HS Algebra IIB Mathematics CC

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1 Scope And Sequence Timeframe Unit Instructional Topics 1 Week(s) 1 Week(s) 1 Week(s) 1 Week(s) 2 Week(s) 1 Week(s) 2 Week(s) 1 Week(s) Unit 1: Linear and Absolute Value Equations and Inequalities Unit 3: Systems and Matrices Unit 4: Polynomials and Rational Functions Unit 5: Rational Exponents and Radical Functions Unit 6: Exponential and Logarithmic Functions Unit 7: Sequences and Series Unit 8: Probability and Statistics Unit 9: Intro to Conic Sections 1. Use unit analysis with examples 2. Represent Relations and Functions 3. Use absolute value functions transformations 1. (+) Perform matrix operations 1. Review properties of exponents 2. Evaluate and graph polynomial equations 3. Analyze graphs of polynomial functions 4. Write polynomial functions and models. 5. Model inverse variation 6. (+) Graph rational functions 7. Multiply and divide rational expressions 8. Add and subtract rational expressions 9. Solve rational equations 1. Evaluate n roots and use rational exponents 2. Perform function operation and composition 3. Inverse functions 4. Graph square root and cube root functions 5. Solve radical equations 1. Graph exponential growth and decay functions 2. Use functions involving e 3. Evaluate logarithms and graph logarithmic functions 4. Apply properties of logarithms 5. Solve exponential and logarithmic equations 6. Write and apply exponential and power functions. 1. Define and use sequences and series 2. Analyze arithmetic sequences and series 3. Analyze geometric sequences and series 4. Use recursive rules with sequences and functions Course This course includes the study of a variety of functions (linear, quadratic higher order polynomials, exponential, absolute value, logarithmic and rational) learning to graph, compare, perform operations and manipulate them in order to solve, analyze and apply to problems. Students will use probability and statistics to evaluate outcomes of decisions. Students develop rigorous problem solving skills, logical reasoning and mathematical communication skills required for success in higher math courses and real life experiences. *Graphing calculators are required. 1. Apply counting principal and permutations 2. Use combinations and the Binomial Theorem 3. Define and use probability 4. Define probabilities of disjunct and overlapping events 5. Find probabilities of independent and dependent events 6. Find measures of central tendencies. 7. Apply transformations to data 8. Use normal distributions 9. Select and draw conclusions from samples 10. (+) Using probability to make decisions 1. Graph and write equations of parabolas. 2. Graph and write equations of circles. 3. Translate and classify conic sections. Page 1

2 Course Rationale In alignment with Common Core State Standards, the Park Hill School District's Mathematics courses provide students with a solid foundation in number sense while building to the application of more demanding math concepts and procedures. The courses focus on procedural skills and conceptual understandings to ensure coherence and depth in mathematical practices and application to real world issues and challenges. 1. Mathematical understanding is built through problem solving and reasoning. 2. There is a direct relationship between Logarithmic and Exponential Functions. 3. Statistics and probability are tools to solve many mathematical situations and make predictions in the real world. 4. Parent functions are the foundation of identifying, analyzing, constructing, and transforming graphs. 5. Appropriate mathematical tools help to investigate, solve and explain real world situations. 6. Problems are solved using both visual (including a graphing calculator) and analytical means. Board Approval Date January 10, 2013 Course Details Unit: Unit 1: Linear and Absolute Value Equations and Inequalities Students will graph and solve linear and absolute value equations and inequalities Duration: 1 Week(s) Equations and inequalities allow graphic representation and problem solving. Equations, tables, graphs and words can all be utilized interchangeably to represent the same event. Equations can provide a model that can be used to analyze, draw conclusions, and make predictions. Why would you choose an inequality to represent a situation instead of an equation? How would you determine which representation would be appropriate for a given event? How would you use a model to analyze, draw conclusions, and make predictions about an event? Given an equation or inequality, make a graph of the function. Given a table of values, write an equation. Given a real world situation, write and solve an equation or inequality to represent the situation. Topic: Use unit analysis with examples Duration: 0 Day(s) The student will 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. The student will choose a level of accuracy appropriate to limitations on measurement when reporting quantities. The student wil perform common conversion. Solve multistep arithemetic problems that involve planning or converting units of measure. ACT College Readiness Standards Mathematics The student will solve routine two or three step arithmetic problems involving concepts such as rate and proportion, tax added, percentage off and computing with a given average. ACT College Readiness Standards Mathematics Represent Relations and Functions The student will 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). Use absolute value functions transformations (+) The student will identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Page 2

3 Unit: Unit 3: Systems and Matrices Duration: 1 Week(s) Students will use mutliple equations and inequalites (called systems) to model and solve real-world problems. Students will use matrices as a method of solving systems of equations. Mulitple equations can be used to simultaneously model and solve sutuations with several variables. Various methods, including matrices, will be used to solve simultaneous equations. Solutions to simulteous equations may include one solution, infitinite solutions, and no solutions. How can simultaneous equations and/or inequalities be used to model real world situations? How can matrices be used to solve simultaneous equations? What types of solutions can arise when solving simultaneous equations and/or inequalities? Write a system of equations and/or inequalties based on information given. Solve simultaneous equations using matrices. Compare and contrast sytems of equations involving intersecting lines, coinciding lines, parallel lines or a common shaded region. Academic Vocabulary Watch the wording of "systems of equations" being called "simultaneons equations". Use both terms when teaching this unit. matrix Topic: (+) Perform matrix operations Duration: 0 Day(s) Add,subtract, multiply by a scalar, and multiply. (+) The student will add, subtract, and multiply matrices of appropriate dimensions. only - otherwise Acc Algebra II (+) The student will find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 3 or greater). (+) The student will multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors. (+) The student will 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 (+) The student will understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties. (+) The student will work with 2 2 matrices as transformations of the plane. (+) The student will use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network. (+) The student will multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. Unit: Unit 4: Polynomials and Rational Functions Duration: 1 Week(s) Page 3

4 Polynomials are explored, generalizing the concepts of quadratic functions to higher degrees with a special focus on zeros and factors. The shape and features of a graph provide valuable information about its corresponding equation. Fractional (rational) equations are used to model real world situations including inverse variation. The terms zero, factor, solution and x-intercept are closely related. How can the features of a polynomial aid in a sketch of its graph? How can higher order polynomials and rational functions be used as tools to best describe and help explain real world situations? How do the terms zero, factor, solution and x-intercept relate? Given the features of a polynomial, sketch its graph. Given a situation, write and solve higher order polynomial (or rational) functions. Use factoring to find the zeros/x-intercepts of a polynomial function. Academic Vocabulary asymptotes, domain/range, parent function, polynomial function, radical equation/function, rational expression, synthetic division, translation of functions, zeros Topic: Review properties of exponents Duration: 0 Day(s) The student will use the properties of exponents to transform expressions for exponential functions. The student will 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. The student will rewrite expressions involving radicals and rational exponents using the properties of exponents. The student will use the properties of exponents to interpret expressions for exponential functions. The student will use the properties of exponents to transform expressions for exponential functions. Evaluate and graph polynomial equations Describe end behavior. Evaluate by direct and synthetic substitution. The student will 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). The student will 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. The student will graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. The student will 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. Analyze graphs of polynomial functions Students will use zeros, factors, solutions, maxima, minima, and end behavior to describe and graph a polynomial. The student will 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. The student will graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. The student will graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. For - otherwise, it will be taught in ACC Algebra II The student will graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. (+) The student will identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Page 4

5 The student will 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. The student will, 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. Write polynomial functions and models. The student will 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. Model inverse variation (+) The student will read values of an inverse function from a graph or a table, given that the function has an inverse. For only - otherwise teach only in ACC Algebra II The student will solve an equation of the form f(x) for a simple function f that has an inverse and write an expression for the inverse. (+) Graph rational functions Find vertical and horizontal asymptotes, zeros, domain and range. The student will 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. The student will graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. For - otherwise, it will be taught in ACC Algebra II The student will 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). The student will produce an invertible function from a non-invertible function by restricting the domain. For only - otherwise only in ACC Algebra II Multiply and divide rational expressions The student will 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. Add and subtract rational expressions The student will 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 and an irrational number is irrational. The student will 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. (+) The student will understand that rational expressions form a system anaologous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. A-APR 7 Page 5

6 Topic: Solve rational equations Duration: 0 Day(s) Check solutions. The student will 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. Unit: Unit 5: Rational Exponents and Radical Functions Duration: 1 Week(s) The properties of exponents are extended to include rational numbers. These properties and function operations are used to graph, solve and/or build new functions from existing ones. Basic mathematical operations can be performed on given functions resulting in an infinite number of possible new functions. Properties of exponents can be used to simplify and solve equations. A function must pass the horizontal line test in order to have an inverse. Parent functions are the foundation for graphing. When might it be useful to combine functions instead of working with two separate functions? How are basic properties of exponents applied? How can the horizontal test be used to determine if a function has an inverse? How are parent functions used to graph translations? Write a composition of two given functions. Use properties of exponents to simplify and solve equations. Determine if a given graph has an inverse. Given a complex radical function, use translations of a parent function to graph. Academic Vocabulary composition of functions, domain/range, function, inverse function, rational exponents Evaluate n roots and use rational exponents Apply properties of rational exponents. Remember to use absolute value when the root is even. The student will choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. The student will 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 and an irrational number is irrational. The student will rewrite expressions involving radicals and rational exponents using the properties of exponents. The student will use the properties of exponents to interpret expressions for exponential functions. The student will use the structure of an expression to identify ways to rewrite it. Perform function operation and composition (+) The student will compose functions. For only - otherwise it will be taught in Acc. Algebra II (+) The student will identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. (+) The student will verify by composition that one function is the inverse of another. Inverse functions Make sure to check domain and range. Page 6

7 (+) The student will read values of an inverse function from a graph or a table, given that the function has an inverse. For only - otherwise teach only in ACC Algebra II The student will solve an equation of the form f(x) for a simple function f that has an inverse and write an expression for the inverse. (+) The student will verify by composition that one function is the inverse of another. (+) The student will produce an invertible function from a non-intervertible function by restricting the domain. F-BF 4d Graph square root and cube root functions The student will 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). The student will compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). The student will 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. The student will graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. (+) The student will identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. The student will observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. The student will recognize the graph of an algebraic function. The student will 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). The student will 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. The student will write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. - 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. The student will, 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. Solve radical equations Check solutions. The student will 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. The student will rewrite expressions involving radicals and rational exponents using the properties of exponents. Unit: Unit 6: Exponential and Logarithmic Functions Duration: 2 Week(s) Page 7

8 Properties and operations of exponential and logarithmic functions are explored. Exponential and logarithms are inverse operations that can be manipulated and re-written. The characteristics of exponential and logarithmic functions and their representations are useful in solving real world problems. Growth and decay are examples of the usefulness of logarithmic and exponential functions. Why do we need both logarithmic and exponential equations? How can exponential and logarithmic functions be used as tools to best describe and help explain real world situations? What characteristics of a function determine exponential growth or decay? Find the inverse of an exponential or logarithmic function. Set up an equation for exponential growth or decay given a situation. Given an equation, determine if it represents exponential growth or decay. Academic Vocabulary domain/range, function, translation of functions Graph exponential growth and decay functions The student will graph exponential and logarithmic functions, showing intercepts and end behavior. Use functions involving e The student will, for exponential models, express as a logarithm the solution to a times b to the (ct) power = d where a, c, and d are numbers and the base b is a real number including e; evaluate the logarithm by properties or technology. a(b)(ct)=d Evaluate logarithms and graph logarithmic functions Introduce by finding the inverse of an exponential function. The student will graph exponential and logarithmic functions, showing intercepts and end behavior. (+) The student will understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. Apply properties of logarithms (+) The student will understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. The student will, for exponential models, express as a logarithm the solution to a times b to the (ct) power = d where a, c, and d are numbers and the base b is a real number including e; evaluate the logarithm by properties or technology. a(b)(ct)=d Solve exponential and logarithmic equations (+) The student will understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. The student will, for exponential models, express as a logarithm the solution to a times b to the (ct) power = d where a, c, and d are numbers and the base b is a real number including e; evaluate the logarithm by properties or technology. a(b)(ct)=d The student will use the properties of exponents to interpret expressions for exponential functions. Page 8

9 Write and apply exponential and power functions. The student will interpret the parameters in a linear or exponential function in terms of a context. The student will observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. The student will recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. Unit: Unit 7: Sequences and Series Students will write equations from patterns of data. Duration: 1 Week(s) Numerical patterns can be quantified and used to make predictions. Linear equations can be expressed recursively and explicitly. Geometric equations can be expressed recursively and explicitly. How can patterns, relations and functions be used as tools to best describe and help explain real world situations. How can a linear sequence be written recursively and explicitly. How can a geometric sequence be written recursively and explicitly? Given a situation involving a set of linear or geometric data, predict a future outcome. Given a linear sequence, express it recursively and explicitly. Given a geometric sequence, express it recursively and explicitly. Topic: Define and use sequences and series Duration: 0 Day(s) The student will 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. The student will write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Analyze arithmetic sequences and series The student will write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Analyze geometric sequences and series The student will 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. The student will write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Use recursive rules with sequences and functions The student will determine an explicit expression, a recursive process, or steps for calculation from a context. The student will write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Unit: Unit 8: Probability and Statistics Duration: 2 Week(s) Page 9

10 Statistics and probability are tools used to solve many mathematical situations and make predictions in the real world. Counting methods can be used to determine possible outcomes as well as the likelihood of an event occurring. Measures of central tendency can be used to quantify sets of data. Normal distributions are used to predict probabilities in real life situations. How can probability be used to make predictions or draw conclusions? How are measures of central tendency used to quantify data? How can normal distribution be used to predict probabilities? Given the probability of P(p) and P(q), find P(p q), P(p U q), etc... Given a set of data, find the five number summary (highest value, lowest value, median, upper and lower quartiles.) Using the empirical rule, find how much data falls within 1 standard deviation of the mean. Topic: Apply counting principal and permutations Duration: 0 Day(s) Permutations is in Accelerated only (+) The student will 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. (+) The student will use permutations and combinations to compute probabilities of compound events and solve problems. Use combinations and the Binomial Theorem Binomial Theorem is in Accelerated only (+) The student will know and apply the Binomial Theorem for the expansion of (x + y) to the n power (where x and y are any numbers and/or coefficents) with coefficients determined for example by Pascal's Triangle. Define and use probability Define probabilities of disjunct and overlapping events The student will 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. Find probabilities of independent and dependent events The student will 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. The student will recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. The student will construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. Find measures of central tendencies. The student will use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling Page 10

11 The student will use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve Apply transformations to data Example: Adding a constant to each data value or multiplying each data value by a constant and how this affects the statistics. Use normal distributions The student will use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve Select and draw conclusions from samples The student will evaluate reports based on data. The student will recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. The student will use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant The student will use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling The student will use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve The student will 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. The student will translate from one representation of data to another (e.g. a bar graph to a circle graph). ACT College Readiness Standards Mathematics (+) Using probability to make decisions Completed in Accelerated (+) The student will define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions. (+) The student will calculate the expected value of a random variable; interpret it as the mean of the probability distribution. (+) The student will develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. (+) The student will develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. (+) The student will weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values. - Find the expected payoff for a game of chance. - Evaluate and compare strategies on the basis of expected values. (+) The student will use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). (+) The student will analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). Page 11

12 Unit: Unit 9: Intro to Conic Sections Students will graph and write equations of parabolas and circles given various parameters. Circles and parabolas can be found from cross sections of a double-napped cone, but have distinctively different characteristics and equations. What's the difference between a parabola and a circle? Given an equation, identify it as a parabola or circle and graph. Duration: 1 Week(s) Graph and write equations of parabolas. The student will derive the equation of a parabola given a focus and directrix. Graph and write equations of circles. The student will 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. Translate and classify conic sections. Cirlces and parabolas only for regular algebra II, add ellipses for acc algebra II The student will complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. (+) The student will identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. The student will 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. Page 12

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