Algebra 2 and Trigonometry

In implementing the Algebra 2 and Trigonometry process and content performance indicators, it is expected that students will identify and justify mathematical relationships, formally and informally.  The intent of both the process and content performance indicators is to provide a variety of ways for students to acquire and demonstrate mathematical reasoning ability when solving problems.  Local curriculum and local/state assessments must support and allow students to use any mathematically correct method when solving a problem.

Throughout this document the performance indicators use the words investigate, explore, discover, conjecture, reasoning, argument, justify, explain, proof, and apply.  Each of these terms is an important component in developing a student’s mathematical reasoning ability.  It is therefore important that a clear and common definition of these terms be understood.  The order of these terms reflects different stages of the reasoning process.

Investigate/Explore - Students will be given situations in which they will be asked to look for patterns or relationships between elements within the setting.

Discover - Students will make note of possible patterns and generalizations that result from investigation/exploration.

Conjecture - Students will make an overall statement, thought to be true, about the new discovery.

Reasoning - Students will engage in a process that leads to knowing something to be true or false.

Argument - Students will communicate, in verbal or written form, the reasoning process that leads to a conclusion.  A valid argument is the end result of the conjecture/reasoning process.

Justify/Explain - Students will provide an argument for a mathematical conjecture.  It may be an intuitive argument or a set of examples that support the conjecture.  The argument may include, but is not limited to, a written paragraph, measurement using appropriate tools, the use of dynamic software, or a written proof.

Proof - Students will present a valid argument, expressed in written form, justified by axioms, definitions, and theorems.

Apply - Students will use a theorem or concept to solve an algebraic or numerical problem.

Problem Solving Strand

Students will build new mathematical knowledge through problem solving.
A2.PS.1 Use a variety of problem solving strategies to understand new mathematical content
A2.PS.2 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.
A2.PS.3

Observe and explain patterns to formulate generalizations and conjectures

A2.PS.4 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.
A2.PS.5

Choose 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.7 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.
A2.PS.8 Determine information required to solve a 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.10 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.
A2.RP.1 Support mathematical ideas using a variety of strategies
Students will make and investigate mathematical conjectures. 
A2.RP.2 Investigate and evaluate conjectures in mathematical terms, using mathematical strategies to reach a conclusion
A2.RP.3 Evaluate 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.5 Develop, verify, and explain an argument, using appropriate mathematical ideas and language
A2.RP.6 Construct logical arguments that verify claims or counterexamples that refute them
A2.RP.7 Present correct mathematical arguments in a variety of forms
A2.RP.8 Evaluate 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.2

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.
A2.CM.3 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
A2.CM.4

Explain relationships among different representations of a problem

A2.CM.5 Communicate logical arguments clearly, showing why a result makes sense and why the reasoning is valid
A2.CM.6 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.
A2.CM.7 Read and listen for logical understanding of mathematical  thinking shared by other students
A2.CM.8

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

A2.CM.9 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.
A2.CM.10 Use correct mathematical language in developing mathematical questions that elicit, extend, or challenge other students’ conjectures
A2.CM.11 Represent word problems using standard mathematical notation
A2.CM.12 Understand and use appropriate language, representations, and terminology when describing objects, relationships, mathematical solutions, and rationale
A2.CM.13 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.
A2.CN.1

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

A2.CN.2 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.
A2.CN.3 Model situations mathematically, using representations to draw conclusions and formulate new situations

A2.CN.4

Understand how concepts, procedures, and mathematical results in one area of mathematics can be used to solve problems in other areas of mathematics
A2.CN.5 Understand how quantitative models connect to various physical models and representations
Students will recognize and apply mathematics in contexts outside of mathematics.
A2.CN.6 Recognize and apply mathematics to situations in the outside world
A2.CN.7 Recognize and apply mathematical ideas to problem situations that develop outside of mathematics
A2.CN.8 Develop 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.1 Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts
A2.R.2 Recognize, compare, and use an array of representational forms
A2.R.3 Use representation as a tool for exploring and understanding mathematical ideas
Students will select, apply, and translate among mathematical representations to solve problems.
A2.R.4 Select appropriate representations to solve problem situations
A2.R.5 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.
A2.R.6 Use mathematics to show and understand physical phenomena (e.g., investigate sound waves using the sine and cosine functions)
A2.R.7 Use mathematics to show and understand social phenomena (e.g., interpret the results of an opinion poll)
A2.R.8

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.

Operations  

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

Algebra Strand

Students will represent and analyze algebraically a wide variety of problem solving situations.
Equations and Inequalities
A2.A.1 Solve absolute value equations and inequalities involving linear expressions in one variable
A2.A.2 Use 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.4 Solve quadratic inequalities in one and two variables, algebraically and graphically
A2.A.5 Use direct and inverse variation to solve for unknown values
A2.A.6 Solve an application which results in an exponential function
Students will perform algebraic procedures accurately.

Variables and Expressions                                          

A2.A.7 Factor polynomial expressions completely, using any combination of the following techniques: common factor extraction, difference of two perfect squares, quadratic trinomials
A2.A.8 Apply the rules of exponents to simplify expressions involving negative and/or fractional exponents
A2.A.9 Rewrite algebraic expressions that contain negative exponents using only positive exponents
A2.A.10 Rewrite algebraic expressions with fractional exponents as radical expressions
A2.A.11 Rewrite algebraic expressions in radical form as expressions with fractional exponents
A2.A.12 Evaluate exponential expressions, including those with base e
A2.A.13 Simplify radical expressions
A2.A.14 Perform addition, subtraction, multiplication, and division of radical expressions
A2.A.15 Rationalize denominators involving algebraic radical expressions
A2.A.16 Perform arithmetic operations with rational expressions and rename to lowest terms
A2.A.17 Simplify complex fractional expressions
A2.A.18 Evaluate logarithmic expressions in any base
A2.A.19 Apply the properties of logarithms to rewrite logarithmic expressions in equivalent forms

Equations and Inequalities

A2.A.20 Determine the sum and product of the roots of a quadratic equation by examining its coefficients
A2.A.21 Determine the quadratic equation, given the sum and product of its roots
A2.A.22 Solve radical equations
A2.A.23 Solve rational equations and inequalities
A2.A.24 Know and apply the technique of completing the square
A2.A.25 Solve quadratic equations, using the quadratic formula
A2.A.26 Find the solution to polynomial equations of higher degree that can be solved using factoring and/or the quadratic formula
A2.A.27 Solve exponential equations with and without common bases
A2.A.28 Solve 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.29 Identify an arithmetic or geometric sequence and find the formula for its nth term
A2.A.30 Determine the common difference in an arithmetic sequence
A2.A.31 Determine the common ratio in a geometric sequence
A2.A.32 Determine a specified term of an arithmetic or geometric sequence
A2.A.33 Specify terms of a sequence, given its recursive definition
A2.A.34 Represent the sum of a series, using sigma notation
A2.A.35 Determine the sum of the first n terms of an arithmetic or geometric series
A2.A.36 Apply the binomial theorem to expand a binomial and determine a specific term of a binomial expansion
A2.A.37 Define a relation and function
A2.A.38 Determine when a relation is a function
A2.A.39 Determine the domain and range of a function from its equation
A2.A.40 Write functions in functional notation
A2.A.41 Use functional notation to evaluate functions for given values in the domain
A2.A.42 Find the composition of functions
A2.A.43 Determine if a function is one-to-one, onto, or both
A2.A.44 Define the inverse of a function
A2.A.45 Determine the inverse of a function and use composition to justify the result
A2.A.46 Perform transformations with functions and relations: , , , ,  

Coordinate Geometry  

A2.A.47 Determine the center-radius form for the equation of a circle in standard form
A2.A.48 Write the equation of a circle, given its center and a point on the circle
A2.A.49 Write the equation of a circle from its graph
A2.A.50 Approximate the solution to polynomial equations of higher degree by inspecting the graph
A2.A.51 Determine the domain and range of a function from its graph
A2.A.52 Identify relations and functions, using graphs
A2.A.53
Graph exponential functions of the form for positive values of b, including
A2.A.54 Graph logarithmic functions, using the inverse of the related exponential function

Trigonometric Functions

A2.A.55 Express and apply the six trigonometric functions as ratios of the sides of a right triangle
A2.A.56 Know the exact and approximate values of the sine, cosine, and tangent of 0º, 30º, 45º, 60º, 90º, 180º, and 270º angles
A2.A.57 Sketch and use the reference angle for angles in standard position
A2.A.58 Know and apply the co-function and reciprocal relationships between trigonometric ratios
A2.A.59 Use 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.60 Sketch the unit circle and represent angles in standard position
A2.A.61 Determine the length of an arc of a circle, given its radius and the measure of its central angle
A2.A.62 Find the value of trigonometric functions, if given a point on the terminal side of angle
A2.A.63 Restrict 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.65 Sketch the graph of the inverses of the sine, cosine, and tangent functions
A2.A.66 Determine the trigonometric functions of any angle, using technology
A2.A.67 Justify the Pythagorean identities
A2.A.68 Solve trigonometric equations for all values of the variable from 0º to 360º
A2.A.69 Determine amplitude, period, frequency, and phase shift, given the graph or equation of a periodic function
A2.A.70 Sketch and recognize one cycle of a function of the form or
A2.A.71
Sketch and recognize the graphs of the functions , , , and
A2.A.72 Write 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.74 Determine the area of a triangle or a parallelogram, given the measure of two sides and the included angle
A2.A.75 Determine the solution(s) from the SSA situation (ambiguous case) 
A2.A.76 Apply the angle sum and difference formulas for trigonometric functions
A2.A.77 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.

Units of Measurement

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

Statistics and Probability Strand

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

Collection of Data     

A2.S.1 Understand the differences among various kinds of studies (e.g., survey, observation, controlled experiment)
A2.S.2 Determine factors which may affect the outcome of a survey

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

Organization and Display of Data

 

A2.S.3 Calculate measures of central tendency with group frequency distributions
A2.S.4 Calculate measures of dispersion (range, quartiles, interquartile range, standard deviation, variance) for both samples and populations
A2.S.5 Know and apply the characteristics of the normal distribution

Students will make predictions that are based upon data analysis.

Predictions from Data 

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

 Probability

A2.S.9    
      Differentiate between situations requiring permutations and those requiring combinations
A2.S.10   
Calculate the number of possible permutations of n items taken r at a time
A2.S.11  
Calculate the number of possible combinations of n items taken r at a time
A2.S.12   
Use 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.13   
Calculate theoretical probabilities, including geometric applications
A2.S.14
Calculate empirical probabilities
A2.S.15
Know and apply the binomial probability formula to events involving the terms exactly, at least, and at most
A2.S.16

Use the normal distribution as an approximation for binomial probabilities

 


                            

                                     

                               

                                    

                                       

                                      

                                       

 

                                   

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Grade 3 Grade 4 Grade 5 Grade 6 Grade 7
Grade 8 Algebra Geometry

04/20/2005