Hoboken Public Schools. College Algebra Curriculum

Transcription

1 Hoboken Public Schools College Algebra Curriculum

2 College Algebra HOBOKEN PUBLIC SCHOOLS Course Description College Algebra reflects the New Jersey learning standards at the high school level and is designed to enable students to solve real world problems. Students will explore writing, solving and graphing linear equations, system of linear functions, and inequalities, polynomial functions, rational functions, exponential and logarithmic functions, sequences, series, and probability. The first unit in this curriculum is an extension and review of learning in Algebra II curriculum. Teachers will formatively assess students throughout the academic year. The graphing calculator will be used as a tool to enhance instruction. Course Resources Pacing Guide Unit Titles Unit One: Polynomials and Number Sense Unit Two: Functions Unit Three: Graphing Inequalities and System of Equations/Inequalities Unit Four: Exponential and Logarithmic Functions Unit Five: Trigonometry and Statistics Time Frame 7-8 Weeks 6-8 Weeks 6-8 Weeks 6-8 Weeks 6-8 Weeks Unit 1 Polynomials and Number Sense Seven to Eight Weeks Unit 1 Overview In this unit, Students will be able to independently solve linear equations, and simplify complex expressions containing exponents. Students will be able to independently combine and solve polynomial

3 functions using operations of addition, subtraction, multiplication, and division as well as the quadratic formula to model and solve real world problems. Students will be able to factor polynomial expressions using grouping, GCF, a>1, difference of two squares, sum and difference of cubes, and complex factors Essential Questions Ø How can polynomials be simplified and applied to solve problems? Ø How are the properties of real numbers related to polynomials? Essential Learning Outcomes Ø Students will be able to independently solve linear equations, and simplify complex expressions containing exponents. Ø Students will be able to independently combine and solve polynomial functions using operations of addition, subtraction, multiplication, and division as well as the quadratic formula to model and solve real world problems. Ø Students will be able to factor polynomial expressions using grouping, GCF, a>1, difference of two squares, sum and difference of cubes, and complex factors Technology Infusion Ø A.1 Create a personal digital portfolio which reflects personal and academic interests, achievements, and career aspirations by using a variety of digital tools and resources Ø A.2 Produce and edit a multi-page digital document for a commercial or professional audience and present it to peers and/or professionals in that related area for review. Standards Addressed: Ø A.APR.B.2: Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x a is p(a), so p(a) = 0 if and only if (x a) is a factor of p(x). Ø A.APR.B.3: Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. Ø A.APR.C.4: Prove polynomial identities and use them to describe numerical relationships. For example, the difference of two squares; the sum and difference of two cubes; the polynomial identity (x2 + y2)2 = (x2 y2)2 + (2xy)2 can be used to generate Pythagorean triples. Ø A.APR.D.6: Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. Ø A.SSE.A.2: Use the structure of an expression to identify ways to rewrite it. For example, see x4 y4 as (x2)2 (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 y2)(x2 + y2). Ø A.REI.A.1: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Differentiation Ø Time: Extra time for assigned tasks, adjust length of assignment, timeline with due dates for reports and projects, communication system between home and school and provide lecture notes/outline.

4 Ø Processing: Extra Response time, verbalize steps, repeat, clarify or reword directions, Minibreaks between tasks, Provide a warning for transitions, and partnering. Ø Recall: Teacher-made checklist, Use visual graphic organizers, reference resources to promote independence and visual/verbal reminders Ø Tests/Quizzes/Grading: Extended time, Study guides, shortened tests, and read directions aloud. Ø Behavior/Attention: Consistent daily structured routine, simple and clear classroom rules, and frequent feedback. Ø Organization: Individual daily planner, display a written agenda, note-taking assistance, and Color code materials. Assessments Ø Describe Learning Vertically Ø Identify Key Building Blocks Ø Make Connections (between and among key building blocks) Ø Short/Extended Constructed Response Items Ø Multiple-Choice Items (where multiple answer choices may be correct) Ø Drag and Drop Items Ø Use of Equation Editor Ø Quizzes/Tests Ø Journal Entries/Reflections/Quick-Writes Ø Accountable talk Ø Projects Ø Portfolio Ø Observation Ø Graphic Organizers/ Concept Mapping Ø Presentations Ø Role Playing Ø Teacher-Student and Student-Student Conferencing Ø Homework 21 st Century Learning Connection Ø A.1 Apply critical thinking and problem- solving strategies during structured learning experiences. Ø A.16 Employ critical thinking skills independently and in teams to solve problems and make decisions, (e.g., analyze, synthesize, and evaluate). Unit 2 Functions Six to Eight Weeks Unit 2 Overview In this unit, Students will be able independently graph and create linear functions (using slope-intercept, point slope, and the standard form), parallel and perpendicular lines, and absolute value equations. Students will be able to independently graph quadratics, write an equation of parabolas from a given graph, and graph inverses and piecewise. Students will be able to successfully use arithmetic and geometric sequences and series.

5 Essential Questions Ø How are the real solutions of a quadratic equation related to the graph of the related quadratic function? Ø How can you find the sum of an infinite geometric series? Essential Learning Outcomes Ø Students will be able independently graph and create linear functions (using slope-intercept, point slope, and the standard form), parallel and perpendicular lines, and absolute value equations. Ø Students will be able to independently graph quadratics, write an equation of parabolas from a given graph, and graph inverses and piecewise. Ø Students will be able to successfully use arithmetic and geometric sequences and series. Technology Infusion Ø A.1 Create a personal digital portfolio which reflects personal and academic interests, achievements, and career aspirations by using a variety of digital tools and resources Ø A.2 Produce and edit a multi-page digital document for a commercial or professional audience and present it to peers and/or professionals in that related area for review. Standards Addressed: Ø F.IF.B.4: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Ø F.IF.B.6: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Ø F.IF.C.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Ø F.IF.C.7c: Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Ø A.CED.A.1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Ø G.GPE.A.2: Derive the equation of a parabola given a focus and direction Ø A.REI.D.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

6 Differentiation Ø Time: Extra time for assigned tasks, adjust length of assignment, timeline with due dates for reports and projects, communication system between home and school and provide lecture notes/outline. Ø Processing: Extra Response time, verbalize steps, repeat, clarify or reword directions, Minibreaks between tasks, Provide a warning for transitions, and partnering. Ø Recall: Teacher-made checklist, Use visual graphic organizers, reference resources to promote independence and visual/verbal reminders Ø Tests/Quizzes/Grading: Extended time, Study guides, shortened tests, and read directions aloud. Ø Behavior/Attention: Consistent daily structured routine, simple and clear classroom rules, and frequent feedback. Ø Organization: Individual daily planner, display a written agenda, note-taking assistance, and Color code materials. Assessments Ø Describe Learning Vertically Ø Identify Key Building Blocks Ø Make Connections (between and among key building blocks) Ø Short/Extended Constructed Response Items Ø Multiple-Choice Items (where multiple answer choices may be correct) Ø Drag and Drop Items Ø Use of Equation Editor Ø Quizzes/Tests Ø Journal Entries/Reflections/Quick-Writes Ø Accountable talk Ø Projects Ø Portfolio Ø Observation Ø Graphic Organizers/ Concept Mapping Ø Presentations Ø Role Playing Ø Teacher-Student and Student-Student Conferencing Ø Homework 21 st Century Learning Connection Ø A.1 Apply critical thinking and problem- solving strategies during structured learning experiences. Ø A.16 Employ critical thinking skills independently and in teams to solve problems and make decisions, (e.g., analyze, synthesize, and evaluate). Unit 3 Graphing Inequalities and System of Equations/Inequalities Six to Eight Weeks Unit 3 Overview In this unit, Students will be able to independently solve system of linear equations by substitution, elimination, and graphing. Students will be able to solve and graph linear inequalities and system of linear inequalities.. Students will be able to graph hyperbola, circles, and ellipses.

7 Essential Questions Ø How do systems of equations model real-world situations? Ø How are different methods of solving systems of equations identified and what are the advantages and disadvantages of each? Ø How might you determine which technique for solving a system of equations is appropriate? Ø How do you graph hyperbolas, circles, and ellipses? How can you recognize them from a standard form equation? Essential Learning Outcomes Ø Students will be able to independently solve system of linear equations by substitution, elimination, and graphing. Ø Students will be able to solve and graph linear inequalities and system of linear inequalities. Ø Students will be able to graph hyperbola, circles, and ellipses. Technology Infusion Ø A.1 Create a personal digital portfolio which reflects personal and academic interests, achievements, and career aspirations by using a variety of digital tools and resources Ø A.2 Produce and edit a multi-page digital document for a commercial or professional audience and present it to peers and/or professionals in that related area for review. Standards Addressed: Ø A.REI.C.6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Ø A.REI.C.7. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x 2 + y 2 = 3. Ø A.REI.D.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. Ø F.LE.B.5. Interpret the parameters in a linear or exponential function in terms of a context. Ø G.GPE.A.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Ø G.GPE.A.3(+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant. Differentiation Ø Time: Extra time for assigned tasks, adjust length of assignment, timeline with due dates for reports and projects, communication system between home and school and provide lecture notes/outline. Ø Processing: Extra Response time, verbalize steps, repeat, clarify or reword directions, Minibreaks between tasks, Provide a warning for transitions, and partnering. Ø Recall: Teacher-made checklist, Use visual graphic organizers, reference resources to promote independence and visual/verbal reminders Ø Tests/Quizzes/Grading: Extended time, Study guides, shortened tests, and read directions aloud. Ø Behavior/Attention: Consistent daily structured routine, simple and clear classroom rules, and frequent feedback. Ø Organization: Individual daily planner, display a written agenda, note-taking assistance, and Color code materials.

8 Assessments Ø Describe Learning Vertically Ø Identify Key Building Blocks Ø Make Connections (between and among key building blocks) Ø Short/Extended Constructed Response Items Ø Multiple-Choice Items (where multiple answer choices may be correct) Ø Drag and Drop Items Ø Use of Equation Editor Ø Quizzes/Tests Ø Journal Entries/Reflections/Quick-Writes Ø Accountable talk Ø Projects Ø Portfolio Ø Observation Ø Graphic Organizers/ Concept Mapping Ø Presentations Ø Role Playing Ø Teacher-Student and Student-Student Conferencing Ø Homework 21 st Century Learning Connection Ø A.1 Apply critical thinking and problem- solving strategies during structured learning experiences. Ø A.16 Employ critical thinking skills independently and in teams to solve problems and make decisions, (e.g., analyze, synthesize, and evaluate). Unit 4 Exponential and Logarithmic Functions Six to Eight Weeks Unit 4 Overview In this unit, Students will be able to identify and use properties of exponents and power functions, rational exponents and roots to solve real life applications. Students will be able to identify and use properties of logarithmic functions to solve real life applications. Essential Questions Ø How do students use properties of rational exponents to simplify and create equivalent forms of numerical expressions? Ø How can you solve exponential and logarithmic equations? How can you use them to solve real life applications? Ø How can you recognize polynomial, exponential, and logarithmic models? Ø What data do you need to write a function to model a given situation? Essential Learning Outcomes Ø Students will be able to identify and use properties of exponents and power functions, rational exponents and roots to solve real life applications. Ø Students will be able to identify and use properties of logarithmic functions to solve real life applications.

9 Technology Infusion Ø A.1 Create a personal digital portfolio which reflects personal and academic interests, achievements, and career aspirations by using a variety of digital tools and resources Ø A.2 Produce and edit a multi-page digital document for a commercial or professional audience and present it to peers and/or professionals in that related area for review. Standards Addressed: Ø A.REI.A.2: Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Ø A.CED.A.1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Ø A.REI.D.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. Ø F.LE.B.5. Interpret the parameters in a linear or exponential function in terms of a context. Ø N.RN.A.1. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3) 3 = 5(1/3) 3 to hold, so (51/3) 3 must equal 5. Ø N.RN.A.2. Rewrite expressions involving radicals and rational exponents using the properties of exponents. Ø F.IF.C.8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function F.IF.C.8b: Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change In functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay. Ø F.LE.A.4. Understand the inverse relationship between exponents and logarithms. For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. Differentiation Ø Time: Extra time for assigned tasks, adjust length of assignment, timeline with due dates for reports and projects, communication system between home and school and provide lecture notes/outline. Ø Processing: Extra Response time, verbalize steps, repeat, clarify or reword directions, Minibreaks between tasks, Provide a warning for transitions, and partnering. Ø Recall: Teacher-made checklist, Use visual graphic organizers, reference resources to promote independence and visual/verbal reminders Ø Tests/Quizzes/Grading: Extended time, Study guides, shortened tests, and read directions aloud. Ø Behavior/Attention: Consistent daily structured routine, simple and clear classroom rules, and frequent feedback. Ø Organization: Individual daily planner, display a written agenda, note-taking assistance, and Color code materials.

10 Assessments Ø Describe Learning Vertically Ø Identify Key Building Blocks Ø Make Connections (between and among key building blocks) Ø Short/Extended Constructed Response Items Ø Multiple-Choice Items (where multiple answer choices may be correct) Ø Drag and Drop Items Ø Use of Equation Editor Ø Quizzes/Tests Ø Journal Entries/Reflections/Quick-Writes Ø Accountable talk Ø Projects Ø Portfolio Ø Observation Ø Graphic Organizers/ Concept Mapping Ø Presentations Ø Role Playing Ø Teacher-Student and Student-Student Conferencing Ø Homework 21 st Century Learning Connection Ø A.1 Apply critical thinking and problem- solving strategies during structured learning experiences. Ø A.16 Employ critical thinking skills independently and in teams to solve problems and make decisions, (e.g., analyze, synthesize, and evaluate). Unit 5 Trigonometry and Statistics Six to Eight Weeks Unit 5 Overview In this unit, Students will be able to successfully use right triangles trigonometric, the Pythagorean Theorem, and the unit circle. Students will be able to graph analysis, find inverse and model with trigonometric functions. Students will be able to independently graph representation of data using box plots, histograms, and scatterplots to interpret and communicate results. Essential Questions Ø How is the Unit Circle significant in Algebra and why do you need it? Ø How does the unit circle let you extend trigonometric functions to all real numbers? Ø How are the characteristics of real-life problems that can be modeled by trigonometry identified? Ø How can you use a quadratic function to model a real-life situation? Ø How is identify trends or associations in a data set important in Algebra? Essential Learning Outcomes Ø Students will be able to successfully use right triangles trigonometric, the Pythagorean Theorem, and the unit circle. Ø Students will be able to graph analysis, find inverse and model with trigonometric functions.

11 Ø Students will be able to independently graph representation of data using box plots, histograms, and scatterplots to interpret and communicate results. Technology Infusion Ø A.1 Create a personal digital portfolio which reflects personal and academic interests, achievements, and career aspirations by using a variety of digital tools and resources Ø A.2 Produce and edit a multi-page digital document for a commercial or professional audience and present it to peers and/or professionals in that related area for review. Standards Addressed: Ø A.APR.B.2: Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x a is p(a), so p(a) = 0 if and only if (x a) is a factor of p(x). Ø A.APR.B.3: Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. Ø A.APR.C.4: Prove polynomial identities and use them to describe numerical relationships. For example, the difference of two squares; the sum and difference of two cubes; the polynomial identity (x2 + y2)2 = (x2 y2)2 + (2xy)2 can be used to generate Pythagorean triples. Ø A.APR.D.6: Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. Ø A.SSE.A.2: Use the structure of an expression to identify ways to rewrite it. For example, see x4 y4 as (x2)2 (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 y2)(x2 + y2). Ø A.REI.A.1: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Differentiation Ø Time: Extra time for assigned tasks, adjust length of assignment, timeline with due dates for reports and projects, communication system between home and school and provide lecture notes/outline. Ø Processing: Extra Response time, verbalize steps, repeat, clarify or reword directions, Minibreaks between tasks, Provide a warning for transitions, and partnering. Ø Recall: Teacher-made checklist, Use visual graphic organizers, reference resources to promote independence and visual/verbal reminders Ø Tests/Quizzes/Grading: Extended time, Study guides, shortened tests, and read directions aloud. Ø Behavior/Attention: Consistent daily structured routine, simple and clear classroom rules, and frequent feedback. Ø Organization: Individual daily planner, display a written agenda, note-taking assistance, and Color code materials. Assessments Ø Describe Learning Vertically Ø Identify Key Building Blocks

12 Ø Make Connections (between and among key building blocks) Ø Short/Extended Constructed Response Items Ø Multiple-Choice Items (where multiple answer choices may be correct) Ø Drag and Drop Items Ø Use of Equation Editor Ø Quizzes/Tests Ø Journal Entries/Reflections/Quick-Writes Ø Accountable talk Ø Projects Ø Portfolio Ø Observation Ø Graphic Organizers/ Concept Mapping Ø Presentations Ø Role Playing Ø Teacher-Student and Student-Student Conferencing Ø Homework 21 st Century Learning Connection Ø A.1 Apply critical thinking and problem- solving strategies during structured learning experiences. Ø A.16 Employ critical thinking skills independently and in teams to solve problems and make decisions, (e.g., analyze, synthesize, and evaluate).