Course Number 432/433 Title Algebra II (A & B) H Grade # of Days 120

Transcription

1 Whitman-Hanson Regional High School provides all students with a high- quality education in order to develop reflective, concerned citizens and contributing members of the global community. Course Number 432/433 Title Algebra II (A & B) H Grade # of Days 120 Course Description Instructional Strategies This intensive course provides an accelerated and more rigorous treatment of the logical development of algebra. Part A topics include systems of equations and inequalities, matrices, and linear, quadratic and polynomial functions. Part B topics include powers, roots and radicals, exponential, logarithmic and rational functions, and sequences and series. Student owned graphing calculators are necessary for this course. This course is recommended for students who have demonstrated exceptional interest and ability in Honors Algebra I and Geometry Honors. All students who plan to take AP Calculus should take this course. This course addresses Whitman-Hanson Student Learning Expectations 1-6. Recommendation: Students should have earned A- or better in Freshman Algebra and Geometry, or B- or better in Geometry H and Honors Algebra I. Instructional Strategies include but may not be limited to the following: 1. Whole class instruction 2. Individual work: homework, classwork, assessments 3. Group work: activities, problem solving 4. Experiments, demonstrations, investigations 5. Video presentations 6. Use of technology (graphing calculators and computers) 7. Projects Student Learning Expectations 1. Read, write and communicate effectively. 2. Utilize technologies appropriately and effectively. 3. Apply critical thinking skills. 4. Explore and express ideas creatively. 5. Participate in learning both individually and collaboratively. 6. Demonstrate personal, social, and civic responsibility. 1

2 Unit of Study Prerequisite Topics MA Standard/Strands: Time Frame: Text (Chapter/Pages) Other Resources: 18 Days Chapters 1-3 Essential Questions Concepts, Content: 1. How can interval notation describe inequalities? Replace x 4, -2 < x < 8, x > 5, x -6 with (, 4], (-2, 8), (5, ), and [-6, ) On a graph, replace closed circles with square brackets, open circles with parentheses and arrows with and parentheses 2. What are the uses for piecewise functions? Use piecewise functions when a function has distinct sections that are best described with different rules, or with step functions, and absolute value functions 3. How can a line of best fit (regression line) be found on a graphing calculator? What is correlation? Enter data through STAT Edit, find the linear regression line through STAT Calc Lin Reg Correlation gives a numeric value to the relationship between the x and y values on a scatterplot 4. How can linear systems be solved? How many solutions can a linear system have? Solve linear systems by graphing, elimination, or substitution. There can be no solution (parallel lines), one solution (intersecting lines) or infinitely many solutions (the same line twice) 5. How do you use linear programming to find the optimum value? Find the objective function, then use the constraints to graph the feasible region. All possible optimum values occur at the vertices of the feasible region. Substitute coordinates of the vertices into the objective function to find the maximum or minimum value. 6. How do you plot and name points in three dimensions? How do you graph a linear equation in three variables in three dimensions? The (x, y, z) points are plotted in 3-d space using the parallelogram or vector method. A linear equation in three variables can be graphed by finding the x, y and z intercepts and connecting them to form a triangular region representing a plane. Targeted Skill(s): 1. Use interval notation to describe inequalities or graph inequalities from interval notation 2. Create step, absolute value and other piecewise functions 2. Find f(x) when f is a piecewise function 2. Write the rules for f(x) and each restricted domain given the graph of a piecewise function 3. Find the linear regression equation on a graphing calculator 2

3 3. Find r, correlation on a graphing calculator 4. Solve a system of 2 or 3 linear equations by graphing, elimination and/or substitution 4. Based on the algebraic solution, determine the number of points of intersection 5. Determine the objective function from a verbal description 5. Determine the constraints from a verbal description 5. Graph the constraints (inequalities) to determine a feasible region 5. Determine when a feasible region is bounded or unbounded 5. Use the vertices of the feasible region to determine the optimum value 6. Draw a 3-d display of intersecting planes 6. Plot points on a 3-d graph using the parallelogram method or vectors 6. Find the x, y, and z intercepts of a linear equation in 3 variables: (x, 0, 0) (0, y, 0) and (0, 0, z) 6. Use intercepts to graph a linear equation in 3 variables Writing: Assessment Practices: Assessment questions such as: Describe and advantages and disadvantages to the methods of solving systems of equations. or Compare the similarities and differences between graphing in two and three dimensions. Answering word problems with labeled numeric answers and/or sentences Quizzes and tests that include vocabulary, graphs, and larger problems where partial credit is available. Some questions require a graphing calculator others must be done without any calculator. 3

4 MA Standard/Strands: Time Frame: Text (Chapter/Pages) Other Resources: Essential Questions Concepts, Content: 11 days Chapter 4 Matrices and Determinants 1. What are matrices and why are they used? A matrix is a rectangular array of numbers, names by the number of rows x the number of columns. It is used to organize a large amount of data. 2. How do you compute with matrices? For addition and subtraction the dimensions of the matrices must be equal. Then the corresponding entries are combined. Scalar multiplication multiplies each entry by a number outside the matrix. With multiplication, the number of columns of the first matrix must equal the number of rows of the second or multiplication is undefined. Matrix multiplication is not commutative. 3. What is the determinant of a matrix and what is it used for? The determinant of A, A, is a real number and can be hand calculated in a 2 x 2 or 3 x 3 matrix. Determinants can be used to find the area of a triangle from its vertices, and in Cramer s Rule to solve systems of equations. 4. What is the Identity matrix and how is it used? The Identity matrix has 1 s on the main diagonal and 0 s everywhere else. Where I = the identity matrix: I x A =A, and A x I = A 5. What is the inverse of a matrix and how is it used? The inverse of matrix A, in symbol is A -1. If A and B are inverses then AB = I and BA = I 6. How are matrices used in the real world? Matrices are used to organize large amounts of data, to code and decode messages and in solving systems of linear equations. Targeted Skill(s): 1. Determine the dimensions of a matrix 1. Name the entries in a matrix 1. Write a matrix with rows and columns labeled with words, that describes a situation presented verbally 2. Add, subtract, and multiply two matrices and multiply a matrix by a scalar. 3. Find the determinant of a 2 x 2 or a 3 x 3 matrix by hand or with a graphing calculator 3. Use determinants to find the area of a triangle from its vertices 3. Use Cramer s Rule to solve a system of linear equations with determinants 4. Write the Identity Matrix 4. Multiply with the Identity Matrix 5. Given A is a 2 x 2 matrix, find A -1 (if it exists) by hand 5. Find the inverse of a matrix (if it exists) with a graphing calculator 6. Write a matrix to organize data presented in a world problem 6. Use matrices to solve problems (especially coding and encoding, and verbal problems solved with linear systems 4

5 Writing: Assessment Practices: Quizzes and tests that include vocabulary, short answer questions and larger problems where partial credit is available. Some questions require a graphing calculator others must be done without any calculator. Students write small messages, put them into code, exchange them and decode the messages. 5

6 MA Standard/Strands: Time Frame: Text (Chapter/Pages) Other Resources: Essential Questions Concepts, Content: Targeted Skill(s): 15 days Chapter 5 Quadratic Functions 1. What are the important aspects of the graph of a quadratic function? The shape is a parabola with a vertex, axis of symmetry, y intercept and sometimes 1 or 2 x intercepts. The equation can be written in standard form, vertex form or intercept form. 2. How can you solve a quadratic equation? Solve quadratic equations by graphing, factoring, by finding square roots, the quadratic formula and completing the square The discriminant indicates the number and the nature of the solutions 3. What are complex numbers and how are they used? Numbers in the form a + bi include the imaginary unit i which is the square root of negative 1 Computation with imaginary numbers differs from real number computation only with powers of i and imaginary components in the denominator of fractions (division) Complex numbers are used in fractals and electric circuits 4. How do you solve quadratic inequalities? The graph of the quadratic equation can be modified to include a solid or dotted line and shading above or below the line 5. How can you create a quadratic model from actual data? Given the vertex substitute into the vertex form, given the x intercepts use the intercept form, and given 3 points use the standard form and a system of 3 equations 1. Graph a parabola including: the axis of symmetry, vertex, any x and y intercepts for a minimum of 5 points 1. Use standard form, vertex form or intercept form to graph a parabola 2. Solve quadratic equations by graphing factoring, taking the square root of both sides, completing the square and the quadratic formula 2. Use the discriminant to determine the number and nature of the solutions 3. Compute with complex numbers, simplify as necessary 3. Plot points on the complex plane 4. Graph quadratic inequalities < and > with a dotted line parabola and shading, and with a solid line parabola and shading 5. Write the equation of a parabola using substitution given special points (vertex or x intercepts) 5. Write the equation of a parabola in standard form using 3 points on the parabola and a system of 3 equations Writing: Assessment Practices: Quizzes and tests that include vocabulary, short answer questions and larger problems where partial credit is available. Some questions require a graphing calculator others must be done without any calculator. 6

7 MA Standard/Strands: Time Frame: Text (Chapter/Pages) Other Resources: Essential Questions Concepts, Content: 10 days Polynomials and Polynomial Functions Chapter 6 1. What are the Properties of Exponents, and how can they be used to simplify computation? The properties are: the Product of Powers, the Power of a Power, the Power of a Product, the Negative Exponent, the Zero Exponent, the Quotient of Powers, and the Power of a Quotient. Numerically, scientific notation is an example of using properties in computation. These are also used in algebraic expressions. 2. How do you name a polynomial by knowing its degree? The polynomials of degree 0-5 have special names: constant, linear, quadratic, cubic, quartic and quintic. 3. How do you evaluate a polynomial? You can evaluate a polynomial directly with substitution or indirectly with synthetic substitution. 4. What attributes of a polynomial should be included (with or without technology) to show a complete graph? A complete graph of a polynomial includes the end behavior, the relative maximum and minimum points, the y-intercept, and any x intercepts. 5. What methods can be used to add, subtract or multiply polynomials? There is a horizontal and a vertical method for addition, subtraction and multiplication. Additionally with multiplication, there is the area model and special product patterns (the sum and difference of two binomials, the square of a binomial and the cube of a binomial). 6. What methods are used to factor higher degree polynomials? Students should factor any greatest common factor first, and use then other factoring techniques including: the sum of two cubes, the difference of two cubes, factoring by grouping, or factoring a polynomial in quadratic form. 7. How do you solve a polynomial equation? If a polynomial equation can be factored, students can use the zero product property to find solutions. All potential solutions should be verified in the original equation. Application problems may only have real number solutions. 8. How can you divide polynomials? Methods for polynomial division include long division and synthetic division (if the divisor is in the form (x k)). The remainder theorem will give the quotient and remainder, and the factor theorem will indicate if the divisor is factor of the polynomial. Using the zero product property, the factor theorem can produce zeros of the polynomial function. You can find rational zeros using the rational zero theorem and synthetic division. 9. What does the Fundamental Theorem of Algebra tell us? An nth-degree polynomial equation has exactly n solutions, and an nth-degree polynomial function has exactly n zeros. Complex zeros occur in conjugate pairs, if a + bi is a zero, then a bi is a zero. 10. Why is knowing the zeros of a function important? Given the zeros of a function, the function s degree and the leading coefficient; the function itself can be determined. 11. What are the connections between the algebraic attributes of a function and its graph? The zero of a function appears as an x-intercept of the graph of the function, the maximum and minimum points of the graph are turning points of the function. 7

8 12. How are the finite differences of a polynomial function related to the function s degree? When the nth-order differences are non-zero and constant, the function is an nth-degree function. 13. How are finite differences used? Finite differences are used to generate polynomials from data tables. Targeted Skill(s): 1. Simplify algebraic and numeric expressions using the properties of positive and negative exponents. 1. Use the properties of exponents and scientific notation to solve application problems. 2. Identify polynomials by name according to degree. 3. Use substitution of x into P(x) and synthetic substitution of x into P(x) to get the same correct answer. 4. Graph a polynomial using the end behavior, the relative maximum and minimum points, the y-intercept, and any x intercepts. 5. Correctly add, subtract and multiply polynomials. 5. Memorize the pattern of the product of the sum and difference of two binomials, the square of a binomial and the cube of a binomial to correctly multiply. 6. Completely factor higher degree polynomials. 7. Solve polynomials equations for all appropriate solutions. 8. Correctly divide polynomials. 8. Use the remainder and factor theorems to solve polynomial equations. 8. Use synthetic division to find rational zeros. 9. Find all real zeros of a polynomial function. 10. Given zeros of a function, its leading coefficient and its minimum degree; find the function. 11. Find all the necessary attributes of a given polynomial function to graph it by hand. 11. Graph a polynomial function on a graphing calculator in an appropriate viewing window. 12. Find the degree of the function by finding the nth-order differences from a table of data. 13. Use finite differences and a table of values to determine a function. Writing: Assessment Practices: Quizzes and tests that include vocabulary, short answer questions and larger problems where partial credit is available. Some questions require a graphing calculator others must be done without any calculator. Find a picture of a parabola and use a superimposed grid system to determine coordinates that can be used to determine the equation of the parabola. 8

9 MA Standard/Strands: Time Frame: Text (Chapter/Pages) Other Resources: 14 days Powers, Roots and Radicals Chapter 7 Essential Questions Concepts, Content: 1. How do you use rational exponents and nth roots of numbers? The rational power (greater than zero) on n in the form p/r can be rewritten as the r th root of n to the p power, and vice versa. A negative rational exponent follows the same pattern as any negative exponent, so n (-p/r) equals 1/ n (p/r). When the nth root of a (if n is greater than 1 and a is real) is odd then a has one real root. If n is even and a > 0, then a has two real roots, if n is even and a = 0, then a has one real root, and if n is even and a < 0, then a has no real roots. 2. How do the properties of exponents compare to the properties of rational exponents? The properties of exponents are the same for integer and rational exponents. 3. What are the restrictions for applying the 4 basic computation rules to all kinds of functions? Computation rules are the same for any kind of function, as long as restricted domains are considered. 4. What is a power function? A power function has the form = where a is a real number and b is a rational number. 5. What is composition of functions and how is composition of functions used? Composition replaces the x of one function with a new function. Composition is generally not commutative, ie. f(g(x)) does not usually equal g(f(x)). In f(g(x)) the domain of g(x) is the set of all x such that x is in the domain of g and g(x) is in the domain of f(x). Other symbolism for f(g(x)) is (f g)(x). Composition of functions f(g(x)) is used when changes in x produce changes in g(x), which in turn cause changes in f(x). 6. What is an inverse of a function and how do you find it? A relation (the set of (x,y)) has a rule to change x values into y values. The inverse of the relation changes the y values back to the x values. If both rules are functions, the rules are called inverse functions. The inverse of f(x) is written f -1 (x). The inverse and the function reflect over the y = x line. The domain of the function is the range of the inverse, and vice versa. To find the inverse of a function, replace the x s with y s and the y s with x s and solve the equation for y. In an application problem, the inverse function is just rewritten in terms of the other variable. 7. How do you know if a function has an inverse? Not all functions have inverses. Some functions have inverses only if you restrict the domain. The horizontal line test determines whether a function has an inverse. 8. How can you graph a square root or cube root function by hand? The general shape of the parent function can be generated by a table of values or memorized. Transformations of = to = h + k or = to = h + k include a horizontal shift h units, a vertical shift k units and a vertical stretch/shrink by a factor of a. 9. How do you solve equations with radicals? Use the properties of equality to isolate the radical, or in the case of two radicals, put one radical on each side. Then raise both sides of the equation to a power that will clear the radicals. Simplify the resulting equation, solve and check for extraneous solutions. 10. How do you solve equations with rational exponents? 9

10 Use the properties of equality to isolate the variable with the rational exponent. Then raise each side of the equation to the power that brings the exponent on the variable to 1. Simplify the resulting equation, solve and check for extraneous solutions. 11. What measures of central tendency or dispersion are used to describe statistical data? The measures of central tendency are mean, median and mode and they are the average, the middle number and most frequent number respectively. The measures of dispersion are range, variance and standard deviation. The range can be found by finding the difference between the greatest and the least data value. The standard deviation is the square root of the variance. The variance is found by adding the squares of the differences between each piece of data and the mean, and then dividing by the number of pieces of data. 12. What graphs are used to show a set of data? The most common graphs to visually display statistical data are the box-and-whisker plot, the frequency distribution and the histogram. Targeted Skill(s): Writing: 1. Find the nth root of an algebraic or numeric expression. 1. Solve equations with nth roots. 1. Evaluate algebraic or numeric expressions with rational exponents. 2. Apply the properties of rational exponents to simplify algebraic and numeric expressions. 3. Add, subtract, multiply and divide functions. 3. State the domain of a function. 4. Recognize a power function by form: = 5. Use f(g(x)) or (f g)(x) to describe composition of functions. 5. Find the composition of two or three functions. 5. Find the domain of a function that has been composed from 2 or 3 functions. 5. Describe the domain of a function with words ex: all real numbers greater than 5, interval notation ex. (0, ), or set notation ex: R Find the inverse (if they exist) of linear and non-linear functions. 7. Use the horizontal line test to determine if a function has an inverse. 7. Use a restricted domain to enable a relation to be a function that has an inverse. 8. Graph a square root or cube root function without a graphing calculator. 8. Graph a square root or cube root function with a graphing calculator. 9. Solve equations with one or two radicals. 9. Find extraneous solutions by verifying potential answers. 10. Solve equations with rational exponents. 10. Find extraneous solutions by verifying potential answers. 11. Analyze data by finding the measures of central tendency and measures of dispersion. 12. Visually display data in box-and-whisker plots, frequency distributions and histograms. Assessment Practices: Quizzes and tests that include vocabulary, short answer questions and larger problems where partial credit is available. Some questions require a graphing calculator others must be done without any calculator. Independently complete a project on statistics. 10

11 MA Standard/Strands: Time Frame: Text (Chapter/Pages) Other Resources: Essential Questions Concepts, Content: 21 days Exponential and Logarithmic Functions Chapter 8 1. What is the equation of an exponential growth function? The exponential growth function has the equation = where a > 0 and b > What are the attributes of the graph of an exponential growth function? The graph passes through ( 0, a ), the x-axis is an asymptote, the domain is all real numbers, the range is y > 0. The end behavior is, f(x) 0 and, f(x). 3. How can you graph an equation of the form = + by hand? The transformation of = to = + translates the graph horizontally h units, and vertically k units. 4. What are the practical applications of exponential growth functions? A real life quantity that increases by a fixed percent per year with the equation = (1+ ) " where a is the initial amount, (1 + r) is the growth factor and t is the time. Compound interest paying interest at an annual rate r compounded n times per year is an advanced form of the same formula: # = $ ( 1+ % & )&". 5. What is the equation of an exponential decay function? The exponential growth function has the equation = where a > 0 and 0 < b < What are the attributes of the graph of an exponential decay function? The graph passes through ( 0, a ), the x-axis is an asymptote, the domain is all real numbers, the range is y > 0. The end behavior is, f(x) and, f(x) 0 7. How can you graph an equation of the form = + by hand? The transformation of = to = + translates the graph horizontally h units, and vertically k units. 8. What are the practical applications of exponential decay functions? A real life quantity that decreases by a fixed percent per year with the equation = (1 ) " where a is the initial amount, (1 r) is the decay factor and t is the time. Depreciation and half life are examples of exponential decay. 9. What is e? e is an irrational number discovered by Leonhard Euler called the natural base. As ', )1+ * & +& approaches e It can be used in computation as if it were a variable. 10. How is e used? e is a base of a natural base exponential function, and the base of a natural logarithm. It is also used in the continuously compounded interest formula: # = $, %" 11. What is a logarithm? A logarithm (or log) is an exponent, often in decimal form, as in -./ 0 = 1,'2 3 = 0 or -./ Common logs have a base of 10 and natural logs have a base of e. -./ 0 = is logarithmic form and 3 = 0 is exponential form. If 1 then -./ 1 = 0 and -./ b = How do you evaluate an expression with a logarithm? Students can change the logarithmic form to the exponential form and solve, or use a calculator to find a log. 13. How do exponential and logarithmic functions relate? 11

12 The logarithmic function /() = -./ x is the inverse of the exponential function ;() =. 14. How do you find the inverse of a logarithmic function? Start with the original function, then change the x s to y s and the y s to x s. Then write the function in exponential form. Finally solve the equation for y. The process can be reversed for finding the inverse of an exponential function. 15. What is the shape of a logarithmic function? A logarithmic curve in the form = -./ ( h)+ has a vertical asymptote at the x = h line, the domain is x > h, and the range is all real numbers. If b >1 the graph moves up to the right. If 0 < b < 1, the graph moves down to the right. 16. What are some common examples of logarithms in use? The ph scale is logarithmic, the Richter scale for earthquakes is logarithmic, and sound is measured on a logarithmic scale. 17. How can logs be used to facilitate computation? The properties of logs: Product Property, Quotient Property and Power Property can be used to expand of condense expressions with logs. The Change of Base formula allows computation on a calculator with bases other than 10 or e. 18. What methods are used to solve exponential and logarithmic equations? Equations can be solved by 1. writing both sides of the equation with the same base and solving by equating exponents, 2. isolating the term with the variable exponent and taking the log of both sides, 3. writing the equation with the same logs on both sides and then equating the expression N (as in -./ 0), or 4. using the definition of a logarithm to change the problem to exponential form. 19. Can extraneous solutions be generated from these solving techniques? Yes, all solutions should be verified. 20. What is the difference between a power function and an exponential function? In a power function the variable has an exponent on it, (y = a ) in an exponential function, the exponent is the variable (y = a ) 21. How do you test to determine if two points model a power function or an exponential function? Use the graph of (,ln) to test for an exponential function (it will be linear) and the graph of (ln,ln) to test for the power function. 22. What is the form of the equation of a logistic growth function? > Logistic growth functions have the form = *?@A BCD where a, c, and r are all positive constants. 23. What are the important attributes of the graph of a logistic growth function? > The important attributes are = E 'F = 0 are the upper and lower bounds (asymptotes), the y intercept is, the graph *?@ increases from left to right, and the rate of growth increases from the left of the point of maximum growth, ) HI@, > + and 4 decreases after (to the right of) that point. 24. What natural occurrences can be modeled by a logistic growth function? One example of a logistic growth model can be population. % Targeted Skill(s): 1. Know the equation of an exponential growth function. 2. Graph an exponential growth function as a function with attributes associated with the graph (y-intercept, asymptotes, domain, range, end behavior). 3. Graph an exponential growth function as a translation of a parent function. 4. Apply exponential growth in fixed percent per year and compound interest problems. 5. Know the equation of an exponential decay function. 12

13 Writing: 6. Graph an exponential decay function as a function with attributes associated with the graph (y-intercept, asymptotes, domain, range, end behavior). 7. Graph an exponential decay function as a translation of a parent function. 8. Apply exponential decay in fixed percent per year depreciation and half-life problems. 9. Define e 10. Compute with e both in natural logs and in the continuously compound interest formula. 11. Define logarithms including specialized common and natural logs. 12. Apply the definition of logs (put them in exponential form) to find logs. 13. Know the logarithmic function /() = -./ x is the inverse of the exponential function ;() = and use that in graphing and solving. 14. Find the inverse of a log function or an exponential function. 15. Recognize the graph of a log function and graph a log function. 16. Apply knowledge of logs in applications of sound intensity, earthquakes, ph, etc. 17. Condense and expand expressions that include logs using the properties of logs. 18. Solve equations with logs and exponents 19. Identify and eliminate extraneous solutions. 20. Differentiate between an exponential function and a power function. 21. Test using (,ln) and (ln,ln) for an exponential or power function respectively. 22. Recognize the logistic growth function. 23. Graph logistic growth curves using attributes including: asymptotes, y-intercept, point of maximum growth. 24. Apply the logistic growth function in population problems. Assessment Practices: Quizzes and tests that include vocabulary, short answer questions and larger problems where partial credit is available. Some questions require a graphing calculator others must be done without any calculator. 13

14 MA Standard/Strands: Time Frame: Text (Chapter/Pages) Other Resources: Essential Questions Concepts, Content: Targeted Skill(s): 13 days Rational Equations and Functions Chapter 9 1. What is inverse variation and when is it used? Inverse variation models situations where =. = What is joint variation? Joint variation is direct variation with more than 2 variables as in situations where = J. = K L. 3. What is a rational function? A rational function is a ratio of two polynomials M() where () 0. N() 4. How do you graph a rational function? If p(x) and q(x) are linear, then the graph of the rational function is a hyperbola. Find the horizontal and vertical asymptotes and then make a table of values to complete the branches. For other polynomials p(x) and q(x), find m and n (the degree of the numerator and denominator respectively) and compare them to find the horizontal asymptote or end behavior. Find the vertical asymptote(s) where O() = 0, and any x or y intercepts. Graphing points near the vertical asymptotes will clarify the nature of the curves. 5. How do you simplify rational expressions? Rewriting rational expressions in factored form makes it possible to see common factors (one each from a numerator and a denominator) to eliminate by division. 6. What is a complex fraction, and how is it simplified? A complex fraction has a numerator and/or denominator which is an algebraic expression of other fractions. To simplify a complex fraction, the numerator and denominator gets simplified to single fractions which then get divided and simplified. 7. What are the methods of solving rational equations? Rational equations are solved by: Multiplying by the Least Common Denominator or by Cross Multiplying 1. Given the type of variation, students will write the equation of variation using x, y and k (the constant of variation). Then using values for x and y, students will find k. 1. Given an equation of x, y and k, students will solve for the missing value. 2. Given the information is a model of joint variation, students will write the equation of variation using x, y, z and k (the constant of variation). 2. Given an equation of x, y, z and k, students will solve for the missing value. 4. Use asymptotes, intercepts, end behavior, and a table of values to graph a rational function. 5. Factor completely all algebraic expressions, change the problem to multiplication, and divide out any common factors. 6. Simplify the numerator and denominator to single fractions and then divide. 7. Solve rational equations by multiplying both sides by the least common denominator and solving the remaining equation. 14

15 7. Solve rational equations in the form M() N() = P() Q() by cross multiplying. Writing: Assessment Practices: Quizzes and tests that include vocabulary, short answer questions and larger problems where partial credit is available. Some questions require a graphing calculator others must be done without any calculator. Graphs will be done by hand and on the graphing calculator. Students will create a graph using given attributes, asymptotes and table values but no written function. 15

16 MA Standard/Strands: Time Frame: Text (Chapter/Pages) Other Resources: Essential Questions Concepts, Content: Targeted Skill(s): 7 days Sequences and Series Chapter What is a sequence and how does it compare to a series? A sequence is a list of terms generated by a rule, separated by commas, while a series is a set of terms generated by a rule that are added to each other. 2. What is the vocabulary needed to work with sequences and series? Term, finite, infinite, sequence, series, summation notation, sigma notation, infinity, common difference, common ratio, arithmetic, geometric, explicit and recursive 3. How do you graph a sequence? The domain is the position in the sequence and the range is the value of the term. This is discrete data. 4. What is sigma notation and how is it used? The symbol Σ (sigma) implies summation with the limits of summation above and below the symbol and the rule to be used to the right of the symbol. Each number beginning with the one on the bottom and continuing through the one on the top is used in the rule on the right. Then all the values are added together. 5. What is an arithmetic sequence? An arithmetic sequence has a common difference, d. F = & &* 6. How do you find the nth term of an arithmetic sequence? Use the formula: & = * + (' 1)F 7. How do you find the sum of the first n terms of an arithmetic series? Use the formula: R & = ' T What is an geometric sequence? A geometric sequence has a common ratio, r. = & &* 9. How do you find the nth term of a geometric sequence? Use the formula: & = &* * 10. How do you find the sum of the first n terms of a finite geometric series? Use the formula: R & = * ) *%T + *% 11. How do you find the sum of an infinite geometric series? Use the formula: R & = S + if V < 1 *% 12. How do explicit and recursive rules differ for sequences? The explicit formula depends on n, where the recursive formula depends on one or more previous terms. 1. Differentiate between arithmetic and geometric sequences and a series 2. Use correct vocabulary for sequence and series 3. Graph a sequence with the domain values the position in the sequence and the range the value of the term 4. Find the sum of a series written in sigma notation 4. Define a given series using summation notation 16

17 5. Determine if a sequence is arithmetic by looking for d 6. Find a given term of an arithmetic sequence using the explicit formula 7. Find the sum of an arithmetic sequence using the R & formula 8. Determine if a sequence is geometric by looking for d 9. Find a given term of an geometric sequence using the explicit formula 10. Find the sum of a finite geometric sequence using the R & formula 11. Find the sum of an infinite geometric sequence using the R & formula 12. Write both explicit and recursive rules for arithmetic and geometric sequences Writing: Assessment Practices: Quizzes and tests that include vocabulary, short answer questions and larger problems where partial credit is available. Some questions require a calculator others must be done without any calculator. 17