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ALGEBRA 2 AND TRIGONOMETRY STRANDS

Problem Solving   Reasoning and Proof   Communication   Connections   Representation
 Number Sense and Operations   Algebra   Measurement   Statistics and Probability

Problem Solving Strand
 

Students will build new mathematical knowledge through problem solving.

A2.PS.1Use a variety of problem solving strategies to understand new mathematical content
A2.PS.2Recognize and understand equivalent representations of a problem situation or a mathematical concept

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

A2.PS.3Observe and explain patterns to formulate generalizations and conjectures
A2.PS.4Use 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.

A2.PS.5Choose an effective approach to solve a problem from a variety of strategies (numeric, graphic, algebraic)
A2.PS.6

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

A2.PS.7Work 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.

A2.PS.8Determine information required to solve the problem, choose methods for obtaining the information, and define parameters for acceptable solutions
A2.PS.9

Interpret solutions within the given constraints of a problem

A2.PS.10Evaluate 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.

A2.RP.1

Support mathematical ideas using a variety of strategies

Students will make and investigate mathematical conjectures.

A2.RP.2Investigate and evaluate conjectures in mathematical terms, using mathematical strategies to reach a conclusion
A2.RP.3Evaluate conjectures and recognize when an estimate or approximation is more appropriate than an exact answer
A2.RP.4

Recognize when an approximation is more appropriate than an exact answer

Students will develop and evaluate mathematical arguments and proofs.

A2.RP.5Develop, verify, and explain an argument, using appropriate mathematical ideas and language
A2.RP.6Construct logical arguments that verify claims or counterexamples that refute claims
A2.RP.7Present correct mathematical arguments in a variety of forms
A2.RP.8Evaluate written arguments for validity

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

A2.RP.9

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

A2.RP.10

Devise ways to verify results, using counterexamples and informal indirect proof

A2.RP.11

Extend specific results to more general cases

A2.RP.12

Apply inductive reasoning in making and supporting  mathematical conjectures

 

Communication Strand
 

Students will organize and consolidate their mathematical thinking through communication.

A2.CM.1

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

A2.CM.2Use 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.

A2.CM.3Present 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
A2.CM.4Explain relationships among different representations of a problem
A2.CM.5Communicate logical arguments clearly, showing why a result makes sense and why the reasoning is valid
A2.CM.6Support 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.

A2.CM.7Read and listen for logical understanding of mathematical thinking shared by other students
A2.CM.8Reflect on strategies of others in relation to one’s own strategy
A2.CM.9Formulate 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.

A2.CM.10Use correct mathematical language in developing mathematical questions that elicit, extend, or challenge other students’ conjectures
A2.CM.11Represent word problems using standard mathematical notation
A2.CM.12Understand and use appropriate language, representations, and terminology when describing objects, relationships, mathematical solutions, and rationale
A2.CM.13Draw 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.

A2.CN.1Understand and make connections among multiple representations of the same mathematical idea
A2.CN.2Understand 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.

A2.CN.3Model situations mathematically, using representations to draw conclusions and formulate new situations
A2.CN.4Understand how concepts, procedures, and mathematical results in one area of mathematics can be used to solve problems in other areas of mathematics
A2.CN.5Understand how quantitative models connect to various physical models and representations

Students will recognize and apply mathematics in contexts outside of mathematics.

A2.CN.6Recognize and apply mathematics to situations in the outside world
A2.CN.7Recognize and apply mathematical ideas to problem situations that develop outside of mathematics
A2.CN.8Develop an appreciation for the historical development of mathematics

 

Representation Strand
 

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

A2.R.1Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts
A2.R.2Recognize, compare, and use an array of representational forms
A2.R.3Use representation as a tool for exploring and understanding mathematical ideas

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

A2.R.4Select appropriate representations to solve problem situations
A2.R.5Investigate relationships among different representations and their impact on a given problem

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

A2.R.6Use mathematics to show and understand physical phenomena (e.g., investigate sound waves using the sine and cosine functions)
A2.R.7Use mathematics to show and understand social phenomena (e.g., interpret the results of an opinion poll)
A2.R.8Use 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.

Operations
A2.N.1Evaluate numerical expressions with negative and/or fractional exponents, without the aid of a calculator (when the answers are rational numbers)
A2.N.2Perform arithmetic operations (addition, subtraction, multiplication, division) with expressions containing irrational numbers in radical form
A2.N.3Perform arithmetic operations with polynomial expressions containing rational coefficients
A2.N.4Perform arithmetic operations on irrational expressions
A2.N.5Rationalize a denominator containing a radical expression
A2.N.6Write square roots of negative numbers in terms of i
A2.N.7Simplify powers of i
A2.N.8Determine the conjugate of a complex number
A2.N.9Perform arithmetic operations on complex numbers and write the answer in the form   Note: This includes simplifying expressions with complex denominators
A2.N.10Know and apply sigma notation

 

Algebra Strand
 

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

Equations and Inequalities
A2.A.1Solve absolute value equations and inequalities involving linear expressions in one variable
A2.A.2Use the discriminant to determine the nature of the roots of a quadratic equation
A2.A.3 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
A2.A.4Solve quadratic inequalities in one and two variables, algebraically and graphically
A2.A.5Use direct and inverse variation to solve for unknown values
A2.A.6Solve an application which results in an exponential function

Students will perform algebraic procedures accurately.

Variables and Expressions
A2.A.7Factor polynomial expressions completely, using any combination of the following techniques: common factor extraction, difference of two perfect squares, quadratic trinomials
A2.A.8Apply the rules of exponents to simplify expressions involving negative and/or fractional exponents
A2.A.9Rewrite algebraic expressions that contain negative exponents using only positive exponents
A2.A.10Rewrite algebraic expressions with fractional exponents as radical expressions
A2.A.11Rewrite algebraic expressions in radical form as expressions with fractional exponents
A2.A.12Evaluate exponential expressions, including those with base e
A2.A.13Simplify radical expressions
A2.A.14Perform addition, subtraction, multiplication, and division of radical expressions
A2.A.15 Rationalize denominators involving algebraic radical expressions
A2.A.16Perform arithmetic operations with rational expressions and rename to lowest terms
A2.A.17Simplify complex fractional expressions
A2.A.18Evaluate logarithmic expressions in any base
A2.A.19Apply the properties of logarithms to rewrite logarithmic expressions in equivalent forms
Equations and Inequalities
A2.A.20Determine the sum and product of the roots of a quadratic equation by examining its coefficients
A2.A.21Determine the quadratic equation, given the sum and product of its roots
A2.A.22Solve radical equations
A2.A.23Solve rational equations and inequalities
A2.A.24Know and apply the technique of completing the square
A2.A.25Solve quadratic equations, using the quadratic formula
A2.A.26Find the solution to polynomial equations of higher degree that can be solved using factoring and/or the quadratic formula
A2.A.27Solve exponential equations with and without common bases
A2.A.28Solve a logarithmic equation by rewriting as an exponential equation

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

Patterns, Relations, and Functions
A2.A.29Identify an arithmetic or geometric sequence and find the formula for its nth term
A2.A.30Determine the common difference in an arithmetic sequence
A2.A.31 Determine the common ratio in a geometric sequence
A2.A.32Determine a specified term of an arithmetic or geometric sequence
A2.A.33Specify terms of a sequence, given its recursive definition
A2.A.34Represent the sum of a series, using sigma notation
A2.A.35Determine the sum of the first n terms of an arithmetic or geometric series
A2.A.36Apply the binomial theorem to expand a binomial and determine a specific term of a binomial expansion
A2.A.37Define a relation and function
A2.A.38Determine when a relation is a function
A2.A.39Determine the domain and range of a function from its equation
A2.A.40Write functions in functional notation
A2.A.41Use functional notation to evaluate functions for given values in the domain
A2.A.42Find the composition of functions
A2.A.43Determine if a function is one-to-one, onto, or both
A2.A.44Define the inverse of a function
A2.A.45Determine the inverse of a function and use composition to justify the result
A2.A.46Perform transformations with functions and relations:
f(x + a),  f(x) + a,  f(-x),  af(x)
Coordinate Geometry
A2.A.47Determine the center-radius form for the equation of a circle in standard form
A2.A.48Write the equation of a circle, given its center and a point on the circle
A2.A.49Write the equation of a circle from its graph
A2.A.50Approximate the solution to polynomial equations of higher degree by inspecting the graph
A2.A.51Determine the domain and range of a function from its graph
A2.A.52Identify relations and functions, using graphs
A2.A.53Graph exponential functions of the form y = bx for positive values of b, including b = x
A2.A.54Graph logarithmic functions, using the inverse of the related exponential function
Trigonometric Functions
A2.A.55Express and apply the six trigonometric functions as ratios of the sides of a right triangle
A2.A.56Know the exact and approximate values of the sine, cosine, and tangent of 0º, 30º, 45º, 60º, 90º, 180º, and 270º angles
A2.A.57Sketch and use the reference angle for angles in standard position
A2.A.58Know and apply the co-function and reciprocal relationships between trigonometric ratios
A2.A.59Use the reciprocal and co-function relationships to find the value of the secant, cosecant, and cotangent of 0º, 30º, 45º, 60º, 90º, 180º, and 270º angles
A2.A.60Sketch the unit circle and represent angles in standard position
A2.A.61Determine the length of an arc of a circle, given its radius and the measure of its central angle
A2.A.62Find the value of trigonometric functions, if given a point on the terminal side of angle θ
A2.A.63Restrict the domain of the sine, cosine, and tangent functions to ensure the existence of an inverse function
A2.A.64 Use inverse functions to find the measure of an angle, given its sine, cosine, or tangent
A2.A.65Sketch the graph of the inverses of the sine, cosine, and tangent functions
A2.A.66Determine the trigonometric functions of any angle, using technology
A2.A.67Justify the Pythagorean identities
A2.A.68Solve trigonometric equations for all values of the variable from 0º to 360º
A2.A.69Determine amplitude, period, frequency, and phase shift, given the graph or equation of a periodic function
A2.A.70Sketch and recognize one cycle of a function of the form y = AsinBx or y = AcosBx
A2.A.71Sketch and recognize the graphs of the functions y = sec(x), y = csc(x), y = tan(x), and  y = cot(x)
A2.A.72Write the trigonometric function that is represented by a given periodic graph
A2.A.73 Solve for an unknown side or angle, using the Law of Sines or the Law of Cosines
A2.A.74Determine the area of a triangle or a parallelogram, given the measure of two sides and the included angle
A2.A.75Determine the solution(s) from the SSA situation (ambiguous case)
A2.A.76Apply the angle sum and difference formulas for trigonometric functions
A2.A.77Apply 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.

Units of Measurement
A2.M.1Define radian measure
A2.M.2Convert between radian and degree measures

 

Statistics and Probability Strand
 

Students will collect, organize, display, and analyze data.

Collection of Data
A2.S.1Understand the differences among various kinds of studies (e.g., survey, observation, controlled experiment)
A2.S.2Determine factors which may affect the outcome of a survey
Organization and Display of Data
A2.S.3Calculate measures of central tendency with group frequency distributions
A2.S.4Calculate measures of dispersion (range, quartiles, interquartile range, standard deviation, variance) for both samples and populations
A2.S.5Know and apply the characteristics of the normal distribution

Students will make predictions that are based upon data analysis.

Predictions from Data
A2.S.6Determine from a scatter plot whether a linear, logarithmic, exponential, or power regression model is most appropriate
A2.S.7Determine the function for the regression model, using appropriate technology, and use the regression function to interpolate and extrapolate from the data
A2.S.8Interpret 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.

Probability
A2.S.9Differentiate between situations requiring permutations and those requiring combinations
A2.S.10Calculate the number of possible permutations of n items taken r at a time
A2.S.11Calculate the number of possible combinations of n items taken r at a time
A2.S.12Use permutations, combinations, and the Fundamental Principle of Counting to determine the number of elements in a sample space and a specific subset (event)
A2.S.13Calculate theoretical probabilities, including geometric applications
A2.S.14Calculate empirical probabilities
A2.S.15Know and apply the binomial probability formula to events involving the terms exactly, at least, and at most
A2.S.16Use the normal distribution as an approximation for binomial probabilities

 

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