1. 2009 Mathematics Standards of Learning Training Institutes Algebra II Virginia Department of Education

2. Algebra II SOL Structure Previous Reporting Categories Expressions and Operations Relations and Functions Equations and Inequalities Analytical Geometry Systems of Equations and Inequalities New Reporting Categories Expressions and Operations Equations and Inequalities Functions Statistics

3. Major Changes to Algebra II Content moved to Mathematical Analysis  matrices  conic sections Content eliminated from Algebra II  linear programming (it resides still in AFDA) Content moved to Algebra I  Inverse variation  Systems of Linear Equations and Inequalities Content added  permutations and combinations  the normal distribution

4. Algebra II Expressions and Operations STANDARD AII.1 The student, given rational, radical, or polynomial expressions, will a) add, subtract, multiply, divide, and simplify rational algebraic expressions; b) add, subtract, multiply, divide, and simplify radical expressions containing rational numbers and variables, and expressions containing rational exponents; c) write radical expressions as expressions containing rational exponents and vice versa; and d) factor polynomials completely.

5. Factor polynomials by applying general patterns. Factor polynomials completely over the integers. Curriculum Framework Highlights for AII.1

6. Examples for AII.1 Factor Completely Simplify

7. Expressions and Operations STANDARD AII.2 The student will investigate and apply the properties of arithmetic and geometric sequences and series to solve real- world problems, including writing the first n terms, finding the nth term, and evaluating summation formulas. Notation will include  and a n. Algebra II

8. Curriculum Framework Highlights for AII.2 Model real-world situations using sequences and series. Generalize patterns in a sequence using explicit and recursive formulas. Find sum of the first n terms of a finite geometric or arithmetic series. Find the sum of a convergent infinite series.

9. AII.2 Example The number can be written as Find the value, S, as a convergent infinite series.

10. Explicitly vs. Recursively Defined Sequences Express the following sequence using a recursive formula and an explicit formula. Recursive Explicit

11. STANDARD AII.3 The student will perform operations on complex numbers, express the results in simplest form using patterns of the powers of i, and identify field properties that are valid for the complex numbers. Expressions and Operations Algebra II

12. Curriculum Framework Highlights for AII.3 Determine which field properties apply to the set of complex numbers. Simplify radical expressions containing negative numbers and express in a+bi form. Simplify powers of i.

13. STANDARD AII.4 The student will solve, algebraically and graphically, a)absolute value equations and inequalities; b)quadratic equations over the set of complex numbers; c)equations containing rational algebraic expressions; and d)equations containing radical expressions. Graphing calculators will be used for solving and for confirming the algebraic solutions. Algebra II Equations and Inequalities

14. Curriculum Framework Highlights for AII.4 Emphasis on quadratic equations solved over the set of complex numbers. Calculate the discriminant of a quadratic equation to determine the number of real and complex solutions The process of solving rational and radical equations can sometimes lead to extraneous solutions. Verify solutions to radical and rational equations. Absolute value equations and inequalities solved algebraically and graphically. Rational equations with monomial or binomial denominators.

15. Solving Absolute Value Equations and Inequalities Graphically Two functions are graphed on the axis below: Use the graphs to solve i. ii. iii.

16. Equations and Inequalities STANDARD AII.5 The student will solve nonlinear systems of equations, including linear-quadratic and quadratic- quadratic, algebraically and graphically. Graphing calculators will be used as a tool to visualize graphs and predict the number of solutions. Algebra II

17. Curriculum Framework Highlights for AII.5 Predict the number of solutions to a non- linear system of equations.

18. Functions STANDARD AII.6 The student will recognize the general shape of function (absolute value, square root, cube root, rational, polynomial, exponential, and logarithmic) families and will convert between graphic and symbolic forms of functions. A transformational approach to graphing will be employed. Graphing calculators will be used as a tool to investigate the shapes and behaviors of these functions. Algebra II

19. Curriculum Framework Highlights for AII.6 Recognize graphs of parent functions. Using a transformational approach graph a function and write the equation of a function given its graph. Given a transformation of a parent function, identify the graph of the transformed function.

20. Functions STANDARD AII.7 The student will investigate and analyze functions algebraically and graphically. Key concepts include a) domain and range, including limited and discontinuous domains and ranges; b) zeros; c) x- and y-intercepts; d) intervals in which a function is increasing or decreasing; e) asymptotes; f) end behavior; g) inverse of a function; and h) composition of multiple functions. Graphing calculators will be used as a tool to assist in investigation of functions. Algebra II

21. Curriculum Framework Highlights for AII.7 Identify intervals of increasing and decreasing. Describe end behavior. Find equations for vertical and horizontal asymptotes. Find the inverse of a function. Graph the inverse of a function. Verify two functions are inverses using composition of functions.

22. STANDARD AII.8 The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Functions Algebra II

23. Curriculum Framework Highlights for AII.8 Define a polynomial function given its zeros. Identify zeros of multiplicity greater than 1 and the effect of those zeros on the graph of a polynomial. Determined factored form of a polynomial from its graph. Given a polynomial equation determine the number of real and non-real solutions

24. STANDARD AII.9 The student will collect and analyze data, determine the equation of the curve of best fit, make predictions, and solve real-world problems, using mathematical models. Mathematical models will include polynomial, exponential, and logarithmic functions. Algebra II Statistics

25. Curriculum Framework Highlights for AII.9 Polynomial models replace just “linear and quadratic”. Find curve of best fit for data using a graphing calculator. Models include exponential, logarithmic and polynomial functions.

26. STANDARD AII.10 The student will identify, create, and solve real-world problems involving inverse variation, joint variation, and a combination of direct and inverse variations. Algebra II Statistics

27. Curriculum Framework Highlights for AII.10 Set up and solve problems, including real- world problems, involving inverse variation, joint variation and a combination of direct and inverse variation.

28. Statistics STANDARD AII.11 The student will identify properties of a normal distribution and apply those properties to determine probabilities associated with areas under the standard normal curve. Algebra II

29. Related SOL in Algebra I and AFDA ALGEBRA I STANDARD A.9 The student, given a set of data, will interpret variation in real-world contexts and calculate and interpret mean absolute deviation, standard deviation, and z-scores. ALGEBRA, FUNCTIONS AND DATA ANALYSIS STANDARD AFDA.7 The student will analyze the normal distribution. Key concepts include: a) characteristics of normally distributed data; b) percentiles; c) normalizing data using z-scores; and d) area under the standard normal curve and probability

30. Proposed Curriculum Framework for AII.11 TOPIC: STATISTICS ALGEBRA II STANDARD AII.11 The student will identify properties of a normal distribution and apply those properties to determine probabilities associated with areas under the standard normal curve. ESSENTIAL UNDERSTANDINGSESSENTIAL KNOWLEDGE AND SKILLS A normal distribution curve is a symmetrical, bell-shaped curve defined by the mean and the standard deviation of a data set. The mean is located on the line of symmetry of the curve. Areas under the curve represent probabilities associated with continuous distributions. The normal curve is a probability distribution and the total area under the curve is 1. For a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, approximately 95% of the data falls within two standard deviations of the mean, and approximately 99.7% of the data falls within three standard deviations of the mean. The mean of the data in a standard normal distribution is 0 and the standard deviation is 1. The standard normal curve allows for the comparison of data from different normal distributions. A z-score is a measure of position derived from the mean and standard deviation of data. A z-score expresses, in standard deviation units, how far an element falls from the mean of the data set. A z-score is a derived score from a given normal distribution. A standard normal distribution is the set of all z-scores. The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to  Identify the properties of a normal probability distribution.  Describe how the standard deviation and the mean affect the graph of the normal distribution.  Compare two sets of normally distributed data using a standard normal distribution and z-scores.  Represent probability as area under the curve of a standard normal probability distribution.  Use the graphing calculator or a standard normal probability table to determine probabilities or percentiles based on z- scores.

31. The graph of a normal distribution is a normal curve and every normal curve has the following characteristics: - The mean, median and mode are equal. - They are bell-shaped and symmetrical about the mean. - The curve never touches, but it closer and closer to the y-axis as it gets farther from the mean. - The total area under the curve is equal to 1. NORMAL CURVES

32. Area under the curve within 1 standard deviation of the mean. Area under the curve within 2 standard deviations of the mean. Area under the curve within 3 standard deviations of the mean. A normal distribution with mean equal to 0 and standard deviation equal to 1 is called a Standard Normal Distribution.

33. Example from AFDA Enhanced Scope and Sequence 1. Mean = 5 Standard Deviation =1 2. Mean = 4 Standard Deviation = 1 3. Mean = 6 Standard Deviation =.5 4. Mean = 5 Standard Deviation = 1.5 a. b. c. d. Each graph in the figure to the right is a normal curve. Match the curve to the corresponding information below. 1 2 3 4 5 6 7 8 9

34. STANDARD AII.12 The student will compute and distinguish between permutations and combinations and use technology for applications. Algebra II Statistics

35. TOPIC: STATISTICS ALGEBRA II STANDARD AII.12 The student will compute and distinguish between permutations and combinations and use technology for applications. ESSENTIAL UNDERSTANDINGSESSENTIAL KNOWLEDGE AND SKILLS The Fundamental Counting Principle states that if one decision can be made n ways and another can be made m ways, then the two decisions can be made nm ways. Permutations are used to calculate the number of possible arrangements of objects. Combinations are used to calculate the number of possible selections of objects without regard to the order selected. The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Compare and contrast permutations and combinations. Calculate the number of permutations of n objects taken r at a time. Calculate the number of combinations of n objects taken r at a time. Use permutations and combinations as counting techniques to solve real-world problems. Proposed Curriculum Framework for AII.12