Jump to Scope and Sequence Map Units of Study Correlation of Standards Special Notes Scope and Sequence Map Conceptual Categories, Domains, Content Clusters, & Standard Numbers NUMBER AND QUANTITY (N) The Real Number System (RN) Extend the properties of exponents to rational exponents: 1, 2 Use properties of rational and irrational numbers: 3 Quantities (Q) Reason quantitatively and use units to solve problems: 4*, 5*, 6* ALGEBRA (A) Seeing Structure in Expressions (SSE) Interpret the structure of expressions: 7*, 8 Write expressions in equivalent forms to solve problems: 9* Arithmetic With Polynomials and Rational Expressions (APR) Perform arithmetic operations on polynomials : 10 Creating Equations (CED) Create equations that describe numbers or relationships: 11*, 12*, 13*, 14* Reasoning With Equations and Inequalities (REI) Understand solving equations as a process of reasoning and explain the reasoning: 15 Solve equations and inequalities in one variable: 16, 17 Solve systems of equations: 18, 19, 20 Represent and solve equations and inequalities graphically: 21, 22*, 23 FUNCTIONS (F) Interpreting Functions (IF) Understand the concept of a function and use function notation: 24, 25, 26 Interpret functions that arise in applications in terms of the context: 27*, 28*, 29* Analyze functions using different representations: 30*, 31, 32 Units of Study 1 st nwks 2 nd nwks 3 rd nwks 4 th nwks 1 7 1 1 5 7, 8, 9 9 8, 9 2, 3, 4 4, 5, 6 7, 9 9 2, 3, 4 4, 5, 6 7, 9 9, 10, 11 4 4, 5 7, 9 9, 10, 11 Page 1 of 27
Building Functions (BF) Build a function that models a relationship between two quantities: 33*, 34* 4 4, 5 7, 9 9, 10 Build new functions from existing functions: 35, 36 Linear, Quadratic, and Exponential Models (LE) Construct and compare linear, quadratic, and exponential models and solve problems: 37*, 38*, 39* 4 4, 5 7, 9 9 Interpret expressions for functions in terms of the situation they model: 40* STATISTICS AND PROBABILITY (S) Interpreting Categorical and Quantitative Data (ID) Summarize, represent, and interpret data on a single count or measurement variable: 41, 42, 43 Summarize, represent, and interpret data on two categorical and quantitative variables: 5 7, 9 9, 11 44, 45 Interpret linear models: 46, 47, 48 Conditional Probability and the Rules of Probability (CP) Understand independence and conditional probability and use them to interpret data: 49, 50 11 Note: See the Special Notes section at the end of this document for a more detailed nine-week breakdown of content by ACOS standard number. Units of Study Unit 1- Basics of Algebra 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. 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. Define appropriate quantities for the purpose of 3 N-RN3 NA 4 N-Q1 5 N-Q2 Page 2 of 27 Answers appropriate problems with units. Answers appropriate problems with units. Know the Closure Property for Real Numbers. Uses correct units consistently throughout the problem. Uses correct units consistently throughout Explain the Closure Property for Real Numbers. Extends the use of units when finding area or volume (square or cube). Extends the use of units when finding area or
Unit 1- Basics of Algebra descriptive modeling. the problem. volume (square or cube). Choose a level of accuracy appropriate to limitations Uses correct units Extends the use of units Answers appropriate on measurement when reporting quantities. 6 N-Q3 consistently throughout when finding area or problems with units. the problem. volume (square or cube). Interpret expressions that represent a quantity in terms of its context.* a) Interpret parts of an expression such as terms, factors, and coefficients. b) Interpret complicated expressions by viewing one or more of their parts as a single entity. Instructional Recommendations / Resources: 7 A-SSE1 NA Identify terms, factors, and coefficients. Write an expression given the verbal definition. Unit 2- Equations in One Variable Create equations and inequalities in one variable, and Defines variables and uses use them to solve problems. Include equations arising Uses given variables to Create word problems and 11 A-CED1 equations to solve word from linear and quadratic functions, and simple create equations. solve. problems. rational and exponential functions.* Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.* 14 A-CED4 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 linear equations and inequalities in one variable, including equations with coefficients represented by letters. Instructional Recommendations / Resources: 15 A-REI1 16 A-REI3 Solve one-step literal equations for a given variable. Uses basic number operations to solve an equation. Solve one-step equations for a given variable. Solve multi-step literal equations for a given variable. Uses the logic of number operations to find errors and correct. Solve multi-step equations for a given variable. Transforms formulas and uses them to find a value. Explains verbally and algebraically the steps used to solve an equation. Solve multi-step equations with fractional coefficients. Page 3 of 27
Unit 3- Inequalities in One Variable Create equations and inequalities in one variable, and Defines variables and uses use them to solve problems. Include equations arising Uses given variables to Create word problems and 11 A-CED1 inequalities to solve word from linear and quadratic functions, and simple create inequalities. solve. problems. rational and exponential functions.* Represent constraints by equations or inequalities, and by systems of equations and/or inequalities and interpret solutions as viable or non-viable options in a modeling context. * Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Instructional Recommendations / Resources: 13 A-CED3 16 A-REI3 Give solutions based on constraints. Solve one-step inequalities for a given variable. Write constraint inequalities. Solve multi-step inequalities for a given variable. Predict possible answers from written constraints. Solve multi-step inequalities with fractional coefficients. Unit 4- Graphs and Functions Create equations in two or more variables to represent Defines variables and uses Uses given variables to Create word problems and relationships between quantities; graph equations on 12 A-CED2 equations to solve word create equations. solve. coordinate axes with labels and scales.* problems. 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). 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 21 A- REI10 24 F-IF1 Page 4 of 27 Know type of equation from graph. Determine whether a relation is a function or not. Identify solutions from a graph or equation. State the domain and range of a given function. Create an equation from a graph. State the domain and range given the graph of a function.
Unit 4- Graphs and Functions corresponding to the input x. The graph of f is the graph of the equation y = f(x). Use function notation, evaluate functions for inputs in Recognize function Interpret statements that Model real-life problems their domains, and interpret statements that use 25 F-IF2 notation and evaluate use function notation in with function notation. function notation in terms of a context. functions given a domain. terms of a context. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. 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.* Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. * 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.* 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 piecewisedefined functions, including step functions and absolute value functions. c) Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. 26 F-IF3 NA 27 F-IF4 Identify the graph given a verbal description of a relationship between two quantities. 28 F-IF5 NA 29 F-IF6 Calculate the average rate of change of a function. 30 F-IF7 NA Given a sequence, be able to recognize that it is a function and find its domain. Given a verbal description of a relationship between two quantities, sketch its graph and find the key features. Know from the domain if the graph will be discrete or continuous. Calculate and interpret the average rate of change of a function. Graph functions and determine intercepts. NA Given a graph, create a relationship between the two quantities. Able to determine discrete or continuous domain from verbal description. Estimate the rate of change from a graph. Use technology to find intercepts. Page 5 of 27
Unit 4- Graphs and Functions 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 algebraic operations. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.* 33 F-BF1 34 F-BF2 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. Interpret the parameters in a linear or exponential function in terms of a context.* 40 F-LE5 Instructional Recommendations / Resources: Write a function rule from a table. Write arithmetic sequences from a formula. 36 F-BF4 NA Identify that parameters are related to domain and range from a word problem. Write a function rule from a table or graph. Write arithmetic sequences from a formula and use them to model situations. Know that horizontal and vertical lines are inverses. Identify that parameters are related to domain and range from a word problem. Write a function rule from a table, graph, or word problem. Write arithmetic sequences from a formula and use them to model situations and translate between the two. NA Identify that parameters are related to domain and range from a word problem. Unit 5- Linear Equations and Graphs Interpret expressions that represent a quantity in terms of its context.* a) Interpret parts of an expression such as terms, factors, and coefficients. b) Interpret complicated expressions by viewing one or more of their parts as a single entity. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.* 7 A-SSE1 NA 12 A-CED2 Page 6 of 27 Uses given variables to create equations. Identify terms, factors, and coefficients. Defines variables and uses equations to solve word problems. Write an expression given the verbal definition. Create word problems and solve.
Unit 5- Linear Equations and Graphs Represent constraints by equations or inequalities, and by systems of equations and/or inequalities and Give solutions based on Write constraint equations/ Predict possible answers 13 A-CED3 interpret solutions as viable or non-viable options in a constraints. inequalities. from written constraints. modeling context. * 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). 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 piecewisedefined functions, including step functions and absolute value functions. c) Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), 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. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of the relationship, or two input-output pairs (include reading these from a table).* 21 A- REI10 30 F-IF7 32 F-IF9 35 F-BF3 38 F-LE2 Know type of equation from graph. Graph lines using slope and y-intercept or two points on the line. Graph lines from slope-intercept form, standard form, and pointslope form. Identify slopes and y- intercepts from equation, graph, or table. Determine the effects of f(x) + k and f(x+k) for specific values of k. Also, determine the value of k given the graph. Write a function rule from a table in a given word problem. Identify solutions of the graph or equation. Graph lines using slope and y-intercept or two points on the line. Graph lines from slope-intercept form, standard form, and pointslope form. Compare slopes and y- intercepts from equation, graph, or table. Experiment with cases involving f(x) + k and f(x+k) using technology. Write a function rule from a table or graph in a given word problem. NA Graph lines using slope and y-intercept or two points on the line using technology. Graph lines from slope-intercept form, standard form, and pointslope form. Model slopes and y- intercepts from created equation, graph, or table. Illustrate cases with technology. Write a function rule from a table, graph, or verbal description in a given word problem. Page 7 of 27
Unit 5- Linear Equations and Graphs Interpret the parameters in a linear or exponential function in terms of a context.* 40 F-LE5 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. b) Informally assess the fit of a function by plotting and analyzing residuals. 45 S-ID6 c) Fit a linear function for a scatter plot that suggests a linear association. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the 46 S-ID7 data. Compare (using technology) and interpret the correlation coefficient of a linear fit. 47 S-ID8 Distinguish between correlation and causation. Instructional Recommendations / Resources: 48 S-ID9 Identify that parameters are related to domain and range from a word problem. Create scatterplot from data and draw a trend line. Interpret slope as rate of change and y-intercept as beginning point. Given the steps, calculate the correlation coefficient on a graphing calculator. Knows the difference between correlation and causation by definition. Identify that parameters are related to domain and range from a word problem. Create scatterplot from data and draw a trend line. Interpret slope as rate of change and y-intercept as beginning point. Given the steps, calculate the correlation coefficient on a graphing calculator. Determine if there is correlation and causation from a graph or word problem. Identify that parameters are related to domain and range from a word problem. Find the equation of a trend line. Estimate new y-values after determining slope and y- intercept. Know how to calculate the correlation coefficient on a graphing calculator. Create a real-life word problem in order to demonstrate both causation and correlation. Unit 6- Systems of Equations and Inequalities Represent constraints by equations or inequalities, and Give solutions based on Write constraint equations/ Predict possible answers by systems of equations and/or inequalities and 13 A-CED3 constraints. inequalities. from written constraints. interpret solutions as viable or non-viable options in a Page 8 of 27
Unit 6- Systems of Equations and Inequalities modeling context. * Prove that, given a system of two equations in two Use the linear combination Use the linear combination variables, replacing one equation by the sum of that Create a system and use 18 A-REI5 (elimination) method to (elimination) method to equation and a multiple of the other produces a elimination to solve. solve a system and check. solve a system and check. system with the same solutions. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. 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.* Graph the solution 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. Instructional Recommendations / Resources: 19 A-REI6 22 23 A- REI11 A- REI12 Use graphing, substitution, and elimination to solve a system. Show that the point of intersection is the solution of the system. Graph the solution to a linear inequality in two variables. Use graphing, substitution, and elimination to solve a system. Graph systems by hand and find solutions. Use graphing calculator by following steps to determine intersections. Graph the solution to a system of linear inequalities in two variables. Find solutions to a system using a graphing calculator to locate intersections. Graph systems by hand and find solutions. Use graphing calculator to determine intersections. Graph the solution to a systems of linear inequalities in two variables using technology. Unit 7- Exponents and Exponential Functions Explain how the definition of the meaning of rational Use exponential laws with Extend exponential laws to Define rational exponents exponents follows from extending the properties of integer exponents. rational exponents. in terms of a radical. 1 N-RN1 integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. Rewrite expressions involving radicals and rational exponents using the properties of exponents. 2 N-RN2 Convert between radical form and rational exponent Page 9 of 27 Simplify radicals using properties of exponents. Solve a word problem that involves the use of rational
Unit 7- Exponents and Exponential Functions form. exponents. Interpret expressions that represent a quantity in terms of its context.* a) Interpret parts of an expression such as terms, factors, and coefficients. b) Interpret complicated expressions by viewing one or more of their parts as a single entity. 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. c) Determine a quadratic equation when given its graph or roots. d) Use the properties of exponents to transform expressions for exponential functions. 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.* 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). 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.* 7 A-SSE1 NA 9 A-SSE3 11 A-CED1 21 22 A- REI10 A- REI11 Convert between radical form and rational exponent form. Uses given variables to create equations. Know type of equation from graph. Show that the point of intersection is the solution of the system. Identify terms, factors, and coefficients. Simplify radicals using properties of exponents. Defines variables and uses equations to solve word problems. Identify solutions of the graph or equation. Graph systems by hand and find solutions. Use graphing calculator by following steps to determine intersections. Write an expression given the verbal definition. Solve a word problem that involves the use of rational exponents. Create word problems and solve. NA Graph systems by hand and find solutions. Use graphing calculator to determine intersections. Page 10 of 27
Unit 7- Exponents and Exponential Functions 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). 24 F-IF1 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. 25 F-IF2 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 27 F-IF4 where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.* Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. * 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 piecewisedefined functions, including step functions and absolute value functions. c) Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Write a function defined by an expression in different but equivalent forms to reveal and explain different 31 F-IF8 Page 11 of 27 Determine whether a relation is a function or not. Evaluate functions given a domain. Identify the graph given a verbal description of a relationship between two quantities. 28 F-IF5 NA 30 F-IF7 NA Identify the different values of variables in exponential growth or State the domain and range of a given function. Interpret statements that use function notation involving exponents. Given a verbal description of a relationship between two quantities, sketch its graph and find the key features. Know from the domain if the graph will be discrete or continuous. Graph functions and determine intercepts and asymptotes. Write exponential growth or decay, including compound interest, State the domain and range given the graph of a function. Model real-life problems with function notation involving exponents. Given a graph, create a relationship between the two quantities. Able to determine discrete or continuous domain from verbal description. Use technology to find intercepts and asymptotes. Model a real-life problem involving exponential growth or decay, including
Unit 7- Exponents and Exponential Functions 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. b) Use the properties of exponents to interpret expressions for exponential functions. decay, including compound interest, functions. functions. compound interest, functions. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). 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 algebraic operations. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.* Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), 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. 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 32 F-IF9 33 F-BF1 34 F-BF2 35 F-BF3 37 F-LE1 Page 12 of 27 Identify initial values and growth rates. Write a function rule from a table. Write geometric sequences from a formula. Determine the effects of f(x) + k and f(x+k) for specific values of k. Also, determine the value of k given the graph. Determine if the function is linear or exponential by examining a table. Compare initial values and growth rates from equation, graph, or table. Write a function rule from a table or graph. Write geometric sequences from a formula and use them to model situations. Experiment with cases involving f(x) + k and f(x+k) using technology. Determine if the function is linear or exponential by examining a table or equation. Model initial values and growth rates from created equation, graph, or table. Write a function rule from a table, graph, or word problem. Write geometric sequences from a formula and use them to model situations and translate between the two. Illustrate cases with technology. From a word problem, determine which type of function should be used.
Unit 7- Exponents and Exponential Functions 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 Construct linear and exponential functions, including Write a function rule from Distinguish between linear Write a function rule from arithmetic and geometric sequences, given a graph, a a table, graph, or verbal 38 F-LE2 and exponential from a a table or graph in a given description of the relationship, or two input-output description in a given word graph. word problem. pairs (include reading these from a table).* problem. Observe, using graphs and tables, that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.* 39 F-LE3 Interpret the parameters in a linear or exponential function in terms of a context.* 40 F-LE5 Instructional Recommendations / Resources: Determine that an exponential function will exceed a linear function by their graphs or tables. Identify that parameters are related to domain and range from a word problem. Determine the point at which an exponential function will exceed a linear function. Identify that parameters are related to domain and range from a word problem. Using interval notation, describe where one graph exceeds the other in interval notation. Identify that parameters are related to domain and range from a word problem. Unit 8- Polynomials and Factoring Use the structure of an expression to identify ways to rewrite it. 8 A-SSE2 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. Instructional Recommendations / Resources: 10 A-APR1 Factor quadratic expressions where leading coefficient is 1 and special cases. Add, subtract, and multiply polynomials. Factor quadratic expressions. Add, subtract, and multiply polynomials. Factor higher degree polynomials by grouping. Add, subtract, and multiply polynomials. Page 13 of 27
Unit 9- Quadratic Equations and Functions Interpret expressions that represent a quantity in terms of its context.* a) Interpret parts of an expression such as terms, factors, and coefficients. b) Interpret complicated expressions by viewing one or more of their parts as a single entity. 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. c) Determine a quadratic equation when given its graph or roots. d) Use the properties of exponents to transform expressions for exponential functions. 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.* Solve quadratic equations in one variable. a) Use the method of completing the square to transform any quadratic equation in x into an equation in 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 x 2 = 49), taking square roots, completing the square and the quadratic formula, and 7 A-SSE1 NA 9 A-SSE3 11 A-CED1 17 A-REI4 Find zeros from factoring a quadratic equation. Page 14 of 27 Uses given variables to create equations/ inequalities. Solve quadratic equations by completing the square where b is an even integer and a is one. Solve quadratic equations by taking square roots or quadratic formula. (Solutions will yield real number solutions.) Determine the number and nature or roots from Identify terms, factors, and coefficients. Find zeros from factoring a quadratic equation and by completing the square. Find maximum and minimum of the function. Defines variables and uses equations to solve word problems. Solve quadratic equations by completing the square, by taking square roots or quadratic formula. Determine the number and nature or roots from discriminant. Write an expression given the verbal definition. Determine a quadratic equation from its roots or graph. Create word problems and solve. Derive quadratic formula from standard form.
Unit 9- Quadratic Equations and Functions factoring as appropriate to the initial form of discriminant. the equation. Recognize when the quadratic formula gives complex solutions, and write them as a +- bi for real numbers a and b. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. 20 A-REI7 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.* 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.* Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. * 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 piecewisedefined functions, including step functions 22 A- REI11 27 F-IF4 Solve a system consisting of a linear equation and a quadratic equation by graphing. Show that the point of intersection is the solution of the system. Identify the graph given a verbal description of a relationship between two quantities. 28 F-IF5 NA 30 F-IF7 NA Page 15 of 27 Solve a system consisting of a linear equation and a quadratic equation by graphing and algebraically. Graph systems by hand and find solutions. Use graphing calculator by following steps to determine intersections. Given a verbal description of a relationship between two quantities, sketch its graph and find the key features. Know from the domain if the graph will be discrete or continuous. Graph functions and determine intercepts and maximum or minimum. Solve a system consisting of a linear equation and a quadratic equation by graphing, algebraically, and by using a graphing calculator. Graph systems by hand and find solutions. Use graphing calculator to determine intersections. Given a graph, create a relationship between the two quantities. Able to determine discrete or continuous domain from verbal description. Use technology to find intercepts and maximum or minimum.
Unit 9- Quadratic Equations and Functions and absolute value functions. c) Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. 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. b) Use the properties of exponents to interpret expressions for exponential functions. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). 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 algebraic operations. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), 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. 31 F-IF8 32 F-IF9 33 F-BF1 35 F-BF3 Re-write a function in standard form to factored form in order to find the zeros. Page 16 of 27 Identify maximum or minimum from graph. Compare the effects of changing a. Write a function rule from a table. Determine the effects of f(x) + k and f(x+k) for specific values of k. Also, determine the value of k given the graph. Re-write a function in standard form to vertex form in order to find the vertex and zeros. Identify maximum or minimum from graph or equation. Compare the effects of changing a. Write a function rule from a table or graph. Experiment with cases involving f(x) + k and f(x+k) using technology. Determine a quadratic equation from its roots or graph. Use graphing calculator to determine the maximum or minimum and to compare the effects of changing a. Write a function rule from a table, graph, or word problem. Illustrate cases with technology. Observe, using graphs and tables, that a quantity 39 F-LE3 Determine that an Determine the point at Using interval notation,
Unit 9- Quadratic Equations and Functions increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.* exponential function will exceed a quadratic function by their graphs or tables. which an exponential function will exceed a quadratic function. describe where one graph exceeds the other in interval notation. Instructional Recommendations / Resources: Unit 10- Radicals Expressions and Equations Relate the domain of a function to its graph and, Know from the domain if Able to determine discrete State the domain from the where applicable, to the quantitative relationship it 28 F-IF5 the graph will be discrete or continuous domain from graph. describes. * or continuous. verbal description. 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 exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), 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 30 F-IF7 NA 32 F-IF9 35 F-BF3 Page 17 of 27 Identify horizontal and vertical shifts. Determine the effects of f(x) + k and f(x+k) for specific values of k. Also, determine the value of k given the graph. Graph functions and determine intercepts. Identify horizontal and vertical shifts. Experiment with cases involving f(x) + k and f(x+k) using technology. Use technology to find intercepts. Create an equation given horizontal and vertical shifts. Illustrate cases with technology.
Unit 10- Radicals Expressions and Equations technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Recommendations / Resources: Unit 11- Statistics and Probability Represent data with plots on the real number line (dot plots, histograms, and box plots). 41 S-ID1 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 differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). Summarize categorical data for two categories in twoway 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. 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. b) Informally assess the fit of a function by 42 S-ID2 43 S-ID3 44 S-ID5 45 S-ID6 Graph dot plots, histograms, and box plots from data. Create a double box-andwhiskers or a double line graph or double histogram. Page 18 of 27 Determine why a graph is skewed. Construct a relative frequency table. Create scatterplot from data and draw a trend line. Interpret data from dot plots, histograms, and box plots. Interpret data from double graphs. Interpret skewed data. Construct and interpret the relative frequencies in the context of the data. Create scatterplot from data and draw a trend line. Collect a set of data and decide which graph is best to use. Collect a set of data (double) and decide which graph is best to use. Collect data and determine how the outliers affect the spread. Recognize the possible associations and trends in the data. Find the equation of a trend line.
Unit 11- Statistics and Probability plotting and analyzing residuals. c) Fit a linear function for a scatter plot that suggests a linear association. Compare (using technology) and interpret the correlation coefficient of a linear fit. 47 S-ID8 Distinguish between correlation and causation. 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 ). 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. Instructional Recommendations / Resources: Correlation of Standards 48 S-ID9 49 S-CP1 50 S-CP2 Given the steps, calculate the correlation coefficient on a graphing calculator. Knows the difference between correlation and causation by definition. Create Venn diagrams from data. Page 19 of 27 Identify and find probability independent events. Given the steps, calculate the correlation coefficient on a graphing calculator. Determine if there is correlation and causation from a graph or word problem. Understand and interpret a Venn diagram. Create two events that are independent. Know how to calculate the correlation coefficient on a graphing calculator. Create a real-life word problem in order to demonstrate both causation and correlation. Calculate missing values in a Venn diagram. Model a problem in which two events are independent. Standards Key AL COS # CCSS # HCS Unit # NUMBER AND QUANTITY: The Real Number System 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. 1 N-RN1 7 Rewrite expressions involving radicals and rational exponents using the properties of exponents. 2 N-RN2 7 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 3 N-RN3 1 irrational. NUMBER AND QUANTITY: Quantities 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. 4 N-Q1 1 Define appropriate quantities for the purpose of descriptive modeling. 5 N-Q2 1
Standards Key AL COS # CCSS # HCS Unit # Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. 6 N-Q3 1 ALGEBRA: Seeing Structure in Expressions Interpret expressions that represent a quantity in terms of its context.* a) Interpret parts of an expression such as terms, factors, and coefficients. 7 A-SSE1 1, 5, 7, 9 b) Interpret complicated expressions by viewing one or more of their parts as a single entity. Use the structure of an expression to identify ways to rewrite it. 8 A-SSE2 8 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. c) Determine a quadratic equation when given its graph or roots. d) Use the properties of exponents to transform expressions for exponential functions. ALGEBRA: Arithmetic With Polynomials and Rational Expressions 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. ALGEBRA: Creating Equations 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.* 9 A-SSE3 7, 8, 9 10 A-APR1 8, 9 11 A-CED1 2, 3, 7, 9 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.* 12 A-CED2 4, 5 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities and interpret solutions as viable or non-viable options in a modeling context. * 13 A-CED3 3, 5, 6 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.* 14 A-CED4 2 ALGEBRA: Reasoning With Equations and Inequalities 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 15 A-REI1 2 solution method. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. 16 A-REI3 2, 3 Solve quadratic equations in one variable. a) Use the method of completing the square to transform any quadratic equation in x into an equation in 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 x 2 = 49), taking square roots, completing the square and 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. 17 A-REI4 9 Page 20 of 27
Standards Key AL COS # CCSS # HCS Unit # 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. 18 A-REI5 6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. 19 A-REI6 6 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. 20 A-REI7 9 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). 21 A-REI10 4, 5, 7 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, 22 A-REI11 6, 7, 9, 10 rational, absolute value, exponential, and logarithmic functions.* Graph the solution 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 23 A-REI12 5, 6 the corresponding half-planes. FUNCTIONS: Interpreting Functions 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) 24 F-IF1 4, 5, 7 denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. 25 F-IF2 4, 5, 7 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. 26 F-IF3 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 27 F-IF4 4, 5, 7, 9 maximums and minimums; symmetries; end behavior; and periodicity.* Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. * 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.* 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 exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric Page 21 of 27 28 F-IF5 4, 5, 7, 9, 10 29 F-IF6 4, 5 30 F-IF7 4, 5, 7, 9, 10