Introduction

            Every teacher of mathematics, whether at the elementary, middle, or high school level, has an individual goal to provide students with the knowledge and understanding of the mathematics necessary to function in a world very dependent upon the application of mathematics.  Instructionally, this goal translates into three components:

v     conceptual understanding

v     procedural fluency

v     problem solving

Conceptual understanding consists of those relationships constructed internally and connected to already existing ideas.  It involves the understanding of mathematical ideas and procedures and includes the knowledge of basic arithmetic facts.  Students use conceptual understanding of mathematics when they identify and apply principles, know and apply facts and definitions, and compare and contrast related concepts.  Knowledge learned with understanding provides a foundation for remembering or reconstructing mathematical facts and methods, for solving new and unfamiliar problems, and for generating new knowledge. 

Procedural fluency is the skill in carrying out procedures flexibly, accurately, efficiently, and appropriately.  It includes, but is not limited to, algorithms (the step-by-step routines needed to perform arithmetic operations).  Although the word procedural may imply an arithmetic procedure to some, it also refers to being fluent with procedures from other branches of mathematics, such as measuring the size of an angle using a protractor.  The use of calculators need not threaten the development of students’ computational skills.  On the contrary, calculators can enhance both understanding and computing if used properly and effectively.  Accuracy and efficiency with procedures are important, but they should be developed through understanding.  When students learn procedures through understanding, they are more likely to remember the procedures and less likely to make common computational errors.

Problem solving is the ability to formulate, represent, and solve mathematical problems.  Problems generally fall into three types:

v     one-step problems

v     multi-step problems

v     process problems

Most problems that students will encounter in the real world are multi-step or process problems.  Solution of these problems involves the integration of conceptual understanding and procedural knowledge.  Students need to have a broad range of strategies upon which to draw.  Selection of a strategy for finding the solution to a problem is often the most difficult part of the solution.  Therefore, mathematics instruction must include the teaching of many strategies to empower all students to become successful problem solvers.   A concept or procedure in itself is not useful in problem solving unless one recognizes when and where to use it as well as when and where it does not apply.  Many textbook problems are not typical of those that students will meet in real life.  Therefore, students need to be able to have a general understanding of how to analyze a problem and how to choose the most useful strategy for solving the problem. 

Individually, each of these components (conceptual understanding, procedural fluency, and problem solving) is necessary but not sufficient for a student to be mathematically proficient. They are not, however, independent of each other.  They are integrally related, need to be taught simultaneously, and should be a component of every lesson.

The mathematics standard presented in this document states that students will:

v     understand the concepts of and become proficient with the skills of mathematics;

v     communicate and reason mathematically;

v     become problem solvers by using appropriate tools and strategies;

through the integrated study of number sense and operations, algebra, geometry, measurement, and statistics and probability.  Mathematics should be viewed as a whole body of knowledge, not as a set of individual components.  Therefore, local mathematics curriculum, instruction, and assessment should be designed to support and sustain the components of this standard. 

 In this document conceptual understanding, procedural fluency, and problem solving are represented as process strands and content strands.  These strands help to define what students should know and be able to do as a result of their engagement in the study of mathematics.

 Process Strands: The process strands (Problem Solving, Reasoning and Proof, Communication, Connections, and Representation) highlight ways of acquiring and using content knowledge.  These process strands help to give meaning to mathematics and help students to see mathematics as a discipline rather than a set of isolated skills.  Student engagement in mathematical content is accomplished through these process strands.  Students will gain a better understanding of mathematics and have longer retention of mathematical knowledge as they solve problems, reason mathematically, prove mathematical relationships, participate in mathematical discourse, make mathematical connections, and model and represent mathematical ideas in a variety of ways. 

Content Strands: The content strands (Number Sense and Operations, Algebra, Geometry, Measurement, and Statistics and Probability) explicitly describe the content that students should learn. Each school’s mathematics curriculum developed from these strands should include a broad range of content.  This broad range of content, taught in an integrated fashion, allows students to see how various mathematics knowledge is related, not only within mathematics, but also to other disciplines and the real world as well.  The performance indicators listed under each band within a strand are intended to assist teachers in determining what the outcomes of instruction should be.  The instruction should engage students in the construction of this knowledge and should integrate conceptual understanding and problem solving with these performance indicators.  The performance indicators should not be viewed as a checklist of skills void of understanding and application.

 Students will only become successful in mathematics if they see mathematics as a whole, not as isolated skills and facts.  As school districts develop their own mathematics curriculum based upon the statements in this standards document, attention must be given to both content and process strands.  Likewise, as teachers develop their instructional plans and their assessment techniques, they also must give attention to the integration of process and content.  To do otherwise would produce students who have temporary knowledge and who are unable to apply mathematics in realistic settings.  Curriculum, instruction, and assessment are intricately related and must be designed with this in mind.  All three domains must address conceptual understanding, procedural fluency, and problem solving.  If this is accomplished, school districts will produce students who will (1) have mathematical knowledge, (2) have an understanding of mathematical concepts, and (3) be able to apply mathematics in the solution of problems.

School districts and individual teachers should be aware that this document is a standards document that guides the development of local curriculum.  Local school districts remain responsible for developing curriculum aligned to the New York State standards.  In this document the mathematics standard is succinctly stated.  The standard outlines what students should know and be able to do in mathematics.  The content strands, consisting of bands and performance indicators within each band, and the performance indicators of the process strands help to define how the standard will be met.  Each school district’s mathematics curriculum should be developed to assure that all students achieve the performance indicators for both the process and content strands. 

 Helping all students become proficient in mathematics is an imperative goal for every school.  It is the hope that this standards document will assist schools and individual teachers in meeting this goal.  For additional information visit the New York State Education Department mathematics website http://www.emsc.nysed.gov/ciai/mst/math.html .

Proposed Mathematics Standard, Content Strands, Process Strands, Bands within the Content Strands, and Grade-By-Grade Performance Indicators

Mathematics, Science, and Technology - Standard 3

Students will:     

•understand the concepts of and become proficient with the skills of mathematics;

•communicate and reason mathematically;

•become problem solvers by using appropriate tools and strategies;

through the integrated study of number sense and operations, algebra, geometry, measurement, and statistics and probability.

The Five Content Strands

Number Sense and Operations Strand

Students will:

•understand numbers, multiple ways of representing numbers, relationships

  among numbers, and number systems;

•understand meanings of operations and procedures, and how they relate to one

  another;

•compute accurately and make reasonable estimates.

Algebra Strand

Students will:

•represent and analyze algebraically a wide variety of problem solving situations;

•perform algebraic procedures accurately;

•recognize, use, and represent algebraically patterns, relations, and functions.

Geometry Strand

Students will:

•use visualization and spatial reasoning to analyze characteristics and properties

  of geometric shapes;

•identify and justify geometric relationships, formally and informally;

•apply transformations and symmetry to analyze problem solving situations;

•apply coordinate geometry to analyze problem solving situations.

Measurement Strand

Students will:

•determine what can be measured and how, using appropriate methods and

  formulas;

•use units to give meaning to measurements;

•understand that all measurement contains error and be able to determine its

  significance;

•develop strategies for estimating measurements.

Statistics and Probability Strand

Students will:

•collect, organize, display, and analyze data;

•make predictions that are based upon data analysis;

•understand and apply concepts of probability.

The Five Process Strands

Problem Solving Strand

Students will:

 •build new mathematical knowledge through problem solving;

•solve problems that arise in mathematics and in other contexts

•apply and adapt a variety of appropriate strategies to solve problems;

•monitor and reflect on the process of mathematical problem solving.

Reasoning and Proof Strand

Students will:

 •recognize reasoning and proof as fundamental aspects of mathematics;

•make and investigate mathematical conjectures;

 •develop and evaluate mathematical arguments and proofs;

 •select and use various types of reasoning and methods of proof.

Communication Strand

 Students will:

•organize and consolidate their mathematical thinking through communication;               

•communicate their mathematical thinking coherently and clearly to peers,

  teachers, and others;                                                                          

•analyze and evaluate the mathematical thinking and strategies of others;

                        •use the language of mathematics to express mathematical ideas precisely.

Connections Strand

Students will:

•recognize and use connections among mathematical ideas;

            •understand how mathematical ideas interconnect and build on one another to    

  produce a coherent whole;

 •recognize and apply mathematics in contexts outside of mathematics.

Representation Strand

Students will:

•create and use representations to organize, record, and communicate

  mathematical ideas;

•select, apply, and translate among mathematical representations to solve

  problems;

•use representations to model and interpret physical, social, and mathematical

  phenomena.

Bands Within the Content Strands

Number Sense and Operations

     Number Systems

Number Theory

Operations

Estimation

 Algebra

Variables and Expressions

Equations and Inequalities

Patterns, Relations, and Functions

Coordinate Geometry

Trigonometric Functions

Geometry

•Shapes

•Geometric Relationships

•Transformational Geometry

•Coordinate Geometry

•Constructions

•Locus

•Informal Proofs

•Formal Proofs

Measurement

                         •Units of Measurement

•Tools and Methods

•Units

•Error and Magnitude

 •Estimation

Statistics and Probability

•Collection of Data

•Organization and Display of Data

•Analysis of Data

•Predictions from Data

 •Probability

 In implementing the Algebra process and content performance indicators, it is expected that students will identify and justify mathematical relationships.  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.

                                    A.PS.1       Use a variety of problem solving strategies to understand new mathematical content

                                    A.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.

                                    A.PS.3       Observe and explain patterns to formulate generalizations and conjectures

                                    A.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.

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

                                    A.PS.6       Use a variety of strategies to extend solution methods to other problems     

                                    A.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.

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

                                    A.PS.9       Interpret solutions within the given constraints of a problem

                                    A.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.

                                    A.RP.1       Recognize that mathematical ideas can be supported by a variety of strategies

Students will make and investigate mathematical conjectures.

                                    A.RP.2       Use mathematical strategies to reach a conclusion and provide supportive arguments for a conjecture

                                    A.RP.3       Recognize when an approximation is more appropriate than an exact answer

Students will develop and evaluate mathematical arguments and proofs.

                                    A.RP.4       Develop, verify, and explain an argument, using appropriate mathematical ideas and language

                                    A.RP.5       Construct logical arguments that verify claims or counterexamples that refute them

                                    A.RP.6       Present correct mathematical arguments in a variety of forms

                                    A.RP.7       Evaluate written arguments for validity

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

                                    A.RP.8       Support an argument by using a systematic approach to test more than one case

                                    A.RP.9       Devise ways to verify results or use counterexamples to refute incorrect statements  

                                    A.RP.10     Extend specific results to more general cases

                                    A.RP.11    Use a Venn diagram to support a logical argument

A.RP.12    Apply inductive reasoning in making and supporting

      mathematical conjectures

Communication Strand

Students will organize and consolidate their mathematical thinking through communication.

                                    A.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

                                    A.CM.2     Use mathematical representations to communicate with appropriate accuracy, including numerical tables, formulas, functions, equations, charts, graphs, Venn diagrams, and other diagrams

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

                                    A.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

                                    A.CM.4     Explain relationships among different representations of a problem

                                    A.CM.5     Communicate logical arguments clearly, showing why a result makes sense and why the reasoning is valid

                                    A.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.

                                    A.CM.7     Read and listen for logical understanding of mathematical

                                                      thinking shared by other students

                                    A.CM.8     Reflect on strategies of others in relation to one’s own strategy

                                    A.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.

                                    A.CM.10   Use correct mathematical language in developing mathematical questions that elicit, extend, or challenge other students’ conjectures

                                    A.CM.11   Represent word problems using standard mathematical notation

                                    A.CM.12   Understand and use appropriate language, representations, and terminology when describing objects, relationships, mathematical solutions, and rationale

                                    A.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.

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

                                    A.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.

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

                                    A.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

                                    A.CN.5      Understand how quantitative models connect to various physical models and representations

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

                                    A.CN.6      Recognize and apply mathematics to situations in the outside world

                                    A.CN.7      Recognize and apply mathematical ideas to problem situations that develop outside of mathematics

                                    A.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.

                                    A.R.1         Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts

                                    A.R.2         Recognize, compare, and use an array of representational forms

                                    A.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.

                                    A.R.4         Select appropriate representations to solve problem situations

                                    A.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.

A.R.6         Use mathematics to show and understand physical

                  phenomena (e.g., find the height of a building if a ladder of

                  a given length forms a given angle of elevation with the

                  ground)

                                    A.R.7         Use mathematics to show and understand social phenomena (e.g., determine profit from student and adult ticket sales)

A.R.8         Use mathematics to show and understand mathematical phenomena (e.g., compare the graphs of the functions represented by the equations and )

Number Sense and Operations Strand

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

Number Theory           A.N.1        Identify and apply the properties of real numbers (closure, commutative, associative, distributive, identity, inverse)  Note: Students do not need to identify groups and fields, but students should be engaged in the ideas.

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

Operations                  A.N.2        Simplify radical terms (no variable in the radicand)

                                    A.N.3        Perform the four arithmetic operations using like and unlike radical terms and express the result in simplest form

                                    A.N.4        Understand and use scientific notation to compute products and quotients of numbers

                                    A.N.5        Solve algebraic problems arising from situations that involve fractions, decimals, percents (decrease/increase and discount), and proportionality/direct variation

                                    A.N.6        Evaluate expressions involving factorial(s), absolute value(s), and exponential expression(s)

                                    A.N.7        Determine the number of possible events, using counting techniques or the Fundamental Principle of Counting

                                    A.N.8        Determine the number of possible arrangements (permutations) of a list of items

Algebra Strand

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

Variables and              A.A.1         Translate a quantitative verbal phrase into an algebraic

Expressions                                   expression

A.A.2         Write a verbal expression that matches a given

mathematical expression

Equations and             A.A.3         Distinguish the difference between an algebraic

Inequalities                                    expression and an algebraic equation

                                    A.A.4         Translate verbal sentences into mathematical equations or

                                                      inequalities

                                    A.A.5         Write algebraic equations or inequalities that represent a situation

                                    A.A.6         Analyze and solve verbal problems whose solution requires solving a linear equation in one variable or linear inequality in one variable

                                    A.A.7         Analyze and solve verbal problems whose solution requires solving systems of linear equations in two variables

                                    A.A.8         Analyze and solve verbal problems that involve quadratic equations

                                    A.A.9         Analyze and solve verbal problems that involve exponential growth and decay

                                    A.A.10       Solve systems of two linear equations in two variables     algebraically (See A.G.7)

A.A.11       Solve a system of one linear and one quadratic equation

in two variables, where only factoring is required  Note: The quadratic equation should represent a parabola and the solution(s) should be integers.

Students will perform algebraic procedures accurately.

Variables and              A.A.12       Multiply and divide monomial expressions with a common

Expressions                                   base, using the properties of exponents  Note: Use integral exponents only.

                                    A.A.13       Add, subtract, and multiply monomials and polynomials

                                    A.A.14       Divide a polynomial by a monomial or binomial, where the quotient has no remainder

                                    A.A.15       Find values of a variable for which an algebraic fraction is undefined

                                    A.A.16       Simplify fractions with polynomials in the numerator and denominator by factoring both and renaming them to lowest terms

                                    A.A.17       Add or subtract fractional expressions with monomial or like binomial denominators

                                    A.A.18       Multiply and divide algebraic fractions and express the product or quotient in simplest form

                                    A.A.19       Identify and factor the difference of two perfect squares

                                    A.A.20       Factor algebraic expressions completely, including trinomials with a lead coefficient of one (after factoring a GCF)

Equations and             A.A.21       Determine whether a given value is a solution to a given

Inequalities                                    linear equation in one variable or linear inequality in one

                                                      variable

                                    A.A.22       Solve all types of linear equations in one variable

                                    A.A.23       Solve literal equations for a given variable

                                    A.A.24       Solve linear inequalities in one variable 

                                    A.A.25       Solve equations involving fractional expressions  Note: Expressions which result in linear equations in one variable.

                                    A.A.26       Solve algebraic proportions in one variable which result in    linear or quadratic equations

                                    A.A.27       Understand and apply the multiplication property of zero to solve quadratic equations with integral coefficients and integral roots

                                    A.A.28       Understand the difference and connection between roots of a quadratic equation and factors of a quadratic expression

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

Patterns,                     A.A.29      Use set-builder notation and/or interval notation to

Relations,                                      illustrate the elements of a set, given the elements in

and Functions                               roster form

                                    A.A.30       Find the complement of a subset of a given set, within a given universe

                                    A.A.31       Find the intersection of sets (no more than three sets) and/or union of sets (no more than three sets)

Coordinate                  A.A.32       Explain slope as a rate of change between dependent and

Geometry                                      independent variables

                                    A.A.33       Determine the slope of a line, given the coordinates of two points on the line

                                    A.A.34       Write the equation of a line, given its slope and the coordinates of a point on the line

                                    A.A.35       Write the equation of a line, given the coordinates of two points on the line

                                    A.A.36       Write the equation of a line parallel to the x- or y-axis

                                    A.A.37       Determine the slope of a line, given its equation in any form

                                    A.A.38       Determine if two lines are parallel, given their equations in any form

                                    A.A.39       Determine whether a given point is on a line, given the equation of the line

                                    A.A.40       Determine whether a given point is in the solution set of a system of linear inequalities

                                    A.A.41       Determine the vertex and axis of symmetry of a parabola, given its equation  (See A.G.10 )

Trigonometric             A.A.42       Find the sine, cosine, and tangent ratios of an angle of Functions                 a right triangle, given the lengths of the sides

                                    A.A.43       Determine the measure of an angle of a right triangle, given the length of any two sides of the triangle

                                    A.A.44       Find the measure of a side of a right triangle, given an acute angle and the length of another side

                                    A.A.45       Determine the measure of a third side of a right triangle using the Pythagorean theorem, given the lengths of any two sides

Geometry Strand

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

Shapes                         A.G.1         Find the area and/or perimeter of figures composed of polygons and circles or sectors of a circle  Note: Figures may include triangles, rectangles, squares, parallelograms, rhombuses, trapezoids, circles, semi-circles, quarter-circles, and regular polygons (perimeter only).

                                    A.G.2         Use formulas to calculate volume and surface area of rectangular solids and cylinders

Students will apply coordinate geometry to analyze problem solving situations.

Coordinate                  A.G.3         Determine when a relation is a function, by examining

Geometry                                      ordered pairs and inspecting graphs of relations

                                    A.G.4         Identify and graph linear, quadratic (parabolic), absolute value, and exponential functions

                                    A.G.5         Investigate and generalize how changing the coefficients of a function affects its graph

                                    A.G.6         Graph linear inequalities

                                    A.G.7         Graph and solve systems of linear equations and inequalities with rational coefficients in two variables (See A.A.10)

                                    A.G.8         Find the roots of a parabolic function graphically  Note: Only quadratic equations with integral solutions.

                                    A.G.9         Solve systems of linear and quadratic equations graphically Note: Only use systems of linear and quadratic equations that lead to solutions whose coordinates are integers.

                                    A.G.10       Determine the vertex and axis of symmetry of a parabola, given its graph (See A.A.41)  Note: The vertex will have an ordered pair of integers and the axis of symmetry will have an integral value.

Measurement Strand

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

Units of                       A.M.1        Calculate rates using appropriate units  (e.g., rate of a Measurement                                               space ship versus the rate of a snail)

                                    A.M.2        Solve problems involving conversions within measurement

                                                      systems, given the relationship between the units

Students will understand that all measurement contains error and be able to determine its significance.

Error and                    A.M.3        Calculate the relative error in measuring square and cubic

Magnitude                                     units, when there is an error in the linear measure

Statistics and Probability Strand

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

Organization and        A.S.1         Categorize data as qualitative or quantitative

Display of Data

                                    A.S.2         Determine whether the data to be analyzed is univariate or

                                                      bivariate

                                    A.S.3         Determine when collected data or display of data may be biased

                                    A.S.4         Compare and contrast the appropriateness of different measures of central tendency for a given data set

                                    A.S.5         Construct a histogram, cumulative frequency histogram, and a box-and-whisker plot, given a set of data

                                    A.S.6         Understand how the five statistical summary (minimum, maximum, and the three quartiles) is used to construct a box-and-whisker plot

                                    A.S.7         Create a scatter plot of bivariate data

                                    A.S.8         Construct manually a reasonable line of best fit for a scatter plot and determine the equation of that line

Analysis of Data          A.S.9         Analyze and interpret a frequency distribution table or histogram, a cumulative frequency distribution table or histogram, or a box-and-whisker plot

                                    A.S.10       Evaluate published reports and graphs that are based on data by considering: experimental design, appropriateness of the data analysis, and the soundness of the conclusions

                                    A.S.11       Find the percentile rank of an item in a data set and identify the point values for first, second, and third quartiles

                                    A.S.12       Identify the relationship between the independent and

                                                      dependent variables from a scatter plot (positive, negative, or none)

                                    A.S.13       Understand the difference between correlation and causation

                                    A.S.14       Identify variables that might have a correlation but not a   causal relationship

Students will make predictions that are based upon data analysis.

Predictions from         A.S.15       Identify and describe sources of bias and its effect,

Data                                              drawing conclusions from data

                                    A.S.16       Recognize how linear transformations of one-variable data affect the data’s mean, median, mode, and range

                                    A.S.17       Use a reasonable line of best fit to make a prediction involving interpolation or extrapolation

Students will understand and apply concepts of probability.

Probability                  A.S.18       Know the definition of conditional probability and use it to solve for probabilities in finite sample spaces 

                                    A.S.19       Determine the number of elements in a sample space and the number of favorable events

                                    A.S.20       Calculate the probability of an event and its complement

                                    A.S.21       Determine empirical probabilities based on specific sample data

                                    A.S.22       Determine, based on calculated probability of a set of events, if:

o       some or all are equally likely to occur

o       one is more likely to occur than another

o       whether or not an event is certain to happen or not  to happen

                                    A.S.23       Calculate the probability of:

o       a series of independent events

o       a series of dependent events

o       two mutually exclusive events

o       two events that are not mutually exclusive

STOP HERE!

FYI:  I have also included the  next two regents course objectives as

 well,  just in case you are interested in looking ahead!

In implementing the Geometry process and content performance indicators, it is expected that students will identify and justify geometric relationships, formally and informally.   For example, students will begin with a definition of a figure and from that definition students will be expected to develop a list of conjectured properties of the figure and to justify each conjecture informally or with formal proof.  Students will also be expected to list the assumptions that are needed in order to justify each conjectured property and present their findings in an organized manner.

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.  The variety of approaches to verification and proof is what gives curriculum developers and teachers the flexibility to adapt strategies to address these performance indicators in a manner that meets the diverse needs of our students.  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 relationships of perpendicularity, parallelism, congruence, and/or similarity after 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 using properties of perpendicularity, parallelism, congruence, and similarity with polygons and circles.

Apply - Students will use a theorem or concept to solve a geometric problem.

Problem Solving Strand

Students will build new mathematical knowledge through problem solving.

                                    G.PS.1       Use a variety of problem solving strategies to understand new mathematical content

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

                                    G.PS.2       Observe and explain patterns to formulate generalizations and conjectures

                                    G.PS.3       Use multiple representations to represent and explain problem situations (e.g., spatial, geometric, verbal, numeric, algebraic, and graphical representations)

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

                                    G.PS.4       Construct various types of reasoning, arguments, justifications and methods of proof for problems

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

                                    G.PS.6       Use a variety of strategies to extend solution methods to other problems     

                                    G.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.

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

                                    G.PS.9       Interpret solutions within the given constraints of a problem

                                    G.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.

                                    G.RP.1       Recognize that mathematical ideas can be supported by a variety of strategies

                                    G.RP.2       Recognize and verify, where appropriate, geometric relationships of perpendicularity, parallelism, congruence, and similarity, using algebraic strategies

Students will make and investigate mathematical conjectures.

                                    G.RP.3       Investigate and evaluate conjectures in mathematical terms, using mathematical strategies to reach a conclusion

Students will develop and evaluate mathematical arguments and proofs.

                                    G.RP.4       Provide correct mathematical arguments in response to other students’ conjectures, reasoning, and arguments

                                    G.RP.5       Present correct mathematical arguments in a variety of forms

                                    G.RP.6       Evaluate written arguments for validity

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

                                    G.RP.7       Construct a proof using a variety of methods (e.g., deductive, analytic, transformational)

                                    G.RP.8       Devise ways to verify results or use counterexamples to refute incorrect statements  

G.RP.9       Apply inductive reasoning in making and supporting                         mathematical conjectures

Communication Strand

Students will organize and consolidate their mathematical thinking through communication.

                                    G.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

                                    G.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.

                                    G.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

                                    G.CM.4     Explain relationships among different representations of a problem

                                    G.CM.5     Communicate logical arguments clearly, showing why a result makes sense and why the reasoning is valid

                                    G.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.

                                    G.CM.7     Read and listen for logical understanding of mathematical

                                                      thinking shared by other students

                                    G.CM.8     Reflect on strategies of others in relation to one’s own strategy

                                    G.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.

                                    G.CM.10   Use correct mathematical language in developing mathematical questions that elicit, extend, or challenge other students’ conjectures

                                    G.CM.11   Understand and use appropriate language, representations, and terminology when describing objects, relationships, mathematical solutions, and geometric diagrams

                                    G.CM.12   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.

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

                                    G.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.

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

                                    G.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

                                    G.CN.5      Understand how quantitative models connect to various physical models and representations

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

                                    G.CN.6      Recognize and apply mathematics to situations in the outside world

                                    G.CN.7      Recognize and apply mathematical ideas to problem situations that develop outside of mathematics

                                    G.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.

                                    G.R.1         Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts

                                    G.R.2         Recognize, compare, and use an array of representational forms

                                    G.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.

                                    G.R.4         Select appropriate representations to solve problem situations

                                    G.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.

                                    G.R.6         Use mathematics to show and understand physical phenomena (e.g., determine the number of gallons of water in a fish tank)

                                    G.R.7         Use mathematics to show and understand social phenomena (e.g., determine if conclusions from another person’s argument have a logical foundation)

                                    G.R.8         Use mathematics to show and understand mathematical phenomena (e.g., use investigation, discovery, conjecture, reasoning, arguments, justification and proofs to validate that the two base angles of an isosceles triangle are congruent)

Algebra Strand

Note:  The algebraic skills and concepts within the Algebra process and content performance indicators must be maintained and applied as students are asked to investigate, make conjectures, give rationale, and justify or prove geometric concepts.

Geometry Strand

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

Geometric                   Note: Two-dimensional geometric relationships are addressed

Relationships               in the Informal and Formal Proofs band.

                                    G.G.1         Know and apply that if a line is perpendicular to each of two intersecting lines at their point of intersection, then the line is perpendicular to the plane determined by them

                                    G.G.2         Know and apply that through a given point there passes one and only one plane perpendicular to a given line

                                    G.G.3         Know and apply that through a given point there passes one and only one line perpendicular to a given plane

                                    G.G.4         Know and apply that two lines perpendicular to the same plane are coplanar

                                    G.G.5         Know and apply that two planes are perpendicular to each other if and only if one plane contains a line perpendicular to the second plane

                                    G.G.6         Know and apply that if a line is perpendicular to a plane, then any line perpendicular to the given line at its point of intersection with the given plane is in the given plane 

                                    G.G.7         Know and apply that if a line is perpendicular to a plane, then every plane containing the line is perpendicular to the given plane

                                    G.G.8         Know and apply that if a plane intersects two parallel planes, then the intersection is two parallel lines

                                    G.G.9         Know and apply that if two planes are perpendicular to the same line, they are parallel

                                    G.G.10       Know and apply that the lateral edges of a prism are congruent and parallel

                                    G.G.11       Know and apply that two prisms have equal volumes if their bases have equal areas and their altitudes are equal

                                    G.G.12       Know and apply that the volume of a prism is the product of the area of the base and the altitude

                                    G.G.13       Apply the properties of a regular pyramid, including:

o       lateral edges are congruent

o       lateral faces are congruent isosceles triangles 

o       volume of a pyramid equals one-third the product of the area of the base and the altitude

                                    G.G.14       Apply the properties of a cylinder, including:

o       bases are congruent

o       volume equals the product of the area of the base and the altitude

o       lateral area of a right circular cylinder equals the product of an altitude and the circumference of the base

                                    G.G.15       Apply the properties of a right circular cone, including:

o       lateral area equals one-half the product of the slant height and the circumference of its base

o       volume is one-third the product of the area of its base and its altitude

                                    G.G.16       Apply the properties of a sphere, including:

o       the intersection of a plane and a sphere is a circle

o       a great circle is the largest circle that can be drawn on a sphere

o       two planes equidistant from the center of the sphere and intersecting the sphere do so in congruent circles

o       surface area is  

o       volume is

Constructions              G.G.17       Construct a bisector of a given angle, using a straightedge and compass, and justify the construction

                                    G.G.18       Construct the perpendicular bisector of a given segment, using a straightedge and compass, and justify the construction

                                    G.G.19       Construct lines parallel (or perpendicular) to a given line through a given point, using a straightedge and compass, and justify the construction

                                    G.G.20       Construct an equilateral triangle, using a straightedge and compass, and justify the construction

Locus                           G.G.21       Investigate and apply the concurrence of medians, altitudes, angle bisectors, and perpendicular bisectors of triangles

                                    G.G.22       Solve problems using compound loci

                                    G.G.23       Graph and solve compound loci in the coordinate plane

Students will identify and justify geometric relationships formally and informally.

Informal and               G.G.24       Determine the negation of a statement and establish its

Formal Proofs                             truth value

                                    G.G.25       Know and apply the conditions under which a compound statement (conjunction, disjunction, conditional, biconditional) is true 

                                    G.G.26       Identify and write the inverse, converse, and contrapositive of a given conditional statement and note the logical equivalences

                                    G.G.27       Write a proof arguing from a given hypothesis to a given

                                                      conclusion

                                    G.G.28       Determine the congruence of two triangles by using one of the five congruence techniques (SSS, SAS, ASA, AAS, HL), given sufficient information about the sides and/or angles of two congruent triangles

                                    G.G.29       Identify corresponding parts of congruent triangles

G.G.30       Investigate, justify, and apply theorems about the sum of the measures of the angles of a triangle

                                    G.G.31       Investigate, justify, and apply the isosceles triangle theorem and its converse

                                    G.G.32       Investigate, justify, and apply theorems about geometric inequalities, using the exterior angle theorem

                                    G.G.33       Investigate, justify, and apply the triangle inequality theorem

                                    G.G.34       Determine either the longest side of a triangle given the three angle measures or the largest angle given the lengths of three sides of a triangle

                                    G.G.35       Determine if two lines cut by a transversal are parallel, based on the measure of given pairs of angles formed by the transversal and the lines

                                    G.G.36       Investigate, justify, and apply theorems about the sum of the measures of the interior and exterior angles of polygons

                                    G.G.37       Investigate, justify, and apply theorems about each interior and exterior angle measure of regular polygons

                                    G.G.38       Investigate, justify, and apply theorems about parallelograms involving their angles, sides, and diagonals

                                    G.G.39       Investigate, justify, and apply theorems about special parallelograms (rectangles, rhombuses, squares) involving their angles, sides, and diagonals

                                    G.G.40       Investigate, justify, and apply theorems about trapezoids (including isosceles trapezoids) involving their angles, sides, medians, and diagonals

                                    G.G.41       Justify that some quadrilaterals are parallelograms, rhombuses, rectangles, squares, or trapezoids

                                    G.G.42       Investigate, justify, and apply theorems about geometric relationships, based on the properties of the line segment joining the midpoints of two sides of the triangle

                                    G.G.43       Investigate, justify, and apply theorems about the centroid of a triangle, dividing each median into segments whose lengths are in the ratio 2:1   

                                    G.G.44       Establish similarity of triangles, using the following theorems: AA, SAS, and SSS

                                    G.G.45       Investigate, justify, and apply theorems about similar triangles

                                    G.G.46       Investigate, justify, and apply theorems about proportional relationships among the segments of the sides of the triangle, given one or more lines parallel to one side of a triangle and intersecting the other two sides of the triangle

                                    G.G.47       Investigate, justify, and apply theorems about mean proportionality:

o       the altitude to the hypotenuse of a right triangle is the mean proportional between the two segments along the hypotenuse

o       the altitude to the hypotenuse of a right triangle divides the hypotenuse so that either leg of the right triangle is the mean proportional between the hypotenuse and segment of the hypotenuse adjacent to that leg

                                    G.G.48       Investigate, justify, and apply the Pythagorean theorem and its converse

                                    G.G.49       Investigate, justify, and apply theorems regarding chords of a circle:

o       perpendicular bisectors of chords

o       the relative lengths of chords as compared to their distance from the center of the circle

                                    G.G.50       Investigate, justify, and apply theorems about tangent lines to a circle:

o       a perpendicular to the tangent at the point of tangency

o       two tangents to a circle from the same external point

o       common tangents of two non-intersecting or tangent circles

                                    G.G.51       Investigate, justify, and apply theorems about the arcs determined by the rays of angles formed by two lines intersecting a circle when the vertex is:

o       inside the circle (two chords)

o       on the circle (tangent and chord)

o       outside the circle (two tangents, two secants, or tangent and secant)

                                    G.G.52       Investigate, justify, and apply theorems about arcs of a circle cut by two parallel lines

                                    G.G.53       Investigate, justify, and apply theorems regarding segments intersected by a circle:

o       along two tangents from the same external point

o       along two secants from the same external point

o       along a tangent and a secant from the same external point

o       along two intersecting chords of a given circle

Students will apply transformations and symmetry to analyze problem solving situations.

Transformational        G.G.54       Define, investigate, justify, and apply isometries in the Geometry                 plane (rotations, reflections, translations, glide reflections)

                                                      Note: Use proper function notation.

                                    G.G.55       Investigate, justify, and apply the properties that remain invariant under translations, rotations, reflections, and glide reflections

                                    G.G.56       Identify specific isometries by observing orientation, numbers of invariant points, and/or parallelism

                                    G.G.57       Justify geometric relationships (perpendicularity, parallelism, congruence) using transformational techniques (translations, rotations, reflections)

                                    G.G.58       Define, investigate, justify, and apply similarities (dilations and the composition of dilations and isometries)

                                    G.G.59       Investigate, justify, and apply the properties that remain invariant under similarities

                                    G.G.60       Identify specific similarities by observing orientation, numbers of invariant points, and/or parallelism

                                    G.G.61       Investigate, justify, and apply the analytical representations for translations, rotations about the origin of 90º and 180º, reflections over the lines , , and , and dilations centered at the origin

Students will apply coordinate geometry to analyze problem solving situations.

Coordinate                  G.G.62       Find the slope of a perpendicular line, given the

Geometry                                      equation of a line

                                    G.G.63       Determine whether two lines are parallel, perpendicular, or neither, given their equations

                                    G.G.64       Find the equation of a line, given a point on the line and the equation of a line perpendicular to the given line

                                    G.G.65       Find the equation of a line, given a point on the line and the equation of a line parallel to the desired line

                                    G.G.66       Find the midpoint of a line segment, given its endpoints

                                    G.G.67       Find the length of a line segment, given its endpoints

                                    G.G.68       Find the equation of a line that is the perpendicular bisector of a line segment, given the endpoints of the line segment

                                    G.G.69       Investigate, justify, and apply the properties of triangles and quadrilaterals in the coordinate plane, using the distance, midpoint, and slope formulas

                                    G.G.70       Solve systems of equations involving one linear equation and one quadratic equation graphically

                                    G.G.71       Write the equation of a circle, given its center and radius or given the endpoints of a diameter 

                                    G.G.72       Write the equation of a circle, given its graph 

                                                      Note: The center is an ordered pair of integers and the radius is an integer.

                                    G.G.73       Find the center and radius of a circle, given the equation of the circle in center-radius form

                                    G.G.74       Graph circles of the form  

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 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.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 claims

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 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.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             A2.A.1      Solve absolute value equations and inequalities involving Inequalities                                         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              A2.A.7       Factor polynomial expressions completely, using any    Expressions                                                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             A2.A.20     Determine the sum and product of the roots of a quadratic

Inequalities                                    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,                     A2.A.29     Identify an arithmetic or geometric sequence and find the

Relations,                                      formula for its nth term

and Functions

                                    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                  A2.A.47     Determine the center-radius form for the equation of a

Geometry                                      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             A2.A.55     Express and apply the six trigonometric functions as ratios Functions                      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                       A2.M.1      Define radian measure

Measurement                               

                                    A2.M.2     Convert between radian and degree measures

Statistics and Probability Strand

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

Collection of               A2.S.1       Understand the differences among various kinds of

Data                                              studies (e.g., survey, observation, controlled experiment)

                                    A2.S.2       Determine factors which may affect the outcome of a survey

Organization and        A2.S.3       Calculate measures of central tendency with group

Display of Data                             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         A2.S.6       Determine from a scatter plot whether a linear, logarithmic,   Data               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