Algebra II Standards of Learning Curriculum Guide

Transcription

1 Expressions Operations AII.1 identify field properties, axioms of equality inequality, properties of order that are valid for the set of real numbers its subsets, complex numbers, matrices. be able to: Identify examples of the field properties of real numbers the properties of equality that occur while solving equations. Identify examples of the properties of inequality order that occur while solving inequalities. Place the following sets of numbers in a hierarchy: complex numbers, pure imaginary numbers, real numbers, rational irrational numbers, integers, whole numbers natural numbers. Venn diagrams may be used. AII.4 AII.11 Must be supplemented when using Heath Algebra II Group Mathematics Teacher Resource Test Blueprints Test Released Items.ht Danville Public Schools, 2003 (Revised 8/14/03) 1 Algebra II

2 Equations Inequalities AII.4 solve absolute value equations inequalities graphically algebraically. Graphing calculators will be used as a primary method of solution to verify algebraic solutions. Define absolute value. Evaluate expressions that contain absolute value. Recognize that the value of he argument for absolute value must be greater than or equal to zero explain why. Solve absolute value equations algebraically graphically. Solve absolute value inequalities algebraically graphically. AII.1 AII.13 Must be supplemented when using Heath Algebra II Group Algebra Instructional Modules - Square Patio Patterns p Matrix Multiplications p Matrices (Beginning) p Solving Systems with Matrices p Linear Programming p. 100 Danville Public Schools, 2003 (Revised 8/14/03) 2 Algebra II

3 Relations Functions AII.19 collect analyze data to make predictions solve practical problems. Graphing calculators will be used to investigate scatterplots to determine the equation for a curve of best fit. Models will include linear, quadratic, exponential, logarithmic functions. be able to: Collect data display it in a scattergraph. AII.19 (Sec. 2.7) Analyze data using measures of central tendency (mean, median, mode), the range of the data, box--whisker plots. AII.19 (Sec. 6.7) Determine the equation of the curve of best fit using the graphing calculator. Consider the graphs of the parent functions when determining which curve might be appropriate. Use the equation to make predictions. AII.19 (supplement) AII.8 Demonstrate the formation of the conic sections using slices of double geometric cones. Review the distance midpoint formulas. Develop a set of rules that can be used to determine the equation of a conic (not in stard form) Stress the relationship between the concepts of solutions, intercepts, zeroes factors. Group Algebra Instructional Modules - Square Patio Patterns p Matrix Multiplications p Matrices (Beginning) p Solving Systems with Matrices p Linear Programming p. 100 Danville Public Schools, 2003 (Revised 8/14/03) 3 Algebra II

4 Systems of Equations/ Inequalities (Linear inequalities) AII.13 solve practical problems, using systems of linear inequalities linear programming, describe the results both orally in writing. A graphing calculator will be used to facilitate solutions to linear programming problems. be able to: Solve a system of linear equalities by graphing. AII.13 (Sec.3.4) Identify examples of the properties of inequality order that occur while solving inequalities. AII.13 (Sec.3.5) Find the maximum minimum values of a function over a region (linear programming). AII.13 (Sec.3.5) Identify the constraints in a practical situation model them as inequalities. Graph the system of inequalities identify the area of intersection as the feasible region. The feasible region contains all solutions possible. AII.4 Mathematics Teacher Resource Test Blueprints Test Released Items.ht Danville Public Schools, 2003 (Revised 8/14/03) 4 Algebra II

5 Systems of Equations/ Inequalities (Linear inequalities) (con t) AII.13 (con t) The maximum minimum values of the function occur at the vertices of the feasible region. Substitute the coordinates of each vertex of the feasible region into the function to determine which vertex yields the maximum (or minimum) value of the function. Describe the results of a linear programming problem orally in writing. AII.13 (Sec.3.5) Algebra Instructional Modules - Square Patio Patterns p Matrix Multiplications p Matrices (Beginning) p Solving Systems with Matrices p Linear Programming p. 100 Danville Public Schools, 2003 (Revised 8/14/03) 5 Algebra II

6 Systems of Equations/ Inequalities (Matrices) AII.11 use matrix multiplication to solve practical problems. Graphing calculators or computer programs with matrix capabilities will be used to find the product. be able to: Organize data into matrices identify the dimensions of the matrix. Investigate commutativity associativity of matrix addition. Compare contrast matrix addition of real numbers. Multiply matrices using a calculator or a computer with matrix capability. Solve problems that require matrix multiplication. Investigate commutativity associativity of matrix multiplication. Compare contrast matrix multiplication with multiplication of real numbers. AII.1 AII.12 Using graphing calculator to - Verify properties of addition subtraction of matrices - Find the inverse of a matrix - Find the determinant of a square matrix - Solve systems of equations using the inverse matrix - Solve systems of linear inequalities Solve problems using matrix logic Use overhead projector colored manipulatives to graph linear inequalities solve systems of linear inequalities. Review evaluating a function. Use overhead projector manipulatives to determine the solutions of quadratic systems. Group Mathematics Teacher Resource.ht Algebra Instructional Modules - Square Patio Patterns p Matrix Multiplications p Matrices (Beginning) p Solving Systems with Matrices p Linear Programming p. 100 Danville Public Schools, 2003 (Revised 8/14/03) 6 Algebra II

7 Systems of Equations/ Inequalities (Matrices) AII.12 represent problem situations with a system of linear equations solve the system, using the inverse matrix method. Graphing calculators or computer programs with matrix capability will be used to perform computations. be able to: Find the determinant of a square matrix. AII.12 (Sec. 4.3) Identify the identity matrix. (i). AII.12 (Sec. 4.4) For a matrix A, find the inverse matrix A -1 (if it exists) such that A * A -1 = A -1 * A = 1. AII.12 (Sec. 4.4) Use an inverse matrix to solve matrix equations. Use the graphing calculator or a computer application with matrix capabilities. AII.12 (Sec. 4.4 & 4.5) Compare contrast solving matrix equations linear equations. AII.1 (Sec. 4.4) Represent a system of equations as a matrix equation where the coefficient matrix times the variable matrix equals the constant matrix. AII.11 (Sec. 4.5) Using graphing calculator to - Verify properties of addition subtraction of matrices - Find the inverse of a matrix - Find the determinant of a square matrix - Solve systems of equations using the inverse matrix - Solve systems of linear inequalities Solve problems using matrix logic Use overhead projector colored manipulatives to graph linear inequalities solve systems of linear inequalities. Review evaluating a function. Use overhead projector manipulatives to determine the solutions of quadratic systems. Group Mathematics Teacher Resource Test Blueprints Test Released Items.ht Danville Public Schools, 2003 (Revised 8/14/03) 7 Algebra II

8 Systems of Equations/ Inequalities (Matrices) (con t) AII.12 (con t) Solve systems of linear equations using inverse matrices. Use the graphing calculator or a computer application with matrix capabilities. AII.12 (Sec. 4.5) Danville Public Schools, 2003 (Revised 8/14/03) 8 Algebra II

9 Relations Functions AII.8 recognize multiple re of functions (linear, quadratic, absolute value, step, exponential functions) convert between a graph, a table, symbolic form. A transformationa l approach to graphing will be employed through the use of graphing calculators. be able to: Recognize the graphs equations of parent functions such as y = x, y = x 2, y = x 3, y = [x], y = a x, step functions, other polynomial functions. AII.8 (Sec. 6.4) AII.15 (Sec. 8.1) Apply transformations (translations, reflections, dilations, rotations) combinations of transformations to parent graphs. AII.8 (Sec. 6.5) AII.15 (Sec ) Given an image graph of a function, describe the transformations that were performed on the pre-image the order in which they could have occurred. Given the equation of a parent graph, vary the coefficients constants of the equation, observe the changes in the graph of the parent, generalize the changes to the graphs of other functions. AII.15 Use graphing calculators extensively to - Perform calculations - Determine how a parameter affects a parent graph - Find domain range of graphs Use overhead transparencies to help visualize conics. Review slope-intercept form of a line Demonstrate the composition of functions by ordered pairs, mappings, Venn diagrams substitution. Make a chart of patterns of factoring to use as a study guide. Group Mathematics Teacher Resource Test Blueprints Test Released Items.ht Danville Public Schools, 2003 (Revised 8/14/03) 9 Algebra II

10 Relations Functions (con t) AII.8 (con t) Build a strong connection between the algebraic geometric re of functions. Investigate the commutativity associativity of combinations of transformations. AII.1 (Sec. 6.2) Danville Public Schools, 2003 (Revised 8/14/03) 10 Algebra II

11 Relations Functions AII.9 find the domain, range, zeros, inverse of a function; the value of a function for a given element in its domain; the composition of multiple functions. Functions will include exponential, logarithmic, those that have domains ranges that are limited /or discontinuous. The graphing calculator will be used as a tool to assist in investigation of functions. be able to: Given the graph or the equation of a function, identify the domain range of the function. Include functions with discontinuities. AII.9 (Sec. 6.1 & 6.2) AII.15 (Sec. 9.2) Find the value of a function for a given element in the domain. AII.9 (Sec. 6.1) Find the composition of functions algebraically graphically. AII.9 (Sec. 6.2) Find the inverse of a function algebraically graphically. AII.9 (Sec. 6.3) Explain how composition of functions finding he inverse of a function affects the domain range of the functions. AII.9 (Sec. 6.2 & 6.3) Demonstrate that the exponential logarithmic functions are inverse functions. AII.9 (Sec. 8.1) AII.8 AII.10 AII.15 Stress algebraic vocabulary symbols Use graphing calculators wherever applicable. Use graphing calculator to determine domain range horizontal vertical asymptotes - By referring to the displayed graph - By using the table feature - By using the trace feature - By employing the zoom feature to verify asymptotes Review factoring. Group Mathematics Teacher Resource Test Blueprints Test Released Items.ht Danville Public Schools, 2003 (Revised 8/14/03) 11 Algebra II

12 Expressions Operations AII.5 identify factor completely polynomials representing the difference of squares, perfect square trinomials, the sum difference of cubes, general trinomials. be able to: Factor polynomials completely (difference of two squares, perfect square trinomials, general trinomials, sum difference of cubes). AII.5 (Sec. 9.3) AII.6 AII.15 Group Mathematics Teacher Resource Test Blueprints Test Released Items.ht Danville Public Schools, 2003 (Revised 8/14/03) 12 Algebra II

13 Equations Inequalities AII.6 select, justify, apply a technique to solve a quadratic equation over the set of complex numbers. Graphing calculators will be used for solving for confirming the algebraic solutions. be able to: Find the roots of a function algebraically graphically. Quadratic equations may be solved using a variety of techniques which include but are not limited to factoring to use the zero product property, square roots, the quadratic formula, completing the square, graphing. AII.6 (Sec. 9.3) AII.15 (Sec , ) AII.5 AII.15 AII.8 Group Mathematics Teacher Resource Test Blueprints Test Released Items.ht Danville Public Schools, 2003 (Revised 8/14/03) 13 Algebra II

14 Expressions Operations AII.3 a) add, subtract, multiply, divide, simplify radical expressions containing positive rational numbers variables expressions containing rational exponents; b) write radical expressions as expressions containing rational exponents vice versa. be able to: Simplify radical expressions with a variety of indices (rational integral) until: AII.3 (Sec. 7.4) The index, n, is as small as possible The radic contains no factors that are the nth powers of an integer or polynomial The radic contains no fractions No radicals appear in the denominator. Rationalize the denominator of a rational expression that contains a radical expression in the denominator. AII.3 (Sec.7.4 supplement) Add, subtract, multiply, divide radical expressions. AII.3 Write expressions in radical form as expressions with rational exponents. AII.3 (Sec ) Evaluate expressions in either exponential or radical form. AII.3 (Sec. 7.3) AII.7 Use graphing calculators to find factors roots to investigate rational exponents. Use pictorial representation (Venn diagrams) to demonstrate relationships. Review laws of exponents. Use geometric dot paper to show the sum of two like radicals. Use power of i to develop higher powers of i. Parallel adding subtracting of complex numbers to combining like terms. Group Mathematics Teacher Resource Test Blueprints Test Released Items.ht Danville Public Schools, 2003 (Revised 8/14/03) 14 Algebra II

15 Equations Inequalities AII.7 solve equations containing rational expressions equations containing radical expressions algebraically graphically. Graphing calculators will be used for solving for confirming the algebraic solutions. be able to: Solve equations containing radical expressions equations of nth roots algebraically graphically. AII.7 (Sec. 7.5) Check solutions for extraneous roots. AII.7 (Sec ) AII.3 Group Mathematics Teacher Resource Test Blueprints Test Released Items.ht Danville Public Schools, 2003 (Revised 8/14/03) 15 Algebra II

16 Expressions Operations AII.17 perform operations on complex numbers express the results in simplest form. Simplifying results will involve using patterns of the powers of i. be able to: Recognize a complex number as a number that can be written as a + bi where a b are real number i is the principal square root of 1. AII.17 (Sec. 5.5) Recognize pure imaginary numbers. Simplify square roots with negative arguments. Represent a complex number geometrically in the coordinate plane. Add, subtract, multiply complex numbers. Compare contrast adding, subtracting, multiplying complex numbers with operating on real numbers. Simplify powers of i generalize the pattern. AII.17 (Sec. 5.5 supplement) Simplify rational expressions with complex numbers in the denominator by using complex conjugates. AII.17 (supplement) AII.6 AII.15 AII.1 Review laws of exponents. Use geometric dot paper to show the sum of two like radicals. Use power of i to develop higher powers of i. Parallel adding subtracting of complex numbers to combining like terms. Group Mathematics Teacher Resource Test Blueprints Test Released Items.ht Danville Public Schools, 2003 (Revised 8/14/03) 16 Algebra II

17 Expressions Operations (con t) AII.17 (con t) Place the following sets of numbers in a hierarchy: complex numbers, pure imaginary numbers, real numbers, rational irrational numbers, integers, whole numbers natural numbers. Venn diagrams may be used. AII (Sec. 5.5) Danville Public Schools, 2003 (Revised 8/14/03) 17 Algebra II

18 Conic Sections AII.18 identify conic sections (circle, ellipse, parabola, hyperbola) from his/her equations. Given the equations in (h, k) form, the student will sketch graphs of conic sections, using transformations. be able to: Recognize the graphs of the conic sections defined as any figure that can be formed by slicing a double cone (parabola, ellipse, circle, hyperbola). AII.18 (Sec. 11.6) Given an equation, identify the conic section graph it transformationally. AII.18 (Sec. 11.6) AII.8 Use hs-on manipulatives to draw conics using their definitions. Demonstrate the formation of the conic sections using slices of double geometric cones. Develop a set of rules that can be used to determine the equation of a conic (not in stard form). Stress the relationship between the concepts of solutions, intercepts, zeroes factors. Group Mathematics Teacher Resource Test Blueprints Test Released Items.ht Danville Public Schools, 2003 (Revised 8/14/03) 18 Algebra II

19 Conic Sections AII.14 solve nonlinear systems of equations, including linearquadratic quadraticquadratic, algebraically graphically. The graphing calculator will be used as a tool to visualize graphs predict the number of solutions. be able to: Solve linear-quadratic systems of equations algebraically identify the set of ordered pairs that is the solution to the system. Solve linear-quadratic systems of equations graphically identify the set of ordered pairs that is the solution to the system. Solve quadratic-quadratic systems of equations algebraically identify the set of ordered pairs that is the solution to the system. Solve quadratic -quadratic systems of equations graphically identify the set of ordered pairs that is the solution to the system. AII.18 Group Mathematics Teacher Resource Test Blueprints Test Released Items.ht Danville Public Schools, 2003 (Revised 8/14/03) 19 Algebra II

20 Analytical Geometry AII.10 investigate describe through the use of graphs the relationships between the solution of an equation, zero of a function, x- intercept of a graph, factors of a polynomial expression. be able to: Investigate the relationship between the solutions of an equation, zeros of a function, x- intercepts, factors of a polynomial. AII.10 (Sec. 9.6) Given the roots of a polynomial, write an equation for the polynomial function. AII.6/AII.15 (Sec. 9.6) AII.9 AII.6 AII.15 Group Mathematics Teacher Resource Test Blueprints Test Released Items.ht Danville Public Schools, 2003 (Revised 8/14/03) 20 Algebra II

21 Relations Functions AII.15 recognize the general shape of polynomial, exponential, logarithmic functions. The graphing calculator will be used as a tool to investigate the shape behavior of these functions. be able to: Given an equation of a function, sketch the graph of the function. AII.9 Group Mathematics Teacher Resource Test Blueprints Test Released Items.ht Danville Public Schools, 2003 (Revised 8/14/03) 21 Algebra II

22 Expressions Operations AII.2 add, subtract, multiply, divide, simplify rational expressions, including complex fractions. be able to: Add, subtract, multiply, divide rational algebraic expressions including complex fractions. AII.2 (Sec. 10.5) AII.7 AII.9 Review operations with fractions compare to problems involving rational expressions. Compare rational equations to simple equations in one variable to equations containing fractions. Group Mathematics Teacher Resource Test Blueprints Test Released Items.ht Danville Public Schools, 2003 (Revised 8/14/03) 22 Algebra II

23 Equations Inequalities AII.7 solve equations containing rational expressions equations containing radical expressions algebraically graphically. Graphing calculators will be used for solving for confirming the algebraic solutions. be able to: Simplify rational algebraic expressions. AII.7 (Sec. 10.3) Compare simplifying rational algebraic expressions to simplifying fractions. AII.7 (Sec. 10.3) Solve rational equations algebraically graphically. The graphing calculator will be used as a primary tool for solution for checking the algebraic solution. AII.7 (Sec. 10.3) AII.2 AII.9 Group Mathematics Teacher Resource Test Blueprints Test Released Items.ht Danville Public Schools, 2003 (Revised 8/14/03) 23 Algebra II

24 Relations Functions AII.9 find the domain, range, zeros, inverse of a function; the value of a function for a given element in its domain; the composition of multiple functions. Functions will include exponential, logarithmic, those that have domains ranges that are limited /or discontinuous. The graphing calculator will be used as a tool to assist in investigation of functions. be able to: State the domain range of a rational function. AII.9 (Sec. 10.1) Identify the vertical horizontal asymptotes of a rational function. AII.9 (Sec. 10.1) Sketch the graph of a rational function. AII.9 (Sec. 10.1) Use graphing calculators wherever applicable. Use graphing calculator to determine domain range horizontal vertical asymptotes - By referring to the displayed graph - By using the table feature - By using the trace feature - By employing the zoom feature to verify asymptotes Review factoring. Group Mathematics Teacher Resource Test Blueprints Test Released Items.ht Danville Public Schools, 2003 (Revised 8/14/03) 24 Algebra II

25 Relations Functions AII.20 identify, create, solve practical problems involving inverse variation a combination of direct inverse variations. be able to: Identify a variation as direct, inverse, or joint. AII.20 (Sec. 2.4 & 10.2) Solve practical problems involving joint variation. AII.20 (Sec. 10.2) Group Mathematics Teacher Resource Test Blueprints Test Released Items.ht Danville Public Schools, 2003 (Revised 8/14/03) 25 Algebra II

26 Sequences Series AII.16 investigate apply the properties of arithmetic geometric sequences series to solve practical problems, including writing the first n terms, finding the n th term, evaluating summation formulas. Notation will include a n. be able to: Find the next term in a sequence by looking for a pattern. AII.16 (Sec. 12.2) Find the nth term of an arithmetic sequence find the position of a given term in an arithmetic sequence. AII.16 (Sec & supplement) Find arithmetic means. AII.16 (Sec. 12.2) Differentiate between a sequence a series. AII.16 (Sec. 12.3) Find the sum of an arithmetic series. AII.16 (Sec. 12.3) Find specific terms in an arithmetic series. AII.16 (Sec. 12.3) Use sigma ( ) notation to denote sums. AII.16 (Sec. 12.3) Explore patterns relationships to determine how sequence terms are generated. Use the iterative feature of the graphing calculator to find a specified term of sequence. Group Mathematics Teacher Resource Test Blueprints Test Released Items.ht Danville Public Schools, 2003 (Revised 8/14/03) 26 Algebra II

27 Sequences Series (con t) AII.16 (con t) Compare contrast arithmetic geometric sequences. AII.16 (Sec. 12.3) Find the nth term of a geometric sequence the position of a given term in a geometric sequence. AII.16 (Sec. 12.3) Find geometric means. AII.16 (Sec & supplement) Find the sum of a geometric series. AII.16 (Sec. 12.3) Find specific terms in a geometric series. AII.16 (Sec. 12.2) Danville Public Schools, 2003 (Revised 8/14/03) 27 Algebra II

28 Additional Topics in Algebra II These topics are not assessed on the Algebra II end-of-course assessment, but they may be used to enhance knowledge of some topics in Algebra II to bring closure to the course. Find the sum of an infinite geometric series. Count the number of permutations combinations possible in a given situation. Use counting techniques in binomial experiments to determine binomial probabilities. Graph points equations in three dimensions. Solve exponential logarithmic equations. Use synthetic divisions synthetic substitution to find all complex roots of a polynomial function. Study parametric equations the graphs of parametric equations. Cover as time permits only after s have been tested. Danville Public Schools, 2003 (Revised 8/14/03) 28 Algebra II