Algebra 2/ Trigonometry Extended Scope and Sequence (revised )

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1 Algebra 2/ Trigonometry Extended Scope and Sequence (revised ) Unit 1: Operations with Radicals and Complex Numbers 9 days 1. Operations with radicals (p.88, 94, 98, 101) a. Simplifying radicals i. Determining nth roots ii. Simplifying radicals with fractions b. Operations with radical expressions i. Addition and subtraction 1. With irrational numbers in radical form 2. With irrational variable expressions ii. Multiplication and division (same index) c. Rationalize a denominator i. Conjugate pairs 2. Powers of i (p.203, 205, 209) a. Definition of i b. Powers of i c. Definition of complex number i. Include graphical representation as vector 1. Absolute value of a complex number d. Operations with complex numbers Unit 2: Rational Expressions 7 days 1. Factoring (p.22) a. Identify and factor the difference of two perfect squares b. Factoring algebraic expressions using the GCF c. Factoring a trinomial into two binomials, a = 1 d. Factoring a trinomial into two binomials, a > 1 e. Factoring completely 2. Operations with Rational Expressions (p. 44, 48, 61) a. Simplify a rational expression and identify when it is undefined b. Addition and Subtraction i. Like denominators ii. Unlike denominators c. Multiplying and dividing d. Complex fractions

2 Unit 3: Quadratics 10 days 1. Quadratic Equations (p.187, 193, 198, 219, 224) a. Review factoring b. Solving a quadratic equation using factoring c. Solving polynomial equations of a degree higher than two that can be solved by factoring and quadratic formula and factor by grouping d. Solving quadratic equations by completing the square e. Solving quadratic equations using quadratic formula f. Determining the nature of the roots i. Discriminant ii. Graphically g. Writing the equation of a quadratic equation given information i. Sum and product of the roots Unit 4: Solving Equations 7 days 1. Solving radical equations (p.108) 2. Solving absolute value equations (p.13) 3. Solving absolute value inequalities (p.14) a. and vs. or 4. Solving quadratic inequalities (p.30) a. Algebraically b. Graphically 5. Solving rational equations (p.61) 6. Solving rational inequalities (p.64) e. Sign analysis using critical values 1 st Quarter Cumulative Exam (Units 1-4) Unit 5: Relations and Functions 9 days 1. Relations and functions (p.120) a. Know and apply definitions of relations, functions, domain, and range through sets, equations, and graphs b. Applying the vertical and horizontal line tests c. Expressing functions algebraically, graphically, numerically, and verbally d. Evaluating function using function notation and graphs e. Determine if a function is one to one, onto, or both f. Restricted domains 2. Composition and inverses (p.155) a. Notation for composition of functions b. Evaluating the composition of functions c. Write a composition of functions as a single function d. Find the inverse of a function algebraically and graphically e. Reflection in the line y = x and the relation to inverses

3 Unit 6: Systems of Equations 10 days 1. Graphing parabolas a. Axis of symmetry b. Turning points c. Identifying roots d. Relate graph of higher degree polynomials to its roots 2. Circles (p.167) a. Identify center as ordered pair b. Convert an equation of a circle in general form to center radius form by completing square c. Write the equation of a circle given information (graph, center/radius, or center/point) 3. Direct and inverse variations (p.174) a. Standard form/graph b. Word problems 4. Perform/Identify transformations in form f (x + a), f (x) + a, f ( x), f (x), a f (x) (p.149) 5. Systems of equations and graphing (p. 229) a. Systems of quadratic/linear systems of equations i. Algebraically ii. Graphically iii. Word Problems Unit 7: Exponential Functions 9 days 1. Properties of exponents (p. 287) a. Product of powers b. Power to a power c. Quotient of powers d. Zero power e. Power of a product f. Power of a quotient g. Negative exponents 2. Fractional exponents (p.293) a. Conversion between fractional exponents and radical form b. Evaluate an expression/solve an equation with fractional exponents with/without use of calculator 3. Exponential expressions/functions a. Evaluate exponential expressions b. Introduce base e c. Evaluate exponential expressions involving base e d. Graph exponential functions in the form y = ab x, including base e e. Solve an application involving an exponential function that does not require logarithms 4. Exponential equations (p.306) a. Express both sides of an equation in terms of the same base

4 Unit 8: Logarithms 11 days 1. Introduction to logarithms and natural logarithms (p.320) a. Convert between exponential and logarithmic form b. Solve basic equations by converting to exponential form c. Graph logarithmic functions as the inverse of an exponential functions 2. Properties of logarithms and natural logarithms (p.327) a. Product property b. Quotient property c. Power property d. Change of base formula 3. Equations and applications (p.340) a. Solve exponential equations without common bases b. Solve logarithmic equations using the properties of logarithms c. Solve application problems that require the use of logarithms (half-life, growth/decay, compound interest) Midterm Exam (Units 1-8) Unit 9: Trigonometry Functions (9-11 days) 1. The unit circle (p.362, 378, 357, 400) a. The unit circle i. Quadrantal angles ii. Special angles(30, 45, 60) in all 4 quadrants iii. Sin, cos, tan of quadrantal angles and special angles(30,45,60) iv. (x,y) (cosѳ, sinѳ) v. Determine the sign of the trig values for each function in each quadrant b. Co-terminal angles with positive and negative angle measures (clockwise and counter-clockwise) c. Convert between degrees and radians (refer back to unit circle) d. Reference angles(refer back to unit circle) 2. Co-Function & Reciprocal Functions (p.425) a. Define co-functions & reciprocal functions b. Find all trig function values: 1. given 1 trig function & a quadrant OR 2. given 2 trig functions 3. solve for simplest radical form or decimal equivalents, incorporating the use of the graphing calculator c. Use inverse functions to find the measure of an angle given the sine, cosine, or tangent and a specific quadrant 3. s = ѳr (p.403) a. Use s=ѳr, using degree and radian angle measure (degree to radian) b. Arc length vs. arc measures

5 Unit 10a/10b: Trig Equations and Identities (12-13 days) Unit 10a: Trig equations, no replacement (4 days) 1. Solve Trig Equations (p.519) a. Linear equations b. Quadratic equations Unit 10b: Identities and Equations with replacement (8 9 days) 1. Justify the 3 Pythagorean identities (p.411) 2. Sum and Difference angles formulas 3. Double-angle and Half-angle formulas 4. Trig Equations a. Equations with substitutions using Pythagorean and double angle formulas. Answers expressed in degrees and radians in a given interval Unit 11: Trigonometry Graphs (9-10 days) 1. Graphing Sine and Cosine (p.434, 447, 455, 451) a. Graph y = sin x and y = cos x based on unit circle values b. Define and apply changes in amplitude, period and frequency c. Connect to concepts of domain & range(max and min) d. Given a graph write the equation e. Given an equation, create the graph f. Explore y = AsinB(x+h)+C and y = AcosB(x+h)+C with vertical and horizontal phase shifts g. Apply concepts from trig graphs to real-life word problems 2. Tangent Graph and inverse trig functions (p.460, 468) a. Tangent curve 1. Graph y = tan x 2. Discuss properties of the tangent curve b. Inverse trig functions 1. Graph the inverses of sin, cos & tan 2. Use various inverse notations, such as arcos, Arccos, cos 1 3. Discuss properties of inverse trig graphs 4. Determine which inverse graphs are functions, and any necessary restrictions on domain to insure the existence of the inverse function 3. Reciprocal trig functions (p.463) a. Sketch y = sec x, y = csc x, and y = cot x from unit circle values and/or using the graphing calculator b. Compare and contrast the graphs, including discussion of asymptotes, domain, period, & cycles. Include comparisons between the original and reciprocal graphs

6 Unit 12: Trigonometry Applications (10-11 days) 1. Area of Triangle or Parallelogram (p.559) a. Area of a triangle = ½ absinc 2. Law of Sines& Law of Cosines (p.552, 564) a. Law of Sines 1. find missing side given 2 angles and a side 2. find a missing angle given 2 sides and a corresponding angle b. Law of Cosines 1. find an angle given 3 sides 2. find a side given 2 sides and an included angle c. Forces and Vectors(application of law of cosines) 1. Find resultant force given 2 forces 2. Find the angle between two forces given the resultanat d. Angles of Elevation and Depression (application of law of sines, double triangle) e. Ambiguous Case (p.569) 3rd Quarter Cumulative Exam (Units 9-12) Unit 13: Sequences and Series (8 9 days) 1. Sigma Notation (p.257) b a. n a f ( x) 2. Sequences and Series (p.252) a. recognize the common diff. in arithmetic sequences and series b. Recognize the common ratio in a geometric sequence and series c. Determine the type of sequence or series d. Write the explicit formula for a sequence e. Find the terms in a sequence defined recursively f. Find the position of a term in a series or sequence g. Find any term in a sequence or series explicitly h. Recognize what information is needed to complete a sum of a series i. Determine the sum of finite arithmetic and geometric series using formulas j. Write a series using sigma notation k. Solve word problems involving sequences and series

7 Unit 14: Probability (8 9 days) 1. Binomial Probabilities (p.701) a. Determine prob. Of independent events b. Expand binomial using the binomial theorem and find specific terms of the expansion c. Find the probability of exactly, at least, at most of r success in n-trials d. Binomial probabilities based on geometric relationships(i.e. area) e. Determine theoretical, empirical, and geometric probabilities 2. Combinatorics (p.678) a. Determine a sample space b. Determine the number of permutations with and without repetition c. Determine the number of events in which order matters or does not matter d. Determine the number of permutation with identical objects, such as the number of different arrangements of the letters in the word bubble e. Understand the formula notation and apply formulas f. Use permutations and combinations of compound events g. Analyze and solve verbal problems Unit 15: Statistics (8 9 days) 1. Studies and Surveys (p.588, 596) a. Define: Survey, Observation, Controlled Experiment b. Determine the factors which may affect the outcome of a survey c. Median, Mode, Quartiles, Range, Standard Deviation, Variance, Interquartile range 2. Normal Distribution (p.628) a. Standard Deviation b. The percentage under the curve c. Percentiles d. Apply the normal distribution curve, and its properties, to word problems involving real life data e. Use normal distribution as an approximation for binomial prob. 3. Regression Models (p.641) a. Construct a scatter plot using a given table of data without calc b. Determine the appropriate regression model from a scatter plot(linear, exponential, power, logarithmic) c. Use the graphing calculator to determine the function for the regression model d. Use the regression function to evaluate and predict real world situations e. Determine the correlation coefficient using the graphing cal f. Determine the type of correlation(positive, negative, none) g. Use the correlation coefficient to determine the strength of a linear relationship Regents Exam

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