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COURSE CONTENT MATHEMATICS ALGEBRA II QUALITY CCRS EVIDENCE OF STUDENT ATTAINMENT CONTENT STANDARDS RESOURCES CORE FIRST SIX WEEKS 27 Explain why the x-coordinates of the points where the graphs of equations y = f(x) and y = g(x) intersect are solutions of the equations 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. [AREI11] 28 [F-IF.7d] Create graphs of conic sections, including parabolas, hyperbolas, ellipses, circles, and degenerate conics, from second-degree equations. (Alabama) Example: Graph x2 - 6x +y2 - 12y + 41 = 0 or y2 - 4x + 2y + 5 = 0. a. 21 D.2.a D.2.b E.2.c Formulate equations of conic sections from their determining characteristics. (Alabama) Example: Write the equation of an ellipse with center (5, -3), a horizontal major axis of length 10, and a minor axis of length 4. Answer: (x - 5)/25 +(y + 3)/4 = 1. Create equations in two or more variables to represent relations between quantities; graph equations on coordinate axes with labels and scales. [A-CED2] Students:Given two functions (linear, polynomial, rational, absolute value, exponential, and logarithmic) that intersect (e.g., y= 3x and y= 2x), - Graph each function and identify the intersection point(s), - Explain solutions for f(x) = g(x) as the x-coordinate of the points of intersection of the graphs, and explain solution paths (e.g., the values that make 3x = 2xtrue, are the xcoordinate intersection points of y=3x and y=2x, - Use technology, tables, and successive approximations to produce the graphs, as well as to determine the approximation of solutions. Students: Given a second degree conic equation, Graph parabolas. Graph hyperbolas. Graph ellipses. Graph circles. Graph degenerate conics. Given the determining characteristics of a conic section, formulate its equation. Students: Given a contextual situation expressing a relationship between quantities with two or more variables, Pages 640-645 Pages 83-89 Model the relationship with 1 COURSE CONTENT MATHEMATICS ALGEBRA II equations and graph the relationship on coordinate axes with labels and scales. 20 D.2.a D.2.b D.1.c E.1.d E.2.a E.2.c Create equations in one variable and use them to solve problems.[A-CED1] (Please Note: This standard must be taught in conjunction with the preceding standard). Students: Given a contextual situation that may include linear, quadratic, exponential, or rational functional relationships in one variable, 7 Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network. [N-VM6] 8 Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs of a game are doubled. [N-VM7] Model the relationship with equations or inequalities and solve the problem presented in the contextual situation for the given variable. (Please Note: This standard must be taught in conjunction with the standard that follows). Students: Given a contextual situation, Pages 171-178 Represent the data in a matrix and interpret the value of each entry. Students: Given a matrix, Pages 282-288 Pages 171-178 Pages 179-186 Use scalar multiplication to produce a new matrix and interpret the value of the new entries. 2 COURSE CONTENT MATHEMATICS ALGEBRA II 9 Add, subtract, and multiply matrices of appropriate dimensions. [N-VM8] Students: Given two matrices, 10 Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties. [N-VM9] Determine whether arithmetic operations (add, subtract, multiply) are defined. Perform arithmetic (add, subtract, multiply) operations to form new matrices. Students: Given two square matrices, Pages 171-178 Pages 179-186Pages 171-178 Pages 171-178 Demonstrate that multiplication is not commutative. Given a matrix expression with square matrices, 11 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. [NVM10] Show that the associative and distributive properties are satisfied. Students: Given a matrix, Pages 171-178 Add the zero matrix to show that the matrix does not change. Multiply by the identity matrix to show that the matrix does not change. 3 COURSE CONTENT MATHEMATICS ALGEBRA II Given a square matrix, 21 D.2.a D.2.b E.2.c Create equations in two or more variables to represent relations between quantities; graph equations on coordinate axes with labels and scales. [A-CED2] Students: Given a contextual situation expressing a relationship between quantities with two or more variables, 26 I.1.e 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 x 3 or greater). [A-REI9] Find the determinant. If the determinant is non-zero, find the multiplicative inverse. Find the multiplicative inverse, if it is defined, and show that the determinant is not zero. If the multiplicative inverse is not defined, show that the determinant is equal to zero. Pages 83-89 Model the relationship with equations and graph the relationship on coordinate axes with labels and scales. (Please Note: This standard must be taught in conjunction with the preceding standard). Students: Given a variety of matrices, Determine the inverse of the matrix, if it exists, and explain the conditions under which the inverse does not exist. Given a system of linear equations, Convert the system to a matrix equation and use this equation 4 COURSE CONTENT MATHEMATICS ALGEBRA II to find the solution to the original system, if one exists, using technology for dimension 3 x 3 or greater. SECOND SIX WEEKS 33 [F-BF.1] Write a function that describes a relationship between two quantities.* a. Combine standard function types using arithmetic operations. 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. Students: Given two functions represented differently (algebraically, graphically, numerically in tables, or by verbal descriptions), 20 21 D.2.a D.2.b D.1.c E.1.d E.2.a E.2.c D.2.a D.2.b E.2.c [A-CED.1T] 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. Create equations in two or more variables to represent relations between quantities; graph equations on coordinate axes with labels and scales. [A-CED2] Use key features to compare the functions, Explain and justify the similarities and differences of the functions. Students: Given a contextual situation that may include linear, quadratic, exponential, or rational functional relationships in one variable, Pages 385-392 Pages 18-25 Pages 27-32 Pages 33-39 Model the relationship with equations or inequalities and solve the problem presented in the contextual situation for the given variable. (Please Note: This standard must be taught in conjunction with the standard that follows). Students: Given a contextual situation expressing a relationship between quantities with two or more variables, Pages 83-89 5 COURSE CONTENT MATHEMATICS ALGEBRA II 29 34 F.2.d E.2.a G.2.a G.3.d G.3.e G.3.f [F-IF.5T] Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Example: If the functionh(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.* Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), 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. [F-BF3] Model the relationship with equations and graph the relationship on coordinate axes with labels and scales. (Please Note: This standard must be taught in conjunction with the preceding standard). Students: Given a contextual situation that is functional, Model the situation with a graph and construct the graph based on the parameters given in the domain of the context. Students: Given a function in algebraic form, Pages 61-67 Pages 109-116 Graph the function, f(x), conjecture how the graph of f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k(both positive and negative) will change from f(x), and test the conjectures, Describe how the graphs of the functions were affected (e.g., horizontal and vertical shifts, horizontal and vertical stretches, or reflections), Use technology to explain possible effects on the graph from adding or multiplying the input or output of a function by a constant value, 6 COURSE CONTENT MATHEMATICS ALGEBRA II Recognize if a function is even or odd. Given the graph of a function and the graph of a translation, stretch, or reflection of that function, 29 22 F.2.d E.2.a G.2.a G.3.d G.3.e G.3.f Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. [F-IF5] Students: Given a contextual situation that is functional, Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. [A-CED3] Determine the value which was used to shift, stretch, or reflect the graph, Recognize if a function is even or odd. Model the situation with a graph and construct the graph based on the parameters given in the domain of the context. Students: Given a contextual situation involving constraints, Pages 61-67 Write equations or inequalities or a system of equations or inequalities that model the situation and justify each part of the model in terms of the context, Solve the equation, inequalities or systems and interpret the solution in the original context including discarding solutions to the mathematical model that Pages 146-152 Pages 154-160 Pages 161-167 Pages 189-197 Pages 198-204 Pages 282-288 7 COURSE CONTENT MATHEMATICS ALGEBRA II 13 14 H.2.a H.2.b H.2.c H.2.d H.2.e Use the structure of an expression to identify ways to rewrite it. [A-SSE2] Students: [A-SSE.4T] 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. For example, calculate mortgage payments.* cannot fit the real world situation (e.g., distance cannot be negative), Solve a system by graphing the system on the same coordinate grid and determine the point(s) or region that satisfies all members of the system, Determine the point(s) of the region satisfying all members of the system that maximizes or minimizes the variable of interest in the case of a system of inequalities. Pages 238-245 Make sense of algebraic expressions by identifying structures within the expression which allow them to rewrite it in useful ways. Students: Present and defend the derivation of the formula for the sum of a finite geometric series. One approach: S = a + ar + ar2 + ar3 + . . . + arn rS = ar + ar2 + ar3+ . . . + arn+1 rS - S = arn+1 - a = a(rn+1 - 1) S = a(rn+1 - 1)/(r - 1), Recognize geometric series 8 COURSE CONTENT MATHEMATICS ALGEBRA II which exist in problem situations and use the sum formula to simplify and solve problems. 33 Combine standard function types using arithmetic operations. [F-BF1b] Students: Given two functions represented differently (algebraically, graphically, numerically in tables, or by verbal descriptions), 12 F.1.b G.1.c [A-SSE.1T] Interpret expressions that represent a quantity in terms of its context.* a. b. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret complicated expressions by viewing one or more of their parts as a single entity.For example, interpret P(1+r)nas the product of P and a factor not depending on P. Use key features to compare the functions, Explain and justify the similarities and differences of the functions. Students: Given a contextual situation and an expression that does model it, Pages 385-392 ALGEBRA II Pages 5-10 Pages 27-32 Pages 83-89 Connect each part of the expression to the corresponding piece of the situation, Interpret parts of the expression such as terms, factors, and coefficients. THIRD SIX WEEKS 13 H.2.a H.2.b H.2.c H.2.d H.2.e Use the structure of an expression to identify ways to rewrite it. [A-SSE2] Students: Pages 11-17 Make sense of algebraic expressions by identifying structures within the expression which allow them 9 COURSE CONTENT MATHEMATICS ALGEBRA II to rewrite it in useful ways. 32 C.1.d Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). [F-IF9] Students: Given two functions represented differently (algebraically, graphically, numerically in tables, or by verbal descriptions), 30 [F-IF.7T] Graph functions expressed symbolically, and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* a. b. c. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Use key features to compare the functions, Explain and justify the similarities and differences of the functions. Students: Given a symbolic representation of a function (including linear, quadratic, square root, cube root, piecewise-defined functions, polynomial, exponential, logarithmic, trigonometric, and (+) rational), a. b. Pages 69-74 Pages 101-107 Produce an accurate graph (by hand in simple cases and by technology in more complicated cases) and justify that the graph is an alternate representation of the symbolic function, Identify key features of the graph and connect these graphical features to the symbolic function, specifically for special functions: quadratic or linear (intercepts, maxima, and minima), square root, cube root, and 10 COURSE CONTENT MATHEMATICS ALGEBRA II c. d. e. f. 31 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. [F-IF8] piecewise-defined functions, including step functions and absolute value functions (descriptive features such as the values that are in the range of the function and those that are not), polynomial (zeros when suitable factorizations are available, end behavior), (+) rational (zeros and asymptotes when suitable factorizations are available, end behavior), exponential and logarithmic (intercepts and end behavior), trigonometric functions (period, midline, and amplitude). Students: Given a contextual situation containing a function defined by an expression, Pages 451-458 Pages 509-515 Use algebraic properties to rewrite the expression in a form that makes key features of the function easier to find, Manipulate a quadratic function by factoring and completing the square to show zeros, extreme values, and symmetry of the graph, Explain and justify the meaning of zeros, extreme values, and symmetry of the graph in terms of the contextual situation, Apply exponential properties to expressions and explain and 11 COURSE CONTENT MATHEMATICS ALGEBRA II justify the meaning in a contextual situation. 35 [F-BF.4] Find inverse functions. a. Solve an equation of the form f(x) =c for a simple function f that has an inverse and write an expression for the inverse. Example: f(x) =2x3 or f(x) = (x+1)/(x-1) for x≠ 1. Students: Given a function in algebraic form, Graph the function, f(x), conjecture how the graph of f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k(both positive and negative) will change from f(x), and test the conjectures, Describe how the graphs of the functions were affected (e.g., horizontal and vertical shifts, horizontal and vertical stretches, or reflections), Use technology to explain possible effects on the graph from adding or multiplying the input or output of a function by a constant value, Recognize if a function is even or odd. Given the graph of a function and the graph of a translation, stretch, or reflection of that function, 1 C.1.a Know that there is a complex number i such that i2 = –1, and every complex number Students: Determine the value which was used to shift, stretch, or reflect the graph, Recognize if a function is even or odd. Pages 246-252 12 COURSE CONTENT MATHEMATICS ALGEBRA II has the form a + bi with a and b real. [N-CN1] Given an equation wherex2 is less than zero, 2 C.1.b Use the relationship i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. [N-CN2] Students: 3 Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex number. [N-CN3] Explain by repeated reasoning from square roots in the positive numbers what conditions a solution must satisfy, how defining a number iby the equation i2= 1 would satisfy those conditions, and extend the real numbers to a set called the complex numbers, Explain how adding and/or multiplying i by any real number results in a complex number and is real when the multiplier is zero. Pages 246-252 Produce equivalent expressions in the form a + bi, where a and b are real for combinations of complex numbers by using addition, subtraction, and multiplication and justify that these expressions are equivalent through the use of properties of operations and equality (Tables 3 and 4). Students: Given a complex number, Pages 245-252 Find the conjugate and the modulus. 13 COURSE CONTENT MATHEMATICS ALGEBRA II Given a quotient of complex numbers, 4 E.1.c 5 Solve quadratic equations with real coefficients that have complex solutions. [N-CN7] Students: Given a contextual situation in which a quadratic solution is necessary find all solutions real or complex. [N-CN.8] (+) Extend polynomial identities to the complex numbers. Example: Rewrite x2 + 4 as (x + 2i)(x - 2i). Students: 6 F.2.c Multiply by 1 by multiplying the numerator and denominator by the conjugate of the denominator to write the indicated quotient as a single complex number. Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. [N-CN9] Use properties of operations on polynomials and the definition of ito justify that: a2+ b2 = (a+bi)(a-bi) Students: Given a polynomial, Pages 256-262 Pages 264-272 Pages 358-365 determine the number of possible roots realizing that some of them may be complex or used more than once. Given a quadratic polynomial, Show that it has two roots (real or complex) and find them. 14 COURSE CONTENT MATHEMATICS ALGEBRA II 20 D.2.a D.2.b D.1.c E.1.d E.2.a E.2.c Create equations in one variable and use them to solve problems.[A-CED1] Students: Given a contextual situation that may include linear, quadratic, exponential, or rational functional relationships in one variable, Pages 342-349 Model the relationship with equations or inequalities and solve the problem presented in the contextual situation for the given variable. (Please Note: This standard must be taught in conjunction with the standard that follows). FOURTH SIX WEEKS 21 D.1.a D.2.b E.2.c Create equations in two or variables to represent relationships between quantities; graph equations or coordinate axes with labels and scales. [A-CED2] Students: Given a contextual situation expressing a relationship between quantities with two or more variables, 15 F.1.a F.1.b F.2.c 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. [A-APR1] Model the relationship with equations and graph the relationship on coordinate axes with labels and scales. (Please Note: This standard must be taught in conjunction with the preceding standard). Students: Pages 229-236 Pages 303-309 Use the repeated reasoning from prior knowledge of properties of arithmetic on integers to progress consistently to rules for arithmetic on polynomials, 15 COURSE CONTENT MATHEMATICS ALGEBRA II 16 E.1.a F.1.b F.2.a F.2.b 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)= o if and only of (x – a) is a factor of p(x). [A-APR2] Students: Given a polynomial p(x): 17 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. [A-APR3] [A-APR.4T] Prove polynomial identities and use them to describe numerical relationships. Example: The polynomial identity (x2 + y2)2 = (x2 -y2)2 + (2xy)2 can be used to generate Pythagorean triples. Pages 352-357 Identify when (x - a) is a factor of the given polynomial p(x), Identify when (x - a)is not a factor, then the remainder when p(x) is divided by (x - a) is p(a). Students: Given any polynomial, 18 Accurately perform combinations of operations on various polynomials. Pages 358-365 Analyze and determine if suitable factorizations exist, Use the root determined by these factorizations to construct a graph of the given polynomials. The graph should include all real roots, Utilize techniques such as plotting points between and outside roots or use technology to find the general shape of the graph. Students: Use properties of operations on polynomials to justify identities such as: 16 COURSE CONTENT MATHEMATICS ALGEBRA II 1. 2. 3. 4. 5. 19 6 D.2.b E.1.a E.1.d E.2.a G.1.a Rewrite simple ration 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 that the degree of b(x), using inspection, long division, or for the more complicated examples, a computer algebra system. [A-APR6] F.2.c Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. [N-CN9] (a+b)2 = a2 + 2ab + b2 (a+b)(c+d) = ac + ad + bc + bd a2 - b2 = (a+b)(a-b) x2 + (a+b)x + ab = (x + a)(x + b) (x2 + y2)2 = (x2 - y2)2 + (2xy)2, Use these identities to describe numerical relationships (e.g., identity 3 can be used to mentally compute 79 x 81, or identity 5 can be used as a generator for Pythagorean triples). Students: Given a rational expression in the form a(x)/b(x), Rewrite rational expressions of the form q(x) + r(x)/b(x) with the degree of r(x) less than the degree of b(x),choosing the most appropriate technique from inspection when b(x)is a common factor of the terms of a(x), long division for other examples and a computer algebra system for more complicated examples. Students: Given a polynomial, Pages 311-317 Pages 358-365 determine the number of possible roots realizing that some of them may be complex 17 COURSE CONTENT MATHEMATICS ALGEBRA II or used more than once. Given a quadratic polynomial, 16 E.1.a F.1.b F.2.a F.2.b 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)= o if and only of (x – a) is a factor of p(x). [A-APR2] Students: Given a polynomial p(x): 17 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. [A-APR3] F.1.b G.1.c [A-SSE.1T] Interpret expressions that represent a quantity in terms of its context.* Pages 352-357 Identify when (x - a) is a factor of the given polynomial p(x), Identify when (x - a)is not a factor, then the remainder when p(x) is divided by (x - a) is p(a). Students: Given any polynomial, 12 Show that it has two roots (real or complex) and find them. Pages 358-365 Analyze and determine if suitable factorizations exist, Use the root determined by these factorizations to construct a graph of the given polynomials. The graph should include all real roots, Utilize techniques such as plotting points between and outside roots or use technology to find the general shape of the graph. Students: Given a contextual situation and an Pages 264-272 Pages 219-227 18 COURSE CONTENT MATHEMATICS ALGEBRA II expression that does model it, a. b. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret complicated expressions by viewing one or more of their parts as a single entity.For example, interpret P(1+r)nas the product of P and a factor not depending on P. Connect each part of the expression to the corresponding piece of the situation, Interpret parts of the expression such as terms, factors, and coefficients. FIFTH SIX WEEKS 13 23 H.2.a H.2.b H.2.c H.2.d H.2.e Use the structure of an expression to identify ways to rewrite it. [A-SSE2] Students: [A-CED.4T] Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Example: Rearrange Ohm's law V = IR to highlight resistance R. Make sense of algebraic expressions by identifying structures within the expression which allow them to rewrite it in useful ways. Pages 407-414 Pages 415-421 Pages 429-435 Students: Rearrange formulas which arise in contextual situations to isolate variables that are of interest for particular problems. For example, if the electric company charges for power by the formula COST = 0.03 KWH + 15, a consumer may wish to determine how many kilowatt hours they may use to keep the cost under particular amounts, by considering KWH< (COST 15)/0.03 which would yield to keep the monthly cost under $75, they need to use less than 2000 KWH. 19 COURSE CONTENT MATHEMATICS ALGEBRA II 24 Solve simple rational and radical equations in one variable, and give examples showing how extraneous roots may arise. [A-REI2] Students: 25 [A-REI.4b] Recognize when the quadratic formula gives complex solutions and write them as a ± bi form for real numbers a and b. (Alabama) Pages 570-578 Solve problems involving rational and radical equations in one variable, Identify extraneous solutions to these equations if any, Produce examples of equations that would or would not have extraneous solutions and communicate the conditions that lead to the extraneous solutions. Students: Solve quadratic equations where both sides of the equation have evident square roots by inspection, Transform quadratic equations to a form where the square root of each side of the equation may be taken, including completing the square, Use the method of completing the square on the equation in standard form (ax2+bx+c=0) to derive the quadratic formula, Identify quadratic equations which may be solved efficiently by factoring, and then use factoring to solve the equation, Use the quadratic formula to solve quadratic equations, Explain when the roots are real or complex for a given quadratic equation, and when 20 COURSE CONTENT MATHEMATICS ALGEBRA II 32 C.1.d Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). [F-IF9] Students: Given two functions represented differently (algebraically, graphically, numerically in tables, or by verbal descriptions), 30 Graph exponential and logarithmic functions , showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. [F-IF7e] complex write them as a ± bi for real numbersa and b, Demonstrate that a proposed solution to a quadratic equation is truly a solution by making the original true. Use key features to compare the functions, Explain and justify the similarities and differences of the functions. Students: Given a symbolic representation of a function (including linear, quadratic, square root, cube root, piecewise-defined functions, polynomial, exponential, logarithmic, trigonometric, and (+) rational), Pages 219-227 Pages 837-843 Pages 845-852 Produce an accurate graph (by hand in simple cases and by technology in more complicated cases) and justify that the graph is an alternate representation of the symbolic function, Identify key features of the graph and connect these graphical features to the symbolic function, specifically 21 COURSE CONTENT MATHEMATICS ALGEBRA II for special functions: a. b. c. d. e. f. 36 G.3.c 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. [F-LE4] quadratic or linear (intercepts, maxima, and minima), square root, cube root, and piecewise-defined functions, including step functions and absolute value functions (descriptive features such as the values that are in the range of the function and those that are not), polynomial (zeros when suitable factorizations are available, end behavior), (+) rational (zeros and asymptotes when suitable factorizations are available, end behavior), exponential and logarithmic (intercepts and end behavior), trigonometric functions (period, midline, and amplitude). Students: Given a contextual situation involving exponential growth or decay, Pages 461-467 Pages 509-515 Develop an exponential function which models the situation, Rewrite the exponential function as an equivalent logarithmic function, Use logarithmic properties to rearrange the logarithmic function, to isolate the variable, and use technology to find an approximation of the 22 COURSE CONTENT MATHEMATICS ALGEBRA II solution. 37 G.3.b Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. [F-TF1] Students: Given a unit circle and an angle that is defined in terms of a fractional part of a revolution, 38 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measure of angles traversed counterclockwise around the unit circle. [F-TF2] 39 Define the six trigonometric functions using ratios of the sides of a right triangle, coordinates on the unit circle, and the reciprocal of other functions. Use the definition of one radian as the measure of the central angle of a unit circle which subtends (cuts off) an arc of length one to determine measures of other central angles as a fraction of a complete revolution (2π for the unit circle), Create a circle in the coordinate plane other than a unit circle, and show that an arc equal in length to the radius defines a triangle inside the circle, similar to one in the unit circle for an arc of length one, so the angle must have the same measure. Students: Given a contextual situation in which a decision needs to be made, Pages 799-806 Pages 830-836 Pages 830-836 Use probability concepts to analyze, justify, and make objective decisions. Students: Given scenarios involving chance, Pages 790-798 Pages 799-806 Pages 830-836 23 COURSE CONTENT MATHEMATICS ALGEBRA II 40 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. [F-TF5] Determine the sample space and a variety of simple and compound events that may be defined from the sample space, Use the language of union, intersection, and complement appropriately to define events. Students: Given a contextual situation of a periodic phenomenon that may be modeled by a trigonometric function, Pages 837-843 Create a trigonometric function to model the phenomena, Use features such as the specified amplitude, frequency, and midline of the function to justify the model. SIXTH SIX WEEKS 41 [S-MD.6T] (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). Students: Given a contextual situation in which a decision needs to be made, Pages P9-P12 Use a random probability selection model to produce unbiased decisions. . 42 Analyze decisions and strategies using probability concepts (e.g. product testing, medical testing, pulling a hockey goalie from the end of a game). [S-MD7] Students: Given a contextual situation and scenarios involving two events, Pages 742-750 Pages 752-759 24 COURSE CONTENT MATHEMATICS ALGEBRA II 43 H.1.a-c, e Describe events as subsets of a sample space(the set of outcomes, using characteristics or categories of the outcomes, or as unions, intersections, or compliments of other events) “or”, “and”, not” Students: Given scenarios involving chance, 44 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying 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. [S-CP4] Explain the meaning of independence from a formula perspective P(A∩B) = P(A) x P(B) and from the intuitive notion that A occurring has no impact on whether B occurs or not, Compare these two interpretations within the context of the scenario. Determine the sample space and a variety of simple and compound events that may be defined from the sample space, Use the language of union, intersection, and complement appropriately to define events. Students: Given scenarios involving two events A and B both when A and B are independent and when Aand B are dependent, Pages P13-P15 Pages P16-P19 Determine the probability of each individual event, then limit the sample space to those outcomes where B has occurred and calculate the probability of A, compare the P(A)and the P(A given B), and explain the equality or difference in the original 25 COURSE CONTENT MATHEMATICS ALGEBRA II context of the problem, Justify that P(A given B) = P(A∩B)/P(B). . 45 Construct and interpret two-way frequency tables of data when two categories are associated each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. [S-CP4] Students: Given a contextual situation consisting of two events A and B, 46 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. [S-CP5] 47 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. [S-CP6] Use the definition of conditional probability P(B|A) = P(A and B)/P(A) to determine the probability of the compound event (A and B) when the P(A|B) and the P(A) are known or may be determined. Interpret the probability as it relates to the context. Students: Given a contextual situation, Pages 742-750 Pages P16-P19 Choose the appropriate counting technique (permutation or combination), Find the number of ways an event(s) can occur, Use these counts to determine probabilities of the event, including compound events. Students: Given a contextual situation consisting of two events, Pages P16-P19 26 COURSE CONTENT MATHEMATICS ALGEBRA II 48 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. [S-CP7] Determine the probability of each individual event, then limit the sample space to those outcomes where B has occurred and calculate the probability of A, compare the P(A)and the P(A given B), and explain the equality or difference in the original context of the problem, Determine the probability of each individual event, then limit the sample space to those outcomes where B has occurred and calculate the P(A and B), compare the ratio of P(A and B) and P(B) to P(A given B), and explain the equality or difference in the original context of the problem. Students: Given a contextual situation consisting of two events, Pages P13-P15 Determine the simple probability of each event, Determine the P(A or B) and P(A and B), Interpret the Addition Rule by counting outcomes in the four events A, B, A and B, A or B and showing the relationship to P(A or B) = P(A) + P(B) - P(A and B), Interpret the Addition Rule in the case that the P(A and B) = 27 COURSE CONTENT MATHEMATICS ALGEBRA II 0. 49 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. [S-CP8] Students: Given a contextual situation consisting of two events A and B, 50 Use permutations and combinations to compute probabilities of compound events and solve problems. [S-CP9] Use the definition of conditional probability P(B|A) = P(A and B)/P(A) to determine the probability of the compound event (A and B) when the P(A|B) and the P(A) are known or may be determined. Interpret the probability as it relates to the context. Students: Given a contextual situation, Pages P16-P19 Pages P9-P12 Choose the appropriate counting technique (permutation or combination), Find the number of ways an event(s) can occur, Use these counts to determine probabilities of the event, including compound events. 28