# Algebra II Pacing Guide Last Updated: August, Guiding Question & Key Topics

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1 1-14 Unit 1 Investigations & AS I investigate functions, am I analyzing the function thoroughly and clearly communicating my reasoning to others? Solving puzzles in Teams Using a Graphing Calculator to Explore a Domain & Range Points of Intersection in Multiple Representations Modeling a Geometric Relationship Investigation The Family of Linear 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. quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers F-IF.7B: Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions Unit 2 Transformations of Parent Graphs How can I model this everyday situation with mathematics? Modeling Non-Linear Data Parabola Investigation Graphing a Parabola Without a Table Rewriting in Graphing Form Mathematical Modeling with Parabolas Transforming Other Parent Graphs Describing (h, k) for Each Family of Transformations of Transforming Non- Transforming Piecewise-Defined A-SSE.1A: Interpret parts of an expression, such as terms, factors, and coefficients. as a single entity. For example, interpret P(1+r) n as the product of P and a factor not F-IF.7B: Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. F-IF.8A: Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. 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 Unit 3 Equivalent Forms How can I show that these forms are equivalent? Equivalent Expressions Rewriting Expressions and Determining Equivalence Solving by Rewriting Investigating Rational Simplifying Rational Expressions Multiplying & Dividing Rational Expressions Adding & Subtracting Rational Expressions Creating New A-APR.1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. A-APR.4: Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x 2 + y 2 ) 2 = (x 2 y 2 ) 2 + (2xy) 2 can be used to generate Pythagorean triples. A-SSE.1A: Interpret parts of an expression, such as terms, factors, and coefficients. as a single entity. For example, interpret P(1+r) n as the product of P and a factor not 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.

2 48-60 Unit 4 Solving & Intersections Which tools can I use to solve the problems and verify my solutions? Strategies for Solving Equations Solving Equations & Systems Graphically Finding Multiple Solutions to Systems of Equations Using Systems of Equations to Solve Problems Solving Inequalities with One or Two Variables Using Systems to Solve a Problem Application of Systems of Linear Inequalities Using Graphs to Find Solutions as a single entity. For example, interpret P(1+r) n as the product of P and a factor not example, see x 4 y 4 as (x 2 ) 2 (y 2 ) 2, thus recognizing it as a difference of squares that can be factored as (x 2 y 2 )(x 2 + y 2 ). A-REI.2: Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. 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. between quantities; graph equations on coordinate axes with labels and scales. A-CED.3: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods Unit 5 Inverses & How can I use the reflective nature of inverse graphs to find the equations for inverses? Undo Equations Using a Graph to Find an Inverse Finding Inverses & Justifying Algebraically Finding the Inverse of an Exponential Defining the Inverse of an Exponential Investigating the Family of Logarithmic Transformations of Logarithmic Investigating Compositions of F-BF.4A: 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-CED.4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm s law V = IR to highlight resistance R. where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. 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. example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. 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-IF.8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. F-BF.1B: Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

3 76-81 Unit 6 3-D Graphing & How can I apply what I learned about graphs and equations in two dimensions to threedimensional situations? Creating a Three-Dimensional Model Graphing Equations in Three Dimensions Systems of Three-Variable Equations Solving Systems of Three Equations with Three Unknowns Using Systems of Three Equations for Curve Fitting Using to Solve Exponential Equations Investigating the Properties of Writing Equations of Exponential An Application of First Semester Review & Final Exam where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. F-LE.5: Interpret the parameters in a linear or exponential function in terms of a context. F-BF.1B: Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model Unit 6 3-D Graphing & How can I apply what I learned about graphs and equations in two dimensions to threedimensional situations? Creating a Three-Dimensional Model Graphing Equations in Three Dimensions Systems of Three-Variable Equations Solving Systems of Three Equations with Three Unknowns Using Systems of Three Equations for Curve Fitting Using to Solve Exponential Equations Investigating the Properties of Writing Equations of Exponential An Application of where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. F-LE.5: Interpret the parameters in a linear or exponential function in terms of a context. F-BF.1B: Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model Unit 7 Trigonometric How can I use what I know about right-triangle trigonometry to describe functions determined by rotations about a circle? Introduction to Cyclic Models Graphing the Sine Unit Circle <-> Graph Graphing & Interpreting the Cosine Defining a Radian Building a Unit Circle The Tangent Transformations of y = sin x One More Parameter for a Cyclic Period of a Cyclic Graph <-> Equation F-TF.5: Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. F-TF.8: Prove the Pythagorean identity sin 2 (θ) + cos 2 (θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle. 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-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.

4 Unit 8 Polynomials Unit 9 Randomization & Normal Distributions Unit 10 Series How can the degree of a polynomial help me determine the nature of its graph or a possible equation? Sketching Graphs of Polynomial More Graphs of Polynomials Stretch Factors for Polynomial Introducing Imaginary Numbers Complex Roots More Complex Numbers & Equations Polynomial Division Factors & Integral Roots An Application of Polynomials How can I use the appropriate degree of precision for this problem in my mathematics, in my vocabulary, and in how I ask survey questions? Comparing Single-Variable Data Survey Design Samples & the Role of Randomness Bias in Convenience Samples Testing Cause & Effect With Experiments Conclusions From Studies Relative Frequency Histograms The Normal Probability Density Percentiles How can I extend the pattern from smaller sums to larger sums? Introduction to Arithmetic Series More Arithmetic Series General Arithmetic Series Summation Notation & Combinations of Series Geometric Series Infinite Series Pascal s Triangle & the Binomial Theorem The Number e 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. 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.7C: Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. N-CN.9: (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. 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. N-CN.7: Solve quadratic equations with real coefficients that have complex solutions. N-CN.2: Use the relation i 2 = 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. 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). 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. N-CN.8: Extend polynomial identities to the complex numbers. For example, rewrite x as (x + 2i)(x 2i). quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers S-IC.6: Evaluate reports based on data. 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.2: Decide if a specified model is consistent with results from a given datagenerating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model? 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. 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. as a single entity. For example, interpret P(1+r) n as the product of P and a factor not A-APR.4: Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x 2 + y 2 ) 2 = (x 2 y 2 ) 2 + (2xy) 2 can be used to generate Pythagorean triples. A-APR.5: Know and apply the Binomial Theorem for the expansion of (x + y) n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal s Triangle. F-IF.8B: 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. where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

5 Unit 11 Simulating Sampling Variability How can I model this situation with a simulation in order to understand it better and to solve the problem? Simulations of Probability More Simulation of Probability Simulating Sampling Variability Statistical Test Using Sampling Variability Variability in Experimental Results Quality Control Statistical Process Control Analyzing Decisions & Strategies S-IC.2: Decide if a specified model is consistent with results from a given datagenerating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model? 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.1: Understand statistics as a process for making inferences about population parameters based on a random sample from that population. S-IC.5: Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. S-MD.6: (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). S-MD.7: (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). S-IC.6: Evaluate reports based on data Second Semester Review & Final Exam *** Each unit includes approximately 2 class periods for review, and 1 class period for an exam. Quizzes will be mixed in as deemed appropriate. *** There is about a week of semester review scheduled prior to each week of finals.

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