West Windsor-Plainsboro Regional School District Algebra II Grades 9-12 Page 1 of 22
Content Area: Mathematics Course & Grade Level: Algebra II, 9 12 Unit 1: Analyzing Equations (Chapter 1) Summary and Rationale Algebra 2 is a second year algebra course intended for students with an average 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, series, probability and statistics. This unit will focus on evaluating and simplifying expressions, applying geometric formulas, and translating verbal expressions and equations, and solving linear and absolute value equations. Equations and formulas are necessary to set up and solve real life problems. For mathematics to be a precise language, the order in which operations are performed must be uniform. 10 days HSA.SSE.A. Interpret the structure of expressions Recommended Pacing State Standards 1 Interpret expressions that represent a quantity in terms of its context HSA.CED.A Create equations that describe numbers or relationships 1 Create equations and inequalities in one variable and use them 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 2 of 22
Objectives Students will know: Terms: Order of operations, Pythagorean Theorem, Solution set for equations, Absolute value equations Students will be able to: Use order of operations to evaluate expressions Simplify algebraic expressions Apply the Pythagorean Theorem and geometric formulas Translate verbal expressions and equations into algebraic expressions and equations Solve equations including solving for a specific variable and absolute value equations Evidence of Learning Assessment Common Assessment 1.1 Competencies for 21 st Century Learners Collaborative Team Member Effective Communicator Globally Aware, Active, & Responsible Student/Citizen Information Literate Researcher Innovative & Practical Problem Solver Self Directed Learner Resources Core Text: Algebra II: Foster, Rath, Winters; McGraw Hill Publishing Co., (1998) Suggested Resources: Page 3 of 22
Content Area: Mathematics Course & Grade Level: Algebra II, 9 12 Unit 2: Graphing Linear Relations & Functions (Chapter 2) Summary and Rationale Algebra 2 is a second year algebra course intended for students with an average 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, series, probability and statistics. This unit focuses on identifying and graphing relations and functions, finding the slope of a line, identifying parallel and perpendicular lines, generalizing slope as a rate of change, writing equations of lines, and solving problems involving direct and inverse variation. The importance of functions is that they model many real world phenomena. The key component of them, the slope, is a foundation concept for the realm of calculus. 10 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 1 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 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 Page 4 of 22
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: Graph a relation, state its domain and range, and determine whether it is a function. Find values of functions for given elements of the domain Using a graphing calculator to graph linear equations. Identify equations that are lines and graph them. Write linear equations in standard form. Determine the intercepts of a line and use them to graph an equation Determine the slope of a line. Use slope and a point to graph an equation. Determine whether two lines are parallel, perpendicular, or neither. Solve problems by identifying and using a pattern Write an equation of a line in slope intercept form given the slope and one of two points. Write an equation of a line that is parallel or perpendicular to the graph of a given equation Identify and graph special functions Determine intervals where the function is increasing, decreasing and/or negative. Solve problems involving direct and inverse variation. Students will be able to: Graph a relation, identify its domain and range, and determine if it s a function Evaluate a function for a given domain Identify and graph linear equations in two variables, both on the coordinate plane and using the graphing calculator Write linear equations in standard and slope intercept form Identify parallel and perpendicular lines, as well as being able to write linear equations for these types of lines Identify and graph special functions such as absolute value functions Page 5 of 22
Assessment Common Assessment 2.1 Competencies for 21 st Century Learners Collaborative Team Member Evidence of Learning Effective Communicator Globally Aware, Active, & Responsible Student/Citizen Innovative & Practical Problem Solver Information Literate Researcher Self Directed Learner Core Text: Algebra 2, Foster, Rath, Winters, 1998 Suggested Resources: Resources Page 6 of 22
Content Area: Mathematics Course & Grade Level: Algebra II, 9 12 Unit 3: Solving Systems of Linear Equations (Chapter 3) Summary and Rationale Algebra 2 is a second year algebra course intended for students with an average 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, series, probability and statistics. This unit focuses on solving systems of linear equations and applying skills to model real life applications. Many problems lend themselves to being solved with systems of linear equations. In real life, these problems can be increasingly complex. The study of linear systems and related concepts is its own branch of mathematics. 7 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. Page 7 of 22
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: system of equations, Substitution Method, Elimination Method Students will be able to: Examine and solve systems of linear equations Utilize both graphing and algebraic methods to solve the systems. Utilize the best method to solve a problem Apply these skills while solving application problems Optional: Use a graphing calculator to solve systems of equations Evidence of Learning Assessment Common Assessment 3.1 Competencies for 21 st Century Learners Collaborative Team Member Effective Communicator Globally Aware, Active, & Responsible Student/Citizen Information Literate Researcher Innovative & Practical Problem Solver Resources Core Text: Algebra 2, Foster, Rath, Winters, 1998 Suggested Resources: Self Directed Learner Page 8 of 22
Unit 4: Exploring Polynomials & Radical Expressions (Chapter 5) Content Area: Mathematics Course & Grade Level: Algebra II, 9 12 Summary and Rationale Algebra 2 is a second year algebra course intended for students with an average 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, series, probability and statistics. This unit focuses on operations with polynomials and radicals, simplifying expressions with rational exponents, complex numbers, and radicals, factoring polynomials, and solving equations with real or complex solutions. Polynomials are a combination of several terms resulting in a particular pattern. Polynomial patterns appear in numbers we use every day. Polynomials are used to model and study numbers. Mathematical problems that include radical equations arise in a number of professions, from engineering to nursing. Complex numbers have essential and concrete applications in a variety of scientific, engineering, and related areas. 26 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. 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. Page 9 of 22
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: Polynomials, radicals, rationalized denominator, rational exponents, imaginary numbers, and complex numbers Students will be able to: Do all operations with polynomials and radicals Factor polynomials Simplify expressions with rational exponents Simplify radical expressions Rationalize denominators of rational expressions Simplify expressions with imaginary and complex numbers Solve equations with imaginary and complex numbers as solutions Evidence of Learning Assessment Common Assessment 4.1 Competencies for 21 st Century Learners Collaborative Team Member Effective Communicator Globally Aware, Active, & Responsible Student/Citizen Information Literate Researcher Innovative & Practical Problem Solver Core Text: Algebra 2, Foster, Rath, Winters, 1998 Suggested Resources: Resources Self Directed Learner Page 10 of 22
Content Area: Mathematics Course & Grade Level: Algebra II, 9 12 Unit 5: Quadratic Functions (Chapter 6) Summary and Rationale Algebra 2 is a second year algebra course intended for students with an average 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, series, probability and statistics. This unit focuses on graphing quadratic functions, solving quadratic equations by factoring, completing the square, and quadratic formula. Quadratic models will be used to solve real life problems. Quadratic functions are applicable to everyday life. Examples include projectile motion (a ball being thrown into the air), prices, areas, and speeds. Recommended Pacing 15 days 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 11 of 22
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 Applications of quadratic functions include acceleration, trajectory, and normal distributions 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 quadratic functions different from linear functions? Objectives Students will know: Terms: Factor, Completing the Square, Discriminant, Vertex, Axis of Symmetry Formulas: Quadratic formula, Discriminant, Standard Form of a Parabola Students will be able to: Write a function in quadratic form, to graph quadratic functions, and to solve quadratic equations by graphing Solve quadratic equations by factoring Solve quadratic equations by completing the square Solve quadratic equations by using the quadratic formula Use discriminants to determine the nature of the roots of quadratic equations Put a quadratic function into standard form and graph Evidence of Learning Assessment Common Assessment 5.1 Competencies for 21 st Century Learners Collaborative Team Member Effective Communicator Globally Aware, Active, & Responsible Student/Citizen Information Literate Researcher Innovative & Practical Problem Solver Resources Core Text: Algebra 2, Foster, Rath, Winters, 1998 Suggested Resources: Self Directed Learner Page 12 of 22
Content Area: Mathematics Course & Grade Level: Algebra II, 9 12 Unit 6: Exploring Polynomial Functions (Chapter 8) Summary and Rationale Algebra 2 is a second year algebra course intended for students with an average 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, series, probability and statistics. This unit focuses on compositions of functions, identifying general shapes of polynomial functions, determining the degree, end behavior, and inverse of functions, and the remainder, factor and rational zero theorems. A polynomial function is a relation between some variable to another variable with a few restrictions to limit its behavior. These functions are needed to demonstrate real life situations, such as banking trends. Composition of functions is useful when one quantity depends on a second quantity, and in turn that second quantity depends on a third quantity. This is an extremely general situation with lots of real world applications. 8 days Recommended Pacing State Standards 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 13 of 22
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 The concept of zeros, intercepts, and solutions to equations all reflect the same mathematical idea 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 does the degree of the polynomial impact the family of graphs? Objectives Students will know: Terms: Composition of functions, Inverse function, Degree, leading coefficient, End behavior of a function Students will be able to: Complete a composition of two functions Identify general shapes of the graphs of polynomial functions Determine the degree of a polynomial Determine the inverse of a function or relation Determine the end behavior of a function Introduce the Remainder, Factor and Rational Zero Theorems Evidence of Learning Assessment Common Assessment 7.1 Competencies for 21 st Century Learners Collaborative Team Member Effective Communicator Globally Aware, Active, & Responsible Student/Citizen Information Literate Researcher Innovative & Practical Problem Solver Core Text: Algebra 2, Foster, Rath, Winters, 1998 Suggested Resources: Resources Self Directed Learner Page 14 of 22
Content Area: Mathematics Course & Grade Level: Algebra II, 9 12 Unit 7: Exploring Rational Expressions (Chapter 9) Summary and Rationale Algebra 2 is a second year algebra course intended for students with an average 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, series, probability and statistics. This unit focuses on simplifying rational expressions and complex fractions, adding and subtracting rational expressions, and solving rational equations. Rational expressions and rational equations can be useful tools for representing real life situations and for finding answers to real problems. For example, they are necessary for describing distance rate time and modeling multiperson work problems. Recommended Pacing 6 days 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 15 of 22
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? Objectives Students will know: Terms: Rational Function, Complex Fractions, Least Common Denominator (LCD) Students will be able to: Simplify rational expressions and simplify complex fractions (multiplying and dividing rational expressions) Find the least common denominator of two or more algebraic expressions Add and subtract rational expressions Solve rational equations Assessment Common Assessment 8.1 Competencies for 21 st Century Learners Collaborative Team Member Evidence of Learning Effective Communicator Globally Aware, Active, & Responsible Student/Citizen Information Literate Researcher Innovative & Practical Problem Solver Core Text: Algebra 2, Foster, Rath, Winters, 1998 Suggested Resources: Resources Self Directed Learner Page 16 of 22
Content Area: Mathematics Course & Grade Level: Algebra II, 9 12 Unit 8: Exponential & Logarithmic Functions (Chapter 10) Summary and Rationale Algebra 2 is a second year algebra course intended for students with an average 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, series, probability and statistics. This unit focuses on graphing exponential functions, finding the domain and range of exponential functions, writing exponential expressions and equations in logarithmic form and vice versa, and using the properties of logarithms to simplify expressions and solve equations. Exponential functions are useful in real world situations. They are used to model populations, carbon date artifacts, and compute investments. Logarithms are necessary to describe very large or very small numbers in terms of their powers of 10. Recommended Pacing 10 days 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 1.012 12t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. HSF IF.C. Analyze functions using different representations 7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. 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. 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 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. 4b Verify by composition that one function is the inverse of another. 4c Read values of an inverse function from a graph or a table, given that the function has an inverse. 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. Page 17 of 22
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. 5 Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. 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: Exponential Function, Logarithmic Function, Exponential Growth & Decay, Common Logarithms Formulas: General Formula for Growth & Decay Students will be able to: Graph exponential functions with and without calculator Find the domain and range of exponential functions Simplify expressions and solve equations involving real exponents To write exponential equations in logarithmic form and vice versa Use the properties of logarithms to simplify expressions and solve equations Optional: Use logarithms to solve problems involving growth and decay Use logarithms to solve equations with variable exponents Evidence of Learning Assessment Common Assessment 8.1 Competencies for 21 st Century Learners Collaborative Team Member Effective Communicator Globally Aware, Active, & Responsible Student/Citizen Information Literate Researcher Innovative & Practical Problem Solver Self Directed Learner Resources Core Text: Algebra 2, Foster, Rath, Winters, 1998 Suggested Resources: Practice and Study Guide worksheets provided by the publisher, Exploration and Practice worksheets provided in curriculum packet, graphing calculator activities provided with textbook by publisher Websites: www.glencoe.com www.regentsprep.org/regents/math/algebra/ae7/indexae7.htm Page 18 of 22
Content Area: Mathematics Course & Grade Level: Algebra II, 9 12 Unit 9: Probability and Statistics Summary and Rationale Algebra 2 is a second year algebra course intended for students with an average 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, series, probability and statistics. This unit focuses on using measures of central tendency and measures of dispersion to describe data sets, using boxand whisker plots and histograms to represent data graphically, using theoretical and experimental probabilities, and calculating probabilities using normal distributions. Measures of central tendency are very useful in statistics. They are used to find representative value, to condense data, to make comparisons, and to analyze data. Probability and statistics are useful in many ways. Knowing the likelihood of an event happening is important information in decision making that is used in nearly every field. The normal distribution produces the normal curve. The bell shaped normal curve is created when large number of measurements are taken and shown to be approximately distributed in this pattern. Height, IQ, or examination scores of a large number of people follow the normal distribution. Recommended Pacing 10 days 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. Page 19 of 22
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 Applications of quadratic functions include acceleration, trajectory, and normal distributions 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, probability, compound event, complement, independent and dependent events, 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. Find theoretical and experimental probabilities. Find the probability of independent and dependent events. Calculate probabilities using normal distributions. Assessment Common Assessment 9.1 Competencies for 21 st Century Learners Collaborative Team Member Evidence of Learning Globally Aware, Active, & Responsible Student/Citizen Innovative & Practical Problem Solver Core Text: Algebra 2, Foster, Rath, Winters, 1998 Suggested Resources: Resources Effective Communicator Information Literate Researcher Self Directed Learner Page 20 of 22
Content Area: Mathematics Course & Grade Level: Algebra II, 9 12 Unit 10: Series and Sequences (Chapter 11) Summary and Rationale Algebra 2 is a second year algebra course intended for students with an average 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, series, probability and statistics. This unit focuses on finding terms in arithmetic and geometric sequences, and finding the sums of arithmetic and geometric series. Series and sequences describe algebraic patterns. Arithmetic and geometric series and sequences play an important role in the early development of calculus and are used throughout mathematics. They have important applications in physics engineering, biology, economics, computer science, and finance. 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 Applications of quadratic functions include acceleration, trajectory, and normal distributions Page 21 of 22
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: series, sequences, arithmetic and geometric sequences Students will be able to: Find the next number in a sequence by looking for a pattern Find terms in arithmetic and geometric sequences Find sums of arithmetic and geometric series Assessment Common Assessment 9.1 Competencies for 21 st Century Learners Collaborative Team Member Evidence of Learning Effective Communicator Globally Aware, Active, & Responsible Student/Citizen Information Literate Researcher Innovative & Practical Problem Solver Core Text: Algebra 2, Foster, Rath, Winters, 1998 Suggested Resources: Resources Self Directed Learner Page 22 of 22