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New Math TEKS Side by Sides New Math TEKS High School - Side by Sides Table of Contents Algebra I Process Standards ...................................................................................................................................................................................................... 3 Linear Functions, Equations, and Inequalities ........................................................................................................................................................... 5 Quadratic Functions & Equations .............................................................................................................................................................................. 9 Exponential Functions & Equations ......................................................................................................................................................................... 12 Number & Algebraic Methods ................................................................................................................................................................................. 13 Algebra II Process Standards .................................................................................................................................................................................................... 16 Attributes of Functions & their Inverses .................................................................................................................................................................. 17 Systems of Equations & Inequalities ........................................................................................................................................................................ 18 Quadratic & Square Root Functions, Equations, & Inequalities .............................................................................................................................. 19 Exponential & Logarithmic Functions & Equations .................................................................................................................................................. 22 Cubic, Cube Root, Absolute Value, and Rational Functions, Equations, & Inequalities .......................................................................................... 23 Number & Algebraic Methods ................................................................................................................................................................................. 25 Data .......................................................................................................................................................................................................................... 26 Geometry Process Standards .................................................................................................................................................................................................... 27 Coordinate & Transformational Geometry .............................................................................................................................................................. 29 Logical Argument & Constructions .......................................................................................................................................................................... 31 Proof & Congruence ................................................................................................................................................................................................. 34 Analysis based on Texas Essential Knowledge and Skills for Mathematics – Texas Education Agency. April 2012 New Math TEKS High School - Side by Sides Geometry (cont.) Similarity Proof Trigonometry .................................................................................................................................................................................. 36 Two‐dimensional & Three‐dimensional Figures ...................................................................................................................................................... 38 Circles ....................................................................................................................................................................................................................... 40 Probability ................................................................................................................................................................................................................ 41 Analysis based on Texas Essential Knowledge and Skills for Mathematics – Texas Education Agency. April 2012 Algebra I Mathematical Process Standards +A.1 The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to: Change New Standard (Implementation Year 2015‐2016) A.1A apply mathematics to problems arising in everyday life, society, and the workplace A.1B use a problem‐solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem‐solving process and the reasonableness of the solution A.1F analyze mathematical relationships to connect and communicate mathematical ideas A.1G display, explain and justify mathematical ideas and arguments using precise mathematical language in written or oral communications Current Standard Cognitive Change Content Change a(6) Underlying mathematical processes. Many processes underlie all content areas in mathematics. As they do mathematics, students continually use problem‐solving, language and communication, and reasoning (justification and proof) to make connections within and outside mathematics… changed “make connections” to “apply mathematics” changed “use communication” to “display” and “communicate ideas” changed “use reasoning (justification and proof)” to “explain and justify” added “analyze mathematical relationships” A.1E interpret and make decisions, predictions, and critical judgments from functional relationships changed “interpret” to “analyze” and “display” moved “Underlying mathematical processes” from introductory “Basic Understandings” to the “Mathematical Process Standards” changed “within and outside mathematics” to “in everyday life, society, and the workplace” changed “language” to “precise mathematical language in written or oral communications” expanded the problem‐solving process to include: analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem‐solving process and the reasonableness changed strand from “Foundations for Functions” to “Mathematical Process Standards” changed “functional relationships” to “mathematical relationships” changed “predictions and critical judgments” to “mathematical ideas and arguments” Key: Change of grade level Change of strand NEW Standard 3 © 2012 lead4ward Algebra I Mathematical Process Standards A.1 The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to: Change New Standard (Implementation Year 2015‐2016) A.1C select tools, including real objects, manipulatives, paper/pencil, and technology, as appropriate and techniques, including mental math, estimation, and number sense, as appropriate to solve problems A.1D communicate mathematical ideas, reasoning, and their implications using multiple representation, including symbols, diagrams, graphs, and language as appropriate A.1E create and use representations to organize, record and communicate mathematical ideas Current Standard Cognitive Change Content Change a(6) Underlying mathematical processes… Students also use multiple representations, technology, applications and modeling, and numerical fluency in problem‐solving contexts. a(5) Tools for algebraic thinking. Techniques for working with functions and equations are essential in understanding underlying relationships. Students use a variety of representations (concrete, pictorial, numerical, symbolic, graphical, and verbal), tools, and technology (including, but not limited to, calculators with graphing capabilities, data collection devices, and computers) to model mathematical situations to solve meaningful problems. changed “use tools” to “select tools” changed “use a variety of representations” to “communicate ideas, reasoning, and their implications using multiple representation” deleted “use modeling” and “model mathematical situations” A.1D represent relationships among quantities using concrete models, tables, graphs, diagrams, verbal descriptions, equations, and inequalities A.2C interpret situations in terms of given graphs or creates situations that fit given graphs A.3A use symbols to represent unknowns and variables a(2) Algebraic thinking and symbolic reasoning. Symbolic reasoning plays a critical role in algebra; symbols provide powerful ways to represent mathematical situations and to express generalizations. Students use symbols in a variety of ways to study relationships among quantities. changed “represent,” “interpret” and “use” to “communicate” moved “Underlying mathematical processes” and “Tools for algebraic thinking” from introductory “Basic Understandings” to the “Mathematical Process Standards” expanded “numerical fluency” to include mental math, estimation, and number sense added use of paper/pencil deleted list of specific types of technology (calculators with graphing capabilities, data collection devices, and computers) changed “concrete” representations to “real objects [and] manipulatives” changed “pictorial” representations to “diagrams” changed “verbal” representations to “language” changed strand from “Foundations for Functions” to “Mathematical Process Standards” changed “represent situations” to “create and use representations” changed “study relationships” and “express generalizations” to “organize, record, and communicate ideas” moved “Algebraic thinking and symbolic reasoning” from introductory “Basic Understandings” to the “Mathematical Process Standards” deleted “symbols;” however, included in “representations” deleted “relationships;” however, included in A.1F Key: Change of grade level Change of strand NEW Standard 4 © 2012 lead4ward Algebra I Linear Functions, Equations, and Inequalities A.2 The student applies the mathematical process standards when using properties of linear functions to write and represent in multiple ways, with and without technology, linear equations, inequalities, and systems of equations. The student is expected to: Change New Standard (Implementation Year 2015‐2016) A.2A determine the domain and range of a linear function in mathematical problems; determine reasonable domain and range values for real‐world situations, both continuous and discrete; and represent domain and range using inequalities A.2B write linear equations in two variables in various forms, including y = mx + b, Ax + By = C, and y – y1 = m(x – x1), given one point and the slope and given two points Current Standard Cognitive Change A.2B identify mathematical domains and ranges and determine reasonable domain and range values for given situations, both continuous and discrete A.5B determine the domain and range for linear functions in given situations A.6D graph and write equations of lines given characteristics such as two points, a point and a slope, or a slope and y‐intercept Content Change specified the use of inequalities to represent domain and range deleted “graph;” however, included in A.3C deleted “slope and y‐intercept,” as it can be included in “one point and the slope” specified the various forms of linear equations (y = mx + b, Ax + By = C, and y – y1 = m(x – x1)) A.5C use, translate, and make connections among algebraic, tabular, graphical, or verbal descriptions of linear functions A.6G relate direct variation to linear functions and solve problems involving proportional change G.7B use slopes and equations of lines to investigate geometric relationships, including parallel lines, perpendicular lines, and special segments of triangles and other polygons changed “use, translate, and make connections” to “write” changed “relate direct variation to linear functions” to “write equations” changed “use” and “investigate” to “write” deleted reference to “proportional change” moved equations of parallel and perpendicular lines from Geometry to Algebra I did not include “special segments of triangles and other polygons” A.2C write linear equations in two variables given a table of values, a graph, and a verbal description A.2D write and solve equations involving direct variation A.2E write the equation of a line that contains a given point and is parallel to a given line A.2F write the equation of a line that contains a given point and is perpendicular to a given line A.2G write an equation of a line that is parallel or perpendicular to the X or Y axis and determine whether the slope of the line is zero or undefined A.2H write linear inequalities in two variables given a table of values, a graph, and a verbal description A.7A analyze situations involving linear functions and formulate linear equations or inequalities to solve problems specified that inequalities involve two variables added the use of tables and graphs in writing inequalities A.2I write systems of two linear equations given a table of values, a graph, and a verbal description A.8A analyze situations and formulate systems of linear equations in two unknowns to solve problems deleted “analyze situations;” however, included in “verbal descriptions” deleted “solve problems;” however, included in Process Standards (see TEKS A.1) deleted “analyze situations;” however, included in “verbal descriptions” deleted “solve problems;” however, included in Process Standards (see TEKS A.1) added the use of tables and graphs in writing systems Key: Change of grade /course Change of strand NEW Standard 5 © 2012 lead4ward Algebra I Linear Functions, Equations, and Inequalities A.3 The student applies the mathematical process standards when using graphs of linear functions, key features, and related transformations to represent in multiple ways and solve, with and without technology, equations, inequalities, and systems of equations. The student is expected to: Change New Standard (Implementation Year 2015‐2016) A.3A determine the slope of a line given a table of values, a graph, two points on the line, and an equation written in various forms, including y = mx + b, Ax + By = C, and y – y1 = m(x – x1) A.3B calculate the rate of change of a linear function represented tabularly, graphically, or algebraically in context of mathematical and real‐world problems A.3C graph linear functions on the coordinate plane and identify key features, including x‐ intercept, y‐intercept, zeros, and slope, in mathematical and real‐world problems A.3D graph the solution set of linear inequalities in two variables on the coordinate plane A.3E determine the effects on the graph of the parent function f(x) = x when f(x) is replaced by af(x), f(x) + d , f(x – c), f(bx) for specific values of a, b, c, and d A.3F graph systems of two linear equations in two variables on the coordinate plane and determine the solutions if they exist Current Standard Cognitive Change A.6A develop the concept of slope as rate of change and determine slopes from graphs, tables, and algebraic representations changed “develop the concept of slope” to “determine the slope” A.6B interpret the meaning of slope and intercepts in situations using data, symbolic representations, or graphs changed “interpret the meaning of slope” to “calculate the rate of change” A.6D graph and write equations of lines given characteristics such as two points, a point and a slope, or a slope and y‐intercept deleted “write equations;” however, included in A.2B A.7B investigate methods for solving linear equations and inequalities using concrete models, graphs, and the properties of equality, select a method, and solve the equations and inequalities A.2A identify and sketch the general forms of 2 linear (y = x) and quadratic (y = x ) parent functions A.6C investigate, describe, and predict the effects of changes in m and b on the graph of y = mx + b A.8B solve systems of linear equations using concrete models, graphs, tables, and algebraic methods deleted “investigate methods” and “select a method” changed “solving” linear inequalities to “graphing” the ”solution set” deleted “identify and sketch” changed “investigate, describe, and predict the effects” to “determine the effects” A.8C interpret and determine the reasonableness of solutions to systems of linear equations changed “interpret” and “determine reasonableness” to “estimate graphically” A.3G estimate graphically the solutions to systems of two linear equations with two variables in real‐world problems A.3H graph the solution set of systems of two linear inequalities in two variables on the coordinate plane 2 A.3B use algebraic methods, graphs, tables, or matrices, to solve systems of equations or inequalities Content Change added “two points on the line” specified the various forms of linear equations (y = mx + b, Ax + By = C, and y – y1 = m(x – x1)) deleted “intercepts” changed “using data” to “represented tabularly” added “in context of mathematical and real‐world problems” added “x‐intercept” and “zeros” added “in mathematical and real‐ world problems” deleted the use of concrete models and properties of equality in finding the solution set to inequalities; however, included in A.5B changed the use of m and b (in y = mx + b) to the use of a, b, c, and d (when f(x) is replaced by af(x), f(x) + d , f(x – c), or f(bx)) deleted the use of concrete models, tables, and algebraic methods to solve systems; however, included in the Process Standards (see A.1B, A.1C, A.1D) included reference to systems where solutions may not exist added “in real‐world problems” moved the graphing of systems of inequalities from Algebra II to Algebra I did not include tables or matrices Key: Change of grade /course Change of strand NEW Standard 6 © 2012 lead4ward Algebra I Linear Functions, Equations, and Inequalities A.4 The student applies the mathematical process standards to formulate statistical relationships and evaluate their reasonableness based on real‐world data. The student is expected to: Change New Standard Current Standard (Implementation Year 2015‐2016) A.4A calculate, using technology, the correlation coefficient between two quantitative variables and interpret this quantity as a measure of the strength of the linear association Cognitive Change Content Change P.3C use regression to determine the appropriateness of a linear function to model real‐ life data (including using technology to determine the correlation coefficient) changed “use” and “determine” to “calculate” and “interpret” moved linear regression and correlation coefficients from Precalculus to Algebra I changed the “appropriateness of a linear function” to “measure of strength of the linear association” A.2D collect and organize data, make and interpret scatterplots (including recognizing positive, negative, or no correlation for data approximating linear situations), and model, predict, and make decisions and critical judgments in problem situations deleted reference to specific types of correlation; however, skills still necessary to “provide a reasonable fit to data” A.4B compare and contrast association and causation in real‐world problems A.4C write, with and without technology, linear functions that provide a reasonable fit to data to estimate solutions and make predictions for real‐ world problems deleted “collect and organize [data]” deleted “make and interpret scatterplots” changed “model” to “write functions” deleted “make decisions and critical judgments” Key: Change of grade /course Change of strand NEW Standard 7 © 2012 lead4ward Algebra I Linear Functions, Equations, and Inequalities A.5 The student applies the mathematical process standards to solve, with and without technology, linear equations and evaluate the reasonableness of their solutions. The student is expected to: Change New Standard (Implementation Year 2015‐2016) A.5A solve linear equations in one variable, including those for which the application of the distributive property is necessary and for which variables are included on both sides Current Standard Cognitive Change A.5B solve linear inequalities in one variable, including those for which the application of the distributive property is necessary and for which variables are included on both sides A.4A find specific function values, simplify polynomial expressions, transform and solve equations, and factor as necessary in problem situations A.4B use the commutative, associative, and distributive properties to simplify algebraic expressions A.7B investigate methods for solving linear equations and inequalities using concrete models, graphs, and the properties of equality, select a method, and solve the equations and inequalities A.5C solve systems of two linear equations with two variables for mathematical and real‐world problems A.8B solve systems of linear equations using concrete models, graphs, tables, and algebraic methods Content Change specified that equations use one variable; however, other literal equations are addressed in A.12E specified that solutions involve the use of the distributive property, and variables on both sides of the equation deleted “investigate methods” and “select a method” deleted the use of concrete models; however, included in the Process Standards (see A.1C) deleted the use of graphs; however, included (for two‐variable inequalities) in A.3D specified the use of one variable inequalities specified that solutions involve the use of the distributive property and variables on both sides of the inequality deleted “using concrete models, graphs, tables, and algebraic methods;” however, included in the Process Standards (see A.1B, A.1C, A.1D) added “for mathematical and real‐ world problems” Key: Change of grade /course Change of strand NEW Standard 8 © 2012 lead4ward Algebra I Quadratic Functions and Equations A.6 The student applies the mathematical process standards when using properties of quadratic functions to write and represent in multiple ways, with and without technology, quadratic equations. The student is expected to: Change New Standard (Implementation Year 2015‐2016) A.6A determine the domain and range of quadratic functions and represent the domain and range using inequalities A.6B write equations of quadratic functions given the vertex and another point on the graph, write the equation in vertex form (f(x) = 2 a(x – h) + k), and rewrite the equation from 2 vertex form to standard form (f(x) = ax + bx + c) A.6C write quadratic functions when given real solutions and graphs of their related equations Current Standard Cognitive Change A.9A determine the domain and range for quadratic functions in given situations 2A.7A use characteristics of the quadratic parent function to sketch the related graphs and connect 2 2 between the y = ax + bx + c and the y = a (x – h) + k symbolic representations of quadratic functions changed “connect symbolic representations” to “write (and rewrite) equations” deleted “sketch the graphs;” however, included in A.7A A.10B make connections among the solutions (roots) of quadratic equations, the zeros of their related functions, and the horizontal intercepts (x‐ intercepts) of the graph of the function changed “make connections” to “write functions” Content Change added “using inequalities” deleted “in given situations;” however, included in the Process Standards (see A.1A) moved vertex and standard forms of quadratic functions from Algebra II to Algebra I specified the use of the vertex and another point on the graph deleted the terms roots, zeros, and x‐ intercepts; however, included in A.7A and A.7B Key: Change of grade/course Change of strand NEW Standard 9 © 2012 lead4ward Algebra I Quadratic Functions and Equations A.7 The student applies the mathematical process standards when using graphs of quadratic functions and their related transformations to represent in multiple ways and determine, with and without technology, the solutions to equations. The student is expected to: Change New Standard (Implementation Year 2015‐2016) A.7A graph quadratic functions on the coordinate plane and use the graph to identify key attributes, if possible, including x‐intercept, y‐intercept, zeros, maximum value, minimum values, vertex, and the equation of the axis of symmetry Current Standard Cognitive Change 2A.7A use characteristics of the quadratic parent function to sketch the related graphs and connect 2 2 between the y = ax + bx + c and the y = a (x – h) + k symbolic representations of quadratic functions A.7B describe the relationship between the linear factors of quadratic expressions and the zeros of their associated quadratic functions A.7C determine the effects on the graph of the 2 parent function f(x) = x when f(x) is replaced by af(x), f(x) + d, f(x – c), f(bx) for specific values of a, b, c, and d A.10B make connections among the solutions (roots) of quadratic equations, the zeros of their related functions, and the horizontal intercepts (x‐ intercepts) of the graph of the function A.2A identify and sketch the general forms of 2 linear (y=x) and quadratic (y=x ) parent functions A.9B investigate, describe, and predict the effects 2 of changes in a on the graph of y = ax + c A.9C investigate, describe, and predict the effects 2 of changes in c on the graph of y = ax + c 2 A.7B use the parent function to investigate, describe, and predict the effects of changes in a, 2 h, and k on the graphs of y = a (x – h) + k form of a function in applied and purely mathematical situations changed “make connections among” to “describe the relationship between” deleted “identify and sketch” changed “investigate, describe, and predict” to “determine” Content Change moved graphing quadratic functions from Algebra II to Algebra I changed “characteristics” to “key attributes” added list of “key attributes,” including x‐intercept(s), y‐intercept, zeros, maximum value, minimum values, vertex, and the equation of the axis of symmetry deleted reference to vertex and standard forms; however, included in A.6B deleted the term “roots” added connections to “linear factors” moved horizontal translations (f(x – c)) of quadratic functions from Algebra II to Algebra I changed the use of h and k (in vertex form of a quadratic function) to c and d (in the forms f(x – c) and f(x) + d) added the use of b in f(bx) Key: Change of grade/course Change of strand NEW Standard 10 © 2012 lead4ward Algebra I Quadratic Functions and Equations A.8 The student applies the mathematical process standards to solve, with and without technology, quadratic equations and evaluate the reasonableness of their solutions. The student formulates statistical relationships and evaluates their reasonableness based on real‐world data. The student is expected to: Change New Standard (Implementation Year 2015‐2016) A.8A solve quadratic equations having real solutions by factoring, taking square roots, completing the square, and applying the quadratic formula A.8B write, using technology, quadratic functions that provide a reasonable fit to data to estimate solutions and make predictions for real‐world problems Current Standard Cognitive Change Content Change A.4A find specific function values, simplify polynomial expressions, transform and solve equations, and factor as necessary in problem situations A.10A solve quadratic equations using concrete models, tables, graphs, and algebraic methods specified that quadratic equations have “real solutions” deleted “using concrete models, tables, and graphs;” however, included in the Process Standards (see A.1B, A.1C, A.1D provided a list of specific “algebraic methods,” such as factoring, taking square roots, completing the square, and applying the quadratic formula A.9D analyze graphs of quadratic functions and draw conclusions deleted reference to analyzing graphs and drawing conclusions; however, included in the Process Standards (see TEKS A.1) Key: Change of grade/course Change of strand NEW Standard 11 © 2012 lead4ward Algebra I Exponential Functions and Equations A.9 The student applies the mathematical process standards when using properties of exponential functions and their related transformations to write, graph, and represent in multiple ways exponential equations and evaluate, with and without technology, the reasonableness of their solutions. The student formulates statistical relationships and evaluates their reasonableness based on real‐world data. The student is expected to: Change Key: New Standard (Implementation Year 2015‐2016) A.9A determine the domain and range of x exponential functions of the form f(x) = ab and represent the domain and range using inequalities Current Standard Cognitive Change 2A.11C determine the reasonable domain and range values of exponential and logarithmic functions, as well as interpret and determine the reasonableness of solutions to exponential and logarithmic equations and inequalities A.9B interpret the meaning of the values of a and b in exponential functions of the form f(x) = x ab in real‐world problems A.9D graph exponential functions that model growth and decay and identify key features, including y‐intercept and asymptote, in mathematical and real‐world problems 2A.11B use the parent functions to investigate, describe, and predict the effects of parameter changes on the graphs of exponential and logarithmic functions, describe limitations on the domains and ranges, and examine asymptotic behavior changed “use,” “investigate, describe, and predict,” “describe,” and “examine” to “graph” added “interpret the meaning” of a and b x in f(x) = ab A.9C write exponential functions in the form x f(x) = ab (where b is a rational number) to describe problems arising from mathematical and real‐world situations, including growth and decay A.11C analyze data and represent situations involving exponential growth and decay using concrete models, tables, graphs, or algebraic methods changed “analyze data and represent situations” to “write functions to describe problems” A.9E write, using technology, exponential functions that provide a reasonable fit to data and make predictions for real‐world problems 2A.11F analyze a situation modeled by an exponential function, formulate an equation or inequality, and solve the problem deleted “analyze a situation;” however, included in the Process Standards (see TEKS A.1) changed “solve the problem” to “make predictions for real‐world problems” Change of grade /course Change of strand NEW Standard 12 Content Change moved finding domains and ranges of exponential functions from Algebra II to Algebra I did not move logarithmic functions from Algebra II deleted “reasonable” and “reasonableness of solutions;” however, included in the Process Standards (see A.1B) specified the form of exponential x functions (f(x) = ab ) added “using inequalities” moved properties of exponential functions from Algebra II to Algebra I did not move logarithmic functions from Algebra II specified the form of exponential x functions (f(x) = ab ) added “that model growth and decay” added y‐intercept as a “key feature” of exponential functions deleted domain and range; however, included in A.9A changed strand from “Quadratic and Other Nonlinear Functions” to “Exponential Functions and Equations” specified the form of exponential x functions (f(x) = ab ) deleted use of concrete models, tables and graphs; however, included in the Process Standards (see A.1B, A.1C, A.1D) moved writing exponential functions from Algebra II to Algebra I added “using technology” added providing “a reasonable fit to data” © 2012 lead4ward Algebra I Number and Algebraic Methods A.10 The student applies the mathematical process standards and algebraic methods to rewrite in equivalent forms and perform operations on polynomial expressions. The student is expected to: Change New Standard (Implementation Year 2015‐2016) A.10A add and subtract polynomials of degree one and degree two A.10B multiply polynomials of degree one and degree two A.10E factor, if possible, trinomials with real 2 factors in the form ax + bx + c, including perfect square trinomials of degree two A.10C determine the quotient of a polynomial of degree one and polynomial of degree two when divided by a polynomial of degree one and polynomial of degree two when the degree of the divisor does not exceed the degree of the dividend A.10D rewrite polynomial expressions of degree one and degree two in equivalent forms using the distributive property Current Standard Cognitive Change Content Change A.4A find specific function values, simplify polynomial expressions, transform and solve equations, and factor as necessary in problem situations changed “simplify” to add, subtract, and multiply changed strand from “Properties and Attributes of Functions” to “Number and Algebraic Methods” deleted “find specific function values;” however, included in A.12B deleted “solve equations;” however, included in A.5A changed “polynomial expressions” to “polynomials of degree one and degree two” specified that factoring is only “if possible” and “with real factors” specified that factoring is only for 2 trinomials of the form ax + bx + c, including perfect square trinomials deleted “in problem situations;” however, included in the Process Standards (see TEKS A.1) A.4B use the commutative, associative, and distributive properties to simplify algebraic expressions changed “simplify” to “rewrite” changed strand from “Properties and Attributes of Functions” to “Number and Algebraic Methods” deleted specific reference to commutative and associative properties; however, included in the process of rewriting expressions changed “algebraic expressions” to “polynomial expressions of degree one and degree two” A.10F decide if a binomial can be written as the difference of two squares and, if possible, use the structure of a difference of two squares to rewrite the binomial Key: Change of grade /course Change of strand NEW Standard 13 © 2012 lead4ward Algebra I Number and Algebraic Methods A.11 The student applies the mathematical process standards and algebraic methods to rewrite algebraic expressions into equivalent forms. The student is expected to: Change New Standard Current Standard Cognitive Change Content Change (Implementation Year 2015‐2016) A.11A simplify numerical radical expressions involving square roots A.11B simplify numeric and algebraic expressions using the laws of exponents, including integral and rational exponents A.11A use patterns to generate the laws of exponents and apply them in problem‐solving situations changed “use” and “apply” patterns to “simplify” expressions changed strand from “Quadratic and Other Nonlinear Functions” to “Number and Algebraic Methods” specified the use of both numeric and algebraic expressions specified the use of both integral and rational exponents deleted “in problem‐solving situations;” however, included in the Process Standards (see TEKS A.1) Key: Change of grade /course Change of strand NEW Standard 14 © 2012 lead4ward Algebra I Number and Algebraic Methods A.12 The student applies the mathematical process standards and algebraic methods to write, solve, analyze, and evaluate equations, relations, and functions. The student is expected to: Change New Standard (Implementation Year 2015‐2016) A.12A decide whether relations represented verbally, tabularly, graphically, and symbolically define a function Key: Cognitive Change A.1B gather and record data and use data sets to determine functional relationships between quantities deleted “gather and record data” changed “determine functional relationships” to “decide whether relations define a function” A.1A describe independent and dependent quantities in functional relationships A.4C connect equation notation with function notation, such as y = x + 1 and f(x) = x + 1 A.12B evaluate functions, expressed in function notation, given one or more elements in their domains A.12E solve mathematic and scientific formulas, and other literal equations, for a specified variable A.4A find specific function values, simplify polynomial expressions, transform and solve equations, and factor as necessary in problem situations changed “find specific function values” to “evaluate functions” A.12C identify terms of arithmetic and geometric sequences when the sequences are given in function form using recursive processes th A.12D write a formula for the n term of arithmetic and geometric sequences, given the value of several of their terms 8.5B find and evaluate an algebraic expression to determine any term in an arithmetic sequence (with a constant rate of change) A.3B look for patterns and represent generalizations algebraically P.4B use arithmetic, geometric and other sequences and series to solve real‐life problems changed “evaluate an expression” to “identify terms” from “function form” deleted “look for patterns” Current Standard Change of grade /course Change of strand NEW Standard 15 Content Change changed strand from “Functional Relationships” to “Number and Algebraic Methods” changed “data sets” to “relations represented tabularly and graphically” added verbal and symbolic representations of relations moved independent and dependent quantities from Algebra I to grade 6 (see 6.6A) deleted specific connection between equation and function notations; however, included in the Process Standards (see TEKS A.1) as well as in the linear functions strand (see TEKS A.3) changed strand from “Properties and Attributes of Functions” to “Number and Algebraic Methods” specified that evaluating functions uses “one or more elements in their domains” specified solving literal equations for a variable; however, solving other one‐ variable equations is included in A.5A and A.8A deleted “simplify polynomial expressions;” however, included in A.10A and A.10B deleted factoring; however, included in A.10E moved arithmetic sequences from grade 8 to Algebra I moved geometric sequences from Precalculus to Algebra I did not include series from Precalculus th changed “any term” to “the n term” added using “recursive processes” deleted “real‐life problems;” however, included in the Process Standards (see TEKS A.1) © 2012 lead4ward Algebra II Mathematical Process Standards 2A.1 The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to: Change Key: New Standard (Implementation Year 2015‐2016) 2A.1A apply mathematics to problems arising in everyday life, society, and the workplace 2A.1B use a problem‐solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem‐solving process and the reasonableness of the solution 2A.1F analyze mathematical relationships to connect and communicate mathematical ideas 2A.1G display, explain and justify mathematical ideas and arguments using precise mathematical language in written or oral communications 2A.1C select tools, including real objects, manipulatives, paper/pencil, and technology, as appropriate and techniques, including mental math, estimation, and number sense, as appropriate to solve problems 2A.1D communicate mathematical ideas, reasoning, and their implications using multiple representation, including symbols, diagrams, graphs, and language as appropriate 2A.1E create and use representations to organize, record and communicate mathematical ideas Change of grade level Current Standard Cognitive Change Content Change a(6) Underlying mathematical processes. Many processes underlie all content areas in mathematics. As they do mathematics, students continually use problem‐solving, language and communication, and reasoning (justification and proof) to make connections within and outside mathematics… changed “make connections” to “apply mathematics” changed “use communication” to “display” and “communicate ideas” changed “use reasoning (justification and proof)” to “explain and justify” added “analyze mathematical relationships” moved “Underlying mathematical processes” from introductory “Basic Understandings” to the “Mathematical Process Standards” changed “within and outside mathematics” to “in everyday life, society, and the workplace” changed “language” to “precise mathematical language in written or oral communications” expanded the problem‐solving process to include: analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem‐ solving process and the reasonableness a(6) Underlying mathematical processes… Students also use multiple representations, technology, applications and modeling, and numerical fluency in problem‐solving contexts. a(5) Tools for algebraic thinking. Techniques for working with functions and equations are essential in understanding underlying relationships. Students use a variety of representations (concrete, pictorial, numerical, symbolic, graphical, and verbal), tools, and technology (including, but not limited to, calculators with graphing capabilities, data collection devices, and computers) to model mathematical situations to solve meaningful problems. changed “use tools” to “select tools” changed “use a variety of representations” to “communicate ideas, reasoning, and their implications using multiple representation” deleted “use modeling” and “model mathematical situations” a(2) Algebraic thinking and symbolic reasoning. Symbolic reasoning plays a critical role in algebra; symbols provide powerful ways to represent mathematical situations and to express generalizations. Students study algebraic concepts and the relationships among them to better understand the structure of algebra. changed “represent situations” to “create and use representations” changed “study relationships” and “express generalizations” to “organize, record, and communicate ideas” moved “Underlying mathematical processes” and “Tools for algebraic thinking” from introductory “Basic Understandings” to the “Mathematical Process Standards” expanded “numerical fluency” to include mental math, estimation, and number sense deleted list of specific types of technology (calculators with graphing capabilities, data collection devices, and computers) changed “concrete” representations to “real objects [and] manipulatives” added use of paper/pencil changed “pictorial” representations to “diagrams” changed “verbal” representations to “language” moved “Algebraic thinking and symbolic reasoning” from introductory “Basic Understandings” to the “Mathematical Process Standards” deleted “symbols;” however, included in “representations” deleted “relationships;” however. Included in 2A.1F Change of strand NEW Standard 16 © 2012 lead4ward Algebra II Attributes of Functions and their Inverses 2A.2 The student applies mathematical processes to understand that functions have distinct key attributes and understand the relationship between a function and its inverse. The student is expected to: Change New Standard Current Standard Cognitive Change Content Change (Implementation Year 2015‐2016) 2A.2A graph the functions f(x)= x , f(x)=1/x, 3 x f(x)=x , f(x)= 3 x , f(x)=b , f(x)=|x|, and f(x)=logb(x) where b is 2, 10, and e, and, when applicable, analyze the key attributes such as domain, range, intercepts, symmetries, asymptotic behavior, and maximum and minimum given an interval 2A.4A identify and sketch graphs of parent functions, including linear (f (x) = x), quadratic (f(x) 2 x = x ), exponential (f(x) = a ), and logarithmic (f(x) = loga x) functions, absolute value of x (f(x) = |x|), 2A.2B graph and write the inverse of a function –1 using notation such as f (x) 2A.2C describe and analyze the relationship between a function and its inverse (quadratic and square root, logarithmic and exponential), including the restriction(s) on domain, which will restrict its range 2A.4C describe and analyze the relationship between a function and its inverse changed “describe and analyze” to “graph and write” 2A.9G connect inverses of square root functions with quadratic functions 2A.11A develop the definition of logarithms by exploring and describing the relationship between exponential functions and their inverses changed “connect” and “develop” to “describe and analyze” square root of x (f (x) = x ), and reciprocal of x (f(x) = 1/x) changed “identify and sketch graphs” to “graph” added “analyze key attributes” moved linear (f (x) = x), quadratic (f(x) = 2 x x ), and exponential (f(x) = a ) functions from Algebra II to Algebra I (see A.3C, A.7A, and A.9D) deleted the term “parent” functions 3 added cubic (f(x)=x ) and cube root (f(x)= 3 x ) functions specified the form of logarithmic functions as f(x)=logb(x) where b is 2, 10, and e added a list of “key attributes” to include (when applicable) domain, range, intercepts, symmetries, asymptotic behavior, and maximum and minimum added the use of inverse function –1 notation (f (x)) combined skills from two different strands (“Properties of Square Root Functions” and “Properties of Exponential and Logarithmic Functions”) into “Attributes of Functions and their Inverses” added restrictions on the domains and ranges of functions and inverses 2A.2D use the composition of two functions, including the necessary restrictions on the domain, to determine if the functions are inverses of each other Key: Change of grade /course Change of strand NEW Standard 17 © 2012 lead4ward Algebra II Systems of Equations and Inequalities 2A.3 The student applies mathematical processes to formulate systems of equations and inequalities, use a variety of methods to solve, and analyze reasonableness of solutions. The student is expected to: Change New Standard (Implementation Year 2015‐2016) 2A.3A formulate systems of equations, including systems consisting of three linear equations in three variables and systems consisting of two equations, the first linear and the second quadratic 2A.3E formulate systems of at least two linear inequalities in two variables Current Standard Cognitive Change 2A.3A analyze situations and formulate systems of equations in two or more unknowns or inequalities in two unknowns to solve problems 2A.3B solve systems of three linear equations in three variables by using Gaussian elimination, technology with matrices, and substitution 2A.3C solve, algebraically, systems of two equations in two variables consisting of a linear equation and a quadratic equation 2A.3F solve systems of two or more linear inequalities in two variables 2A.3B use algebraic methods, graphs, tables, or matrices, to solve systems of equations or inequalities 2A.3D determine the reasonableness of solutions to systems of a linear equation and a quadratic equation in two variables 2A.3G determine possible solutions in the solution set of systems of two or more linear inequalities in two variables 2A.3C interpret and determine the reasonableness of solutions to systems of equations or inequalities for given contexts deleted “analyze situations;” however, included in the Process Standards (see TEKS 2A.1) deleted “interpret;” however, included in the Process Standards (see TEKS 2A.1) Content Change changed strand from “Foundations for Functions” to “Systems of Equations and Inequalities” changed systems “in two or more unknowns” to “three linear equations in three variables” added systems of two equations where the first is linear and the second is quadratic changed strand from “Foundations for Functions” to “Systems of Equations and Inequalities” specified that systems involve “three linear equations in three variables” deleted the use of graphs and tables to solve systems specified “algebraic methods” as Gaussian elimination and substitution added solving systems of two equations where the first is linear and the second is quadratic changed strand from “Foundations for Functions” to “Systems of Equations and Inequalities” specified that solutions involve systems of two equations where the first is linear and the second is quadratic changed “reasonableness of solutions to systems of inequalities” to “solutions in the solution set of systems of linear inequalities” specified that systems of inequalities can have “two or more linear inequalities” Key: Change of grade /course Change of strand NEW Standard 18 © 2012 lead4ward Algebra II Quadratic and Square Root Functions, Equations, and Inequalities 2A.4 The student applies mathematical processes to understand that quadratic and square root functions, equations, and quadratic inequalities can be used to model situations, solve problems, and make predictions. The student is expected to: Current Standard Change New Standard (Implementation Year 2015‐2016) 2A.4A write the quadratic function given three specified points in the plane Cognitive Change changed “relate representations” to “write the function” 2A.6B relate representations of quadratic functions, such as algebraic, tabular, graphical, and verbal descriptions 2A.4B write the equation of a parabola using given attributes, including vertex, focus, directrix, axis of symmetry, and direction of opening 2A.6C determine a quadratic function from its roots (real and complex) or a graph 2A.4C determine the effect on the graph of 2A.9A use the parent function to investigate, describe, and predict the effects of parameter changes on the graphs of square root functions and describe limitations on the domains and ranges 2A.9B relate representations of square root functions, such as algebraic, tabular, graphical, and verbal descriptions deleted “investigate” and “describe” changed “predict the effects” to “determine the effect” 2A.5E use the method of completing the square 2A.7A use characteristics of the quadratic parent function to sketch the related graphs and connect 2 2 between the y = ax + bx + c and the y = a (x ‐ h) + k symbolic representations of quadratic functions 2A.7B use the parent function to investigate, describe, and predict the effects of changes in a, h, 2 and k on the graphs of y = a (x ‐ h) + k form of a function in applied and purely mathematical situations changed “connecting between” forms to “transforming” one to another changed “sketch the graph” to “identify the key attributes” f(x)= x when f(x) is replaced by af(x), f(x) + d, f(bx), and f(x – c) for specific positive and negative values of a, b, c, and d 2A.4D transform a quadratic function f(x) = ax 2 + bx + c to the form f(x) = a(x – h) + k to identify the different attributes of f(x) 2 Content Change specified “tabular and graphical descriptions” to be three points in the plane deleted “verbal descriptions;” however, included in the Process Standards (see TEKS 2A.1) changed “quadratic function” to “equation of a parabola” specified “graph” to include “given attributes” such as vertex, focus, directrix, axis of symmetry, and direction of opening deleted the use of roots in writing quadratic equations; skill moved from Algebra II to Algebra I (see A.6C) specified “parameter changes” on square root functions to be of the forms af(x), f(x) + d, f(bx), and f(x – c) for specific positive and negative values of a, b, c, and d deleted descriptions of the limitations on the domains and ranges of square root functions deleted tabular and verbal descriptions of square root functions; however, included in the Process Standards (see TEKS 2A.1) deleted specific reference to the method of completing the square; however, the skill is included in transforming standard form to vertex form moved transformations of quadratic functions (in vertex form) from Algebra II to Algebra I (see A.7C) Key: Change of grade /course Change of strand NEW Standard 19 © 2012 lead4ward Algebra II Quadratic and Square Root Functions, Equations, and Inequalities 2A.4 The student applies mathematical processes to understand that quadratic and square root functions, equations, and quadratic inequalities can be used to model situations, solve problems, and make predictions. The student is expected to: Change New Standard (Implementation Year 2015‐2016) 2A.4E formulate quadratic and square root equations using technology given a table of data 2A.4F solve quadratic and square root equations Key: Current Standard Cognitive Change deleted “analyze situations,” “select a method,” and “solve problems;” however, included in the Process Standards (see TEKS 2A.1) 2A.6B relate representations of quadratic functions, such as algebraic, tabular, graphical, and verbal descriptions 2A.8A analyze situations involving quadratic functions and formulate quadratic equations or inequalities to solve problems 2A.9F analyze situations modeled by square root functions, formulate equations or inequalities, select a method, and solve problems 2A.6A determine the reasonable domain and range values of quadratic functions, as well as interpret and determine the reasonableness of solutions to quadratic equations and inequalities 2A.9C determine the reasonable domain and range values of square root functions, as well as interpret and determine the reasonableness of solutions to square root equations and inequalities 2A.8D solve quadratic equations and inequalities using graphs, tables, and algebraic methods 2A.9D determine solutions of square root equations using graphs, tables, and algebraic methods 2A.8B analyze and interpret the solutions of quadratic equations using discriminants and solve quadratic equations using the quadratic formula 2A.8C compare and translate between algebraic and graphical solutions of quadratic equations Content Change deleted square root inequalities deleted quadratic inequalities; however, included in 2A.4H added “using technology” deleted graphical descriptions; however, closely related to “table of data” deleted specific references to finding domains and ranges of quadratic and square root functions; however, included in 2A.2A and 2A.7I deleted interpreting the reasonableness of solutions; however, included in the Process Standards (see TEKS 2A.1) deleted quadratic inequalities; however, included in 2A.4H deleted square root inequalities deleted quadratic inequalities; however, included in 2A.4H deleted reference to specific ways of solving equations (graphs, tables, algebraic methods) moved solving quadratic equations from Algebra II to Algebra I (see A.8A) moved graphs of quadratic functions from Algebra II to Algebra I (see TEKS A.7) 2A.4G identify extraneous solutions of square root equations 2A.4H solve quadratic inequalities Change of grade /course deleted quadratic equations; however, included in 2A.4F deleted reference to specific ways of solving inequalities (graphs, tables, algebraic methods) 2A.8D solve quadratic equations and inequalities using graphs, tables, and algebraic methods Change of strand NEW Standard 20 © 2012 lead4ward Algebra II Quadratic and Square Root Functions, Equations, and Inequalities 2A.4 The student applies mathematical processes to understand that quadratic and square root functions, equations, and quadratic inequalities can be used to model situations, solve problems, and make predictions. The student is expected to: Change New Standard Current Standard Cognitive Change Content Change (Implementation Year 2015‐2016) 2A.5A describe a conic section as the intersection of a plane and a cone 2A.5B sketch graphs of conic sections to relate simple parameter changes in the equation to corresponding changes in the graph 2A.5C identify symmetries from graphs of conic sections 2A.5D identify the conic section from a given equation moved descriptions of conic sections from Algebra II to Precalculus (see P.3F) moved graphs of ellipses and hyperbolas from Algebra II to Precalculus (see P.3G, P.3H, and P.3I) moved graphs of circles (as conic sections) from Algebra II to Geometry (see G.12E) however , graphs of parabolas (with properties of conic sections) are included in 2A.4B Key: Change of grade /course Change of strand NEW Standard 21 © 2012 lead4ward Exponential and Logarithmic Functions and Equations Algebra II 2A.5 The student applies mathematical processes to understand that exponential and logarithmic functions can be used to model situations and solve problems. The student is expected to: Change New Standard (Implementation Year 2015‐2016) 2A.5A determine the effects on the key attributes x on the graphs of f(x) = b and f(x) = logb(x) where b is 2, 10, and e when f(x) is replaced by af(x), f(x) + d, and f(x – c) for specific positive and negative real values of a, c, and d 2A.5B formulate exponential and logarithmic equations that model real‐world situations, including exponential relationships written in recursive notation 2A.5C rewrite exponential equations as their corresponding logarithmic equations and logarithmic equations as their corresponding exponential equations 2A.5D solve exponential equations of the form y = x ab where a is a nonzero real number and b is greater than zero and not equal to one and single logarithmic equations having real solutions 2A.5E determine the reasonableness of a solution to a logarithmic equation Current Standard Cognitive Change 2A.11B use the parent functions to investigate, describe, and predict the effects of parameter changes on the graphs of exponential and logarithmic functions, describe limitations on the domains and ranges, and examine asymptotic behavior changed “investigate, describe, and predict” to “determine” 2A.11F analyze a situation modeled by an exponential function, formulate an equation or inequality, and solve the problem deleted “solve the problem;” however, included in 2A.5D and 2A.5E 2A.11A develop the definition of logarithms by exploring and describing the relationship between exponential functions and their inverses changed “develop by exploring and describing” to “rewrite” 2A.11D determine solutions of exponential and logarithmic equations using graphs, tables, and algebraic methods 2A.11C determine the reasonable domain and range values of exponential and logarithmic functions, as well as interpret and determine the reasonableness of solutions to exponential and logarithmic equations and inequalities 2A.11E determine solutions of exponential and logarithmic inequalities using graphs and tables deleted “interpret” Content Change deleted reference to “parent functions” added general forms of exponential x and logarithmic functions (f(x) = b and f(x) = logb(x)) specified that logarithmic functions use bases of 2, 10, and e expanded “parameter changes” to include af(x), f(x) + d, and f(x – c) for specific positive and negative real values of a, c, and d deleted effects on domain and range; however, included in 2A.2A and 2A.7I deleted asymptotic behavior; however, included in 2A.2A deleted writing inequalities added exponential relationships written in recursive notation specified the form of exponential x equations as y=ab , where a 0, b > 0, and b 1 specified that logarithmic equations must involve a single logarithm and have real solutions deleted solving inequalities deleted finding reasonable domain and range values of exponential and logarithmic functions; however, included in 2A.2A and 2A.7I deleted solving exponential and logarithmic inequalities Key: Change of grade /course Change of strand NEW Standard 22 © 2012 lead4ward Cubic, Cube Root, Absolute Value, and Rational Functions, Equations, and Inequalities Algebra II 2A.6 The student applies mathematical processes to understand that cubic, cube root, absolute value and rational functions, equations, and inequalities can be used to model situations, solve problems, and make predictions. The student is expected to: Change New Standard (Implementation Year 2015‐2016) 3 2A.6A analyze the effect on the graphs of f(x)=x 3 and f(x)= x when f(x) is replaced by af(x), f(bx), f(x – c), and f(x) + d for specific positive and negative real values of a, b, c, and d 2A.6C analyze the effect on the graphs of f(x)=|x| when f(x) is replaced by af(x), f(bx), f(x – c), and f(x) + d for specific positive and negative real values of a, b, c, and d Current Standard Cognitive Change P.1A (A) describe parent functions symbolically n and graphically, including f(x) = x , f(x) = ln x, f(x) = x x loga x, f(x) = 1/x, f(x) = e , f(x) = |x|, f(x) = a , f(x) = sin x, f(x) = arcsin x, etc. P.2A apply basic transformations, including a•f(x), f(x) + d, f(x – c), f(b • x), and compositions with absolute value functions, including |f(x)|, and f(|x|), to the parent functions changed “describe” and “apply” to “analyze” Content Change moved cubic, cube root, and absolute value functions from Precalculus to Algebra II did not move trigonometric functions (f(x) = sin x, f(x) = arcsin x, etc.) to Algebra II x deleted f(x) = ln x, f(x) = loga x, f(x) = e , x f(x) = a ; however, included in 2A.5A deleted f(x) = 1/x; however, included in 2A.6G deleted references to “parent functions” n limited the general form of f(x) = x to 3 3 f(x)=x (when n = 3) and f(x)= x (when n = 1/3) moved transformations with cubic, cube root, and absolute value functions from Precalculus to Algebra II did not move compositions with absolute value (|f(x)|, and f(|x|)) to Algebra II 2A.6B solve cube root equations that have real roots 2A.6D formulate absolute value linear equations 2A.6E solve absolute value linear equations 2A.6F solve absolute value linear inequalities Key: Change of grade /course Change of strand NEW Standard 23 © 2012 lead4ward Cubic, Cube Root, Absolute Value, and Rational Functions, Equations, and Inequalities Algebra II 2A.6 The student applies mathematical processes to understand that cubic, cube root, absolute value and rational functions, equations, and inequalities can be used to model situations, solve problems, and make predictions. The student is expected to: Change New Standard (Implementation Year 2015‐2016) 2A.6G analyze the effect on the graphs of f(x) = 1/x when f(x) is replaced by af(x), f(bx), f(x – c), and f(x) + d for specific positive and negative real values of a, b, c, and d 2A.6H formulate rational equations that model real‐world situations 2A.6I solve rational equations that have real solutions 2A.6J determine the reasonableness of a solution to a rational equation 2A.6K determine the asymptotic restrictions on the domain of a rational function and represent domain and range using interval notation, inequalities, and set notation 2A.6L formulate and solve equations involving inverse variation Key: Change of grade /course Current Standard Cognitive Change 2A.4B extend parent functions with parameters such as a in f(x) = a/x and describe the effects of the parameter changes on the graph of parent functions 2A.10A use quotients of polynomials to describe the graphs of rational functions, predict the effects of parameter changes, describe limitations on the domains and ranges, and examine asymptotic behavior changed “predict” to “analyze” deleted “extend” and “describe” 2A.10B analyze various representations of rational functions with respect to problem situations 2A.10F analyze a situation modeled by a rational function, formulate an equation or inequality composed of a linear or quadratic function, and solve the problem 2A.10D determine the solutions of rational equations using graphs, tables, and algebraic methods deleted “analyze” 2A.10C determine the reasonable domain and range values of rational functions, as well as interpret and determine the reasonableness of solutions to rational equations and inequalities deleted “interpret” 2A.10G use functions to model and make predictions in problem situations involving direct and inverse variation 2A.10E determine solutions of rational inequalities using graphs and tables changed “use, model and make predictions” to “formulate and solve equations” Change of strand NEW Standard 24 Content Change deleted reference to “parent functions” deleted description of rational functions as quotients of polynomials expanded “parameter changes” to include af(x), f(bx), f(x – c), and f(x) + d for specific positive and negative real values of a, b, c, and d deleted references to domain, range, and asymptotic behavior; however, included in 2A.6K deleted description of rational functions as compositions of linear or quadratic functions deleted “solve the problem;” however, included in 2A.6I specified that rational equations have real solutions deleted list of specific solution methods (graphs, tables and algebraic methods); however, included in the Process Standards (see TEKS 2A.D) deleted inequalities added “asymptotic restrictions” when determining domain and range specified the use of interval notation, inequalities, and set notation to describe domains and ranges deleted “in problem situations;” however, included in the Process Standards (see TEKS 2A.1) moved solving rational inequalities from Algebra II to Precalculus (see P.5L) © 2012 lead4ward Number and Algebraic Methods Algebra II 2A.7 The student applies mathematical processes to simplify and perform operations on expressions and to solve equations. The student is expected to: Change New Standard (Implementation Year 2015‐2016) 2A.7A add, subtract, and multiply complex numbers Current Standard Cognitive Change Content Change 2A.2B use complex numbers to describe the solutions of quadratic equations changed “use” to “add, subtract, multiply, and divide” changed strand from “Foundations for Functions” to “Number and Algebraic Methods” deleted connection of complex numbers to solutions of quadratic equations 2A.2A use tools including factoring and properties of exponents to simplify expressions and to transform and solve equations changed “use” and “simplify” to “determine” changed strand from “Foundations for Functions” to “Number and Algebraic Methods” deleted properties of exponents deleted connection between factoring and solving equations specified factoring for polynomial functions of degree three and degree four specified several types of factoring: algebraic methods, grouping, and the sum and difference of two cubes 2A.1A identify the mathematical domains and ranges of functions and determine reasonable domain and range values for continuous and discrete situations changed “identify” and “determine” to “write” deleted “reasonable domain and range values;” however, included in the Process Standards (see TEKS 2A.1) specified that domains and ranges can be written in interval notation, inequalities, and set notation deleted reference to “continuous and discrete situations;” however, topics are implied by the use of intervals and inequalities (continuous) and set notation (discrete) for the domains and ranges of functions 2A.7B add, subtract, and multiply polynomials 2A.7C determine the quotient of a polynomial of degree three and of degree four when divided by a polynomial of degree one and of degree two 2A.7D determine the linear factors of a polynomial function of degree three and of degree four using algebraic methods 2A.7E determine linear and quadratic factors of a polynomial expression of degree three and of degree four, including factoring the sum and difference of two cubes and factoring by grouping 2A.7F determine the sum, difference, product, and quotient of rational expressions with integral exponents of degree one and of degree two 2A.7G rewrite radical expressions that contain variables to equivalent forms 2A.7H solve equations involving rational exponents 2A.7I write the domain and range of a function in interval notation, inequalities, and set notation Key: Change of grade /course Change of strand 25 NEW Standard © 2012 lead4ward Data Algebra II 2A.8 The student applies mathematical processes to analyze data, select appropriate models, write corresponding functions, and make predictions. The student is expected to: Change New Standard Current Standard Cognitive Change (Implementation Year 2015‐2016) 2A.8 A analyze data to select the appropriate model from among linear, quadratic, and exponential models P.3B use functions such as logarithmic, exponential, trigonometric, polynomial, etc. to model real‐life data 2A.8 B use regression methods available through technology to write a linear function, a quadratic function, and an exponential function from a given set of data P.3C use regression to determine the appropriateness of a linear function to model real‐life data (including using technology to determine the correlation coefficient) 2A.8 C predict and make decisions and critical judgments from a given set of data using linear, quadratic, and exponential models 2A.1B collect and organize data, make and interpret scatterplots, fit the graph of a function to the data, interpret the results, and proceed to model, predict, and make decisions and critical judgments changed “use” and “model” to “analyze” and “select” deleted “collect and organize data” and “make and interpret scatterplots;” however, included in the Process Standards (see TEKS 2A.1) deleted “fit the graph of a function to the data;” however, included in 2A.8A and 2A.8B Content Change moved the use of linear, quadratic, and exponential functions to model data from Precalculus to Algebra II did not move logarithmic or trigonometric modeling to Algebra II specified “polynomial” as linear or quadratic moved the use of regression from Precalculus to Algebra II expanded regression techniques to include quadratic and exponential functions deleted determining the “appropriateness” of a model deleted reference to correlation coefficient; however, included in Algebra I (see A.4A) changed strand from “Foundations for Functions” to “Data” Key: Change of grade /course Change of strand NEW Standard 26 © 2012 lead4ward Geometry Mathematical Process Standards G.1 The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to: Change Key: New Standard (Implementation Year 2015‐2016) G.1A apply mathematics to problems arising in everyday life, society, and the workplace G.1B use a problem‐solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem‐solving process and the reasonableness of the solution G.1F analyze mathematical relationships to connect and communicate mathematical ideas G.1G display, explain and justify mathematical ideas and arguments using precise mathematical language in written or oral communications G.1C select tools, including real objects, manipulatives, paper/pencil, and technology, as appropriate and techniques, including mental math, estimation, and number sense, as appropriate to solve problems G.1D communicate mathematical ideas, reasoning, and their implications using multiple representation, including symbols, diagrams, graphs, and language as appropriate Change of grade level Current Standard Cognitive Change Content Change a(6) Underlying mathematical processes. Many processes underlie all content areas in mathematics. As they do mathematics, students continually use problem‐solving, language and communication, and reasoning (justification and proof) to make connections within and outside mathematics… changed “make connections” to “apply mathematics” changed “use communication” to “display” and “communicate ideas” changed “use reasoning (justification and proof)” to “explain and justify” added “analyze mathematical relationships” moved “Underlying mathematical processes” from introductory “Basic Understandings” to the “Mathematical Process Standards” changed “within and outside mathematics” to “in everyday life, society, and the workplace” changed “language” to “precise mathematical language in written or oral communications” expanded the problem‐solving process to include: analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem‐solving process and the reasonableness a(6) Underlying mathematical processes… Students also use multiple representations, technology, applications and modeling, and numerical fluency in problem‐solving contexts. a(5) Tools for geometric thinking. Techniques for working with spatial figures and their properties are essential in understanding underlying relationships. Students use a variety of representations (concrete, pictorial, numerical, symbolic, graphical, and verbal), tools, and technology (including, but not limited to, calculators with graphing capabilities, data collection devices, and computers) to solve meaningful problems by representing and transforming figures and analyzing relationships. changed “use tools” to “select tools” changed “use a variety of representations” to “communicate ideas, reasoning, and their implications using multiple representation” deleted “use modeling” deleted “representing and transforming figures” deleted “analyzing relationships;” however, included in G.1F moved “Underlying mathematical processes” and “Tools for geometric thinking” from introductory “Basic Understandings” to the “Mathematical Process Standards” expanded “numerical fluency” to include mental math, estimation, and number sense added use of paper/pencil deleted list of specific types of technology (calculators with graphing capabilities, data collection devices, and computers) changed “concrete” representations to “real objects [and] manipulatives” changed “pictorial” representations to “diagrams” changed “verbal” representations to “language” Change of strand NEW Standard 27 © 2012 lead4ward Geometry Mathematical Process Standards G.1 The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to: Change New Standard (Implementation Year 2015‐2016) G.1E create and use representations to organize, record and communicate mathematical ideas Current Standard Cognitive Change a(2) Geometric thinking and spatial reasoning. Spatial reasoning plays a critical role in geometry; geometric figures provide powerful ways to represent mathematical situations and to express generalizations about space and spatial relationships. Students use geometric thinking to understand mathematical concepts and the relationships among them. changed “represent situations” to “create and use representations” changed “understand concepts” and “express generalizations” to “organize, record, and communicate ideas” Content Change moved “Geometric thinking and spatial reasoning” from introductory “Basic Understandings” to the “Mathematical Process Standards” deleted references to spatial reasoning and geometric figures deleted “relationships;” however, Included in G.1F Key: Change of grade level Change of strand NEW Standard 28 © 2012 lead4ward Geometry Coordinate and Transformational Geometry G.2 The student uses the process skills to understand the connections between algebra and geometry and uses the one‐ and two‐dimensional coordinate systems to verify geometric conjectures. The student is expected to: Change New Standard (Implementation Year 2015‐2016) G.2A determine the coordinates of a point that is a given fractional distance less than one from one end of a line segment to the other in one‐ and two‐dimensional coordinate systems, including finding the midpoint G.2B derive and use the distance, slope, and midpoint formulas to verify geometric relationships, including congruence of segments and parallelism or perpendicularity of pairs of lines Current Standard Cognitive Change G.7C derive and use formulas involving length, slope, and midpoint changed “use” and “investigate” to “determine” G.2C determine an equation of a line parallel or perpendicular to a given line that passes through a given point G.7B use slopes and equations of lines to investigate geometric relationships, including parallel lines, perpendicular lines, and special segments of triangles and other polygons G.7A use one‐ and two‐dimensional coordinate systems to represent points, lines, rays, line segments and figures Content Change changed strand from “Dimensionality and the Geometry of Location” to “Coordinate and Transformational Geometry” added “to verify geometric relationships” listed specific relationships, including congruence of segments and parallelism or perpendicularity of pairs of lines changed strand from “Dimensionality and the Geometry of Location” to “Coordinate and Transformational Geometry” deleted special segments of triangles and other polygons; however, included in G.5A deleted specific reference to using coordinate systems for geometric figures; however, included throughout the Geometry course (for example, see TEKS G.2 and G.3) as well as in the Process Standards (see G.1E) Key: Change of grade level Change of strand NEW Standard 29 © 2012 lead4ward Geometry Coordinate and Transformational Geometry G.3 The student uses the process skills to generate and describe rigid transformations (translation, reflection, and rotation) and non‐rigid transformations (dilations that preserve similarity and reductions and enlargements that do not preserve similarity). The student is expected to: Change New Standard (Implementation Year 2015‐2016) G.3A describe and perform transformations of figures in a plane using coordinate notation G.3B determine the image or pre‐image of a given two‐dimensional figure under a composition of rigid transformations, a composition of non‐rigid transformations, and a composition of both, including dilations where the center can be any point in the plane G.3C identify the sequence of transformations that will carry a given pre‐image onto an image on and off the coordinate plane Current Standard Cognitive Change Content Change G.10A use congruence transformations to make conjectures and justify properties of geometric figures including figures represented on a coordinate plane G.11A use and extend similarity properties and transformations to explore and justify conjectures about geometric figures changed “use” and “extend” to “describe” and “perform” deleted “make, explore, and justify conjectures” deleted “justify properties” combined skills from two strands (“Congruence and the Geometry of Size” and “Similarity and the Geometry of Shape”) into “Coordinate and Transformational Geometry” added the use of “coordinate notation” G.5C use properties of transformations and their compositions to make connections between mathematics and the real world, such as tessellations changed “use properties” and “make connections” to “identify” changed strand from “Geometric Patterns” to “Coordinate and Transformational Geometry” changed “compositions” to a “sequence” of transformations added terms “image” and “pre‐image” deleted tessellations deleted real‐world connections; however, included in the Process Standards (see G.1A) G.3D identify and distinguish between reflectional and rotational symmetry in a plane figure Key: Change of grade level Change of strand NEW Standard 30 © 2012 lead4ward Geometry Logical Argument and Constructions G.4 The student uses the process skills with deductive reasoning to understand geometric relationships. The student is expected to: Change New Standard (Implementation Year 2015‐2016) G.4A distinguish between undefined terms, definitions, postulates, conjectures, and theorems Current Standard Cognitive Change G.1A develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems changed “develop” and “connect” to “distinguish between” G.4B identify and determine the validity of the converse, inverse, and contrapositive of a conditional statement and recognize the connection between a biconditional statement and a true conditional statement with a true converse G.4C verify that a conjecture is false using a counterexample G.3A determine the validity of a conditional statement, its converse, inverse, and contrapositive added “identify” G.3C use logical reasoning to prove statements are true and find counter examples to disprove statements that are false changed “use,” “prove” and “find” to “verify” G.4D compare geometric relationships between Euclidean and spherical geometries, including parallel lines and the sum of the angles in a triangle G.1C compare and contrast the structures and implications of Euclidean and non‐Euclidean geometries deleted “contrast” Content Change changed strand from “Geometric Structure” to “Logical Argument and Constructions” deleted “structure of a mathematical system” and “logical reasoning” added “undefined terms” and “conjectures” changed strand from “Geometric Structure” to “Logical Argument and Constructions” added biconditional statements (where the conditional statement and its converse are both true) changed strand from “Geometric Structure” to “Logical Argument and Constructions” deleted the use of logical reasoning to prove statements are true; however, included throughout the Geometry TEKS (for example, see G.6B and G.8A) changed strand from “Geometric Structure” to “Logical Argument and Constructions” specified “non‐Euclidean geometries” to include on spherical geometry listed specific relationships for comparison, including parallel lines and the sum of the angles in a triangle Key: Change of grade level Change of strand NEW Standard 31 © 2012 lead4ward Geometry Logical Argument and Constructions G.5 The student uses constructions to validate conjectures about geometric figures. The student is expected to: Change Key: New Standard (Implementation Year 2015‐2016) G.5A investigate patterns to make conjectures about geometric relationships, including angles formed by parallel lines cut by a transversal, criteria required for triangle congruence, special segments of triangles, diagonals of quadrilaterals, interior and exterior angles of polygons, and special segments and angles of circles choosing from a variety of tools G.5B construct congruent segments, congruent angles, a segment bisector, an angle bisector, perpendicular lines, the perpendicular bisector of a line segment, and a line parallel to a given line through a point not on a line using a compass and a straightedge G.5C use the constructions of congruent segments, congruent angles, angle bisectors, and perpendicular bisectors to make conjectures about geometric relationships G.5D verify the Triangle Inequality theorem using constructions and apply the theorem to solve problems Change of grade level Current Standard Cognitive Change G.5B use numeric and geometric patterns to make generalizations about geometric properties, including properties of polygons, ratios in similar figures and solids, and angle relationships in polygons and circles G.9A formulate and test conjectures about the properties of parallel and perpendicular lines based on explorations and concrete models G.9B formulate and test conjectures about the properties and attributes of polygons and their component parts based on explorations and concrete models G.9C formulate and test conjectures about the properties and attributes of circles and the lines that intersect them based on explorations and concrete models G.10B justify and apply triangle congruence relationships changed “use” to “investigate” deleted “formulate,” “test,” and “justify” G.2A use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships added “construct” Change of strand NEW Standard 32 Content Change changed strand from “Geometric Patterns” to “Logical Argument and Constructions” changed (in some cases) “generalizations” to “conjectures” deleted several references to “explorations and concrete models;” however, included in the Process Standards (see G.1C) deleted reference to “ratios in similar figures;” however, included in TEKS G.7 and G.8 changed “properties of parallel lines” to “angles formed by parallel lines cut by a transversal” expanded “properties and attributes” and “angle relationships” in polygons to include “special segments of triangles, diagonals of quadrilaterals, interior and exterior angles of polygons” changed “circles and the lines that intersect them” to “special segments and angles of circles” changed strand from “Geometric Structure” to “Logical Argument and Constructions” specified the use of a compass and straight edge provided a list of several constructions: congruent segments, congruent angles, a segment bisector, an angle bisector, perpendicular lines, the perpendicular bisector of a line segment, and a line parallel to a given line through a point not on a line © 2012 lead4ward Geometry Logical Argument and Constructions G.5 The student uses constructions to validate conjectures about geometric figures. The student is expected to: Change New Standard Current Standard Cognitive Change Content Change (Implementation Year 2015‐2016) deleted specific reference to developing algebraic expression for geometric properties; however, included in the Process Standards (see G.1D) G.5A use numeric and geometric patterns to develop algebraic expressions representing geometric properties Key: Change of grade level Change of strand NEW Standard 33 © 2012 lead4ward Proof and Congruence Geometry G.6 The student uses the process skills with deductive reasoning to prove and apply theorems by using a variety of methods such as coordinate, transformational, and axiomatic and formats such as two‐column, paragraph, and flow chart. The student is expected to: Change Key: New Standard (Implementation Year 2015‐2016) G.6A verify theorems about angles formed by the intersection of lines and line segments, including vertical angles, and angles formed by parallel lines cut by a transversal and prove equidistance between the endpoints of a segment and points on its perpendicular bisector and apply these relationships to solve problems Current Standard Cognitive Change G.2B make conjectures about angles, lines, polygons, circles, and three‐dimensional figures and determine the validity of the conjectures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic G.9A formulate and test conjectures about the properties of parallel and perpendicular lines based on explorations and concrete models deleted the making or formulating of conjectures; however, included in G.4A and G.4C changed “test” and “determine the validity” to “verify,” “prove” and “apply” G.6B prove two triangles are congruent by applying the Side‐Angle‐Side, Angle‐Side‐Angle, Side‐Side‐Side, Angle‐Angle‐Side, and Hypotenuse‐Leg congruence conditions G.6C apply the definition of congruence, in terms of rigid transformations, to identify congruent figures and their corresponding sides and angles G.10B justify and apply triangle congruence relationships changed “justify” to “prove” G.10A use congruence transformations to make conjectures and justify properties of geometric figures including figures represented on a coordinate plane changed “make conjectures” and “justify properties” to “apply” and “identify” G.6D verify theorems about the relationships in triangles, including proof of the Pythagorean Theorem, the sum of interior angles, base angles of isosceles triangles, midsegments, and medians, and apply these relationships to solve problems G.8C derive, extend, and use the Pythagorean Theorem G.9B formulate and test conjectures about the properties and attributes of polygons and their component parts based on explorations and concrete models deleted “formulate conjectures;” however, included in G.4A and G.4C changed “derive” and “test” to “verify” added “apply” Change of grade level Change of strand NEW Standard 34 Content Change provided a list of specific relationships involving angles and lines (vertical angles, and angles formed by parallel lines cut by a transversal, equidistance between the endpoints of a segment and points on its perpendicular bisector) deleted “a variety of approaches;” however, included in the Process Standards (see TEKS G.1) deleted “explorations and concrete models;” however, included in the Process Standards (see G.1C) provided a list of the specific conditions for triangle congruence (Side‐Angle‐ Side, Angle‐Side‐Angle, Side‐Side‐Side, Angle‐Angle‐Side, and Hypotenuse‐Leg) added the identification of the corresponding sides and angles in congruent figures deleted “on a coordinate plane;’ however, included in G.3A changed “polygons” to triangles provided a list of specific relationships to verify (Pythagorean Theorem, the sum of interior angles, base angles of isosceles triangles, midsegments, and medians) deleted “use” of Pythagorean Theorem; however, included in G.9B deleted “explorations and concrete models;” however, included in the Process Standards (see G.1C) © 2012 lead4ward Proof and Congruence Geometry G.6 The student uses the process skills with deductive reasoning to prove and apply theorems by using a variety of methods such as coordinate, transformational, and axiomatic and formats such as two‐column, paragraph, and flow chart. The student is expected to: Change New Standard (Implementation Year 2015‐2016) G.6E prove a quadrilateral is a parallelogram, rectangle, square, or rhombus using opposite sides, opposite angles, or diagonals and apply these relationships to solve problems Current Standard Cognitive Change G.9B formulate and test conjectures about the properties and attributes of polygons and their component parts based on explorations and concrete models deleted “formulate conjectures;” however, included in G.4A and G.4C changed “test” to “prove” added “apply” Content Change changed “polygons” to quadrilaterals (parallelogram, rectangle, square, or rhombus) changed “component parts” to “opposite sides, opposite angles, or diagonals” deleted “explorations and concrete models;” however, included in the Process Standards (see G.1C) Key: Change of grade level Change of strand NEW Standard 35 © 2012 lead4ward Similarity, Proof, and Trigonometry Geometry G.7 The student uses the process skills in applying similarity to solve problems. The student is expected to: New Standard Current Standard (Implementation Year 2015‐2016) G.7A apply the definition of similarity in terms of a dilation to identify similar figures and their proportional sides and the congruent corresponding angles G.11A use and extend similarity properties and transformations to explore and justify conjectures about geometric figures changed “use,” “explore and justify” to “apply” and “identify” G.11B use ratios to solve problems involving similar figures G.11C develop, apply, and justify triangle similarity relationships, such as right triangle ratios, trigonometric ratios, and Pythagorean triples using a variety of methods deleted “use,” “develop” and “justify” added “apply” Angle‐Angle criterion and proportionality added “solve problems” Change G.7B apply the Angle‐Angle criterion to verify similar triangles and apply the proportionality of the corresponding sides to solve problems Cognitive Change Content Change changed “transformations” to “dilation” defined “similarity properties” as “proportional sides and the congruent corresponding angles” changed “ratios” and “similarity relationships” to “the proportionality of corresponding sides” deleted trigonometric ratios; however, included in G.9A deleted Pythagorean triples; however, included in G.9B G.8 The student uses the process skills with deductive reasoning to prove and apply theorems by using a variety of methods such as coordinate, transformational, and axiomatic and formats such as two‐column, paragraph, and flow chart. The student is expected to: Change New Standard (Implementation Year 2015‐2016) G.8A prove theorems about similar triangles, including the Triangle Proportionality theorem, and apply these theorems to solve problems G.8B identify and apply the relationships that exist when an altitude is drawn to the hypotenuse of a right triangle, including the geometric mean, to solve problems Current Standard Cognitive Change G.11C develop, apply, and justify triangle similarity relationships, such as right triangle ratios, trigonometric ratios, and Pythagorean triples using a variety of methods deleted “develop” changed “justify” to “prove” added “identify” added “solve problems” Content Change identified a specific “similarity relationship” to be the “Triangle Proportionality theorem” identified a specific “right triangle ratio” to be “the relationships that exist when an altitude is drawn to the hypotenuse of a right triangle” (geometric mean) deleted trigonometric ratios; however, included in G.9A deleted Pythagorean triples; however, included in G.9B Key: Change of grade level Change of strand NEW Standard 36 © 2012 lead4ward Geometry Similarity, Proof, and Trigonometry G.9 The student uses the process skills to understand and apply relationships in right triangles. The student is expected to: Change New Standard Current Standard Cognitive Change (Implementation Year 2015‐2016) G.9A determine the lengths of sides and measures of angles in a right triangle by applying the trigonometric ratios sine, cosine, and tangent to solve problems G.11C develop, apply, and justify triangle similarity relationships, such as right triangle ratios, trigonometric ratios, and Pythagorean triples using a variety of methods changed “develop, apply, and justify” to “determine” added “solve problems” G.9B apply the relationships in special right triangles 30‐60‐90 and 45‐45‐90 and the Pythagorean theorem, including Pythagorean triples, to solve problems G.5D identify and apply patterns from right triangles to solve meaningful problems, including special right triangles (45‐45‐90 and 30‐60‐90) and triangles whose sides are Pythagorean triples deleted “identify” Content Change delineated the trigonometric ratios as sine, cosine, and tangent specified the use of the trigonometric ratios in finding side lengths and angle measures deleted Pythagorean triples; however, included in G.9B changed strand from “Geometric Patterns” to “Similarity, Proof, and Trigonometry” changed “patterns” to “relationships” added the general use of the Pythagorean theorem Key: Change of grade level Change of strand NEW Standard 37 © 2012 lead4ward Geometry Two-dimensional and Three-dimensional Figures G.10 The student uses the process skills to recognize characteristics and dimensional changes of two‐ and three‐dimensional figures. The student is expected to: Change New Standard (Implementation Year 2015‐2016) G.10A identify the shapes of two‐dimensional cross‐sections of prisms, pyramids, cylinders, cones, and spheres and identify three‐ dimensional objects generated by rotations of two‐dimensional shapes G.10B determine and describe how changes in the linear dimensions of a shape affect its perimeter, area, surface area, or volume, including proportional and non‐proportional dimensional change Current Standard Cognitive Change Content Change G.6A describe and draw the intersection of a given plane with various three‐dimensional geometric figures G.6C use orthographic and isometric views of three‐dimensional geometric figures to represent and construct three‐dimensional geometric figures and solve problems G.9D analyze the characteristics of polyhedra and other three‐dimensional figures and their component parts based on explorations and concrete models changed “describe and draw” to “identify” deleted “use,” “represent,” and “construct” G.11D describe the effect on perimeter, area, and volume when one or more dimensions of a figure are changed and apply this idea in solving problems added “determine” deleted “apply” combined two strands (“Dimensionality and the Geometry of Location” and “Congruence and the Geometry of Size”) into “Two‐ dimensional and Three‐dimensional Figures” provided a list of specific “three‐ dimensional geometric figures” (prisms, pyramids, cylinders, cones, and spheres) deleted specific reference to “ isometric views” of three‐dimensional figures changed “orthographic views” to “cross‐sections” added the rotation of two‐ dimensional shapes to generate three‐dimensional objects deleted “explorations and concrete models;” however, included in the Process Standards (see G.1C) changed strand from “Similarity and the Geometry of Shape” to “Two‐ dimensional and Three‐dimensional Figures” specified that dimensional changes include both “proportional and non‐ proportional” change deleted “solving problems;” however, included in the Process Standards (see G.1A) Key: Change of grade level Change of strand NEW Standard 38 © 2012 lead4ward Geometry Two-dimensional and Three-dimensional Figures G.11 The student uses the process skills in the application of formulas to determine measures of two‐ and three‐dimensional figures. The student is expected to: Change New Standard (Implementation Year 2015‐2016) G.11A apply the formula for the area of regular polygons to solve problems using appropriate units of measure G.11B determine the area of composite two‐ dimensional figures comprised of a combination of triangles, parallelograms, trapezoids, kites, regular polygons, or sectors of circles to solve problems using appropriate units of measure G.11C apply the formulas for the total and lateral surface area of three‐dimensional figures, including prisms, pyramids, cones, cylinders, spheres, and composite figures, to solve problems using appropriate units of measure G.11D apply the formulas for the volume of three‐dimensional figures, including prisms, pyramids, cones, cylinders, spheres, and composite figures, to solve problems using appropriate units of measure Current Standard Cognitive Change G.8A find areas of regular polygons, circles, and composite figures G.8B find areas of sectors and arc lengths of circles using proportional reasoning changed “find” to “apply,” “solve” and “determine” G.8D find surface areas and volumes of prisms, pyramids, spheres, cones, cylinders, and composites of these figures in problem situations changed “find” to “apply” and “solve” G.6B use nets to represent and construct three‐ dimensional geometric figures G.8F use conversions between measurement systems to solve problems in real‐world situations Content Change changed strand from “Congruence and the Geometry of Size” to “Two‐ dimensional and Three‐dimensional Figures” specified that “composite figures” can be “comprised of a combination of triangles, parallelograms, trapezoids, kites, regular polygons, or sectors of circles” deleted arc length; however, included in G.12B added “using appropriate units of measure” changed strand from “Congruence and the Geometry of Size” to “Two‐ dimensional and Three‐dimensional Figures” changed “surface areas” to “total and lateral surface areas” added “using appropriate units of measure” deleted specific reference to nets; however, concept is included in determining surface area (see G.11C) deleted unit conversions; however, included use of “appropriate units of measure” in G.11A, G.11B, G.11C, and G.11D Key: Change of grade level Change of strand NEW Standard 39 © 2012 lead4ward Circles Geometry G.12 The student uses the process skills to understand geometric relationships and apply theorems and equations about circles. The student is expected to: Change New Standard (Implementation Year 2015‐2016) G.12A apply theorems about circles, including relationships among angles, radii, chords, tangents, and secants, to solve non‐contextual problems G.12B apply the proportional relationship between the measure of an arc length of a circle and the circumference of the circle to solve problems G.12C apply the proportional relationship between the measure of the area of a sector of a circle and the area of the circle to solve problems Current Standard Cognitive Change Content Change G.9C formulate and test conjectures about the properties and attributes of circles and the lines that intersect them based on explorations and concrete models changed “formulate and test conjectures” to “apply theorems” G.8B find areas of sectors and arc lengths of circles using proportional reasoning changed “find” to “apply” and “solve” changed strand from “Congruence and the Geometry of Size” to “Circles” changed “circles and the lines that intersect them” to “relationships among angles, radii, chords, tangents, and secants” in circles deleted “explorations and concrete models;” however, included in the Process Standards (see G.1C) added “in non‐contextual problems” changed strand from “Congruence and the Geometry of Size” to “Circles” for arc length, specified “proportional reasoning” to be the “relationship between the measure of an arc length of a circle and the circumference of the circle” for sector area, specified “proportional reasoning” to be the “relationship between the measure of the area of a sector of a circle and the area of the circle” 2A.5B sketch graphs of conic sections to relate simple parameter changes in the equation to corresponding changes in the graph changed “sketch” and “relate” to “show” and “determine” G.12D describe radian measure of an angle as the ratio of the length of an arc intercepted by a central angle and the radius of the circle G.12E show that the equation of a circle with 2 2 2 center at the origin and radius r is x + y = r and determine the equation for the graph of a circle 2 2 with radius r and center (h, k), (x – h) + (y – k) 2 =r moved the equation of a circle (as a conic section) from Algebra II to Geometry provided the general form for the equation of a circle centered at the 2 2 2 origin (x + y = r ) specified “parameter changes” to be values for the radius r and 2 2 2 center (h, k) in (x – h) + (y – k) =r Key: Change of grade level Change of strand NEW Standard 40 © 2012 lead4ward Probability Geometry G.13 The student uses the process skills to understand probability in real‐world situations and how to apply independence and dependence of events. The student is expected to: Change New Standard Current Standard Cognitive Change Content Change (Implementation Year 2015‐2016) G.13A develop strategies to use permutations and combinations to solve contextual problems G.13B determine probabilities based on area to solve contextual problems G.8E use area models to connect geometry to probability and statistics changed “use” and “connect” to “determine” and “solve” G.13C identify whether two events are independent and compute the probability of the two events occurring together with or without replacement 8.11A find the probabilities of dependent and independent events added “identify” changed “find” to “compute” changed strand from “Congruence and the Geometry of Size” to “Probability” deleted reference to “statistics” moved finding probability from grade 8 to Geometry changed “dependent events” to “two events occurring together” added “with or without replacement” G.13D apply conditional probability in contextual problems G.13E apply independence in contextual problems Key: Change of grade level Change of strand NEW Standard 41 © 2012 lead4ward
