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Swampscott High School Math Department Algebra II Curriculum Algebra 2 331: Book: Algebra 2 Common Core - Pearson Chapter 1: Expressions, Equations, and Inequalities (Review of Algebra 1) Essential Question(s) • How do variables help you model real world situations? (You can use variables to represent variable quantities in real-world situations and patterns) • How can you use the properties of real numbers to simplify algebraic expressions? (The properties that apply to real numbers also apply to variables that represent them) • How do you solve an equation or inequality? (You can use properties of numbers and equality (or inequality) to solve an equation (or inequality) by finding increasingly simpler equations (or inequalities) which have the same solution as the original equation or inequality) CC Standard A.SSE.3 Lesson: Objective 1.1 Patterns and Expressions (This is review of Algebra 1) To Identify and Describe patterns N.RN.3 1.2 Properties of Real Numbers (This is review of Algebra 1) 1.3 Algebraic Expressions (This is review of Algebra 1) To graph and order real numbers To identify properties of real numbers To model words with an algebraic equation To evaluate algebraic expressions To simplify algebraic expressions To solve equations: one step, multi-step, equations with no solutions and identities To solve literal equations To solve problems by writing equations To solve and graph inequalities To write and solve compound inequalities A.SSE.1a A.SSE.1a A.CED.4 1.4 Solving Equations A.CED.1 1.5 Solving Inequalities A.SSE.1b A.CED.1 1.6 Absolute Value Equation and Inequalities To write and solve equations and inequalities involving absolute value Chapter 2: Functions, Equations, and Graphs (Review of Algebra 1) Essential Question(s) • Does it matter which form of a linear equation you use? You can use either slope-intercept, point-slope, or standard form to represent linear functions. You can transform one version to another as needed) • How do you use transformations to help graph absolute value functions? (You can use the values of a, h, and k in the form = | − ℎ| + to determine how the parent function = || has been transformed • How can you model data with a linear function? (You can use the equation of a trend line or line of best fit to model data that cluster in a linear pattern.) CC Standard Lesson Objective F.1F.1 F.1F.2 2.1 Relations and Functions (Algebra 1 - F.1F.2) A.CED.2 2.2 Direct Variation A.CED.2 F.1F.4 2.3 Linear Functions and Slope Intercept Form F.1F.8 F.1F.9 F.1F.2 2.4 More about Linear Equations To graph relations, To find the domain and range of a function, To identify functions To write and evaluate a function To identify direct variation from tables To identify direct variation from equations To solve direct variation equations using proportions To use direct variation equations to solve a problem To find the slope of a line To write equations of lines in slope-intercept form To graph a linear equation To write an equation of a line given its slope and a point on the line To write an equation of a line given two points To write an equation of a line in standard form A.CED.2 Graph an equation of a line using intercepts To write equations of parallel and perpendicular lines To graph piecewise-defined functions, including step functions and absolute value functions F.1F.7.b Piecewise Functions F.1F.4 A.CED.2 F.1F.6 2.5 Using Linear Models To write linear equations that model real-world data To make predictions from linear models To describe the correlation of a scatter plot To write an equation of a trend line To find the line of best fit using a calculator F.BF.3 F.BF.3 F.1F.7.b 2.6 Families of Functions 2.7 Absolute Value Functions and Graphs A.CED.2 F.1F.7b 2.8 Two-Variable Inequalities To analyze transformations of functions: vertical, horizontal, reflections To graph absolute value functions To identify transformations of an absolute value function from a graph To write an absolute value function given the graph To graph two variable inequalities To Graph an absolute value inequality To write an absolute value inequality based on a graph Chapter 3 : Linear Systems (Review of Algebra 1 except 3.4 and 3.5) Essential Question(s) • How does representing functions graphically help you solve a system of equations? (Find a point of intersection (x,y) of the graphs of functions f and g and you have found a solution to the system y = f(x), y = g(x) • How does writing equivalent equations help you solve a system of equations? (If the equations of two systems are equivalent, then a solution of the system that is easier to solve is also a solution of the more difficult system) CC Standard Lesson A.CED.2 3.1 Solving Systems Using tables and Graphs A.REI.6 A.REI.11 Objectives To solve a linear system using a graph or table To solve a system using linear regression on the calculator To classify a system as independent, dependent, or inconsistent A.CED.3 A.REI.6 A.REI.5 A.CED.2 A.REI12 A.REI.6 A.CED.3 3.2 Solving Systems algebraically To solve linear systems algebraically using substitution and elimination 3.3 Systems of Inequalities To solve systems of linear inequalities To solve a linear/absolute value system A.CED.3 3.4 Linear Programming To solve problems using linear programming To solve linear programming using the graphing calculator A.REI.6 3.5 Systems with three variables To solve systems in three variables using elimination To solve systems in three variables using substitution Chapter 4: Quadratic Functions and Equations Essential Question(s) • What are the advantages of a quadratic function in vertex form? (Vertex form of a quadratic function shows the vertex of the parabola. Standard form is “calculator ready.” Both forms give additional information.) • How is any quadratic function related to the parent quadratic function? (Any quadratic function is possibly a stretch or compression, a reflection, and a translation of y = x2) • How are the real solutions of a quadratic equation related to the graph of the related quadratic function? ( The real solutions of a quadratic equation show the zeros of the related quadratic function and the x-intercepts of its graph) CC Standard F.BF.3 A.CED.1 F.1F.4 F.1F.6 Lesson Objective 4.1 Quadratic Functions and Transformations To identify and graph quadratic functions To identify vertex form To use vertex form to identify transformations of a quadratic function To write a quadratic function in vertex form given the vertex and a point A.CED.2 F.1F.4 4.2 Standard Form of a Quadratic Function To graph quadratic functions in standard form F.1F.6 F.1F.8 F.1f.9 F.1F.5 F.1F.4 4.3 Modeling with Quadratic Functions F.1F.6 Concept Byte: Identifying Quadratic Data 4.4 Factoring Quadratic Expressions Square Roots and Radicals A.SSE.2 Reviews N.RN.2 A.CED.1 A.APR.3 A.SSE.1.a A.REI.4.b To identify the features of a quadratic function including the axis of symmetry, the minimum or maximum value, the domain and range, and the y-intercept To convert standard form to vertex form To interpret a quadratic graph To model data with quadratic functions To write an equation of a parabola given three points To use quadratic regression to find a quadratic model To determine if data represents perfect quadratic data To find common and binomial factors of quadratic expression To factor special quadratic expressions: perfect square trinomials, difference of two squares To simplify radical expressions 4.5 Quadratic Equations To solve quadratic equations by factoring To solve quadratic equations by graphing 4.6 Completing the Square To solve by finding the square roots To solve equations by completing the square To re-write a quadratic function in standard form to vertex form by completing the square To solve quadratic equations using the Quadratic formula To determine the number of solutions by using the discriminant To identify, graph, and perform operations with complex numbers To find complex number solutions of quadratic equations A.REI.4.b 4.7 The Quadratic Formula N.CN.1 N.CN.2 N.CN.7 N.CN.8(+) A.CED.3 A.REI.7 A.REI.11 4.8 Complex Numbers 4.9 Quadratic Systems To solve and graph systems of linear and quadratic equations To solve and graph systems of quadratic inequalities Chapter 5: Polynomials and Polynomial Functions Essential Question(s) • What does the degree of a polynomial tell you about its related polynomial function? (A polynomial of degree n has n linear factors. The graph of the related function crosses the x-axis an even or odd number of times depending on whether n is even or odd.) • For a polynomial function, how are the factors, zeros, and x-intercepts related? ( (x - a) is a linear factor if and only if a is a zero, and if and on if (a , 0) is an x-intercept when a is a real number.) • For a polynomial equation, how area the factors and roots related? ((x - a) is a linear factor if and only if a is a root of the related polynomial equation. CC Standard F.1F.7.c A.SSE.1.a Lesson Objective 5.1 Polynomial Functions F.1F.7.c A.APR.3 5.2 Polynomials, Linear Factors, and Zeros A.REI.11 A.SSE.2 A.APR.2 A.APR.1 A.APR.6 N.CN.7 N.CN.8(+) N.CN.8 (+) N.CN.9 (+) 5.3 Solving Polynomial Equations To classify polynomials by degree and number of terms To describe the end behavior of polynomial functions To graph polynomial functions To look at a table of values and using differences to determine the degree of the polynomial To write a polynomial in factored form To find the zeros of a polynomial function in factored form To write a polynomial function from its zeros To find the multiplicity of a zero To identify a relative maximum and minimum To solve polynomial equations by factoring To solve polynomial equations by graphing To divide polynomials using long division To divide polynomials using synthetic division 5.4 Dividing Polynomials 5.5 Theorems About Roots of Polynomial Equations 5.6 The Fundamental Theorem of Algebra To solve equations using the Rational Root Theorem To use the conjugate root theorem (+) To use the Fundamental Theorem of Algebra to solve polynomial equations with complex solutions A.APR.3 A.APR.5 (+) F.1F.5 F.1F.4 F.1F.6 F.BF.3 F.1F.7.c F.1F.8 F.1F.9 Concept Byte: Graphing Polynomials Using Zeros 5.7 The Binomial Theorem (+) 5.8 Polynomial Models in the Real World 5.9 Transforming Polynomial Functions To identify zeros when suitable factorizations are available and use the zeros to construct a rough graph of the function defined by the polynomial To expand a binomial using Pascal’s Triangle To use the binomial theorem To fit data to linear, quadratic, cubic or quartic models To use interpolation and Extrapolation To apply transformations to graphs of polynomials Chapter 6: Radical Functions and Rational Exponents Essential Questions To simplify the nth root of an expression, what must be true about the expression? (You can simplify the nth root of an expression that contains an nth power as a factor √n = = x if n is odd or || if n is even. ) When you square each side of an equation, is the resulting equation equivalent to the original? ( When you square each side of an equation, the resulting equation may have more solutions than the original equaltion.) How are a function and its inverse function related? ( If f and f—1 are inverse functions and if one maps a to b, then the other maps b to a, i.e., (° −1 )() = ( −1 °)() = . ) CC Standard Lesson Objective Prepares for N.RN.1 A.SSE.2 Concept Byte: Properties of Exponents Simplify expressions using the properties of exponents 6.1 Roots and Radical Expressions A.SSE.2 6.2 Multiplying and Dividing Radical Expressions A.SSE.2 6.3 Binomial Radical Expressions To find nth roots To simplify radical expressions To multiply and divide radical expressions To rationalize the denominator when simplifying a radical expression To add and subtract radical expressions To multiply binomial radical expressions To multiply conjugates Reviews N.RN.2 N.RN.1 A.REI.2 A.CED.4 F.BF.1.b F.BF.1.c F.BF.4.a F.BF.4.c F.1F.7.b F.1F.8 6.4 Rational Exponents To simplify expressions with rational exponents To convert between exponential and radical forms To combine radical expressions To simplify numbers with rational exponents 6.5 Solving Square Root and Other Radical Equations To solve square root and other radical equations To check for extraneous solutions 6.6 Function Operations To add, subtract, multiply and divide functions To find the composite of two functions 6.7 Inverse Relations and Functions To find the inverse of a relation or function To graph a function and its inverse To identify the domain and range of a function and its inverse To find the composition of inverse functions 6.8 Graphing Radical Functions To graph square root and other radical functions To solve a radical equation by graphing To re-write a radical function so you can graph it using transformations Chapter 7: Exponential and Logarithmic Functions Essential Questions: • How do you model a quantity that changes regularly over time by the same percentage? (The function y= , > 0, > 1, represents exponential growth. = models exponential decay if 0 < b < 1. ) How are exponents and logarithms related? (Logarithms are exponents. In fact, = if and only if = ) How are exponential and logarithmic functions related? (The exponential function y = bx and the Logarithmic function y = logbx are inverse functions CC Standard Lesson F.1F.7.e 7.1 Exploring Exponential Models A.CED.2 F.1F.8 Objective To graph an exponential function To identify exponential growth and decay To model exponential growth and decay A.SSE.1.b F.1F.8 F.1F.7.e F.BF.1.b A.CED.2 A.SSE.1.b 7.2 Properties of Exponential Functions To write an exponential function To explore the properties of the form y = abx To graph translations of the parent function To graph exponential functions with base e To calculate continuously compounded interest problems F.BF.4.a F.1F.7.e ASSE.1.b F.1F.8 F.1F.9 7.3 Logarithmic Functions as Inverses To write exponential equations in logarithmic form To evaluate a logarithmic expression To graph a logarithmic function To graph translations of logarithmic functions Prepares for F.LE.4 7.4 Properties of Logarithms To use the properties of logarithms to simplify and expand logarithmic expressions To use the change of base formula F.LE.4 A.REI.11 7.5 Exponential and Logarithmic Equations To solve an exponential equation with a common base To solve an exponential equation with different bases To solve an exponential equation with a graph or table To solve a logarithmic equation using exponents, graph or table 7.6 Natural Logarithms To evaluate and simplify natural logarithmic expressions To solve equations using natural logarithms F.LE.4 Chapter 8 Rational Functions Essential Questions: Are two quantities inversely proportional if an increase in one corresponds to a decrease in the other? Quantities x and y are inversely 1 proportional only if growing x by the factor k (k > 1) means shrinking y by the factor . What kinds of asymptotes are possible for a rational function? A rational function may have no asymptote, one horizontal or oblique asymptote, and any number of vertical asymptotes + 1 Are a rational expression and its simplified form equivalent? () = 2 −2 , ≠ ± and () = − , ≠ ± , are equivalent CC Standard A.CED.2 Lesson Objective 8.1 Inverse Variation F.BF.3 A.CED.2 A.APR.1 A.APR.3 F.BF.1.b 8.2 The Reciprocal Function Family To identify direct and inverse variations To recognize and use inverse variation To use joint and other variations To graph an inverse variation function To identify Reciprocal function transformations To graph translations of reciprocal functions To identify properties of rational functions – to find points of discontinuity, to identify vertical and horizontal asymptotes To graph rational functions A.SSE.2 A.SSE.1.b A.SSE.1.a A.APR.7 8.4 Rational Expressions A.SSE.1a 8.8 Factoring by Grouping 8.3 Rational Functions and Their Graphs 8.5 Adding and Subtracting Rational Expressions To simplify rational expressions To multiply and divide rational expressions To use rational expressions to solve a problem To find the least common multiple To add and subtract rational expressions To simplify a complex fraction To use rational expressions to solve problems Factor higher degree polynomials by grouping Chapter 9: Sequences and Series Essential Question(s) How can you model a geometric sequence? How can you model its sum? CC Standards Lesson Objective Prepares for ASSE.4 9.1Mathematical Patterns To identify mathematical patterns found in a sequence A.SSE.4 A.SSE.4 9.3 Geometric Series 9.5 Geometric Series To use a formula to find the nth term of a sequence To define, identify, and apply geometric series To define geometric series and find their sums Chapter 11: Probability and Statistics Essential Question(s) What is the difference between experimental and theoretical probability? (You base experimental probability on past – and theoretical probability on possible occurrences.) How are measures of central tendency different from standard deviation? (Probability concepts can be used to analyze data and make decisions. Standard deviation describes how data spread out from a particular middle, central tendency, value.) Prepares for S.IC.2 S.MD.6 S.MD.7 S.MD.6 S.MD.7 S.ID.4 S.IC.6 S.IC.1 S.IC.3 S.IC.4 S.IC.6 S.ID.4 S.IC.4 S.IC.5 11.2 Probability To find the probability of an event using theoretical , experimental, and simulation models 11.5 Probability Models To use probabilities to make fair decisions and analyze decisions 11.6 Analyzing Data To calculate measures of central tendency To draw and interpret box and whisker plots To find the standard deviation and variance of a set of values To apply standard deviation and variance To identify sampling methods To recognize bias in samples and surveys 11.7 Standard Deviation 11.8 Samples and Surveys 11.10 Normal Distributions Concept Byte: Margin of Error Concept Byte: Drawing Conclusions from Samples To use a normal distribution Activities Compare samples to determine if the difference in mean or proportion for a large population, based on a given confidence level, is significant Chapter 13: Periodic Functions and Trigonometry Essential Questions: F.IF.4 How can you model periodic behavior? ( You can use combinations of circular functions (sine and cosine) to model natural periodic behavior). What function has as its graph a sine curve with amplitude 4, period π, and a minimum at the origin? (The graph of = 42 ( − 4 ) + 4 has amplitude 4, period π, midline = 4, and a minimum at the origin) If you know the value of sin , how can you find cos, , , , and ? (If you know the value of , find an angle with measure in standard position on the unit circle to find values of the other trigonometric functions) 13.1 Exploring Periodic Data To identify cycles and periods and periodic functions To find the amplitude of periodic functions Prepares for 13.2 Angles and The Unit Circle To work with angles in standard position F.TF.2 To find coordinates of points on the unit circle Identify Coterminal Angles Find cosines and sines of angles Determine exact values of cosine and sine F.TF.1 13.3 Radian Measure To use radian measure for angles To find the length of an arc of a circle F.TF.2 13.4 The Sine Function To identify properties of the sine function F.TF.5 To graph sine curves F.TF.5 13.5 The Cosine Functions To graph and write cosine functions F.TF.4 To solve trigonometric equations