NORTHERN VALLEY REGIONAL HIGH SCHOOL Office of Curriculum and Instruction

Mathematics Department Demarest and Old Tappan

Algebra 2

Unit I Functions Unit II Irrational and Complex Numbers Unit III Quadratic Expression, Equations and Functions Unit IV Polynomials Unit V Statistics Unit VI Exponential and Logarithmic Functions Unit VII Rational Functions Unit VIII Sequences and Series Unit IX Trigonometric Functions May 2015

Algebra II Unit I- Functions Time Frame: 5 weeks

Stage 1 – Desired Results Content Standard(s): F-BF-1a Write a function that describes a relationship between two quantities. Determine an explicit expression, a recursive process, or steps for calculation from a context. F-BF- 3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. F-BF- 4a Find inverse functions.  Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x ≠  1. F-BF- 4b Verify by composition that one function is the inverse of another. F-BF- 4c Read values of an inverse function from a graph or table, given that the function has an inverse. F-IF-4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. F-IF- 6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. F-IF-7c Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. F-IF-7e Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. F-IF-9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. F-LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Understanding (s)/goals:

Students will understand that: • Functions are relations between two variables that have

special properties. • Functions can be represented numerically, graphically,

analytically and verbally. • Functions are used to model real life phenomena. • Transformations and function operations can be used to

build new functions.

Essential Question(s): • How can real phenomena be modeled by functions? • How and when can technology help solve problems involving

functions? • How can we analyze and interpret the domain, range, intercepts,

asymptotes, local and global behaviors, and increasing/decreasing nature of a function?

• How can reasonable conclusions be drawn from modeling with functions?

Student objectives (outcomes): Students will be able to: • Use function notation. F-BF-1a, F-LE.2

• Introduce parent graphs of - linear: f (x) = mx + b , power: f (x) = xn , exponential: f (x) = abx , rational: f (x) = 1x

, radical: f (x) = x ,

absolute value: f (x) = x , greatest integer: f (x) = x[ ] , and piecewise defined functions. F-IF-4, F-IF-7c, F-IF-7e , F-IF-9 • Identify relations that are functions using numerical (table), graphical and algebraic methods. F-IF-9 • Find the domain and range of functions when represented numerically, graphically, and algebraically.

(Use inequality and interval notation) F-IF-4, F-IF-9 • Interpret functions in terms of a real application context. F-LE.2 • Find intercepts and explain their significance. F-IF-4 • Describe the effects of transformations on the graph of a function. Explore the graphs of f(x+a), f(x)+a, af(x), f(ax), f(|x|), |f(x)| for any

a≠0. F-BF- 3 • Identify even and odd functions algebraically, graphically and numerically. F-BF- 3, F-IF-9 • Identify intervals in which the function is increasing, decreasing, or is constant. F-IF-4 • Compare properties of two functions each represented in a different way: numerically, graphically, or algebraically. F-IF-9 • Perform algebraic operations on functions, and compute compositions. F-LE.2 • Find inverses algebraically, numerically and graphically, and identify functions that have inverses that are also functions. F-BF- 4a,b,c • Compute average rate of change of a function, using a table, graph and algebraic representations. F-IF- 6, F-IF-9 • Use functions to model real life phenomena. F-LE.2

**This unit should be assessed without a graphing calculator.

Algebra II Unit II- Irrational and Complex Numbers Time Frame: 3 weeks

Stage 1 – Desired Results Content Standard(s): A-SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2). m A-REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. m N-RN.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)*3 to hold, so (51/3)3 must equal 5. m N-RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents. m N-CN.1 Know there is a complex number i such that i 2= –1, and every complex number has the form a + bi with a and b real. a N-CN.2 Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. A N-CN.3 Find the conjugate of a complex number; Use conjugates to find moduli and quotients of complex numbers. N-CN.7 Solve quadratic equations with real coefficients that have complex solutions. A N-CN.8 (+) Extend polynomial identities to the complex numbers. For example, write 𝑥! + 4  as (x+2i)(x-2i). Understanding (s)/goals:

Students will understand that: • Rational exponents and radicals are equivalent ways of representing

irrational numbers • The properties of integer exponents can be extended to rational

exponents • Complex numbers are an extension of the real numbers • Adding, subtracting, multiplying and dividing complex numbers

produce new complex numbers

Essential Question(s):

• Why do we need rational exponents? • Can the real number system be extended? • Why do we need complex numbers?

Student objectives (outcomes): Students will be able to: • Solve radical equations with various indices and at most two radical terms. A-REI.2 • Identify extraneous solutions and determine why they exist. A-REI.2 • Simplify expressions involving rational exponents and radicals using various indices. A-SSE.2, N-RN.2 • Use polynomial identities using the (a + bi) form of complex numbers. N-CN.1, 8 • Compute powers of i. N-CN.2 • Add, subtract, and multiply complex numbers; use conjugates. A-SSE.2, N-RN.2 • Use conjugates to simplify and divide complex numbers. A-SSE.2, N-CN.3

Algebra II Unit III- Quadratic Expression, Equations and Functions Time Frame: 6 weeks

Stage 1 – Desired Results Content Standard(s): A-CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. s Α .CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. s

A-REI.4b Solve quadratic equations in one variable. 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. s

N-CN.7 Solve quadratic equations with real coefficients that have complex solutions. a F-BF.1 Write a function that describes a relationship between two quantities. m S-ID.6a Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. s G-GPE.2 Derive the equation of a parabola given a focus and directrix. a A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.a

A-REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3. a

A-REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. m-BF.4a Find inverse functions.  Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x ≠  1. a

F-IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. m F-IF.7c Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. S Understanding (s)/goals:

Students will understand: • Solving systems as a process of reasoning.

Essential Question(s): • How can we use quadratic functions to model real life phenomena? • Why do we need the different but equivalent forms of a quadratic function?

• The meaning of the solution to a system in context. • That systems of inequalities can be used to solve real

life problems. • That the solution obtained through graphing a system

of equations is often an approximation. • That visualizing the graphs of the lines helps determine

the number of solutions to a system of equations. • When each method is most effective in solving a

system. • How to classify solutions to a quadratic equation.

• How can we decide that the quadratic function will be the best fit for a real life situation?

Student objectives (outcomes): Students will be able to: • Distinguish between quadratic expressions and quadratic equations. • Solve quadratic equations by factoring, completing the square, the quadratic formula, and graphing a related function using technology.

A-REI.4b, A-REI.11, N-CN.7 • Extend the knowledge of the quadratics to find complex solutions of quadratic equations. A-REI.4b • Use the discriminant to identify the nature of solutions of a quadratic equation. A-REI.4b • Derive and use the sum and product formulas to find a quadratic equation given the roots with emphasis on complex roots. A-CED.1 • Interpret real solutions graphically. F-IF.4 • Interpret complex solutions graphically. F-IF.4 • Solve three-variable systems of equations by elimination and substitution. A.REI.6, A.REI.7, Α .CED.2 • Determine the number of real solutions to a system involving quadratics. • Solve systems of quadratic equations and inequalities both algebraically and graphically. A-REI.7, A-REI.11 • Given 3 points, find the equation of the quadratic function in the form, f (x) = ax2 + bx + c • Rewrite quadratic functions in standard, vertex, and factored form and identify advantages of each form. • Manually, graph parabolas using standard forms 𝑦 = 𝑎 𝑥 − ℎ ! + 𝑘    𝑎𝑛𝑑    𝑥 = 𝑎 𝑦 − ℎ ! + 𝑘  . Use completing the square to write the

equations in standard form. Produce the invertible function from a non-invertible function by altering the domain. F-IF.7c, F-BF.4a • Find the vertex, focus, directrix and axis of symmetry of a parabola, using the forms 𝑦 = 𝑎 𝑥 − ℎ ! + 𝑘    𝑎𝑛𝑑    𝑥 = 𝑎 𝑦 − ℎ ! + 𝑘  .. • Use completing the square to identify the vertex, focus, directrix and axis of symmetry of a parabola. F-IF.7c • Find the equation of a parabola given the focus and directrix, focus and vertex, vertex and directrix, or vertex and any point on the

parabola. F-IF.7c, G-GPE.2 • Use a given quadratic function that models a real life situation to solve problems. F-IF.4, F-BF.1 • Create quadratic equations, from data and using regression, to model and solve real life application problems. F-IF.4, F-BF.1, S-ID.6a

Algebra II Unit IV – Polynomials Time Frame: 4 weeks

Stage 1 – Desired Results Content Standard(s): Polynomial equations and functions

A.APR.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x). m A.APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. m A.APR.4 Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples. a A.APR.6 Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. s A.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2). F-IF.7c Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. s Understanding (s)/goals:

Students will understand: • The definition of a polynomial function. • The algebraic and graphical properties of polynomial

functions. • The relationship between the zeros of a polynomial

function and the roots of a polynomial equation.

Essential Question(s): • How can polynomial functions be used to model real life problems? • How can properties of linear and quadratic functions be generalized to

polynomial functions? • What is the Fundamental Theorem of Algebra? • What are advantages of the factored form of a polynomial?

Student objectives (outcomes): Students will be able to:

• Identify the degree, terms, and leading coefficient of a polynomial function. F-IF.7c • Recognize the shape of a polynomial function given its degree and leading coefficient. F-IF.7c • Identify even and odd polynomial functions. F-IF.7c • Graph polynomial functions using the TI graphing calculator and identify points of relative maximum and relative minimum. F-IF.7c • Use the graph of polynomial functions to determine intervals where the function in increasing and decreasing. F-IF.7c • Identify the end behavior of a polynomial function. F-IF.7c • Identify the real zeros of a polynomial function, and use them to write an equation of the function. A.APR.3 • Use polynomial functions to model real life optimization problems. F-IF.7c • Use the structure of a polynomial expression to identify ways to rewrite that expression (factor sum and difference of cubes, factor by grouping, difference of quadratics) ASSE.2

• Solve polynomial equations by factoring and graphing. A.APR.2 • Divide polynomials with binomials using long division and synthetic division. A.APR.2, A.APR.6 • Use the remainder and factor theorems to factor polynomials with degree greater than 2. A.APR.2 • List all the possible roots using the rational root theorem. A.APR.2 • Find the rational and imaginary roots of a polynomial function using the graphing calculator and the quadratic formula. A.APR.3, F-IF.7c

Algebra II Unit V - Statistics Time Frame: 3 weeks Stage 1 – Desired Results

Content Standard(s): S-ID 4 Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. S-IC.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population. S-IC.3 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. S-IC.4 Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling. S-IC.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. S-IC.6 Evaluate reports based on data. Understanding (s)/goals:

Students will understand how to: • Calculate probabilities using normal distributions. • Recognize data sets that are normal. • Distinguish between populations and samples. • Analyze hypotheses and methods of collecting data • Identify types of sampling methods in studies • Recognize bias in sampling and survey questions. • Interpret and use z-scores.

Essential Question(s): • In a normal distribution, about what percent of the data lies within one,

two, and three standard deviations of the mean? • How can you test theoretical probability using sample data? • What are some considerations when undertaking a statistical study?

Student objectives (outcomes): Students will be able to: • Use the 68-95-99.7 rule to determine whether a set of data is normal. S-ID.4 • Use z-scores and the graphing calculator to find probabilities and to compare data from two distributions. S-ID.4 • Use mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. S-ID.4 • Estimate and interpret areas under a normal curve. S-ID.4 • Determine whether something is a population or a sample. S-IC.1 • Determine whether a statistic from a random sample approximates a parameter. S-IC.1 • Analyze a hypothesis using simulation. S-IC.2 • Estimate a population mean or proportion from a random sample. S-IC.4 • Compare parameters for two populations using simulations from two samples. S-IC.5

Algebra II Unit VI- Exponential and Logarithmic Functions Time Frame: 4 weeks

Stage 1 – Desired Results Content Standard(s): Exponential and Logarithmic Functions A-SSE.3c Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12t 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. m A-CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. s A-REI. 1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. m F-IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. m F-IF.7e Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Graph exponential and logarithmic functions, showing intercepts and end behavior. s F- IF.8b Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay. s F-IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. s F-BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. A F-BF.5 Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. F-LE.4 For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. s F-LE.5 Interpret the parameters in a linear or exponential function in terms of a context. a S-ID.6a Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. s Understanding (s)/goals:

Students will understand: • the difference between exponential functions, quadratic

functions, polynomial functions and linear functions. • the relationship between exponential and logarithmic

Essential Question(s):

• How can exponential functions be used to model real life problems? • How do exponential function graphs compare to linear and quadratic

≈

functions. • the end behavior that occurs around the horizontal and

vertical asymptotes

function graphs?

Student objectives (outcomes): Students will be able to:

• Define, graph, and explore exponential functions, including domain, range, intercepts, and asymptotes. F-IF.4 , F-IF.7e , F-LE.5 • Define and use the natural base e. F-IF.7e • Translate exponential functions. F-BF.3 • Compare exponential graphs to quadratic, linear, and polynomial graphs. Use average rate of change to compare linear, quadratic, and

exponential functions over a given interval. F-IF.4, F-IF.9 • Define, graph, and explore the logarithmic function, including domain, range, intercepts, and asymptotes. F-IF.4 , F-IF.7e • Translate logarithmic functions. F-BF.3 • Use a graphing calculator to evaluate logs and natural logs. F-LE.4 • Prove and use the change of base theorem. F-LE.4 • Use the exponential property of logarithms. F- IF.8b • Solve exponential and logarithmic equations. A-CED.1, A-REI. 1, F-LE.4, F-BF.5 • Given data from real-life exponential applications, calculate the exponential regression equation by hand and with the calculator. Use

equation to solve problems. S-ID.6a

• Solve exponential application problems involving interest, growth, and decay. Use the formulas y = P(1+ r)t , y = P 1+ rn

⎛⎝⎜

⎞⎠⎟nt

, y = Pert ,

F-LE.4, A-SSE.3c, A-CED.1, A-REI. 1, F-LE.5, F- IF.8b

Algebra II Unit VII- Rational Functions Time Frame: 4 weeks

Stage 1 – Desired Results Content Standard(s): Rational Functions A.APR-7 – (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions

A.CED-1 - Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. s A.CED-2 - Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. s A.APR.6 - Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. A.REI-1 - Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. m A.REI-2 - Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. m A.REI-11 - Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. m F.BF.4a Find inverse functions. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x 1. F.IF-4 - For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. F.IF-5 – Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. m F.IF-7d – (+) Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

Understanding (s)/goals: Students will understand:

• The local and global behaviors of rational functions.

• That operations on rational functions are extensions of operations with rational numbers.

Essential Question(s):

• How can rational functions be used to model real life problems? • How are inverse variation and rational functions are related? • What are equivalent forms of rational functions and how are they

useful? • What do the vertical asymptotes of a rational function signify?

• What do the horizontal asymptotes of a rational function signify? Student objectives (outcomes):

Students will be able to:

• Identify direct and inverse variation between two variables, numerically and graphically. A.CED-2, A.REI-1 • Graph a rational function using transformations, with and without a graphing calculator. F.IF-7d, A.APR-6 • Find the domain and range of a rational function. F.IF-5 • Identify horizontal asymptotes of rational functions using end behavior. F.IF-7d • Identify the vertical asymptotes of a rational function. F.IF-7d • Identify the x and y-intercepts algebraically and using the graphing calculator. A.REI-11, F.IF-4 • Add, subtract, multiply, and divide rational expressions with polynomial parts. A.APR-6 • Rewrite complex rational expressions in simplest form. A.APR-6 • Solve rational equations algebraically and graphically. A.CED-1, A.REI-1,2 • Find the inverse of a rational function. F.BF-4a • Use rational functions to model real life application problems. A.REI-2, F.IF-4

Algebra II Unit VIII -Sequences and Series Time Frame: 3 weeks

Stage 1 – Desired Results Content Standard(s): F-IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥  1. s F-BF.1a Write a function that describes a relationship between two quantities.  Determine an explicit expression, a recursive process, or steps for  calculation from a context. m F-­‐BF.2  Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. m F-LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). s F-IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. m A-­‐SSE.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. m Understanding (s)/goals:

Students will understand: • That the recursive and explicit definitions of arithmetic and geometric sequences are equivalent. • That sequences are discrete functions. • The relationship between sequences and series.

Essential Question(s): • How can sequences be used to model real life situations? • What is the difference between arithmetic and geometric sequences? • How can sequences be graphed on the TI-84? • What is a series?

Student objectives (outcomes): Students will be able to:

• Define an arithmetic sequence using a table of values, a verbal description, a recursive description and an explicit formula. F-BF.1a, F-BF.2 • Define a geometric sequence using a table of values, a verbal description, a recursive description and an explicit formula. F-BF.1a, F-BF.2 • Recognize that an arithmetic sequence is a linear function whose domain is a subset of the integers. F-IF.3 • Recognize that a geometric sequence is an exponential function whose domain is a subset of the integers. F-IF.3 • Translate between recursive and explicit forms. F-BF.2 • Construct and compare arithmetic and geometric sequences using tables, discrete graphs or verbal descriptions. F-LE.2 • Find and interpret the common ratio and common difference given a recursive or explicit formula, a graph, or a table. F-IF.4 • Use arithmetic and geometric sequences to model real life situation. F-BF.2 • Use sigma notation to represent partial sums of arithmetic and geometric series. • Derive the formulas for the partial sums of arithmetic and geometric series (when the common ratio is not 1). A-SSE.4 • Use the partial sum formulas for arithmetic and geometric series to solve real life problems. A-SSE.4

Standards for Mathematical Practices 6. Attend to precision. 7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

Algebra II Unit IX - Trigonometric Functions Time Frame: 3 weeks

Stage 1 – Desired Results Content Standard(s): F-TF.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. F-TF.2 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. F-TF.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. F-TF.8 Prove the Pythagorean identity sin! θ + cos! θ = 1 and use it to calculate trigonometric ratios and the quadrant of the angle. F-IF.7e Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Graph trigonometric functions, showing period, midline, and amplitude. Understanding (s)/goals:

Students will understand: • The length of an arc subtended by an angle in standard form on

the unit circle represents radian measure of the central angle • The ordered pairs for the points around the unit circle are given

by (𝑐𝑜𝑠𝜃, 𝑠𝑖𝑛𝜃) • How the unit circle generates the parent graphs of 𝑦 = 𝑠𝑖𝑛𝜃

and 𝑦 = 𝑐𝑜𝑠𝜃. • How a, b, and d values from 𝑦 = 𝑎𝑠𝑖𝑛𝑏 𝜃 + 𝑑 and 𝑦 =

𝑎𝑐𝑜𝑠𝑏 𝜃 + 𝑑 effect the parent graph • How the graphs of trigonometric functions can model real

world periodic phenomena • How to draw triangles in the given quadrant to determine the

exact value of a trigonometric function with the appropriate sign

Essential Question(s): • What is the connection between arc length of a circle and

radian measure? • When is it better to use radians? Degrees? • How can you use the unit circle to define the trigonometric

functions of quadrantal angles? • What is the connection between the unit circle values and the

critical points of the graphs of the sine and cosine functions? • What are the similarities and differences between the graphs of

the sine and cosine function? • What are the characteristics of real-life problems that can be

modeled by trigonometric functions?

Student objectives (outcomes): Students will be able to: • Identify measures for angles that are in standard position in degrees. • Use the unit circle to prove the Pythagorean identity sin! 𝜃 + cos! 𝜃 = 1 F-TF.8 • Find the length of an arc subtended by an angle in the unit circle. F-TF.1 • Identify measures for angles that are in standard position in radian measure. F-TF.1 • Derive the parent graphs of 𝑦 = sin  (𝜃) and 𝑦 = cos  (𝜃) from the unit circle. • Graph sine and cosine functions that result from transforming the respective parent graphs (specifically: amplitude, frequency/change in

period, and vertical shifts/midline) F-IF.7e • Identify the amplitude, period, and vertical shift (midline) for a sine or cosine graph. F-IF.7e • Graph sine and cosine functions that model real world phenomena and derive a sine or cosine function that models periodic real world

phenomena. F-TF.5 • Find the exact value of 𝑠𝑖𝑛𝜃, 𝑐𝑜𝑠𝜃, or 𝑡𝑎𝑛𝜃 given the value of a trig function and the angle’s quadrant. F-TF.2

• Create sinusoidal equations, from data, using regression, to model and solve real life application problems. F-IF.4, F-BF.1b, S-ID.6a