New York Mathematics Standards Alignment to: the Common Core State Standards (CCSS) the National Essential Skills Study (NESS) Please note that the National Essential Skills Study (NESS) is only aligned to the New York Integrated Algebra, Geometry, and Algebra II/Trigonometry Performance Indicators. The NESS descriptors are not intentionally aligned to the Common Core State Standards (CCSS) or their subparts. Any alignment between NESS descriptors and CCSS is coincidental. New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra Mathematics Domains/Clusters/ Common Core State Standards High School National Essential Skills Study (NESS) National Rankings Rank Problem Solving Strand Students will new mathematical knowledge through problem solving. A.PS.1 Use a variety of problem solving strategies to understand new mathematical content Number & Quantity: Quantities Reason quantitatively and use units to solve problems. 1. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Algebra: Reasoning with Equations & Inequalities Solve equations and inequalities in one variable. 3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. 4. Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Functions: Interpreting Functions Interpret functions that arise in applications in terms of the context. 6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. 2011 International Center for Leadership in Education Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 1 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra Mathematics Domains/Clusters/ Common Core State Standards High School A.PS.2 Recognize and understand equivalent representations of a problem situation or a mathematical concept Number & Quantity: Quantities Reason quantitatively and use units to solve problems. 1. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Algebra: Seeing Structure in Expressions Interpret the structure of expressions. 2. Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2). Functions: Interpreting Functions Analyze functions using different representations. 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. 9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Statistics & Probability: Interpreting Categorical & Quantitative Data Summarize, represent and interpret data on a single count or measurement variable. 1. Represent data with plots on the real number line (dot plots, histograms, and box plots). Summarize, represent and interpret data on two categorical and quantitative variables. 6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 2 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra Students will solve problems that arise in mathematics and in other contexts. A.PS.3 Observe and explain patterns to formulate generalizations and conjectures Mathematics Domains/Clusters/ Common Core State Standards High School Algebra: Seeing Structure in Expressions Write expressions in equivalent forms to solve problems. 4. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. Functions: Building Functions Build a function that models a relationship between two quantities. 1. Write a function that describes a relationship between two quantities. a. Determine an explicit expression, a recursive process, or steps for calculation from a context. b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. 2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Apply pattern recognition in data sets and series to reason or solve problems involving arithmetic, geometry, exponents, etc. M16 Mathematics – Page 3 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra Mathematics Domains/Clusters/ Common Core State Standards High School A.PS.4 Use multiple representations to represent and explain problem situations (e.g., verbally, numerically, algebraically, graphically) Number & Quantity: Quantities Reason quantitatively and use units to solve problems. 1. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Algebra: Creating Equations Create equations that describe numbers or relationships. 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. 2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R. Functions: Interpreting Functions Analyze functions using different representations. 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. 8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. 9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank M10 M11 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Apply variables in expressions and equations to solve problems (i.e., write mathematical equations for given situation, create a mathematical model to understand the relationships between variables, or make connections between the structures of mathematically abstract concepts and the real world). Evaluate and employ accurate and appropriate procedures for statistical data collection, organization, analysis, and display including making estimates and predictions, critiquing data, and drawing inferences (e.g., using the normal curve and z-scores, line of best fit). M21 Mathematics – Page 4 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra Students will apply and adapt a variety of appropriate strategies to solve problems. A.PS.5 Choose an effective approach to solve a problem from a variety of strategies (numeric, graphic, algebraic) Mathematics Domains/Clusters/ Common Core State Standards High School Number & Quantity: Quantities Reason quantitatively and use units to solve problems. 1. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. 2. Define appropriate quantities for the purpose of descriptive modeling. 3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. Algebra: Seeing Structure in Expressions Write expressions in equivalent forms to solve problems. 3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. a. Factor a quadratic expression to reveal the zeros of the function it defines. b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. Functions: Interpreting Functions Analyze functions using different representations. 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. 8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. 9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 5 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra Mathematics Domains/Clusters/ Common Core State Standards High School A.PS.5 (Continued from previous page) (Continued from previous page) Statistics & Probability: Interpreting Categorical & Quantitative Data Summarize, represent and interpret data on a single count or measurement variable. 1. Represent data with plots on the real number line (dot plots, histograms, and box plots). Summarize, represent and interpret data on two categorical and quantitative variables. 6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. Functions: Interpreting Functions Analyze functions using different representations. 8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Functions: Building Functions Build a function that models a relationship between two quantities. 2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. There is no New York Mathematics Performance Indicator–Common Core alignment. A.PS.6 Use a variety of strategies to extend solution methods to other problems A.PS.7 Work in collaboration with others to propose, critique, evaluate, and value alternative approaches to problem solving Students will monitor and reflect on the process of mathematical problem solving A.PS.8 Determine information required to solve a problem, choose methods for obtaining the information, and define parameters for acceptable solutions National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 M10 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. There is no New York Mathematics Performance Indicator–Common Core alignment. 2011 International Center for Leadership in Education Mathematics – Page 6 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra A.PS.9 Interpret solutions within the given constraints of a problem A.PS.10 Evaluate the relative efficiency of different representations and solution methods of a problem Mathematics Domains/Clusters/ Common Core State Standards High School Algebra: Creating Equations Create equations that describe numbers or relationships. 3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Algebra: Reasoning with Equations & Inequalities Understand solving equations as a process of reasoning and explain the reasoning. 2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. There is no New York Mathematics Performance Indicator–Common Core alignment. National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 M10 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Reasoning and Proof Strand Students will recognize reasoning and proof as fundamental aspects of mathematics. A.RP.1 Recognize that mathematical ideas can be supported by a variety of strategies Number & Quantity: Quantities Reason quantitatively and use units to solve problems. 1. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Algebra: Reasoning with Equations & Inequalities Understand solving equations as a process of reasoning and explain the reasoning. 1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. 2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. 2011 International Center for Leadership in Education Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 7 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra Mathematics Domains/Clusters/ Common Core State Standards High School A.RP.1 (Continued from previous page) (Continued from previous page) Statistics & Probability: Interpreting Categorical & Quantitative Data Summarize, represent and interpret data on a single count or measurement variable. 2. Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. 3. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). 4. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. Students will make and investigate mathematical conjectures. A.RP.2 Use mathematical strategies to reach a conclusion and provide supportive arguments for a conjecture Number & Quantity: Quantities Reason quantitatively and use units to solve problems. 1. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Algebra: Reasoning with Equations & Inequalities Understand solving equations as a process of reasoning and explain the reasoning. 1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. 2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Statistics & Probability: Interpreting Categorical & Quantitative Data Summarize, represent and interpret data on a single count or measurement variable. 2. Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. 3. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 8 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra Mathematics Domains/Clusters/ Common Core State Standards High School A.RP.2 (Continued from previous page) (Continued from previous page) 4. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. Summarize, represent and interpret data on two categorical and quantitative variables. 6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. Number & Quantity: Quantities Reason quantitatively and use units to solve problems. 1. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. 3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. A.RP.3 Recognize when an approximation is more appropriate than an exact answer Students will develop and evaluate mathematical arguments and proofs. A.RP.4 Develop, verify, and explain an argument, using appropriate mathematical ideas and language Algebra: Seeing Structure in Expressions Write expressions in equivalent forms to solve problems. 4. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. Algebra: Reasoning with Equations & Inequalities Understand solving equations as a process of reasoning and explain the reasoning. 1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. 2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Functions: Building Functions Build a function that models a relationship between two quantities. 2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank M10 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 9 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra A.RP.5 Construct logical arguments that verify claims or counterexamples that refute them Mathematics Domains/Clusters/ Common Core State Standards High School Algebra: Seeing Structure in Expressions Write expressions in equivalent forms to solve problems. 4. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. Algebra: Reasoning with Equations & Inequalities Understand solving equations as a process of reasoning and explain the reasoning. 1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. 2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Functions: Building Functions Build a function that models a relationship between two quantities. 2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 10 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra A.RP.6 Present correct mathematical arguments in a variety of forms Mathematics Domains/Clusters/ Common Core State Standards High School Algebra: Seeing Structure in Expressions Write expressions in equivalent forms to solve problems. 4. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. Algebra: Reasoning with Equations & Inequalities Understand solving equations as a process of reasoning and explain the reasoning. 1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. 2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Functions: Building Functions Build a function that models a relationship between two quantities. 2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Statistics & Probability: Interpreting Categorical & Quantitative Data Summarize, represent and interpret data on a single count or measurement variable. 3. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). 4. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 11 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra Mathematics Domains/Clusters/ Common Core State Standards High School A.RP.7 Evaluate written arguments for validity Algebra: Seeing Structure in Expressions Write expressions in equivalent forms to solve problems. 4. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. Algebra: Reasoning with Equations & Inequalities Understand solving equations as a process of reasoning and explain the reasoning. 1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. 2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Functions: Building Functions Build a function that models a relationship between two quantities. 2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Statistics & Probability: Interpreting Categorical & Quantitative Data Summarize, represent and interpret data on a single count or measurement variable. 3. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). 4. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. Students will select and use various types of reasoning and methods of proof. A.RP.8 Support an argument by using a systematic approach to test more than one case National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 There is no New York Mathematics Performance Indicator–Common Core alignment. 2011 International Center for Leadership in Education M10 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Mathematics – Page 12 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra Mathematics Domains/Clusters/ Common Core State Standards High School A.RP.9 Devise ways to verify results or use counterexamples to refute incorrect statements There is no New York Mathematics Performance Indicator–Common Core alignment. A.RP.10 Extend specific results to more general cases Algebra: Reasoning with Equations & Inequalities Solve equations and inequalities in one variable. 3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. 4. Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. There is no New York Mathematics Performance Indicator–Common Core alignment. A.RP.11 Use a Venn diagram to support a logical argument National Essential Skills Study (NESS) National Rankings Rank M10 M10 M10 M21 A.RP.12 Apply inductive reasoning in making and supporting mathematical conjectures There is no New York Mathematics Performance Indicator–Common Core alignment. 2011 International Center for Leadership in Education M10 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Evaluate and employ accurate and appropriate procedures for statistical data collection, organization, analysis, and display including making estimates and predictions, critiquing data, and drawing inferences (e.g., using the normal curve and z-scores, line of best fit). Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Mathematics – Page 13 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra Mathematics Domains/Clusters/ Common Core State Standards High School National Essential Skills Study (NESS) National Rankings Rank Communication Strand Students will organize and consolidate their mathematical thinking through communication A.CM.1 Communicate verbally and in writing a correct, complete, coherent, and clear design (outline) and explanation for the steps used in solving a problem A.CM.2 Use mathematical representations to communicate with appropriate accuracy, including numerical tables, formulas, functions, equations, charts, graphs, Venn diagrams, and other diagrams Algebra: Reasoning with Equations & Inequalities Understand solving equations as a process of reasoning and explain the reasoning. 1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Number & Quantity: Quantities Reason quantitatively and use units to solve problems. 2. Define appropriate quantities for the purpose of descriptive modeling. 3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. Algebra: Creating Equations Create equations that describe numbers or relationships. 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. 2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Algebra: Reasoning with Equations & Inequalities Represent and solve equations and inequalities graphically. 10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). 11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. 2011 International Center for Leadership in Education M10 M10 M11 M21 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Apply variables in expressions and equations to solve problems (i.e., write mathematical equations for given situation, create a mathematical model to understand the relationships between variables, or make connections between the structures of mathematically abstract concepts and the real world). Evaluate and employ accurate and appropriate procedures for statistical data collection, organization, analysis, and display including making estimates and predictions, critiquing data, and drawing inferences (e.g., using the normal curve and z-scores, line of best fit). Mathematics – Page 14 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra A.CM.2 (Continued from previous page) Mathematics Domains/Clusters/ Common Core State Standards High School National Essential Skills Study (NESS) National Rankings Rank (Continued from previous page) 12. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. Functions: Interpreting Functions Analyze functions using different representations. 77. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. 8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. 9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Functions: Linear, Quadratic & Exponential Construct and compare linear, quadratic, and exponential models and solve problems. 1. Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. Statistics & Probability: Interpreting Categorical & Quantitative Data Summarize, represent and interpret data on a single count or measurement variable. 1. Represent data with plots on the real number line (dot plots, histograms, and box plots). 2011 International Center for Leadership in Education Mathematics – Page 15 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra A.CM.2 (Continued from previous page) Students will communicate their mathematical thinking coherently and clearly to peers, teachers, and others. A.CM.3 Present organized mathematical ideas with the use of appropriate standard notations, including the use of symbols and other representations when sharing an idea in verbal and written form Mathematics Domains/Clusters/ Common Core State Standards High School National Essential Skills Study (NESS) National Rankings Rank (Continued from previous page) Summarize, represent and interpret data on two categorical and quantitative variables. 6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. Number & Quantity: Quantities Reason quantitatively and use units to solve problems. 1. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. 2. Define appropriate quantities for the purpose of descriptive modeling. 3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. Functions: Interpreting Functions Understand the concept of a function and use function notation. 2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. 3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1. Functions: Building Functions Build a function that models a relationship between two quantities. 2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. 2011 International Center for Leadership in Education Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 16 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra A.CM.4 Explain relationships among different representations of a problem A.CM.5 Communicate logical arguments clearly, showing why a result makes sense and why the reasoning is valid Mathematics Domains/Clusters/ Common Core State Standards High School Functions: Interpreting Functions Interpret functions that arise in applications in terms of the context. 4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Analyze functions using different representations. 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. 8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Algebra: Creating Equations Create equations that describe numbers or relationships. 4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R. Algebra: Reasoning with Equations & Inequalities Understand solving equations as a process of reasoning and explain the reasoning. 1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Solve equations and inequalities in one variable. 4. Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 17 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra A.CM.6 Support or reject arguments or questions raised by others about the correctness of mathematical work Students will analyze and evaluate the mathematical thinking and strategies of others. A.CM.7 Read and listen for logical understanding of mathematical thinking shared by other students A.CM.8 Reflect on strategies of others in relation to one’s own strategy Mathematics Domains/Clusters/ Common Core State Standards High School National Essential Skills Study (NESS) National Rankings Rank There is no New York Mathematics Performance Indicator–Common Core alignment. M10 There is no New York Mathematics Performance Indicator–Common Core alignment. M10 There is no New York Mathematics Performance Indicator–Common Core alignment. M10 A.CM.9 Formulate mathematical questions that elicit, extend, or challenge strategies, solutions, and/or conjectures of others Students will use the language of mathematics to express mathematical ideas precisely A.CM.10 Use correct mathematical language in developing mathematical questions that elicit, extend, or challenge other students’ conjectures There is no New York Mathematics Performance Indicator–Common Core alignment. M10 There is no New York Mathematics Performance Indicator–Common Core alignment. 2011 International Center for Leadership in Education M10 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Mathematics – Page 18 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra A.CM.11 Represent word problems using standard mathematical notation Mathematics Domains/Clusters/ Common Core State Standards High School Number & Quantity: Quantities Reason quantitatively and use units to solve problems. 1. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Algebra: Creating Equations Create equations that describe numbers or relationships. 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. 2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Functions: Interpreting Functions Interpret functions that arise in applications in terms of the context. 5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Functions: Building Functions Build a function that models a relationship between two quantities. 1. Write a function that describes a relationship between two quantities. a. Determine an explicit expression, a recursive process, or steps for calculation from a context. b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. 2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank M10 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Apply variables in expressions and equations to solve problems (i.e., write mathematical equations for given situation, create a mathematical model to understand the relationships between variables, or make connections between the structures of mathematically abstract concepts and the real world). M11 Mathematics – Page 19 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra A.CM.12 Understand and use appropriate language, representations, and terminology when describing objects, relationships, mathematical solutions, and rationale Mathematics Domains/Clusters/ Common Core State Standards High School Number & Quantity: Quantities Reason quantitatively and use units to solve problems. 1. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Algebra: Creating Equations Create equations that describe numbers or relationships. 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. 2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Algebra: Reasoning with Equations & Inequalities Understand solving equations as a process of reasoning and explain the reasoning. 1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Functions: Interpreting Functions Interpret functions that arise in applications in terms of the context. 5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Functions: Building Functions Build a function that models a relationship between two quantities. 1. Write a function that describes a relationship between two quantities. a. Determine an explicit expression, a recursive process, or steps for calculation from a context. b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. 2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 20 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra A.CM.13 Draw conclusions about mathematical ideas through decoding, comprehension, and interpretation of mathematical visuals, symbols, and technical writing Mathematics Domains/Clusters/ Common Core State Standards High School Number & Quantity: Quantities Reason quantitatively and use units to solve problems. 3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. Algebra: Seeing Structure in Expressions Interpret the structure of expressions. 1. Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. Algebra: Creating Equations Create equations that describe numbers or relationships. 4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R. National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Connections Strand Students will recognize and use connections among mathematical ideas. A.CN.1 Understand and make connections among multiple representations of the same mathematical idea Functions: Interpreting Functions Interpret functions that arise in applications in terms of the context. 4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Analyze functions using different representations. 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. 8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. 2011 International Center for Leadership in Education Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 21 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra Mathematics Domains/Clusters/ Common Core State Standards High School A.CN.2 Understand the corresponding procedures for similar problems or mathematical concepts Functions: Interpreting Functions Interpret functions that arise in applications in terms of the context. 4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Analyze functions using different representations. 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. 8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 22 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra Students will understand how mathematical ideas interconnect and build on one another to produce a coherent whole. A.CN.3 Model situations mathematically, using representations to draw conclusions and formulate new situations Mathematics Domains/Clusters/ Common Core State Standards High School Algebra: Creating Equations Create equations that describe numbers or relationships. 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. 2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Functions: Interpreting Functions Interpret functions that arise in applications in terms of the context. 5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. 6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Functions: Linear, Quadratic & Exponential Construct and compare linear, quadratic, and exponential models and solve problems. 1. Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 23 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra Mathematics Domains/Clusters/ Common Core State Standards High School A.CN.4 Understand how concepts, procedures, and mathematical results in one area of mathematics can be used to solve problems in other areas of mathematics Algebra: Reasoning with Equations & Inequalities Solve equations and inequalities in one variable. 4. Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. Functions: Interpreting Functions Analyze functions using different representations. 8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Number & Quantity: Quantities Reason quantitatively and use units to solve problems. 2. Define appropriate quantities for the purpose of descriptive modeling. Algebra: Creating Equations Create equations that describe numbers or relationships. 4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R. A.CN.5 Understand how quantitative models connect to various physical models and representations Students will recognize and apply mathematics in contexts outside of mathematics. A.CN.6 Recognize and apply mathematics to situations in the outside world Algebra: Seeing Structure in Expressions Write expressions in equivalent forms to solve problems. 4. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. Functions: Building Functions Build a function that models a relationship between two quantities. 1. Write a function that describes a relationship between two quantities. a. Determine an explicit expression, a recursive process, or steps for calculation from a context. b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 M10 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 24 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra A.CN.7 Recognize and apply mathematical ideas to problem situations that develop outside of mathematics A.CN.8 Develop an appreciation for the historical development of mathematics Mathematics Domains/Clusters/ Common Core State Standards High School Algebra: Creating Equations Create equations that describe numbers or relationships. 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. 2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Functions: Building Functions Build a function that models a relationship between two quantities. 1. Write a function that describes a relationship between two quantities. a. Determine an explicit expression, a recursive process, or steps for calculation from a context. b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. There is no New York Mathematics Performance Indicator–Common Core alignment. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 M10 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Mathematics – Page 25 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra Mathematics Domains/Clusters/ Common Core State Standards High School National Essential Skills Study (NESS) National Rankings Rank Representation Strand Students will create and use representations to organize, record, and communicate mathematical ideas. A.R.1 Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts Algebra: Creating Equations Create equations that describe numbers or relationships. 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. 2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Algebra: Reasoning with Equations & Inequalities Represent and solve equations and inequalities graphically. 10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). 12. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. Functions: Interpreting Functions Analyze functions using different representations. 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. 8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. 2011 International Center for Leadership in Education M10 M11 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Apply variables in expressions and equations to solve problems (i.e., write mathematical equations for given situation, create a mathematical model to understand the relationships between variables, or make connections between the structures of mathematically abstract concepts and the real world). Evaluate and employ accurate and appropriate procedures for statistical data collection, organization, analysis, and display including making estimates and predictions, critiquing data, and drawing inferences (e.g., using the normal curve and z-scores, line of best fit). M21 Mathematics – Page 26 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra Mathematics Domains/Clusters/ Common Core State Standards High School A.R.1 (Continued from previous page) (Continued from previous page) 9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Statistics & Probability: Interpreting Categorical & Quantitative Data Summarize, represent and interpret data on a single count or measurement variable. 1. Represent data with plots on the real number line (dot plots, histograms, and box plots). Summarize, represent and interpret data on two categorical and quantitative variables. 6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. Algebra: Creating Equations Create equations that describe numbers or relationships. 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. 2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Algebra: Reasoning with Equations & Inequalities Represent and solve equations and inequalities graphically. 10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). 12. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. A.R.2 Recognize, compare, and use an array of representational forms 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 27 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra Mathematics Domains/Clusters/ Common Core State Standards High School A.R.2 (Continued from previous page) (Continued from previous page) Functions: Interpreting Functions Analyze functions using different representations. 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. 8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. 9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Statistics & Probability: Interpreting Categorical & Quantitative Data Summarize, represent and interpret data on a single count or measurement variable. 1. Represent data with plots on the real number line (dot plots, histograms, and box plots). Summarize, represent and interpret data on two categorical and quantitative variables. 6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Mathematics – Page 28 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra Mathematics Domains/Clusters/ Common Core State Standards High School A.R.3 Use representation as a tool for exploring and understanding mathematical ideas Algebra: Creating Equations Create equations that describe numbers or relationships. 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. 2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Algebra: Reasoning with Equations & Inequalities Represent and solve equations and inequalities graphically. 10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). 12. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. Functions: Interpreting Functions Analyze functions using different representations. 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. 8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. 9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 29 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra Mathematics Domains/Clusters/ Common Core State Standards High School A.R.3 (Continued from previous page) (Continued from previous page) Statistics & Probability: Interpreting Categorical & Quantitative Data Summarize, represent and interpret data on a single count or measurement variable. 1. Represent data with plots on the real number line (dot plots, histograms, and box plots). Summarize, represent and interpret data on two categorical and quantitative variables. 6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. Students will select, apply, and translate among mathematical representations to solve problems A.R.4 Select appropriate representations to solve problem situations Algebra: Creating Equations Create equations that describe numbers or relationships. 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. 2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Algebra: Reasoning with Equations & Inequalities Represent and solve equations and inequalities graphically. 10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). 12. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. Functions: Interpreting Functions Analyze functions using different representations. 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 30 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra Mathematics Domains/Clusters/ Common Core State Standards High School A.R.4 (Continued from previous page) (Continued from previous page) 8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. 9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Algebra: Creating Equations Create equations that describe numbers or relationships. 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. 2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Algebra: Reasoning with Equations & Inequalities Represent and solve equations and inequalities graphically. 11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. Functions: Interpreting Functions Analyze functions using different representations. 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. 8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. A.R.5 Investigate relationships between different representations and their impact on a given problem 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 31 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra A.R.5 (Continued from previous page) Students will use representations to model and interpret physical, social, and mathematical phenomena. A.R.6 Use mathematics to show and understand physical phenomena (e.g., find the height of a building if a ladder of a given length forms a given angle of elevation with the ground) A.R.7 Use mathematics to show and understand social phenomena (e.g., determine profit from student and adult ticket sales) A.R.8 Use mathematics to show and understand mathematical phenomena (e.g., compare the graphs of the functions represented by the equations y x 2 and y x 2 ) Mathematics Domains/Clusters/ Common Core State Standards High School National Essential Skills Study (NESS) National Rankings Rank (Continued from previous page) a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. 9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Geometry: Similarity, Right Triangles, & Trigonometry Define trigonometric ratios and solve problems involving right triangles. 8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Algebra: Creating Equations Create equations that describe numbers or relationships. 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Algebra: Reasoning with Equations & Inequalities Solve equations and inequalities in one variable. 3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Functions: Building Functions Build new functions from existing functions. 3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. 2011 International Center for Leadership in Education M10 M10 M11 M10 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Apply variables in expressions and equations to solve problems (i.e., write mathematical equations for given situation, create a mathematical model to understand the relationships between variables, or make connections between the structures of mathematically abstract concepts and the real world). Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Mathematics – Page 32 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra Mathematics Domains/Clusters/ Common Core State Standards High School National Essential Skills Study (NESS) National Rankings Rank Number Sense and Operations Strand Students will understand numbers, multiple ways of representing numbers, relationships among numbers, and number systems. Number Theory A.N.1 Identify and apply the properties of real numbers (closure, commutative, associative, distributive, identity, inverse) Note: Students do not need to identify groups and fields, but students should be engaged in the ideas Students will understand meanings of operations and procedures, and how they relate to one another. Operations Number & Quantity: The Real Number System Use properties of rational and irrational numbers. 3. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. A.N.2 Simplify radical terms (no variable in the radicand) Number & Quantity: The Real Number System Extend the properties of exponents to rational exponents. 2. Rewrite expressions involving radicals and rational exponents using the properties of exponents. M2 M20 M29 M33 A.N.3 Perform the four arithmetic operations using like and unlike radical terms and express the result in simplest form Algebra: Reasoning with Equations & Inequalities Understand solving equations as a process of reasoning and explain the reasoning. 2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. 2011 International Center for Leadership in Education Understand and apply basic algebraic properties (commutative and associative laws of addition and multiplication, distributive law of multiplication over addition, and identities and inverses). Understand and apply the basic properties and laws of exponents and scientific notation to solve problems, including those with fractional, negative, and zero exponents. Factor a composite number into its prime components and use least common denominators or least common multiples to solve equations. Perform operations with radicals, such as addition, subtraction, and multiplication. Perform operations with radicals, such as addition, subtraction, and multiplication. M33 Mathematics – Page 33 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra Mathematics Domains/Clusters/ Common Core State Standards High School A.N.4 Understand and use scientific notation to compute products and quotients of numbers There is no New York Mathematics Performance Indicator–Common Core alignment. A.N.5 Solve algebraic problems arising from situations that involve fractions, decimals, percents (decrease/increase and discount), and proportionality/direct variation Number & Quantity: Quantities Reason quantitatively and use units to solve problems. 1. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Algebra: Arithmetic with Polynomials & Rational Expressions Rewrite rational expressions. 7. (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. Algebra: Reasoning with Equations & Inequalities Understand solving equations as a process of reasoning and explain the reasoning. 2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Number & Quantity: The Real Number System Extend the properties of exponents to rational exponents. 1. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5. 2. Rewrite expressions involving radicals and rational exponents using the properties of exponents. Algebra: Arithmetic with Polynomials & Rational Expressions Use polynomial identities to solve problems. 5. (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle. [The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.] A.N.6 Evaluate expressions involving factorial(s), absolute value(s), and exponential expression(s) National Essential Skills Study (NESS) National Rankings Rank M20 2011 International Center for Leadership in Education M1 M3 Understand and apply the basic properties and laws of exponents and scientific notation to solve problems, including those with fractional, negative, and zero exponents. Perform operations fluently with positive and negative numbers, including decimals, ratios, percents, and fractions, and show reasoning to justify results. Use proportional reasoning to solve realworld problems. Solve and graphically sketch problems involving two variables that exhibit direct and indirect variation. M57 M20 Understand and apply the basic properties and laws of exponents and scientific notation to solve problems, including those with fractional, negative, and zero exponents. Use the properties of real (rational and irrational) numbers and demonstrate understanding of ordering and absolute value. M35 Mathematics – Page 34 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra A.N.7 Determine the number of possible events, using counting techniques or the Fundamental Principle of Counting A.N.8 Determine the number of possible arrangements (permutations) of a list of items Mathematics Domains/Clusters/ Common Core State Standards High School Statistics & Probability: Conditional Probability & the Rules of Probability Use the rules of probability to compute probabilities of compound events in a uniform probability model. 9. (+) Use permutations and combinations to compute probabilities of compound events and solve problems. Statistics & Probability: Conditional Probability & the Rules of Probability Use the rules of probability to compute probabilities of compound events in a uniform probability model. 9. (+) Use permutations and combinations to compute probabilities of compound events and solve problems. National Essential Skills Study (NESS) National Rankings Rank M32 M51 Determine the probability of single and compound events and use the Counting Principle to determine the probability of independent events occurring jointly. Determine combinations (the various groupings a set may be arranged in without regard to order) and permutations (arrangements of a set where order matters). Algebra Strand Students will represent and analyze algebraically a wide variety of problem solving situations. Variables and Expressions A.A.1 Translate a quantitative verbal phrase into an algebraic expression Algebra: Creating Equations Create equations that describe numbers or relationships. 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. 2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Functions: Building Functions Build a function that models a relationship between two quantities. 1. Write a function that describes a relationship between two quantities. a. Determine an explicit expression, a recursive process, or steps for calculation from a context. b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. 2011 International Center for Leadership in Education M11 Apply variables in expressions and equations to solve problems (i.e., write mathematical equations for given situation, create a mathematical model to understand the relationships between variables, or make connections between the structures of mathematically abstract concepts and the real world). Mathematics – Page 35 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra A.A.1 (Continued from previous page) A.A.2 Write a verbal expression that matches a given mathematical expression Mathematics Domains/Clusters/ Common Core State Standards High School National Essential Skills Study (NESS) National Rankings Rank (Continued from previous page) 2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. There is no New York Mathematics Performance Indicator–Common Core alignment. M11 Apply variables in expressions and equations to solve problems (i.e., write mathematical equations for given situation, create a mathematical model to understand the relationships between variables, or make connections between the structures of mathematically abstract concepts and the real world). Equations and Inequalities A.A.3 Distinguish the difference between an algebraic expression and an algebraic equation There is no New York Mathematics Performance Indicator–Common Core alignment. M7 M11 A.A.4 Translate verbal sentences into mathematical equations or inequalities Algebra: Creating Equations Create equations that describe numbers or relationships. 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. M7 M27 2011 International Center for Leadership in Education Simplify and solve algebraic equations by identifying and using the correct order of operations and techniques necessary to carry out the solution. Apply variables in expressions and equations to solve problems (i.e., write mathematical equations for given situation, create a mathematical model to understand the relationships between variables, or make connections between the structures of mathematically abstract concepts and the real world). Simplify and solve algebraic equations by identifying and using the correct order of operations and techniques necessary to carry out the solution. Find the solution of linear equations and inequalities where the variable appears on either or both sides and in which one or both sides must be simplified before solving the equation (e.g., solve x + 2(x - 3) = -4x + 5 for x). Mathematics – Page 36 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra A.A.5 Write algebraic equations or inequalities that represent a situation Mathematics Domains/Clusters/ Common Core State Standards High School Algebra: Creating Equations Create equations that describe numbers or relationships. 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. National Essential Skills Study (NESS) National Rankings Rank M7 M11 M27 A.A.6 Analyze and solve verbal problems whose solution requires solving a linear equation in one variable or linear inequality in one variable A.A.7 Analyze and solve verbal problems whose solution requires solving systems of linear equations in two variables Algebra: Creating Equations Create equations that describe numbers or relationships. 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Algebra: Reasoning with Equations & Inequalities Solve equations and inequalities in one variable. 3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Algebra: Reasoning with Equations & Inequalities Solve systems of equations. 5. Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. 6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. 2011 International Center for Leadership in Education M7 M27 Simplify and solve algebraic equations by identifying and using the correct order of operations and techniques necessary to carry out the solution. Apply variables in expressions and equations to solve problems (i.e., write mathematical equations for given situation, create a mathematical model to understand the relationships between variables, or make connections between the structures of mathematically abstract concepts and the real world). Find the solution of linear equations and inequalities where the variable appears on either or both sides and in which one or both sides must be simplified before solving the equation (e.g., solve x + 2(x - 3) = -4x + 5 for x). Simplify and solve algebraic equations by identifying and using the correct order of operations and techniques necessary to carry out the solution. Find the solution of linear equations and inequalities where the variable appears on either or both sides and in which one or both sides must be simplified before solving the equation (e.g., solve x + 2(x - 3) = -4x + 5 for x). Solve systems of linear equations algebraically or graphically. M40 Mathematics – Page 37 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra A.A.8 Analyze and solve verbal problems that involve quadratic equations A.A.9 Analyze and solve verbal problems that involve exponential growth and decay Mathematics Domains/Clusters/ Common Core State Standards High School Algebra: Creating Equations Create equations that describe numbers or relationships. 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Algebra: Reasoning with Equations & Inequalities Solve equations and inequalities in one variable. 4. Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Algebra: Seeing Structure in Expressions Write expressions in equivalent forms to solve problems. 3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. Functions: Interpreting Functions Analyze functions using different representations. 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Functions: Linear, Quadratic & Exponential Construct and compare linear, quadratic, and exponential models and solve problems. 1. Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Solve quadratic equations by applying various tools or techniques. M47 Express, graph, and interpret exponential and logarithmic functions. M48 Mathematics – Page 38 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra A.A.10 Solve systems of two linear equations in two variables algebraically (See A.G.7) A.A.11 Solve a system of one linear and one quadratic equation in two variables, where only factoring is required Note: The quadratic equation should represent a parabola and the solution(s) should be integers. Mathematics Domains/Clusters/ Common Core State Standards High School Algebra: Reasoning with Equations & Inequalities Solve systems of equations. 5. Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. 6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Algebra: Reasoning with Equations & Inequalities Solve systems of equations. 7. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3. National Essential Skills Study (NESS) National Rankings Rank Solve systems of linear equations algebraically or graphically. M40 M36 M47 Simplify polynomials by performing operations (addition, subtraction, multiplication, and division) to simplify expressions (e.g., (2a + 2) + (3a - 1) = 5a + 1). Solve quadratic equations by applying various tools or techniques. Students will perform algebraic procedures accurately. Variables and Expressions A.A.12 Multiply and divide monomial expressions with a common base, using the properties of exponents Note: Use integral exponents only. A.A.13 Add, subtract, and multiply monomials and polynomials Number & Quantity: The Real Number System Extend the properties of exponents to rational exponents. 1. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5. Algebra: Arithmetic with Polynomials & Rational Expressions Perform arithmetic operations on polynomials. 1. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. 2011 International Center for Leadership in Education M20 M36 Understand and apply the basic properties and laws of exponents and scientific notation to solve problems, including those with fractional, negative, and zero exponents. Simplify polynomials by performing operations (addition, subtraction, multiplication, and division) to simplify expressions (e.g., (2a + 2) + (3a - 1) = 5a + 1). Mathematics – Page 39 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra A.A.14 Divide a polynomial by a monomial or binomial, where the quotient has no remainder Mathematics Domains/Clusters/ Common Core State Standards High School Algebra: Arithmetic with Polynomials & Rational Expressions Rewrite rational expressions. 6. Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. National Essential Skills Study (NESS) National Rankings Rank M20 M62 A.A.15 Find values of a variable for which an algebraic fraction is undefined Functions: Interpreting Functions Understand the concept of a function and use function notation. 2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. M7 M37 A.A.16 Simplify fractions with polynomials in the numerator and denominator by factoring both and renaming them to lowest terms Algebra: Arithmetic with Polynomials & Rational Expressions Rewrite rational expressions. 6. Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. 7. (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. M29 M36 M62 2011 International Center for Leadership in Education Understand and apply the basic properties and laws of exponents and scientific notation to solve problems, including those with fractional, negative, and zero exponents. Perform division of a polynomial by a monomial by dividing powers with like bases, using the rules for the division of powers with like bases to simplify fractions with monomial denominators and reducing fractions to lowest terms. Simplify and solve algebraic equations by identifying and using the correct order of operations and techniques necessary to carry out the solution. Define and apply the properties of relations and functions (domain, range, function composition, and inverses) and use algebraic and graphic methods to determine if a relation is a function. Factor a composite number into its prime components and use least common denominators or least common multiples to solve equations. Simplify polynomials by performing operations (addition, subtraction, multiplication, and division) to simplify expressions (e.g., (2a + 2) + (3a - 1) = 5a + 1). Perform division of a polynomial by a monomial by dividing powers with like bases, using the rules for the division of powers with like bases to simplify fractions with monomial denominators and reducing fractions to lowest terms. Mathematics – Page 40 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra A.A.17 Add or subtract fractional expressions with monomial or like binomial denominators A.A.18 Multiply and divide algebraic fractions and express the product or quotient in simplest form Mathematics Domains/Clusters/ Common Core State Standards High School Algebra: Arithmetic with Polynomials & Rational Expressions Rewrite rational expressions. 7. (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. Algebra: Arithmetic with Polynomials & Rational Expressions Rewrite rational expressions. 6. Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. National Essential Skills Study (NESS) National Rankings Rank M36 M36 M62 A.A.19 Identify and factor the difference of two perfect squares Algebra: Seeing Structure in Expressions Interpret the structure of expressions. 2. Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2). M36 M47 2011 International Center for Leadership in Education Simplify polynomials by performing operations (addition, subtraction, multiplication, and division) to simplify expressions (e.g., (2a + 2) + (3a - 1) = 5a + 1). Simplify polynomials by performing operations (addition, subtraction, multiplication, and division) to simplify expressions (e.g., (2a + 2) + (3a - 1) = 5a + 1). Perform division of a polynomial by a monomial by dividing powers with like bases, using the rules for the division of powers with like bases to simplify fractions with monomial denominators and reducing fractions to lowest terms. Simplify polynomials by performing operations (addition, subtraction, multiplication, and division) to simplify expressions (e.g., (2a + 2) + (3a - 1) = 5a + 1). Solve quadratic equations by applying various tools or techniques. Mathematics – Page 41 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra A.A.20 Factor algebraic expressions completely, including trinomials with a lead coefficient of one (after factoring a GCF) Mathematics Domains/Clusters/ Common Core State Standards High School Algebra: Seeing Structure in Expressions Interpret the structure of expressions. Write expressions in equivalent forms to solve problems. 3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. a. Factor a quadratic expression to reveal the zeros of the function it defines. National Essential Skills Study (NESS) National Rankings Rank M29 M36 M62 Factor a composite number into its prime components and use least common denominators or least common multiples to solve equations. Simplify polynomials by performing operations (addition, subtraction, multiplication, and division) to simplify expressions (e.g., (2a + 2) + (3a - 1) = 5a + 1). Perform division of a polynomial by a monomial by dividing powers with like bases, using the rules for the division of powers with like bases to simplify fractions with monomial denominators and reducing fractions to lowest terms. Equations and Inequalities A.A.21 Determine whether a given value is a solution to a given linear equation in one variable or linear inequality in one variable Algebra: Reasoning with Equations & Inequalities Solve equations and inequalities in one variable. 3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. M7 M27 M45 2011 International Center for Leadership in Education Simplify and solve algebraic equations by identifying and using the correct order of operations and techniques necessary to carry out the solution. Find the solution of linear equations and inequalities where the variable appears on either or both sides and in which one or both sides must be simplified before solving the equation (e.g., solve x + 2(x - 3) = -4x + 5 for x). Solve linear inequalities and graph the solution set on a number line. Mathematics – Page 42 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra A.A.22 Solve all types of linear equations in one variable Mathematics Domains/Clusters/ Common Core State Standards High School Algebra: Seeing Structure in Expressions Write expressions in equivalent forms to solve problems. 3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. National Essential Skills Study (NESS) National Rankings Rank M7 M27 A.A.23 Solve literal equations for a given variable A.A.24 Solve linear inequalities in one variable A.A.25 Solve equations involving fractional expressions Note: Expressions which result in linear equations in one variable. Algebra: Creating Equations Create equations that describe numbers or relationships. 4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R. Algebra: Seeing Structure in Expressions Write expressions in equivalent forms to solve problems. 3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. Algebra: Reasoning with Equations & Inequalities Understand solving equations as a process of reasoning and explain the reasoning. 2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. M7 M27 M7 M27 M29 2011 International Center for Leadership in Education Simplify and solve algebraic equations by identifying and using the correct order of operations and techniques necessary to carry out the solution. Find the solution of linear equations and inequalities where the variable appears on either or both sides and in which one or both sides must be simplified before solving the equation (e.g., solve x + 2(x - 3) = -4x + 5 for x). Simplify and solve algebraic equations by identifying and using the correct order of operations and techniques necessary to carry out the solution. Find the solution of linear equations and inequalities where the variable appears on either or both sides and in which one or both sides must be simplified before solving the equation (e.g., solve x + 2(x - 3) = -4x + 5 for x). Simplify and solve algebraic equations by identifying and using the correct order of operations and techniques necessary to carry out the solution. Find the solution of linear equations and inequalities where the variable appears on either or both sides and in which one or both sides must be simplified before solving the equation (e.g., solve x + 2(x - 3) = -4x + 5 for x). Factor a composite number into its prime components and use least common denominators or least common multiples to solve equations. Mathematics – Page 43 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra Mathematics Domains/Clusters/ Common Core State Standards High School A.A.26 Solve algebraic proportions in one variable which result in linear or quadratic equations Algebra: Reasoning with Equations & Inequalities Solve equations and inequalities in one variable. 3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. 4. Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Algebra: Reasoning with Equations & Inequalities Solve equations and inequalities in one variable. 4. Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Algebra: Seeing Structure in Expressions Write expressions in equivalent forms to solve problems. 3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. a. Factor a quadratic expression to reveal the zeros of the function it defines. Algebra: Reasoning with Equations & Inequalities Solve equations and inequalities in one variable. 4. Solve quadratic equations in one variable. b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. A.A.27 Understand and apply the multiplication property of zero to solve quadratic equations with integral coefficients and integral roots A.A.28 Understand the difference and connection between roots of a quadratic equation and factors of a quadratic expression 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank M3 M7 Use proportional reasoning to solve realworld problems. Simplify and solve algebraic equations by identifying and using the correct order of operations and techniques necessary to carry out the solution. Solve quadratic equations by applying various tools or techniques. M47 Solve quadratic equations by applying various tools or techniques. M47 M36 Simplify polynomials by performing operations (addition, subtraction, multiplication, and division) to simplify expressions (e.g., (2a + 2) + (3a - 1) = 5a + 1). Solve quadratic equations by applying various tools or techniques. M47 Mathematics – Page 44 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra Mathematics Domains/Clusters/ Common Core State Standards High School National Essential Skills Study (NESS) National Rankings Rank Students will recognize, use, and represent algebraically patterns, relations, and functions. Patterns, Relations, and Functions A.A.29 Use set-builder notation and/or interval notation to illustrate the elements of a set, given the elements in roster form A.A.30 Find the complement of a subset of a given set, within a given universe Functions: Interpreting Functions Understand the concept of a function and use function notation. 1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). Statistics & Probability: Conditional Probability & the Rules of Probability Understand independence and conditional probability and use them to interpret data. 1. Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”). M37 M21 M37 A.A.31 Find the intersection of sets (no more than three sets) and/or union of sets (no more than three sets) Statistics & Probability: Conditional Probability & the Rules of Probability Understand independence and conditional probability and use them to interpret data. 1. Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”). M21 M49 2011 International Center for Leadership in Education Define and apply the properties of relations and functions (domain, range, function composition, and inverses) and use algebraic and graphic methods to determine if a relation is a function. Evaluate and employ accurate and appropriate procedures for statistical data collection, organization, analysis, and display including making estimates and predictions, critiquing data, and drawing inferences (e.g., using the normal curve and z-scores, line of best fit). Define and apply the properties of relations and functions (domain, range, function composition, and inverses) and use algebraic and graphic methods to determine if a relation is a function. Evaluate and employ accurate and appropriate procedures for statistical data collection, organization, analysis, and display including making estimates and predictions, critiquing data, and drawing inferences (e.g., using the normal curve and z-scores, line of best fit). Analyze the truth value of compound sentences by creating truth tables. Mathematics – Page 45 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra Mathematics Domains/Clusters/ Common Core State Standards High School National Essential Skills Study (NESS) National Rankings Rank Coordinate Geometry A.A.32 Explain slope as a rate of change between dependent and independent variables A.A.33 Determine the slope of a line, given the coordinates of two points on the line A.A.34 Write the equation of a line, given its slope and the coordinates of a point on the line A.A.35 Write the equation of a line, given the coordinates of two points on the line Functions: Interpreting Functions Interpret functions that arise in applications in terms of the context. 6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Functions: Linear, Quadratic & Exponential Construct and compare linear, quadratic, and exponential models and solve problems. 1. Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. Functions: Linear, Quadratic & Exponential Construct and compare linear, quadratic, and exponential models and solve problems. 1. Distinguish between situations that can be modeled with linear functions and with exponential functions. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. Functions: Linear, Quadratic & Exponential Construct and compare linear, quadratic, and exponential models and solve problems. 2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Functions: Linear, Quadratic & Exponential Construct and compare linear, quadratic, and exponential models and solve problems. 2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). 2011 International Center for Leadership in Education Know the equation for the slope of a line and compute slope given the coordinates of two points. M46 Know the equation for the slope of a line and compute slope given the coordinates of two points. M46 M44 M44 M46 Know the equation of a line and interpret graphically using the slope-intercept form (y = mx+b) and the point-slope form (y-b = m(x-a)). Know the equation of a line and interpret graphically using the slope-intercept form (y = mx+b) and the point-slope form (y-b = m(x-a)). Know the equation for the slope of a line and compute slope given the coordinates of two points. Mathematics – Page 46 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra A.A.36 Write the equation of a line parallel to the x- or y-axis A.A.37 Determine the slope of a line, given its equation in any form A.A.38 Determine if two lines are parallel, given their equations in any form Mathematics Domains/Clusters/ Common Core State Standards High School Geometry: Expressing Geometric Properties with Equations Use coordinates to prove simple geometric theorems algebraically. 5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). Functions: Linear, Quadratic & Exponential Construct and compare linear, quadratic, and exponential models and solve problems. 1. Distinguish between situations that can be modeled with linear functions and with exponential functions. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. Geometry: Expressing Geometric Properties with Equations Use coordinates to prove simple geometric theorems algebraically. 5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). National Essential Skills Study (NESS) National Rankings Rank M4 M44 M44 M46 M4 M44 M46 A.A.39 Determine whether a given point is on a line, given the equation of the line Functions: Interpreting Functions Understand the concept of a function and use function notation. 1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). 2011 International Center for Leadership in Education M44 Understand the properties of and apply parallel, perpendicular, and intersecting lines in problem-solving situations. Know the equation of a line and interpret graphically using the slope-intercept form (y = mx+b) and the point-slope form (y-b = m(x-a)). Know the equation of a line and interpret graphically using the slope-intercept form (y = mx+b) and the point-slope form (y-b = m(x-a)). Know the equation for the slope of a line and compute slope given the coordinates of two points. Understand the properties of and apply parallel, perpendicular, and intersecting lines in problem-solving situations. Know the equation of a line and interpret graphically using the slope-intercept form (y = mx+b) and the point-slope form (y-b = m(x-a)). Know the equation for the slope of a line and compute slope given the coordinates of two points. Know the equation of a line and interpret graphically using the slope-intercept form (y = mx+b) and the point-slope form (y-b = m(x-a)). Mathematics – Page 47 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra A.A.40 Determine whether a given point is in the solution set of a system of linear inequalities A.A.41 Determine the vertex and axis of symmetry of a parabola, given its equation (See A.G.10 ) Mathematics Domains/Clusters/ Common Core State Standards High School Algebra: Reasoning with Equations & Inequalities Represent and solve equations and inequalities graphically. 12. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. Functions: Interpreting Functions Analyze functions using different representations. 8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. National Essential Skills Study (NESS) National Rankings Rank M65 M47 M66 Find the graphic solution of systems of linear inequalities (i.e., graph the solution set or region of the coordinate plane common to both inequalities). Solve quadratic equations by applying various tools or techniques. Know how to sketch basic conic sections (e.g., circles, parabolas) by using their equations and solve systems of non-linear equations graphically. Trigonometric Functions A.A.42 Find the sine, cosine, and tangent ratios of an angle of a right triangle, given the lengths of the sides A.A.43 Determine the measure of an angle of a right triangle, given the length of any two sides of the triangle A.A.44 Find the measure of a side of a right triangle, given an acute angle and the length of another side A.A.45 Determine the measure of a third side of a right triangle using the Pythagorean theorem, given the lengths of any two sides Geometry: Similarity, Right Triangles, & Trigonometry Define trigonometric ratios and solve problems involving right triangles. 6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Geometry: Similarity, Right Triangles, & Trigonometry Define trigonometric ratios and solve problems involving right triangles. 8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Geometry: Similarity, Right Triangles, & Trigonometry Define trigonometric ratios and solve problems involving right triangles. 8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Geometry: Similarity, Right Triangles, & Trigonometry Define trigonometric ratios and solve problems involving right triangles. 8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. 2011 International Center for Leadership in Education M28 M28 M28 M23 Know and apply the six basic trigonometric functions and ratios and solve right triangles using basic trigonometric ratios (sine, cosine, tangent). Know and apply the six basic trigonometric functions and ratios and solve right triangles using basic trigonometric ratios (sine, cosine, tangent). Know and apply the six basic trigonometric functions and ratios and solve right triangles using basic trigonometric ratios (sine, cosine, tangent). Apply the Pythagorean Theorem to right triangles. Mathematics – Page 48 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra Mathematics Domains/Clusters/ Common Core State Standards High School National Essential Skills Study (NESS) National Rankings Rank Geometry Strand Students will use visualization and spatial reasoning to analyze characteristics and properties of geometric shapes. Shapes A.G.1 Find the area and/or perimeter of figures composed of polygons and circles or sectors of a circle Note: Figures may include triangles, rectangles, squares, parallelograms, rhombuses, trapezoids, circles, semicircles, quarter-circles, and regular polygons (perimeter only). A.G.2 Use formulas to calculate volume and surface area of rectangular solids and cylinders Geometry: Circles Find arc length and areas of sectors of circles. 5. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Geometry: Geometric Measurement & Dimensions Explain volume formulas and use them to solve problems. 1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. Geometry: Geometric Measurement & Dimensions Explain volume formulas and use them to solve problems. 1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. 3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. 2011 International Center for Leadership in Education M9 M18 M26 Compute the perimeter and area of common two-dimensional figures. Understand the properties of circles (radius, arc, diameter, chord, secant, and tangent) and apply circle quantities (lengths of line segments, angle measure within a circle, circumference, and area) in problem-solving situations. Know the classification and properties of three-dimensional figures (prisms, rectangular solids, pyramids, right circular cylinders, cones, and spheres) and be able to compute the volume and surface area of common solids. Mathematics – Page 49 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra Mathematics Domains/Clusters/ Common Core State Standards High School National Essential Skills Study (NESS) National Rankings Rank Students will apply coordinate geometry to analyze problem solving situations. Coordinate Geometry A.G.3 Determine when a relation is a function, by examining ordered pairs and inspecting graphs of relations A.G.4 Identify and graph linear, quadratic (parabolic), absolute value, and exponential functions Functions: Interpreting Functions Understand the concept of a function and use function notation. 1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). Functions: Interpreting Functions Analyze functions using different representations. 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Functions: Linear, Quadratic & Exponential Construct and compare linear, quadratic, and exponential models and solve problems. 1. Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. 2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). 2011 International Center for Leadership in Education M30 M37 M48 Know and apply the components and properties of the rectangular coordinate system: x–y axis, origin, quadrants, abscissa (x-coordinate) and ordinate (y-coordinate), and general representation of a point (x,y). Define and apply the properties of relations and functions (domain, range, function composition, and inverses) and use algebraic and graphic methods to determine if a relation is a function. Express, graph, and interpret exponential and logarithmic functions. Express, graph, and interpret polynomial functions (linear, quadratic, cubic, etc.). M53 Mathematics – Page 50 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra Mathematics Domains/Clusters/ Common Core State Standards High School A.G.5 Investigate and generalize how changing the coefficients of a function affects its graph Functions: Building Functions Build new functions from existing functions. 3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Algebra: Reasoning with Equations & Inequalities Represent and solve equations and inequalities graphically. 12. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. A.G.6 Graph linear inequalities A.G.7 Graph and solve systems of linear equations and inequalities with rational coefficients in two variables (See A.A.10) A.G.8 Find the roots of a parabolic function graphically Note: Only quadratic equations with integral solutions. Algebra: Reasoning with Equations & Inequalities Solve systems of equations. 5. Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. 6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Represent and solve equations and inequalities graphically. 12. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. Functions: Interpreting Functions Analyze functions using different representations. 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank M48 Express, graph, and interpret exponential and logarithmic functions. Express, graph, and interpret polynomial functions (linear, quadratic, cubic, etc.). M53 M44 M53 M40 Know the equation of a line and interpret graphically using the slope-intercept form (y = mx+b) and the point-slope form (y-b = m(x-a)). Express, graph, and interpret polynomial functions (linear, quadratic, cubic, etc.). Solve systems of linear equations algebraically or graphically. Find the graphic solution of systems of linear inequalities (i.e., graph the solution set or region of the coordinate plane common to both inequalities). M65 M66 Know how to sketch basic conic sections (e.g., circles, parabolas) by using their equations and solve systems of non-linear equations graphically. Mathematics – Page 51 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra A.G.9 Solve systems of linear and quadratic equations graphically Note: Only use systems of linear and quadratic equations that lead to solutions whose coordinates are integers. A.G.10 Determine the vertex and axis of symmetry of a parabola, given its graph (See A.A.41) Note: The vertex will have an ordered pair of integers and the axis of symmetry will have an integral value. Mathematics Domains/Clusters/ Common Core State Standards High School Algebra: Reasoning with Equations & Inequalities Solve systems of equations. 6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. 7. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3. Functions: Interpreting Functions Analyze functions using different representations. 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. National Essential Skills Study (NESS) National Rankings Rank M40 M66 M30 M54 M66 Solve systems of linear equations algebraically or graphically. Know how to sketch basic conic sections (e.g., circles, parabolas) by using their equations and solve systems of non-linear equations graphically. Know and apply the components and properties of the rectangular coordinate system: x–y axis, origin, quadrants, abscissa (x-coordinate) and ordinate (y-coordinate), and general representation of a point (x,y). Apply transformations (reflection, rotation, translation, and dilation) of 2-dimensional figures graphically to interpret, analyze, and illustrate the concepts of congruency, similarity, and symmetry. Know how to sketch basic conic sections (e.g., circles, parabolas) by using their equations and solve systems of non-linear equations graphically. Measurement Strand Students will determine what can be measured and how, using appropriate methods and formulas. Units of Measurement A.M.1 Calculate rates using appropriate units (e.g., rate of a space ship versus the rate of a snail) Number & Quantity: Quantities Reason quantitatively and use units to solve problems. 1. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. 2. Define appropriate quantities for the purpose of descriptive modeling. 2011 International Center for Leadership in Education M13 Use the technique of dimensional analysis to convert units of measure (e.g., kilometers/hour to meters/minute) and apply ratios in real-world situations (e.g., scale drawings). Mathematics – Page 52 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra Mathematics Domains/Clusters/ Common Core State Standards High School A.M.2 Solve problems involving conversions within measurement systems, given the relationship between the units Number & Quantity: Quantities Reason quantitatively and use units to solve problems. 1. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. 2. Define appropriate quantities for the purpose of descriptive modeling. National Essential Skills Study (NESS) National Rankings Rank M8 M13 Solve problems using units of metric measure and convert between metric and English/customary units. Use the technique of dimensional analysis to convert units of measure (e.g., kilometers/hour to meters/minute) and apply ratios in real-world situations (e.g., scale drawings). Students will understand that all measurement contains error and be able to determine its significance. Error and Magnitude A.M.3 Calculate the relative error in measuring square and cubic units, when there is an error in the linear measure Number & Quantity: Quantities Reason quantitatively and use units to solve problems. 3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. M12 Understand accuracy and precision of measurement, round off numbers according to the correct number of significant figures, and determine percent error. Statistics and Probability Strand Students will collect, organize, display, and analyze data Organization and Display of Data A.S.1 Categorize data as qualitative or quantitative Statistics & Probability: Interpreting Categorical & Quantitative Data Summarize, represent and interpret data on a single count or measurement variable. 1. Represent data with plots on the real number line (dot plots, histograms, and box plots). Summarize, represent and interpret data on two categorical and quantitative variables. 6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. 2011 International Center for Leadership in Education M21 Evaluate and employ accurate and appropriate procedures for statistical data collection, organization, analysis, and display including making estimates and predictions, critiquing data, and drawing inferences (e.g., using the normal curve and z-scores, line of best fit). Mathematics – Page 53 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra Mathematics Domains/Clusters/ Common Core State Standards High School A.S.2 Determine whether the data to be analyzed is univariate or bivariate Statistics & Probability: Interpreting Categorical & Quantitative Data Summarize, represent and interpret data on a single count or measurement variable. 1. Represent data with plots on the real number line (dot plots, histograms, and box plots). Summarize, represent and interpret data on two categorical and quantitative variables. 6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. A.S.3 Determine when collected data or display of data may be biased A.S.4 Compare and contrast the appropriateness of different measures of central tendency for a given data set Statistics & Probability: Making Inferences & Justifying Conclusions Understand and evaluate random processes underlying statistical experiments. 1. Understand statistics as a process for making inferences about population parameters based on a random sample from that population. Make inferences and justify conclusions from sample surveys, experiments and observational studies. 3. Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. 4. Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling. 5. Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. 6. Evaluate reports based on data. Statistics & Probability: Interpreting Categorical & Quantitative Data Summarize, represent and interpret data on a single count or measurement variable. 2. Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. 3. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank M21 M31 M14 M17 M21 M31 Evaluate and employ accurate and appropriate procedures for statistical data collection, organization, analysis, and display including making estimates and predictions, critiquing data, and drawing inferences (e.g., using the normal curve and z-scores, line of best fit). Understand and apply measures of dispersion (range, mean deviation, variance, and standard deviation). Understand and apply measures of central tendency (mean, median, and mode, and representative sampling of a population). Understand the importance of random sampling and sample size in generating representative data. Evaluate and employ accurate and appropriate procedures for statistical data collection, organization, analysis, and display including making estimates and predictions, critiquing data, and drawing inferences (e.g., using the normal curve and z-scores, line of best fit). Understand and apply measures of dispersion (range, mean deviation, variance, and standard deviation). Understand and apply measures of central tendency (mean, median, and mode, and representative sampling of a population). M14 Mathematics – Page 54 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra Mathematics Domains/Clusters/ Common Core State Standards High School A.S.5 Construct a histogram, cumulative frequency histogram, and a box-and-whisker plot, given a set of data Statistics & Probability: Interpreting Categorical & Quantitative Data Summarize, represent and interpret data on a single count or measurement variable. 1. Represent data with plots on the real number line (dot plots, histograms, and box plots). A.S.6 Understand how the five statistical summary quartiles) is used to construct a box-and-whisker plot Statistics & Probability: Interpreting Categorical & Quantitative Data Summarize, represent and interpret data on a single count or measurement variable. 1. Represent data with plots on the real number line (dot plots, histograms, and box plots). National Essential Skills Study (NESS) National Rankings Rank M21 M14 M21 M43 A.S.7 Create a scatter plot of bivariate data Statistics & Probability: Interpreting Categorical & Quantitative Data Summarize, represent and interpret data on two categorical and quantitative variables. 6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. M21 M31 2011 International Center for Leadership in Education Evaluate and employ accurate and appropriate procedures for statistical data collection, organization, analysis, and display including making estimates and predictions, critiquing data, and drawing inferences (e.g., using the normal curve and z-scores, line of best fit). Understand and apply measures of central tendency (mean, median, and mode, and representative sampling of a population). Evaluate and employ accurate and appropriate procedures for statistical data collection, organization, analysis, and display including making estimates and predictions, critiquing data, and drawing inferences (e.g., using the normal curve and z-scores, line of best fit). Understand and apply the concepts and applications of quartiles (distributing groups into four equal sizes), percentiles (distributing individuals into 100 groups of equal size), and random distribution to understand and interpret data. Evaluate and employ accurate and appropriate procedures for statistical data collection, organization, analysis, and display including making estimates and predictions, critiquing data, and drawing inferences (e.g., using the normal curve and z-scores, line of best fit). Understand and apply measures of dispersion (range, mean deviation, variance, and standard deviation). Mathematics – Page 55 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra Mathematics Domains/Clusters/ Common Core State Standards High School A.S.8 Construct manually a reasonable line of best fit for a scatter plot and determine the equation of that line Statistics & Probability: Interpreting Categorical & Quantitative Data Summarize, represent and interpret data on two categorical and quantitative variables. 6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. c. Fit a linear function for a scatter plot that suggests a linear association. National Essential Skills Study (NESS) National Rankings Rank M21 Evaluate and employ accurate and appropriate procedures for statistical data collection, organization, analysis, and display including making estimates and predictions, critiquing data, and drawing inferences (e.g., using the normal curve and z-scores, line of best fit). Analysis of Data A.S.9 Analyze and interpret a frequency distribution table or histogram, a cumulative frequency distribution table or histogram, or a box-and-whisker plot A.S.10 Evaluate published reports and graphs that are based on data by considering: experimental design, appropriateness of the data analysis, and the soundness of the conclusions Statistics & Probability: Interpreting Categorical & Quantitative Data Summarize, represent and interpret data on a single count or measurement variable. 1. Represent data with plots on the real number line (dot plots, histograms, and box plots). Summarize, represent and interpret data on two categorical and quantitative variables. 5. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. Statistics & Probability: Making Inferences & Justifying Conclusions Make inferences and justify conclusions from sample surveys, experiments and observational studies. 6. Evaluate reports based on data. M21 M17 M21 2011 International Center for Leadership in Education Evaluate and employ accurate and appropriate procedures for statistical data collection, organization, analysis, and display including making estimates and predictions, critiquing data, and drawing inferences (e.g., using the normal curve and z-scores, line of best fit). Understand the importance of random sampling and sample size in generating representative data. Evaluate and employ accurate and appropriate procedures for statistical data collection, organization, analysis, and display including making estimates and predictions, critiquing data, and drawing inferences (e.g., using the normal curve and z-scores, line of best fit). Mathematics – Page 56 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra Mathematics Domains/Clusters/ Common Core State Standards High School A.S.11 Find the percentile rank of an item in a data set and identify the point values for first, second, and third quartiles Statistics & Probability: Interpreting Categorical & Quantitative Data Summarize, represent and interpret data on a single count or measurement variable. 2. Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. 4. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. Statistics & Probability: Interpreting Categorical & Quantitative Data Summarize, represent and interpret data on two categorical and quantitative variables. 6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. b. Informally assess the fit of a function by plotting and analyzing residuals. Interpret linear models. 8. Compute (using technology) and interpret the correlation coefficient of a linear fit. Statistics & Probability: Interpreting Categorical & Quantitative Data Interpret linear models. 9. Distinguish between correlation and causation. Statistics & Probability: Interpreting Categorical & Quantitative Data Interpret linear models. 9. Distinguish between correlation and causation. A.S.12 Identify the relationship between the independent and dependent variables from a scatter plot (positive, negative, or none) A.S.13 Understand the difference between correlation and causation A.S.14 Identify variables that might have a correlation but not a causal relationship National Essential Skills Study (NESS) National Rankings Rank M43 M21 M22 M21 M22 2011 International Center for Leadership in Education Understand and apply the concepts and applications of quartiles (distributing groups into four equal sizes), percentiles (distributing individuals into 100 groups of equal size), and random distribution to understand and interpret data. Evaluate and employ accurate and appropriate procedures for statistical data collection, organization, analysis, and display including making estimates and predictions, critiquing data, and drawing inferences (e.g., using the normal curve and z-scores, line of best fit). Interpret data to determine correlation and distinguish between correlation and cause and effect. Evaluate and employ accurate and appropriate procedures for statistical data collection, organization, analysis, and display including making estimates and predictions, critiquing data, and drawing inferences (e.g., using the normal curve and z-scores, line of best fit). Interpret data to determine correlation and distinguish between correlation and cause and effect. Mathematics – Page 57 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra Mathematics Domains/Clusters/ Common Core State Standards High School National Essential Skills Study (NESS) National Rankings Rank Students will make predictions that are based upon data analysis. Predictions from Data A.S.15 Identify and describe sources of bias and its effect, drawing conclusions from data A.S.16 Recognize how linear transformations of one-variable data affect the data’s mean, median, mode, and range A.S.17 Use a reasonable line of best fit to make a prediction involving interpolation or extrapolation Statistics & Probability: Making Inferences & Justifying Conclusions Make inferences and justify conclusions from sample surveys, experiments and observational studies. 3. Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. 6. Evaluate reports based on data. Statistics & Probability: Interpreting Categorical & Quantitative Data Summarize, represent and interpret data on a single count or measurement variable. 2. Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. Statistics & Probability: Interpreting Categorical & Quantitative Data Interpret linear models. 7. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. 2011 International Center for Leadership in Education M17 M21 M14 M31 M21 Understand the importance of random sampling and sample size in generating representative data. Evaluate and employ accurate and appropriate procedures for statistical data collection, organization, analysis, and display including making estimates and predictions, critiquing data, and drawing inferences (e.g., using the normal curve and z-scores, line of best fit). Understand and apply measures of central tendency (mean, median, and mode, and representative sampling of a population). Understand and apply measures of dispersion (range, mean deviation, variance, and standard deviation). Evaluate and employ accurate and appropriate procedures for statistical data collection, organization, analysis, and display including making estimates and predictions, critiquing data, and drawing inferences (e.g., using the normal curve and z-scores, line of best fit). Mathematics – Page 58 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra Mathematics Domains/Clusters/ Common Core State Standards High School National Essential Skills Study (NESS) National Rankings Rank Students will understand and apply concepts of probability. Probability A.S.18 Know the definition of conditional probability and use it to solve for probabilities in finite sample spaces Statistics & Probability: Conditional Probability & the Rules of Probability Understand independence and conditional probability and use them to interpret data. 1. Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”). 2. Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. 3. Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. 4. Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. 5. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. Use the rules of probability to compute probabilities of compound events in a uniform probability model. 6. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. 7. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model. 8. (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model. 2011 International Center for Leadership in Education M5 Examine problem-solving situations involving simple probability and use probabilistic reasoning to compare and communicate the theoretical or empirical likelihood of events. Determine the probability of single and compound events and use the Counting Principle to determine the probability of independent events occurring jointly. M32 Mathematics – Page 59 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra A.S.19 Determine the number of elements in a sample space and the number of favorable events A.S.20 Calculate the probability of an event and its complement A.S.21 Determine empirical probabilities based on specific sample data Mathematics Domains/Clusters/ Common Core State Standards High School Statistics & Probability: Conditional Probability & the Rules of Probability Use the rules of probability to compute probabilities of compound events in a uniform probability model. 9. (+) Use permutations and combinations to compute probabilities of compound events and solve problems. Statistics & Probability: Conditional Probability & the Rules of Probability Understand independence and conditional probability and use them to interpret data. 1. Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”). Use the rules of probability to compute probabilities of compound events in a uniform probability model. 6. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. 7. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model. 8. (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model. Statistics & Probability: Conditional Probability & the Rules of Probability Use the rules of probability to compute probabilities of compound events in a uniform probability model. 9. (+) Use permutations and combinations to compute probabilities of compound events and solve problems. National Essential Skills Study (NESS) National Rankings Rank M5 Examine problem-solving situations involving simple probability and use probabilistic reasoning to compare and communicate the theoretical or empirical likelihood of events. M5 M5 M32 2011 International Center for Leadership in Education Examine problem-solving situations involving simple probability and use probabilistic reasoning to compare and communicate the theoretical or empirical likelihood of events. Examine problem-solving situations involving simple probability and use probabilistic reasoning to compare and communicate the theoretical or empirical likelihood of events. Determine the probability of single and compound events and use the Counting Principle to determine the probability of independent events occurring jointly. Mathematics – Page 60 New York Mathematics Strands/Bands/ Performance Indicators Integrated Algebra Mathematics Domains/Clusters/ Common Core State Standards High School A.S.22 Determine, based on calculated probability of a set of events, if: some or all are equally likely to occur one is more likely to occur than another whether or not an event is certain to happen or not to happen A.S.23 Calculate the probability of: a series of independent events a series of dependent events two mutually exclusive events two events that are not mutually exclusive Statistics & Probability: Conditional Probability & the Rules of Probability Use the rules of probability to compute probabilities of compound events in a uniform probability model. 9. (+) Use permutations and combinations to compute probabilities of compound events and solve problems. Statistics & Probability: Conditional Probability & the Rules of Probability Understand independence and conditional probability and use them to interpret data. 2. Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. 3. Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. 4. Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. Use the rules of probability to compute probabilities of compound events in a uniform probability model. 6. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. 7. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model. 8. (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank M5 M5 Examine problem-solving situations involving simple probability and use probabilistic reasoning to compare and communicate the theoretical or empirical likelihood of events. Examine problem-solving situations involving simple probability and use probabilistic reasoning to compare and communicate the theoretical or empirical likelihood of events. Determine the probability of single and compound events and use the Counting Principle to determine the probability of independent events occurring jointly. M32 Mathematics – Page 61 New York Mathematics Standards Alignments New York Mathematics Strands/Bands/ Performance Indicators Geometry Mathematics Domains/Clusters/ Common Core State Standards High School National Essential Skills Study (NESS) National Rankings Rank Problem Solving Strand Students will new mathematical knowledge through problem solving. G.PS.1 Use a variety of problem solving strategies to understand new mathematical content Geometry: Similarity, Right Triangles, & Trigonometry Define trigonometric ratios and solve problems involving right triangles. 6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. 7. Explain and use the relationship between the sine and cosine of complementary angles. 8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Geometry: Expressing Geometric Properties with Equations Use coordinates to prove simple geometric theorems algebraically. 4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). 5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). 6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. 7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. 2011 International Center for Leadership in Education Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 62 New York Mathematics Strands/Bands/ Performance Indicators Geometry Students will solve problems that arise in mathematics and in other contexts. G.PS.2 Observe and explain patterns to formulate generalizations and conjectures Mathematics Domains/Clusters/ Common Core State Standards High School Geometry: Congruence Understand congruence in terms of rigid motions. 6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. 7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. 8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Geometry: Similarity, Right Triangles, & Trigonometry Understand similarity in terms of similarity transformations. 2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. 3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Apply pattern recognition in data sets and series to reason or solve problems involving arithmetic, geometry, exponents, etc. M16 Mathematics – Page 63 New York Mathematics Strands/Bands/ Performance Indicators Geometry Mathematics Domains/Clusters/ Common Core State Standards High School G.PS.3 Use multiple representations to represent and explain problem situations (e.g., spatial, geometric, verbal, numeric, algebraic, and graphical representations) Geometry: Congruence Experiment with transformations in the plane. 2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). 5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Understand congruence in terms of rigid motions. 8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Make geometric constructions. 12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. 13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Geometry: Similarity, Right Triangles, & Trigonometry Understand similarity in terms of similarity transformations. 1. Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. 3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Define trigonometric ratios and solve problems involving right triangles. 6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank M10 M11 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Apply variables in expressions and equations to solve problems (i.e., write mathematical equations for given situation, create a mathematical model to understand the relationships between variables, or make connections between the structures of mathematically abstract concepts and the real world). Evaluate and employ accurate and appropriate procedures for statistical data collection, organization, analysis, and display including making estimates and predictions, critiquing data, and drawing inferences (e.g., using the normal curve and z-scores, line of best fit). M21 Mathematics – Page 64 New York Mathematics Strands/Bands/ Performance Indicators Geometry Mathematics Domains/Clusters/ Common Core State Standards High School G.PS.3 (Continued from previous page) (Continued from previous page) 7. Explain and use the relationship between the sine and cosine of complementary angles. Geometry: Expressing Geometric Properties with Equations Translate between the geometric description and the equation for a conic section. 1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Use coordinates to prove simple geometric theorems algebraically. 4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). 5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). 6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. 7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Students will apply and adapt a variety of appropriate strategies to solve problems. G.PS.4 Construct various types of reasoning, arguments, justifications and methods of proof for problems Geometry: Congruence Understand congruence in terms of rigid motions. 8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Prove geometric theorems. 9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. 10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 65 New York Mathematics Strands/Bands/ Performance Indicators Geometry Mathematics Domains/Clusters/ Common Core State Standards High School G.PS.4 (Continued from previous page) (Continued from previous page) 11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Geometry: Similarity, Right Triangles, & Trigonometry Understand similarity in terms of similarity transformations. 3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove theorems involving similarity. 4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. 5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Geometry: Circles Understand and apply theorems about circles. 1. Prove that all circles are similar. 2. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Geometry: Congruence Experiment with transformations in the plane. 3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. 4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Understand congruence in terms of rigid motions. 6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. G.PS.5 Choose an effective approach to solve a problem from a variety of strategies (numeric, graphic, algebraic) 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 66 New York Mathematics Strands/Bands/ Performance Indicators Geometry Mathematics Domains/Clusters/ Common Core State Standards High School G.PS.5 (Continued from previous page) (Continued from previous page) Geometry: Similarity, Right Triangles, & Trigonometry Understand similarity in terms of similarity transformations. 1. Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Define trigonometric ratios and solve problems involving right triangles. 6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. 7. Explain and use the relationship between the sine and cosine of complementary angles. 8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Geometry: Geometric Measurement & Dimensions Explain volume formulas and use them to solve problems. 1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. 3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Geometry: Modeling with Geometry Apply Geometric Concepts in modeling situations 1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). 2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). There is no New York Mathematics Performance Indicator–Common Core alignment. G.PS.6 Use a variety of strategies to extend solution methods to other problems G.PS.7 Work in collaboration with others to propose, critique, evaluate, and value alternative approaches to problem solving 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank M10 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Mathematics – Page 67 New York Mathematics Strands/Bands/ Performance Indicators Geometry Students will monitor and reflect on the process of mathematical problem solving. G.PS.8 Determine information required to solve a problem, choose methods for obtaining the information, and define parameters for acceptable solutions Mathematics Domains/Clusters/ Common Core State Standards High School Geometry: Congruence Experiment with transformations in the plane. 3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. 4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Understand congruence in terms of rigid motions. 6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Geometry: Similarity, Right Triangles, & Trigonometry Understand similarity in terms of similarity transformations. 1. Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. 3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Define trigonometric ratios and solve problems involving right triangles. 6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. 7. Explain and use the relationship between the sine and cosine of complementary angles. 8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 68 New York Mathematics Strands/Bands/ Performance Indicators Geometry Mathematics Domains/Clusters/ Common Core State Standards High School G.PS.8 (Continued from previous page) (Continued from previous page) Geometry: Geometric Measurement & Dimensions Explain volume formulas and use them to solve problems. 1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. 3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. There is no New York Mathematics Performance Indicator–Common Core alignment. G.PS.9 Interpret solutions within the given constraints of a problem National Essential Skills Study (NESS) National Rankings Rank M10 G.PS.10 Evaluate the relative efficiency of different representations and solution methods of a problem There is no New York Mathematics Performance Indicator–Common Core alignment. M10 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Reasoning and Proof Strand Students will recognize reasoning and proof as fundamental aspects of mathematics. G.RP.1 Recognize that mathematical ideas can be supported by a variety of strategies Geometry: Congruence Understand congruence in terms of rigid motions. 8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Prove geometric theorems. 9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. 10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. 2011 International Center for Leadership in Education Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 69 New York Mathematics Strands/Bands/ Performance Indicators Geometry Mathematics Domains/Clusters/ Common Core State Standards High School G.RP.1 (Continued from previous page) (Continued from previous page) 11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Geometry: Similarity, Right Triangles, & Trigonometry Understand similarity in terms of similarity transformations. 3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove theorems involving similarity. 4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. 5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Geometry: Circles Understand and apply theorems about circles. 1. Prove that all circles are similar. 2. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Geometry: Expressing Geometric Properties with Equations Use coordinates to prove simple geometric theorems algebraically. 5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). G.RP.2 Recognize and verify, where appropriate, geometric relationships of perpendicularity, parallelism, congruence, and similarity, using algebraic strategies National Essential Skills Study (NESS) National Rankings Rank M4 M10 M54 2011 International Center for Leadership in Education Understand the properties of and apply parallel, perpendicular, and intersecting lines in problem-solving situations. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Apply transformations (reflection, rotation, translation, and dilation) of 2-dimensional figures graphically to interpret, analyze, and illustrate the concepts of congruency, similarity, and symmetry. Mathematics – Page 70 New York Mathematics Strands/Bands/ Performance Indicators Geometry Students will make and investigate mathematical conjectures. G.RP.3 Investigate and evaluate conjectures in mathematical terms, using mathematical strategies to reach a conclusion Mathematics Domains/Clusters/ Common Core State Standards High School Geometry: Congruence Experiment with transformations in the plane. 2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). 3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. 4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Understand congruence in terms of rigid motions. 6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. 7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. 8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Prove geometric theorems. 9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 71 New York Mathematics Strands/Bands/ Performance Indicators Geometry Mathematics Domains/Clusters/ Common Core State Standards High School G.RP.3 (Continued from previous page) (Continued from previous page) 10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. 11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Geometry: Similarity, Right Triangles, & Trigonometry Understand similarity in terms of similarity transformations. 1. Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. 2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. 3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove theorems involving similarity. 4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. 5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Students will develop and evaluate mathematical arguments and proofs. G.RP.4 Provide correct mathematical arguments in response to other students’ conjectures, reasoning, and arguments National Essential Skills Study (NESS) National Rankings Rank There is no New York Mathematics Performance Indicator–Common Core alignment. 2011 International Center for Leadership in Education M10 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Mathematics – Page 72 New York Mathematics Strands/Bands/ Performance Indicators Geometry G.RP.5 Present correct mathematical arguments in a variety of forms Mathematics Domains/Clusters/ Common Core State Standards High School Geometry: Congruence Experiment with transformations in the plane. 2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). 3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. 4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Understand congruence in terms of rigid motions. 6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. 7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. 8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Prove geometric theorems. 9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. 10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 73 New York Mathematics Strands/Bands/ Performance Indicators Geometry Mathematics Domains/Clusters/ Common Core State Standards High School G.RP.5 (Continued from previous page) (Continued from previous page) 11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Geometry: Similarity, Right Triangles, & Trigonometry Understand similarity in terms of similarity transformations. 1. Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. 2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. 3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove theorems involving similarity. 4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. 5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Mathematics – Page 74 New York Mathematics Strands/Bands/ Performance Indicators Geometry Mathematics Domains/Clusters/ Common Core State Standards High School G.RP.6 Evaluate written arguments for validity Geometry: Congruence Understand congruence in terms of rigid motions. 8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Prove geometric theorems. 9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. 10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. 11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Geometry: Similarity, Right Triangles, & Trigonometry Understand similarity in terms of similarity transformations. 2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. 3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove theorems involving similarity. 4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. 5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 75 New York Mathematics Strands/Bands/ Performance Indicators Geometry Students will select and use various types of reasoning and methods of proof. G.RP.7 Construct a proof using a variety of methods (e.g., deductive, analytic, transformational) Mathematics Domains/Clusters/ Common Core State Standards High School Geometry: Congruence Experiment with transformations in the plane. 2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). 3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. 4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Understand congruence in terms of rigid motions. 6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. 7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. 8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Prove geometric theorems. 9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 76 New York Mathematics Strands/Bands/ Performance Indicators Geometry Mathematics Domains/Clusters/ Common Core State Standards High School G.RP.7 (Continued from previous page) (Continued from previous page) 10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. 11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Geometry: Similarity, Right Triangles, & Trigonometry Understand similarity in terms of similarity transformations. 1. Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. 2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. 3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove theorems involving similarity. 4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. 5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Mathematics – Page 77 New York Mathematics Strands/Bands/ Performance Indicators Geometry G.RP.8 Devise ways to verify results or use counterexamples to refute incorrect statements Mathematics Domains/Clusters/ Common Core State Standards High School Geometry: Congruence Experiment with transformations in the plane. 2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). 3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. 4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Understand congruence in terms of rigid motions. 6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. 7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. 8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Prove geometric theorems. 9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. 10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 78 New York Mathematics Strands/Bands/ Performance Indicators Geometry Mathematics Domains/Clusters/ Common Core State Standards High School G.RP.8 (Continued from previous page) (Continued from previous page) 11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Geometry: Similarity, Right Triangles, & Trigonometry Understand similarity in terms of similarity transformations. 1. Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. 2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. 3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove theorems involving similarity. 4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. 5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. There is no New York Mathematics Performance Indicator–Common Core alignment. G.RP.9 Apply inductive reasoning in making and supporting mathematical conjectures 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank M10 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Mathematics – Page 79 New York Mathematics Strands/Bands/ Performance Indicators Geometry Mathematics Domains/Clusters/ Common Core State Standards High School National Essential Skills Study (NESS) National Rankings Rank Communication Strand Students will organize and consolidate their mathematical thinking through communication. G.CM.1 Communicate verbally and in writing a correct, complete, coherent, and clear design (outline) and explanation for the steps used in solving a problem Geometry: Congruence Experiment with transformations in the plane. 2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). 3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. 4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Understand congruence in terms of rigid motions. 6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. 7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. 8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Prove geometric theorems. 9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. 2011 International Center for Leadership in Education Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 80 New York Mathematics Strands/Bands/ Performance Indicators Geometry G.CM.1 (Continued from previous page) Mathematics Domains/Clusters/ Common Core State Standards High School National Essential Skills Study (NESS) National Rankings Rank (Continued from previous page) 10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. 11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Geometry: Similarity, Right Triangles, & Trigonometry Understand similarity in terms of similarity transformations. 1. Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. 2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. 3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove theorems involving similarity. 4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. 5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 2011 International Center for Leadership in Education Mathematics – Page 81 New York Mathematics Strands/Bands/ Performance Indicators Geometry Mathematics Domains/Clusters/ Common Core State Standards High School G.CM.2 Use mathematical representations to communicate with appropriate accuracy, including numerical tables, formulas, functions, equations, charts, graphs, and diagrams Number & Quantity: Quantities Reason quantitatively and use units to solve problems. 1. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. 2. Define appropriate quantities for the purpose of descriptive modeling. 3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. Geometry: Expressing Geometric Properties with Equations Translate between the geometric description and the equation for a conic section. 1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Use coordinates to prove simple geometric theorems algebraically. 4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). 5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). 6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. 7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Geometry: Geometric Measurement & Dimensions Explain volume formulas and use them to solve problems. 1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. 2. (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. 3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank M10 M11 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Apply variables in expressions and equations to solve problems (i.e., write mathematical equations for given situation, create a mathematical model to understand the relationships between variables, or make connections between the structures of mathematically abstract concepts and the real world). Evaluate and employ accurate and appropriate procedures for statistical data collection, organization, analysis, and display including making estimates and predictions, critiquing data, and drawing inferences (e.g., using the normal curve and z-scores, line of best fit). M21 Mathematics – Page 82 New York Mathematics Strands/Bands/ Performance Indicators Geometry Students will communicate their mathematical thinking coherently and clearly to peers, teachers, and others. G.CM.3 Present organized mathematical ideas with the use of appropriate standard notations, including the use of symbols and other representations when sharing an idea in verbal and written form Mathematics Domains/Clusters/ Common Core State Standards High School Geometry: Congruence Experiment with transformations in the plane. 2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). 3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. 4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Understand congruence in terms of rigid motions 6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. 7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. 8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Prove geometric theorems 9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. 10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 83 New York Mathematics Strands/Bands/ Performance Indicators Geometry G.CM.3 (Continued from previous page) Mathematics Domains/Clusters/ Common Core State Standards High School National Essential Skills Study (NESS) National Rankings Rank (Continued from previous page) 11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Geometry: Similarity, Right Triangles, & Trigonometry Understand similarity in terms of similarity transformations. 1. Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. 2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. 3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove theorems involving similarity 4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. 5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 2011 International Center for Leadership in Education Mathematics – Page 84 New York Mathematics Strands/Bands/ Performance Indicators Geometry G.CM.4 Explain relationships among different representations of a problem Mathematics Domains/Clusters/ Common Core State Standards High School Geometry: Congruence Experiment with transformations in the plane. 2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). 3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. 4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Understand congruence in terms of rigid motions 6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. 7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. 8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Geometry: Similarity, Right Triangles, & Trigonometry Understand similarity in terms of similarity transformations. 1. Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. 2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 85 New York Mathematics Strands/Bands/ Performance Indicators Geometry G.CM.4 (Continued from previous page) G.CM.5 Communicate logical arguments clearly, showing why a result makes sense and why the reasoning is valid Mathematics Domains/Clusters/ Common Core State Standards High School (Continued from previous page) 3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Geometry: Congruence Experiment with transformations in the plane. 2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). 3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. 4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Understand congruence in terms of rigid motions 6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. 7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. 8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Prove geometric theorems 9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 86 New York Mathematics Strands/Bands/ Performance Indicators Geometry G.CM.5 (Continued from previous page) G.CM.6 Support or reject arguments or questions raised by others about the correctness of mathematical work Mathematics Domains/Clusters/ Common Core State Standards High School National Essential Skills Study (NESS) National Rankings Rank (Continued from previous page) 10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. 11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Geometry: Similarity, Right Triangles, & Trigonometry Understand similarity in terms of similarity transformations. 1. Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. 2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. 3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove theorems involving similarity 4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. 5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. There is no New York Mathematics Performance Indicator–Common Core alignment. 2011 International Center for Leadership in Education M10 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Mathematics – Page 87 New York Mathematics Strands/Bands/ Performance Indicators Geometry Students will analyze and evaluate the mathematical thinking and strategies of others G.CM.7 Read and listen for logical understanding of mathematical thinking shared by other students G.CM.8 Reflect on strategies of others in relation to one’s own strategy Mathematics Domains/Clusters/ Common Core State Standards High School National Essential Skills Study (NESS) National Rankings Rank There is no New York Mathematics Performance Indicator–Common Core alignment. M10 There is no New York Mathematics Performance Indicator–Common Core alignment. M10 G.CM.9 Formulate mathematical questions that elicit, extend, or challenge strategies, solutions, and/or conjectures of others Students will use the language of mathematics to express mathematical ideas precisely G.CM.10 Use correct mathematical language in developing mathematical questions that elicit, extend, or challenge other students’ conjectures There is no New York Mathematics Performance Indicator–Common Core alignment. M10 There is no New York Mathematics Performance Indicator–Common Core alignment. 2011 International Center for Leadership in Education M10 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Mathematics – Page 88 New York Mathematics Strands/Bands/ Performance Indicators Geometry G.CM.11 Understand and use appropriate language, representations, and terminology when describing objects, relationships, mathematical solutions, and geometric diagrams Mathematics Domains/Clusters/ Common Core State Standards High School Geometry: Congruence Experiment with transformations in the plane. 1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. 2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). 3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. 4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Understand congruence in terms of rigid motions. 6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. 7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. 8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Geometry: Similarity, Right Triangles, & Trigonometry Understand similarity in terms of similarity transformations. 1. Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 89 New York Mathematics Strands/Bands/ Performance Indicators Geometry G.CM.11 (Continued from previous page) Mathematics Domains/Clusters/ Common Core State Standards High School National Essential Skills Study (NESS) National Rankings Rank (Continued from previous page) 2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. 3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Define trigonometric ratios and solve problems involving right triangles. 6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. 7. Explain and use the relationship between the sine and cosine of complementary angles. 8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Geometry: Expressing Geometric Properties with Equations Translate between the geometric description and the equation for a conic section. 1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Use coordinates to prove simple geometric theorems algebraically. 4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). 5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). 6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. 7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. 2011 International Center for Leadership in Education Mathematics – Page 90 New York Mathematics Strands/Bands/ Performance Indicators Geometry G.CM.12 Draw conclusions about mathematical ideas through decoding, comprehension, and interpretation of mathematical visuals, symbols, and technical writing Mathematics Domains/Clusters/ Common Core State Standards High School Geometry: Expressing Geometric Properties with Equations Translate between the geometric description and the equation for a conic section. 1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Use coordinates to prove simple geometric theorems algebraically. 4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). 5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). 6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. 7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Connections Strand Students will recognize and use connections among mathematical ideas. G.CN.1 Understand and make connections among multiple representations of the same mathematical idea Geometry: Congruence Experiment with transformations in the plane. 2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). 3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. 4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. 2011 International Center for Leadership in Education Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 91 New York Mathematics Strands/Bands/ Performance Indicators Geometry Mathematics Domains/Clusters/ Common Core State Standards High School G.CN.1 (Continued from previouspage) (Continued from previous page) Understand congruence in terms of rigid motions. 6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. 7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. 8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Geometry: Circles Understand and apply theorems about circles. 1. Prove that all circles are similar. Find arc length and areas of sectors of circles. 5. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Geometry: Expressing Geometric Properties with Equations Translate between the geometric description and the equation for a conic section. 1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Use coordinates to prove simple geometric theorems algebraically. 5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). 6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. 7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Geometry: Geometric Measurement & Dimensions Explain volume formulas and use them to solve problems. 1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. 3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Mathematics – Page 92 New York Mathematics Strands/Bands/ Performance Indicators Geometry G.CN.2 Understand the corresponding procedures for similar problems or mathematical concepts Students will understand how mathematical ideas interconnect and build on one another to produce a coherent whole. G.CN.3 Model situations mathematically, using representations to draw conclusions and formulate new situations G.CN.4 Understand how concepts, procedures, and mathematical results in one area of mathematics can be used to solve problems in other areas of mathematics Mathematics Domains/Clusters/ Common Core State Standards High School Geometry: Expressing Geometric Properties with Equations Translate between the geometric description and the equation for a conic section. 1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Use coordinates to prove simple geometric theorems algebraically. 4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). Geometry: Modeling with Geometry Apply Geometric Concepts in modeling situations 1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). 2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). Geometry: Expressing Geometric Properties with Equations Translate between the geometric description and the equation for a conic section. 1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Use coordinates to prove simple geometric theorems algebraically. 4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). 5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). 6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank M10 M10 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 93 New York Mathematics Strands/Bands/ Performance Indicators Geometry G.CN.4 (Continued from previous page) G.CN.5 Understand how quantitative models connect to various physical models and representations Students will recognize and apply mathematics in contexts outside of mathematics. G.CN.6 Recognize and apply mathematics to situations in the outside world Mathematics Domains/Clusters/ Common Core State Standards High School (Continued from previous page) 7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Geometry: Similarity, Right Triangles, & Trigonometry Understand similarity in terms of similarity transformations. 1. Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Define trigonometric ratios and solve problems involving right triangles. 6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. 7. Explain and use the relationship between the sine and cosine of complementary angles. 8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Geometry: Modeling with Geometry Apply Geometric Concepts in modeling situations 1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). 2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). Geometry: Modeling with Geometry Apply Geometric Concepts in modeling situations 1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). 2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 M10 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Mathematics – Page 94 New York Mathematics Strands/Bands/ Performance Indicators Geometry G.CN.7 Recognize and apply mathematical ideas to problem situations that develop outside of mathematics G.CN.8 Develop an appreciation for the historical development of mathematics Mathematics Domains/Clusters/ Common Core State Standards High School Geometry: Modeling with Geometry Apply Geometric Concepts in modeling situations 1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). 2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). National Essential Skills Study (NESS) National Rankings Rank M10 There is no New York Mathematics Performance Indicator–Common Core alignment. M10 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Representation Strand Students will create and use representations to organize, record, and communicate mathematical ideas. G.R.1 Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts Geometry: Congruence Experiment with transformations in the plane. 2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). 3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. 4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. 2011 International Center for Leadership in Education M7 M10 M21 Simplify and solve algebraic equations by identifying and using the correct order of operations and techniques necessary to carry out the solution. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Evaluate and employ accurate and appropriate procedures for statistical data collection, organization, analysis, and display including making estimates and predictions, critiquing data, and drawing inferences (e.g., using the normal curve and z-scores, line of best fit). Mathematics – Page 95 New York Mathematics Strands/Bands/ Performance Indicators Geometry Mathematics Domains/Clusters/ Common Core State Standards High School G.R.1 (Continued from previous page) (Continued from previous page) Understand congruence in terms of rigid motions. 8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Make geometric constructions. 12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. 13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Geometry: Similarity, Right Triangles, & Trigonometry Understand similarity in terms of similarity transformations. 1. Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. 3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Geometry: Expressing Geometric Properties with Equations Translate between the geometric description and the equation for a conic section. 1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Use coordinates to prove simple geometric theorems algebraically. 5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). 6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. 7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Mathematics – Page 96 New York Mathematics Strands/Bands/ Performance Indicators Geometry Mathematics Domains/Clusters/ Common Core State Standards High School G.R.2 Recognize, compare, and use an array of representational forms Geometry: Congruence Experiment with transformations in the plane. 2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). 3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. 4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Understand congruence in terms of rigid motions. 8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Make geometric constructions. 12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. 13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Geometry: Similarity, Right Triangles, & Trigonometry Understand similarity in terms of similarity transformations. 1. Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. 3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 97 New York Mathematics Strands/Bands/ Performance Indicators Geometry Mathematics Domains/Clusters/ Common Core State Standards High School G.R.2 (Continued from previous page) (Continued from previous page) Geometry: Expressing Geometric Properties with Equations Translate between the geometric description and the equation for a conic section. 1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Use coordinates to prove simple geometric theorems algebraically. 5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). 6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. 7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Geometry: Congruence Experiment with transformations in the plane. 2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). 3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. 4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Understand congruence in terms of rigid motions. 8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. G.R.3 Use representation as a tool for exploring and understanding mathematical ideas 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 98 New York Mathematics Strands/Bands/ Performance Indicators Geometry Mathematics Domains/Clusters/ Common Core State Standards High School G.R.3 (Continued from previous page) (Continued from previous page) Make geometric constructions. 12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. 13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Geometry: Similarity, Right Triangles, & Trigonometry Understand similarity in terms of similarity transformations. 1. Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. 3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Geometry: Expressing Geometric Properties with Equations Translate between the geometric description and the equation for a conic section. 1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Use coordinates to prove simple geometric theorems algebraically. 5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). 6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. 7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Mathematics – Page 99 New York Mathematics Strands/Bands/ Performance Indicators Geometry Students will select, apply, and translate among mathematical representations to solve problems. G.R.4 Select appropriate representations to solve problem situations Mathematics Domains/Clusters/ Common Core State Standards High School Geometry: Congruence Experiment with transformations in the plane. 2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). 3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. 4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Understand congruence in terms of rigid motions. 8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Make geometric constructions. 12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. 13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Geometry: Similarity, Right Triangles, & Trigonometry Understand similarity in terms of similarity transformations. 1. Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 100 New York Mathematics Strands/Bands/ Performance Indicators Geometry Mathematics Domains/Clusters/ Common Core State Standards High School G.R.4 (Continued from previous page) (Continued from previous page) 3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Geometry: Expressing Geometric Properties with Equations Translate between the geometric description and the equation for a conic section. 1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Use coordinates to prove simple geometric theorems algebraically. 5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). 6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. 7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Geometry: Congruence Experiment with transformations in the plane. 2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). 3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. 4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Geometry: Expressing Geometric Properties with Equations Use coordinates to prove simple geometric theorems algebraically. 5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). G.R.5 Investigate relationships between different representations and their impact on a given problem 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 101 New York Mathematics Strands/Bands/ Performance Indicators Geometry Students will use representations to model and interpret physical, social, and mathematical phenomena. G.R.6 Use mathematics to show and understand physical phenomena (e.g., determine the number of gallons of water in a fish tank) G.R.7 Use mathematics to show and understand social phenomena (e.g., determine if conclusions from another person’s argument have a logical foundation) Mathematics Domains/Clusters/ Common Core State Standards High School Number & Quantity: Quantities Reason quantitatively and use units to solve problems. 1. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. 3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. Geometry: Geometric Measurement & Dimensions Explain volume formulas and use them to solve problems. 1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. 3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. There is no New York Mathematics Performance Indicator–Common Core alignment. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 M10 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Mathematics – Page 102 New York Mathematics Strands/Bands/ Performance Indicators Geometry Mathematics Domains/Clusters/ Common Core State Standards High School G.R.8 Use mathematics to show and understand mathematical phenomena (e.g., use investigation, discovery, conjecture, reasoning, arguments, justification and proofs to validate that the two base angles of an isosceles triangle are congruent) Geometry: Congruence Experiment with transformations in the plane. 2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). 3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. 4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Understand congruence in terms of rigid motions. 6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. 7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. 8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Prove geometric theorems 9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. 10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 103 New York Mathematics Strands/Bands/ Performance Indicators Geometry Mathematics Domains/Clusters/ Common Core State Standards High School G.R.8 (Continued from previous page) (Continued from previous page) 11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Geometry: Similarity, Right Triangles, & Trigonometry Understand similarity in terms of similarity transformations. 1. Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. 2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. 3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove theorems involving similarity. 4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. 5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Geometry: Expressing Geometric Properties with Equations Use coordinates to prove simple geometric theorems algebraically. 4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). 5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Mathematics – Page 104 New York Mathematics Strands/Bands/ Performance Indicators Geometry Mathematics Domains/Clusters/ Common Core State Standards High School G.R.8 (Continued from previous page) (Continued from previous page) 6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. 7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. National Essential Skills Study (NESS) National Rankings Rank Algebra Strand Note: The algebraic skills and concepts within the Algebra process and content performance indicators must be maintained and applied as students are asked to investigate, make conjectures, give rationale, and justify or prove geometric concepts. Geometry Strand Students will use visualization and spatial reasoning to analyze characteristics and properties of geometric shapes. Geometric Relationships Note: Two-dimensional geometric relationships are addressed in the Informal and Formal Proofs band. G.G.1 Know and apply that if a line is perpendicular to each of two intersecting lines at their point of intersection, then the line is perpendicular to the plane determined by them There is no New York Mathematics Performance Indicator–Common Core alignment. 2011 International Center for Leadership in Education M4 M41 Understand the properties of and apply parallel, perpendicular, and intersecting lines in problem-solving situations. Understand the properties and applications of the undefined terms of geometry (point, line, and plane) and their relationship with intuitive concepts (i.e., collinear points, coplanar points, opposite rays, and parallel lines). Mathematics – Page 105 New York Mathematics Strands/Bands/ Performance Indicators Geometry G.G.2 Know and apply that through a given point there passes one and only one plane perpendicular to a given line Mathematics Domains/Clusters/ Common Core State Standards High School There is no New York Mathematics Performance Indicator–Common Core alignment. National Essential Skills Study (NESS) National Rankings Rank M4 M41 G.G.3 Know and apply that through a given point there passes one and only one line perpendicular to a given plane There is no New York Mathematics Performance Indicator–Common Core alignment. M4 M41 G.G.4 Know and apply that two lines perpendicular to the same plane are coplanar G.G.5 Know and apply that two planes are perpendicular to each other if and only if one plane contains a line perpendicular to the second plane G.G.6 Know and apply that if a line is perpendicular to a plane, then any line perpendicular to the given line at its point of intersection with the given plane is in the given plane There is no New York Mathematics Performance Indicator–Common Core alignment. There is no New York Mathematics Performance Indicator–Common Core alignment. There is no New York Mathematics Performance Indicator–Common Core alignment. M4 M4 M4 M41 2011 International Center for Leadership in Education Understand the properties of and apply parallel, perpendicular, and intersecting lines in problem-solving situations. Understand the properties and applications of the undefined terms of geometry (point, line, and plane) and their relationship with intuitive concepts (i.e., collinear points, coplanar points, opposite rays, and parallel lines). Understand the properties of and apply parallel, perpendicular, and intersecting lines in problem-solving situations. Understand the properties and applications of the undefined terms of geometry (point, line, and plane) and their relationship with intuitive concepts (i.e., collinear points, coplanar points, opposite rays, and parallel lines). Understand the properties of and apply parallel, perpendicular, and intersecting lines in problem-solving situations. Understand the properties of and apply parallel, perpendicular, and intersecting lines in problem-solving situations. Understand the properties of and apply parallel, perpendicular, and intersecting lines in problem-solving situations. Understand the properties and applications of the undefined terms of geometry (point, line, and plane) and their relationship with intuitive concepts (i.e., collinear points, coplanar points, opposite rays, and parallel lines). Mathematics – Page 106 New York Mathematics Strands/Bands/ Performance Indicators Geometry G.G.7 Know and apply that if a line is perpendicular to a plane, then every plane containing the line is perpendicular to the given plane G.G.8 Know and apply that if a plane intersects two parallel planes, then the intersection is two parallel lines Mathematics Domains/Clusters/ Common Core State Standards High School There is no New York Mathematics Performance Indicator–Common Core alignment. There is no New York Mathematics Performance Indicator–Common Core alignment. National Essential Skills Study (NESS) National Rankings Rank M4 M4 M41 G.G.9 Know and apply that if two planes are perpendicular to the same line, they are parallel G.G.10 Know and apply that the lateral edges of a prism are congruent and parallel G.G.11 Know and apply that two prisms have equal volumes if their bases have equal areas and their altitudes are equal There is no New York Mathematics Performance Indicator–Common Core alignment. Geometry: Geometric Measurement & Dimensions Visualize relationships between two-dimensional and three-dimensional objects. 4. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of twodimensional objects. Geometry: Geometric Measurement & Dimensions Explain volume formulas and use them to solve problems. 2. (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. 2011 International Center for Leadership in Education M4 M4 M26 M26 Understand the properties of and apply parallel, perpendicular, and intersecting lines in problem-solving situations. Understand the properties of and apply parallel, perpendicular, and intersecting lines in problem-solving situations. Understand the properties and applications of the undefined terms of geometry (point, line, and plane) and their relationship with intuitive concepts (i.e., collinear points, coplanar points, opposite rays, and parallel lines). Understand the properties of and apply parallel, perpendicular, and intersecting lines in problem-solving situations. Understand the properties of and apply parallel, perpendicular, and intersecting lines in problem-solving situations. Know the classification and properties of three-dimensional figures (prisms, rectangular solids, pyramids, right circular cylinders, cones, and spheres) and be able to compute the volume and surface area of common solids. Know the classification and properties of three-dimensional figures (prisms, rectangular solids, pyramids, right circular cylinders, cones, and spheres) and be able to compute the volume and surface area of common solids. Mathematics – Page 107 New York Mathematics Strands/Bands/ Performance Indicators Geometry Mathematics Domains/Clusters/ Common Core State Standards High School G.G.12 Know and apply that the volume of a prism is the product of the area of the base and the altitude Geometry: Geometric Measurement & Dimensions Explain volume formulas and use them to solve problems. 1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. G.G.13 Apply the properties of a regular pyramid, including: lateral edges are congruent lateral faces are congruent isosceles triangles volume of a pyramid equals onethird the product of the area of the base and the altitude G.G.14 Apply the properties of a cylinder, including: bases are congruent volume equals the product of the area of the base and the altitude lateral area of a right circular cylinder equals the product of an altitude and the circumference of the base G.G.15 Apply the properties of a right circular cone, including: lateral area equals one-half the product of the slant height and the circumference of its base volume is one-third the product of the area of its base and its altitude Geometry: Geometric Measurement & Dimensions Explain volume formulas and use them to solve problems. 1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. 3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Geometry: Geometric Measurement & Dimensions Explain volume formulas and use them to solve problems. 1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. 3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Geometry: Geometric Measurement & Dimensions Explain volume formulas and use them to solve problems. 1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. 3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank M26 M26 M26 M26 Know the classification and properties of three-dimensional figures (prisms, rectangular solids, pyramids, right circular cylinders, cones, and spheres) and be able to compute the volume and surface area of common solids. Know the classification and properties of three-dimensional figures (prisms, rectangular solids, pyramids, right circular cylinders, cones, and spheres) and be able to compute the volume and surface area of common solids. Know the classification and properties of three-dimensional figures (prisms, rectangular solids, pyramids, right circular cylinders, cones, and spheres) and be able to compute the volume and surface area of common solids. Know the classification and properties of three-dimensional figures (prisms, rectangular solids, pyramids, right circular cylinders, cones, and spheres) and be able to compute the volume and surface area of common solids. Mathematics – Page 108 New York Mathematics Strands/Bands/ Performance Indicators Geometry G.G.16 Apply the properties of a sphere, including: the intersection of a plane and a sphere is a circle a great circle is the largest circle that can be drawn on a sphere two planes equidistant from the center of the sphere and intersecting the sphere do so in congruent circles surface area is 4 r volume is Mathematics Domains/Clusters/ Common Core State Standards High School National Essential Skills Study (NESS) National Rankings Rank Know the classification and properties of three-dimensional figures (prisms, rectangular solids, pyramids, right circular cylinders, cones, and spheres) and be able to compute the volume and surface area of common solids. Geometry: Geometric Measurement & Dimensions Explain volume formulas and use them to solve problems. 2. (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. M26 2 4 3 r 3 Constructions G.G.17 Construct a bisector of a given angle, using a straightedge and compass, and justify the construction G.G.18 Construct the perpendicular bisector of a given segment, using a straightedge and compass, and justify the construction Geometry: Congruence Make geometric constructions. 12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Geometry: Congruence Make geometric constructions. 12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. 2011 International Center for Leadership in Education M15 M42 M4 M42 Classify angles by measure (acute, right, obtuse, and straight) and understand angle relationships (supplementary, complementary, and vertical). Use geometric methods, such as using an unmarked straightedge and compass, to complete basic geometric constructions. Understand the properties of and apply parallel, perpendicular, and intersecting lines in problem-solving situations. Use geometric methods, such as using an unmarked straightedge and compass, to complete basic geometric constructions. Mathematics – Page 109 New York Mathematics Strands/Bands/ Performance Indicators Geometry Mathematics Domains/Clusters/ Common Core State Standards High School G.G.19 Construct lines parallel (or perpendicular) to a given line through a given point, using a straightedge and compass, and justify the construction Geometry: Congruence Make geometric constructions. 12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Geometry: Congruence Make geometric constructions. 13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. G.G.20 Construct an equilateral triangle, using a straightedge and compass, and justify the construction National Essential Skills Study (NESS) National Rankings Rank M4 M42 M34 M42 Understand the properties of and apply parallel, perpendicular, and intersecting lines in problem-solving situations. Use geometric methods, such as using an unmarked straightedge and compass, to complete basic geometric constructions. Understand the properties and classification of polygons (triangles, the family of quadrilaterals, pentagon, hexagon, etc.) and apply knowledge of angle and side relationships of geometric shapes in problem-solving situations. Use geometric methods, such as using an unmarked straightedge and compass, to complete basic geometric constructions. Locus G.G.21 Investigate and apply the concurrence of medians, altitudes, angle bisectors, and perpendicular bisectors of triangles Geometry: Congruence Prove geometric theorems. 10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. M15 M34 G.G.22 Solve problems using compound loci There is no New York Mathematics Performance Indicator–Common Core alignment. 2011 International Center for Leadership in Education M66 Classify angles by measure (acute, right, obtuse, and straight) and understand angle relationships (supplementary, complementary, and vertical). Understand the properties and classification of polygons (triangles, the family of quadrilaterals, pentagon, hexagon, etc.) and apply knowledge of angle and side relationships of geometric shapes in problem-solving situations. Know how to sketch basic conic sections (e.g., circles, parabolas) by using their equations and solve systems of non-linear equations graphically. Mathematics – Page 110 New York Mathematics Strands/Bands/ Performance Indicators Geometry G.G.23 Graph and solve compound loci in the coordinate plane Mathematics Domains/Clusters/ Common Core State Standards High School There is no New York Mathematics Performance Indicator–Common Core alignment. National Essential Skills Study (NESS) National Rankings Rank M66 Know how to sketch basic conic sections (e.g., circles, parabolas) by using their equations and solve systems of non-linear equations graphically. Students will identify and justify geometric relationships formally and informally. Informal and Formal Proofs G.G.24 Determine the negation of a statement and establish its truth value G.G.25 Know and apply the conditions under which a compound statement (conjunction, disjunction, conditional, biconditional) is true There is no New York Mathematics Performance Indicator–Common Core alignment. There is no New York Mathematics Performance Indicator–Common Core alignment. M49 M10 M49 G.G.26 Identify and write the inverse, converse, and contrapositive of a given conditional statement and note the logical equivalences G.G.27 Write a proof arguing from a given hypothesis to a given conclusion There is no New York Mathematics Performance Indicator–Common Core alignment. Geometry: Congruence Prove geometric theorems. 9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. 10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. 11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. 2011 International Center for Leadership in Education M49 Analyze the truth value of compound sentences by creating truth tables. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Analyze the truth value of compound sentences by creating truth tables. Analyze the truth value of compound sentences by creating truth tables. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 111 New York Mathematics Strands/Bands/ Performance Indicators Geometry Mathematics Domains/Clusters/ Common Core State Standards High School G.G.27 (Continued from previous page) (Continued from previous page) Geometry: Similarity, Right Triangles, & Trigonometry Prove theorems involving similarity. 4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. 5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Geometry: Congruence Understand congruence in terms of rigid motions. 8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. G.G.28 Determine the congruence of two triangles by using one of the five congruence techniques (SSS, SAS, ASA, AAS, HL), given sufficient information about the sides and/or angles of two congruent triangles National Essential Skills Study (NESS) National Rankings Rank M10 M54 G.G.29 Identify corresponding parts of congruent triangles Geometry: Congruence Understand congruence in terms of rigid motions. 8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. M34 M54 2011 International Center for Leadership in Education Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Apply transformations (reflection, rotation, translation, and dilation) of 2-dimensional figures graphically to interpret, analyze, and illustrate the concepts of congruency, similarity, and symmetry. Understand the properties and classification of polygons (triangles, the family of quadrilaterals, pentagon, hexagon, etc.) and apply knowledge of angle and side relationships of geometric shapes in problem-solving situations. Apply transformations (reflection, rotation, translation, and dilation) of 2-dimensional figures graphically to interpret, analyze, and illustrate the concepts of congruency, similarity, and symmetry. Mathematics – Page 112 New York Mathematics Strands/Bands/ Performance Indicators Geometry Mathematics Domains/Clusters/ Common Core State Standards High School G.G.30 Investigate, justify, and apply theorems about the sum of the measures of the angles of a triangle Geometry: Congruence Prove geometric theorems. 10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. National Essential Skills Study (NESS) National Rankings Rank M10 M34 G.G.31 Investigate, justify, and apply the isosceles triangle theorem and its converse Geometry: Congruence Prove geometric theorems. 10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. M10 M34 G.G.32 Investigate, justify, and apply theorems about geometric inequalities, using the exterior angle theorem Geometry: Congruence Prove geometric theorems. 10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. M10 M34 2011 International Center for Leadership in Education Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand the properties and classification of polygons (triangles, the family of quadrilaterals, pentagon, hexagon, etc.) and apply knowledge of angle and side relationships of geometric shapes in problem-solving situations. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand the properties and classification of polygons (triangles, the family of quadrilaterals, pentagon, hexagon, etc.) and apply knowledge of angle and side relationships of geometric shapes in problem-solving situations. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand the properties and classification of polygons (triangles, the family of quadrilaterals, pentagon, hexagon, etc.) and apply knowledge of angle and side relationships of geometric shapes in problem-solving situations. Mathematics – Page 113 New York Mathematics Strands/Bands/ Performance Indicators Geometry G.G.33 Investigate, justify, and apply the triangle inequality theorem Mathematics Domains/Clusters/ Common Core State Standards High School National Essential Skills Study (NESS) National Rankings Rank There is no New York Mathematics Performance Indicator–Common Core alignment. M10 M34 G.G.34 Determine either the longest side of a triangle given the three angle measures or the largest angle given the lengths of three sides of a triangle There is no New York Mathematics Performance Indicator–Common Core alignment. M10 M34 G.G.35 Determine if two lines cut by a transversal are parallel, based on the measure of given pairs of angles formed by the transversal and the lines Geometry: Congruence Prove geometric theorems. 9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. 2011 International Center for Leadership in Education M4 M15 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand the properties and classification of polygons (triangles, the family of quadrilaterals, pentagon, hexagon, etc.) and apply knowledge of angle and side relationships of geometric shapes in problem-solving situations. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand the properties and classification of polygons (triangles, the family of quadrilaterals, pentagon, hexagon, etc.) and apply knowledge of angle and side relationships of geometric shapes in problem-solving situations. Understand the properties of and apply parallel, perpendicular, and intersecting lines in problem-solving situations. Classify angles by measure (acute, right, obtuse, and straight) and understand angle relationships (supplementary, complementary, and vertical). Mathematics – Page 114 New York Mathematics Strands/Bands/ Performance Indicators Geometry Mathematics Domains/Clusters/ Common Core State Standards High School G.G.36 Investigate, justify, and apply theorems about the sum of the measures of the interior and exterior angles of polygons Geometry: Congruence Prove geometric theorems. 10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. 11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. G.G.37 Investigate, justify, and apply theorems about each interior and exterior angle measure of regular polygons G.G.38 Investigate, justify, and apply theorems about parallelograms involving their angles, sides, and diagonals Geometry: Congruence Prove geometric theorems. 10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. 11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Geometry: Congruence Prove geometric theorems. 11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. National Essential Skills Study (NESS) National Rankings Rank M10 M34 M10 M34 M10 M34 2011 International Center for Leadership in Education Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand the properties and classification of polygons (triangles, the family of quadrilaterals, pentagon, hexagon, etc.) and apply knowledge of angle and side relationships of geometric shapes in problem-solving situations. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand the properties and classification of polygons (triangles, the family of quadrilaterals, pentagon, hexagon, etc.) and apply knowledge of angle and side relationships of geometric shapes in problem-solving situations. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand the properties and classification of polygons (triangles, the family of quadrilaterals, pentagon, hexagon, etc.) and apply knowledge of angle and side relationships of geometric shapes in problem-solving situations. Mathematics – Page 115 New York Mathematics Strands/Bands/ Performance Indicators Geometry Mathematics Domains/Clusters/ Common Core State Standards High School G.G.39 Investigate, justify, and apply theorems about special parallelograms (rectangles, rhombuses, squares) involving their angles, sides, and diagonals Geometry: Congruence Prove geometric theorems. 11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. National Essential Skills Study (NESS) National Rankings Rank M10 M34 G.G.40 Investigate, justify, and apply theorems about trapezoids (including isosceles trapezoids) involving their angles, sides, medians, and diagonals There is no New York Mathematics Performance Indicator–Common Core alignment. M10 M34 G.G.41 Justify that some quadrilaterals are parallelograms, rhombuses, rectangles, squares, or trapezoids Geometry: Congruence Prove geometric theorems. 11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. 2011 International Center for Leadership in Education M34 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand the properties and classification of polygons (triangles, the family of quadrilaterals, pentagon, hexagon, etc.) and apply knowledge of angle and side relationships of geometric shapes in problem-solving situations. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand the properties and classification of polygons (triangles, the family of quadrilaterals, pentagon, hexagon, etc.) and apply knowledge of angle and side relationships of geometric shapes in problem-solving situations. Understand the properties and classification of polygons (triangles, the family of quadrilaterals, pentagon, hexagon, etc.) and apply knowledge of angle and side relationships of geometric shapes in problem-solving situations. Mathematics – Page 116 New York Mathematics Strands/Bands/ Performance Indicators Geometry Mathematics Domains/Clusters/ Common Core State Standards High School G.G.42 Investigate, justify, and apply theorems about geometric relationships, based on the properties of the line segment joining the midpoints of two sides of the triangle Geometry: Congruence Prove geometric theorems. 10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. National Essential Skills Study (NESS) National Rankings Rank M10 M34 G.G.43 Investigate, justify, and apply theorems about the centroid of a triangle, dividing each median into segments whose lengths are in the ratio 2:1 Geometry: Congruence Prove geometric theorems. 10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. M3 M10 M34 G.G.44 Establish similarity of triangles, using the following theorems: AA, SAS, and SSS Geometry: Similarity, Right Triangles, & Trigonometry Understand similarity in terms of similarity transformations. 3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove theorems involving similarity. 5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. M10 M54 2011 International Center for Leadership in Education Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand the properties and classification of polygons (triangles, the family of quadrilaterals, pentagon, hexagon, etc.) and apply knowledge of angle and side relationships of geometric shapes in problem-solving situations. Use proportional reasoning to solve realworld problems. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand the properties and classification of polygons (triangles, the family of quadrilaterals, pentagon, hexagon, etc.) and apply knowledge of angle and side relationships of geometric shapes in problem-solving situations. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Apply transformations (reflection, rotation, translation, and dilation) of 2-dimensional figures graphically to interpret, analyze, and illustrate the concepts of congruency, similarity, and symmetry. Mathematics – Page 117 New York Mathematics Strands/Bands/ Performance Indicators Geometry Mathematics Domains/Clusters/ Common Core State Standards High School G.G.45 Investigate, justify, and apply theorems about similar triangles Geometry: Similarity, Right Triangles, & Trigonometry Prove theorems involving similarity. 4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. 5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. National Essential Skills Study (NESS) National Rankings Rank M10 M54 G.G.46 Investigate, justify, and apply theorems about proportional relationships among the segments of the sides of the triangle, given one or more lines parallel to one side of a triangle and intersecting the other two sides of the triangle Geometry: Similarity, Right Triangles, & Trigonometry Prove theorems involving similarity. 4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. M3 M4 M34 G.G.47 Investigate, justify, and apply theorems about mean proportionality: the altitude to the hypotenuse of a right triangle is the mean proportional between the two segments along the hypotenuse the altitude to the hypotenuse of a right triangle divides the hypotenuse so that either leg of the right triangle is the mean proportional between the hypotenuse and segment of the hypotenuse adjacent to that leg Geometry: Similarity, Right Triangles, & Trigonometry Prove theorems involving similarity. 4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. 2011 International Center for Leadership in Education M3 M10 M34 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Apply transformations (reflection, rotation, translation, and dilation) of 2-dimensional figures graphically to interpret, analyze, and illustrate the concepts of congruency, similarity, and symmetry. Use proportional reasoning to solve realworld problems. Understand the properties of and apply parallel, perpendicular, and intersecting lines in problem-solving situations. Understand the properties and classification of polygons (triangles, the family of quadrilaterals, pentagon, hexagon, etc.) and apply knowledge of angle and side relationships of geometric shapes in problem-solving situations. Use proportional reasoning to solve realworld problems. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand the properties and classification of polygons (triangles, the family of quadrilaterals, pentagon, hexagon, etc.) and apply knowledge of angle and side relationships of geometric shapes in problem-solving situations. Mathematics – Page 118 New York Mathematics Strands/Bands/ Performance Indicators Geometry Mathematics Domains/Clusters/ Common Core State Standards High School G.G.48 Investigate, justify, and apply the Pythagorean theorem and its converse Geometry: Similarity, Right Triangles, & Trigonometry Define trigonometric ratios and solve problems involving right triangles. 8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. National Essential Skills Study (NESS) National Rankings Rank M10 M23 G.G.49 Investigate, justify, and apply theorems regarding chords of a circle: perpendicular bisectors of chords the relative lengths of chords as compared to their distance from the center of the circle Geometry: Circles Understand and apply theorems about circles. 2. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. M10 M18 G.G.50 Investigate, justify, and apply theorems about tangent lines to a circle: a perpendicular to the tangent at the point of tangency two tangents to a circle from the same external point common tangents of two nonintersecting or tangent circles Geometry: Circles Understand and apply theorems about circles. 2. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. M10 M18 M61 2011 International Center for Leadership in Education Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Apply the Pythagorean Theorem to right triangles. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand the properties of circles (radius, arc, diameter, chord, secant, and tangent) and apply circle quantities (lengths of line segments, angle measure within a circle, circumference, and area) in problem-solving situations. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand the properties of circles (radius, arc, diameter, chord, secant, and tangent) and apply circle quantities (lengths of line segments, angle measure within a circle, circumference, and area) in problem-solving situations. Use derivatives and the process of differentiation to determine slopes of tangent lines, maxima and minima, velocity, and acceleration. Mathematics – Page 119 New York Mathematics Strands/Bands/ Performance Indicators Geometry Mathematics Domains/Clusters/ Common Core State Standards High School G.G.51 Investigate, justify, and apply theorems about the arcs determined by the rays of angles formed by two lines intersecting a circle when the vertex is: inside the circle (two chords) on the circle (tangent and chord) outside the circle (two tangents, two secants, or tangent and secant) There is no New York Mathematics Performance Indicator–Common Core alignment. G.G.52 Investigate, justify, and apply theorems about arcs of a circle cut by two parallel lines There is no New York Mathematics Performance Indicator–Common Core alignment. National Essential Skills Study (NESS) National Rankings Rank M10 M18 M4 M10 M18 G.G.53 Investigate, justify, and apply theorems regarding segments intersected by a circle: along two tangents from the same external point along two secants from the same external point along a tangent and a secant from the same external point along two intersecting chords of a given circle Geometry: Circles Understand and apply theorems about circles. 2. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. 2011 International Center for Leadership in Education M10 M18 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand the properties of circles (radius, arc, diameter, chord, secant, and tangent) and apply circle quantities (lengths of line segments, angle measure within a circle, circumference, and area) in problem-solving situations. Understand the properties of and apply parallel, perpendicular, and intersecting lines in problem-solving situations. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand the properties of circles (radius, arc, diameter, chord, secant, and tangent) and apply circle quantities (lengths of line segments, angle measure within a circle, circumference, and area) in problem-solving situations. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand the properties of circles (radius, arc, diameter, chord, secant, and tangent) and apply circle quantities (lengths of line segments, angle measure within a circle, circumference, and area) in problem-solving situations. Mathematics – Page 120 New York Mathematics Strands/Bands/ Performance Indicators Geometry Mathematics Domains/Clusters/ Common Core State Standards High School National Essential Skills Study (NESS) National Rankings Rank Students will apply transformations and symmetry to analyze problem solving situations Transformational Geometry G.G.54 Define, investigate, justify, and apply isometries in the plane (rotations, reflections, translations, glide reflections) Note: Use proper function notation. G.G.55 Investigate, justify, and apply the properties that remain invariant under translations, rotations, reflections, and glide reflections Geometry: Congruence Experiment with transformations in the plane. 1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. 2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). 3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. 4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Geometry: Congruence Experiment with transformations in the plane. 1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. 2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). 3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. 4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 2011 International Center for Leadership in Education M10 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Apply transformations (reflection, rotation, translation, and dilation) of 2-dimensional figures graphically to interpret, analyze, and illustrate the concepts of congruency, similarity, and symmetry. M54 M10 M54 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Apply transformations (reflection, rotation, translation, and dilation) of 2-dimensional figures graphically to interpret, analyze, and illustrate the concepts of congruency, similarity, and symmetry. Mathematics – Page 121 New York Mathematics Strands/Bands/ Performance Indicators Geometry Mathematics Domains/Clusters/ Common Core State Standards High School G.G.55 (Continued from previous page) (Continued from previous page) 5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Understand congruence in terms of rigid motions. 6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Geometry: Congruence Experiment with transformations in the plane. 2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). 3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. 4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Geometry: Congruence Experiment with transformations in the plane. 4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. G.G.56 Identify specific isometries by observing orientation, numbers of invariant points, and/or parallelism G.G.57 Justify geometric relationships (perpendicularity, parallelism, congruence) using transformational techniques (translations, rotations, reflections) National Essential Skills Study (NESS) National Rankings Rank Apply transformations (reflection, rotation, translation, and dilation) of 2-dimensional figures graphically to interpret, analyze, and illustrate the concepts of congruency, similarity, and symmetry. M54 M4 M10 M54 2011 International Center for Leadership in Education Understand the properties of and apply parallel, perpendicular, and intersecting lines in problem-solving situations. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Apply transformations (reflection, rotation, translation, and dilation) of 2-dimensional figures graphically to interpret, analyze, and illustrate the concepts of congruency, similarity, and symmetry. Mathematics – Page 122 New York Mathematics Strands/Bands/ Performance Indicators Geometry Mathematics Domains/Clusters/ Common Core State Standards High School G.G.58 Define, investigate, justify, and apply similarities (dilations and the composition of dilations and isometries) Geometry: Similarity, Right Triangles, & Trigonometry Understand similarity in terms of similarity transformations. 1. Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. 2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Geometry: Similarity, Right Triangles, & Trigonometry Understand similarity in terms of similarity transformations. 2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. G.G.59 Investigate, justify, and apply the properties that remain invariant under similarities National Essential Skills Study (NESS) National Rankings Rank M10 M54 M10 M54 G.G.60 Identify specific similarities by observing orientation, numbers of invariant points, and/or parallelism Geometry: Similarity, Right Triangles, & Trigonometry Prove theorems involving similarity. 4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. M10 M54 2011 International Center for Leadership in Education Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Apply transformations (reflection, rotation, translation, and dilation) of 2-dimensional figures graphically to interpret, analyze, and illustrate the concepts of congruency, similarity, and symmetry. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Apply transformations (reflection, rotation, translation, and dilation) of 2-dimensional figures graphically to interpret, analyze, and illustrate the concepts of congruency, similarity, and symmetry. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Apply transformations (reflection, rotation, translation, and dilation) of 2-dimensional figures graphically to interpret, analyze, and illustrate the concepts of congruency, similarity, and symmetry. Mathematics – Page 123 New York Mathematics Strands/Bands/ Performance Indicators Geometry Mathematics Domains/Clusters/ Common Core State Standards High School G.G.61 Investigate, justify, and apply the analytical representations for translations, rotations about the origin of 90º and 180º, reflections over the lines x 0 , y 0 , and y x , and dilations centered at the origin Geometry: Congruence Experiment with transformations in the plane. 5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. National Essential Skills Study (NESS) National Rankings Rank M10 M54 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Apply transformations (reflection, rotation, translation, and dilation) of 2-dimensional figures graphically to interpret, analyze, and illustrate the concepts of congruency, similarity, and symmetry. Students will apply coordinate geometry to analyze problem solving situations. Coordinate Geometry G.G.62 Find the slope of a perpendicular line, given the equation of a line G.G.63 Determine whether two lines are parallel, perpendicular, or neither, given their equations Geometry: Expressing Geometric Properties with Equations Use coordinates to prove simple geometric theorems algebraically. 5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). Geometry: Expressing Geometric Properties with Equations Use coordinates to prove simple geometric theorems algebraically. 5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). 2011 International Center for Leadership in Education M4 M44 M4 M44 Understand the properties of and apply parallel, perpendicular, and intersecting lines in problem-solving situations. Know the equation of a line and interpret graphically using the slope-intercept form (y = mx+b) and the point-slope form (y-b = m(x-a)). Understand the properties of and apply parallel, perpendicular, and intersecting lines in problem-solving situations. Know the equation of a line and interpret graphically using the slope-intercept form (y = mx+b) and the point-slope form (y-b = m(x-a)). Mathematics – Page 124 New York Mathematics Strands/Bands/ Performance Indicators Geometry G.G.64 Find the equation of a line, given a point on the line and the equation of a line perpendicular to the given line Mathematics Domains/Clusters/ Common Core State Standards High School Geometry: Expressing Geometric Properties with Equations Use coordinates to prove simple geometric theorems algebraically. 5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). National Essential Skills Study (NESS) National Rankings Rank M4 M44 M46 G.G.65 Find the equation of a line, given a point on the line and the equation of a line parallel to the desired line Geometry: Expressing Geometric Properties with Equations Use coordinates to prove simple geometric theorems algebraically. 5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). M4 M44 M46 G.G.66 Find the midpoint of a line segment, given its endpoints G.G.67 Find the length of a line segment, given its endpoints Geometry: Expressing Geometric Properties with Equations Use coordinates to prove simple geometric theorems algebraically. 6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Geometry: Expressing Geometric Properties with Equations Use coordinates to prove simple geometric theorems algebraically. 7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. 2011 International Center for Leadership in Education M19 M19 Understand the properties of and apply parallel, perpendicular, and intersecting lines in problem-solving situations. Know the equation of a line and interpret graphically using the slope-intercept form (y = mx+b) and the point-slope form (y-b = m(x-a)). Know the equation for the slope of a line and compute slope given the coordinates of two points. Understand the properties of and apply parallel, perpendicular, and intersecting lines in problem-solving situations. Know the equation of a line and interpret graphically using the slope-intercept form (y = mx+b) and the point-slope form (y-b = m(x-a)). Know the equation for the slope of a line and compute slope given the coordinates of two points. Compute the distance between two points on a coordinate plane (length of a line segment) and determine the midpoint of a line segment between two points. Compute the distance between two points on a coordinate plane (length of a line segment) and determine the midpoint of a line segment between two points. Mathematics – Page 125 New York Mathematics Strands/Bands/ Performance Indicators Geometry G.G.68 Find the equation of a line that is the perpendicular bisector of a line segment, given the endpoints of the line segment Mathematics Domains/Clusters/ Common Core State Standards High School Geometry: Expressing Geometric Properties with Equations Use coordinates to prove simple geometric theorems algebraically. 5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). National Essential Skills Study (NESS) National Rankings Rank M4 M44 M46 G.G.69 Investigate, justify, and apply the properties of triangles and quadrilaterals in the coordinate plane, using the distance, midpoint, and slope formulas Geometry: Expressing Geometric Properties with Equations Use coordinates to prove simple geometric theorems algebraically. 7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. M19 M34 M46 G.G.70 Solve systems of equations involving one linear equation and one quadratic equation graphically G.G.71 Write the equation of a circle, given its center and radius or given the endpoints of a diameter Algebra: Reasoning with Equations & Inequalities Solve systems of equations. 7. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3. Geometry: Expressing Geometric Properties with Equations Translate between the geometric description and the equation for a conic section. 1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. 2011 International Center for Leadership in Education M40 M66 M66 Understand the properties of and apply parallel, perpendicular, and intersecting lines in problem-solving situations. Know the equation of a line and interpret graphically using the slope-intercept form (y = mx+b) and the point-slope form (y-b = m(x-a)). Know the equation for the slope of a line and compute slope given the coordinates of two points. Compute the distance between two points on a coordinate plane (length of a line segment) and determine the midpoint of a line segment between two points. Understand the properties and classification of polygons (triangles, the family of quadrilaterals, pentagon, hexagon, etc.) and apply knowledge of angle and side relationships of geometric shapes in problem-solving situations. Know the equation for the slope of a line and compute slope given the coordinates of two points. Solve systems of linear equations algebraically or graphically. Know how to sketch basic conic sections (e.g., circles, parabolas) by using their equations and solve systems of non-linear equations graphically. Know how to sketch basic conic sections (e.g., circles, parabolas) by using their equations and solve systems of non-linear equations graphically. Mathematics – Page 126 New York Mathematics Strands/Bands/ Performance Indicators Geometry G.G.72 Write the equation of a circle, given its graph. Note: The center is an ordered pair of integers and the radius is an integer. G.G.73 Find the center and radius of a circle, given the equation of the circle in center-radius form Mathematics Domains/Clusters/ Common Core State Standards High School Geometry: Expressing Geometric Properties with Equations Translate between the geometric description and the equation for a conic section. 1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Geometry: Expressing Geometric Properties with Equations Translate between the geometric description and the equation for a conic section. 1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. National Essential Skills Study (NESS) National Rankings Rank M66 M18 M66 G.G.74 Graph circles of the form ( x h) 2 ( j k ) 2 r 2 Geometry: Expressing Geometric Properties with Equations Translate between the geometric description and the equation for a conic section. 1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. 2011 International Center for Leadership in Education M66 Know how to sketch basic conic sections (e.g., circles, parabolas) by using their equations and solve systems of non-linear equations graphically. Understand the properties of circles (radius, arc, diameter, chord, secant, and tangent) and apply circle quantities (lengths of line segments, angle measure within a circle, circumference, and area) in problem-solving situations. Know how to sketch basic conic sections (e.g., circles, parabolas) by using their equations and solve systems of non-linear equations graphically. Know how to sketch basic conic sections (e.g., circles, parabolas) by using their equations and solve systems of non-linear equations graphically. Mathematics – Page 127 New York Mathematics Standards Alignments New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry Mathematics Domains/Clusters/ Common Core State Standards High School National Essential Skills Study (NESS) National Rankings Rank Problem Solving Strand Students will new mathematical knowledge through problem solving. A2.PS.1 Use a variety of problem solving strategies to understand new mathematical content Number & Quantity: Quantities Reason quantitatively and use units to solve problems. 1. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. 2. Define appropriate quantities for the purpose of descriptive modeling. 3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. Algebra: Creating Equations Create equations that describe numbers or relationships. 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Algebra: Reasoning with Equations & Inequalities Understand solving equations as a process of reasoning and explain the reasoning. 1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. 2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Solve equations and inequalities in one variable. 3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. 4. Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. 2011 International Center for Leadership in Education Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 128 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry A2.PS.1 (Continued from previous page) A2.PS.2 Recognize and understand equivalent representations of a problem situation or a mathematical concept Mathematics Domains/Clusters/ Common Core State Standards High School (Continued from previous page) b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Functions: Interpreting Functions Interpret functions that arise in applications in terms of the context. 5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Analyze functions using different representations. 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. 8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 129 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry Students will solve problems that arise in mathematics and in other contexts. A2.PS.3 Observe and explain patterns to formulate generalizations and conjectures A2.PS.4 Use multiple representations to represent and explain problem situations (e.g., verbally, numerically, algebraically, graphically) Mathematics Domains/Clusters/ Common Core State Standards High School Algebra: Seeing Structure in Expressions Write expressions in equivalent forms to solve problems. 4. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. Algebra: Creating Equations Create equations that describe numbers or relationships. 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. 2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Functions: Building Functions Build a function that models a relationship between two quantities. 1. Write a function that describes a relationship between two quantities. a. Determine an explicit expression, a recursive process, or steps for calculation from a context. 2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Functions: Interpreting Functions Interpret functions that arise in applications in terms of the context. 5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Analyze functions using different representations. 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Apply pattern recognition in data sets and series to reason or solve problems involving arithmetic, geometry, exponents, etc. M16 M10 M21 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Evaluate and employ accurate and appropriate procedures for statistical data collection, organization, analysis, and display including making estimates and predictions, critiquing data, and drawing inferences (e.g., using the normal curve and z-scores, line of best fit). Mathematics – Page 130 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry A2.PS.4 (Continued from previous page) Mathematics Domains/Clusters/ Common Core State Standards High School National Essential Skills Study (NESS) National Rankings Rank (Continued from previous page) c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. 8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Statistics & Probability: Interpreting Categorical & Quantitative Data Summarize, represent and interpret data on a single count or measurement variable. 1. Represent data with plots on the real number line (dot plots, histograms, and box plots). Summarize, represent and interpret data on two categorical and quantitative variables. 6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. 2011 International Center for Leadership in Education Mathematics – Page 131 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry Students will apply and adapt a variety of appropriate strategies to solve problems. A2.PS.5 Choose an effective approach to solve a problem from a variety of strategies (numeric, graphic, algebraic) Mathematics Domains/Clusters/ Common Core State Standards High School Functions: Interpreting Functions Interpret functions that arise in applications in terms of the context. 5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Analyze functions using different representations. 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. 8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Statistics & Probability: Interpreting Categorical & Quantitative Data Summarize, represent and interpret data on a single count or measurement variable. 1. Represent data with plots on the real number line (dot plots, histograms, and box plots). Summarize, represent and interpret data on two categorical and quantitative variables. 6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 132 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry A2.PS.6 Use a variety of strategies to extend solution methods to other problems Mathematics Domains/Clusters/ Common Core State Standards High School Functions: Interpreting Functions Interpret functions that arise in applications in terms of the context. 5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Analyze functions using different representations. 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. 8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Statistics & Probability: Interpreting Categorical & Quantitative Data Summarize, represent and interpret data on a single count or measurement variable. 1. Represent data with plots on the real number line (dot plots, histograms, and box plots). Summarize, represent and interpret data on two categorical and quantitative variables. 6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 133 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry A2.PS.7 Work in collaboration with others to propose, critique, evaluate, and value alternative approaches to problem solving Students will monitor and reflect on the process of mathematical problem solving. A2.PS.8 Determine information required to solve the problem, choose methods for obtaining the information, and define parameters for acceptable solutions A2.PS.9 Interpret solutions within the given constraints of a problem A2.PS.10 Evaluate the relative efficiency of different representations and solution methods of a problem Mathematics Domains/Clusters/ Common Core State Standards High School National Essential Skills Study (NESS) National Rankings Rank There is no New York Mathematics Performance Indicator–Common Core alignment. M10 There is no New York Mathematics Performance Indicator–Common Core alignment. M10 Algebra: Reasoning with Equations & Inequalities Understand solving equations as a process of reasoning and explain the reasoning. 2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Functions: Interpreting Functions Interpret functions that arise in applications in terms of the context. 5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Functions: Interpreting Functions Interpret functions that arise in applications in terms of the context. 5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Analyze functions using different representations. 8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. 2011 International Center for Leadership in Education Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 134 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry Mathematics Domains/Clusters/ Common Core State Standards High School National Essential Skills Study (NESS) National Rankings Rank Reasoning and Proof Strand Students will recognize reasoning and proof as fundamental aspects of mathematics. A2.RP.1 Support mathematical ideas using a variety of strategies Algebra: Reasoning with Equations & Inequalities Understand solving equations as a process of reasoning and explain the reasoning. 1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. 2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Functions: Trigonometric Functions Prove and apply trigonometric identities. 8. Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to calculate trigonometric ratios. 9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. Geometry: Similarity, Right Triangles, & Trigonometry Apply trigonometry to general triangles. 9. (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. 10. (+) Prove the Laws of Sines and Cosines and use them to solve problems. 2011 International Center for Leadership in Education Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 135 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry Students will make and investigate mathematical conjectures. A2.RP.2 Investigate and evaluate conjectures in mathematical terms, using mathematical strategies to reach a conclusion A2.RP.3 Evaluate conjectures and recognize when an estimate or approximation is more appropriate than an exact answer Mathematics Domains/Clusters/ Common Core State Standards High School Algebra: Reasoning with Equations & Inequalities Understand solving equations as a process of reasoning and explain the reasoning. 1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. 2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Functions: Trigonometric Functions Prove and apply trigonometric identities. 8. Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to calculate trigonometric ratios. 9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. Geometry: Similarity, Right Triangles, & Trigonometry Apply trigonometry to general triangles. 9. (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. 10. (+) Prove the Laws of Sines and Cosines and use them to solve problems. Number & Quantity: Quantities Reason quantitatively and use units to solve problems. 1. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. 2. Define appropriate quantities for the purpose of descriptive modeling. 3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. Statistics & Probability: Interpreting Categorical & Quantitative Data Summarize, represent and interpret data on a single count or measurement variable. 3. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). 4. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 M10 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Use the technique of dimensional analysis to convert units of measure (e.g., kilometers/hour to meters/minute) and apply ratios in real-world situations (e.g., scale drawings). M13 Mathematics – Page 136 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry Mathematics Domains/Clusters/ Common Core State Standards High School A2.RP.4 Recognize when an approximation is more appropriate than an exact answer Number & Quantity: Quantities Reason quantitatively and use units to solve problems. 1. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. 2. Define appropriate quantities for the purpose of descriptive modeling. 3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. Statistics & Probability: Interpreting Categorical & Quantitative Data Summarize, represent and interpret data on a single count or measurement variable. 3. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). 4. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. Students will develop and evaluate mathematical arguments and proofs. A2.RP.5 Develop, verify, and explain an argument, using appropriate mathematical ideas and language Algebra: Seeing Structure in Expressions Write expressions in equivalent forms to solve problems. 4. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. Functions: Trigonometric Functions Prove and apply trigonometric identities. 8. Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to calculate trigonometric ratios. 9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. Geometry: Similarity, Right Triangles, & Trigonometry Apply trigonometry to general triangles. 9. (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. 10. (+) Prove the Laws of Sines and Cosines and use them to solve problems. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank M10 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Use the technique of dimensional analysis to convert units of measure (e.g., kilometers/hour to meters/minute) and apply ratios in real-world situations (e.g., scale drawings). M13 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 137 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry A2.RP.5 (Continued from previous page) A2.RP.6 Construct logical arguments that verify claims or counterexamples that refute claims Mathematics Domains/Clusters/ Common Core State Standards High School (Continued from previous page) Geometry: Expressing Geometric Properties with Equations Translate between the geometric description and the equation for a conic section. 1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Algebra: Seeing Structure in Expressions Write expressions in equivalent forms to solve problems. 4. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. Functions: Trigonometric Functions Prove and apply trigonometric identities. 8. Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to calculate trigonometric ratios. 9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. Geometry: Similarity, Right Triangles, & Trigonometry Apply trigonometry to general triangles. 9. (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. 10. (+) Prove the Laws of Sines and Cosines and use them to solve problems. Geometry: Expressing Geometric Properties with Equations Translate between the geometric description and the equation for a conic section. 1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 138 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry A2.RP.7 Present correct mathematical arguments in a variety of forms A2.RP.8 Evaluate written arguments for validity Students will select and use various types of reasoning and methods of proof. A2.RP.9 Support an argument by using a systematic approach to test more than one case Mathematics Domains/Clusters/ Common Core State Standards High School Algebra: Seeing Structure in Expressions Write expressions in equivalent forms to solve problems. 4. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. Functions: Trigonometric Functions Prove and apply trigonometric identities. 8. Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to calculate trigonometric ratios. 9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. Geometry: Similarity, Right Triangles, & Trigonometry Apply trigonometry to general triangles. 9. (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. 10. (+) Prove the Laws of Sines and Cosines and use them to solve problems. Geometry: Expressing Geometric Properties with Equations Translate between the geometric description and the equation for a conic section. 1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. There is no New York Mathematics Performance Indicator–Common Core alignment. National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 M10 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. There is no New York Mathematics Performance Indicator–Common Core alignment. 2011 International Center for Leadership in Education Mathematics – Page 139 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry Mathematics Domains/Clusters/ Common Core State Standards High School A2.RP.10 Devise ways to verify results, using counterexamples and informal indirect proof There is no New York Mathematics Performance Indicator–Common Core alignment. A2.RP.11 Extend specific results to more general cases There is no New York Mathematics Performance Indicator–Common Core alignment. National Essential Skills Study (NESS) National Rankings Rank M10 M10 A2.RP.12 Apply inductive reasoning in making and supporting mathematical conjectures Algebra: Arithmetic with Polynomials & Rational Expressions Use polynomial identities to solve problems. 5. (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle. [The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.] M10 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Communication Strand Students will organize and consolidate their mathematical thinking through communication. A2.CM.1 Communicate verbally and in writing a correct, complete, coherent, and clear design (outline) and explanation for the steps used in solving a problem Algebra: Seeing Structure in Expressions Interpret the structure of expressions. 1. Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. Algebra: Reasoning with Equations & Inequalities Understand solving equations as a process of reasoning and explain the reasoning. 1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. 2011 International Center for Leadership in Education Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 140 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry Mathematics Domains/Clusters/ Common Core State Standards High School A2.CM.2 Use mathematical representations to communicate with appropriate accuracy, including numerical tables, formulas, functions, equations, charts, graphs, and diagrams Number & Quantity: Quantities Reason quantitatively and use units to solve problems. 1. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. 2. Define appropriate quantities for the purpose of descriptive modeling. 3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. Algebra: Seeing Structure in Expressions Write expressions in equivalent forms to solve problems. 3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. 4. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. Students will communicate their mathematical thinking coherently and clearly to peers, teachers, and others. A2.CM.3 Present organized mathematical ideas with the use of appropriate standard notations, including the use of symbols and other representations when sharing an idea in verbal and written form Algebra: Arithmetic with Polynomials & Rational Expressions Use polynomial identities to solve problems. 5. (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle. [The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.] Statistics & Probability: Interpreting Categorical & Quantitative Data Summarize, represent and interpret data on a single count or measurement variable. 2. Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. Interpret linear models. 8. Compute (using technology) and interpret the correlation coefficient of a linear fit. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank M7 M10 M21 Simplify and solve algebraic equations by identifying and using the correct order of operations and techniques necessary to carry out the solution. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Evaluate and employ accurate and appropriate procedures for statistical data collection, organization, analysis, and display including making estimates and predictions, critiquing data, and drawing inferences (e.g., using the normal curve and z-scores, line of best fit). Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 141 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry Mathematics Domains/Clusters/ Common Core State Standards High School A2.CM.4 Explain relationships among different representations of a problem Functions: Trigonometric Functions Extend the domain of trigonometric functions using the unit circle. 2. Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. Algebra: Reasoning with Equations & Inequalities Understand solving equations as a process of reasoning and explain the reasoning. 1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Functions: Trigonometric Functions Extend the domain of trigonometric functions using the unit circle. 2. Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. There is no New York Mathematics Performance Indicator–Common Core alignment. A2.CM.5 Communicate logical arguments clearly, showing why a result makes sense and why the reasoning is valid A2.CM.6 Support or reject arguments or questions raised by others about the correctness of mathematical work Students will analyze and evaluate the mathematical thinking and strategies of others. A2.CM.7 Read and listen for logical understanding of mathematical thinking shared by other students A2.CM.8 Reflect on strategies of others in relation to one’s own strategy National Essential Skills Study (NESS) National Rankings Rank M10 M10 M10 There is no New York Mathematics Performance Indicator–Common Core alignment. M10 There is no New York Mathematics Performance Indicator–Common Core alignment. M10 2011 International Center for Leadership in Education Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Mathematics – Page 142 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry A2.CM.9 Formulate mathematical questions that elicit, extend, or challenge strategies, solutions, and/or conjectures of others Students will use the language of mathematics to express mathematical ideas precisely. A2.CM.10 Use correct mathematical language in developing mathematical questions that elicit, extend, or challenge other students’ conjectures A2.CM.11 Represent word problems using standard mathematical notation A2.CM.12 Understand and use appropriate language, representations, and terminology when describing objects, relationships, mathematical solutions, and rationale Mathematics Domains/Clusters/ Common Core State Standards High School National Essential Skills Study (NESS) National Rankings Rank There is no New York Mathematics Performance Indicator–Common Core alignment. M10 There is no New York Mathematics Performance Indicator–Common Core alignment. M10 Algebra: Creating Equations Create equations that describe numbers or relationships. 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Algebra: Creating Equations Create equations that describe numbers or relationships. 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Algebra: Reasoning with Equations & Inequalities Understand solving equations as a process of reasoning and explain the reasoning. 1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. 2011 International Center for Leadership in Education M10 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 143 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry A2.CM.13 Draw conclusions about mathematical ideas through decoding, comprehension, and interpretation of mathematical visuals, symbols, and technical writing Mathematics Domains/Clusters/ Common Core State Standards High School Functions: Linear, Quadratic & Exponential Construct and compare linear, quadratic, and exponential models and solve problems. 1. Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. 2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). 3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Connections Strand Students will recognize and use connections among mathematical ideas. A2.CN.1 Understand and make connections among multiple representations of the same mathematical idea Algebra: Reasoning with Equations & Inequalities Solve equations and inequalities in one variable. 4. Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. Functions: Interpreting Functions Analyze functions using different representations. 8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. 2011 International Center for Leadership in Education Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 144 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry A2.CN.2 Understand the corresponding procedures for similar problems or mathematical concepts Students will understand how mathematical ideas interconnect and build on one another to produce a coherent whole. A2.CN.3 Model situations mathematically, using representations to draw conclusions and formulate new situations Mathematics Domains/Clusters/ Common Core State Standards High School Algebra: Reasoning with Equations & Inequalities Solve equations and inequalities in one variable. 4. Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. Functions: Interpreting Functions Analyze functions using different representations. 8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Functions: Building Functions Build a function that models a relationship between two quantities. 1. Write a function that describes a relationship between two quantities. a. Determine an explicit expression, a recursive process, or steps for calculation from a context. b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. c. (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time. 2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 145 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry A2.CN.4 Understand how concepts, procedures, and mathematical results in one area of mathematics can be used to solve problems in other areas of mathematics A2.CN.5 Understand how quantitative models connect to various physical models and representations Students will recognize and apply mathematics in contexts outside of mathematics. A2.CN.6 Recognize and apply mathematics to situations in the outside world Mathematics Domains/Clusters/ Common Core State Standards High School Algebra: Arithmetic with Polynomials & Rational Expressions Understand the relationship between zeros and factors of polynomials. 3. Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. Geometry: Modeling with Geometry Apply Geometric Concepts in modeling situations 1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Geometry: Modeling with Geometry Apply Geometric Concepts in modeling situations 2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). A2.CN.7 Recognize and apply mathematical ideas to problem situations that develop outside of mathematics Geometry: Modeling with Geometry Apply Geometric Concepts in modeling situations 1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). A2.CN.8 Develop an appreciation for the historical development of mathematics There is no New York Mathematics Performance Indicator–Common Core alignment. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank M10 M10 M10 M10 M10 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Mathematics – Page 146 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry Mathematics Domains/Clusters/ Common Core State Standards High School National Essential Skills Study (NESS) National Rankings Rank Representation Strand Students will create and use representations to organize, record, and communicate mathematical ideas. A2.R.1 Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts Algebra: Seeing Structure in Expressions Write expressions in equivalent forms to solve problems. 3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. a. Factor a quadratic expression to reveal the zeros of the function it defines. b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. Algebra: Creating Equations Create equations that describe numbers or relationships. 2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Algebra: Reasoning with Equations & Inequalities Represent and solve equations and inequalities graphically. 10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Functions: Interpreting Functions Understand the concept of a function and use function notation. 3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1. Analyze functions using different representations. 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. 2011 International Center for Leadership in Education M7 M10 Simplify and solve algebraic equations by identifying and using the correct order of operations and techniques necessary to carry out the solution. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Evaluate and employ accurate and appropriate procedures for statistical data collection, organization, analysis, and display including making estimates and predictions, critiquing data, and drawing inferences (e.g., using the normal curve and z-scores, line of best fit). M21 Mathematics – Page 147 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry Mathematics Domains/Clusters/ Common Core State Standards High School A2.R.1 (Continued from previous page) (Continued from previous page) e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Statistics & Probability: Interpreting Categorical & Quantitative Data Summarize, represent and interpret data on a single count or measurement variable. 1. Represent data with plots on the real number line (dot plots, histograms, and box plots). Summarize, represent and interpret data on two categorical and quantitative variables. 6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. Algebra: Seeing Structure in Expressions Write expressions in equivalent forms to solve problems. 3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. a. Factor a quadratic expression to reveal the zeros of the function it defines. b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. Algebra: Creating Equations Create equations that describe numbers or relationships. 2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Algebra: Reasoning with Equations & Inequalities Represent and solve equations and inequalities graphically. 10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Functions: Interpreting Functions Understand the concept of a function and use function notation. 3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1. A2.R.2 Recognize, compare, and use an array of representational forms 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 148 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry Mathematics Domains/Clusters/ Common Core State Standards High School A2.R.2 (Continued from previous page) (Continued from previous page) Analyze functions using different representations. 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Statistics & Probability: Interpreting Categorical & Quantitative Data Summarize, represent and interpret data on a single count or measurement variable. 1. Represent data with plots on the real number line (dot plots, histograms, and box plots). Summarize, represent and interpret data on two categorical and quantitative variables. 6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. Functions: Interpreting Functions Understand the concept of a function and use function notation. 3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1. Analyze functions using different representations. 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. A2.R.3 Use representation as a tool for exploring and understanding mathematical ideas 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 149 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry Mathematics Domains/Clusters/ Common Core State Standards High School A2.R.3 (Continued from previous page) (Continued from previous page) e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Statistics & Probability: Interpreting Categorical & Quantitative Data Summarize, represent and interpret data on a single count or measurement variable. 1. Represent data with plots on the real number line (dot plots, histograms, and box plots). Summarize, represent and interpret data on two categorical and quantitative variables. 6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. Students will select, apply, and translate among mathematical representations to solve problems. A2.R.4 Select appropriate representations to solve problem situations Algebra: Seeing Structure in Expressions Write expressions in equivalent forms to solve problems. 3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. a. Factor a quadratic expression to reveal the zeros of the function it defines. b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. Statistics & Probability: Interpreting Categorical & Quantitative Data Summarize, represent and interpret data on a single count or measurement variable. 1. Represent data with plots on the real number line (dot plots, histograms, and box plots). Summarize, represent and interpret data on two categorical and quantitative variables. 6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. c. Fit a linear function for a scatter plot that suggests a linear association. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 Mathematics – Page 150 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry Mathematics Domains/Clusters/ Common Core State Standards High School A2.R.5 Investigate relationships among different representations and their impact on a given problem Algebra: Seeing Structure in Expressions Write expressions in equivalent forms to solve problems. 3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. a. Factor a quadratic expression to reveal the zeros of the function it defines. b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. Statistics & Probability: Interpreting Categorical & Quantitative Data Summarize, represent and interpret data on a single count or measurement variable. 1. Represent data with plots on the real number line (dot plots, histograms, and box plots). Summarize, represent and interpret data on two categorical and quantitative variables. 6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. c. Fit a linear function for a scatter plot that suggests a linear association. Students will use representations to model and interpret physical, social, and mathematical phenomena. A2.R.6 Use mathematics to show and understand physical phenomena (e.g., investigate sound waves using the sine and cosine functions) Functions: Trigonometric Functions Model periodic phenomena with trigonometric functions. 5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. M10 M10 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Mathematics – Page 151 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry A2.R.7 Use mathematics to show and understand social phenomena (e.g., interpret the results of an opinion poll) A2.R.8 Use mathematics to show and understand mathematical phenomena (e.g., use random number generator to simulate a coin toss) Mathematics Domains/Clusters/ Common Core State Standards High School Statistics & Probability: Making Inferences & Justifying Conclusions Understand and evaluate random processes underlying statistical experiments. 1. Understand statistics as a process for making inferences about population parameters based on a random sample from that population. Make inferences and justify conclusions from sample surveys, experiments and observational studies. 4. Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling. Statistics & Probability: Making Inferences & Justifying Conclusions Make inferences and justify conclusions from sample surveys, experiments and observational studies. 3. Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. 5. Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. National Essential Skills Study (NESS) National Rankings Rank M10 M10 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Number Sense and Operations Strand Students will understand meanings of operations and procedures, and how they relate to one another. Operations A2.N.1 Evaluate numerical expressions with negative and/or fractional exponents, without the aid of a calculator (when the answers are rational numbers) Number & Quantity: The Real Number System Extend the properties of exponents to rational exponents. 1. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5. 2. Rewrite expressions involving radicals and rational exponents using the properties of exponents. 2011 International Center for Leadership in Education M1 M20 Perform operations fluently with positive and negative numbers, including decimals, ratios, percents, and fractions, and show reasoning to justify results. Understand and apply the basic properties and laws of exponents and scientific notation to solve problems, including those with fractional, negative, and zero exponents. Mathematics – Page 152 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry A2.N.2 Perform arithmetic operations (addition, subtraction, multiplication, division) with expressions containing irrational numbers in radical form A2.N.3 Perform arithmetic operations with polynomial expressions containing rational coefficients Mathematics Domains/Clusters/ Common Core State Standards High School Number & Quantity: The Real Number System Use properties of rational and irrational numbers. 3. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Algebra: Arithmetic with Polynomials & Rational Expressions Perform arithmetic operations on polynomials. 1. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. National Essential Skills Study (NESS) National Rankings Rank Perform operations with radicals, such as addition, subtraction, and multiplication. M33 M36 M62 A2.N.4 Perform arithmetic operations on irrational expressions A2.N.5 Rationalize a denominator containing a radical expression A2.N.6 Write square roots of negative numbers in terms of i A2.N.7 Simplify powers of i Number & Quantity: The Real Number System Use properties of rational and irrational numbers. 3. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Number & Quantity: The Real Number System Extend the properties of exponents to rational exponents. 2. Rewrite expressions involving radicals and rational exponents using the properties of exponents. Number & Quantity: The Complex Number System Perform arithmetic operations with complex numbers. 1. Know there is a complex number i such that i2 = –1, and every complex number has the form a + bi with a and b real. Number & Quantity: The Complex Number System Perform arithmetic operations with complex numbers. 1. Know there is a complex number i such that i2 = –1, and every complex number has the form a + bi with a and b real. 2011 International Center for Leadership in Education M33 M39 M33 M25 M25 Simplify polynomials by performing operations (addition, subtraction, multiplication, and division) to simplify expressions (e.g., (2a + 2) + (3a - 1) = 5a + 1). Perform division of a polynomial by a monomial by dividing powers with like bases, using the rules for the division of powers with like bases to simplify fractions with monomial denominators and reducing fractions to lowest terms. Perform operations with radicals, such as addition, subtraction, and multiplication. Apply techniques to obtain a rational approximation or estimate of a quantity or number (including irrational numbers such as radicals). Perform operations with radicals, such as addition, subtraction, and multiplication. Perform operations and solve equations containing complex numbers. Perform operations and solve equations containing complex numbers. Mathematics – Page 153 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry A2.N.8 Determine the conjugate of a complex number A2.N.9 Perform arithmetic operations on complex numbers and write the answer in the form a bi Note: This includes simplifying expressions with complex denominators. A2.N.10 Know and apply sigma notation Mathematics Domains/Clusters/ Common Core State Standards High School Number & Quantity: The Complex Number System Perform arithmetic operations with complex numbers. 3. (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. Number & Quantity: The Complex Number System Perform arithmetic operations with complex numbers. 2. Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Functions: Building Functions Build new functions from existing functions. 3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. National Essential Skills Study (NESS) National Rankings Rank M25 Perform operations and solve equations containing complex numbers. Perform operations and solve equations containing complex numbers. M25 Apply summation notation to take the sum of an expression using limits (e.g., take the sum of 3i + 1 from i = 1 to 5). M69 Algebra Strand Students will represent and analyze algebraically a wide variety of problem solving situations. Equations and Inequalities A2.A.1 Solve absolute value equations and inequalities involving linear expressions in one variable Algebra: Reasoning with Equations & Inequalities Solve equations and inequalities in one variable. 3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Functions: Interpreting Functions Analyze functions using different representations. 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. 2011 International Center for Leadership in Education M7 M27 Simplify and solve algebraic equations by identifying and using the correct order of operations and techniques necessary to carry out the solution. Find the solution of linear equations and inequalities where the variable appears on either or both sides and in which one or both sides must be simplified before solving the equation (e.g., solve x + 2(x - 3) = -4x + 5 for x). Mathematics – Page 154 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry A2.A.2 Use the discriminant to determine the nature of the roots of a quadratic equation A2.A.3 Solve systems of equations involving one linear equation and one quadratic equation algebraically Note: This includes rational equations that result in linear equations with extraneous roots Mathematics Domains/Clusters/ Common Core State Standards High School Algebra: Reasoning with Equations & Inequalities Solve equations and inequalities in one variable. 4. Solve quadratic equations in one variable. b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Algebra: Reasoning with Equations & Inequalities Solve systems of equations. 7. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3. National Essential Skills Study (NESS) National Rankings Rank Solve quadratic equations by applying various tools or techniques. M47 M40 M47 M66 A2.A.4 Solve quadratic inequalities in one and two variables, algebraically and graphically A2.A.5 Use direct and inverse variation to solve for unknown values Algebra: Reasoning with Equations & Inequalities Solve equations and inequalities in one variable. 3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Represent and solve equations and inequalities graphically. 12. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. There is no New York Mathematics Performance Indicator–Common Core alignment. 2011 International Center for Leadership in Education M47 Solve systems of linear equations algebraically or graphically. Solve quadratic equations by applying various tools or techniques. Know how to sketch basic conic sections (e.g., circles, parabolas) by using their equations and solve systems of non-linear equations graphically. Solve quadratic equations by applying various tools or techniques. Express, graph, and interpret polynomial functions (linear, quadratic, cubic, etc.). M53 M57 Solve and graphically sketch problems involving two variables that exhibit direct and indirect variation. Mathematics – Page 155 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry A2.A.6 Solve an application which results in an exponential function A2.A.6 (Continued from previous page) Mathematics Domains/Clusters/ Common Core State Standards High School Algebra: Seeing Structure in Expressions Write expressions in equivalent forms to solve problems. 3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. 4. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. Functions: Interpreting Functions Analyze functions using different representations. 8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay. (Continued from previous page) Functions: Linear, Quadratic & Exponential Construct and compare linear, quadratic, and exponential models and solve problems. 2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). 4. For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Express, graph, and interpret exponential and logarithmic functions. M48 Mathematics – Page 156 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry Mathematics Domains/Clusters/ Common Core State Standards High School National Essential Skills Study (NESS) National Rankings Rank Students will perform algebraic procedures accurately. Variables and Expressions A2.A.7 Factor polynomial expressions completely, using any combination of the following techniques: common factor extraction, difference of two perfect squares, quadratic trinomials A2.A.8 Apply the rules of exponents to simplify expressions involving negative and/or fractional exponents A2.A.9 Rewrite algebraic expressions that contain negative exponents using only positive exponents A2.A.10 Rewrite algebraic expressions with fractional exponents as radical expressions Algebra: Seeing Structure in Expressions Interpret the structure of expressions. 2. Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2). Write expressions in equivalent forms to solve problems. 3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. a. Factor a quadratic expression to reveal the zeros of the function it defines. Number & Quantity: The Real Number System Extend the properties of exponents to rational exponents. 1. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5. 2. Rewrite expressions involving radicals and rational exponents using the properties of exponents. Number & Quantity: The Real Number System Extend the properties of exponents to rational exponents. 2. Rewrite expressions involving radicals and rational exponents using the properties of exponents. Number & Quantity: The Real Number System Extend the properties of exponents to rational exponents. 1. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5. 2. Rewrite expressions involving radicals and rational exponents using the properties of exponents. 2011 International Center for Leadership in Education M47 Simplify polynomials by performing operations (addition, subtraction, multiplication, and division) to simplify expressions (e.g., (2a + 2) + (3a - 1) = 5a + 1). Solve quadratic equations by applying various tools or techniques. M20 Understand and apply the basic properties and laws of exponents and scientific notation to solve problems, including those with fractional, negative, and zero exponents. M36 M20 M20 M33 Understand and apply the basic properties and laws of exponents and scientific notation to solve problems, including those with fractional, negative, and zero exponents. Understand and apply the basic properties and laws of exponents and scientific notation to solve problems, including those with fractional, negative, and zero exponents. Perform operations with radicals, such as addition, subtraction, and multiplication. Mathematics – Page 157 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry Mathematics Domains/Clusters/ Common Core State Standards High School A2.A.11 Rewrite algebraic expressions in radical form as expressions with fractional exponents Number & Quantity: The Real Number System Extend the properties of exponents to rational exponents. 1. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5. 2. Rewrite expressions involving radicals and rational exponents using the properties of exponents. Functions: Linear, Quadratic & Exponential Construct and compare linear, quadratic, and exponential models and solve problems. 2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). A2.A.12 Evaluate exponential expressions, including those with base e National Essential Skills Study (NESS) National Rankings Rank M20 M33 M11 M48 A2.A.13 Simplify radical expressions A2.A.14 Perform addition, subtraction, multiplication, and division of radical expressions A2.A.15 Rationalize denominators involving algebraic radical expressions Number & Quantity: The Real Number System Extend the properties of exponents to rational exponents. 2. Rewrite expressions involving radicals and rational exponents using the properties of exponents. Number & Quantity: The Real Number System Use properties of rational and irrational numbers. 3. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Number & Quantity: The Real Number System Extend the properties of exponents to rational exponents. 2. Rewrite expressions involving radicals and rational exponents using the properties of exponents. 2011 International Center for Leadership in Education M33 Understand and apply the basic properties and laws of exponents and scientific notation to solve problems, including those with fractional, negative, and zero exponents. Perform operations with radicals, such as addition, subtraction, and multiplication. Apply variables in expressions and equations to solve problems (i.e., write mathematical equations for given situation, create a mathematical model to understand the relationships between variables, or make connections between the structures of mathematically abstract concepts and the real world). Express, graph, and interpret exponential and logarithmic functions. Perform operations with radicals, such as addition, subtraction, and multiplication. Perform operations with radicals, such as addition, subtraction, and multiplication. M33 M33 Perform operations with radicals, such as addition, subtraction, and multiplication. Mathematics – Page 158 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry A2.A.16 Perform arithmetic operations with rational expressions and rename to lowest terms Mathematics Domains/Clusters/ Common Core State Standards High School Algebra: Arithmetic with Polynomials & Rational Expressions Rewrite rational expressions. 7. (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. National Essential Skills Study (NESS) National Rankings Rank M1 M36 A2.A.17 Simplify complex fractional expressions Algebra: Arithmetic with Polynomials & Rational Expressions Rewrite rational expressions. 7. (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. M1 M33 M36 A2.A.18 Evaluate logarithmic expressions in any base Functions: Linear, Quadratic & Exponential Construct and compare linear, quadratic, and exponential models and solve problems. 4. For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. M11 M48 2011 International Center for Leadership in Education Perform operations fluently with positive and negative numbers, including decimals, ratios, percents, and fractions, and show reasoning to justify results. Simplify polynomials by performing operations (addition, subtraction, multiplication, and division) to simplify expressions (e.g., (2a + 2) + (3a - 1) = 5a + 1). Perform operations fluently with positive and negative numbers, including decimals, ratios, percents, and fractions, and show reasoning to justify results. Perform operations with radicals, such as addition, subtraction, and multiplication. Simplify polynomials by performing operations (addition, subtraction, multiplication, and division) to simplify expressions (e.g., (2a + 2) + (3a - 1) = 5a + 1). Apply variables in expressions and equations to solve problems (i.e., write mathematical equations for given situation, create a mathematical model to understand the relationships between variables, or make connections between the structures of mathematically abstract concepts and the real world). Express, graph, and interpret exponential and logarithmic functions. Mathematics – Page 159 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry A2.A.19 Apply the properties of logarithms to rewrite logarithmic expressions in equivalent forms Mathematics Domains/Clusters/ Common Core State Standards High School Functions: Linear, Quadratic & Exponential Construct and compare linear, quadratic, and exponential models and solve problems. 4. For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. National Essential Skills Study (NESS) National Rankings Rank M11 M48 Apply variables in expressions and equations to solve problems (i.e., write mathematical equations for given situation, create a mathematical model to understand the relationships between variables, or make connections between the structures of mathematically abstract concepts and the real world). Express, graph, and interpret exponential and logarithmic functions. Equations and Inequalities A2.A.20 Determine the sum and product of the roots of a quadratic equation by examining its coefficients A2.A.21 Determine the quadratic equation, given the sum and product of its roots A2.A.22 Solve radical equations A2.A.23 Solve rational equations and inequalities There is no New York Mathematics Performance Indicator–Common Core alignment. M47 Solve quadratic equations by applying various tools or techniques. There is no New York Mathematics Performance Indicator–Common Core alignment. M47 Solve quadratic equations by applying various tools or techniques. Algebra: Reasoning with Equations & Inequalities Understand solving equations as a process of reasoning and explain the reasoning. 2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Algebra: Reasoning with Equations & Inequalities Understand solving equations as a process of reasoning and explain the reasoning. 2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. M7 M33 M7 M27 M33 2011 International Center for Leadership in Education Simplify and solve algebraic equations by identifying and using the correct order of operations and techniques necessary to carry out the solution. Perform operations with radicals, such as addition, subtraction, and multiplication. Simplify and solve algebraic equations by identifying and using the correct order of operations and techniques necessary to carry out the solution. Find the solution of linear equations and inequalities where the variable appears on either or both sides and in which one or both sides must be simplified before solving the equation (e.g., solve x + 2(x - 3) = -4x + 5 for x). Perform operations with radicals, such as addition, subtraction, and multiplication. Mathematics – Page 160 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry Mathematics Domains/Clusters/ Common Core State Standards High School A2.A.24 Know and apply the technique of completing the square Algebra: Seeing Structure in Expressions Write expressions in equivalent forms to solve problems. 3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. Algebra: Reasoning with Equations & Inequalities Solve equations and inequalities in one variable. 4. Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. Functions: Interpreting Functions Analyze functions using different representations. 8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Algebra: Reasoning with Equations & Inequalities Solve equations and inequalities in one variable. 4. Solve quadratic equations in one variable. b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Functions: Interpreting Functions Analyze functions using different representations. 8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. A2.A.25 Solve quadratic equations, using the quadratic formula 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank Solve quadratic equations by applying various tools or techniques. M47 Solve quadratic equations by applying various tools or techniques. M47 Mathematics – Page 161 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry A2.A.26 Find the solution to polynomial equations of higher degree that can be solved using factoring and/or the quadratic formula A2.A.27 Solve exponential equations with and without common bases Mathematics Domains/Clusters/ Common Core State Standards High School Algebra: Arithmetic with Polynomials & Rational Expressions Understand the relationship between zeros and factors of polynomials. 3. Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. Algebra: Reasoning with Equations & Inequalities Solve equations and inequalities in one variable. 4. Solve quadratic equations in one variable. b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Algebra: Creating Equations Create equations that describe numbers or relationships. 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Functions: Building Functions Build new functions from existing functions. 5. (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. Functions: Linear, Quadratic & Exponential Construct and compare linear, quadratic, and exponential models and solve problems. 4. For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. Interpret expressions for functions in terms of the situation they model. 5. Interpret the parameters in a linear or exponential function in terms of a context. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank M47 Solve quadratic equations by applying various tools or techniques. Express, graph, and interpret polynomial functions (linear, quadratic, cubic, etc.). M53 M7 Simplify and solve algebraic equations by identifying and using the correct order of operations and techniques necessary to carry out the solution. Express, graph, and interpret exponential and logarithmic functions. M48 Mathematics – Page 162 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry Mathematics Domains/Clusters/ Common Core State Standards High School A2.A.28 Solve a logarithmic equation by rewriting as an exponential equation Algebra: Creating Equations Create equations that describe numbers or relationships. 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Functions: Building Functions Build new functions from existing functions. 5. (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. Functions: Linear, Quadratic & Exponential Construct and compare linear, quadratic, and exponential models and solve problems. 4. For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. National Essential Skills Study (NESS) National Rankings Rank M7 Simplify and solve algebraic equations by identifying and using the correct order of operations and techniques necessary to carry out the solution. Express, graph, and interpret exponential and logarithmic functions. M48 Students will recognize, use, and represent algebraically patterns, relations, and functions. Patterns, Relations, and Functions A2.A.29 Identify an arithmetic or geometric sequence and find the formula for its nth term A2.A.30 Determine the common difference in an arithmetic sequence Algebra: Seeing Structure in Expressions Write expressions in equivalent forms to solve problems. 4. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. Functions: Building Functions Build a function that models a relationship between two quantities. 2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. 2011 International Center for Leadership in Education M16 M60 M16 Apply pattern recognition in data sets and series to reason or solve problems involving arithmetic, geometry, exponents, etc. Evaluate and use finite sequence and series as systematic and useful means of quantifying things. Apply pattern recognition in data sets and series to reason or solve problems involving arithmetic, geometry, exponents, etc. Mathematics – Page 163 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry A2.A.31 Determine the common ratio in a geometric sequence A2.A.32 Determine a specified term of an arithmetic or geometric sequence A2.A.33 Specify terms of a sequence, given its recursive definition A2.A.34 Represent the sum of a series, using sigma notation Mathematics Domains/Clusters/ Common Core State Standards High School Algebra: Seeing Structure in Expressions Write expressions in equivalent forms to solve problems. 4. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. Functions: Building Functions Build a function that models a relationship between two quantities. 2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Algebra: Seeing Structure in Expressions Write expressions in equivalent forms to solve problems. 4. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. Functions: Building Functions Build a function that models a relationship between two quantities. 2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Functions: Interpreting Functions Understand the concept of a function and use function notation. 3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1. Functions: Building Functions Build a function that models a relationship between two quantities. 2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Functions: Building Functions Build a function that models a relationship between two quantities. 2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank M3 Use proportional reasoning to solve realworld problems. Apply pattern recognition in data sets and series to reason or solve problems involving arithmetic, geometry, exponents, etc. M16 M16 Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Apply pattern recognition in data sets and series to reason or solve problems involving arithmetic, geometry, exponents, etc. M24 Understand the concepts of recurrence relations and apply them to solve consumer mathematics problems involving such things as percentage rates, personal loans, simple interest, compound interest, installment buying, mortgage rates, etc. M69 Apply summation notation to take the sum of an expression using limits (e.g., take the sum of 3i + 1 from i = 1 to 5). M10 Mathematics – Page 164 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry Mathematics Domains/Clusters/ Common Core State Standards High School A2.A.35 Determine the sum of the first n terms of an arithmetic or geometric series Functions: Building Functions Build a function that models a relationship between two quantities. 2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Algebra: Arithmetic with Polynomials & Rational Expressions Use polynomial identities to solve problems. 5. (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle. [The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.] Functions: Interpreting Functions Understand the concept of a function and use function notation. 1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). Functions: Interpreting Functions Understand the concept of a function and use function notation. 1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). Functions: Interpreting Functions Understand the concept of a function and use function notation. 1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). 2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. A2.A.36 Apply the binomial theorem to expand a binomial and determine a specific term of a binomial expansion A2.A.37 Define a relation and function A2.A.38 Determine when a relation is a function A2.A.39 Determine the domain and range of a function from its equation 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank M69 Apply summation notation to take the sum of an expression using limits (e.g., take the sum of 3i + 1 from i = 1 to 5). M70 Understand and apply the binomial theorem (e.g., explore the relationship of the binomial theorem with Pascal’s triangle and the Fibonacci sequence). M37 M37 M37 Define and apply the properties of relations and functions (domain, range, function composition, and inverses) and use algebraic and graphic methods to determine if a relation is a function. Define and apply the properties of relations and functions (domain, range, function composition, and inverses) and use algebraic and graphic methods to determine if a relation is a function. Define and apply the properties of relations and functions (domain, range, function composition, and inverses) and use algebraic and graphic methods to determine if a relation is a function. Mathematics – Page 165 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry A2.A.40 Write functions in functional notation Mathematics Domains/Clusters/ Common Core State Standards High School Functions: Interpreting Functions Understand the concept of a function and use function notation. 1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). National Essential Skills Study (NESS) National Rankings Rank M37 M53 M56 A2.A.41 Use functional notation to evaluate functions for given values in the domain Functions: Interpreting Functions Understand the concept of a function and use function notation. 2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. M37 M53 M56 A2.A.42 Find the composition of functions A2.A.43 Determine if a function is oneto-one, onto, or both Functions: Building Functions Build a function that models a relationship between two quantities. 1. Write a function that describes a relationship between two quantities. c. (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time. There is no New York Mathematics Performance Indicator–Common Core alignment. M37 M37 2011 International Center for Leadership in Education Define and apply the properties of relations and functions (domain, range, function composition, and inverses) and use algebraic and graphic methods to determine if a relation is a function. Express, graph, and interpret polynomial functions (linear, quadratic, cubic, etc.). Express a linear function (f(x) = mx + b) with the appropriate notation and determine the ordered pairs. Define and apply the properties of relations and functions (domain, range, function composition, and inverses) and use algebraic and graphic methods to determine if a relation is a function. Express, graph, and interpret polynomial functions (linear, quadratic, cubic, etc.). Express a linear function (f(x) = mx + b) with the appropriate notation and determine the ordered pairs. Define and apply the properties of relations and functions (domain, range, function composition, and inverses) and use algebraic and graphic methods to determine if a relation is a function. Define and apply the properties of relations and functions (domain, range, function composition, and inverses) and use algebraic and graphic methods to determine if a relation is a function. Mathematics – Page 166 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry A2.A.44 Define the inverse of a function A2.A.45 Determine the inverse of a function and use composition to justify the result A2.A.46 Perform transformations with functions and relations: f ( x a) , f ( x) a , f ( x) , f (x) , af (x) Mathematics Domains/Clusters/ Common Core State Standards High School Functions: Building Functions Build new functions from existing functions. 4. Find inverse functions. a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x ≠ 1. Functions: Building Functions Build new functions from existing functions. 4. Find inverse functions. a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x ≠ 1. b. (+) Verify by composition that one function is the inverse of another. Functions: Building Functions Build new functions from existing functions. 3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. National Essential Skills Study (NESS) National Rankings Rank M37 M37 M54 Define and apply the properties of relations and functions (domain, range, function composition, and inverses) and use algebraic and graphic methods to determine if a relation is a function. Define and apply the properties of relations and functions (domain, range, function composition, and inverses) and use algebraic and graphic methods to determine if a relation is a function. Apply transformations (reflection, rotation, translation, and dilation) of 2-dimensional figures graphically to interpret, analyze, and illustrate the concepts of congruency, similarity, and symmetry. Coordinate Geometry A2.A.47 Determine the center-radius form for the equation of a circle in standard form Geometry: Expressing Geometric Properties with Equations Translate between the geometric description and the equation for a conic section. 1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. M18 M66 2011 International Center for Leadership in Education Understand the properties of circles (radius, arc, diameter, chord, secant, and tangent) and apply circle quantities (lengths of line segments, angle measure within a circle, circumference, and area) in problem-solving situations. Know how to sketch basic conic sections (e.g., circles, parabolas) by using their equations and solve systems of non-linear equations graphically. Mathematics – Page 167 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry A2.A.48 Write the equation of a circle, given its center and a point on the circle Mathematics Domains/Clusters/ Common Core State Standards High School Geometry: Expressing Geometric Properties with Equations Translate between the geometric description and the equation for a conic section. 1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. National Essential Skills Study (NESS) National Rankings Rank M18 M66 A2.A.49 Write the equation of a circle from its graph A2.A.50 Approximate the solution to polynomial equations of higher degree by inspecting the graph A2.A.51 Determine the domain and range of a function from its graph A2.A.52 Identify relations and functions, using graphs Geometry: Expressing Geometric Properties with Equations Translate between the geometric description and the equation for a conic section. 1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Functions: Interpreting Functions Analyze functions using different representations. 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Functions: Interpreting Functions Interpret functions that arise in applications in terms of the context. 5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Functions: Interpreting Functions Understand the concept of a function and use function notation. 1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). 2011 International Center for Leadership in Education M66 Understand the properties of circles (radius, arc, diameter, chord, secant, and tangent) and apply circle quantities (lengths of line segments, angle measure within a circle, circumference, and area) in problem-solving situations. Know how to sketch basic conic sections (e.g., circles, parabolas) by using their equations and solve systems of non-linear equations graphically. Know how to sketch basic conic sections (e.g., circles, parabolas) by using their equations and solve systems of non-linear equations graphically. Express, graph, and interpret polynomial functions (linear, quadratic, cubic, etc.). M53 M37 M37 Define and apply the properties of relations and functions (domain, range, function composition, and inverses) and use algebraic and graphic methods to determine if a relation is a function. Define and apply the properties of relations and functions (domain, range, function composition, and inverses) and use algebraic and graphic methods to determine if a relation is a function. Mathematics – Page 168 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry A2.A.53 Graph exponential functions of the form y b x for positive values of b, including b e A2.A.54 Graph logarithmic functions, using the inverse of the related exponential function Mathematics Domains/Clusters/ Common Core State Standards High School Functions: Interpreting Functions Analyze functions using different representations. 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Functions: Interpreting Functions Analyze functions using different representations. 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. National Essential Skills Study (NESS) National Rankings Rank Express, graph, and interpret exponential and logarithmic functions. M48 Express, graph, and interpret exponential and logarithmic functions. M48 Trigonometric Functions A2.A.55 Express and apply the six trigonometric functions as ratios of the sides of a right triangle A2.A.56 Know the exact and approximate values of the sine, cosine, and tangent of 0º, 30º, 45º, 60º, 90º, 180º, and 270º angles A2.A.57 Sketch and use the reference angle for angles in standard position A2.A.58 Know and apply the cofunction and reciprocal relationships between trigonometric ratios Geometry: Similarity, Right Triangles, & Trigonometry Define trigonometric ratios and solve problems involving right triangles. 6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Functions: Trigonometric Functions Extend the domain of trigonometric functions using the unit circle. 3. (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π–x, π+x, and 2π–x in terms of their values for x, where x is any real number. Geometry: Similarity, Right Triangles, & Trigonometry Define trigonometric ratios and solve problems involving right triangles. 6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Geometry: Similarity, Right Triangles, & Trigonometry Define trigonometric ratios and solve problems involving right triangles. 7. Explain and use the relationship between the sine and cosine of complementary angles. 2011 International Center for Leadership in Education M28 M28 M28 M28 Know and apply the six basic trigonometric functions and ratios and solve right triangles using basic trigonometric ratios (sine, cosine, tangent). Know and apply the six basic trigonometric functions and ratios and solve right triangles using basic trigonometric ratios (sine, cosine, tangent). Know and apply the six basic trigonometric functions and ratios and solve right triangles using basic trigonometric ratios (sine, cosine, tangent). Know and apply the six basic trigonometric functions and ratios and solve right triangles using basic trigonometric ratios (sine, cosine, tangent). Mathematics – Page 169 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry A2.A.59 Use the reciprocal and cofunction relationships to find the value of the secant, cosecant, and cotangent of 0º, 30º, 45º, 60º, 90º, 180º, and 270º angles A2.A.60 Sketch the unit circle and represent angles in standard position A2.A.61 Determine the length of an arc of a circle, given its radius and the measure of its central angle Mathematics Domains/Clusters/ Common Core State Standards High School Functions: Trigonometric Functions Extend the domain of trigonometric functions using the unit circle. 3. (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π–x, π+x, and 2π–x in terms of their values for x, where x is any real number. Functions: Trigonometric Functions Extend the domain of trigonometric functions using the unit circle. 2. Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. Functions: Trigonometric Functions Extend the domain of trigonometric functions using the unit circle. 1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. National Essential Skills Study (NESS) National Rankings Rank M28 M64 M18 M64 A2.A.62 Find the value of trigonometric functions, if given a point on the terminal side of angle A2.A.63 Restrict the domain of the sine, cosine, and tangent functions to ensure the existence of an inverse function Functions: Trigonometric Functions Extend the domain of trigonometric functions using the unit circle. 2. Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. Functions: Trigonometric Functions Model periodic phenomena with trigonometric functions. 6. (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed. M64 M28 M37 2011 International Center for Leadership in Education Know and apply the six basic trigonometric functions and ratios and solve right triangles using basic trigonometric ratios (sine, cosine, tangent). Understand the trigonometric properties of the unit circle and sketch the graphs of basic circular functions (y = sin x, y = cos x, and y = tan x, where the measure of the angle x is expressed in radians). Understand the properties of circles (radius, arc, diameter, chord, secant, and tangent) and apply circle quantities (lengths of line segments, angle measure within a circle, circumference, and area) in problem-solving situations. Understand the trigonometric properties of the unit circle and sketch the graphs of basic circular functions (y = sin x, y = cos x, and y = tan x, where the measure of the angle x is expressed in radians). Understand the trigonometric properties of the unit circle and sketch the graphs of basic circular functions (y = sin x, y = cos x, and y = tan x, where the measure of the angle x is expressed in radians). Know and apply the six basic trigonometric functions and ratios and solve right triangles using basic trigonometric ratios (sine, cosine, tangent). Define and apply the properties of relations and functions (domain, range, function composition, and inverses) and use algebraic and graphic methods to determine if a relation is a function. Mathematics – Page 170 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry A2.A.64 Use inverse functions to find the measure of an angle, given its sine, cosine, or tangent Mathematics Domains/Clusters/ Common Core State Standards High School Functions: Trigonometric Functions Model periodic phenomena with trigonometric functions. 7. (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context. National Essential Skills Study (NESS) National Rankings Rank M28 M37 A2.A.65 Sketch the graph of the inverses of the sine, cosine, and tangent functions Functions: Building Functions Build new functions from existing functions. 4. Find inverse functions. c. (+) Read values of an inverse function from a graph or a table, given that the function has an inverse. M28 M37 A2.A.66 Determine the trigonometric functions of any angle, using technology A2.A.67 Justify the Pythagorean identities Functions: Trigonometric Functions Extend the domain of trigonometric functions using the unit circle. 3. (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π–x, π+x, and 2π–x in terms of their values for x, where x is any real number. Functions: Trigonometric Functions Prove and apply trigonometric identities. 8. Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to calculate trigonometric ratios. M28 M10 M55 2011 International Center for Leadership in Education Know and apply the six basic trigonometric functions and ratios and solve right triangles using basic trigonometric ratios (sine, cosine, tangent). Define and apply the properties of relations and functions (domain, range, function composition, and inverses) and use algebraic and graphic methods to determine if a relation is a function. Know and apply the six basic trigonometric functions and ratios and solve right triangles using basic trigonometric ratios (sine, cosine, tangent). Define and apply the properties of relations and functions (domain, range, function composition, and inverses) and use algebraic and graphic methods to determine if a relation is a function. Know and apply the six basic trigonometric functions and ratios and solve right triangles using basic trigonometric ratios (sine, cosine, tangent). Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Perform the general solution of triangles by using the Law of Sines and Law of Cosines to obtain the angle and side length measurements of any triangle. Mathematics – Page 171 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry A2.A.68 Solve trigonometric equations for all values of the variable from 0º to 360º Mathematics Domains/Clusters/ Common Core State Standards High School Functions: Trigonometric Functions Prove and apply trigonometric identities. 9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. National Essential Skills Study (NESS) National Rankings Rank M28 M64 A2.A.69 Determine amplitude, period, frequency, and phase shift, given the graph or equation of a periodic function A2.A.70 Sketch and recognize one cycle of a function of the form y A sin Bx or y A cos Bx A2.A.71 Sketch and recognize the graphs of the functions y sec(x) , y csc(x) , y tan(x) , and y cot(x) A2.A.72 Write the trigonometric function that is represented by a given periodic graph Functions: Trigonometric Functions Extend the domain of trigonometric functions using the unit circle. 4. (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. Model periodic phenomena with trigonometric functions. 5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. Functions: Trigonometric Functions Model periodic phenomena with trigonometric functions. 5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. Functions: Trigonometric Functions Model periodic phenomena with trigonometric functions. 5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. Functions: Trigonometric Functions Extend the domain of trigonometric functions using the unit circle. 4. (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. 2011 International Center for Leadership in Education M64 M64 M64 M64 Know and apply the six basic trigonometric functions and ratios and solve right triangles using basic trigonometric ratios (sine, cosine, tangent). Understand the trigonometric properties of the unit circle and sketch the graphs of basic circular functions (y = sin x, y = cos x, and y = tan x, where the measure of the angle x is expressed in radians). Understand the trigonometric properties of the unit circle and sketch the graphs of basic circular functions (y = sin x, y = cos x, and y = tan x, where the measure of the angle x is expressed in radians). Understand the trigonometric properties of the unit circle and sketch the graphs of basic circular functions (y = sin x, y = cos x, and y = tan x, where the measure of the angle x is expressed in radians). Understand the trigonometric properties of the unit circle and sketch the graphs of basic circular functions (y = sin x, y = cos x, and y = tan x, where the measure of the angle x is expressed in radians). Understand the trigonometric properties of the unit circle and sketch the graphs of basic circular functions (y = sin x, y = cos x, and y = tan x, where the measure of the angle x is expressed in radians). Mathematics – Page 172 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry Mathematics Domains/Clusters/ Common Core State Standards High School A2.A.73 Solve for an unknown side or angle, using the Law of Sines or the Law of Cosines Functions: Trigonometric Functions Prove and apply trigonometric identities. 8. Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to calculate trigonometric ratios. Geometry: Similarity, Right Triangles, & Trigonometry Apply trigonometry to general triangles. 10. (+) Prove the Laws of Sines and Cosines and use them to solve problems. 11. (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). Geometry: Similarity, Right Triangles, & Trigonometry Define trigonometric ratios and solve problems involving right triangles. 8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. A2.A.74 Determine the area of a triangle or a parallelogram, given the measure of two sides and the included angle National Essential Skills Study (NESS) National Rankings Rank Perform the general solution of triangles by using the Law of Sines and Law of Cosines to obtain the angle and side length measurements of any triangle. M55 M9 M28 M55 A2.A.75 Determine the solution(s) from the SSA situation (ambiguous case) Geometry: Similarity, Right Triangles, & Trigonometry Apply trigonometry to general triangles. 11. (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). M10 M28 2011 International Center for Leadership in Education Compute the perimeter and area of common two-dimensional figures. Know and apply the six basic trigonometric functions and ratios and solve right triangles using basic trigonometric ratios (sine, cosine, tangent). Perform the general solution of triangles by using the Law of Sines and Law of Cosines to obtain the angle and side length measurements of any triangle. Understand and apply a systematic methodology or procedure (e.g., direct or indirect measurement, direct or indirect proof, inductive or deductive reasoning) to model and solve problems. Know and apply the six basic trigonometric functions and ratios and solve right triangles using basic trigonometric ratios (sine, cosine, tangent). Mathematics – Page 173 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry A2.A.76 Apply the angle sum and difference formulas for trigonometric functions Mathematics Domains/Clusters/ Common Core State Standards High School Functions: Trigonometric Functions Prove and apply trigonometric identities. 9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. National Essential Skills Study (NESS) National Rankings Rank M28 M64 A2.A.77 Apply the double-angle and half angle formulas for trigonometric functions Geometry: Similarity, Right Triangles, & Trigonometry Apply trigonometry to general triangles. 9. (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. M28 Know and apply the six basic trigonometric functions and ratios and solve right triangles using basic trigonometric ratios (sine, cosine, tangent). Understand the trigonometric properties of the unit circle and sketch the graphs of basic circular functions (y = sin x, y = cos x, and y = tan x, where the measure of the angle x is expressed in radians). Know and apply the six basic trigonometric functions and ratios and solve right triangles using basic trigonometric ratios (sine, cosine, tangent). Measurement Strand Students will determine what can be measured and how, using appropriate methods and formulas. Units of Measurement A2.M.1 Define radian measure A2.M.2 Convert between radian and degree measures Functions: Trigonometric Functions Extend the domain of trigonometric functions using the unit circle. 1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. Functions: Trigonometric Functions Extend the domain of trigonometric functions using the unit circle. 1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. 2. Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. 2011 International Center for Leadership in Education M64 M64 Understand the trigonometric properties of the unit circle and sketch the graphs of basic circular functions (y = sin x, y = cos x, and y = tan x, where the measure of the angle x is expressed in radians). Understand the trigonometric properties of the unit circle and sketch the graphs of basic circular functions (y = sin x, y = cos x, and y = tan x, where the measure of the angle x is expressed in radians). Mathematics – Page 174 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry Mathematics Domains/Clusters/ Common Core State Standards High School National Essential Skills Study (NESS) National Rankings Rank Statistics and Probability Strand Students will collect, organize, display, and analyze data. Collection of Data A2.S.1 Understand the differences among various kinds of studies (e.g., survey, observation, controlled experiment) Statistics & Probability: Making Inferences & Justifying Conclusions Make inferences and justify conclusions from sample surveys, experiments and observational studies. 3. Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. M17 M21 A2.S.2 Determine factors which may affect the outcome of a survey Statistics & Probability: Making Inferences & Justifying Conclusions Make inferences and justify conclusions from sample surveys, experiments and observational studies. 4 M17 Understand the importance of random sampling and sample size in generating representative data. Evaluate and employ accurate and appropriate procedures for statistical data collection, organization, analysis, and display including making estimates and predictions, critiquing data, and drawing inferences (e.g., using the normal curve and z-scores, line of best fit). Understand the importance of random sampling and sample size in generating representative data. Organization and Display of Data A2.S.3 Calculate measures of central tendency with group frequency distributions Statistics & Probability: Making Inferences & Justifying Conclusions Make inferences and justify conclusions from sample surveys, experiments and observational studies. 4. Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling. 2011 International Center for Leadership in Education M14 M31 Understand and apply measures of central tendency (mean, median, and mode, and representative sampling of a population). Understand and apply measures of dispersion (range, mean deviation, variance, and standard deviation). Mathematics – Page 175 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry A2.S.4 Calculate measures of dispersion (range, quartiles, interquartile range, standard deviation, variance) for both samples and populations A2.S.5 Know and apply the characteristics of the normal distribution Mathematics Domains/Clusters/ Common Core State Standards High School Statistics & Probability: Interpreting Categorical & Quantitative Data Summarize, represent and interpret data on a single count or measurement variable. 2. Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. 3. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). 4. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. Statistics & Probability: Interpreting Categorical & Quantitative Data Summarize, represent and interpret data on a single count or measurement variable. 4. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. National Essential Skills Study (NESS) National Rankings Rank Understand and apply measures of dispersion (range, mean deviation, variance, and standard deviation). M31 M36 M43 Simplify polynomials by performing operations (addition, subtraction, multiplication, and division) to simplify expressions (e.g., (2a + 2) + (3a - 1) = 5a + 1). Understand and apply the concepts and applications of quartiles (distributing groups into four equal sizes), percentiles (distributing individuals into 100 groups of equal size), and random distribution to understand and interpret data. Students will make predictions that are based upon data analysis. Predictions from Data A2.S.6 Determine from a scatter plot whether a linear, logarithmic, exponential, or power regression model is most appropriate Statistics & Probability: Interpreting Categorical & Quantitative Data Summarize, represent and interpret data on two categorical and quantitative variables. 6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. 2011 International Center for Leadership in Education M21 Evaluate and employ accurate and appropriate procedures for statistical data collection, organization, analysis, and display including making estimates and predictions, critiquing data, and drawing inferences (e.g., using the normal curve and z-scores, line of best fit). Mathematics – Page 176 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry A2.S.7 Determine the function for the regression model, using appropriate technology, and use the regression function to interpolate and extrapolate from the data A2.S.8 Interpret within the linear regression model the value of the correlation coefficient as a measure of the strength of the relationship Mathematics Domains/Clusters/ Common Core State Standards High School Statistics & Probability: Interpreting Categorical & Quantitative Data Summarize, represent and interpret data on two categorical and quantitative variables. 6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. b. Informally assess the fit of a function by plotting and analyzing residuals. Interpret linear models. 8. Compute (using technology) and interpret the correlation coefficient of a linear fit. Statistics & Probability: Interpreting Categorical & Quantitative Data Interpret linear models. 8. Compute (using technology) and interpret the correlation coefficient of a linear fit. National Essential Skills Study (NESS) National Rankings Rank M21 M31 M21 Evaluate and employ accurate and appropriate procedures for statistical data collection, organization, analysis, and display including making estimates and predictions, critiquing data, and drawing inferences (e.g., using the normal curve and z-scores, line of best fit). Understand and apply measures of dispersion (range, mean deviation, variance, and standard deviation). Evaluate and employ accurate and appropriate procedures for statistical data collection, organization, analysis, and display including making estimates and predictions, critiquing data, and drawing inferences (e.g., using the normal curve and z-scores, line of best fit). Students will understand and apply concepts of probability. Probability A2.S.9 Differentiate between situations requiring permutations and those requiring combinations A2.S.10 Calculate the number of possible permutations ( n Pr ) of n items taken r at a time Statistics & Probability: Conditional Probability & the Rules of Probability Use the rules of probability to compute probabilities of compound events in a uniform probability model. 9. (+) Use permutations and combinations to compute probabilities of compound events and solve problems. Statistics & Probability: Conditional Probability & the Rules of Probability Use the rules of probability to compute probabilities of compound events in a uniform probability model. 9. (+) Use permutations and combinations to compute probabilities of compound events and solve problems. 2011 International Center for Leadership in Education M51 M51 Determine combinations (the various groupings a set may be arranged in without regard to order) and permutations (arrangements of a set where order matters). Determine combinations (the various groupings a set may be arranged in without regard to order) and permutations (arrangements of a set where order matters). Mathematics – Page 177 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry Mathematics Domains/Clusters/ Common Core State Standards High School A2.S.11 Calculate the number of possible combinations ( n C r ) of n items taken r at a time Statistics & Probability: Conditional Probability & the Rules of Probability Use the rules of probability to compute probabilities of compound events in a uniform probability model. 9. (+) Use permutations and combinations to compute probabilities of compound events and solve problems. Statistics & Probability: Conditional Probability & the Rules of Probability Use the rules of probability to compute probabilities of compound events in a uniform probability model. 9. (+) Use permutations and combinations to compute probabilities of compound events and solve problems. A2.S.12 Use permutations, combinations, and the Fundamental Principle of Counting to determine the number of elements in a sample space and a specific subset (event) A2.S.13 Calculate theoretical probabilities, including geometric applications A2.S.14 Calculate empirical probabilities Statistics & Probability: Using Probability to Make Decisions Calculate expected values and use them to solve problems. 3. (+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes. Statistics & Probability: Using Probability to Make Decisions Calculate expected values and use them to solve problems. 4. (+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households? 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank M51 M32 M51 M5 M32 M5 Determine combinations (the various groupings a set may be arranged in without regard to order) and permutations (arrangements of a set where order matters). Determine the probability of single and compound events and use the Counting Principle to determine the probability of independent events occurring jointly. Determine combinations (the various groupings a set may be arranged in without regard to order) and permutations (arrangements of a set where order matters). Examine problem-solving situations involving simple probability and use probabilistic reasoning to compare and communicate the theoretical or empirical likelihood of events. Determine the probability of single and compound events and use the Counting Principle to determine the probability of independent events occurring jointly. Examine problem-solving situations involving simple probability and use probabilistic reasoning to compare and communicate the theoretical or empirical likelihood of events. Mathematics – Page 178 New York Mathematics Strands/Bands/ Performance Indicators Algebra 2/Trigonometry Mathematics Domains/Clusters/ Common Core State Standards High School A2.S.15 Know and apply the binomial probability formula to events involving the terms exactly, at least, and at most Algebra: Arithmetic with Polynomials & Rational Expressions Use polynomial identities to solve problems. 5. (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle. [The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.] A2.S.16 Use the normal distribution as an approximation for binomial probabilities Algebra: Arithmetic with Polynomials & Rational Expressions Use polynomial identities to solve problems. 5. (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle. [The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.] Statistics & Probability: Interpreting Categorical & Quantitative Data Summarize, represent and interpret data on a single count or measurement variable. 4. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. 2011 International Center for Leadership in Education National Essential Skills Study (NESS) National Rankings Rank M5 M70 M31 Examine problem-solving situations involving simple probability and use probabilistic reasoning to compare and communicate the theoretical or empirical likelihood of events. Understand and apply the binomial theorem (e.g., explore the relationship of the binomial theorem with Pascal’s triangle and the Fibonacci sequence). Understand and apply measures of dispersion (range, mean deviation, variance, and standard deviation). Understand and apply the binomial theorem (e.g., explore the relationship of the binomial theorem with Pascal’s triangle and the Fibonacci sequence). M70 Mathematics – Page 179