Unit 1: Foundations of Algebra

Transcription

1 ALGEBRA I COURSE OVERVIEW Unit 1: Foundations of Algebra Unit Overview: Students will continue to increase their understanding of constants, variables, coefficients, and exponents to develop and interpret algebraic expressions and equations in both linear and nonlinear contexts, including dimensional analysis and graphical reasoning. They will interpret data and explore the structure of equivalent algebraic expressions in various representations. Students will build algebraic models to represent real- world scenarios and use these models to solve problems in context. (Focus on MP.2, MP.4, MP.8) Concept 1: Analyze Expressions and Equations Concept Overview: Students build on their knowledge of constants, variables, and exponents to interpret algebraic expressions and equations in both linear and nonlinear contexts. They explore the structure of equivalent algebraic expressions in various representations. Students build algebraic models to represent real- world scenarios and use these models to solve problems in context. Unit 1, Concept 1 Standards HSN- Q.A.1 HSN- Q.A.2 HSA- SSE.A.1a Concept 2: Reason with Expressions and Equations Concept Overview: Students focus on their understanding of algebraic expressions and equations in both linear and nonlinear contexts to include dimensional analysis and graphical reasoning. They interpret data and explore the structure of equivalent algebraic expressions in various representations. Students build algebraic models to represent real- world scenarios and use these models to solve problems in context. Unit 1, Concept 2 Standards HSA- SSE.A.1 HSA- SSE.A.1a HSA- SSE.A.1b HSN- Q.A.1, HSN- Q.A.2 HSA- CED.A.1 Concept 3: Apply and Evaluate Expressions and Equations Concept Overview: Students continue to concentrate on their knowledge of constants, variables, coefficients, and exponents to develop and interpret algebraic expressions and equations of increasing complexity to model real- world problems involving financial decisions. Unit 1, Concept 3 Standards HSN- Q.A.1 HSA- SSE.A.1 HSA- SSE.A.1a HSA- CED.A.1 HSA- CED.A.2 1

2 ALGEBRA I COURSE OVERVIEW Unit 2: Equations and Inequalities Unit Overview: Students will create and solve multistep linear equations and linear inequalities to model and solve a variety of problems. They will interpret the solution sets of equations and inequalities in the context of real- world problems, and distinguish viable from nonviable solutions. Students will begin to explore how to find the points of intersection of two functions. They will investigate compound inequalities in one variable and their relationship to absolute value equations and inequalities. Students will then solve and graph the equations. They will extend their ability to distinguish between situations with 0, 1, and many solutions, and apply this understanding to compound inequalities and absolute value equations and inequalities. Students will recognize that solutions can be verified by substituting them into the original equation and use this strategy to find extraneous solutions. (Focus on MP.3, MP.4, MP.8) Concept 1: Solve Equations and Inequalities Concept Overview: Students extend their understanding by creating and solving multistep linear equations and multistep linear inequalities to model and solve a variety of problems. They also learn to justify their steps when solving equations and inequalities. They interpret the solution sets of equations and inequalities in the context of real- world problems, and they distinguish viable from nonviable solutions. Finally, students begin to explore how to find the points of intersection of two functions and begin to understand what it means for f(x) = g(x). Concept 2: Rewrite Literal Equations Concept Overview: Students build upon their knowledge of variables and their related units though literal equations. They rearrange variables in formulas and are expected to justify each step using mathematical properties while solving for a given variable. Unit 2, Concept 1 Standards HSA.CED.A.1 HSA.CED.A.3 HSA.REI.A.1 HSA.REI.B.3 HSA.REI.D.11 Unit 2, Concept 2 Standards HSN- Q.A.1 HSN- Q.A.2 HSA- SSE.A.1.B HSA- CED.A.4 HSA- REI.B.3 Concept 3: Solve Absolute Value Equations and Inequalities Concept Overview: Students investigate compound inequalities in one variable and their relationship to absolute value equations and inequalities. They translate real- world constraints into compound inequalities and absolute value equations and inequalities, and solve and graph the equations. Students expand their ability to 2 Unit 2, Concept 3 Standards HSA- CED.A.3 HSA- REI.B.3 distinguish between situations with 0, 1, and many solutions, and apply this understanding to compound inequalities and absolute value equations and inequalities. They recall that solutions can be

3 ALGEBRA I COURSE OVERVIEW verified by substituting them into the original equation and use this strategy to find extraneous solutions. Unit 3: Functions Unit Overview: Students will broaden their understanding of functions, use function notation, and interpret function notation in context. They will deepen their understanding of domain and range and analyze functions in context to determine which values for the domain and range make sense in the problem. Students will discover that arithmetic sequences are linear functions defined over a subset of the set of integers and compare properties of functions represented in different ways. They will identify the common difference as the average rate of change either from a table or a graph. Students will create arithmetic sequences from a description of a relationship and interpret the parameters in context. They will learn that geometric sequences are exponential functions defined over a subset of the integers and will write exponential functions in next- now, recursive, implicit, and explicit forms. Students will be able to distinguish between the average rate of change and the growth rate of geometric sequences. (Focus on MP.1, MP.7, MP.8) Concept 1: Understand and Interpret Functions Concept Overview: Students deepen their knowledge of functions, use function notation, and interpret function notation in context. They continue to develop their understanding of domain and range. Students analyze functions in context to determine which values for the domain and range make sense in the problem. Unit 3, Concept 1 Standards HSF- IF.A.1 HSF- IF.A.2 HSF.IF.C.9 HSF- LE.B.5 Concept 2: Analyze Arithmetic Sequences and Linear Functions Concept Overview: Students learn that arithmetic sequences are linear functions defined over a subset of the set of integers and compare properties of functions represented in different ways. They identify the common difference as the average rate of change either from a table or a graph. Students create arithmetic sequences from a description of a relationship and interpret the parameters in context. They identify and write arithmetic sequences as an explicit expression or a recursive process, given a context, and translate between the two forms. Unit 3, Concept 2 Standards HSF- IF.A.3 HSF- IF.B.6 HSF- IF.C.9 HSF- BF.A.1a HSF- BF.A.2 HSF- LE.A.1a HSF- LE.A.1b HSF- LE.A.2 HSF- LE.B.5 3

4 ALGEBRA I COURSE OVERVIEW Concept 3: Analyze Geometric Sequence and Exponential Functions Concept Overview: Students discover that geometric sequences are exponential functions defined over a subset of the integers. They write exponential functions in next- now, recursive, implicit, and explicit forms. Students distinguish between the average rate of change and the growth rate of geometric sequences. They create and analyze geometric sequences from real- world contexts. Unit 3, Concept 3 Standards HSF- IF.A.3 HSF- IF.B.6 HSF- IF.C.9 HSF- LE.A.1 HSF- LE.A.1c HSF- LE.A.2 HSF- LE.A.3 HSF- LE.B.5 Unit 4: Graphs of Functions Unit Overview: Students will expand upon their prior understanding of linear and nonlinear functions. They will represent the functions in different forms and identify and interpret key features of the functions. Students will also combine linear or exponential functions to form new functions. In addition, they will apply their prior experience with transformations of plane figures as they investigate transformations of these functions. Students will determine the average rate of change over an interval for both linear and exponential functions and identify the meaning of various function parameters in context, including the domain, range, and appropriate scale. Students will also represent arithmetic and geometric sequences as linear and exponential relationships in the form of tables of values, equations, and graphs. (Focus on MP.2, MP.3, MP.4) Concept 1: Analyze Graphs of Linear and Exponential Functions Concept Overview: Students build on their prior understanding of linear and nonlinear functions. They represent linear and exponential functions in different forms, and they identify and interpret key features of the functions, including domain and range. Students also combine linear or exponential functions to form new functions. In addition, they apply their prior experience with transformations of plane figures as they investigate transformations of linear and exponential functions. Finally, they extend their understanding of transformations to explore and apply the relationships between the equations of parallel and perpendicular lines. Unit 4, Concept 1 Standards HSA- REI.D.10 HSF- IF.A.2 HSF- IF.B.4 HSF- IF.B.5 HSF- IF.B.6 HSF- IF.C.7a HSF- IF.C.7e HSF- BF.A.1a HSF- BF.A.1b HSF- BF.A.1c HSG- GPE.B.5 4

5 ALGEBRA I COURSE OVERVIEW Concept 2: Compare Graphs of Linear and Exponential Functions Concept Overview: Students distinguish key features of linear and exponential functions using multiple representations. They determine the average rate of change over an interval for both linear and exponential functions and identify the meaning of various function parameters in context, including the domain, range, and appropriate scale. Students also represent arithmetic and geometric sequences as linear and exponential relationships in the form of tables of values, equations, and graphs. Unit 4, Concept 2 Standards HSF- IF.B.4 HSF- IF.C.7a HSF- IF.C.7e HSF- IF.C.9 HSF- LE.A.1 HSF- LE.A.3 HSF- LE.B.5 Unit 5: Systems of Equations and Inequalities Unit Overview: Students will be able to explain the intersection of the two equations on a graph as the ordered- pair solution to the system. They will be able to explain the linear combination method and prove that replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. Students will apply their understanding of systems of linear inequalities to represent real- world contexts involving constraints and interpret whether the solutions are viable or nonviable options in context. They will graph various inequalities, determine the feasible region for solutions, and recognize that the solutions at the vertices represent the extreme points of the solution set. (Focus on MP.1, MP.2, MP.4) Concept 1: Solve Systems of Equations and Inequalities Concept Overview: Students interpret the intersection of the two equations on a graph as the ordered- pair solution to the system. They create and solve systems of equations and inequalities graphically. Students are able to explain the linear combination method and prove that replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. They apply their understanding of systems of linear inequalities to represent real- world contexts involving constraints and interpret whether the solutions are viable or nonviable options in context. Concept 2: Use Systems in Decision Making: Linear Programming Unit 5, Concept 1 Standards HSA- CED.A.2 HSA- CED.A.3 HSA- CED.A.4 HSA- REI.C.5 HSA- REI.C.6 HSA- REI.D.11 HSA- REI.D.12 Concept Overview: Students explore real- world problems involving systems, beginning with estimating solutions. They write systems of inequalities, involving constraints, to represent real- world situations. Students graph the various inequalities, determine the feasible region 5 Unit 5, Concept 2 Standards HSA.CED.A.3 HSN.Q.A.3 for solutions, and recognize that the solutions at the vertices represent the extreme points of the solution set. They interpret each of these solutions in terms of the real- world situation.

6 ALGEBRA I COURSE OVERVIEW Unit 6: Descriptive Statistics Unit Overview: Students will develop their ability to represent data with plots on the real number line and use the appropriate measures of center and spread. They will extend their understanding of measures of variation to include standard deviation and utilize additional statistical data to make more precise inferences. Students will identify the line of best fit and assess the fit of the line using residuals. Students will interpret the slope and y- intercept of the line of best fit in the context of the data and use the correlation coefficient to interpret the models. They will also study correlation and causation, and will distinguish between the two. Students will interpret relative frequencies in the context of data, including joint, marginal, and conditional relative frequencies. They will discover that possible associations and trends in data are best determined using conditional relative frequency. (Focus on MP.1, MP.2, MP.4) Concept 1: Represent and Analyze Data Concept Overview: Students extend their skill in representing data with plots on the real number line and use the appropriate measures of center and spread, building their understanding of measure of variation to include standard deviation and using more statistical data to make more precise inferences. They interpret two sets of data in terms of shape, center, and spread and account for the effect of extreme data points on a normal distribution. Unit 6, Concept 1 Standards HSS- ID.A.1 HSS- ID.A.2 HSS- ID.A.3 Concept 2: Analyze Scatter Plots Concept Overview: Students learn to formally identify the line of best fit and assess the fit of the line using residuals. They interpret the slope and y- intercept of the line of best fit in the context of the data and use the correlation coefficient to interpret the models. Students also explore correlation and causation, and distinguish between the two. Unit 6, Concept 2 Standards HSS.ID.B.6b HSS.ID.B.6c HSS.ID.C.7 HSS.ID.C.8 HSS.ID.C.9 Concept 3: Interpret Two- Way Frequency Tables Concept Overview: Students deepen their understanding of Unit 6, Concept 3 Standards bivariate data and two- way frequency tables by interpreting relative HSS.ID.B.5 frequencies in the context of data, including joint, marginal, and conditional relative frequencies. They discover that possible associations and trends in data are best determined using conditional relative frequency. 6

7 ALGEBRA I COURSE OVERVIEW Unit 7: Nonlinear Functions Unit Overview: Students will write and define piecewise functions. They will rely on their understanding of key features and interpretations of graphs to explore other nonlinear function families, including piecewise, absolute value, and step functions. Students will focus their work on square and cube root functions. They will use approximations of rational and irrational numbers to graph square root and cube root functions and will explore the effects of vertical and horizontal transformations to the functions. Students will relate the importance of restricted domain and range of the functions to its graph and to the context of the problem. Students will extend the properties of rational and irrational numbers, as well as integer exponents, to that of rational exponents. They will rewrite radical expressions using rational exponents. Students will use the properties of exponents to write equivalent expressions, providing insight into the structure of the expression. (Focus on MP.2, MP.4, MP.8) Concept 1: Create and Analyze Piecewise Functions Concept Overview: Students use their previous work with writing linear equations to write and define piecewise functions. They use their knowledge of key features and interpretations of graphs to explore other nonlinear function families, including piecewise, absolute value, and step functions. Students extend their Unit 7, Concept 1 Standards HSA- CED.A.2 HSF- IF.C.7b HSF- IF.C.9 knowledge of interpreting nonlinear functions to compare properties of functions represented in a different way: algebraically, graphically, numerically in tables, or by verbal descriptions. Concept 2: Investigate Square Root and Cube Root Functions Concept Overview: Students extend their understanding of rational and irrational numbers to square and cube root functions. They use approximations of rational and irrational numbers to graph square root and cube root functions. Students explore the effects of vertical and horizontal transformations to the square root and cube root functions, as well as describe the features of the transformed functions. They also relate the importance of restricted domain and range of the functions to its graph and to the context of the problem. Concept 3: Investigate Rational Exponents Concept Overview: Students build on the properties of rational and irrational numbers, as well as integer exponents, to that of rational exponents. They rewrite radical expressions, many of which are irrational, using rational exponents. Students employ the properties of exponents to write equivalent expressions, providing 7 Unit 7, Concept 2 Standards HSF- BF.B.3 HSF- IF.B.4 HSF- IF.B.5 HSF- IF.C.7b Unit 7, Concept 3 Standards HSA- SSE.B.3.C HSN- RN.A.1 HSN- RN.A.2

8 ALGEBRA I COURSE OVERVIEW insight into the structure of the expression. Unit 8: Exponential Functions Unit Overview: Students will broaden their understanding of exponential functions to model real- world scenarios. They will learn how to interpret the domain, range, growth factor, and initial value in an exponential function in context. They will recognize situations that can be represented by exponential functions and will write and graph the equations that model exponential behavior. Students will go further into interpreting the parameters of the equations in the context of real- world problems and use laws of exponents to rewrite the functions. Students will also see complicated expressions by viewing one or more of their parts as a single entity as they explore compound interest. (Focus on MP.2, MP. 4,MP.5) Concept 1: Represent Exponential Functions Concept Overview: Students expand their knowledge of exponential functions to model real- world scenarios. They create exponential equations and use them to solve problems. They learn how to interpret the domain, range, growth factor, and initial value in an exponential function in context. Students graph exponential functions to analyze key features and understand the effects of changing the parameters of these functions. Students calculate and interpret the average rate of change for exponential functions. Unit 8, Concept 1 Standards HSF- BF.A.1 HSF- BF.A.1a HSF- BF.A.1b HSF- BF.B.3 HSF- IF.B.4 HSF- IF.B.5 HSF- IF.B.6 HSF- IF.C.7e Concept 2: Analyze Exponential Growth and Decay Models Concept Overview: Students continue to recognize situations that can be represented by exponential functions and write and graph the equations that model exponential behavior. They go further, interpreting the parameters of the equations in context of real- world problems and use laws of exponents to rewrite the functions in order to aid in their interpretation of them. Students also see complicated expressions by viewing one or more of their parts as a single entity as they explore compound interest. They graphically solve problems related to exponential functions. Unit 8, Concept 2 Standards HSA- CED.A.2 HSA- SSE.A.1b HSA- REI.D.11 HSF- IF.C.7e HSF- IF.C.8b HSF- LE.A.1c HSF- LE.B.5 HAS- SSE.B.3c 8

9 ALGEBRA I COURSE OVERVIEW Unit 9: Polynomials Unit Overview: Students will work with linear expressions and integer exponents as they begin to explore more complex polynomial expressions. They will interpret different parts of polynomials in context and begin to see expressions as sums, products, and factors instead of different entities. Students will add, subtract, and multiply polynomials to create equivalent expressions that will allow them to interpret different forms of quadratic functions. They will deepen their knowledge of properties of rational exponents and will use these properties to simplify variable expressions. Students will further explore algebraic expressions that can be expressed as products of factors. They will discover patterns to identify factors, leading to the examination of the structure of quadratic equations. Students will find different methods for factoring quadratic expressions. (Focus on MP.1, MP.7, MP.8) Concept 1: Perform Operations on Polynomials Concept Overview: Students build on their earlier work with linear expressions and integer exponents as they begin to explore more complex polynomial expressions. They interpret different parts of polynomials in context and begin to see expressions as sums, products, and factors instead of different entities. Students add, subtract, and multiply polynomials to create equivalent expressions that allow them to interpret Unit 9, Concept 1 Standards HSA- SSE.A.1a HSA- SSE.A.1b HSA- APR.A.1 HSN- RN- A.2 different forms of quadratic functions in later investigations. They extend their knowledge of properties of rational exponents and use these properties to simplify variable expressions. Concept 2: Factor Polynomials Concept Overview: Students explore algebraic expressions that can be expressed as products of factors. They rely on patterns observed in previous work to identify factors, leading to the examination of the structure of quadratic equations. Students discover different methods for factoring quadratic expressions, including factoring by graphing and using geometric area models, as well as algebraically. Unit 9, Concept 2 Standards HSA- SSE.A.1a HSA- SSE.A.2 HSA- SSE.B.3a 9

10 ALGEBRA I COURSE OVERVIEW Unit 10: Quadratic Expressions and Equations Unit Overview: Students will use prior knowledge of functions and equations as they solve quadratic equations. They will use the properties of rational and irrational numbers to solve quadratic equations with rational or irrational solutions. In addition, students will begin to investigate some of the properties of quadratic functions. They will apply the quadratic formula to solve quadratic equations, and they will identify the type and number of real solutions given by the formula. Students will continue their exploration of quadratic functions and key features of the functions graphs. (Focus on MP.3, MP.7, MP.8) Concept 1: Solve Quadratics Concept Overview: Students build on previous knowledge of functions and equations as they solve quadratic equations by factoring, taking square roots, completing the square, and graphing. They also use the properties of rational and irrational numbers to solve quadratic equations with rational or irrational solutions. In addition, students begin to investigate some of the properties of quadratic functions, reflecting on how the roots of equations determine the x- intercepts on the graph of the function. They use their new learning to develop the graph of the related function and determine some of the key features. Unit 10, Concept 1 Standards HSN- RN.B.3 HSA- REI.B.4a HSA- REI.B.4b HSA- APR.B.3 HSF- IF.C.8 HSF- IF.C.8a Concept 2: Analyze Quadratic Equations Concept Overview: Students derive the quadratic formula based on their understanding of solving quadratic equations by completing the square. They then apply the quadratic formula to solve quadratic equations, and they identify the type and number of real solutions given by the formula. They also make use of properties of rational and irrational numbers as they solve quadratic equations with rational or irrational solutions. In addition, students continue their exploration of quadratic functions and key features of the functions graphs. Unit 10, Concept 2 Standards HSN- RN.B.3 HSA- REI.B.4a HSA- REI.B.4b HSA- APR.B.3 HSF- IF.C.7a HSF- IF.C.8 HSF- IF.C.8a 10

11 ALGEBRA I COURSE OVERVIEW Unit 11: Graphs of Quadratic Functions Unit Overview: Students will explore quadratic, exponential, and linear behaviors numerically, algebraically, and graphically. They will apply their understanding of completing the square to rewrite quadratic functions into vertex form and interpret quadratic functions in both standard and vertex form. They will graph quadratic functions and interpret intercepts, maxima, and minima. (Focus on MP.2, MP.4, MP.5) Concept 1: Analyze Graphs of Quadratic Functions Concept Overview: Students further analyze and compare quadratic, exponential, and linear behaviors numerically, algebraically, and graphically. They apply their understanding of completing the square to rewrite quadratic functions into vertex form. Students interpret quadratic functions in both standard form and vertex form, identifying the key features by analyzing the characteristics of the functions. They graph quadratic functions and interpret intercepts, maxima, and minima. Unit 11, Concept 1Standards HSA- CED.A1 HSF- IF.B.4 HSF- IF.B.5 HSF- IF.C.7a HSF- IF.C.7c HSF- IF.C.8a HSF- IF.C.9 HSF- BF.A.1 HSF- BF.B.3 HSF- LE.A.3 11

12 ALGEBRA II COURSE OVERVIEW Unit 1: Recursive, Explicit, and Inverse Functions Unit Overview: Students will write recursive and explicit formulas for arithmetic and geometric sequences, and they will use these formulas to solve problems. They will relate arithmetic sequences to linear functions and geometric sequences to exponential functions. They will develop the ideas and notation for composite functions and apply this to the relationship between functions and their inverses. Students will investigate inverses of functions, identifying graphical and algebraic representations by analyzing ordered pairs, domains, and ranges. They will explore the effects of translations, reflections, stretches, and compressions on the graphs of functions. Students will predict the effects of changes in a function rule on the graph of a function and sketch the graphs of transformations of functions. (Focus on MP.2, MP.7, MP.8) Concept 1: Explore Recursive Functions Concept Overview: Students discover how to write recursive and explicit formulas for arithmetic and geometric sequences and use these formulas to solve problems. They relate arithmetic sequences to linear functions and geometric sequences to exponential functions. Students also write and apply the formulas for the sums of finite arithmetic and geometric series and for infinite geometric series when the sum exists. Unit 1, Concept 1 Standards HSF- BF.A.1 HSF- BF.A.1a HSF- BF.A.2 HSF- IF.A.3 HSF- LE.A.2 HSA- SSE.B.4 Concept 2: Explore Inverse Functions Concept Overview: Students develop the ideas and notation for composite functions and apply this understanding to the relationship between functions and their inverses. They explore inverses of functions, identifying graphical and algebraic representations by analyzing ordered pairs, domains, and ranges. Students garner insight about the relationship between functions and their inverses while investigating how to restrict the domain of a quadratic function to create an inverse function. Unit 1, Concept 2 Standards HSF- BF.B.4 HSF- BF.B.4a HSF- BF.B.4c HSF- BF.A.1c 1

13 ALGEBRA II COURSE OVERVIEW Concept 3: Explore Function Transformations Concept Overview: Students explore the effects of translations, reflections, stretches, and compressions on the graphs of functions. They predict the effects of changes in a function rule on the graph of a function and sketch the graphs of transformations of functions. Unit 1, Concept 3 Standards HSF- BF.B.3 HSF.IF.C.7b HSF- BF.A.1b Unit 2: Exponents and Logarithms Unit Overview: Students will expand upon their knowledge of exponential functions, equations, and graphs. They will learn to solve exponential equations by inspection and analyze graphs of exponential functions. Students will interpret the expressions or parameters for an exponential function in terms of the situation that it models. They will apply their understanding of arithmetic and geometric sequences and series to build and compare linear and exponential function models. They will use exponential equations to model real- word problems including population and compounding interest, building their understanding of exponential parameters in context. Students will study exponential patterns within the compound interest formula, compounding the interest in smaller and smaller intervals to develop the number e. They will graph exponential and logarithmic functions and note the key features of the graphs. Students will learn to use the inverse relationship between exponential and logarithmic functions to solve real- world problems, interpreting appropriate parameters for these functions in terms of their contexts. (Focus on MP.1, MP.2, MP.8) Concept 1: Represent Exponential Functions Concept Overview: Students build on their understanding of exponential functions, equations, and graphs. They learn to solve exponential equations by inspection and analyze graphs of exponential functions. Students study the exponential function family, understanding how a change in parameters affects the graph of a function. They graph simple cases by hand and more complex cases using technology. They learn to interpret the expressions or parameters for an exponential function in terms of the situation that it models. Students apply their understanding of Unit 2, Concept 1 Standards HSA- REI.D.11 HSF- IF.C.7 HSF- IF.C.7e HSF- BF.A.2 HSF- LE.B.5 HSF- IF.C.9 arithmetic and geometric sequences and series to build and compare linear and exponential function models. 2

14 ALGEBRA II COURSE OVERVIEW Concept 2: Model Exponential Growth and Decay Concept Overview: Students use exponential equations to model real- word problems including population and compounding interest, building their understanding of exponential parameters in context. They investigate exponential patterns within the compound interest formula, compounding the interest in smaller and smaller intervals to develop the number e. Unit 2, Concept 2 Standards HSA- CED.A.1 HSF- LE.A.1c HSF- LE.B.5 HSF- BF.A.1b HSF- IF.C.8 HSF- IF.C.8b HAS- REI.D.11 HSF- LE.A.2 Concept 3: Discover and Analyze Logarithms Concept Overview: Students learn to graph exponential and logarithmic functions and note the key features of the graphs. They explore the inverse relationship between exponents and logarithms and use this understanding to solve problems. They also learn to solve exponential equations by using properties of logarithms. Unit 2, Concept 3 Standards HSF- IF.C.7e HSF- BF.B.4a HSF- BF.B.5 HSF- BF.B.4c Concept 4: Apply Logarithmic Functions Concept Overview: Students will use the inverse relationship between exponential and logarithmic functions to solve real- world problems, interpreting appropriate parameters for these functions in terms of their contexts. They express logarithmic equations as equivalent exponential functions and evaluate, using technology when appropriate. Unit 2, Concept 4 Standards HSF- BF.A.1a HSF- BF.B.5 HSF- LE.A.4 HSF- LE.B.5 3

15 ALGEBRA II COURSE OVERVIEW Unit 3: Real and Complex Solutions Unit Overview: Students will improve upon their understanding of inverses as they find inverses of quadratic and cubic functions. They will graph radical functions, explore transformations of these functions, and compare their properties. Students will continue to apply the laws of exponents as they work with radical functions and equations. They will explore a new subset of real numbers and how they behave in the system of numbers. Students will recognize the need for this expansion of the system through their study of quadratic equations and solutions that are not represented as real numbers in the quadratic formula or x- intercepts on the graph. (Focus on MP.3, MP.7, MP.8) Concept 1: Analyze Radical Functions Concept Overview: Students build on their understanding of inverses as they find inverses of quadratic and cubic functions. They graph radical functions, explore transformations of these functions, and compare their properties. Students continue to apply the laws of exponents as they work with radical functions and equations. In addition, they write and solve radical equations in one variable, including those with extraneous solutions. Unit 3, Concept 1 Standards HSN- RN.A.1 HSN- RN.A.2 HSA- CED.A.1 HSA- REI.A.2 HSF- IF.C.7b HSF- IF.C.9 HSF- BF.B.3 HSF- BF.B.4a HSF- BF.A.1b Concept 2: Determine Complex Quadratic Roots Concept Overview: Students discover a new subset of real numbers and how they behave in the system of numbers. They recognize the need for this expansion of the system through their study of quadratic equations and solutions that are not represented as real numbers in the quadratic formula or x- intercepts on the graph. For the first time, students discover a Unit 3, Concept 2 Standards HSN- CN.A.1 HSN- CN.A.2 HSN- CN.C.7 number that cannot be plotted on the real number line. Although complex numbers require a new model, students discover that operations on complex numbers are a natural extension of real number operations. The commutative, associative, and distributive properties guide early explorations in complex number operations. 4

16 ALGEBRA II COURSE OVERVIEW Unit 4: Conic Sections Unit Overview: Students will use prior knowledge to graph the equations of circles and ellipses, and they will identify key features of the graphs. They will discover the relationship between the points on an ellipse and the foci of the ellipse, and they will write the equations of ellipses given key information. Students will build on their earlier work with equations in two variables by graphing the equations of hyperbolas and identifying key features of the graphs. Students will learn how to recognize and analyze an equation of a parabola. Key features will be determined from the equation and validated through graphing. Students will derive the equation of a parabola using the graph, distance formula, and algebraic manipulation through several tests and the general form. They will investigate the relationships among different conic sections and extend their understanding of individual conics, discovering how to discern between a variety of algebraic representations and their characteristics. (Focus on MP.4, MP.5, MP.7) Concept 1: Analyze Graphs and Equations of Circles and Ellipses Concept Overview: Students build on their past learning with equations in two variables by graphing the equations of circles and ellipses and identifying key features of the graphs. They write the equations of circles given key information, and they complete the square to determine key information about circles given their equations. Students also discover the relationship between the points on an ellipse and the foci of the ellipse, and they write the equations of ellipses given key information. Unit 4, Concept 1 Standards HSG.GPE.A.1 HSG.GPE.A.2 HSG.GPE.A.3 Concept 2: Analyze Graphs and Equations of Hyperbolas Concept Overview: Students use what they learned in their earlier Unit 4, Concept 2 Standards work with equations in two variables by graphing the equations of HSG- GPE.A.3 hyperbolas and identifying key features of the graphs. They also explore the relationship between the points on a hyperbola and the foci of the hyperbola, and they write the equations of hyperbolas given key information. Concept 3: Analyze Graphs and Equations of Parabolas Concept Overview: Students learn how to recognize and analyze an equation of a parabola. Key features are determined from the equation and validated through graphing. Then, students derive the equation of a parabola using the graph, distance formula, and 5 Unit 4, Concept 3 Standards HSG- GPE.A.2 algebraic manipulation through several tests and the general form. This equation is applied in algebraic and real- world situations during the investigations.

17 ALGEBRA II COURSE OVERVIEW Concept 4: Compare Conic Equations Concept Overview: Students investigate the relationships among different conic sections. They extend their understanding of individual conics, now discovering how to discern between a variety of algebraic representations and their characteristics. Students also explore solving systems of equations that include conic sections. Unit 5: Multivariate Equations and Inequalities Unit 4, Concept 4 Standards HSA- REI.C.7 HSA- REI.D.11 HSG- GPE.A.1 HSG- GPE.A.2 HSG- GPE.A.3 Unit Overview: Students will focus on solving systems of inequalities by various methods, including graphing. They will write equations and inequalities modeling constraints and use the graphs of the inequalities to find optimal solutions. Students will deepen their understanding of systems of equations by including the solving of systems consisting of linear and quadratic equations, and they will extend their understanding of solving a linear system by graphing to encompass simple polynomial, rational, absolute value, exponential, and logarithmic functions. (Focus on MP.1, MP.2, MP.4) Concept 1: Investigate Linear Systems Concept Overview: Students extend their ability to solve systems of inequalities by various methods, including graphing. They apply their knowledge to real- world situations that require a greater degree of scrutiny than they have applied previously; students write equations and inequalities modeling constraints and use the graphs of the inequalities to find optimal solutions. Unit 5, Concept 1 Standards HSA- CED.A.2 HSA- CED.A.3 HSA- REI.C.6 HSN- Q.A.2 HSN- Q.A.3 Concept 2: Solve Nonlinear Systems Concept Overview: Student build on their understanding of systems of equations to include solving systems consisting of linear and quadratic equations, and they extend their understanding of solving a linear system by graphing to encompass simple polynomial, rational, absolute value, exponential, and logarithmic functions. Unit 5, Concept 2 Standards HSA- REI.C.7 HSA- REI.D.11 6

18 ALGEBRA II COURSE OVERVIEW Unit 6: Polynomial Expressions and Equations Unit Overview: Students will write polynomials in different forms, perform operations with polynomials, and interpret parts of polynomial expressions. They will develop an understanding of the relationship between the zeroes of a polynomial function and the factors of its function rule. They will write polynomials in different forms to identify roots of the related equations. Students will apply the rational root theorem, polynomial long division, synthetic division, and factoring to identify rational roots and the quadratic formula to identify irrational and complex roots. They will build on their earlier work with other types of functions as they identify and interpret key features of polynomial functions. They will sketch graphs of polynomial functions using characteristics identified from the equations of the functions. (Focus on MP.6, MP.7, MP.8) Concept 1: Operate with Polynomials Concept Overview: Students use their prior knowledge of quadratic and other types of expressions as they write polynomials in different forms, perform operations with polynomials, and interpret parts of polynomial expressions. They develop an understanding of the relationship between the zeroes of a polynomial function and the factors of its function rule. Students also apply factoring techniques learned when working with quadratic expressions to help them factor polynomials and identify zeroes of polynomial functions. In addition, students learn how they can use Pascal s triangle to help them expand powers of binomials. Unit 6, Concept 1 Standards HSA- SSE.A.1b HSA- SSE.A.2 HSA- APR.A.1 HSA- APR.B.2 HSA- APR.B.3 HSA- APR.C.4 HSA- APR.C.5 Concept 2: Explore Polynomial Factors Concept Overview: Students build on their work with quadratic functions as they develop an understanding of the relationship between the zeroes of a polynomial function and the factors of its function rule. They write polynomials in different forms to identify roots of the related equations. Students apply the rational root Unit 6, Concept 2 Standards HSA- SSE.B.3 HSA- APR.B.2 theorem, polynomial long division, synthetic division, and factoring to identify rational roots and the quadratic formula to identify irrational and complex roots. 7

19 ALGEBRA II COURSE OVERVIEW Concept 3: Analyze Polynomial Functions Concept Overview: Students expand on their earlier work with other types of functions as they identify and interpret key features of polynomial functions. They also sketch graphs of polynomial functions using characteristics identified from the equations of the functions. In addition, students write polynomial equations and develop polynomial functions to model and solve real- world problems. Unit 6, Concept 3 Standards HSA- APR.B.3 HSA- CED.A.1 HSF- IF.B.4 HSF- IF.B.6 HSF- IF.C.7c HSS- ID.B.6a Unit 7: Rational Expressions and Equations Unit Overview: Students will use mathematics concepts learned previously to enable them to explore how operations with rational expressions are analogous to operations with rational numbers. They will use their knowledge of writing rational numbers in different forms to help them write rational expressions. In addition, students will apply their understanding of factoring to identify and generate equivalent rational expressions. They will build on their earlier work with rational expressions as they write rational equations and inequalities in one variable in order to model real- world situations. In addition, they will apply what they have learned about algebraic properties to solve rational equations and inequalities, including those that give rise to extraneous solutions. (Focus on MP.1, MP.2, MP.7) Concept 1: Develop Rational Expressions Concept Overview: Students build on mathematics concepts learned previously, enabling them to explore how operations with rational expressions are analogous to operations with rational numbers. They use their knowledge of writing rational numbers in different forms to help them write rational expressions in different forms. In addition, students apply their understanding of factoring to identify and generate equivalent rational expressions. Unit 7, Concept 1 Standards HSA.SSE.A.2 HSA.APR.D.6 HSA.APR.D.7 Concept 2: Solve Rational Equations Concept Overview: Students use their earlier work with rational Unit 7, Concept 2 Standards expressions as they write rational equations and inequalities in HSA- CED.A.1 one variable in order to model real- world situations. In addition, HSA- REI.A.2 they apply what they have learned about algebraic properties to solve rational equations and inequalities, including those that give rise to extraneous solutions. 8

20 ALGEBRA II COURSE OVERVIEW Unit 8: Rational Functions Unit Overview: Students will use what they have learned about functions and rational expressions as they explore rational functions. They will graph rational functions and identify key features of the graphs. They will determine and interpret the mathematical and reasonable domains and ranges of rational functions. Students will learn to write simple rational functions and use the functions to investigate related one- variable equations and inequalities. They will analyze one- variable rational inequalities by writing functions for the expressions on either side of the inequalities. (Focus on MP.2, MP.4, MP.7) Concept 1: Represent Rational Functions Concept Overview: Students apply what they have learned about functions and rational expressions as they explore rational functions. They graph rational functions and identify key features of the graphs. They determine and interpret the mathematical and reasonable domains and ranges of rational functions. They also solve rational equations graphically or by using tables to find approximate solutions. Unit 8, Concept 1 Standards HSF- IF.C.7d HSF- IF.B.5 HAS- REI.D.10 Concept 2: Compare Rational Functions Concept Overview: Students learn to write simple rational functions and use the functions to investigate related one- variable equations and inequalities. Students analyze one- variable rational inequalities by writing functions for the expressions on either side of the inequalities. Using the zero(s) and the asymptote(s) of the functions, students determine solutions for the original inequalities. Unit 8, Concept 2 Standards HSA- CED.A.1 HSA- REI.D.11 Unit 9: Trigonometry Unit Overview: Students will focus on the relationships between arc lengths on the unit circle and radian measures of angles. They will learn how to convert between radians and degrees, and they will calculate arc lengths for any given angle measure and radius. Students will learn about the unit circle and how it is used to find the trigonometric values for all real numbers. They will use the Pythagorean theorem for ordered pairs on the unit circle to prove the Pythagorean identity, which they will use to find trigonometric quantities. Students will analyze the key features of trigonometric graphs, including period, midline, and amplitude, and they will compare transformations for these functions. They will use trigonometric functions to model real- world, periodic phenomenon. Students will relate the 9

21 ALGEBRA II COURSE OVERVIEW features of graphs of the trigonometric functions to characteristics of real- word scenarios and write equations describing real- world phenomena. (Focus on MP.2, MP.7, MP.8) Concept 1: Explore Angle Measures Concept Overview: Students learn the relationships between arc lengths on the unit circle and radian measures of angles. They learn how to convert between radians and degrees and calculate arc lengths for any given angle measure and radius. Concept 2: Represent Trigonometric Functions Concept Overview: Students focus on the unit circle and how it is used to find the trigonometric values for all real numbers. Students invoke the Pythagorean theorem for ordered pairs on the unit circle to prove the Pythagorean identity, which they use to find trigonometric quantities. Students analyze the key features of trigonometric graphs, including period, midline, and amplitude, and they compare transformations for these functions. Unit 9, Concept 1 Standards HSF- TF.A.1 Unit 9, Concept 2 Standards HSF- IF- C.7e HSF- TF.A.2 HSF- TF.A.3 HSF- TF.C.8 Concept 3: Apply Trigonometric Relationships Concept Overview: Students develop their knowledge of trigonometric functions to model real- world, periodic phenomenon. They relate the features of graphs of the trigonometric functions to characteristics of real- word scenarios and write equations describing real- world phenomena. They analyze real- world data in various forms and convert from one form to another when appropriate. Unit 9, Concept 3 Standards HSF- IF.C.9 HSF- TF.B.5 Unit 10: Probability Unit Overview: Students will describe a sample space using characteristics of the outcomes or as unions, intersections, or complements of other events. They will develop an informal understanding of conditional probability, recognizing that conditional probability is dependent on a primary event that has already occurred. Students will also construct and interpret two- way frequency tables of data and then use the two- way tables to decide if events are independent and to approximate conditional probabilities. They will formalize their understanding of conditional probability and independence. Students will find conditional probabilities, use the addition and multiplication rules, and interpret the probabilities. They will learn to analyze decisions and strategies using probability concepts and use probability to make fair decisions. (Focus on MP.1, MP.2, MP.4) 10

22 ALGEBRA II COURSE OVERVIEW Concept 1: Explore Conditional Probability Concept Overview: Students learn to describe a sample space using characteristics of the outcomes or as unions, intersections, or complements of other events. They understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities. In addition, students develop an informal understanding of conditional probability, recognizing that conditional probability is dependent on a primary event that has already occurred. They apply this understanding to the Monty Hall problem. They Unit 10, Concept 1 Standards HSS- CP.A.1 HSS- CP.A.2 HSS- CP.A.3 HSS- CP.A.4 HSS- CP.A.5 also construct and interpret two- way frequency tables of data and then use the two- way tables to decide if events are independent and to approximate conditional probabilities. Concept 2: Apply the Rules of Probability Concept Overview: Students formalize their understanding of conditional probability and independence. They find conditional probabilities, use the addition and multiplication rules, and interpret the probabilities. They learn to analyze decisions and strategies using probability concepts and use probability to make fair decisions. Unit 11: Data Modeling Unit 10, Concept 2 Standards HSS- CP.B.6 HSS- CP.B.7 HSS- CP.B.8 HSS- MD.B.6 HSS- MD.B.7 Unit Overview: Students will expand their understanding of sampling by exploring and evaluating a variety of sampling methods. They will investigate the use of experiments in testing hypotheses and analyze the ability of experiments to establish causal relationships. Students will focus on the characteristics of a normal distribution. They will use statistical pictures and numerical measures to analyze normally distributed data. Students will investigate the empirical rule and its role in describing normal distributions. (Focus on MP.3, MP.4, MP.5) Concept 1: Collect, Analyze, and Interpret Statistical Data Concept Overview: Students build upon their understanding of sampling by exploring and evaluating a variety of sampling methods. Students use simulations to develop an understanding of variability in sampling and learn to calculate the margin of error for data collected from a sample. Students investigate the use of experiments in testing hypotheses and analyze the ability of experiments to establish causal relationships. Unit 11, Concept 1 Standards HSS- IC.A.1 HSS- IC.A.2 HSS- IC.B.3 HSS- IC.B.4 HSS- IC.B.5 HSS- IC.B.6 11