ƒ(x + a) , ƒ(x) + a, ƒ(-x), - ƒ(x), aƒ(x)

Determine the center-radius form for the equation of a circle in standard form

Write the equation of a circle, given its center and a point on the circle

Write the equation of a circle from its graph

Approximate the solution to polynomial equations of higher degree by inspecting the graph

Determine the domain and range of a function from its graph

Identify relations and functions, using graphs

Graph exponential functions of the form y = bx for positive values of b, including b = e

Graph logarithmic functions, using the inverse of the related exponential function

Express and apply the six trigonometric functions as ratios of the sides of a right triangle

Know the exact and approximate values of the sine, cosine, and tangent of 0°, 30°, 45°, 60°, 90°, 180°, and 270° angles

Sketch and use the reference angle for angles in standard position

Know and apply the co-function and reciprocal relationships between trigonometric ratios

Use the reciprocal and co-function relationships to find the value of the secant, cosecant, and cotangent of 0°, 30°, 45°, 60°, 90°, 180°, and 270° angles

Sketch the unit circle and represent angles in standard position

Determine the length of an arc of a circle, given its radius and the measure of its central angle

Find the value of trigonometric functions, if given a point on the terminal side of angle θ

Restrict the domain of the sine, cosine, and tangent functions to ensure the existence of an inverse function

Use inverse functions to find the measure of an angle, given its sine, cosine, or tangent

Sketch the graph of the inverses of the sine, cosine, and tangent functions

Determine the trigonometric functions of any angle, using technology

Justify the Pythagorean identities

Solve trigonometric equations for all values of the variable from 0° to 360°

Determine amplitude, period, frequency, and phase shift, given the graph or equation of a periodic function

Sketch and recognize one cycle of a function of the form y = Asin Bx or y = Acos Bx

Sketch and recognize the graphs of the functions y = sec(x), y = csc(x), y = tan(x), and y = cot(x)

Write the trigonometric function that is represented by a given periodic graph

Solve for an unknown side or angle, using the Law of Sines or the Law of Cosines

Determine the area of a triangle or a parallelogram, given the measure of two sides and the included angle

Determine the solution(s) from the SSA situation (ambiguous case)

Apply the angle sum and difference formulas for trigonometric functions

Apply the double-angle and half-angle formulas for trigonometric functions

Measurement Strand

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

Define radian measure

Convert between radian and degree measures

Statistics and Probability Strand

Students will collect, organize, display, and analyze data

Understand the differences among various kinds of studies (e.g., survey, observation, controlled experiment)

Determine factors which may affect the outcome of a survey

Calculate measures of central tendency with group frequency distributions

Calculate measures of dispersion (range, quartiles, interquartile range, standard deviation, variance) for both samples and populations

Know and apply the characteristics of the normal distribution

Students will make predictions that are based upon data analysis

Determine from a scatter plot whether a linear, logarithmic, exponential, or power regression model is most appropriate

Determine the function for the regression model, using appropriate technology, and use the regression function to interpolate and extrapolate from the data

Interpret within the linear regression model the value of the correlation coefficient as a measure of the strength of the relationship

Students will understand and apply concepts of probability

Differentiate between situations requiring permutations and those requiring combinations

Calculate the number of possible permutations (nPr) of n items taken r at a time

Calculate the number of possible combinations (nCr) of n items taken r at a time

Use permutations, combinations, and the Fundamental Principle of Counting to determine the number of elements in a sample space and a specific subset (event)

Calculate theoretical probabilities, including geometric applications

Calculate empirical probabilities

Know and apply the binomial probability formula to events involving the terms exactly, at least, and at most

Use the normal distribution as an approximation for binomial probabilities

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