Number and Number Sense MATHEMATICS Mathematics Curriculum---Hanover County Public Schools Revised Summer 2005 Probability & Expressions Equations and Systems Statistics Inequalities Relations & Functions Virginia Standard: AII.1 The student will identify field properties, axioms of equality and inequality, and properties of order that are valid for the set of real numbers and its subsets, complex numbers, and matrices. Hanover Objective(s): same Related Standard(s): Curriculum Framework: Essential Understandings • Complex numbers are organized into a hierarchy of subsets with properties applicable to each subset. • Complex numbers are a superset of real numbers and, as a system, contain solutions for equations that are not solvable over the set of real numbers. Resources: • Essential Knowledge and Skills Identify examples of field properties: commutative, associative, identity, inverse, and distributive. • Identify examples of axioms of equality: reflexive, symmetric, transitive, substitution, addition, and multiplication. • Identify examples of axioms of inequality and order: trichotomy, transitive, addition, and multiplication. • Place the following sets of numbers in a hierarchy of subsets: complex, pure imaginary, real, rational, irrational, integers, whole, and natural. • Add and multiply matrices, and determine which field properties hold. MATHEMATICS Mathematics Curriculum---Hanover County Public Schools Revised Summer 2005 Textbook Correlation Activities “Properties of Real Numbers”, Max A. Sobel, Norbert Lerner, Glencoe Algebra II Chapter 1 p. 6-46 Evan M. Matetsky, Louis s. Cohen, Harper & Row Algebra Two, “Expressions and Formulas” Scribner Educational Publishers, New York, NY: 1985, p. 13-15. “Properties of Real Numbers” [Hereafter referred to as Sobel and number.] “Solving Equations” Norton Juster, Dot and Line, Random House, New York: 1977. “Solving Absolute Value Equations” (Now available in paperback: ISBN 0-394-73352-5) “Solving Inequalities” Using Matrices in Real-Life Settings, Roland E. Larson, Algebra “Solving Compound and Absolute Value Inequalities” 2, D.C. Heath and Company, 1993, p. 176. Chapter 2 p. 63-67 I Have/Who Has(Instructional Activity AII.1a.ACT) “Linear Equations” Patterns from Pascal’s Triangle (Instructional Activity Chapter 5 p. 270-275 AII.1B.ACT). “Complex Numbers” Fibonacci Sequence (Instructional Activity AII.1C.ACT) Worksheet Review WS 1 – AII.1B Noteables Equation WS 1- AII.1C p. 1-21 Equation WS 2 – AII.1D p. 30-32 Math Block Table-AII.1E p. 116-118 Practice 1-3-AII.1F Math Tool Box Skills Review-AII.1G Properties – AII.1H Lesson 3-3, Solving Equations – AII.1I Lesson 3-3, Combining Like Terms – AII.1J Lesson 3-4, Using the Distributive Property, Simplifying Variable Expressions – AII.1K Lesson 3-4, Using the Distributive Property, Solving and Modeling Equations – AII.1L Practice 3-3, Mixed Exercises – AII.1M Practice 3-4, Example Exercises – AII.1N Practice 4-7, Mixed Exercises, AII.1O Lesson 8-7, More Multiplication Properties of Exponents, Raising a Product to a Power – AII.1P Lesson 8-7, More Multiplication Properties of Exponents, Raising a Power to a Power – AII.1Q Lesson 8-6, A Multiplication Property of MATHEMATICS Mathematics Curriculum---Hanover County Public Schools Revised Summer 2005 Exponents, Multiplying Powers – AII.1R Lesson 8-8, Division Properties of Exponents, Dividing Powers with the Same Base – AII.1S Lesson 8-8, Division Properties of Exponents, Raising a Quotient to a Power – AII.1T Powere Properties Chart- AII.1U Properties Table – AII.1V Properties Worksheet-AII.1V Test Review – AII.1X Textbook: “Properties and Formulas of Advanced Algebra”, p. T660, p. 662663. MATHEMATICS Mathematics Curriculum---Hanover County Public Schools Revised Summer 2005 Number and Number Sense Probability & Statistics Expressions Equations and Inequalities Systems Relations & Functions Virginia Standard: AII.2 The student will add, subtract, multiply, divide, and simplify rational expressions, including complex fractions. Hanover Objective(s): same Curriculum Framework: Essential Understandings • Computational skills applicable to numerical fractions also apply to rational expressions involving variables. • • Essential Knowledge and Skills Add, subtract, multiply, and divide rational expressions whose denominators are monomials or polynomial expressions in completely factored form. Simplify a rational expression with common monomial or binomial factors. • Recognize a complex fraction, and simplify it as a quotient or product of simple fractions. Resources: MATHEMATICS Mathematics Curriculum---Hanover County Public Schools Revised Summer 2005 Textbook Correlation Activities Glencoe Algebra II Chapter 1 p. 11-27 “Properties of Real Numbers” “Solving Equations” Chapter 2 p. 81-86 “Modeling Real-World Data: Using Scatter Plots” Chapter 5 p. 222-228 “Monomials” Chapter 9 p. 472-484 “Multiplying and Dividing Rational Expressions” “Adding and Subtracting Rational Expressions” Noteables p. 6-10 p. 38-40 p. 98-100 p. 203-208 Resources “ * and / Rational Expressions”Ronald E. Larson, Timothy D. Kanold, and Lee Stiff, Algebra 2, D. C. Heath and Company, Lesington, MA: 1993, p. 533-540 [Hereafter referred as Larson and page number] “+ and – Rational Expressions” Larson, p. 548-550 Worksheets Factoring Worksheet- AII.2A Operations with Fractions- AII.2B Rational Expressions-AII.2C Assessment: Simplifying Rational Expressions-AII.2A.ASMT Multiply/Divide Rational Exp (3 Forms)- AII.2B.ASMT Simplify Rational Exp with bonus-AII.2C.ASMT Make Up Test – AII.2D.ASMT Operations with Rational Exp- AII.2E.ASMT MATHEMATICS Mathematics Curriculum---Hanover County Public Schools Revised Summer 2005 Number and Number Sense Probability & Statistics Expressions Equations and Inequalities Systems Relations & Functions Virginia Standard: AII.3 The student will: • add, subtract, multiply, divide, and simplify radical expressions containing positive rational numbers and variables and expressions containing rational exponents • write radical expressions as expressions containing rational exponents, and vice versa. Hanover Objective(s): same Curriculum Framework: Essential Understandings The student will: • add, subtract, multiply, divide, and simplify radical expressions containing positive rational numbers and variables and expressions containing rational exponents • write radical expressions as expressions containing rational exponents, and vice versa. • simplify radical expressions containing numbers and variables • add radical expressions • subtract radical expressions • multiply radical expressions • divide radical expressions • write expressions with rational exponents in simplest radical form and vice versa. • Essential Knowledge and Skills Simplify radical expressions containing positive rational numbers and variables. • Convert from radical notation to exponential notation, and vice versa. • Add and subtract radical expressions with like radicands. • Multiply and divide radical expressions not requiring rationalizing the denominators. Resources: Textbook Correlation Activities MATHEMATICS Mathematics Curriculum---Hanover County Public Schools Revised Summer 2005 Glencoe Algebra II Chapter 5 p. 233-238, p. 245-262 “Dividing Polynomials” “Roots of Real Numbers” “Radical Expressions” “Rational Exponents” Noteables: p. 103-104 p. 107-113 Resources: Properties Of Roots of Real Numbers” Larson, p. 368-371 and Max A. Sobel, Norbert Lerner, Evan M. Maletsky, and Louis S. Cohen, Algebra 2, Harper & Row, 1985, p. 174, 214 [Hereafter referred to as Sobel and page number.] Worksheets: Radical WS 1 – AII.3A Radical WS 2 – AII. 3B Radical WS 3 – AII.3C Radical WS 4 – AII.3D Radical WS 5 – AII.3E Lesson 9-4, Simplifying Radicals, Multiplications with Radicals – AII.3F Lesson 9-4, Simplifying Radicals, Division with Radicals – AII.3G Lesson 9-5, Adding and Subtracting Radicals – AII.3H Reteaching 9-4 – AII.3I Lesson 9-5, Adding and Subtracting Radicals – AII.3J Reteaching 9-5 – AII.3K Assessment: Radical Expressions/Equations – AII.3A.ASMT Simplifying Radicals and Solving Equations (2 Forms) – AII.3B.ASMT MATHEMATICS Mathematics Curriculum---Hanover County Public Schools Revised Summer 2005 Number and Number Sense Probability & Statistics Expressions Equations and Inequalities Systems Relations & Functions Virginia Standard: AII.4 The student will solve absolute value equations and inequalities graphically and algebraically. Graphing calculators will be used both as a primary method of solution and to verify algebraic solutions. Hanover Objective(s): same Curriculum Framework: Essential Understandings • Absolute value equations and inequalities can be used to model practical problems. Essential Knowledge and Skills • Solve absolute value equations in one variable algebraically and graphically, using a graphing calculator. • Solve absolute value inequalities in one variable algebraically and graphically. • Express the solutions to absolute value equations and inequalities in one variable graphically and as an algebraic inequality. • Graph absolute value equations in two variables. •Verify solutions to absolute value equations and inequalities in two variables, using a graphing calculator. Resources: MATHEMATICS Mathematics Curriculum---Hanover County Public Schools Revised Summer 2005 Textbook Correlation Activities Glencoe Algebra II Resources Chapter 1 p. 40-46 “Solving Absolute Value Equations and Inequalities” Larson, p. “Solving Compound and Absolute Value Inequalities” 44-46. Chapter 9 p. 499-504 Worksheets: “Classes of Functions” Practice 4-8, Absolute Value Inequalities – AII.4A Noteables: Assessment Sample: p. 16-18 Sobel, p. 78 and p. A8. p. 213-214 MATHEMATICS Mathematics Curriculum---Hanover County Public Schools Revised Summer 2005 Number and Number Sense Probability & Statistics Expressions Equations and Inequalities Systems Relations & Functions Virginia Standard: AII.5 The student will identify and factor completely polynomials representing the difference of squares, perfect square trinomials, the sum and difference of cubes, and general trinomials. Hanover Objective(s): same Curriculum Framework: Essential Understandings • The complete factorization of polynomials has occurred when each factor is a prime polynomial. Essential Knowledge and Skills • Determine the greatest monomial factor as a first step in complete factorization. • Pattern recognition can be used to determine complete factorization of a polynomial. • Recognize squares and cubes of positive integers. • Recognize examples of general patterns: difference of squares, sum and difference of cubes, and perfect square trinomials. • Factor polynomials by applying general patterns. Resources: MATHEMATICS Mathematics Curriculum---Hanover County Public Schools Revised Summer 2005 Textbook Correlation Activities Glencoe Algebra II Chapter 5 p. 239-244 “Factoring Polynomials” Chapter 6 p. 294-305 “Solving Quadratic Equations by Graphing” “Solving Quadratic Equations by Factoring” Noteables p. 105-106 p. 129-133 Resources: “Factoring Polynomials and Solving Polynomial Equations” Larson, p. 473-475. Worksheets: Practice 10-2- AII.5A Practice 10-4, Example Exercises- AII.5B Practice 10-5, Mixed Exercises – AII.5C Practice 10-6, Example Exercises – AII.5D Practice 10-6, Mixed Exercises – AII.5E 10-2,Multiplying and Factoring with Monomials10-4,Factoring Trinomials using Tiles – AII.5G 10-4, Factoring Trinomials – AII.5H 10-5, Factoring Special Cases – AII.5I 10-6, Solve Equations by Factoring – AII.5J AII.5F MATHEMATICS Mathematics Curriculum---Hanover County Public Schools Revised Summer 2005 Number and Number Sense Probability & Statistics Expressions Equations and Inequalities Systems Relations & Functions Virginia Standard: AII.6 The student will select, justify, and apply a technique to solve a quadratic equation over the set of complex numbers. Graphing calculators will be used for solving and confirming algebraic solutions. Hanover Objective(s): same Related Standard(s): Curriculum Framework: Essential Understandings • A quadratic equation whose graph does not intersect the xaxis has only complex solutions. • Complex solutions occur in pairs (conjugates). • The quadratic formula can be used to solve any quadratic equation. Resources: Essential Knowledge and Skills • Recognize a quadratic equation. • Select an appropriate strategy for solving a quadratic equation (factoring, using the quadratic formula, or graphing). • Solve a quadratic equation over the set of complex numbers. • Identify from a graph the real solutions to a quadratic equation. • Find the real roots of a quadratic equation, using a graphing calculator. MATHEMATICS Mathematics Curriculum---Hanover County Public Schools Revised Summer 2005 Textbook Correlation Activities Glencoe Algebra II Worksheets: Chapter 6 p. 294-321 Equation WS 1 – AII.6A “Solving Quadratic Equations by Graphing” Equation WS 2 – AII.6B “Solving Quadratic Equations by Factoring” “Completing the Square” “The Quadratic Formula and the Discriminant” Chapter 9 p. 499-504 “Classes of Functions” Noteables p. 129-139 p. 213-214 MATHEMATICS Mathematics Curriculum---Hanover County Public Schools Revised Summer 2005 Number and Number Sense Probability & Statistics Expressions Equations and Inequalities Systems Relations & Functions Virginia Standard: AII. 7 The student will solve equations containing rational expressions and equations containing radical expressions algebraically and graphically. Graphing calculators will be used for solving and confirming algebraic solutions. Hanover Objective(s): same Curriculum Framework: Essential Understandings • A solution of an equation makes the equation true. • Equations can be solved in a variety of ways. Essential Knowledge and Skills • Solve equations containing rational expressions with monomial denominators algebraically and graphically. • The solution of an equation in one variable can be found by graphing each side of the equation separately and finding the x-coordinate of the point of intersection. • Solve equations containing a radical expression algebraically and graphically. The equation will contain a linear expression under the radical, and all terms outside the radical will be constants. • Practical problems can be interpreted, represented, and solved using equations. • Identify from a graph the solutions to an equation containing rational or radical expressions. • Solve an equation containing rational or radical expressions, using a graphing calculator. • Check possible solutions to an equation containing rational or radical expressions, using a graphing calculator. Resources: MATHEMATICS Mathematics Curriculum---Hanover County Public Schools Revised Summer 2005 Textbook Correlation Activities Glencoe Algebra II Chapter 1 p.20-27 “Solving Equations” Chapter 5 p. 263-267 “Radical Equations and Inequalities” Chapter 9 p. 499-511 “Classes of Functions” “Solving Rational Equations and Inequalities” Noteables p. 8-10 p. 114-115 p. 213-218 Resources: Exploring Rational Functions (Instructional Activity AII.7A). “Solving Radical Equations” , Larson p. 374-378. “Solving Rational Equations” , Larson p. 541-551. Worksheets: Exploring Rational Functions (2 Pages)-AII.7A 9-6,Solving Radical Equations-AII.7B 9-6, Solving Radical Equations w/extraneous solutions- AII.7C MATHEMATICS Mathematics Curriculum---Hanover County Public Schools Revised Summer 2005 Number and Number Sense Probability & Statistics Expressions Equations and Inequalities Systems Relations & Functions Virginia Standard: AII.8 The student will recognize multiple representations of functions (linear, quadratic, absolute value, step, and exponential functions) and convert between a graph, a table, and symbolic form. A transformational approach to graphing will be employed through the use of graphing calculators. Hanover Objective(s): same Curriculum Framework: Essential Understandings • The graphs/equations for a family of functions can be determined using a transformational approach. Essential Knowledge and Skills • Recognize graphs of parent functions for linear, quadratic, absolute value, step, and exponential functions. • Given an equation of a function, identify the function as linear, quadratic, absolute value, step, or exponential. • Write the equation of a linear (slope-intercept form), quadratic ([h, k] form), absolute value, step, or exponential function, given the graph of the parent function or an integral translation of a parent function. • Given an equation, graph a linear, quadratic, absolute value, step, or exponential function with the aid of a graphing calculator. Resources: MATHEMATICS Mathematics Curriculum---Hanover County Public Schools Revised Summer 2005 Textbook Correlation Activities Glencoe Algebra II Worksheets: Chapter 2 p. 56-80, 89-99 Piecewise Functions (Activity AII.8A) “Relations and Functions” Postage and the Greatest Integer Function “Linear Equations’ (Instructional Activity AII.8B) “Slope” Doubling Time Growth Model (Activity AII.8C) “Writing Linear Equations” Investment Banking (Activity AII.8D) “Special Functions” Algebra II Group Activity (Activity AII.8E) “Graphing Inequalities” Rational Functions – AII.8F Chapter 5 p. 229-232 Graphing Step, Absolute Value and Linear “Polynomials” Functions – AII.8G Chapter 6 p. 286-299, 322-328 Building A Bridge (2 Pages) AII.8H “Graphing Quadratic Functions” Relationships and Graphs (Packet) AII.8I “Solving Quadratic Equations by Graphing” “Analyzing Graphs of Quadratic Functions” Assessment: Graphing Calculator Investigation p. 320-321 AII Quiz-Lesson 1.7, 1.3 (4 Forms) AII.8A.ASMT Chapter 7 p. 346-352 Test on Review of Alg I and Chap 1 (4 Forms) “Polynomial Functions” AII.8B.ASMT Chapter 8 p. 419-425 Make up Test, Review of Alg I and Chap 1 (2 “Parabolas” AII.8C.ASMT Chapter 9 p. 499-504 Quiz 2.1, 2.3- AII.8D.ASMT “Classes of Functions” Test 2 (1-5) and Lesson 8.1- AII.8E.ASMT Chapter 10 p. 523-530 “Exponential Functions” Noteables p. 26-37 p. 41-44 p. 101-102 p. 126-131 p. 140-141 p. 152-154 p. 181-183 p. 213-214 p. 226-228 Forms) MATHEMATICS Mathematics Curriculum---Hanover County Public Schools Revised Summer 2005 Number and Number Sense Probability & Statistics Expressions Equations and Inequalities Systems Relations & Functions Virginia Standard: AII.9 The student will find the domain, range, zeros and inverse of a function, the value of a function for a given element in its domain, and the composition of multiple functions. Functions will include exponential, logarithmic, and those that have domains and ranges that are limited those that have domains and ranges that are limited and/or discontinuous. The Graphing calculator will be used as a tool to assist in investigation of functions, including exponential and logarithmic. Hanover Objectives: same Curriculum Framework: Essential Understandings • Functions describe the relationship between two variables. Essential Knowledge and Skills • Identify the domain, range, zeros, and inverse of a function presented algebraically or graphically. • Graphs of functions that are inverses of each other are reflections across the line y = x. • Distinguish between relations and functions that are expressed algebraically and graphically. • The composition of a function and its inverse is the identity function. • Recognize restricted/discontinuous domains and ranges. • Functions arise from practical situations. • If (a, b) is an element of a function, then (b, a) is an element of the inverse of the function. • Use interchange of variables to find the inverse of a function. • Given the graphs, recognize that exponential and logarithmic functions are inverses of each other. • Find the composition of two functions. • Find the value of a function for a given element from the MATHEMATICS Mathematics Curriculum---Hanover County Public Schools Revised Summer 2005 domain. • Investigate exponential and logarithmic functions, using the graphing calculator. Resources: Textbook Correlation Glencoe Algebra II Chapter 2 p. 56-62 “Relations and Functions” Chapter 6 p. 286-293 “Graphing Quadratic Functions” Chapter 7 p. 383-394 “Operations on Functions” “Inverse Functions and Relations” Chapter 9 p. 499-504 “Classes of Functions” Noteables p. 26-29 p. 126-128 p. 167-170 p. 213-214 Activities Worksheets: Domain and Range (Instructional Activity AII.9A) WS – Section 5-4 – AII.9B WS – Section 5-4 – AII.9C WS – Section 5-4 – AII.9D Relations and Functions (Packet) AII.9E Assessment Sample: Restrict your teaching to graphing logs on the calculator (example y = log x y = log 2x y= 2 log x.) Teach how to use the “ log button” on the calculator. Emphasize domain and range on every graphing problem. MATHEMATICS Mathematics Curriculum---Hanover County Public Schools Revised Summer 2005 Number and Number Sense Probability & Statistics Expressions Equations and Inequalities Systems Relations & Functions Virginia Standard: AII.10 The student will investigate and describe the relationships between the solution of an equation, zero of a function, x-intercept of a graph, and factors of a polynomial expression through the use of graphs. Hanover objectives: same Curriculum Framework: Essential Understandings Essential Knowledge and Skills • The Fundamental Theorem of Algebra states that, including complex and repeated solutions, an nth degree polynomial equation has exactly n roots (solutions). • Identify the x-intercept(s) of a graph. • Identify the zero(s) of a function, given a graph. • The following statements are equivalent: k is a zero of the polynomial function f; − (x – k) is a factor of f(x); − k is a solution of the polynomial equation f(x) = 0; and − k is an x-intercept for the graph of the polynomial. • Determine the linear factors of a polynomial expression when the zeros of the corresponding polynomial function are displayed on a graph. Resources: MATHEMATICS Mathematics Curriculum---Hanover County Public Schools Revised Summer 2005 Textbook Correlation Activities Glencoe Algebra II Worksheets: Chapter 2 p. 63-80 Relationships and Graphs (Packet) AII.10A “Linear Equations” “Slope” “Writing Linear Equations” Chapter 5 p. 239-244 “Factoring Polynomials” Chapter 6 p. 294-305 “Solving Quadratic Equations by Graphing” “Solving Quadratic Equations by Factoring” Chapter 7 p. 346-352 “Polynomial Functions” Noteables p. 30-37 p. 105-106 p. 129-133 p. 152-154 MATHEMATICS Mathematics Curriculum---Hanover County Public Schools Revised Summer 2005 Number and Number Sense Probability & Statistics Expressions Equations and Inequalities Systems Relations & Functions Virginia Standard: AII. 11 The student will use matrix multiplication to solve practical problems. Graphing calculators or computer programs with matrix capabilities will be used to find the product. Hanover Objective(s): same Curriculum Framework: Essential Understandings • Matrices can be used to model and solve practical problems. Essential Knowledge and Skills • Recognize matrices that can be multiplied. • Perform matrix multiplication with a graphing calculator or a computer program with matrix capabilities. • Resources: Use matrix multiplication to solve practical problems. MATHEMATICS Mathematics Curriculum---Hanover County Public Schools Revised Summer 2005 Textbook Correlation Activities Glencoe Algebra II Worksheets: Chapter 4 p. 154-174, 182-188, 195-207 Assignment Sheet for Chapter 3 (2 pages) AII.11A “Introduction to Matrices” “Operations with Matrices” Assessment: “Multiplying Matrices” Test 3.1-3.3, 3.6 (2 Forms)-No Calculator-AII.11A.ASMT “Determinants” Quiz 3.1-3.3 AII.11B.ASMT “Identity and Inverse Matrices” Test Chap 3 (w/calc) AII.11C.ASMT “Using Matrices to Solve Systems of Equations” Make Up Test 3 (w/calc) AII.11D.ASMT Noteables p. 72-79 p. 83-84 p. 87-90 MATHEMATICS Mathematics Curriculum---Hanover County Public Schools Revised Summer 2005 Number and Number Sense Probability & Statistics Expression Equations and Inequalities Systems Relations & Functions Virginia Standard: AII.12 The student will represent problem situations with a system of linear equations and solve the system using the inverse matrix method. Graphing calculators of computer programs with matrix capability will be used to perform computations. Hanover Objective(s): same Curriculum Framework: Essential Understandings • Matrices are a convenient shorthand for solving systems of equations. Essential Knowledge and Skills • Model problems with a system of no more than three linear equations. • Matrices can model a variety of linear systems. • Represent a system of no more than three linear equations in matrix form. • Solutions of a linear system are values that satisfy every equation in the system. Resources: • Solve a matrix equation using a graphing calculator or computer program with matrix capability. • Find the inverse of a matrix with a graphing calculator. • Express a system of linear equations as a matrix equation. MATHEMATICS Mathematics Curriculum---Hanover County Public Schools Revised Summer 2005 Textbook Correlation Activities Glencoe Algebra II Chapter 3 p. 110-122 “Solving Systems of Equations by Graphing” “Solving Systems of Equations Algebraically” Noteables p. 52-57 “Linear Systems”, Larson, p. 207-208. Systems of Equations & Inequalities – AII.12A Matrices for Problems in Manufacturing – AII.12B Using Matrices to Organize Data – AII.12C MATHEMATICS Mathematics Curriculum---Hanover County Public Schools Revised Summer 2005 Number and Number Sense Probability & Statistics Expression Equations and Inequalities Systems Relations & Functions Virginia Standard: AII.13 The student will solve systems of linear inequalities and linear programming problems and describe the results both orally and in writing. A graphing calculator will be used to facilitate solutions to linear programming problems. Hanover Objective(s): same Curriculum Framework: Essential Understandings • Linear programming models an optimization process. • A linear programming model consists of a system of constraints and an objective quantity that can be maximized or minimized. • Any maximum or minimum value will occur at a corner point of a feasible region. Resources: Essential Knowledge and Skills • Model practical problems with systems of linear inequalities. • Solve systems of linear inequalities. • Identify the feasibility region of a system of linear inequalities with no more than five constraints. • Identify the coordinates of the corner points of a feasibility region. • Find the maximum or minimum value for the function defined over the feasibility region. • Describe the meaning of the maximum or minimum value. MATHEMATICS Mathematics Curriculum---Hanover County Public Schools Revised Summer 2005 Textbook Correlation Activities Algebra II Activity (Instructional Activity Glencoe Algebra II Chapter 3 p. 123-135 AII.13A) “Solving Systems of Inequalities by Graphing” Linear Programming – AII.13B “Linear Programming” Graphing Systems – AII.13C Noteables p. 58-63 UCSMP: Systems of Linear Inequalities, Section 5.7 Linear Programming I, Section 5.8 Linear Programming II, Section 5.9 “Concepts of Linear Programming”, Bellman, p. 300-304, 313, 561. MATHEMATICS Mathematics Curriculum---Hanover County Public Schools Revised Summer 2005 Number and Number Sense Probability & Statistics Expression Equations and Inequalities Systems Relations & Functions Virginia Standard: AII. 14 The student will solve nonlinear systems of equations, including linear-quadratic and quadratic-quadratic, algebraically and graphically. The graphing calculator will be used as a tool to visualize graphs and predict the number of solutions. Hanover Objective(s): same Curriculum Framework: Essential Understandings • Solutions of a nonlinear system of equations are numerical values that satisfy every equation in the system. Essential Knowledge and Skills • Identify nonlinear systems of equations as linear-quadratic or quadratic-quadratic. • The coordinates of points of intersection in nonlinear systems are solutions to the system. • Visualize a nonlinear system of two equations, and predict the number of solutions, using the graphing calculator. Resources: • Solve a linear-quadratic system of two equations algebraically and graphically. • Solve a quadratic-quadratic system of two equations algebraically and graphically. MATHEMATICS Mathematics Curriculum---Hanover County Public Schools Revised Summer 2005 Textbook Correlation Activities “Nonlinear Systems” , Sobel, p. 348-349, 352-353. Glencoe Algebra II Chapter 8 p. 455-460 Systems – AII.14A “Solving Quadratic Systems” UCSMP: Noteables Quadratic-Linear Systems, Section 12.9 p. 194-196 Quadratic-Quadratic Systems, Section 12.10 “Linear-Quadratic Systems”, Sobel, p. 346-349. “Quadratic Systems”, Sobel, p. 350-353. “Systems with Nonlinear Equations”, Bellman, p. 305-308. MATHEMATICS Mathematics Curriculum---Hanover County Public Schools Revised Summer 2005 Number and Number Sense Probability & Statistics Expression Equations and Inequalities Systems Relations & Functions Virginia Standard: AII. 15 The student will recognize the general shape of polynomial, exponential, and logarithmic functions. The graphing calculator will be used as a tool to investigate the shape and behavior of polynomial functions. Hanover Objective(s): same Curriculum Framework: Essential Understandings • Shapes and behavior of graphs of polynomials can be determined by analyzing transformations of parent functions. Essential Knowledge and Skills • Investigate the shape and behavior of linear, quadratic, and cubic functions. Behaviors will include intercepts, number of turning points, and end behavior. • Using graphing calculators is a strategy for investigating the shape and behavior of polynomial functions. • Investigate the shape and behavior of exponential (ax = y) and logarithmic (log b x = y) functions, including intercepts and end behavior. • Using the general shape of the graph of a function, identify the family of graphs to which a particular graph belongs. Characteristics of a graph may include the xand y-intercepts, number and location of turning points, and end behaviors. • The Fundamental Theorem of Algebra (Carl Fredrich Gauss) states that in the complex number system, an nth degree polynomial equation has n zeros. • Exponential and logarithmic functions are either strictly increasing or strictly decreasing. Resources: MATHEMATICS Mathematics Curriculum---Hanover County Public Schools Revised Summer 2005 Textbook Correlation Activities Circle (Instructional Activity AII.15A) Glencoe Algebra II Chapter 6 p. 286-293, 322-328 Ellipses (Instructional Activity AII.15B) “Graphing Quadratic Functions” Hyperbolas (Instructional Activity AII.15C) “Analyzing Graphs of Quadratic Functions” Parabolas (Instructional Activity AII.15D) Chapter 9 p. 499-504 Projectile Motion (Instructional Activity AII.15E) “Classes of Functions” “Identifying Conics” , Kenneth J. Travers, Leroy C. Dalton and Chapter 10 p. 523-538 Vincent F. Brunner, Using Advanced Algebra, Laidlaw Brothers, River Forest, Illinois, 1975, p. 335. “Exponential Functions” “Logarithms and Logarithmic Functions” UCSMP: Noteables Polynomial Models, Section 11.1 p. 126-128 Functions, Chapter 7 p. 140-141 “The Factor Theorem”, Section 11.4 p. 213-214 “Quadratic-Linear Systems”, Section 12.9 p. 226-230 “Quadratic-Quadratic Systems”, Section 12.10 MATHEMATICS Mathematics Curriculum---Hanover County Public Schools Revised Summer 2005 Number and Number Sense Probability & Statistics Expression Equations and Inequalities Systems Relations & Functions Virginia Standard: AII. 16 The student will investigate and apply the properties of arithmetic and geometric sequences and series to solve problems, including writing the first n terms, finding the nth term, and evaluating summation formulas. Notation will include Σ and an. Hanover Objective(s): same Curriculum Framework: Essential Understandings • Sequences and series arise from practical situations. • The study of sequences and series is an application of investigation of patterns. Essential Knowledge and Skills • Distinguish between a sequence and a series. • Recognize patterns in a sequence. • Distinguish between arithmetic and geometric sequences. • Use and interpret the notations ∑, n, nth term, and an. • Write the first n terms in an arithmetic or geometric sequence. • Given the formula, find an (the nth term) for an arithmetic or a geometric sequence. • Resources: Given formulas, find the sum, Sn, of the first n terms of an arithmetic or geometric series, including infinite series. MATHEMATICS Mathematics Curriculum---Hanover County Public Schools Revised Summer 2005 Textbook Correlation Activities Seating Capacity, Roland E. Larson, Timothy Glencoe Algebra II Chapter 11 p. 578-598 D. Kanold, and Lee Stiff, Algebra 2, D.C. “Arithmetic Sequences” Heath and Company, Lesington, MA: 1993, “Arithmetic Series” p. 632. “Geometric Sequences” Sharing Ideas Larson, p. 640. “Geometric Series” Sharing Ideas Larson, p. 641. Zooming Inn Larson, p. 649. Noteables p. 248-256 UCSMP: Explicit Formulas for Sequences, Section 1-3 Recursive Formulas for Sequences, Section 1-4 Arithmetic Sequences: Explicit Formulas, Section 3.6 Arithmetic Sequences: Recursive Formulas Section 3.7 Geometric Sequences, Section 8.3 Series, Combinations, and Statistics, Sections 13.1-13.4 MATHEMATICS Mathematics Curriculum---Hanover County Public Schools Revised Summer 2005 Number and Number Sense Probability & Statistics Expression Equations and Inequalities Systems Relations & Functions Virginia Standard: AII.17 The student will perform operations on complex numbers and express the results in simplest form. Simplifying results will involve using patterns of the powers of i. Hanover Objective(s): same Curriculum Framework: Essential Understandings • Complex numbers are a superset of real numbers. Essential Knowledge and Skills • Recognize that the square root of –1 is represented as i. • Define and identify a complex number. • Apply the definition of i to simplify square roots of negative numbers. • Simplify powers of i. • Textbook Correlation Glencoe Algebra II Chapter 5 p. 270-275 “Complex Numbers” Noteables p. 116-118 Add, subtract, and multiply complex numbers. Activities Simplifying Expressions Containing Complex Numbers – AII.17A UCSMP: The Imaginary Number i, Section 6..8 Complex Numbers, Section 6.9 “Complex Numbers”, Sobel, p. 296-300. Number and Number Sense MATHEMATICS Mathematics Curriculum---Hanover County Public Schools Revised Summer 2005 Probability & Expression Equations and Systems Statistics Inequalities Relations & Functions Virginia Standard: AII.18 The student will identify conic sections (circle, ellipse, parablola, and hyperbola) from his/her equations. Given the equations in (h, k) form, students will sketch graphs of conic sections, using transformations. Hanover Objective(s): same Curriculum Framework: Essential Understandings • Changing parameters (h, k) affects the graph of a conic section in a predictable pattern. • Conic sections can be distinguished by their equations. Essential Knowledge and Skills • Identify types of conic sections, given (h, k) form of an equation. • Identify types of conic sections from a graph. • Sketch the graph of a conic section in (h, k) form, using knowledge of transformations. Resources: Textbook Correlation Glencoe Algebra II Chapter 8 p. 419-452 “Parabolas”’ “Circles” “Ellipses” “Hyperbolas” “Conic Sections” Noteables p. 181-193 Activities Golden Gate Bridge, Ronald E. Larson, Timothy D. Kanold, and Lee Stiff, Algebra 2, D.C. Heath and Company, Lesington, MA: 1993, p.241. Sleepy Hare Restaurant Larson, p. 275 Gateway Arch Larson, p. 249. Circles, [Instructional Activity AII.15A] Ellipses, [Instructional Activity AII.15B] Hyperbolas, [Instructional Activity AII.15C] Parabolas, [Instructional Activity AII.15D] UCSMP: Parabolas and Quadratic Equations, Chapter 6 (omit 6.3) Quadratic Relations, Chapter 12, Sections 12.1 – 12.8 MATHEMATICS Mathematics Curriculum---Hanover County Public Schools Revised Summer 2005 Number and Number Sense Probability & Statistics Expression Equations and Inequalities Systems Relations & Functions Virginia Standard: AII.19 The student will collect and analyze data to make predictions, write equations, and solve practical problems. Graphing calculators will be used to investigate scatter plots to determine the equation for a curve of best fit. Models will include linear, quadratic, exponential, and logarithmic functions. Hanover Objective(s): same Curriculum Framework: Essential Understandings • Data and scatterplots may indicate patterns that can be modeled with an algebraic equation. • Graphing calculators can be used to collect, organize, picture, and create an algebraic model of the data. • Data that fit linear, quadratic, exponential, and logarithmic models arise from practical situations. Essential Knowledge and Skills • Collect and analyze data. • Investigate scatterplots to determine if patterns exist, and then identify the patterns. • Find an equation for the curve of best fit for data, using a graphing calculator. Models will include linear, quadratic, exponential, and logarithmic functions. • Make predictions, using data, scatterplots, or curve of best fit. • Given a set of data, determine the model that would best describe the data. MATHEMATICS Mathematics Curriculum---Hanover County Public Schools Revised Summer 2005 Resources: Textbook Correlation Glencoe Algebra II Chapter 2 p. 87 Graphing Calculator Investigation Chapter 6 p. 300 Graphing Calculator Investigation Chapter 7 p. 359 Graphing Calculator Investigation Essential Knowledge and Skills Pulse Rate and Child’s Age (Instructional Activity AII.19A) “Scatter Plots”, Rubenstein, p. 39-40 “Scatter Plots II” , Rubenstein, p. 13. TI-82/83 Procedure 23: Fining Best-Fit Models – AII.19B Scatter plot – AII.19C Scatter plot and Line of Best Fit – AII.19D UCSMP: Fitting a Model to Data I, Section 2.8 Fitting a Model to Data II, Section 2.9 MATHEMATICS Mathematics Curriculum---Hanover County Public Schools Revised Summer 2005 Number and Number Sense Probability & Statistics Expression Equations and Inequalities Systems Relations & Functions Virginia Standard: AII.20 The student will identify, create, and solve practical problems involving inverse variation and a combination of direct and inverse variations. Hanover Objective(s): same Curriculum Framework: Essential Understandings • Practical problems can be modeled and solved by using direct and/or inverse variations. • Joint variation is a combination of direct variations. Essential Knowledge and Skills • Translate “y is directly proportional to x” as y = kx. k • Translate “y is inversely proportional to x” as y = x . • Translate “y varies jointly as x and z” as y = kxz. • Determine the value of the constant of proportionality, k, given initial conditions for x and y. • Set up and solve practical problems, using combinations of direct and inverse variation. MATHEMATICS Mathematics Curriculum---Hanover County Public Schools Revised Summer 2005 Resources: Textbook Correlation Glencoe Algebra II Chapter 9 p. 492-504 “Direct, Joint, and Inverse Variation” “Classes of Functions” Noteables: p. 211-214 Activities Distance to a Thunderstorm (Instructional Activity AII.20A) Temperature Determines Chirps (Instructional Activity AII.20B) Direct, Inverse, and Joint Variations – AII.20C Direct, Inverse, and Joint Variations – AII.20D Direct Variation – AII.20E Inverse Variation – AII.20F Variation Review – AII.20G UCSMP: Variation and Graphs, Chapter 2.1-2.6
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