West Windsor-Plainsboro Regional School District Advanced Algebra II Grades 10-12

Transcription

1 West Windsor-Plainsboro Regional School District Advanced Algebra II Grades Page 1 of 23

2 Unit 1: Linear Equations & Functions (Chapter 2) Content Area: Mathematics Course & Grade Level: Advanced Algebra II, Summary and Rationale Advanced Algebra 2 is a second year algebra course intended for students with a strong mathematics background in Algebra I and Geometry. This course extends the topics of Algebra I while covering many additional topics. The course includes linear, quadratic, polynomial, exponential, logarithmic, rational and radical equations and functions, systems of linear and polynomial equations, power and inverse functions, sequences and series, and probability and statistics. This unit will focus on the study of linear functions. Linear functions are used to describe relationships that have a constant rate of change, in terms of a dependent and independent variable. By studying the domain, range and rate of change of a linear function, mathematicians can describe and analyze relationships. This understanding can provide the foundation to make decisions and reasonable predictions. 13 days Recommended Pacing State Standards HSF IF.A Understand the concept of a function and use function notation HSF IF.B Interpret functions that arise in applications and in terms of the context Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its 1 domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). 2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. 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. 5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. HSF BF.A Build a function that models the relationship between two quantities 1 Write a function that describes a relationship between two quantities Instructional Focus Unit Enduring Understandings Functions model real world situations using various equations, graphs and tables. These models help mathematicians understand and explain real life data. Mathematical solutions do not always represent a real or possible solution Algebraic representations can be used to generalize patterns and relationships in order to help us understand and explain the relationship Page 2 of 23

3 Patterns and relationships can be represented graphically, numerically, symbolically, and verbally. Each representation has advantages and disadvantages. Unit Essential Questions How can mathematicians model complex real world situations? What is a function and why are they important when describing data? When is the mathematical solution to a problem not a viable solution? Objectives Students will know: Terms: function, domain, range, function, notation, transformations Students will be able to: Identify functions, domain, and range Graph linear functions by y = mx + b and by intercepts Write functional notation Evaluate functions Find the slope Determine whether lines are parallel, perpendicular, or neither Write equations of lines Find the line of best fit by hand/calculator Graph and solve absolute value functions Apply transformations to all functions Evidence of Learning Assessment Common Assessment 2.1 Resources Core Text: Algebra 2, Larson, Boswell, Kanold, Stiff, 2004 Suggested Resources: Page 3 of 23

4 Unit 2: Systems of Linear Equations & Inequalities (Chapter 3) Content Area: Mathematics Course & Grade Level: Advanced Algebra II, Summary and Rationale Advanced Algebra 2 is a second year algebra course intended for students with a strong mathematics background in Algebra I and Geometry. This course extends the topics of Algebra I while covering many additional topics. The course includes linear, quadratic, polynomial, exponential, logarithmic, rational and radical equations and functions, systems of linear and polynomial equations, power and inverse functions, sequences and series, and probability and statistics. This unit involves the study of systems of linear equations. By learning to solve the systems both algebraically and graphically and to appropriately interpret their solutions students will be able to apply systems to model real world situations. 6 days Recommended Pacing State Standards HSA CED.A Create equations that describe numbers or relationships. 2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. HSA REI.C Solve systems of equations 5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. 6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Instructional Focus Unit Enduring Understandings Utilize technology (graphing calculator) to investigate and extend applications of functions. Develop a problem solving repertoire and be able to choose the appropriate method to solve real world problems. Analyze data to find patterns in order to make decisions based on the data. Unit Essential Questions How well can mathematics model complex real world situations? When is the mathematical solution not the real solution? Objectives Students will know: Terms: system of linear equations, substitution, elimination, ordered triple Graphing Method Substitution Method Elimination Method Page 4 of 23

5 Students will be able to: Solve linear systems by graphing Solve linear systems by substitution and elimination method Solve systems of linear equations in three variables Solve application problems involving two or three variables Evidence of Learning Assessment Common Assessment 3.1 Resources Core Text: Algebra 2, Larson, Boswell, Kanold, Stiff, 2004 Suggested Resources: Page 5 of 23

6 Unit 3: Quadratic Functions (Chapter 5) Content Area: Mathematics Course & Grade Level: Advanced Algebra II, Summary and Rationale Advanced Algebra 2 is a second year algebra course intended for students with a strong mathematics background in Algebra I and Geometry. This course extends the topics of Algebra I while covering many additional topics. The course includes linear, quadratic, polynomial, exponential, logarithmic, rational and radical equations and functions, systems of linear and polynomial equations, power and inverse functions, sequences and series, and probability and statistics. This unit will focus on the study of quadratic functions. Quadratic functions can be used to model data to analyze real world situations. 18 days Recommended Pacing State Standards HSA SSE.B Write expressions in equivalent forms to solve problems 3a 3b Factor a quadratic expression to reveal the zeros of the function it defines. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. HSN CN.A Perform arithmetic operations with complex numbers 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. 2 Use the relation i 2 = 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers HSN CN.C Use complex numbers in polynomial identities and equations 7 Solve quadratic equations with real coefficients that have complex solutions. HSA REI.B Solve equations and inequalities in one variable 4 Solve quadratic equations in one variable. 4a 4b Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x p) 2 = q that has the same solutions. Derive the quadratic formula from this form. Solve quadratic equations by inspection (e.g., for x 2 = 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. Page 6 of 23

7 HSF IF.B. Interpret functions that arise in applications in terms of the context 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. HSF IF.C. Analyze functions using different representations 7a Graph linear and quadratic functions and show intercepts, maxima, and minima. 8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. 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. 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. Instructional Focus Unit Enduring Understandings Quadratic equations can be used to model and solve real life problems. Unit Essential Questions How can we use quadratic equations to model and solve real life problems? How can we solve quadratic equations using a variety of methods? How can we write quadratic functions given characteristics of their graphs? Objectives Students will know: Terms: parabola, vertex, axis of symmetry, standard form, intercept form, factoring, quadratic equation, zero product property, zero of a function, rationalize the denominator, imaginary number, complex number, standard form of a complex numbers, complex plane, conjugates,, completing the square, discriminant Formulas: Quadratic Formula, discriminant Completing the Square Students will be able to: Graph quadratic functions Factor and solve quadratic equations by factoring Find the zeros of a quadratic function Solve quadratic functions by finding square roots Solve quadratic equations with complex solutions Perform operations with complex numbers Solve quadratic equations by completing the square Apply completing the square to write quadratic functions in vertex form Solve quadratic equations using the quadratic formula Solve real life problems involving quadratic formulas Solve systems of polynomial functions (circles, parabolas, lines) Page 7 of 23

8 Evidence of Learning Assessment Common Assessment 5.1 Resources Core Text: Algebra 2, Larson, Boswell, Kanold, Stiff, 2004 Suggested Resources: Page 8 of 23

9 Unit 4: Polynomials & Polynomial Functions (Chapter 6) Content Area: Mathematics Course & Grade Level: Advanced Algebra II, Summary and Rationale Advanced Algebra 2 is a second year algebra course intended for students with a strong mathematics background in Algebra I and Geometry. This course extends the topics of Algebra I while covering many additional topics. The course includes linear, quadratic, polynomial, exponential, logarithmic, rational and radical equations and functions, systems of linear and polynomial equations, power and inverse functions, sequences and series, and probability and statistics. This unit will focus on polynomials and polynomial functions. Students will apply the skills they have learned to solve real life problems. 16 days Recommended Pacing State Standards HSA APR.A Perform arithmetic operations on polynomials 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. HSA APR.B Understand the relationship between zeros and factors of polynomials 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. HSF IF.B Interpret functions that arise in applications in terms of the context 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. 5 Relate the domain of a function to its graph and, where applicable, to the 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 would be an appropriate domain for the function. HSA IF.C Analyze functions using different representations 7c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. 8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. Page 9 of 23

10 Instructional Focus Unit Enduring Understandings The symbolic language of algebra is used to communicate and generalize the patterns in mathematics Algebraic representation can be used to generalize patterns and relationships Patterns and relationships can be represented graphically, numerically, symbolically, or verbally Mathematical models can be used to describe and quantify physical relationships Physical models can be used to clarify mathematical relationships Algebraic and numeric procedures are interconnected and build on one another to produce a coherent whole Reasoning and/or proof can be used to verify or refute conjectures or theorems in algebra Unit Essential Questions How can change be best represented mathematically? How can patterns, relations, and functions be used as tools to best describe and help explain real life situations? How are patterns of change related to the behavior of functions? How can we use mathematical models to describe physical relationships? How can we use physical models to clarify mathematical relationships? What makes an algebraic algorithm both effective and efficient? Objectives Students will know: Terms: polynomial function, exponents, degree of polynomial function, standard form of a polynomial function, synthetic substitution and synthetic division, end behavior, factor by grouping, quadratic form, polynomial long division, remainder theorem, factor theorem, repeated solution, local maximum and minimum, odd/even function Synthetic substitution Synthetic division Factor by grouping Rational Zero Theorem Students will be able to: Use the properties of exponents to evaluate and simplify expressions involving powers Evaluate a polynomial function Graph a polynomial function Determine if a function is even or odd Add, subtract and multiply polynomials Factor polynomial expressions Factoring to solve polynomial equations Divide polynomials Find the rational zeros of a polynomial function Apply the Fundamental Theorem of Algebra to determine the number of zeros Analyze the graph of a polynomial function Use technology to find polynomial models for real life data. Page 10 of 23

11 Evidence of Learning Assessment Common Assessment 6.1 Resources Core Text: Algebra 2, Larson, Boswell, Kanold, Stiff, 2004 Suggested Resources: Page 11 of 23

12 Unit 5: Powers, Roots, & Radicals (Chapter 7) Content Area: Mathematics Course & Grade Level: Advanced Algebra II, Summary and Rationale Advanced Algebra 2 is a second year algebra course intended for students with a strong mathematics background in Algebra I and Geometry. This course extends the topics of Algebra I while covering many additional topics. The course includes linear, quadratic, polynomial, exponential, logarithmic, rational and radical equations and functions, systems of linear and polynomial equations, power and inverse functions, sequences and series, and probability and statistics. This unit will focus on powers, roots, and radicals. Real numbers and algebraic expressions are often written with exponents and radicals. Students will use roots, rational exponents, power functions, function operations, and radical equations to solve real life problems. 13 days Recommended Pacing State Standards HSN RN.B Extend properties of exponents to rational exponents 3 Rewrite expressions involving radicals and rational exponents using the properties of exponents. HSA REI.A Understand solving equations as a process of reasoning and explain the reasoning 2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. HSF BF.B Build new functions from existing functions 4 Find inverse functions HSF IF.C Analyze functions using different representations 7b Graph square root, cube root, and piecewise defined functions, including step functions and absolute value functions. Instructional Focus Unit Enduring Understandings The symbolic language of algebra is used to communicate and generalize the patterns in mathematics Algebraic representation can be used to generalize patterns and relationships Patterns and relationships can be represented graphically, numerically, symbolically, or verbally Mathematical models can be used to describe and quantify physical relationships Physical models can be used to clarify mathematical relationships Algebraic and numeric procedures are interconnected and build on one another to produce a coherent whole Reasoning and/or proof can be used to verify or refute conjectures or theorems in algebra Page 12 of 23

13 Unit Essential Questions How can change be best represented mathematically? How can patterns, relations, and functions be used as tools to best describe and help explain real life situations? How are patterns of change related to the behavior of functions? How can we use mathematical models to describe physical relationships? How can we use physical models to clarify mathematical relationships? What makes an algebraic algorithm both effective and efficient? Objectives Students will know: Terms: nth root of a, index, simplest form, like radicals, power function, composition, inverse relation, inverse function, radical function, extraneous solution Students will be able to: Evaluate nth roots of real numbers using both radical notation and rational exponent notation Use properties of rational exponents to evaluate and simplify expressions Perform operations with functions (including power functions) Find inverses of linear and nonlinear functions Graph square and cube root functions Solve equations that contain radicals or rational exponents Evidence of Learning Assessment Common Assessment 7.1 Resources Core Text: Algebra 2, Larson, Boswell, Kanold, Stiff, 2004 Suggested Resources: Page 13 of 23

14 Unit 6: Exponents & Logarithmic Functions (Chapter 8) Content Area: Mathematics Course & Grade Level: Advanced Algebra II, Summary and Rationale Advanced Algebra 2 is a second year algebra course intended for students with a strong mathematics background in Algebra I and Geometry. This course extends the topics of Algebra I while covering many additional topics. The course includes linear, quadratic, polynomial, exponential, logarithmic, rational and radical equations and functions, systems of linear and polynomial equations, power and inverse functions, sequences and series, and probability and statistics. This unit will focus on exponents and logarithmic functions. Students will write exponential and power functions to model real life problems. 12 days Recommended Pacing State Standards HSA SSE.B Write expressions in equivalent forms to solve problems 3c Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15 t can be rewritten as (1.15 1/12 ) 12t t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. HSF IF.C. Analyze functions using different representations 7e 8b Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. 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. 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. HSF BF.B Build new functions from existing functions 4a 4b 4c 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 x 3 or f(x) = (x+1)/(x 1) for x 1. Verify by composition that one function is the inverse of another. Read values of an inverse function from a graph or a table, given that the function has an inverse. 5 Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. Page 14 of 23

15 HSF LE.A Construct and compare linear, quadratic and exponential models and solve problems 1c Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. 4 For exponential models, express as a logarithm the solution to ab ct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. Instructional Focus Unit Enduring Understandings The symbolic language of algebra is used to communicate and generalize the patterns in mathematics Algebraic representation can be used to generalize patterns and relationships Patterns and relationships can be represented graphically, numerically, symbolically, or verbally Mathematical models can be used to describe and quantify physical relationships Physical models can be used to clarify mathematical relationships Algebraic and numeric procedures are interconnected and build on one another to produce a coherent whole Reasoning and/or proof can be used to verify or refute conjectures or theorems in algebra Unit Essential Questions How can change be best represented mathematically? How can patterns, relations, and functions be used as tools to best describe and help explain real life situations? How are patterns of change related to the behavior of functions? How can we use mathematical models to describe physical relationships? How can we use physical models to clarify mathematical relationships? What makes an algebraic algorithm both effective and efficient? Objectives Students will know: Definitions of exponentiations for rational exponents Definitions of logarithm by finding the logarithm, base, or argument if two are given Students will be able to: Recognize what the graph of an exponential function looks like, by point wise plotting Use the definition of exponentiation to evaluate expressions that have positive integer exponents Explain why the properties of exponentiation are correct and use them to transform expressions Evaluate expressions containing radicals or fractional exponents Solve an exponential equation by trial and error using either calculator or computer Solve an exponential equation with any positive constant base, using base 10 and base e logarithms Recognize the properties of logarithms by transforming expressions and solving equations Find the inverse equation when given the equations of a function Draw the graph of the inverse of a function Use the add multiply property of exponential functions to calculate many values quickly and to tell whether or not an exponential function is a suitable mathematical model for a given set of data Use an exponential, linear, or quadratic function as a mathematical model when given real world situation relating variables Page 15 of 23

16 Evidence of Learning Assessment Common Assessment 8.1 Resources Core Text: Algebra 2, Larson, Boswell, Kanold, Stiff, 2004 Suggested Resources: Page 16 of 23

17 Unit 7: Rational Equations & Functions (Chapter 9) Content Area: Mathematics Course & Grade Level: Advanced Algebra II, Summary and Rationale Advanced Algebra 2 is a second year algebra course intended for students with a strong mathematics background in Algebra I and Geometry. This course extends the topics of Algebra I while covering many additional topics. The course includes linear, quadratic, polynomial, exponential, logarithmic, rational and radical equations and functions, systems of linear and polynomial equations, power and inverse functions, sequences and series, and probability and statistics. This unit focuses on rational equations and functions. Students will use rational expressions or equations to solve real life problems. 11 days Recommended Pacing State Standards HSA APR.A Perform arithmetic operations on polynomials 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. HSA APR.D Rewrite rational expressions 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. 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. HSA REI.A Understand solving equations as a process of reasoning and explain the reasoning 2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. HSF IF.C Analyze functions using different representations 7d Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. 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. Page 17 of 23

18 Instructional Focus Unit Enduring Understandings The symbolic language of algebra is used to communicate and generalize the patterns in mathematics Algebraic representation can be used to generalize patterns and relationships Patterns and relationships can be represented graphically, numerically, symbolically, or verbally Mathematical models can be used to describe and quantify physical relationships Physical models can be used to clarify mathematical relationships Algebraic and numeric procedures are interconnected and build on one another to produce a coherent whole Reasoning and/or proof can be used to verify or refute conjectures or theorems in algebra Asymptotes represents the limiting properties of a rational function Unit Essential Questions How can change be best represented mathematically? How can patterns, relations, and functions be used as tools to best describe and help explain real life situations? How are patterns of change related to the behavior of functions? How can we use mathematical models to describe physical relationships? How can we use physical models to clarify mathematical relationships? What makes an algebraic algorithm both effective and efficient? How are asymptotes the basic structure used in graphing rational functions? Objectives Students will know: Terms: Rational Function, Asymptotes, Complex Fractions, Least Common Denominator (LCD) Students will be able to: Discover by point wise plotting what the graph of a rational function looks like Simplify a rational algebraic function, find the discontinuities, and draw the graph Multiply conjugate binomials, factor a difference of two squares, and multiply polynomials with more than two terms, as well as factor familiar quadratic trinomials Factor the sum or difference of two cubes Factor polynomials Graph rational functions involving sums or differences of two cubes Long divide a polynomial by a polynomial, and use the results to factor and to draw graphs Find linear factors for cubic and higher degree polynomials Multiply or divide several rational expressions and simplify the result Add or subtract rational expressions and simplify the resolute Determine what value of x are excluded from the domain of a given rational function, figure out what happens to the graph at these points, and draw the graph Solve fractional equations Determine what kind of variation function is a reasonable mathematical model and find the equation for the function, and predict values of y or x when given real world situations Appreciate the connections between algebra and the real world Use technology to learn and discover concepts Page 18 of 23

19 Evidence of Learning Assessment Common Assessment 9.1 Resources Core Text: Algebra 2, Larson, Boswell, Kanold, Stiff, 2004 Suggested Resources: Page 19 of 23

20 Unit 8: Sequences and Series (Chapter 11) Content Area: Mathematics Course & Grade Level: Advanced Algebra II, Summary and Rationale Advanced Algebra 2 is a second year algebra course intended for students with a strong mathematics background in Algebra I and Geometry. This course extends the topics of Algebra I while covering many additional topics. The course includes linear, quadratic, polynomial, exponential, logarithmic, rational and radical equations and functions, systems of linear and polynomial equations, power and inverse functions, sequences and series, and probability and statistics. This unit focuses on sequences and series. Students will use infinite geometric series to model real life situations and use recursive rules for arithmetic and geometric sequences to solve real life problems. 10 days Recommended Pacing State Standards HSF BF.A Build a function that models a relationship between two quantities 2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. HSA SSE.B Write expressions in equivalent forms to solve problems 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. Instructional Focus Unit Enduring Understandings The symbolic language of algebra is used to communicate and generalize the patterns in mathematics Algebraic representation can be used to generalize patterns and relationships Patterns and relationships can be represented graphically, numerically, symbolically, or verbally Mathematical models can be used to describe and quantify physical relationships Physical models can be used to clarify mathematical relationships Algebraic and numeric procedures are interconnected and build on one another to produce a coherent whole Reasoning and/or proof can be used to verify or refute conjectures or theorems in algebra Unit Essential Questions How can change be best represented mathematically? How can patterns, relations, and functions be used as tools to best describe and help explain real life situations? How are patterns of change related to the behavior of functions? How can we use mathematical models to describe physical relationships? How can we use physical models to clarify mathematical relationships? What makes an algebraic algorithm both effective and efficient? Page 20 of 23

21 Objectives Students will know: Terms: sequence, series, finite and infinite sequence, summation notation, arithmetic and geometric sequence and series, common difference and ratio, explicit rule, recursive rule Formulas: Explicit rule for arithmetic and geometric sequence, sum of a finite arithmetic and geometric series, sum of an infinite geometric series. Students will be able to: Use and write arithmetic and geometric sequences and their rules Graph sequences Use summation notation to write and find the sum of a series. Write a rule for the nth term of arithmetic and geometric sequences Find the nth term given either a term or the common difference or ratio or two terms Find the sum of arithmetic and geometric sequences and series Evidence of Learning Assessment Common Assessment 10.1 Resources Core Text: Algebra 2, Larson, Boswell, Kanold, Stiff, 2004 Suggested Resources: Page 21 of 23

22 Unit 9: Probability and Statistics (Chapter 7 and 12) Content Area: Mathematics Course & Grade Level: Advanced Algebra II, Summary and Rationale Advanced Algebra 2 is a second year algebra course intended for students with a strong mathematics background in Algebra I and Geometry. This course extends the topics of Algebra I while covering many additional topics. The course includes linear, quadratic, polynomial, exponential, logarithmic, rational and radical equations and functions, systems of linear and polynomial equations, power and inverse functions, sequences and series, and probability and statistics. This unit focuses on probability and statistics. Students will use measures of central tendency and measures of dispersion to describe data sets. Students will learn how to count the number of ways an event can happen, calculate and use probabilities, and use binomial and normal distributions. 17 days Recommended Pacing State Standards HSS.CPA Understand independence and conditional probability and use them to interpret data 1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ("or," "and," "not"). 2 Understand that two events A and B are independent if the probability of A and Boccurring together is the product of their probabilities, and use this characterization to determine if they are independent. 3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. 5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. HSS CP.B Use the rules of probability to compute probabilities of compound events. 6 Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model. 7 Apply the Addition Rule, P(A or B) = P(A) + P(B) P(A and B), and interpret the answer in terms of the model. 8 Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B A) = P(B)P(A B), and interpret the answer in terms of the model. 9 Use permutations and combinations to compute probabilities of compound events and solve problems. Page 22 of 23

23 Instructional Focus Unit Enduring Understandings The symbolic language of algebra is used to communicate and generalize the patterns in mathematics Algebraic representation can be used to generalize patterns and relationships Patterns and relationships can be represented graphically, numerically, symbolically, or verbally Mathematical models can be used to describe and quantify physical relationships Physical models can be used to clarify mathematical relationships Algebraic and numeric procedures are interconnected and build on one another to produce a coherent whole Reasoning and/or proof can be used to verify or refute conjectures or theorems in algebra Unit Essential Questions How can change be best represented mathematically? How can patterns, relations, and functions be used as tools to best describe and help explain real life situations? How are patterns of change related to the behavior of functions? How can we use mathematical models to describe physical relationships? How can we use physical models to clarify mathematical relationships? What makes an algebraic algorithm both effective and efficient? Objectives Students will know: Terms: mean, standard deviation, permutation, combination, binomial theorem, probability, compound event, complement, independent and dependent events, binomial distribution, hypothesis testing, and normal distribution. Students will be able to: Use measures of central tendency and measures of dispersion to describe data sets. Use box and whisker plots and histograms to represent data graphically. Use the fundamental counting principle to count the number of ways an event can happen. Use permutations to count the number of ways an event can happen. Use combinations to count the number of ways an event can happen. Use the binomial theorem to expand a binomial that is raised to a power. Find theoretical and experimental probabilities. Find geometric probabilities. Find probabilities of unions and intersections of two events. Use complements to find the probability of an event. Find the probability of independent and dependent events. Find binomial probabilities and analyze binomial distributions. Calculate probabilities using normal distributions. Evidence of Learning Assessment Common Assessment 10.1 Resources Core Text: Algebra 2, Larson, Boswell, Kanold, Stiff, 2004 Suggested Resources: Page 23 of 23