High School. Algebra 2 Curriculum

Transcription

1 Bethlehem Area School District High School Algebra 2 Curriculum Committee Members Shaunna Knepp, Liberty Barbara Kitchell, Freedom 2008 Julie Victory, Supervisor of Mathematics

2 Name of Course: Honors Algebra 2, Algebra 2, Standards Algebra 2 Grade Level(s): 10 or 11 Unit 1: Numbers and Operations Estimated Instructional Time: 20 days PA Academic Standards: Numbers, Number Systems and Number Relationships Computation and Estimation Mathematical Reasoning and Connections Mathematical Problem Solving and Communication Unit (Strand) Objectives: 1-1 Use the set of real numbers for computation and in problem solving settings 1-2 Represent and use numbers in equivalent forms 1-3 Use estimation 1-4 Solve routine and non-routine problems 1-5 Use communication, including mathematical vocabulary, appropriate to numbers and operations Enabling Objectives (Explore, Develop, Master): (Master unless otherwise indicated. PSSA Eligible Content is indicated where relevant.) From Algebra 1 (continue to master do not spend much time reteaching) Calculate with integers and rational numbers (fractions and decimals, both positive and negative) Solve problems using operations with rational numbers including rates and percents (M11.A.2.1.1) Find the Greatest Common Factor or the Least Common Multiple for sets of numbers Find the square root of an integer to the nearest tenth using a calculator or estimation (M11.A.1.1.1) Express numbers and simplify expressions using scientific notation( M11.A.1.1.2) Simplify square roots of numbers (M11.A.1.1.3) Locate irrational numbers at the approximate location on a number line (M11.A.1.3.1) Compare and order natural, whole, integer. rational and irrational numbers (M11A.1.3.2) Solve problems using direct and inverse proportions (M11.A.2.1.2) Use proportional relationships in problem solving settings M11.A.2.1.3) Simplify or evaluate expressions involving positive and negative exponents (M11.A.2.2.1) Simplify or evaluate expressions involving roots and absolute value (M11.A.2.2.1) Use order of operations to simplify a numerical expression (M11.A.3.1.1) Use order of operations to evaluate an algebraic expression Explore simplifying expressions with negative exponents (M11.A.2.2.1) Explore simplifying expressions with zero exponent Use the property of multiplying powers with the same base to rewrite and simplify expressions (M11.A.2.2.2) Use the property of dividing powers with the same base to rewrite and simplify expressions Use the property of raising a power to a power to rewrite and simplify expressions (M11.A.2.2.2) Use the property of raising a product to a power to rewrite and simplify expressions (M11.A.2.2.2) Use rounding or front-end estimation to check for reasonableness of results Unit 1--Numbers and Operations Page 1

3 Use estimation to solve problems (M11.A.3.2.1) Use the properties of exponents to simplify and rewrite expressions Determine and use the absolute value of a number (M11.A.2.2.1) Vocabulary: absolute value, algebraic expression, base, equation, evaluate, exponent, function, inequality, integer, irrational number, natural number, order of operations, power, rational numbers, real numbers, simplify, variable, whole numbers All Algebra 2 (Honors, Algebra 2, Standards) Calculate with real numbers Identify and use properties of real numbers (closure, commutative, associative, identity, inverse, distributive) Compare and order any real numbers (M11A.1.3.2) Simplify or evaluate expressions involving positive and negative exponents, roots, or absolute value, with all types of real numbers (M11.A.2.2.1) Use order of operations to evaluate an algebraic expression Use estimation to solve problems (M11.A.3.2.1) Algebra 2 and Honors only Identify matrices and their elements Organize data into matrices Add or subtract matrices Multiply a matrix by a scalar Multiply 2 matrices Identify subsets of complex numbers Represent complex numbers in the form a+bi Simplify powers of pure imaginary numbers Perform operations on the set of complex numbers Develop and Master simplifying expressions with negative exponents (M11.A.2.2.1) Develop and Master simplifying expressions with zero exponent Develop and Master simplifying expressions with rational exponents Develop and Master adding and subtracting radical expressions Develop and Master multiplying and dividing binomial radical expressions Vocabulary: additive inverse, multiplicative inverse, complex number, imaginary number, real number, irrational number, rational number, integer, whole number, natural number, matrix, matrix element, dimension Text/Resource Materials: Prentice Hall Mathematics, Algebra 2, 2007 Chapters 1-1, 1-2, 4, 5-6, 7-1, 7-2, 7-3, 7-4 Unit 1--Numbers and Operations Page 2

4 Assessment: Lesson Quiz Web Codes: There is an online quiz for every lesson in the textbook. Access these quizzes with Web Codes aga-0101 through aga Prefix of aga then first 2 digits indicate chapter; next 2 digits indicate lesson number. Vocabulary Quiz Web Codes: Access these quizzes with Web Codes agj-0151 through agj Prefix of agj then first 2 digits indicate chapter; next 2 digits are always 51 Chapter Test Web Codes: Access the chapter tests with Web Codes aga-0152 through aga Prefix of aga then first 2 digits indicate chapter; next 2 digits are always 52. Teacher-made assessments (tests, quizzes, projects) Common district final exam Extensions: Math Careers: miniaturist, landscape architect use Web Code agb-2031 Remediation: Video Tutor Help Web Codes: Instructional videos to help with math concepts can be accessed by using web code age Homework Video Tutor Web Codes: For every lesson in the textbook, there is additional support online for homework help. Access this site with Web Codes age-0101 through age Activities for reading the mathematics textbook including (1) previewing text and (2) previewing a lesson or chapter Activities for effective note taking strategies; learn to use graphic organizers Activities for understanding math vocabulary, including appropriate abbreviations Technology activities Use guided problem solving in PH text for scaffolded support Use Skills Handbook in back of PH text: Percents and Percent Applications, Operations With Fractions, Ratios and Proportions, Simplifying Expressions With Integers, Operations With Exponents, Scientific Notation and Significant Digits, Unit 1--Numbers and Operations Page 3

5 Operations With Radicals Review Problem Solving Strategies: Draw a Diagram; Try, Check, Revise; Look For A Pattern; Make a Table; Solve a Simpler Problem; Use Logical Reasoning; Work Backward Special Education See Appendix I Guided Problem Solving ESOL Three global strategies are essential for supporting the learning of English Language Learners in the mathematics classroom: 1 Creating and Maintaining Access to Lessons Draw students attention to related concepts, skills, and situations that they already understand. Form context for subsequent instruction and conversation by establishing schedules and routines, including an initial announcement of what topic will be studied, what students will do, and how they will be grouped during a lesson. Use cooperative group and partner work. 2 Using Visual Tools Assist students in generating nonlinguistic representations for new knowledge. Use pictures, objects, chalkboard drawings with accompanying text, graphic organizers, posters, manipulative materials, and mathematical diagrams. 3 Providing Language Support Realize that all students can benefit from listening to, and thinking about, a wide variety of questions. Model problem-solving by thinking aloud. Understanding of specific mathematical vocabulary is enhanced by purposeful instruction. Word walls and student glossaries visibly display important math vocabulary. Post graphic organizers to serve as reminders of new concepts and skills. Provide purposeful instruction in understanding mathematical symbols: reading, or understanding the concepts represented by symbols; verbalizing, or translating symbols into spoken language; and, writing symbols, or producing symbols to represent a mathematical relationship. Encourage analytical thinking by having students discuss why incorrect answers are incorrect. Unit 1--Numbers and Operations Page 4

6 See Appendix II: Instructional Adaptations For the ESOL Student in the Regular Education Classroom Use additional activities for understanding mathematics vocabulary (Vocabulary and Study Skills), adapted chapter tests, additional worksheets for reading and writing math symbols. Unit 1--Numbers and Operations Page 5

7 Name of Course: Honors Algebra 2, Algebra 2, Standards Algebra 2 Grade Level(s): 10 or 11 Unit 2: Skills and Concepts of Algebra Estimated Instructional Time: 25 days PA Academic Standards: Mathematical Reasoning and Connections Mathematical Problem Solving and Communication Algebra and Functions Unit (Strand) Objectives: 2-1 Define and use variables, parameters, constants, formulas, expressions, equations, and inequalities 2-2 Solve equations symbolically, graphically, and numerically 2-3 Solve routine and non-routine problems using skills and concepts of algebra 2-4 Use communication, including mathematical vocabulary, appropriate to algebra skills and concepts Enabling Objectives (Explore, Develop, Master): (Master unless otherwise indicated. PSSA Eligible Content is indicated where relevant.) From Algebra 1 (continue to master) Use variables to write algebraic expressions and equations Find the Greatest Common Factor (GCF) or the Least Common Multiple (LCM) for sets of monomials (M11.A.1.2.1) Analyze a set of data for the existence of a pattern (M11.D.1.1.1) Represent a pattern algebraically (M11.D.1.1.1) Determine if a relation is a function (M11.D.1.1.2) Graph linear equations M11.D.2.1.3) Write, solve and apply linear equations (M11.D.2.1.3) Explore solving a system of equations by graphing (M11.D.2.1.4) Add, subtract and multiply polynomial expressions (M11.D.2.2.1) Factor a monomial from a polynomial Simplify algebraic fractions (M11.D.2.2.3) Find the square of a binomial Find the difference of squares (M11.D.2.2.2) Factor trinomials of the type x2+bx+c (leading coefficient=1) (M11.D.2.2.2) Find constant rates of change using tables or graphs (M11.D.3.1.1) Determine how a change in one variable relates to a change in a second variable (M11.D.3.1.2) Find slope of a line using a graph (M11.D.3.2.3) Find slope of a line from its equation (M11.D.3.2.3) Find y-intercept of a line using a graph (M11.D.3.2.3) Find y-intercept of a line from its equation (M11.D.3.2.3) Unit 2--Skills & Concepts of Algebra Page 1

8 Write an equation of a line from its graph Match the graph of a linear function to its table or equation (M11.D.4.1.1) Find slope of a line using formula and coordinates (M11.D.3.2.1) Graph linear equations on a coordinate plane (M11.D.2.1.2) Write a linear equation in slope-intercept form (y=mx+b) (M11.D.3.2.2) Write a linear equation in standard form (Ax+By=C) (M11.D.3.2.2) Graph and write a linear equation using point-slope form (y-y1)=m(x-x1) (M11.D.3.2.2) Graph a line using x- and y-intercepts Write equations for graphs of horizontal (y=k) and vertical (x=k) lines Distinguish between slope of 0 and undefined slope Use 2 points to write an equation for a line Graph and write inequalities in 1 variable; use open dot/closed dot notation Write an inequality in 1 variable from its graph Describe situations that can be represented using specific inequalities Use addition and subtraction to solve inequalities in 1 variable Use multiplication and division to solve inequalities in 1 variable Solve compound inequalities and express solution graphically and symbolically (M11.D.2.1.1) Explore solving equations that involve absolute value Explore solving inequalities that involve absolute value Vocabulary: binomial, coefficient, constant, degree of a polynomial, Distributive Property, factoring, like terms, equivalent equations, linear equation, literal equation, inequality, compound inequality, monomial, negative reciprocal, perfect square, polynomial, point-slope form, principal square root, proportion, rate, ratio, reciprocal, relation, slope, slope-intercept form, solution, standard form, system of equations, term, trinomial, x-intercept, y-intercept All Algebra 2 (Honors, Algebra 2, Standards) Determine if a relation is a function (M11.D.1.1.2) Identify the domain, range, and inverse of a relation (M11.D.1.1.3) Graph linear equations (M11.D.2.1.3) Write and interpret direct variation equations (M11.D.3.1.2) Write, solve and apply linear equations (M11.D.2.1.3) Write linear equations that model real-world data and make predictions (M11.E.4.2.2) Identify quadratic functions and graphs (M11.D.4.1.1) Model data with quadratic functions (M11.D.4.1.1) Factor algebraic expressions, including difference of squares and trinomials of the form ax 2 +bx+c where a!0 (M11.A.1.1.3) Solve quadratic equations using factoring (M11.D.2.1.5) Solve quadratic equations using the quadratic formula Use the discriminant (D=b 2-4ac) to determine the nature of the solutions of the equation ax 2 +bx+c=0 Determine key characteristics (e.g. domain, range, vertex, min/max, intercepts, axis of symmetry, end behavior) of quadratic functions and their graphs(m11.d.4.1.1) Graph functions or linear inequalities on a coordinate plane (M11.D.2.1.2) Describe the effect that changes in the parameters of a quadratic function have on the shape and position of its graph Solve compound inequalities and graph their solution sets (M11.D.2.1.1) Graph absolute value functions Unit 2--Skills & Concepts of Algebra Page 2

9 Develop and Master solving equations that involve absolute value Solve and graph absolute value inequalities Describe translations of functions and draw their graphs Develop and Master solving a system of equations by graphing (M11.D.2.1.4) Solve a system of equations by substitution or elimination (M11.D.2.1.4) Classify systems of equations as independent, dependent, or inconsistent Solve systems of linear inequalities Algebra 2 and Honors only Use the vertex form of the quadratic function Graph piecewise functions Write a piecewise function to represent a graph Represent piecewise-defined functions using tables, graphs, verbal statements, and equations. Graph absolute value functions (M11.D.4.1.1) Analyze translations, sketches, shrinks, and reflections of absolute value graphs (M11.D.4.1.1) Honors only Solve equations by completing the square Solve problems with linear programming Graph points and equations in 3 dimensions Solve systems of equations in 3 variables by elimination or substitution Graph linear and absolute value inequalities Vocabulary: binomial, coefficient, constant, degree of a polynomial, Distributive Property, factoring, like terms, equivalent equations, linear equation, literal equation, inequality, compound inequality, monomial, negative reciprocal, perfect square, polynomial, point-slope form, principal square root, proportion, rate, ratio, reciprocal, relation, slope, slope-intercept form, solution, standard form, system of equations, term, trinomial, x-intercept, y-intercept, parent function, translations Text/Resource Materials: Prentice Hall Mathematics, Algebra Chapters 1-3, 1-4, 1-5, 2, 3 Assessment: Lesson Quiz Web Codes: There is an online quiz for every lesson in the textbook. Access these quizzes with Web Codes aga-0101 through aga Prefix of aga then first 2 digits indicate chapter; next 2 digits indicate lesson number. Vocabulary Quiz Web Codes: Access these quizzes with Web Codes agj-0151 through agj Prefix of agj then first 2 digits indicate chapter; next 2 digits are always 51 Unit 2--Skills & Concepts of Algebra Page 3

10 Chapter Test Web Codes: Access the chapter tests with Web Codes aga-0152 through aga Prefix of aga then first 2 digits indicate chapter; next 2 digits are always 52. Teacher-made assessments (tests, quizzes, projects) Common district final exam for Algebra 2 Extensions: Math Careers: miniaturist, radiologist use Web Code agb-2031 Remediation: Video Tutor Help Web Codes: Instructional videos to help with math concepts can be accessed by using web code age Homework Video Tutor Web Codes: For every lesson in the textbook, there is additional support online for homework help. Access this site with Web Codes age-0101 through age Activities for reading the mathematics textbook including (1) previewing text and (2) previewing a lesson or chapter Activities for effective note taking strategies; learn to use graphic organizers Activities for understanding math vocabulary, including appropriate abbreviations Technology activities Use guided problem solving in PH text for scaffolded support Use Skills Handbook in back of PH text: Evaluating Formulas and Solving Literal Equations, The Coordinate Plane, Slope, and Midpoint, Solving Linear Equations and Inequalities, Absolute value Equations and Inequalities, Graphing Two-Variable Equations and Inequalities, Factoring and Operations with Polynomials Review Problem Solving Strategies: Draw a Diagram; Try, Check, Revise; Look For A Pattern; Make a Table; Solve a Simpler Problem; Use Logical Reasoning; Work Backward Unit 2--Skills & Concepts of Algebra Page 4

11 Special Education See Appendix I Guided Problem Solving Activity Lab: Finding a Line of Best Fit (Technology) Activity Lab: Solving Systems Using Tables ESOL See Appendix II: Instructional Adaptations For the ESOL Student in the Regular Education Classroom Additional activities for understanding mathematics vocabulary (Vocabulary and Study Skills) Adapted chapter tests Additional worksheets for reading and writing math symbols Unit 2--Skills & Concepts of Algebra Page 5

12 Name of Course: Honors Algebra 2, Algebra 2, Standards Algebra 2 Grade Level(s): 10 or 11 Unit 3: Geometry and Measurement Estimated Instructional Time: 6 days PA Academic Standards: Numbers, Number Systems and Number Relationships Computation and Estimation Measurement and Estimation Mathematical Reasoning and Connections Mathematical Problem Solving and Communication Geometry Unit (Strand) Objectives: 3-1 Use symbolic algebra to represent geometric relationships 3-2 Solve routine and non-routine problems using skills and concepts of geometry 3-3 Use communication, including mathematical vocabulary, appropriate to geometry Enabling Objectives (Explore, Develop, Master): (Master unless otherwise indicated. PSSA Eligible Content is indicated where relevant.) From Algebra 1 (continue to master) Use geometric models with proportions and percent Find ratios Solve proportions Find the measure of a side of a right triangle using the Pythagorean Theorem (M11.C.1.4.1) Find the measurement of a missing length of a figure given the perimeter, circumference, or area (M11.B.2.2.4) Explore calculating the distance between 2 points using the Distance Formula (M11.C.3.1.1) : PQ=!(x 2 - x 1 ) 2 +(y 2 -y 1 ) 2 Relate slope to perpendicularity and parallelism (M11.C.3.1.2) Transform geometric formulas to solve for one variable in terms of the others Describe how a change in the linear dimension of a figure affects perimeter, circumference, area, or volume (M11.B.2.3.1) Vocabulary: ratio, proportion, hypotenuse, leg, Pythagorean Theorem, literal equation, perpendicular, parallel, slope All Algebra 2 (Honors, Algebra 2, Standards) Find the measurement of a missing length of a figure given the perimeter, circumference, or area (M11.B.2.2.4) Calculate the distance between 2 points using the Distance Formula (M11.C.3.1.1) : PQ=!(x 2 -x 1 ) 2 +(y 2 -y 1 ) 2 Measure and compare angles in degrees (M11.B.2.1.1) Unit 3--Geometry & Measurement Page 1

13 Algebra 2 and Honors only Use the Pythagorean Theorem to find the length of a leg or hypotenuse when determining trigonometric ratios Transform geometric formulas to solve for one variable in terms of the others Identify cycles of periodic functions Identify periods of periodic functions Find amplitude of periodic functions Work with angles in standard position Find coordinates of points on the unit circle Use positive and negative measures of angles Sketch a positive or negative angle for a given value Use the unit circle to find sine and cosine of an angle Measure and compare angles in radian measure Find the length of an arc of a circle Use six trig ratios with right triangles Honors only (Explore) Graph sine and cosine function Solve trigonometric equations Vocabulary: amplitude, central angle, cosine, coterminal, cycle, initial side, literal equation, period, Pythagorean Theorem, radian, since, standard position, terminal side, unit circle Text/Resource Materials: Prentice Hall Mathematics, Algebra Chapters 13, 14 Assessment: Lesson Quiz Web Codes: There is an online quiz for every lesson in the textbook. Access these quizzes with Web Codes aga-0101 through aga Prefix of aga then first 2 digits indicate chapter; next 2 digits indicate lesson number. Vocabulary Quiz Web Codes: Access these quizzes with Web Codes agj-0151 through agj Prefix of agj then first 2 digits indicate chapter; next 2 digits are always 51 Chapter Test Web Codes: Access the chapter tests with Web Codes aga-0152 through aga Prefix of aga then first 2 digits indicate chapter; next 2 digits are always 52. Teacher-made assessments (tests, quizzes, projects) Common district final examination Unit 3--Geometry & Measurement Page 2

14 Extensions: Math Careers: acoustical physicist, use Web Code agb-2031 Remediation: Video Tutor Help Web Codes: Instructional videos to help with math concepts can be accessed by using web code age Homework Video Tutor Web Codes: For every lesson in the textbook, there is additional support online for homework help. Access this site with Web Codes age-0101 through age Activities for reading the mathematics textbook including (1) previewing text and (2) previewing a lesson or chapter Activities for effective note taking strategies; learn to use graphic organizers Activities for understanding math vocabulary, including appropriate abbreviations Technology activities Use guided problem solving in PH text for scaffolded support Use Skills Handbook in back of PH text: Area and Volume, The Coordinate Plane, Slope, and Midpoint, Operations With Radicals, The Pythagorean Theorem and the Distance Formula Review Problem Solving Strategies: Draw a Diagram; Try, Check, Revise; Look For A Pattern; Make a Table; Solve a Simpler Problem; Use Logical Reasoning; Work Backward Special Education See Appendix I Have each student use index cards to form a literal expression on his/her desk. (Each variable, sign, fraction bar, etc. has its own card.) Give each student a counter. Tell students to place counter on the variable to be solved for. Have students move to different desks and solve the equation at that desk by moving the index cards. For Pythagorean Theorem, use Guided Problem Solving, Reteaching, or Adapted Practice masters Unit 3--Geometry & Measurement Page 3

15 ESOL See Appendix II: Instructional Adaptations For the ESOL Student in the Regular Education Classroom Additional activities for understanding mathematics vocabulary (Vocabulary and Study Skills) Adapted chapter tests Additional worksheets for reading and writing math symbols Unit 3--Geometry & Measurement Page 4

16 Name of Course: Honors Algebra 2, Algebra 2, Standards Algebra 2 Grade Level(s): 10 or 11 Unit 4: Data Analysis and Probability Estimated Instructional Time: 7 days PA Academic Standards: Numbers, Number Systems and Number Relationships Computation and Estimation Mathematical Reasoning and Connections Mathematical Problem Solving and Communication Statistics and Data Analysis Unit (Strand) Objectives: 4-1 Use appropriate statistical methods to analyze data 4-2 Draw conclusions based on data 4-3 Solve routine and non-routine problems using skills and concepts of data analysis 4-4 Use communication, including mathematical vocabulary, appropriate to data analysis 4-5 Identifying mathematical patterns and generating formulas Enabling Objectives (Explore, Develop, Master): (Master unless otherwise noted. PSSA Eligible Content is indicated where relevant.) From Algebra 1(continue to master) Analyze a set of data for the existence of a pattern and represent the pattern algebraically or graphically (M11.D.1.1.1) Create and use appropriate graphical representations of data including scatter plots (M11.E.1.1.1) Use scatter plots to find trends in data and determine whether there is positive correlation, negative correlation, or no correlation (M11.E.1.1.2) Analyze data and answer questions based on displayed data (M11.E.1.1.2) Calculate, select or interpret the appropriate measure of central tendency (mean, median, mode) of a set of data (M11.E.2.1.1) Calculate or interpret the range, quartiles, and interquartile range of data (M11.E.2.1.2) Describe how outliers affect measures of central tendency (M11.E.2.1.3) Write an equation for a line of best fit for a scatter plot (M11.E.4.2.1) Make predictions using the equations or graphs of best-fit lines of scatter plots (M11.E.4.2.2) Graph data from a table and find an equation for a trend line of the data Use a graphing calculator to find the equation of the line of best fit for given data Vocabulary: scatter plot, positive correlation, negative correlation, no correlation, undefined correlation, trend line, measures of central tendency, mean, outlier, median, mode, range, line of best fit, correlation coefficient Unit 4--Data Analysis Page 1

17 All Algebra 2 (Honors, Algebra 2, Standards) Analyze a set of data for the existence of a pattern and represent the pattern algebraically or graphically (M11.D.1.1.1) Create and use appropriate graphical representations of data including scatter plots (M11.E.1.1.1) Use scatter plots to find trends in data and determine whether there is positive correlation, negative correlation, or no correlation (M11.E.1.1.2) Analyze data and answer questions based on displayed data (M11.E.1.1.2) Make predictions based on a circle graph, line graph, or bar graph (M11.E.4.1.1) Calculate, select or interpret the appropriate measure of central tendency (mean, median, mode) of a set of data (M11.E.2.1.1) Draw and interpret box-and-whisker plots Calculate or interpret the range, quartiles, and interquartile range of data (M11.E.2.1.2) Describe how outliers affect measures of central tendency (M11.E.2.1.3) Determine the standard deviation for a set of values Use a normal distribution Write an equation for a line of best fit for a scatter plot (M11.E.4.2.1) Make predictions using the equations or graphs of best-fit lines of scatter plots (M11.E.4.2.2) Graph data from a table and find an equation for a trend line of the data Use a graphing calculator to find the equation of the line of best fit for given data Find probabilities for independent, dependent, and compound events (M11.E.3.1.1) Find and compare odds for simple events (M11.E.3.1.2) Evaluate the number of permutations of n items of a set arranged r items at a time; n P r (M11.E.3.2.1) Evaluate the number of combinations of n items of a set chosen r items at a time; n C r (M11.E.3.2.1) Apply the Multiplication Counting Principle (M11.E.3.2.1) Use probability to predict outcomes (M11.E.4.1.2) Find the probability of A and B when A and B are independent events: P(A and B) = P(A) P(B) Find the probability of A or B when A and B are mutually exclusive events: P(A or B) = P(A) + P(B) Find the probability of A or B when A and B are not mutually exclusive events: P(A or B) = P(A) + P(B) P(A and B) Find the conditional probability of B given A: P(B A) = P(A and B)/P(A) Solve real-life problems involving probabilities and make informed decisions in practical situations Compare theoretical probabilities with the results of simple experiments Create a probability distribution to perform a simulation Identifying mathematical patterns and writing a formula to determine nth term of sequence Generate explicit formulas for arithmetic and geometric sequences and series Unit 4--Data Analysis Page 2

18 Honors only Make and use a probability distribution Find the standard deviation of a set of values and use in real-world situations Use normal distribution and normal curve Vocabulary: scatter plot, positive correlation, negative correlation, no correlation, undefined correlation, trend line, measures of central tendency, mean, outlier, median, mode, range, line of best fit, correlation coefficient, permutation, combination, Multiplication Counting Principle, n! (n factorial), frequency table, cumulative probability, probability distribution, Text/Resource Materials: Prentice Hall Mathematics, Algebra Chapters 1-6, 6-7, 9-7, 12 Assessment: Lesson Quiz Web Codes: There is an online quiz for every lesson in the textbook. Access these quizzes with Web Codes aga-0101 through aga Prefix of aga then first 2 digits indicate chapter; next 2 digits indicate lesson number. Vocabulary Quiz Web Codes: Access these quizzes with Web Codes agj-0151 through agj Prefix of agj then first 2 digits indicate chapter; next 2 digits are always 51 Chapter Test Web Codes: Access the chapter tests with Web Codes aga-0152 through aga Prefix of aga then first 2 digits indicate chapter; next 2 digits are always 52. Teacher-made assessments (tests, quizzes, projects) Common district final examination Extensions: Connection to geometry: Use data such as diameters and circumferences of cylindrical boxes. Graph data. Find equation of trend line. Obtain real data from U.S. Census Bureau Remediation: Video Tutor Help Web Codes: Instructional videos to help with math concepts can be accessed by using web code age Homework Video Tutor Web Codes: For every lesson in the textbook, there is additional support online for homework help. Access this site with Web Codes age-0101 through age Unit 4--Data Analysis Page 3

19 Activities for reading the mathematics textbook including (1) previewing text and (2) previewing a lesson or chapter Activities for effective note taking strategies; learn to use graphic organizers Activities for understanding math vocabulary, including appropriate abbreviations Technology activities Use guided problem solving in PH text for scaffolded support Use Skills Handbook in back of PH text: Bar and Circle Graphs, Descriptive Statistics and Histograms Use Table 5 Random Numbers on page 892 Review Problem Solving Strategies: Draw a Diagram; Try, Check, Revise; Look For A Pattern; Make a Table; Solve a Simpler Problem; Use Logical Reasoning; Work Backward Special Education See Appendix I ESOL See Appendix II: Instructional Adaptations For the ESOL Student in the Regular Education Classroom Additional activities for understanding mathematics vocabulary (Vocabulary and Study Skills) Adapted chapter tests Additional worksheets for reading and writing math symbols Unit 4--Data Analysis Page 4

20 Name of Course: Honors Algebra 2, Algebra 2, Standards Algebra 2 Grade Level(s): 10 or 11 Unit 5: Algebra and Functions Estimated Instructional Time: 40 days PA Academic Standards: Numbers, Number Systems and Number Relationships Computation and Estimation Mathematical Reasoning and Connections Mathematical Problem Solving and Communication Algebra and Functions Unit (Strand) Objectives: 5-1 Write and graph polynomial functions; solve polynomial equations 5-2 Solve radical equations 5-3 Graph and use exponential and logarithmic functions and solve exponential and logarithmic equations 5-4 Graph rational functions and solve rational equations 5-5 Use communication, including mathematical vocabulary, appropriate to functions Enabling Objectives (Explore, Develop, Master): (Master unless otherwise noted. PSSA Eligible Content is indicated where relevant. All topics in this unit are new.) All Algebra 2 (Honors, Algebra 2, Standards) Write polynomial in standard form and classify by degree(m11.d.4.1.1) Write a polynomial in factored form (M11.B.2.2.4) Write a polynomial function from its zeros; find the zeros of a function (M11.B.2.2.2) Use polynomial functions to solve real-world problems Write a function to model direct or inverse variation (M11.A.2.1.2) Solve real-world problems using direct or inverse variation (M11.D.3.1.2) Simplify rational expressions (M11.D.2.2.3) Add and subtract rational expressions (M11.A.1.2.1) Simplify complex fractions (M11.A.1.2.1) Find the least common multiple of a pair of polynomial expressions (M11.A.1.2.1) Multiply and divide rational expressions; state any restrictions on the variables Determine key characteristics (domain, range, intercepts, end behavior) of functions in the form f(x) = ax n, a!0, for positive integral values of n and their graphs Determine key characteristics of absolute value functions, of step and other piecewise functions, of polynomial functions, of simple rational functions, and their graphs Unit 5 Algebra and Functions Page 1

21 Algebra 2 and Honors only Divide polynomials using long division Use the Remainder Theorem Divide polynomials using synthetic division (explore and develop) Find all real number roots Simplify radical expressions Multiply and divide radical expressions Rationalize the denominator Add and subtract radical expressions Simplify expressions with rational exponents Convert to and from radical form Use properties of rational exponents to simplify expressions Solve square root equations Check for extraneous solutions to an equation Find the domain of any function Determine the inverse of a relation Determine the inverse of a function Describe the conditions under which an inverse relation is a function Determine and graph the inverse relation of a function Graph exponential growth Graph exponential decay Use an exponential function, (y = ab x, where x is a real number, a!0, b>0, b!1), to model growth Use an exponential function, (0<b<1), to model decay Write an exponential function for a graph that includes 2 given points Determine whether a function represents exponential growth or exponential decay (without graphing) Solve real-world problems using exponential growth or decay Recognize and use families of exponential functions Understand and use the irrational number e (" ) Graph y = e x Understand and use formula A = Pe rt to determine continuously compounded interest Honors only Solve polynomial equations by graphing and by factoring Add, subtract, multiply, and divide functions Determine the composition of 2 functions, including any necessary restrictions on the domain Graph square root and other radical functions Solve square root equation by graphing Define a logarithm: for base b of a positive number y, if y = b x, then log b y = x. Evaluate a logarithm Use common logarithms in real-life problem solving situations Graph logarithmic functions Recognize and use families of logarithmic functions Unit 5 Algebra and Functions Page 2

22 Use properties of logarithms (Product Property, Quotient Property, Power Property) to solve equations Solve an exponential equation of the form a = b cx by taking the log of each side Use the change of base formula to evaluate a logarithm. For any positive numbers M, b, and c, b!1, c!1, log b M = log c M/log c b Solve a logarithmic equation Solve rational equations Solve real-world problems using rational equations Vocabulary; radicand, index, principal root, radical function, square root function, Text/Resource Materials: Prentice Hall Mathematics, Algebra Chapters 6, 7, 8, 9 Assessment: Lesson Quiz Web Codes: There is an online quiz for every lesson in the textbook. Access these quizzes with Web Codes aga-0101 through aga Prefix of aga then first 2 digits indicate chapter; next 2 digits indicate lesson number. Vocabulary Quiz Web Codes: Access these quizzes with Web Codes agj-0151 through agj Prefix of agj then first 2 digits indicate chapter; next 2 digits are always 51 Chapter Test Web Codes: Access the chapter tests with Web Codes aga-0152 through aga Prefix of aga then first 2 digits indicate chapter; next 2 digits are always 52. Teacher-made assessments (tests, quizzes, projects) Common district final examination Extensions: Math Careers: quality control engineer, demographer, economist use Web Code agb Remediation: Practice, Reteaching, Guided Problem Solving Video Tutor Help Web Codes: Instructional videos to help with math concepts can be accessed by using web code age Homework Video Tutor Web Codes: For every lesson in the textbook, there is additional Unit 5 Algebra and Functions Page 3

23 support online for homework help. Access this site with Web Codes age-0101 through age Activities for reading the mathematics textbook including (1) previewing text and (2) previewing a lesson or chapter Activities for effective note taking strategies; learn to use graphic organizers Activities for understanding math vocabulary, including appropriate abbreviations Technology activities Use guided problem solving in PH text for scaffolded support Review Problem Solving Strategies: Draw a Diagram; Try, Check, Revise; Look For A Pattern; Make a Table; Solve a Simpler Problem; Use Logical Reasoning; Work Backward Special Education See Appendix I Activity Lab: Interpreting Graphs (use with lesson 1-5) Activity Lab: Collecting Linear Data (use with lesson 6-7) ESOL See Appendix II: Instructional Adaptations For the ESOL Student in the Regular Education Classroom Additional activities for understanding mathematics vocabulary (Vocabulary and Study Skills) Adapted chapter tests Additional worksheets for reading and writing math symbols Unit 5 Algebra and Functions Page 4

24 Appendix I Instructional Adaptations For the Special Education Student in the Regular Education Classroom 1. Assign a peer tutor, peer support, or study buddy to each special education student. This would be a reliable student who could help with problems, share notes, explain concepts and check problems that have been completed. This would give immediate help to the student and would mean that they wouldn t have to wait to receive help from the teacher. It would also mean that the teacher would have more time to spend with other students. Also research shows that students increase their understanding of material when they teach it to others so it would benefit the regular education student too. 2. Extended time for assignments, projects and tests. This will enable the special education student the chance to review problems, check more thoroughly for errors and to receive help for problems where they need tutoring. The ultimate goal is for the student to learn the concepts and this may take a little extra time, more than one chance on a test and the opportunity to retake a test or redo a project or homework assignment. 3. Modified assignments. A good example of this would be if the class were assigned 20 problems (#1-20) from a text, then the special education student would only be expected to complete the odd or even numbered questions. This would enable to student the time to thoroughly understand the 10 problems instead of having to rush through 20 problems and make mistakes or become overwhelmed with the workload. 4. Simple graphic organizers that outline a step by step procedure to solve problems. Many math problems require a sequential, step by step approach and it would be beneficial for special education students to have simple graphic organizers that break problem solving down into smaller steps. Examples would also be useful since they serve as a good model for the special education student to utilize. 5. Supplemental materials from textbooks with lower reading levels. These will help the special education student who has difficulty with reading and comprehension. Many times these students cannot even attempt a problem because they do not understand what the question is asking them to do. Once a comfort level has been established then more difficult reading can be incorporated. 6. Access to a calculator, tables and a comprehensive tool kit will be valuable to the special education student and will allow them more time to spend solving the problem than figuring out the arithmetic. It will also reduce the number of simple errors made by the student. 7. Study guides and chapter outline checklists are useful tools for the special education student and help them with organizing their work. 8. Regular notebook checks by the teacher are also helpful to increase organization.

25 9. Close and regular contact with the skill development teacher will help increase the success of the special education student. Supplying the skill development teacher with a condensed packet of notes is another useful tool. 10. Arranging after school tutoring for the special education student will provide a more comfortable atmosphere to ask questions and increase comprehension. 11. Students learn using a variety of modalities. Some special education students will learn more through the use of manipulatives, visual aids and using computer programs as opposed to the traditional pen, pencil, chalkboard approach.

26 Appendix II Instructional Adaptations For the ESOL Student in the Regular Education Classroom 1. Manipulatives/Visuals - Use diagrams, pictures, graphs, and tables to express concepts instead of sentences. These are used very often in every chapter of this book. 2. Cooperative/Peer Support Permit students to work in Groups, helping and learning from each other. There are many activities in which students can be assigned to work in groups. 3. Word Bank/Vocabulary - New terms should go on a word wall which can be a room display or a personal/portable word wall. Have students make posters of glossary terms. 4. Graphic Organizers Try using webs, maps, charts, and diagrams. They provide a visual representation of key concepts and related terms, helping students see relationships among ideas and show how ideas link together. 5. Student/Led discussions - Many discussions and questions motivate students to answer questions individually or as a group. 6. Rephrasing/Clarifying - Short sentences and explanations allow students to understand a concept without relying so much on the language. Students need to rephrase information in their own words. 7. Calculator Support - Calculators should be used throughout the entire book to review or explain concepts and objectives. 8. Individual Support This must be given by the classroom teacher to help students who have problems understanding the language or using information. 9. Extended Completion Time This will be necessary in some instances to assure students have enough time to complete a task or assignment. It usually takes an ESOL student longer to read and understand concepts because of the language barrier. 10. It is the responsibility of the classroom teacher to work on student s prior knowledge by asking questions before introducing a concept for the first time. 11. Modified Lessons - Objectives in every chapter are explained in simple words or phases. Common words are used to describe objectives.

27 12. Word Banks/Vocabulary This adaptation can be used at the end of every chapter or unit of study. 13. Graphs/Charts These should be used frequently 14. Highlight Key Concepts - Key concepts can be highlighted in different colors to attract students attention. 15. Applying Concepts to students personal lives and diversities Use Real World connections. 16. Verbalize Strategies - Students are asked to describe or explain important concepts in their own words. 17. Technology Programs PHSchool.com 18. Alternative Assessments - Different types of alternative assessments may be used. Concept maps, review questions, open ended questions, problem solving, oral responses, using calculators to solve problems, multiple choice questions, self evaluations, group projects, comparing similarities and differences, and oral presentations are some examples. 19. Study Guides/Outlines These should be used to stress important information. 20. Modified Coaching Rubrics- Rubrics are used for projects. Rubrics should be clear, well written, and easy to understand. 22. Mathematical Operations - Division is written differently in certain countries. For example, the division sign is written backwards in the Dominican Republic. Some countries use a comma where we would use a decimal point. Some students learn to subtract by compensating the subtrahend, not borrowing from the minuend. 23. Recognizing role models, in math and other natural sciences, that belong to different countries should be examined Additional warm ups and activities for better understanding of vocabulary should be used when working with ESOL students. 25. Create a glossary of geometry terms that students can review when necessary. 26. Integrate language arts, the sciences, and social studies into the study of data representation. 27. There are many reasons the language and vocabulary used in math can challenge even able readers: conceptual density of math text; overlap between math vocab and vocab used in everyday English; math symbolism; and, concepts embedded within

28 concepts. Marzano asserts that direct instruction on words that are an integral part of new content has a big impact on learning.