MAFS.912.A APR.1.1 MAFS.912.A SSE.1.1 MAFS.912.A SSE.1.2 MAFS.912.N RN.1.1 MAFS.912.N RN.1.2 MAFS.912.N RN.2.3 LAFS.910.SL.1.1 LAFS.910.SL.2.4 LAFS.910.RST.1.3 The School District of Palm Beach County ALGEBRA 1 REGULAR / HONORS (REVISED 9-18-17) Section 1: Expressions ematics Florida August 15 August 30 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. 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. Use the structure of an expression to identify ways to rewrite it. 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. Rewrite expressions involving radicals and rational exponents using the properties of exponents. 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. Come to discussions prepared, having read and researched material under study; explicitly draw on that preparation by referring to evidence from texts and other research on the topic or issue to stimulate a thoughtful, well reasoned exchange of ideas. b. Work with peers to set rules for collegial discussions and decision making (e.g., informal consensus, taking votes on key issues, c. Propel conversations by posing and responding to questions that relate the current discussion to broader themes or larger ideas; actively incorporate others into the discussion; and clarify, verify, or challenge ideas and conclusions. d. Respond thoughtfully to diverse perspectives, summarize points of agreement and disagreement, and, when warranted, qualify or justify their own views and understanding and make new connections in light of the evidence and reasoning presented. Using Expressions to Represent Real World Situations Understanding Polynomial Expressions Algebraic Expressions Using the Distributive Property Algebraic Expressions Using the Commutative and Associative Properties Properties of Exponents Radical Expressions and Expressions with Rational Exponents Adding Expressions with Radicals and Rational Exponents More Operations with Radicals and Rational Exponents Operations with Rational and Irrational Numbers Students will relate the addition, subtraction, and multiplication of integers to the addition, subtraction, and multiplication of polynomials with integral coefficients through application of the distributive property. apply their understanding of closure to adding, subtracting, and multiplying polynomials with integral coefficients. add, subtract, and multiply polynomials with integral coefficients. interpret parts of an expression, such as ters, factors and coefficients. rewrite algebraic expressions in different equivalent forms by recognizing the expression s structure. rewrite algebraic expressions in different equivalent forms by simplifying expressions (e.g., combining like terms, using the distributive property, and other operations with polynomials). use the properties of exponents to rewrite a radical expression as an expression with a rational exponent. use the properties of exponents to rewrite an expression with a rational exponent as a radical expression. apply the properties of operations of integer exponents to expressions with rational exponents. apply the properties of operations of integer exponents to radical expressions. write algebraic proofs that show that a 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. Nation 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1 of 21 School District of Palm Beach County September 2017
LAFS.910.WHST.1.1 c. Use words, phrases, and clauses to link the major sections of the text, create cohesion, and clarify the relationships between claim(s) and reasons, between reasons and evidence, and between claim(s) and counterclaims. Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience. ematics. FSQ Section 1 2 of 21 School District of Palm Beach County September 2017
MAFS.912.A CED.1.1 MAFS.912.A CED.1.2 MAFS.912.A CED.1.4 MAFS.912.A REI.1.1 MAFS.912.A REI.2.3 MAFS.912.A REI.4.10 MAFS.912.A SSE.1.2 LAFS.910.SL.1.1 ematics Florida Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational, absolute, and exponential functions. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. 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. 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). Use the structure of an expression to identify ways to rewrite it. The School District of Palm Beach County ALGEBRA 1 REGULAR / HONORS (REVISED 9-18-17) Section 2: Equations and Inequalities a. Come to discussions prepared, having read and researched material under study; explicitly draw on that preparation by referring to evidence from texts and other research on the topic or issue to stimulate a thoughtful, well reasoned exchange of ideas. b. Work with peers to set rules for collegial discussions and decision making (e.g., informal consensus, taking votes on key issues, c. Propel conversations by posing and responding to questions that relate the current discussion to broader themes or larger ideas; actively incorporate others into the discussion; and clarify, verify, or challenge ideas and conclusions. d. Respond thoughtfully to diverse perspectives, summarize points of agreement and disagreement, and, when warranted, qualify or justify their own views and understanding and make new connections in light of the evidence and reasoning presented. August 31 October 2 Equations: True or False? Identifying Properties When Solving Equations Solving Equations Solving Equations Using the Zero Product Property Solving Inequalities Solving Compound Inequalities Rearranging Formulas Solution Sets to Equations with Two Variables Students will write an equation in one variable that represents a real world context. write an inequality in one variable that represents a realworld context. identify the quantities in a real world situation that should be represented by distinct variables.. solve multi variable formulas or literal equations for a specific variable. solve formulas and equations with coefficients represented by letters. complete an algebraic proof of solving a linear equation. construct a viable argument to justify a solution method. solve a linear equation. solve a linear inequality. verify if a set of ordered pairs is a solution of a function. rewrite algebraic expressions in different equivalent forms by recognizing the expression s structure. rewrite algebraic expressions in different equivalent forms by simplifying expressions (e.g., combining like terms, using the distributive property, and other operations with polynomials). Nation 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 LAFS.910.SL.2.4 LAFS.910.RST.1.3 3 of 21 School District of Palm Beach County September 2017
LAFS.910.WHST.1.1 c. Use words, phrases, and clauses to link the major sections of the text, create cohesion, and clarify the relationships between claim(s) and reasons, between reasons and evidence, and between claim(s) and counterclaims. Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience. ematics. FSQ Section 2 4 of 21 School District of Palm Beach County September 2017
The School District of Palm Beach County ALGEBRA 1 REGULAR / HONORS (REVISED 9-18-17) Section 3: Introduction to Functions ematics Florida October 3 October 20 Studens will MAFS.912.A APR.1.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, relate the addition, subtraction, and multiplication of integers Nation subtraction, and multiplication; add, subtract, and multiply polynomials. Input and Output Values to the addition, subtraction, and multiplication of polynomials 3.1 Interpret expressions that represent a quantity in terms of its context. with integral coefficients through application of the distributive 3.2 MAFS.912.A SSE.1.1 a. Interpret parts of an expression, such as terms, factors, and coefficients. Representing, Naming, and property. 3.3 b. Interpret complicated expressions by viewing one or more of their parts as a single entity. Evaluating Functions apply their understanding of closure to adding, subtracting, 3.4 MAFS.912.A SSE.1.2 and multiplying polynomials with integral coefficients. 3.5 Use the structure of an expression to identify ways to rewrite it. Adding and Subtracting add, subtract, and multiply polynomials with integral 3.6 Functions coefficients. 3.7 Write a function that describes a relationship between two quantities. rewrite algebraic expressions in different equivalent forms by 3.8 MAFS.912.F BF.1.1 a. Determine an explicit expression, a recursive process, or steps for calculation from a context. Multiplying Functions recognizing the expression s structure. 3.9 b. Combine standard function types using arithmetic operations. rewrite algebraic expressions in different equivalent forms by 3.10 c. Compose functions. Closure Property simplifying expressions (e.g., combining like terms, using the 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 distributive property, and other operations with polynomials). MAFS.912.F BF.2.3 negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using Real World Combinations and write a function that combines functions using arithmetic technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Compositions of Functions operations and relate the result to the context of the problem. write a function to model a real world context by composing Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the MAFS.912.F IF.1.1 Key Features of Graphs of functions and the information within the context. 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 Functions determine the value of k when given a graph of the function corresponding to the input x. The graph of f is the graph of the equation y = f(x). and its transformation. MAFS.912.F IF.1.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms Average Rate of Change Over identify differences and similarities between a function and its of a context. an Interval transformation. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the identify a graph of a function given a graph or a table of a MAFS.912.F IF.2.4 quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; Transformations of Functions transformation and the type of transformation that is intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end represented. behavior; and periodicity. graph by applying a given transformation to a function. MAFS.912.F IF.2.5 identify ordered pairs of a transformed graph. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. complete a table for a transformed function. MAFS.912.F IF.2.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. use the definition of a function to determine if a relationship Estimate the rate of change from a graph. is a function, given tables, graphs, mapping diagrams, or sets of ordered pairs. determine the feasible domain of a function that models a real world context. evaluate functions that model a real world context for inputs a. Come to discussions prepared, having read and researched material under study; explicitly draw on that preparation by referring in the domain. to evidence from texts and other research on the topic or issue to stimulate a thoughtful, well reasoned exchange of ideas. interpret the domain of a function within the real world b. Work with peers to set rules for collegial discussions and decision making (e.g., informal consensus, taking votes on key issues, context given. LAFS.910.SL.1.1 interpret statements that use function notation within the c. Propel conversations by posing and responding to questions that relate the current discussion to broader themes or larger ideas; real world context given. actively incorporate others into the discussion; and clarify, verify, or challenge ideas and conclusions. determine and relate the key features of a function within a d. Respond thoughtfully to diverse perspectives, summarize points of agreement and disagreement, and, when warranted, qualify real world context by examining the function s table. or justify their own views and understanding and make new connections in light of the evidence and reasoning presented. determine and relate the key features of a function within a real world context by examining the function s graph. use a given verbal description of the relationship between two quantities to label key features of a graph of a function that model the relationship. calculate the average rate of change of a continuous function that is represented algebraically, in a table of values, on a graph, or as a set of data. LAFS.910.SL.2.4 interpret the average rate of change of a continuous function that is represented algebraicailly, in a table of values, on a h f d ih l ld 5 of 21 School District of Palm Beach County September 2017
LAFS.910.RST.1.3 LAFS.910.WHST.1.1 c. Use words, phrases, and clauses to link the major sections of the text, create cohesion, and clarify the relationships between claim(s) and reasons, between reasons and evidence, and between claim(s) and counterclaims. Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience. ematics. USA Sections 1 3 graph, or as a set of data with a real world context. 6 of 21 School District of Palm Beach County September 2017
MAFS.912.A CED.1.2 MAFS.912.A CED.1.3 MAFS.912.A REI.3.5 MAFS.912.A REI.3.6 MAFS.912.A REI.4.10 MAFS.912.A REI.4.11 MAFS.912.A REI.4.12 MAFS.912.F BF.1.1 MAFS.912.F IF.1.3 MAFS.912.F LE.1.2 MAFS.912.F LE.2.5 MAFS.912.S ID.3.7 LAFS.910.SL.1.1 ematics Florida October 26 November 14 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Arithmetic Sequences 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. 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. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. 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. 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. 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. c. Compose functions. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. 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). Interpret the parameters in a linear or exponential function in terms of a context. The School District of Palm Beach County ALGEBRA 1 REGULAR / HONORS (REVISED 9-18-17) Section 4: Linear Functions and Inequalities Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. a. Come to discussions prepared, having read and researched material under study; explicitly draw on that preparation by referring to evidence from texts and other research on the topic or issue to stimulate a thoughtful, well reasoned exchange of ideas. b. Work with peers to set rules for collegial discussions and decision making (e.g., informal consensus, taking votes on key issues, c. Propel conversations by posing and responding to questions that relate the current discussion to broader themes or larger ideas; actively incorporate others into the discussion; and clarify, verify, or challenge ideas and conclusions. d. Respond thoughtfully to diverse perspectives, summarize points of agreement and disagreement, and, when warranted, qualify or justify their own views and understanding and make new connections in light of the evidence and reasoning presented. Rate of Change of Linear Functions Interpreting Rate of Change and y Intercept in a Real World Context Introduction to Systems of Equations Finding Solution Sets to Systems of Equations Using Substitution and Graphing Using Equivalent Systems of Equations Finding Solution Sets to Systems of Equations Using Elimination Solution Sets to Inequalities with Two Variables Finding Solution Sets to Systems of Linear Inequalities Students Will identify the quantities in a real world situation that should be represented by distinct variables. write a system of equations given a real world situation. graph a system of equations that represents a real world context using appropriate axis labels and scale. write constraints for a real world context using equations, inequalities, a system of equations, or a system of inequalities. interpret the solution of a real world context as viable or not viable. provide steps in an algebraic proof that shows one equation being replaced with another to find a solution for a system of equations. solve systems of linear equations. identify systems whose solutions would be the same through examination of the coefficients. graph a system of equations that represents a real world context using appropriate axis labels and scale. verify if a set of ordered pairs is a solution of a function. find a solution or an approximate solution for f(x) = g(x) using a graph, table of values, or successive approximations that give the solution to a given place value. justify why the intersection of two functions is a solution to f(x) = g(x). identify the graph that represents a linear inequality. graph a linear inequality. identify the solution set to a system of inequalities. identify ordered pairs that are in the solution set of a system of inequalities. graph the solution set to a system of inequalities. write an explicit function, define a recursive process, or complete a table of calculations that can be used to mathematically define a real world context. write a function that combines functions using arithmetic operations and relate the result to the context of the problem. write a function to model a real world context by composing functions and the information within the context. write a recursive definition for a sequence that is presented as a sequence, graph, or table. write a linear function, an arithmetic sequence, an exponential function, or a geometric sequence when given a graph, a verbal description, a table of values or a set of ordered pairs that models a real world context. interpret the y intercept of a linear model that represents a set of data with a real world context. interpret the rate of change and intercepts of a linear function when given an equation that models a real world context. Nation 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 7 of 21 School District of Palm Beach County September 2017
LAFS.910.SL.2.4 LAFS.910.RST.1.3 LAFS.910.WHST.1.1 c. Use words, phrases, and clauses to link the major sections of the text, create cohesion, and clarify the relationships between claim(s) and reasons, between reasons and evidence, and between claim(s) and counterclaims. Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience. ematics. FSQ Section 4 8 of 21 School District of Palm Beach County September 2017
MAFS.912.A REI.2.4 MAFS.912.A SSE.1.2 MAFS.912.A SSE.2.3 MAFS.912.F IF.2.4 MAFS.912.F IF.3.8 LAFS.910.SL.1.1 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. Use the structure of an expression to identify ways to rewrite it. The School District of Palm Beach County ALGEBRA 1 REGULAR / HONORS (REVISED 9-18-17) Section 5: Quadratic Equations and Functions Part 1 ematics Florida 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. Use the properties of exponents to transform expressions for exponential 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. 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. a. Come to discussions prepared, having read and researched material under study; explicitly draw on that preparation by referring to evidence from texts and other research on the topic or issue to stimulate a thoughtful, well reasoned exchange of ideas. b. Work with peers to set rules for collegial discussions and decision making (e.g., informal consensus, taking votes on key issues, c. Propel conversations by posing and responding to questions that relate the current discussion to broader themes or larger ideas; actively incorporate others into the discussion; and clarify, verify, or challenge ideas and conclusions. d. Respond thoughtfully to diverse perspectives, summarize points of agreement and disagreement, and, when warranted, qualify or justify their own views and understanding and make new connections in light of the evidence and reasoning presented. November 15 January 12 Real World Examples of Quadratic Functions Factoring Quadratic Expressions Solving Quadratic Equations by Factoring Solving Other Quadratic Equations by Factoring Solving Quadratic Equations by Factoring Special Cases Solving Quadratic Equations by Taking Square Roots Solving Quadratic Equations by Completing the Square Deriving the Quadratic Formula Solving Quadratic Equations Using the Quadratic Formula Quadratics in Action Students will rewrite a quadratic equation in vertex form by completing the square. use the vertex form of a quadratic equation to complete steps in the derivation of the quadratic formula. solve a simple quadratic equation by inspection or by taking square roots. solve a quadratic equation by choosing an appropriate method (i.e., completing the square, the quadratic formula, or factoring). validate why taking the square root of both sides when solving a quadratic equation wil yield two solutions. recognize that the quadratic formula can be used to find complex solutions. rewrite algebraic expressions in different equivalent forms by recognizing the expression's structure. use equivalent forms of a quadratic expression to interpret the expression s terms, factors, zeros, maximum, minimum, coefficients, or parts in terms of the real world situation the expression represents. rewrite algebraic expressions in different equivalent forms using factoring techniques (e.g., common factors, grouping, the difference of two squares, the sum or difference of two cubes, or a combination of methods to factor completely) or simplifying expressions (e.g., combining like terms, using the distributive property, and other operations with polynomials). determine and relate the key features of a function within a real world context by examining the function s table. determine and relate the key features of a function within a real world context by examining the function s graph. use a given verbal description of the relationship between two quantities to label key features of a graph of a function that models the relationship. identify zeros, extreme values, and symmetry of a quadratic function written symbolically. Nation 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 LAFS.910.SL.2.4 LAFS.910.RST.1.3 9 of 21 School District of Palm Beach County September 2017
LAFS.910.WHST.1.1 c. Use words, phrases, and clauses to link the major sections of the text, create cohesion, and clarify the relationships between claim(s) and reasons, between reasons and evidence, and between claim(s) and counterclaims. Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience. ematics. FSQ Section 5 10 of 21 School District of Palm Beach County September 2017
MAFS.912.A CED.1.2 MAFS.912.A REI.2.4 MAFS.912.A REI.4.11 MAFS.912.A SSE.1.1 MAFS.912.F BF.2.3 MAFS.912.F IF.2.4 MAFS.912.F IF.3.7 MAFS.912.F IF.3.8 MAFS.912.F IF.3.9 The School District of Palm Beach County ALGEBRA 1 REGULAR / HONORS (REVISED 9-18-17) Section 6: Quadratic Equations and Functions Part 2 ematics Florida Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 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. 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. 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. 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. 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. 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. d. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. 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). January 16 January 29 Observations from a Graph of a Quadratic Function Nature of the Solutions of Quadratic Equations and Functions Graphing Quadratic Functions Using a Table Graphing Quadratic Functions Using the Vertex and Intercepts Graphing Quadratic Functions Using Vertex Form Transformations of the Dependent Variable of Quadratic Functions Transformations of the Independent Variable of Quadratic Functions Finding Solution Sets to Systems of Equations Using Tables of Values and Successive Approximations Students will identify the quantities in a real world situation that should be represented by distinct variables. write a system of equations given a real world situation. graph a system of equations that represents a real world context using appropriate axis labels and scale. rewrite a quadratic equation in vertex form by completing the square. use the vertex form of a quadratic equation to complete steps in the derivation of the quadratic formula. solve a simple quadratic equation by inspection or by taking square roots. solve a quadratic equation by choosing an appropriate method (i.e., completing the square, the quadratic formula, or factoring). validate why taking the square root of both sides when solving a quadratic equation wil yield two solutions. recognize that the quadratic formula can be used to find complex solutions. find a solution or an approximate solution for f(x) = g(x) using a graph, table of values, or successive approximations that give the solution to a given place value. justify why the intersection of two functions is a solution to f(x) = g(x). use equivalent forms of a quadratic expression to interpret the expression s terms, factors, zeros, maximum, minimum, coefficients, or parts in terms of the real world situation the expression represents. determine the value of k when given a graph of the function and its transformation. identify differences and similarities between a function and its transformation. identify a graph of a function given a graph or a table of a transformation and the type of transformation that is represented. graph by applying a given transformation to a function. identify ordered pairs of a transformed graph. complete a table for a transformed function. determine and relate the key features of a function within a real world context by examining the function s table or graph. use a given verbal description of the relationship between two quantities to label key features of a graph of a function that Larson Nation 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 11 of 21 School District of Palm Beach County September 2017
LAFS.910.SL.1.1 LAFS.910.SL.2.4 LAFS.910.RST.1.3 LAFS.910.WHST.1.1 a. Come to discussions prepared, having read and researched material under study; explicitly draw on that preparation by referring to evidence from texts and other research on the topic or issue to stimulate a thoughtful, well reasoned exchange of ideas. b. Work with peers to set rules for collegial discussions and decision making (e.g., informal consensus, taking votes on key issues, c. Propel conversations by posing and responding to questions that relate the current discussion to broader themes or larger ideas; actively incorporate others into the discussion; and clarify, verify, or challenge ideas and conclusions. d. Respond thoughtfully to diverse perspectives, summarize points of agreement and disagreement, and, when warranted, qualify or justify their own views and understanding and make new connections in light of the evidence and reasoning presented. c. Use words, phrases, and clauses to link the major sections of the text, create cohesion, and clarify the relationships between claim(s) and reasons, between reasons and evidence, and between claim(s) and counterclaims. Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience. ematics. USA Sections 4 6 q y g p model the relationship. use the x intercepts of a polynomial function and end behavior to graph the function. identify the x and y intercepts and the slope of the graph of a linear function. identify zeros, extreme values, and symmetry of the graph of a quadratic function. graph a linear function using key features. graph a quadratic function using key features. identify and interpret key features of a graph within the real world context that the function represents. 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. differentiate between different types of functions using a variety of descriptors (e.g., graphically, verbally, numerically, and algebraically). compare and contrast properties of two functions using a variety of function representations (e.g., algebraic, graphic, numeric in tables, or verbal descriptions). 12 of 21 School District of Palm Beach County September 2017
MAFS.912.A SSE.2.3 MAFS.912.F BF.2.3 MAFS.912.F IF.1.3 MAFS.912.F IF.2.4 MAFS.912.F IF.3.7 MAFS.912.F IF.3.8 MAFS.912.F LE.1.2 MAFS.912.F LE.1.3 MAFS.912.F LE.2.5 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. Use the properties of exponents to transform expressions for exponential functions. 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. 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. 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. d. Graph rational functions, identifying zeros and asymptotes 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, and using phase shift. 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. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input output pairs (including reading these from a table). Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. Interpret the parameters in a linear or exponential function in terms of a context. The School District of Palm Beach County ALGEBRA 1 REGULAR / HONORS (REVISED 9-18-17) Section 7: Exponential Functions ematics Florida February 2 February 14 Geometric Sequences Comparing Arithmetic and Geometric Sequences Exponential Functions Graphs of Exponential Functions Growth and Decay Rates of Exponential Functions Transformations of Exponential Functions Students will use equivalent forms of an exponential expression to interpret the expression's terms, factors, zeros, maximum, minimum, coefficients, or parts in terms of the real world situation the expression represents. determine the value of k when given a graph of the function and its transformation. identify differences and similarities between a function and its transformation. identify a graph of a function given a graph or a table of a transformation and the type of transformation that is represented. graph by applying a given transformation to a function. identify ordered pairs of a transformed graph. complete a table for a transformed function. write a recursive definition for a sequence that is presented as a sequence, a graph, or a table. determine and relate the key features of a function within a real world context by examining the function s table. determine and relate the key features of a function within a real world context by examining the function s graph. use a given verbal description of the relationship between two quantities to label key features of a graph of a function that model the relationship. identify and interpret key features of a graph within the real world context that the function represents. graph an exponential function using key features. classify the exponential function as exponential growth or decay by examining the base, and give the rate of growth or decay. use the properties of exponents to interpret exponential expressions in a real world context. write an exponential function defined by an expression in different but equivalent forms to reveal and explain different properties of the function, and determine which form of the function is the most appropriate for interpretation for a realworld context. compare a linear function and an exponential function given in real world context by interpreting the functions graphs. compare a linear function and an exponential function given in a real world context through tables. Nation 7.1 7.2 7.3 7.4 7.5 7.6 13 of 21 School District of Palm Beach County September 2017
LAFS.910.SL.1.1 LAFS.910.SL.2.4 LAFS.910.RST.1.3 LAFS.910.WHST.1.1 a. Come to discussions prepared, having read and researched material under study; explicitly draw on that preparation by referring to evidence from texts and other research on the topic or issue to stimulate a thoughtful, well reasoned exchange of ideas. b. Work with peers to set rules for collegial discussions and decision making (e.g., informal consensus, taking votes on key issues, c. Propel conversations by posing and responding to questions that relate the current discussion to broader themes or larger ideas; actively incorporate others into the discussion; and clarify, verify, or challenge ideas and conclusions. d. Respond thoughtfully to diverse perspectives, summarize points of agreement and disagreement, and, when warranted, qualify or justify their own views and understanding and make new connections in light of the evidence and reasoning presented. c. Use words, phrases, and clauses to link the major sections of the text, create cohesion, and clarify the relationships between claim(s) and reasons, between reasons and evidence, and between claim(s) and counterclaims. Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience. ematics. FSQ Section 7 g compare a quadratic function and an exponential function given in real world context by interpreting the functions graphs. compare a quadratic function and an exponential function given in a real world context through tables. write a linear function, an arithmetic sequence, an exponential function, or a geometric sequence when given a graph that models a real world context. write a linear function, an arithmetic sequence, an exponential function, or a geometric sequence when given a verbal description of a real world context. write a linear function, an arithmetic sequence, an exponential function, or a geometric sequence when given a table of values or a set of ordered pairs that model a real world context. interpret the x intercept, y intercept, and/or rate of growth or decay of an exponential function given in a real world context. 14 of 21 School District of Palm Beach County September 2017
MAFS.912.A APR.2.3 MAFS.912.A CED.1.1 MAFS.912.A CED.1.2 MAFS.912.A CED.1.3 MAFS.912.A REI.2.4 MAFS.912.A REI.4.11 MAFS.912.F BF.1.1 MAFS.912.F BF.2.3 MAFS.912.F IF.1.1 MAFS.912.F IF.1.2 MAFS.912.F IF.1.3 MAFS.912.F IF.2.4 MAFS.912.F IF.2.5 MAFS.912.F IF.2.6 MAFS.912.F IF.3.7 The School District of Palm Beach County ALGEBRA 1 REGULAR / HONORS (REVISED 9-18-17) Section 8: Summary of Functions ematics Florida 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. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational, absolute, and exponential functions. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 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 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. 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. 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. c. Compose functions. 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. 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). Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in 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 piecewise defined functions, including step functions and absolute value functions. c. Graph polynomials, identifying zeros when suitable factorizations are available, & showing end behavior. d. Graph rational functions, identifying zeros and asymptotes 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, and using phase shift. February 15 March 7 Comparing Linear, Quadratic, and Exponential Functions Comparing Arithmetic and Geometric Sequences Exploring non Arithmetic, non Geometric Sequences Modeling with Functions Understanding Piecewise Defined Functions Absolute Value Functions Graphing Power Functions Finding Zeros of Polynomial Funtions of Higher Degrees End Behavior of Graphs of Polynomials Graphing Polynomial Functions of Higher Degrees Recognizing Even and Odd Functions Solutions to Systems of Functions Students will identify zeros, extreme values, and symmetry of a quadratic function written symbolically. find the zeros of a polynomial function when the polynomial is in factored form. write an equation in one variable that represents a real world context. identify the quantities in a real world situation that should be represented by distinct variables. write a system of equations given a real world situation. graph a system of equations representing a real world context using appropriate axis labels and scale. write constraints for a real world context using equations, inequalities, a system of equations, or a system of inequalities. interpret the solution of a real world context as viable or not viable. solve a simple quadratic equation by inspection or by taking square roots. solve a quadratic equation by choosing an appropriate method (i.e., completing the square, the quadratic formula, or factoring). validate why taking the square root of both sides when solving a quadratic equation wil yield two solutions. recognize that the quadratic formula can be used to find complex solutions. create a rough graph of a polynomial function in factored form by examining the zeros of the function. find a solution or an approximate solution for f(x) = g(x) using a graph. find a solution or an approximate solution for f(x) = g(x) using a table of values. find a solution or an approximate solution for f(x) = g(x) using successive approximations that give the solution to a given place value. justify why the intersection of two functions is a solution to f(x) = g(x). write an explicit function, define a recursive process, or complete a table of calculations that can be used to mathematically define a real world context. write a function that combines functions using arithmetic operations and relate the result to the context of the problem. write a function to model a real world context by composing functions. determine the value of k when given a graph of the function and its transformation. identify differences and similarities between a function and its transformation. identify a graph of a function given a graph or a table of a transformation and the type of transformation that is represented. graph by applying a given transformation to a function. identify ordered pairs of a transformed graph. complete a table for a transformed function. use the definition of a function to determine if a relationship 15 of 21 School District of Palm Beach County September 2017 Nation 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 8.14