College Prep Algebra III Course #340. Course of Study. Findlay City School

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1 College Prep Algebra III Course #340 Course of Study Findlay City School

2 Algebra III Table of Contents 1. Findlay City Schools Mission Statement and Beliefs 2. Algebra III Curriculum Map 3. Algebra III Standards and Course of Study Algebra III Course of Study Writing Team Ellen Laube Karen Ouwenga

3 Algebra III CURRICULUM MAP WEEK CHAPTER TOPIC STANDARDS 1 0 Prerequisites and Review 2 0 Prerequisites and Review 3 0 Prerequisites and Review 4 1 Equations and Inequalities 5 1 Equations and Inequalities 6 1 Equations and Inequalities 7 1 Equations and Inequalities See Ch 0 Details See Ch 0 Details See Ch 0 Details See Ch 1 Details See Ch 1 Details See Ch 1 Details See Ch 1 Details 8 2 Graphs See Ch 2 Details 9 2 Graphs See Ch 2 Details 10 3 Functions and Their Graphs 11 3 Functions and Their Graphs 12 3 Functions and Their Graphs 13 3 Functions and Their Graphs 14 4 Polynomials and Rational Functions 15 4 Polynomials and Rational Functions 16 4 Polynomials and Rational Functions 17 4 Polynomials and Rational Functions 18 EXAMS 19 5 Exponential and Logarithmic Functions 20 5 Exponential and Logarithmic Functions See Ch 3 Details See Ch 3 Details See Ch 3 Details See Ch 3 Details See Ch 4 Details See Ch 4 Details See Ch 4 Details See Ch 4 Details See Ch 5 Details See Ch 5 Details

4 21 5 Exponential and Logarithmic Functions 22 6 Systems of Linear Equations and Inequalities 23 6 Systems of Linear Equations and Inequalities See Ch 5 Details See Ch 6 Details See Ch 6 Details 24 7 Matrices See Ch 7 Details 25 7 Matrices See Ch 7 Details 26 7 Matrices See Ch 7 Details 27 8 Conics and Systems of Nonlinear Equations and Inequalities 28 8 Conics and Systems of Nonlinear Equations and Inequalities 29 8 Conics and Systems of Nonlinear Equations and Inequalities 30 9 Sequences, Series, and Probability 31 9 Sequences, Series, and Probability 32 9 Sequences, Series, and Probability Trigonometric Functions Trigonometric Functions See Ch 8 Details See Ch 8 Details See Ch 8 Details See Ch 9 Details See Ch 9 Details See Ch 9 Details See Ch 10 Details See Ch 10 Details 35 EXAM REVIEW All 36 EXAMS All Textbook Name: Intermediate Algebra 2 nd Edition Publisher: Wiley Authors: Cynthia Young ISBN #

5 TOPIC/UNIT: Chapter 0 Prerequisites and Review Time Line: 3 Weeks CONCEPTUAL CATEGORY: Number and Quantity DOMAIN: Real Number System CLUSTER: Extend the properties of exponents to rational exponents 1. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. 2. Rewrite expressions involving radicals and rational exponents using the properties of exponents. CLUSTER: Use properties of rational and irrational numbers. 3. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number is an irrational number is irrational. DOMAIN: The Complex Number System CLUSTER: Perform arithmetic operations with complex numbers 1. Know there is a complex number i such as 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. 3. Find the conjugate of complex number CONCEPTUAL CATEGORY: Algebra DOMAIN: Seeing Structure in Expressions CLUSTER: Interpret the structure of expressions 1. Interpret expressions that represent a quantity in terms of its context. (a) Interpret parts of an expression, such as terms, factors, and coefficients. (b) Interpret complicated expressions by viewing one or more of their parts as a single entity. 2. Use the structure of an expression to identify ways to rewrite it. DOMAIN: Arithmetic with Polynomials and Rational Expressions CLUSTER: 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. CLUSTER: Rewrite rational expressions

6 6. Rewrite rational expressions in different forms. 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. KNOW Understand that rational and irrational numbers are mutually exclusive and complementary subsets of real numbers Learn the order of operations Visualize negative exponents as reciprocals Understand that scientific notation is an effective way to represent very large or very small numbers Recognize like terms Learn formulas for special products Identify common factors Understand that factoring has its basis in the distributive property Identify prime polynomials Develop a general strategy for factoring polynomials Understand why rational expressions have domain restrictions Understand the least common denominator method for rational expressions Understand that radicals are equivalent to rational exponents Understand that a radical implies one number, not two Understand that real numbers and imaginary numbers are DO Understand that rational and irrational numbers together constitute real numbers Round and truncate real numbers Apply properties of exponents Use scientific notation to represent a real number Simplify expressions that contain rational exponents Simplify radicals Perform operations on polynomials Multiply two binomials Factor polynomials Perform operations on rational expressions Write complex numbers in standard form

7 subsets of complex numbers Understand how to eliminate imaginary numbers in denominators PRE-ASSESSMENT: ASSESSMENT: Chapter 0 Practice Test page 86 Teacher Created Pre-Assessment GRAPHIC ORGANIZER & OR TECHNOLOGY: Chapter 0 Test Banks Chapter 0 Practice Test page 86 Teacher created assessments Chapter 0 Review Exercises pgs TESTING SKILL(S) & OR SAMPLE OGT TYPE QUESTIONS: Video Tutorials available in every section Technology Tips to demonstrate graphing calculator key strokes Catch the Mistake pgs. 17, 26, 36, 47, 62, 72, 80 BEST PRACTICES: RESOURCES: Cooperative Learning Technology Integration Lecture with guided practice Group work Intermediate Algebra Second Edition WileyPLUS Printed Test Banks PowerPoint Reviews Online Student Resources Answer Documents Additional Technology Tips TESTING VOCABULARY: HISTORICAL/MODERN LINK: Chapter 0 Review- pgs Technology Based Exercises Application Based Exercises

8 TOPIC/UNIT: Chapter 1 Equations and Inequalities Time Line: 4 Weeks CONCEPTUAL CATEGORY: Number and Quantity DOMAIN: Quantities CLUSTER: Reason quantitatively and use units to solve problems. 1. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. 2. Define appropriate quantities for the purpose of descriptive modeling. 3. Choose a level of accuracy appropriate to limitations on measurement reporting quantities. CONCEPTUAL CATEGORY: Algebra DOMAIN: Creating Equations CLUSTER: Create equations that describe numbers or relationships. 1. Create equations and inequalities in one variable and use them to solve problems. 2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable in a modeling context. 4. Rearrange formulas to highlight a quantity of interest, using same reasoning as in solving equations. DOMAIN: Reasoning with Equations and Inequalities CLUSTER: Understand solving equations as a process of reasoning and explain the reasoning 1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. 2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. 3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. 4. Solve quadratic equations in one variable.

9 KNOW Eliminate values that result in a denominator being equal to zero Understand the mathematical modeling process Choose appropriate methods for solving quadratic equations Interpret different types of solution sets Derive the quadratic formula Transform a difficult equation into a simpler linear or quadratic equation Recognize the need to check solutions when the transformation process may produce extraneous solutions Realize that not all polynomial equations are factorable Apply intersection and union concepts Compare and contrast equations and inequalities Understand that linear inequalities may have one solution, no solution, or an integral solution Understand zeros and test intervals Realize that a rational inequality has an implied domain restriction on the variable Understand absolute value in terms of distance on the number line DO Classify equations as linear, quadratic, polynomial, rational, radical, quadratic in form, or absolute value Solve linear equations Solve quadratic equations Solve rational equations Solve polynomial equations Solve radical equations Solve equations involving rational exponents Solve equations that are quadratic in form Solve absolute value equations Eliminate extraneous solutions Eliminate solutions that are not possible values of the independent variable Solve linear, quadratic, polynomial, rational, and absolute value inequalities Solve application problems involving equations and inequalities

10 PRE-ASSESSMENT: ASSESSMENT: Chapter 1 Practice Test page 170 Teacher Created Pre-Assessment GRAPHIC ORGANIZER & OR TECHNOLOGY: Chapter 1 Test Banks Chapter 1 Practice Test page 170 Teacher created assessments Chapter 1 Review Exercises pgs Chapter 1 Cumulative Test- page 171 TESTING SKILL(S) & OR SAMPLE OGT TYPE QUESTIONS: Video Tutorials available in every section Technology Tips to demonstrate graphing calculator key strokes Catch the Mistake pgs. 98, 126, , , 162 BEST PRACTICES: RESOURCES: Cooperative Learning Technology Integration Lecture with guided practice Group work Compare and contrast Error analysis Intermediate Algebra Second Edition WileyPLUS Printed Test Banks PowerPoint Reviews Online Student Resources Answer Documents Additional Technology Tips TESTING VOCABULARY: HISTORICAL/MODERN LINK: Chapter 1 Review- pgs Technology Based Exercises Application Based Exercises Modeling Your World Page

11 TOPIC/UNIT: Chapter 2 Graphs Time Line: 2 Weeks CONCEPTUAL CATEGORY: Functions DOMAIN: Linear, Quadratic, and Exponential Models CLUSTER: Interpret expressions for functions in terms of the situation they model. 5. Interpret the parameters in a linear or exponential function in terms of a context. DOMAIN: Building Functions CLUSTER: Build new functions from existing functions. 1. Write a function that describes a relationship between two quantities. (a) Determine an explicit expression, a recursive process, or steps for calculation from a context. (b) Combine standard function types using arithmetic operations. (c) (+) Compose functions. 2. Write arithmetic and geometric sequences both recursively and with an explicit formula; use them to model situations, and translate between the two forms. CONCEPTUAL CATEGORY: Algebra DOMAIN: Reasoning with Equations and Inequalities CLUSTER: Solve systems of equations. 5. Prove that 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. 7. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x 2 + y2 = 3.

12 KNOW Expand the concept of a onedimensional number line to a twodimensional plane Relate symmetry graphically and algebraically Classify lines as rising, falling, horizontal and vertical Understand slope as a rate of change Associate two lines having the same slope with the graph of parallel lines Associate two lines having negative reciprocal slopes with the graph of perpendicular lines Understand algebraic and graphical representations of circles DO Plot points in the Cartesian plane Calculate the distance between two points Find the midpoint of a line segment joining two points Sketch the graph of an equations by points-plotting Find the intercepts of a graph Determine whether a graph has symmetry Use intercepts and symmetry as graphing aids Find the slope of a line Graph lines and circles Determine equations of lines and circles PRE-ASSESSMENT: Chapter 2 Practice Test page 222 Teacher Created Pre-Assessment ASSESSMENT: Chapter 2 Test Banks Chapter 2 Practice Test page 222 Teacher created assessments Chapter 2 Review Exercises pgs Chapters 1-2 Cumulative Testpage 223 GRAPHIC ORGANIZER & OR TECHNOLOGY: Video Tutorials available in every section Technology Tips to demonstrate graphing calculator key strokes TESTING SKILL(S) & OR SAMPLE OGT TYPE QUESTIONS: Catch the Mistake pgs. 180, 192, 208, 216

13 BEST PRACTICES: Cooperative Learning Technology Integration Lecture with guided practice Group work Compare and contrast Error analysis RESOURCES: Intermediate Algebra Second Edition WileyPLUS Printed Test Banks PowerPoint Reviews Online Student Resources Answer Documents Additional Technology Tips TESTING VOCABULARY: Chapter 2 Review- pgs. 219 HISTORICAL/MODERN LINK: Technology Based Exercises Application Based Exercises Modeling Your World Page

14 TOPIC/UNIT: Chapter 3 Functions and Their Graphs Time Line: 4 Weeks CONCEPTUAL CATEGORY: Functions DOMAIN: Interpreting Functions CLUSTER: Understand the concept of a function and use function notation. 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, the 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. 3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. DOMAIN: Interpreting Functions CLUSTER: 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. 5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. 6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. DOMAIN: Interpreting Functions CLUSTER: Analyze functions using different representations. 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima.

15 b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. d. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. 8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. 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. b. Use the properties of exponents to interpret expressions for exponential functions. 9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). DOMAIN: Building Functions CLUSTER: Build new functions from existing functions. 4. Find inverse functions. a. 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. b. (+) Verify by composition that one function is the inverse of another. c. (+) Read values of an inverse function from a graph or table, given that the function has an inverse. d. (+) Produce an invertible function from a non-invertible function by restricting the domain.

16 KNOW Think of functional notation as a placeholder or mapping Understand that all functions are relations, but not all relations are functions Identify common functions Develop and graph piecewisedefined functions Understand that even functions have graphs that are symmetric about the y axis Understand that odd functions have graphs that are symmetric about the origin Identify the common functions by their graphs Apply multiple transformations of common functions to obtain graphs of functions Understand that domain and range are also transformed Understand domain restrictions when dividing functions Realize that the domain of a composition of functions excludes values that are not in the domain of the inside function Visualize the relationships between domain and range of a function and the domain and range of its inverse Understand why functions and their inverses are symmetric about y = x Understand the difference between direct variation and inverse variation Understand the difference between combined variation and DO Determine whether a relations if a function Use function notation Find the domain and range of a function Perform operations on functions Sketch the graphs of common functions Determine whether a functions is increasing, decreasing, or constant on an interval Find the average rate of change of a function Classify a function as even, odd, or neither Sketch graphs of general functions employing translations of common functions Determine whether functions are one-to-one Find the inverse of a function Model applications with functions using variation

17 joint variation PRE-ASSESSMENT: Chapter 3 Practice Test page 321 Teacher Created Pre-Assessment ASSESSMENT: Chapter 3 Test Banks Chapter 3 Practice Test page 321 Teacher created assessments Chapter 3 Review Exercises pgs Chapters 1-3 Cumulative Testpage 323 GRAPHIC ORGANIZER & OR TECHNOLOGY: Video Tutorials available in every section Technology Tips to demonstrate graphing calculator key strokes TESTING SKILL(S) & OR SAMPLE OGT TYPE QUESTIONS: Catch the Mistake pgs. 242, 262, 278, 288, 302, 311 BEST PRACTICES: Cooperative Learning Technology Integration Lecture with guided practice Group work Compare and contrast Error analysis RESOURCES: Intermediate Algebra Second Edition WileyPLUS Printed Test Banks PowerPoint Reviews Online Student Resources Answer Documents Additional Technology Tips

18 TESTING VOCABULARY: Chapter 3 Review- pgs HISTORICAL/MODERN LINK: Technology Based Exercises Application Based Exercises Modeling Your World Page 314

19 TOPIC/UNIT: Chapter 4 Polynomial and Rational Functions Time Line: 4 Weeks CONCEPTUAL CATEGORY: Algebra DOMAIN: Arithmetic with Polynomials and Rational Expressions CLUSTER: Understand the relationship between zeros and factors of polynomials. 2. Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x a is p(a), so p(a) = 0 if and only if (x a) is a factor of p(x). 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. DOMAIN: Arithmetic with Polynomials and Rational Expressions CLUSTER: 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. DOMAIN: Reasoning with Equations and Inequalities CLUSTER: Solve equations and inequalities in one variable. 3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. 4. Solve quadratic equations in one variable. a. 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. b. 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 in a ± bi for real numbers a and b. DOMAIN: Reasoning with Equations and Inequalities CLUSTER: Represent and solve equations and inequalities graphically.

20 10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). 11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. CONCEPTUAL CATEGORY: Number and Quantity DOMAIN: The Complex Number System CLUSTER: Perform arithmetic operations with complex numbers. 1. Know there is a complex number i such as 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. 3. (+) Find the conjugate of complex number; use conjugates to find moduli and quotients of complex numbers. DOMAIN: The Complex Number System CLUSTER: Use complex numbers in polynomial identities and equations. 7. Solve quadratic equations with real coefficients that have complex solutions 8. (+) Extend polynomial identities to the complex numbers. For example, rewrite (x 2 + 4) as (x +2i)(x - 2i) CONCEPTUAL CATEGORY: Functions DOMAIN: Interpreting Functions CLUSTER: Understand the concept of a function and use function notation. 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, the 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).

21 2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. DOMAIN: Interpreting Functions CLUSTER: 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. 5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. 6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. DOMAIN: Interpreting Functions CLUSTER: Analyze functions using different representations. 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. d. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. 8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. 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. b. Use the properties of exponents to interpret expressions for exponential functions.

22 9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). DOMAIN: Linear, Quadratic, and Exponential Models CLUSTER: Construct and compare linear, quadratic, and exponential models and solve problems. 1. Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. KNOW Recognize characteristics of graphs of quadratic functions (parabolas) Understand that real zeros of polynomial functions correspond to x-intercepts Understand the intermediate value theorem and how it assists in graphing polynomial functions Realize that end behavior is a result of the leading term dominating Understand that zeros correspond to factors of the polynomial Extend long division of real numbers to polynomials Understand when synthetic division can be used Understand that a polynomial of DO Find the vertex (maximum or minimum) of the graph of a quadratic function. Graph quadratic functions Divide polynomials using long division Divide a polynomial by linear factor with synthetic division Understand that real zeros of a polynomials functions correspond to x-intercepts on its graph Develop strategies for searching for zeros (both real and complex) of a polynomial function Understand that complex zeros come in conjugate pairs Graph polynomials functions Find the asymptotes on the graph of a rational function Graph rational functions

23 degree n has at most n real zeros Understand that a real zero can be either rational or irrational and that irrational zeros will not be listed as possible zeros through the rational zero test Realize that rational zeros can be found exactly, whereas irrational zeros must be approximated Extend the domain of polynomials functions to complex numbers Solve application problems involving polynomial and rational functions Understand how the fundamental theorem of algebra guarantees at least one zero Understand why complex zeros occur in conjugate pairs Understand arrow notation Interpret the behavior of a rational function approaching an asymptote PRE-ASSESSMENT: Chapter 4 Practice Test page 418 Teacher Created Pre-Assessment ASSESSMENT: Chapter 4 Test Banks Chapter 4 Practice Test page 418 Teacher created assessments Chapter 4 Review Exercises pgs Chapters 1-4 Cumulative Testpage 419

24 GRAPHIC ORGANIZER & OR TECHNOLOGY: Video Tutorials available in every section Technology Tips to demonstrate graphing calculator key strokes TESTING SKILL(S) & OR SAMPLE OGT TYPE QUESTIONS: Catch the Mistake pgs. 341, 356, 365, 381, 389, 407 BEST PRACTICES: Cooperative Learning Technology Integration Lecture with guided practice Group work Compare and contrast Error analysis RESOURCES: Intermediate Algebra Second Edition WileyPLUS Printed Test Banks PowerPoint Reviews Online Student Resources Answer Documents Additional Technology Tips TESTING VOCABULARY: Chapter 4 Review- pgs HISTORICAL/MODERN LINK: Technology Based Exercises Application Based Exercises Modeling Your World pgs

25 TOPIC/UNIT: Chapter 5 Exponential and Logarithmic Functions Time Line: 3 Weeks CONCEPTUAL CATEGORY: Functions DOMAIN: Interpreting Functions CLUSTER: Understand the concept of a function and use function notation. 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, the 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. DOMAIN: Interpreting Functions CLUSTER: 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. 5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. DOMAIN: Interpreting Functions CLUSTER: Analyze functions using different representations. 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. 8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. b. Use the properties of exponents to interpret expressions for exponential functions. 9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

26 DOMAIN: Building Functions CLUSTER: Build new functions from existing functions. 4. Find inverse functions. a. 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. b. (+) Verify by composition that one function is the inverse of another. c. (+) Read values of an inverse function from a graph or table, given that the function has an inverse. d. (+) Produce an invertible function from a non-invertible function by restricting the domain. 5. (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. DOMAIN: Linear, Quadratic, and Exponential Models CLUSTER: Construct and compare linear, quadratic, and exponential models and solve problems. 1. Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. 2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table.) 3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. 4. For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

27 DOMAIN: Linear, Quadratic, and Exponential Models CLUSTER: Interpret expressions for functions in terms of the situation they model. 5. Interpret the parameters in a linear or exponential function in terms of a context. CONCEPTUAL CATEGORY: Algebra DOMAIN: Seeing Structure in Expressions CLUSTER: Write expressions in equivalent forms to solve problems. 3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression c. Use the properties of exponents to transform expressions for exponential functions.

28 KNOW Understand the difference between algebraic and exponential functions. Understand that irrational exponents lead to approximations. Interpret logarithmic functions as inverses of exponential functions. Understand that logarithmic functions allow very large ranges of numbers in science and engineering applications to be represented on a smaller scale. Derive the seven basic logarithmic properties. Derive the change-of-base formula. Understand how exponential and logarithmic equations are solved using properties of one-to-one functions and inverses. Recognize exponential growth, exponential decay, Gaussian distributions, logistic growth, and logarithmic models. DO Evaluate exponential functions Graph exponential functions Evaluate logarithmic functions Graph logarithmic functions Understand that exponential functions and logarithmic functions are inverses of each other Solve exponential and logarithmic equations. Understand domain restrictions on logarithmic functions Check for extraneous solutions when solving logarithmic equations Use exponential growth models and exponential decay models to represent a variety of real-world phenomena Use logarithmic models to represent ph values in chemistry and the Richter scale and decibels in engineering Use logistic grown models to represent conservation biology populations. PRE-ASSESSMENT: Chapter 5 Practice Test page 488 Teacher Created Pre-Assessment ASSESSMENT: Chapter 5 Test Banks Chapter 5 Practice Test page 488 Teacher created assessments Chapter 5 Review Exercises pgs Chapters 1-5 Cumulative Testpage 489

29 GRAPHIC ORGANIZER & OR TECHNOLOGY: Video Tutorials available in every section Technology Tips to demonstrate graphing calculator key strokes TESTING SKILL(S) & OR SAMPLE OGT TYPE QUESTIONS: Catch the Mistake pgs. 435, 449, 459, 469, 480 BEST PRACTICES: Cooperative Learning Technology Integration Lecture with guided practice Group work Compare and contrast Error analysis RESOURCES: Intermediate Algebra Second Edition WileyPLUS Printed Test Banks PowerPoint Reviews Online Student Resources Answer Documents Additional Technology Tips TESTING VOCABULARY: Chapter 5 Review- pgs HISTORICAL/MODERN LINK: Technology Based Exercises Application Based Exercises Modeling Your World - Page

30 TOPIC/UNIT: Chapter 6 Systems of Linear Equations and Inequalities Time Line: 2 Weeks CONCEPTUAL CATEGORY: Algebra DOMAIN: Creating Equations CLUSTER: Create equations that describe numbers or relationships. 1. Create equations and inequalities in one variable and use them to solve problems. 2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable in a modeling context. DOMAIN: Reasoning with Equations and Inequalities CLUSTER: Solve equations and inequalities in one variable. 3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. DOMAIN: Reasoning with Equations and Inequalities CLUSTER: Solve systems of equations. 5. Prove that 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. 7. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x 2 + y 2 = (+) Represent a system of linear equations as a single matrix equation in a vector variable. 9. (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3x3 or greater) DOMAIN: Reasoning with Equations and Inequalities CLUSTER: Represent and solve equations and inequalities graphically.

31 10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). 11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. 12. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. CONCEPTUAL CATEGORY: Functions DOMAIN: Interpreting Functions CLUSTER: Analyze functions using different representations. 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. d. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. 9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

32 KNOW Understand that a system of linear equations has either one solution, no solution, or infinitely many solutions Visualize two lines that intersect at one point, no points (parallel lines), or infinitely many points (same line) Understand that a graph of linear equation in three variables corresponds to a plane Identify three types of solutions: one solution (point), no solution, or infinitely many solutions (a single line in three dimensional space) Understand the connections between partial fraction decomposition and systems of linear equations Interpret the difference between solid and dashed lines Interpret an overlapped shaded region as a solution Understand that linear programming is a graphical method that solve optimization problems Understand why vertices represent maxima or minima DO Solve systems of linear equations in two variables with the substitution method and the elimination method Graph systems of linear equations (lines) Understand that systems of linear equations may have on solution, no solution, or infinitely many solutions Solve systems of linear equations in three variables employing a combination of the elimination and substitution methods Understand that graphs of linear equations in three variables correspond to planes Perform partial-fraction decomposition Solve a system of linear inequalities by finding the overlapping shaded regions Use the linear programming model to solve optimization problems subject to constraints

33 PRE-ASSESSMENT: Chapter 6 Practice Test page 558 Teacher Created Pre-Assessment ASSESSMENT: Chapter 6 Test Banks Chapter 6 Practice Test page 558 Teacher created assessments Chapter 6 Review Exercises pgs Chapters 1-6 Cumulative Testpage 559 GRAPHIC ORGANIZER & OR TECHNOLOGY: Video Tutorials available in every section Technology Tips to demonstrate graphing calculator key strokes TESTING SKILL(S) & OR SAMPLE OGT TYPE QUESTIONS: Catch the Mistake pgs. 507, 519, 531, 543, 550, BEST PRACTICES: Cooperative Learning Technology Integration Lecture with guided practice Group work Compare and contrast Error analysis RESOURCES: Intermediate Algebra Second Edition WileyPLUS Printed Test Banks PowerPoint Reviews Online Student Resources Answer Documents Additional Technology Tips TESTING VOCABULARY: Chapter 6 Review- pg. 554 HISTORICAL/MODERN LINK: Technology Based Exercises Application Based Exercises Modeling Your World pgs

34 TOPIC/UNIT: Chapter 7 Matrices Time Line: 3 Weeks CONCEPTUAL CATEGORY: Number and Quantity DOMAIN: Vector and Matrix Quantities CLUSTER: Perform operations on matrices and use matrices in applications. 6. (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network. 7. (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. 8. (+) Add, subtract, and multiply matrices of appropriate dimensions. 9. (+) Understand that, unlike multiplications of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties. 10. (+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse. 11. (+) Multiply a vector (regarded as a matrix with one column) by matrix suitable dimensions to produce another vector. Work with matrices as transformations of vectors. 12. (+) Work with 2x2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area. CONCEPTUAL CATEGORY: Algebra DOMAIN: Reasoning with Equations and Inequalities CLUSTER: Solve systems of equations. 5. Prove that 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. 8. (+) Represent a system of linear equations as a single matrix equation in a vector variable. 9. (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3x3 or greater)

35 KNOW Visualize an augmented matrix as a system of linear equations. Understand that solving systems with augmented matrices is equivalent to solving by the method of elimination. Recognize matrices that correspond to inconsistent and dependent systems. Understand what is meant by equal matrices. Understand why multiplication of some matrices is undefined. Realize that matrix multiplication is not commutative. Visualize a system of linear equations as a matrix equation. Understand that only a square matrix can have an inverse. Realize that not every square matrix has an inverse. Derive Cramer s Rule Understand that if a determinant of a matrix is equal to zero, then that matrix does not have an inverse. Understand that Cramer s rule can be used to find only a unique solution. DO Represent a system of linear equations with an augmented matrix. Preform row operations in order to present augmented matrices in row-echelon form and reduced row-echelon form. Apply Gaussians elimination with back-substitution to solve systems of linear equations. Apply Gauss-Jordan elimination to solve systems of linear equations Perform matrix operations: addition, subtraction, and multiplication. Find the inverse of a square matrix Utilize matrix algebra and inverses of matrices to solve systems of linear equations. Find the determinant of a square matrix. Use Cramer s rule (determinants) to solve systems of linear equations.

36 PRE-ASSESSMENT: Chapter 7 Practice Test page 634 Teacher Created Pre-Assessment GRAPHIC ORGANIZER & OR TECHNOLOGY: Video Tutorials available in every section Technology Tips to demonstrate graphing calculator key strokes ASSESSMENT: Chapter 7 Test Banks Chapter 7 Practice Test page 634 Teacher created assessments Chapter 7 Review Exercises pgs Chapters 1-7 Cumulative Testpage 635 TESTING SKILL(S) & OR SAMPLE OGT TYPE QUESTIONS: Catch the Mistake pgs. 582, 597, 610, 624 BEST PRACTICES: Cooperative Learning Technology Integration Lecture with guided practice Group work Compare and contrast Error analysis RESOURCES: Intermediate Algebra Second Edition WileyPLUS Printed Test Banks PowerPoint Reviews Online Student Resources Answer Documents Additional Technology Tips TESTING VOCABULARY: Chapter 7 Review- pgs HISTORICAL/MODERN LINK: Technology Based Exercises Application Based Exercises Modeling Your World - Pgs

37 TOPIC/UNIT: Chapter 8 Conics and Systems of Nonlinear Equations and Inequalities Time Line: 3 Weeks CONCEPTUAL CATEGORY: Geometry DOMAIN: Expressing Geometric Properties with Equations CLUSTER: Translate between the geometric description and the equation for a conic section. 1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. 2. Derive the equation of a parabola given a focus and directrix. 3. (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant. CONCEPTUAL CATEGORY: Algebra DOMAIN: Reasoning with Equations and Inequalities CLUSTER: Solve systems of equations. 5. Prove that 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. 7. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x 2 + y 2 = 3. DOMAIN: Reasoning with Equations and Inequalities CLUSTER: Represent and solve equations and inequalities graphically. 10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

38 KNOW Understand each conic as an intersection of a plane and a cone. Understand how the three equations of the conic sections are related to the general form of a second-degree equation in two variables. Derive the general equation of a parabola. Identify, draw, and use the focus, directrix, and axis of symmetry. Derive the general equation of an ellipse. Understatnd the meaning of major and minor axes are foci. Understand the properties of an ellipse that result in a circle. Interpret eccentricity in terms of the shape of the ellipse. Derive the general equation of a hyperbola. Identify, apply, and graph the transverse axis, vertices, and foci. Use asymptotes to determine the shape of a hyperbola. Interpret the algebraic solution graphically. Understand the types of solutions: distinct number of solutions, no solution, and infinitely many solutions. DO Visualize conics in terms of a plane intersecting a right double cone. Define the three conics. Develop the equations of the three conics from the general second-degree equation in two variables. Graph parabolas, ellipses, and hyperbolas. Solve application problems involving all three conics. Solve systems of nonlinear equations an interpret some graphically in terms of conics. Solve systems of nonlinear inequalities and interpret some graphically in terms of conics. Understand that equations of conic sections are nonlinear equations. Understand that a nonlinear inequality in two variables may be

39 represented by either a bounded or an unbounded region. Interpret an overlapping shaded region as a solution. PRE-ASSESSMENT: Chapter 8 Practice Test page 702 Teacher Created Pre-Assessment ASSESSMENT: Chapter 8 Test Banks Chapter 8 Practice Test page 702 Teacher created assessments Chapter 8 Review Exercises pgs Chapters 1-8 Cumulative Testpage 703 GRAPHIC ORGANIZER & OR TECHNOLOGY: Video Tutorials available in every section Technology Tips to demonstrate graphing calculator key strokes TESTING SKILL(S) & OR SAMPLE OGT TYPE QUESTIONS: Catch the Mistake pgs. 653, 665, 677, 688, 696 BEST PRACTICES: Cooperative Learning Technology Integration Lecture with guided practice Group work Compare and contrast Error analysis RESOURCES: Intermediate Algebra Second Edition WileyPLUS Printed Test Banks PowerPoint Reviews Online Student Resources Answer Documents Additional Technology Tips