F.IF.1 F.IF.2 F.IF.4 F.IF.5 A.SSE.2 A.CED.1 A.CED.2 G.CO.2 F.IF.7 F.IF.8 F.IF.9 F.BF.3 F.BF.4 F.BF.4a

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1 CURRICULUM MAP TEMPLATE Priority Standards = Approximately 70% Supporting Standards = Approximately 20% Additional Standards = Approximately 10% HONORS/ACCELERATED PRECALCULUS Essential Questions & Content How do you define real life data verbally, numerically, graphically and algebraically? What are functions and how can they be defined? How can you represent a function and what are the key features of each representation? When changes are made to an equation, what changes are made to the graph? Framework Standard Skills Assessment & Learner Expectations Sept/ Oct Prerequisite Concepts Students should already be able to: Graph linear equations and inequalities Systems of Equations Basic Complex Number Operations Simple Log Operations Solving Quadratic Equations Factoring Quadratics Graphing Quadratics Domain and Range Simple Operations on Functions Unit 1 Concepts: Distance/midpoint formulas Vertical/horizontal line test Linear/quadratic/cubic/qu artic functions F.IF.1 F.IF.2 F.IF.4 F.IF.5 A.SSE.2 A.CED.1 A.CED.2 G.CO.2 F.IF.7 F.IF.8 F.IF.9 F.BF.3 F.BF.4 F.BF.4a MP.1 MP.2 MP.3 MP.4 MP.5 MP.6 MP.7 UNIT 1: Function Preview Identify functions from a variety of representations. Distinguish between linear, exponential and quadratic functions from multiple representations Relate the domain of a function to its graph and to the context. Relate values of a function back to the original context. 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 function in increasing, decreasing, positive, or negative, relative maximums and minimums, symmetries, end behavior, and periodicity Transform graphs based on changes in equations and write equations based on a transformed parent graph. Explain the features of a function in relation to its context and to its mathematical structure. 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), and 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 TI Nspire Labs: Functions and Graphs Lab Folder GSP Examples Internet Research Ch ,

2 Absolute value/radical functions Greatest integer function Piecewise functions Circle equations Parent functions Horizontal and Vertical shrinks and stretches Increasing/decreasing functions Asymptotes and Domain restrictions with rational functions Relative maximum/minimums Vocabulary Function Domain/range/zeroes Independent variable Dependent variable Inverse Maximum/minimum Even/odd Composite Reflection/symmetry Phase shifts Rigid/nonrigid transformation Asymptotes expressions for them. Identify asymptotes and relate them to the restrictions of a function in algebraic form. Graph a function using basic transformations. Compare transformations that preserve distance and angle to those that do not. Create inverse functions, and 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. and composite functions Create compositions of functions by accurately using function notation. Use compositions to prove inverse relationships. 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. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales

3 Nov/ Dec What is a polynomial? How does the degree of a polynomial affect its graph? How do the factors of a polynomial affect its graph? How is the domain of a rational function related to its graph? How can rewriting the equation of a rational function (using long division of polynomials) give further information about its graph? Prerequisite Concepts: Students should already be able to: Factor a quadratic expression over the integers. Determine zeros from a factored form of a quadratic. Manipulate a quadratic function between various forms to determine key features, including zeros, the vertex, maximum/minimum value, and end behavior. Determine domain and range of functions involving simple rational expressions. Unit 2 Concepts: Synthetic substitution/division Rational zero test Factor/remainder theorems Intermediate value theorem Leading coefficient test Upper/lower bound rules A.SSE.2 A.CED.1 A.CED.2 A.REI.1 A.REI.2 A.APR.2 A.APR.3 A.REI.11 F.IF.4 A.APR.6 S.ID.6 MP.1 MP.2 MP.4 MP.5 MP.6 MP.7 UNIT 2: Polynomial and Rational Functions Prove and use polynomial identities Graph a polynomial given in factored form, indicating all intercepts and directions of end behaviors. Use the structure of an expression to identify ways to rewrite it. Create equations in one, two, or more variables and use them to solve problems. Construct viable arguments to justify a solution method. Write the equation of a polynomial function given its graph or defining characteristics of its graph. Solve rational equations in one variable, checking for extraneous solutions. Perform the long division algorithm for polynomials in order to rewrite simple rational expressions in different forms; use computer algebra systems to perform the same on complicated examples. Perform the synthetic division method to solve for zeros, rewrite polynomials to find binomial factors. Use direct/synthetic substitution to evaluate polynomials Use technology (graphs, tables) to solve the equation f(x) = g(x), where f(x) and/or g(x) are polynomial or rational functions. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Apply the Remainder/Rational Root 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). Identify zeros of polynomials when suitable factorizations are available, and use the CIM- Make Open Box GSP- Polynomial Surprises TI Nspire Labs Polynomial, Power and Rational Functions Labs Folder Ch

4 Asymptotes Sign analysis Vocabulary: Polynomial Degree Constant Cubic Quartic Quintic Rational/complex Synthetic/direct Intermediate value Integral coefficients Odd/even multiplicity Critical points zeros to construct a rough graph of the function defined by the polynomial. 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. Simplify complex numbers by using magnitudes and conjugates. Recognize visual representations of higher degree polynomials and their characteristics. Apply Descartes Rule of Signs to determine upper and lower bound. Design polynomial functions to represent real- world applications Analyze graphs of rational functions Solve polynomial/rational inequalities Write a polynomial equation and/or function to model a real- life situation. Use a model of a polynomial function to interpret information about a real- life situation. Translate among representations of polynomial functions including tables, graphs, equations and real- life situations. Rewrite polynomial and rational equations to reveal new information. Create a scatterplot from data and interpret the relationship of the quantities represented. Appropriately fit a model to data

5 Dec/ Jan What is a logarithm? How are logarithms and exponentials related? How can exponential equations be solved for an unknown in the exponent? What are the key features of the graph of a logarithmic function? How can exponents and logarithms apply to real- life situations? Prerequsite Concepts: Students should already be able to: Write, graph, and interpret exponential functions. Solve equations with unknowns in the exponent by inspection or use of exponent rules. Identify the effects of the transformations f(x) + k, k f(x), f(kx), and f(x + k). Unit 3 Concepts: Properties of exponents The definition of a logarithm. The graph of f(x) = log (x) has a domain of x>0, a vertical asymptote at x=0, and an x- intercept at x = 1. Change of base formula. Exponential functions Growth and decay formula Common/natural logarithms Base e Vocabulary: Integral and Rational exponents Transcendental A.CED.1 A.CED.2 A.REI.1 F.IF.4 F.IF.6 A.SSE.3c F.IF.7e A.SSE.4 F.LE.4 F.LE.5 S.ID.6 MP.2 MP.4 MP.5 MP.7 UNIT 3: Exponentials ad Logarithms Review exponential laws and simplifications, including integral and rational exponents Define and identify transcendental numbers Use logarithms to solve exponential equations in base 2, 10, or e. Evaluate logarithms based on the definition for simple cases. Evaluate logarithms using the change of base formula with technology. Graph exponential functions, identifying intercepts and end behavior. Graph logarithmic functions, identifying intercepts and end behavior. Construct a viable argument to justify a solution method when solving exponential equations Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. Derive the formula for the sum of a finite geometric series Determine the best function to fit a certain situation or set of data. Use technology to fit exponential models to data. Model applied situations using exponential and logarithmic functions and answer questions using those models. Apply exponents to model population growth, business/finance, health/medicine, and physics/science Reason quantitatively and use units to solve problems. 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. TI Nspire Labs: Exponential and Logarithmic Functions Labs Folder (230): Sequences and Series Labs Folder GSP Examples Internet Research Ch Ch

6 Feb/ March Asymptote Exponential Logarithmic Inverse Growth/decay Half life Natural logarithms Common logarithms Change of base Limit Sequence Series How is the probability of events found? How is statistics used? Prerequisite Concepts: Students should already be able to: Represent sample spaces. Apply basic properties of probability. Use two- way frequency tables. Use P( A B) as the probability of A and B occurring together. Create visual displays of data sets. Analyze data using statistical measures The definition of event, sample space, union, intersection, and complement. Identify independent events. The definition of dependent events and conditional probability Addition Rule with unions Permutations and Combinations Unit 4 Concepts: Multiplication Rule with S.CP.8 S.CP.9 S.IC.1 S.IC.2 S.IC.3 S.IC.4 S.IC.5 S.IC.6 S.ID.4 S.ID.6 MP.2 MP.3 MP.4 MP.5 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the instantaneous rate of change from a graph. Interpret the parameters in a log or exponential function in terms of a context and by using inequalities UNIT 4: Probability and Statistics Probability: Review probabilities from Integrated Algebra 2 curriculum, Unit 7 Apply the general Multiplication Rule in a uniform probability model, P(A and B), and interpret the answer in terms of the model. Use permutations and combinations to compute probabilities of compound events and solve problems. Statistics: Review statistics from Integrated Algebra 2, Unit 7 Use the mean and standard deviation for a data set to fit the data to a normal curve (normal distribution) and estimate population percentages. Use Z scores. 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 function or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models Estimate the area under a normal curve using technology and explain the significance of this value in terms of probability and the original TI Nspire Labs: Exploring the Normal Curve Family Z Scores Assessing Normality Normal Probability Plot Looking Normal GSP Examples Internet Research Ch ,9.7 (also used Pearson Text here)

7 intersections Standard Deviation Normal distributions and Z scores Scatterplots Normal distributions are only appropriate for some data. Scatter plots can only be used to represent quantitative variables. The role of randomization in sample surveys, experiments, and observational studies. The difference between variables as quantitative or categorical Vocabulary Critical Terms: Intersection Inference Population parameter Random sample Statistics Randomization Population mean/sample mean Margin of error Confidence interval Standard deviation Bias Supplemental Terms: Sample space Subset Outcome Union Complement Set notation Joint probability Event and Independent events Conditional probability Inference Simulation Model context. Sketch the function of best fit on a scatter plot and find the function using technology when necessary. Choose a probability model for a problem context. Conduct a simulation of a model and determine which results are typical or considered outliers. Calculate a sample mean or population. Determine how often the true population mean or proportion is within the margin of error of each sample mean or proportion. Conduct a simulation for each group using sample results as the parameters for the distributions.

8 April How do you use/read a unit circle (using radians)? How does the Pythagorean theorem and the unit circle relate to the identity sin 2 (θ) + cos 2 (θ) = 1? What do the key features of a trigonometric function represent? What are the different ways you can represent a trigonometric function? What transformations can occur to a trigonometric function/graph and how do you know which one is which? Prerequisite Concepts: Students should already be able to: Define a function and the different parts in expressions, equations, and functions. Define trigonometric ratios and solve problems involving right triangles. Graph, evaluate, and solve trigonometric functions. Should be able to work in degrees and radians. F.TF.1 F.TF.2 F.TF.8 A.SSE.2 F.IF.7e F.IF.9 F.BF.3 F.IF.6 MP.3 MP.6 MP.7 MP.8 UNIT 5: Unit Circle Trigonometry Rewrite trigonometric expressions/equations to solve trigonometric identities. Prove the Pythagorean Identity. Graph a trigonometric function Extract information from trigonometric equations and graphs Convert radians to degree and degrees to radians Find reference angles, and use them to calculate trig values Complete a unit circle and use it to determine trigonometric values of special angles. Find all six trig function values Translate among representations of trigonometric functions including tables, graphs, equations, and real- life situations. Calculate the area of a sector, and linear and angular velocities. Calculate or interpret the rate of change of a trig function (230) Find inverse trig values TI Nspire Labs: Trigonometry(Triangular and Circular) Labs Folder CIM- Trigonometry GSP Examples Internet Research Ch , 4.7 Unit 5 Concepts: Students will understand that The unit circle allows all real numbers to work in trigonometric functions. Trigonometric identities can be proven and used to solve problems with specified context. Key features in graphs and tables shed light on relationships between two quantities with trig

9 Trigonometric functions can be represented by a table, graph, verbal description or equation, and each representation can be transferred to another representation Specific transformations occur to trigonometric functions based on a value k and its manipulation to the function May/ June Vocabulary: Angle Initial/terminal ray Unit circle Degree/minute/second Radian Standard position Amplitude Frequency Period Midline Coterminal/quadrantal Arc/sector Revolution Sine/cosine/tangent Secant/cosecant/cot Reference angle How does the study of trigonometry relate to real- world periodic phenomena? What are key characteristics to identify when choosing a function to model a given situation? How can the graphs of trigonometric functions be modified to best fit the situations being modeled. How do factors such as A.CED.1 A.CED.2 N.Q.2 F.IF.4 F.IF.6 A.REI.11 S.ID.6a F.TF.3 F.TF.4 F.TF.5 F.TF.6 F.TF.7 F.TF.9 UNIT 6: Applications of Trigonometry Modify f(x)=asin[b(x+c)] +D to model various real life situations and contexts. They should also be able to explain the contextual meaning of points on these functions and the significance of values and solutions. For a trig function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs TI Nspire Labs: (230): Trig Laws and Identities Labs Folder (230): Applications of Trig Labs Folder GSP Examples Internet Research

10 amplitude, period, midline, and horizontal shift affect these functions and relate to the phenomena being modeled? Prerequisite Concepts: Graph, evaluate, and solve trigonometric functions. Should be able to work in degrees and radians Unit 6 Concepts: Students will understand that The trigonometric functions sin(x) and cos(x) can be used to model real- life situations that exhibit periodic behavior. Changing parameters such as amplitude, period, and midline of a function will alter its graph and that these parameters are related to the context or phenomena being modeled. Understand there are trigonometric/geometric relationships, and apply them 230: Manipulate trig equations to prove identities by using formulas Solve with inverse concept Identify even/odd functions Use, graph and apply parametric equations Rectangular and polar conversions Vocabulary: Right triangle trigonometry Angles of elevation/depression Navigation/compass G.SRT.11 G.SRT.10 MP.1 MP.4 MP.6 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. Calculate and interpret the average rate of change of a trig function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. 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 trig 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. Define appropriate quantities for the purpose of descriptive modeling and create equations in one, two, or more variables Model real- world applications including navigation, aviation, and surveying Sketch appropriate trigonometric models (230) Restrict a trig function to a domain that allows for an inverse. (230) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context (230) Use special triangles to determine geometrically the values of sine, cosine, tangent for ð/3, ð/4 and ð/6, and use the unit circle to express the values of sine, cosine, and tangent for ð x, ð+x, and 2ð x in terms of their values for x, where x is any real number. Ch ,4.6,4.8

11 bearings SOHCAHTOA Inverses Regular Period/amplitude Phase shift Trigonometric identity Periodic Trigonometric models Reciprocal/Pythagorean/ Cofunction/Negative Trigonometric Relationships Sum and difference formulas Double angle formulas Rectangular coordinates Polar coordinates Parametrics (230) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. (230) Apply trig formulas, including identities, angle formulas, to rewrite trig functions (230) Generate double angle formulas (230) Interpret more advanced trig graphs (230) Convert between rectangular and polar coordinates (230) Input data into graphing calculator for parametric problems