Westbrook Public Schools Algebra II Curriculum Grades 10-12

Transcription

1 Westbrook Public Schools Algebra II Curriculum Grades Unit 1: Functions and Inverse Functions Mathematical Practices: (Practices in bold are to be emphasized in the unit.) Anticipated Length: 4 weeks 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Standards (priority standards in bold, supporting standards in italics): CC.9-12.F.IF.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. CC.9-12.F.IF.7b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. CC.9-12.F.BF.1 Write a function that describes a relationship between two quantities. CC.9-12.F.BF.1c (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time. CC.9-12.F.BF.4 Find inverse functions CC.9-12.F.BF.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 ) for x > 0 or f(x) = (x+1)/(x-1) for x = 1 (x not equal to 1). CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9-12.F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, kf(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. Include recognizing even and odd functions from their graphs and algebraic expressions for them. CC.9-12.F.IF.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. CC.9-12.F.IF.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. CC.9-12.F.BF.4d (+) Produce an invertible function from a non-invertible function by restricting the domain. 480

2 CC.9-12.F.BF.4b (+) Verify by composition that one function is the inverse of another. CC.9-12.A.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. CC.9-12.F.IF.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. CC.9-12.F.BF.4c (+) Read values of an inverse function from a graph or a table, given that the function has an inverse. CC.9-12.F.BF.4d (+) Produce an invertible function from a non-invertible function by restricting the domain. Knowledge: Students will know/understand: Functions (expressed symbolically) o Square root o o Key Features o Intercepts o Cube root Piecewise-defined (includes step and absolute value) Intervals Increasing or decreasing, Positive or negative Relative maximums and minimums Symmetries End behavior / endpoints Technology (graphing complicated functions) Technology (graphing complicated functions) Inverse functions Equation (of form f(x)=c) Essential Questions: When you square each side of an equation, is the resulting equation equivalent to the original? How are a function and its inverse related? Skills: Students will be able to: GRAPH functions SHOW key features of functions USE technology to identify key features of a function WRITE functions COMPOSE functions and understand composition in terms of context of the problem FIND inverse functions and attend to domain e.g. restrictions SOLVE equations WRITE expressions Unit Questions: What is the domain and range of a function? How are inverse functions identified? What is meant by a relative maximum or minimum? Core Belief Addressed: The Westbrook High School student will Think critically to solve problems and reach well reasoned judgments. Assessment Type Questions: 1. For the following functions: A. Graph the function B. State the domain and range. C. Find the inverse of the function. D. State the domain and range of the inverse function. a.) y = x 2 b.) y = x

3 1 c.) y = x d.) y = 2 x+ 1 4 e.) y = 3 x For 1 f( x) = x 1 what is each of the following: a.) f 1 ( x) b.) 1 ( f f )(1) 3. For 2 f ( x) = x and g( x) = x 3find the following: a.) ( g f)( 2) 7 b.) ( f g)( ) 2 1 c.) 3 f( 2) + g(4) 2 4. What are the relative maximum or minimum of 3 2 f( x) x 3x 24x = +? What are the zeros of at these zeros? f( x) x 2x 8x =? What are their multiplicities? How does the graph behave Possible Assessments: Teacher created assessments Unit tests Quizzes Student presentations Entrance/Exit slips Group projects Benchmark Assessments: Common unit assessments Star assessments (October, January, May) Midterm exam Final exam 482

4 Performance Task: Technology Integration: SMART Board SMART Math tools Texas Instrument virtual calculator Resources/Materials: Texts: o Pearson Algebra 2, Chapters 2, 5, and 6 Internet Sites: o PowerAlgebra.com Other: o Graphing calculators 483

5 Sample Lessons/Learning Activities: CC.9-12.F.IF.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. For each of the following functions, state the zeros, end-behavior, and relative maximum or minimum ) ( ) = f x x x x ) f( x) = 4x + 12x + 4x 12 CC.9-12.F.IF.7b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. 2x + 3, if x < 2 1. Use a graphing calculator to graph the function f( x) = { x 1, if x Graph the absolute function f( x) = 4x 6 + 3and state the vertex. 2 CC.9-12.F.BF.1 Write a function that describes a relationship between two quantities. CC.9-12.F.BF.1c (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time. CC.9-12.F.BF.4 Find inverse functions For each function, find the inverse and the domain and range of the function and its inverse. Determine whether the inverse is a function. 1. f( x) = x f( x) = x + 1 CC.9-12.F.BF.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 ) for x > 0 or f(x) = (x+1)/(x-1) for x = 1 (x not equal to 1). 484

6 Westbrook Public Schools Algebra II Curriculum Grades Unit 2: Polynomial Functions Mathematical Practices: (Practices in bold are to be emphasized in the unit.) Anticipated Length: 5 weeks 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Standards (priority standards in bold, supporting standards in italics): CC.9-12.N.CN.7 Solve quadratic equations with real coefficients that have complex solutions. CC.9-12.A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. CC.9-12.F.IF.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. CC.9-12.F.IF.7c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior CC.9-12.N.CN.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. CC.9-12.N.CN.2: Use the relation i 2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers CC.9-12.N.CN.8 (+) Extend polynomial identities to the complex numbers. For example, rewrite x as (x + 2i)(x - 2i). CC.9-12.N.CN.9 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials CC.9-12.A.REI.4b 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 CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9-12.A.REI.4 Solve quadratic equations in one variable CC.9-12.A.REI.4b 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 485

7 CC.9-12.A.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x 4 - y 4 as (x 2 ) 2 - (y 2 ) 2, thus recognizing it as a difference of squares that can be factored as (x 2 - y 2 )(x 2 + y 2 ). CC.9-12.A.APR.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. CC.9-12.A.APR.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). CC.9-12.A.APR.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 CC.9-12.A.APR.4 Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x 2 + y 2 ) 2 = (x 2 - y 2 ) 2 + (2xy) 2 can be used to generate Pythagorean triples CC.9-12.A.APR.5 (+) Know and apply that the Binomial Theorem gives the expansion of (x + y) n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal s Triangle. (The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.) CC.9-12.N.CN.9 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. CC.9-12.F.1F.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. CC.9-12.F.LE.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. Knowledge: Students will know/understand: Quadratic equations Root of quadratic equation Solution of a quadratic equation Quadratic inequality Problem solving with quadratics Polynomial functions Zeroes of a polynomial function Factoring x- /y- intercepts Intervals of increase/decrease Relative maxima/minima Symmetry Imaginary numbers Complex numbers Properties of complex numbers Operations on complex numbers Skills: Students will be able to: SOLVE quadratic equations over the set of complex numbers. SOLVE a quadratic equation by completing the square over the set of complex numbers SOLVE a quadratic equation by the quadratic formula over the set of complex numbers. DETERMINE the nature of the roots of a quadratic equation CREATE a polynomial equation from its roots. GRAPH polynomial functions using a calculator LABEL important points of a polynomial function. CALCULATE the x- /y- intercepts of a polynomial function DETERMINE the intervals of increase/decrease of a polynomial function APPLY properties of polynomial functions to solve real world problems FIND the roots of a polynomial function 486

8 USE a graphing utility to find relative maxima/minima of a polynomial function SIMPLIFY an imaginary number SIMPLIFY a complex number CALCULATE the sum of two or more complex numbers. FIND the difference of two or more complex numbers. CALCULATE the product of two or more complex numbers. FIND the quotient of two or more complex numbers. RATIONALIZE the denominator of a complex rational number. Essential Questions: How do patterns and functions help us describe data and physical phenomena and solve a variety of problems? How are zeroes important? Unit Questions: What are the zeroes of a quadratic equation? What are the different ways a quadratic equation can be solved? How can a function be created from its roots? What information does the roots of a function give? Where are complex numbers used? Why are complex numbers important? Core Belief Addressed: The Westbrook High School student will Think critically to solve problems and reach well reasoned judgments. Assessment Type Questions: 2 1. For the equation y 3( x 4) 2 minimum value, the domain, and the range? 2. Write a quadratic equation in vertex form of the following graphs. = what are the vertex, the axis of symmetry, the maximum or a) 487

9 b) 3. Solve the quadratic equation 4. Solve the quadratic equation 2 3x 16x 12 = + by factoring. 2 3x 12x =. 5. Solve the quadratic equation 2 2x 3x= 5 using the quadratic equation. 6. Simplify: a) (8 i)( 4 i)(9 i) b) 5(1+ 2) i + 3(3 i 4) i c) 2+ 3i 1 4i Possible Assessments: Teacher created assessments Unit tests Quizzes Students presentations Entrance/Exit slips Group projects Benchmark Assessments: Common unit assessments Star assessments (October, January, May) Midterm exam Final exam 488

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12 Technology Integration: SMART Board SMART math tools Texas Instrument virtual calculator Resources/Materials: Texts: o Pearson Algebra 2, Chapters 4 and 5 Internet Sites: o PowerAlgebra.com Other: o Graphing calculators Sample Lessons/Learning Activities: CC.9-12.N.CN.7 Solve quadratic equations with real coefficients that have complex solutions. Find all the solutions to each quadratic equation x + x 3= x = 4x 7 491

13 CC.9-12.A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. CC.9-12.F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. CC.9-12.F.IF.7c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior 492

14 Westbrook Public Schools Algebra II Curriculum Grades Unit 3: Rational Expressions and Functions Mathematical Practices: (Practices in bold are to be emphasized in the unit.) Anticipated Length: 3 weeks 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Standards (priority standards in bold, supporting standards in italics): CC.9-12.A.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise CC.9-12.F.IF.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* CC.9-12.F.IF.7d (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior CC.9-12.A.SSE.1 Interpret expressions that represent a quantity in terms of its context.* CC.9-12.A.SSE.1b Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r) n as the product of P and a factor not depending on P CC.9-12.A.APR.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 CC.9-12.A.APR.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 CC.9-12.A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. CC.9-12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. CC.9-12.A.REI.11Explain 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.* 493

15 CC.9-12.F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, kf(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. Include recognizing even and odd functions from their graphs and algebraic expressions for them Knowledge: Students will know/understand: Root Radicand Radical expression Product and quotient rules for radicals Rationalize Sum and difference rules for radicals Rational exponent Radical equation Extraneous solutions Essential Questions: How can you determine and use the relationships between models to further investigate future relationships? What is the correlation between the graphic and algebraic representations of the solutions to a function? Skills: Students will be able to: FIND the nth root SIMPLIFY a radical expression PERFORM operations with radical expressions RATIONALIZE a denominator SIMPLIFY rational exponents SOLVE a radical equation CHECK FOR extraneous solutions Unit Questions: Why do negative radicands have no real roots? How does a radicand and its simplest form compare? How are rational exponents related to radicals? Core Belief Addressed: The Westbrook HS student will Think critically to solve problems and reach well reasoned judgments. Assessment Type Questions: Solve. Check for extraneous solutions x x = ( x+ 5) (5 2 x) = 0 Find the domain, points of discontinuity, x- and y- intercepts for each rational function. Determine whether the discontinuities are removable or non-removable and then find the vertical asymptotes y = y = 2x x 2x x x + x 3 494

16 Possible Assessments: Teacher created assessments Unit tests Quizzes Students presentations Entrance/Exit slips Group projects Benchmark Assessments: Common unit assessments Star assessments (October, January, May) Midterm exam Final exam 495

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20 Technology Integration: SMART Board SMART Math tools Texas Instrument virtual calculator Resources/Materials: Texts: o Pearson Algebra 2, Chapters 6 and 8 Internet Sites: o PowerAlgebra.com Other: o Graphing calculators 499

21 Sample Lessons/Learning Activities: CC.9-12.A.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise What is the solution of 3 ( 1) 1 25 x + + = and check for extraneous solutions. CC.9-12.F.IF.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* CC.9-12.F.IF.7d (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior 500

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24 Westbrook Public Schools Algebra II Curriculum Grades Unit 4: Trigonometric Functions Mathematical Practices: (Practices in bold are to be emphasized in the unit.) Anticipated Length: 4 weeks 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Standards (priority standards in bold, supporting standards in italics): CC.9-12.F.IF.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. CC.9-12.F.IF.7e Graph trigonometric functions, showing period, midline, and amplitude CC.9-12.F.TF.2 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. CC.9-12.F.TF.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. CC.9-12.F.1F.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.* CC.9-12.F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, kf(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. Include recognizing even and odd functions from their graphs and algebraic expressions for them. CC.9-12.F.TF.4 (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. CC.9-12.F.TF.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. CC.9-12.F.TF.3 (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π - x, π + x, and 2π - x in terms of their values for x, where x is any real number CC.9-12.F.TF.8 Prove the Pythagorean identity (sin A) 2 + (cos A) 2 = 1 and use it to calculate trigonometric ratios. 503

25 CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales Knowledge: Students will know/understand: Sine function Cosine function Tangent function x- intercepts Maxima Minima Symmetry Axis Asymptotes Period Intervals of increase/decrease Unit circle Special angles Radian measure Transformations Positive/negative angles Even/odd function Skills: Students will be able to: GRAPH the sine function on graph paper GRAPH the cosine function on graph paper GRAPH the tangent function on graph paper IDENTIFY zeros of trig functions on the graph DETERMINE the equations of asymptotes on the graph. Manually GRAPH all special angles of the unit circle for sine function Manually GRAPH all special angles of the unit circle for the cosine function. Manually GRAPH all special angles of the unit circle for the tangent function DETERMINE the axi of the sine and cosine function. USE technology to create a good graph of the sine function USE technology to create a good graph of the cosine function USE technology to create a good graph of the tangent function DETERMINE the amplitude of a trig function STATE all transformations of a graph of a trigonometric function APPLY the properties of even/odd functions to the graphs of trigonometric functions. CALCULATE the period of any trigonometric function DEFINE radian CONVERT any angle in degrees to radian measure STATE the values of the nspecial angles of the unit circle EVALUATE the sine ratio for any angle of a right triangle CALCULATE the cosine function for any angle of a right triangle FIND the value of the tangent eratio for any angle of a tight triangle STATE any angle as a positive value of the unit circle SHOW key features and end behavior USE unit technology EXPLAIN extension MODEL periodic phenomena 504

26 Essential Questions: How do patterns and functions help describe data and physical phenomena and solve a variety of problems? What patterns do angles follow? Unit Questions: How are the graphs of the sine and cosine function related? What is amplitude? How do transformations affect the shape of the graph? What is the role of asymptotes? How is the unit circle related to any right triangle? What is period? How is period affected? Core Belief Addressed: The Westbrook HS student will Think critically to solve problems and reach well reasoned judgments. Assessment Type Questions: 505

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28 Possible Assessments: Teacher created assessments Unit tests Quizzes Students presentations Entrance/Exit slips Group projects Benchmark Assessments: Common unit assessments Star assessments (October, January, May) Midterm exam Final exam Performance Task: 507

29 Technology Integration: SMART Board SMART Math tools Texas Instrument virtual calculator Resources/Materials: Texts: o Pearson Algebra 2, Chapter 13 Internet Sites: o PowerAlgebra.com Other: o Graphing calculators Sample Lessons/Learning Activities: CC.9-12.F.IF.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. 508

30 CC.9-12.F.IF.7e Graph trigonometric functions, showing period, midline, and amplitude 509

31 CC.9-12.F.TF.2 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. 510

32 CC.9-12.F.TF.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. 511

33 Westbrook Public Schools Algebra II Curriculum Grades Unit 5: Exponential and Logarithmic Functions Mathematical Practices: (Practices in bold are to be emphasized in the unit.) Anticipated Length: 5 weeks 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Standards (priority standards in bold, supporting standards in italics): CC.9-12.F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. CC.9-12.F.IF.8b Use the properties of exponents to interpret expressions for exponential functions. CC.9-12.F.IF.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. CC.9-12.F.IF.7e Graph exponential and logarithmic functions, showing intercepts and end behavior. CC.9-12.F.LE.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. CC.9-12.A.SSE.1 Interpret expressions that represent a quantity in terms of its context. CC.9-12.A.SSE.1b Interpret complicated expressions by viewing one or more of their parts as a single entity. CC.9-12.A.SSE.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. CC.9-12.A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. CC.9-12.F.BF.1 Write a function that describes a relationship between two quantities. CC.9-12.F.BF.1b Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9-12.F.1F.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 512

34 relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. CC.9-12.F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, kf(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. Include recognizing even and odd functions from their graphs and algebraic expressions for them CC.9-12.A.REI.11Explain 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 CC.9-12.F.BF.5 (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents Knowledge: Students will know/understand: Exponential function Logarithmic function Inverse function Graph of exponential function Graph of logarithmic equation Domain Range Transformation of exponential equation Transformation of logarithmic equation Equation of asymptote of exponential equation Equation of asymptote of logarithmic equation Review of exponent properties Simplify a logarithmic expression Expand a logarithmic expression Condense a logarithmic expression Natural logarithm Logarithmic equations Exponential equation Change of base Applications of exponential growth Application of exponential decay Skills: Students will be able to: WRITE an equation in exponential form WRITE an equation in logarithmic form FIND the inverse of an exponential equation FIND the inverse of a logarithmic equation GRAPH an exponential function manually GRAPH an exponential function using technology GRAPH a logarithmic function manually. GRAPH an exponential function using technology DETERMINE the domain of an exponential function DETERMINE the range of an exponential function DETERMINE the domain of a logarithmic equation DETERMINE the range of a logarithmic equation CONVERT an exponential expression/equation to logarithmic form CONVERT a logarithmic expression/equation to exponential form WRITE a function to model growth/decay. EXPAND a logarithmic expression using properties of exponents CONDENSE a logarithmic expression using properties of exponents SOLVE a logarithmic equation SOLVE an exponential function DISCARD extraneous roots of a logarithmic function DISCARD extraneous roots of an exponential function. SOLVE real word problems with logarithms. CALCULATE the logarithm of any base using technology. 513

35 Essential Questions: How do patterns and functions help describe data and physical phenomena and solve a variety of problems? When did logarithms become important? How are exponents important in our world? Unit Questions: What is a logarithm? What is an exponential equation? How do logarithms make calculation easier? Why are there extraneous roots with logarithms? What is a natural logarithm? How are exponents related to logarithms? Where are logarithms applied? Core Belief Addressed: The Westbrook High School student will Think critically to solve problems and reach well reasoned judgments Assessment Type Questions: 1. Identify the following as exponential growth or exponential decay: a.) y = 12(0.95) x b.) 1 (4) x y = c.) y = ( ) 4 5 x 2. Write a function for a car purchase of $22,500 car that depreciates 9% each year and find the value of the car in five years. Round your answer to the nearest dollar amount. 3. Graph the logarithmic function y = 3log ( x) 2 4. Write the expression 3log x+ 2log 4 4 x 5. Solve each equation: 3 a.) 6. (27) x = 81 b.) 2log x = 4 c.) 3x 2e = 16 Possible Assessments Teacher created assessments Unit tests Quizzes Students presentations Entrance/Exit slips Group projects 514

36 Benchmark Assessments: Common unit assessments Star assessments (October, January, May) Midterm exam Final exam Performance Task: Technology Integration: SMART Board SMART Math tools Texas Instrument virtual calculator Resources/Materials: Texts: o Pearson Algebra 2, Chapter 7 515

37 Internet Sites: o PowerAlgebra.com Other: o Graphing calculators Sample Lessons/Learning Activities: CC.9-12.F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. Use the Change of Base Formula to rewrite each expression using common logarithms. 1.log316 2.log 9 4 CC.9-12.F.IF.8b Use the properties of exponents to interpret expressions for exponential functions. 1.ln 2 + ln x = 1 2.ln( x+ 1) + ln( x 1) = ln(2x 1) = 7 CC.9-12.F.IF.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Graph each logarithmic function. Compare each graph to the parent function. List the domain, range, y-intercept, and asymptotes. 1. y= log 3( x 1) 2.1 log x 2 CC.9-12.F.IF.7e Graph exponential and logarithmic functions, showing intercepts and end behavior. CC.9-12.F.LE.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. 516

38 517

39 Westbrook Public Schools Algebra II Curriculum Grades Unit 7: Higher Order Equations Mathematical Practices: (Practices in bold are to be emphasized in the unit.) Anticipated Length: 4 weeks 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Standards (priority standards in bold, supporting standards in italics): CC.9-12.F.IF.7c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. CC.9-12.A.REI.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. CC.9-12.A.APR.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). CC.9-12.A.APR.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. CC.9-12.A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. CC.9-12.A.APR.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. CC.9-12.N.CN.7 Solve quadratic equations with real coefficients that have complex solutions. CC.9-12.N.CN.8 (+) Extend polynomial identities to the complex numbers. CC.9-12.A.SSE.1a Interpret parts of an expression, such as terms, factors, and coefficients. CC.9-12.A.SSE.1b Interpret complicated expressions by viewing one or more of their parts as a single entity. CC.9-12.A.SSE.2 Use the structure of an expression to identify ways to rewrite it. CC.9-12.N.CN.3 (+) Find the conjugate of a complex number. CC.9-12.N.CN.9 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. 518

40 CC.9-12.F.IF.8a Use the process of factoring and completing the square in a quadratic function to show zeros. Knowledge: Students will know/understand: Polynomial function Linear factor Roots of polynomial function Zeroes of polynomial function Factor theorem Degree of multiplicity Polynomial equation Solution of a polynomial equation Long division Synthetic division Depressed polynomial Remainder theorem for polynomials Factor remainder theorem for polynomials Rational root theorem One test Negative one test Irrational roots Complex roots Conjugate pairs Sum of roots Product of roots Descartes rule of signs Fundamental theorem of algebra Graphs of polynomial functions Essential Questions: How are mathematical relationships related? How do patterns and functions help us describe data and physical phenomena and solve a variety of problems? Skills: Students will be able to: DEFINE a polynomial function FACTOR a polynomial into a product of linear factors. FIND the roots of a polynomial function. CALCULATE the zeroes of a polynomial function. DIVIDE a polynomial by another polynomial using long division. DIVIDE a polynomial by another polynomial using synthetic division. EXPRESS a polynomial as a quotient polynomial times a factor polynomial plus a remainder polynomial. WRITE the depressed polynomial from the results of synthetic division. DETERMINE all possible roots of a polynomial function. DETERMINE the degree of a polynomial from its roots. WRITE the factor of a polynomial from its root. FIND the polynomial from its factors. CALCULATE the coefficients of a quadratic equation using the sum/product of the roots. CREATE a quadratic polynomial from two of its roots using sum and product of roots. DETERMINE if one is a root of a polynomial by the one test. DETERMINE if negative one is a root of a polynomial by the negative one test. FIND all possible combinations of roots using Descartes rule of signs. GRAPH a polynomial using a graphing calculator. Unit Questions: How are roots, zeroes and factors related? How is synthetic division different from long division? Why do irrational and complex roots come in conjugate pairs? What are conjugate pairs? What is a factor? What is a root? How is the one test related to a polynomial? How is the negative one test related to a polynomial? How does Descartes rule of signs help to find roots? 519

41 Core Belief Addressed: The Westbrook High School student will. Think critically to solve problems and reach well reasoned judgments. Assessment Type Questions: How are the coefficients of a quadratic polynomial related to each other? What is a root of multiplicity? What does the degree of a polynomial give? Describe the shape of the graph of each cubic function including end behavior, turning points, and increasing/decreasing intervals. 3 a.) f( x) = 3x x 3 b.) f( x) = 4x 5x The graphs of yy = gg(xx) and yy = ff(xx) are shown below. Mark a point that will satisfy each given condition. A point (A) on the graph of gg where xx = 0 A point (B) on the graph of gg where ff(xx) > gg(xx) A point (C) on the graph of ff where ff(xx) = 0 A point (D) on the graph of f where ff(xx) = gg(xx) 520

42 4. 5. Given that P x x x x what is P ( ) = 2 + 2, (3)? Given that P x x x x x what is P 6. Divide using long division: 7. Divide using synthetic division: Possible Assessments: Teacher created Assessments Unit tests Quizzes Student presentations Entrance/Exit slips Group projects Benchmark Assessments: Common Unit assessments Star assessments (October, January, May) Midterm exam Final exam Performance Task: ( ) = , ( 4)? 3 2 (3x + 9x + 8x+ 4) ( x+ 2) 3 2 ( x 3x 5x 25) ( x 5) 521

43 Sample Lessons/Learning Activities: CC.9-12.F.IF.7c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. CC.9-12.A.REI.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. CC.9-12.A.APR.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) What are all the zeroes of f( x) = x + x 7x 9x 18? CC.9-12.A.APR.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. Identify the vertex, the axis of symmetry, whether the vertex is a maximum or minimum, and sketch the graph: f( x) = ( x 1) f( x) = 3( x+ 7) 8 CC.9-12.A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. CC.9-12.A.APR.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. 2 x + 7x What is in simplest form? State any restrictions on the variable x. 2 x 2. What is the product variable x. 3x x + x 6 x 25 2 x 5 x + 4x+ 3 in simplest form? State any restrictions on the CC.9-12.N.CN.7 Solve quadratic equations with real coefficients that have complex solutions. CC.9-12.N.CN.8 (+) Extend polynomial identities to the complex numbers Find all the roots of the polynomial: x x 3x + 3x = 4x 4 522

44 Westbrook Public Schools Algebra II Curriculum Grades Unit 8: Probability Mathematical Practices: (Practices in bold are to be emphasized in the unit.) Anticipated Length: 3 weeks 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Standards (priority standards in bold, supporting standards in italics): CC.9-12.S.CP.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. CC.9-12.S.CP.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. CC.9-12.S.CP.9 (+) Use permutations and combinations to compute probabilities of compound events and solve problems. CC.9-12.S.IC.2 Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. CC.9-12.S.MD.6 (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). CC.9-12.S.MD.7 (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). CC.9-12.S.CP.1Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as union, intersections, or compliments of other events. CC.9-12.S.CP.2 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. CC.9-12.S.CP.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. CC.9-12.S.CP.4 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. CC.9-12.S.CP.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. 523

45 CC.9-12.S.CP.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. Knowledge: Students will know/understand: Permutation Combination Factorial Fundamental counting principal Experimental probability Theoretical probability Simulation Dependent events Independent events Mutually exclusive events Non mutually exclusive events Conditional probability Frequency table Fair decision Binomial probability Binomial theorem Essential Questions: How do patterns and functions help us to describe data and physical phenomena and solve a variety of problems? Why use probability? Skills: Students will be able to: CALCULATE the number of ways something happens using permutations CALCULATE the number of ways something happens using combinations. USE technology to calculate n factorial. FIND the number of permutations of a model using a calculator. FIND the number of combinations of a model using a calculator. IDENTIFY when order is important. Determine if events are equally likely. COMPUTE the probability of events in a simulation. CALCULATE the probability of independent events. COMPUTE the probability of dependent events. DETERMINE if any two events are mutually exclusive. DETERMINE if any two events are non-mutually exclusive. CALCULATE the conditional probability of a model. CREATE a model for a fair decision. EXPAND a binomial using the binomial theorem CALCULATE the binomial probability of an experiment Unit Questions: What is the difference between a permutation and a combination? What is a trial? What is an event? How are experimental and theoretical probability related? What is equally likely? Why use simulations? What makes a model a fair decision? Core Belief Addressed: The Westbrook High School student will Think critically to solve problems and reach well reasoned judgments. Assessment Type Questions: 1. How many different telephone numbers are available just to the town of Westbrook given every number has the same area code (860) and three-digit town identifier (399)? In other words, the only thing that can be different is the four-digit home identifier? 2. How many ways can the top 5 places be awarded to the division that consists of twelve teams? 524

46 3. A local union representative is trying to fill six leadership positions. Eight men and four women are applying for these positions. Find the probability these positions will be filled so that they consist of: a.) Two men b.) All women c.) Three women 4. An experiment consists of drawing one card from a deck of fifty-two cards. Find the probability that one card is a : a.) Ace or Club b.) Face or Spade c.) Black or Jack d.) Heart or Red e.) King or Face 5. According to a study, Americans face a 5% chance of acquiring an infection while hospitalized. If the records of 15 randomly selected hospitalized patients are examined, find the probability that a.) None acquired an infection. b.) Exactly twelve acquired an infection. c.) At most eight did not acquire an infection. d.) Find the expected number of patients that would acquire an infection. e.) Find the standard deviation of this distribution. f.) How many patients would have to be selected at random to be 96.5% sure at least two of them did not acquire an infection? Possible Assessments: Teacher created assessments Unit tests Quizzes Student presentations Entrance/Exit slips Group projects Benchmark Assessments: Common unit assessments Star assessments (October, January, May) Midterm exam Final exam Performance Task: 525

47 Sample Lessons/Learning Activities: CC.9-12.S.CP.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. The following table represents the number of active duty military personnel by rank in the four major branches of the military as of February 28, Complete the table and answer the questions below. Officers Enlisted Total Army 80, ,990 Navy 310, ,595 Air Force 73, , ,495 Marine Corps 18, ,328 Total 1,166,069 1,392, If an active duty military person is selected at random, what is the probability that the individual is an officer? a) b) c) d) e) If an active duty military person is selected at random, what is the probability that the individual is an enlisted person or is in the Marine Corps? a) b) c) d) e) Given that the individual selected at random is an officer, what is the probability that the individual is in the Navy? a) b) c) d) e) CC.9-12.S.CP.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. CC.9-12.S.CP.9 (+) Use permutations and combinations to compute probabilities of compound events and solve problems. The probability that Congress will keep the government in shutdown status for the next six months is 50% while the probability that they will get re-elected the next time is 25%. The probability that both will occur is 12.5% while the probability that the government will stay shutdown or they will get re-elected is 62.5%. What statement about these events is true? a.) The events are independent and non-mutually exclusive b.) The events are dependent and non-mutually exclusive c.) The events are independent and mutually exclusive d.) The events are dependent and mutually exclusive CC.9-12.S.IC.2 Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. 526

48 CC.9-12.S.MD.6 (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). CC.9-12.S.MD.7 (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). 527

49 Westbrook Public Schools Algebra II Curriculum Grades Unit 9: Inferential Statistics Mathematical Practices: (Practices in bold are to be emphasized in the unit.) Anticipated Length: 3 weeks 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Standards (priority standards in bold, supporting standards in italics): CC.9-12.S.ID.4 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. CC.9-12.S.IC.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population. CC.9-12.S.IC.4 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. CC.9-12.S.IC.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. CC.9-12.S.IC.2 Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. CC.9-12.S.IC.3Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. CC.9-12.S.IC.6 Evaluate reports based on data. Knowledge: Students will know/understand: Functions (expressed symbolically) Mean Median Mode Quartiles Outlier Box and Whisker plot Percentile Standard deviation Variance Normal distribution Skills: Students will be able to: CALCULATE the mean of a data set DETERMINE the median of a data set FIND the mode of a data set CALCULATE five-number summary of a data set IDENTIFY outliers of a data set Manually CREATE a box and whisker plot from a data set CREATE a box and whisker plot using technology CALCULATE percentile values from a data 528

50 Confidence interval Margin of error Area under normal curve Population Empirical Rule Area under the normal curve Sample Random Sampling Essential Questions: How do patterns and functions help us to describe data and physical phenomena and solve a variety of problems? Why is data interpreted? Where is statistics used? set. DETERMINE the variance of a data set FIND the standard deviation of a data set using technology PREDICT outcomes of a situation using statistics. FIT data to a normal distribution model using the mean and standard deviation. CALCULATE the area under the curve of a normal distribution. ANALYZE data from the graph of a normal distribution. CREATE specific confidence intervals with margin of error to interpret data. DETERMINE if the data fits a normal distribution model. ANALYZE results of two versions of the same experiment to predict success. Unit Questions: How are measures of central tendency different from standard deviation? What is a percentile? What information does standard deviation give? How is a sample different from a survey? What is a normal distribution? What is a random sample? Why are simulations used? Why is margin of error important? Why are samples used to analyze data? Why use simulations? Core Belief Addressed: The Westbrook High School student will Think critically to solve problems and reach well reasoned judgments Assessment Type Questions:

51 2. 3. a.) Find the standard deviation and variance for the number of hurricane strikes. b.) Are there any decades which were outliers? 4. Use the Empirical Rule and the following graph for the following: 530

52 5. The heights of American males are approximately normally distributed with a mean of 69.5 inches and a standard deviation of 2.5 inches. If a random sample of 1,200 males was taken, how many could we expect to have heights between 64.5 and 72 inches? Possible Assessments: Teacher created assessment Unit tests Quizzes Student presentations Entrance/Exit slips Group projects Benchmark Assessments: Common unit assessments Star assessments (October, January, May) Midterm exam Final exam Performance Task: a.) Find the Standard Error for a 95% confidence interval for the difference between the proportion of night students to day students that would favor capital punishment. b.) Construct a 95% confidence interval for the difference between the proportion of night students to day students that would favor capital punishment. c.) Use the CI to make a conclusion about the differences between night and day students. 531

53 Sample Lessons/Learning Activities: CC.9-12.S.ID.4 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. CC.9-12.S.IC.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population. CC.9-12.S.IC.4 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. In a learning experiment, untrained mice are placed in a maze and the time required for each mouse to exit the maze is recorded. The average time for untrained mice to exit the maze is µ = 50 seconds and the standard deviation of their times is σ = 16 seconds. 64 randomly selected untrained mice are placed in the maze and the time necessary to exit the maze is recorded for each. a. Find the mean and standard deviation of the sampling distribution of x. b. What is the probability that the sample mean differs from the population mean by more than 3? CC.9-12.S.IC.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. A university investigation was conducted to determine whether women and men complete medical school in significantly different amounts of time, on the average. Two independent random samples were selected and the following summary information concerning times to completion of medical school computed: Women Men Sample Size Sample Mean 8.4 years 8.5 years Sample Standard Deviation 0.6 years 0.5 years Perform the appropriate test of hypothesis to determine whether there is a significant difference in time to completion of medical school between women and men. Test using α =