Learning Power Course: AP Calculus AB

Transcription

1 Learning Power Course: AP Calculus AB Sections: Instructor: Online AP Calculus AB: Students view class segments as streamed digital video via a direct link to the NSU server Mary M. Cundy Address: Northern State University Center for Statewide E-learning 1200 South Jay Street Aberdeen, South Dakota Office Phone: Office Fax: Home Phone: cundym@northern.edu Office Hours: Students are encouraged to call or any time; I will respond as quickly as possible. I also take questions in the evenings and on weekends. Individual students can schedule study time with me. To call, please use the toll free Help Desk number ( ) and ask for me. Required Text: Company: McDougall Littell A Division of Houghton Mifflin Customer Service Center 1900 S. Batavia Ave. Geneva, Il To order by Phone/Fax: Customer Service (phone): Customer Service (Fax): Larson, Hostetler, Edwards, Calculus of a Single Variable, Seventh Edition (2002) (with Student Learning Tools CD-ROM) / Order Number: / ISBN: Required Calculator: TI89 Course Prerequisites: Algebra I and II, Geometry, Trigonometry

2 Course Description: The course is designed to build a strong foundation in the basics of calculus: differentiation and integration. Students are introduced to new ways of thinking about math, including graphical, numerical, and analytical approaches; modeling; problem solving; and analysis emphasizing real-life data. Student Expectations: Students are expected to have a strong commitment to learning evidenced through spirited explorations and narratives, online discussions and attention to homework. Homework is a basic component of the class. Students may submit homework electronically, by fax, and by mail. Course Support: Academic assistance is available via a toll free help line, also via the Internet with Desire2Learn resources including access to course materials, discussion boards, , and homework chat opportunities. Live, synchronous net meeting studying is also available. Course Policies: All homework and projects must be submitted in a timely manner. Late papers will result in a reduced grade. Make-up work should be addressed immediately with arrangements being made with the instructor. Academic Honesty: All students are expected to conduct themselves in a manner consistent with academic pursuits. Cheating will not be tolerated. Specifically, students may not use or give unauthorized assistance during quizzes or tests; students may not use the work of others or claim it as their own. Course Goals (as stated in the College Board Course Description): 1. Students should be able to work with functions represented in a variety of ways: graphical, numerical, analytical, or verbal. They should understand the connections among these representations. 2. Students should understand the meaning of the derivative in terms of a rate of change and local linear approximation and should be able to use derivatives to solve a variety of problems. 3. Students should understand the meaning of the definite integral both as a limit of Riemann sums and as the net accumulation of change and should be able to use integrals to solve a variety of problems. 4. Students should understand the relationship between the derivative and the indefinite integral as expressed in both parts of the Fundamental Theorem of Calculus. 5. Students should be able to communicate mathematics both orally and in wellwritten sentences and should be able to explain solutions to problems. 6. Students should be able to model a written description of a physical situation with a function, a differential equation, or an integral. 7. Students should be able to use technology to help solve problems, experiment, interpret results, and verify conclusions.

3 8. Students should be able to determine the reasonableness of solutions, including sign, size, relative accuracy, and units of measurement. 9. Students should develop an understanding of calculus as a coherent body of knowledge and as a human accomplishment. Primary Text: Larson, R., Hostetler, R. P., Edwards, B. (2002). Calculus of a Single Variable (Seventh Edition). New York: Houghton Mifflin Company. Secondary Text: Larson, R., Hostetler, R. P., Edwards, B. (2002). Calculus of a Single Variable Early Transcendental Functions (Third Edition). New York: Houghton Mifflin Company. Technology: I have a Desire2Learn course site for managing course materials. On that site, the discussion board provides a forum for posting problems as well as discussing concepts and issues (a part of most assignments). I use animations whenever possible to assist students in understanding topics and concepts. I use a tablet pc for lesson design, to enhance power point presentations and animations, and to post problems for discussion or assistance. Students share access to these technologies. Assessment: I use online homework quizzes as content embedded assessment for learning. The immediacy of the format gives me formative information during the course of chapters. I use performance assessment to culminate the studies of each unit. I use format assessment in the form of chapter tests that mimic the AP exam. Other assessments involve homework assignment as well as class explorations and activities. Course Grading Scale: 90%-100% A 80%-89% B 70%-79% C 60%-69% D 0%-50% F A note about transcendental functions: Historically, my students have struggled with transcendental functions. To overcome this deficiency, I have integrated transcendental function problems into every assignment from the beginning of the school year. Thus, there is no separate section for transcendental function problems in the Topic Outlines because they are included in every section. A note about calculator skills: All students have Ti89 calculators used in conjunction with class work. The students graph functions (in appropriate windows for optimal viewing), find zeros and the intersection points of graphs, find numerical values for derivatives and definite integrals, and use the symbolic capabilities for practice with equivalent forms of answers. The calculator is the student s tool for solving problems, experimenting, interpreting results, and verifying conclusions.

4 The Course Review of Functions and Their Graphs (about 1 week): Goal 1 Narrative: This is a general review of all functions, graphs, models, and rates of change with particular emphasis on transcendental functions. The students should be able to Demonstrate a knowledge of and proficiency with all functions, graphs, models, and rates of change with particular emphasis on transcendental functions. Limits and Their Properties (about 4 weeks) Narrative: We open this chapter with a graphing activity Making a Curve Behave Like a Line. The students are given a common function graph and the equation of the tangent line to that function at x = 1. After using calculators to zoom in on the point of tangency, the students come to the conclusion that the curve does, indeed, behave like a line. They repeat the process with other curves and tangent lines and report results in paragraph format. With this wonderful combination of local linearity and limits, we begin our study of calculus. In this chapter we focus on what calculus is and how it compares to pre-calculus. Our study of limits begins with an intuitive understanding of the concept; students find examples of limits from sports records and other common sources. Then we estimate limits numerically (from a table) and graphically. In each case, we use a verbal approach to report our findings and share ideas. In repeating our opening activity with an absolute value graph and the tangent line to the vertex, students to the see that local linearity is not always possible and conclude that some limits fail to exist. At this point we are ready for the formal definition of a limit and the appropriate terminology and symbols. Our next approach to limits is analytic. The students use graphical, numerical, and analytical analysis and report the results in a short paragraph (opportunities for writing are embedded in many assignments.) Our discussion of limits leads us to the formal definition of continuity at a point. With analytic work, tables and graphs, we examine and classify discontinuities. Our first interval work is based on the domain of functions and leads to continuity on open intervals. One-sided limits and continuity on a closed interval follow. To gain an intuitive understanding of the Intermediate Value Theorem, we begin with a graph and look for intercepts. Then we look at a table and talk about what the table tells us about intercepts. Then I go to the formal discussion of the Intermediate Value Theorem. We conclude our study of limits with infinite limits and limits at infinity. Infinite limits are introduced graphically and numerically with an emphasis on the domain of the

5 function. Limits at infinity are introduced graphically and numerically, also. In both cases, we follow up with analytical work because it is the combination of the three approaches that gives us an accurate picture. Plus, only the analytic approach offers definitive answers. I use a culminating activity to wrap up this important transition to calculus. In the activity, students discuss the various aspects and implications of limits and continuity, then illustrate the discussion with examples from a given piece wise-defined function that (conveniently) offers examples of all of the chapter topics! Topic Outline: The students should be able to I. A Preview of Calculus (Goal 9) A. Compare calculus and pre-calculus B. Understand the tangent line problem is basic to calculus C. Understand the area problem is basic to calculus II. Finding the Limit Graphically and Numerically (Goals 1, 5, 7) A. Estimate the limit from a graph or a table B. Understand the role of domain in vertical asymptotes of a graph (an intuitive understanding of continuity) C. Understand the role of domain in creating holes in graphs (an intuitive understanding of continuity) D. Examine and understand the behavior of piecewise-defined functions (with and without gaps) and step functions E. Recognize ways that a limit fails to exist F. Use the formal definition of a limit III. Evaluating Limits Analytically (Goals 1, 5, 7) A. Use limit properties to evaluate a limit B. Develop and use different strategies for finding limits C. Use dividing out and rationalizing techniques to find a limit D. Find the limit of a composite function E. Use the Squeeze Theorem to find a limit IV. Continuity and One-sided Limits (Goals 1, 5, 7) A. Determine continuity at a point and on an open interval B. Classify discontinuities as removable and nonremovable C. Understand continuity in respect to limits and vice versa D. Use the definition of continuity E. Find one-sided limits F. Determine continuity on a closed interval G. Use the properties of continuity H. Understand and use the Intermediate Value Theorem V. Infinite Limits (Goals 1, 5, 7) A. Find infinite limits from the left and the right B. Examine the domain of functions for vertical asymptotes C. Understand the role of domain and vertical asymptotes in unbounded behavior

6 D. Sketch graphs with vertical asymptotes and using the sketch to examine infinite limits E. Understand and use the properties of infinite limits VI. Limits at Infinity (Goals 1, 5, 7) A. Find limits at infinity B. Find the horizontal asymptotes of a graph numerically, graphically and analytically C. Associate limits at infinity with horizontal asymptotes D. Use the definition of infinite limits at infinity Differentiation (about 8 weeks) Narrative: Our transition to calculus begins with a review of slope. We look at the slope of a secant line joining two points on a curve (pre-calculus). From the slope of that secant line joining two distinct points with x- and y-values, together we change the labels and f ( x+ Δx) f( x) the slope expression until we arrive at the difference quotient as our Δx expression of the slope. Then I use an animation to allow delta x to approach 0 and let my students see the secant line become a tangent line. Tacking a limit statement in front of that difference quotient and reminding the students about Making a Curve Behave Like a Line completes our transition to calculus! We use the limit definition of the derivative extensively; we identify the two end points of the secant line and then talk about how the limit statement in front of the difference quotient turned our secant line into a tangent line. I encourage the students to use the term Local Linearity often in describing this dynamic process of using a tangent line to approximate the slope of a curve at a point. The introduction to derivatives is complete when we have added continuity to the conversation in light of the fact that continuity affects limits and, thus, affects differentiability. I stress the fact that differentiability implies continuity, while continuity does not imply differentiability. From the formal definition of the derivative, we proceed to derivative rules. In each section there are problems for writing the equation of tangent lines at a point, finding points where horizontal tangent lines occur, using graphical, numerical and analytic analysis, and writing descriptions of functions graphs, tangent lines and derivatives. We examine derivatives from a verbal standpoint as rates of change. Vertical motion is another topic we study as part of many assignments. We look at instantaneous rate of change and average rate of change through the lens of limits, with an emphasis on the fact that an average rate of change is the slope between two points (the secant line) and the instantaneous rate of change is the slope at a point (the tangent line). Our study of the derivative and the tangent line problem ends with equations that cannot be explicitly expressed as a function of x, requiring implicit differentiation techniques.

7 I add related rates to this chapter because students find related rates a logical extension of implicit differentiation. Plus, some real life problems are a nice way to remind students that calculus is a powerful way to describe events in the real world. One culminating activity for this chapter asks students to discuss and illustrate (analytic, graphical, numerical examples) the difference between continuity and differentiability. A second culminating activity requires students to choose a function (from a given list) and graph the function. Then students choose any point on that function to physically draw a tangent line. Using the tangent line, the student must approximate the slope of the function graphically (by observing the slope of the tangent line) and numerically (by developing a table of values to approximate slope using change in y over change in x). The students then find the derivative of the function, write the equation of the tangent line to the function at the chosen point, and use the tangent line to approximate the function. The narrative requires the students to use conceptual terms to discuss and analyze the implications of the process. Topic Outline: The students should be able to I. The Derivative and the Tangent Line Problem (Goals 1, 2, 5, 9) A. Find the slope of the tangent line to a curve at a point B. Use the limit definition of the derivative C. Understand the conditions where derivatives fail to exist D. Examine continuity with the graphs of functions and discussing the intuitive definition of continuity E. Examine the relationship between differentiability and continuity II. Basic Differentiation Rules and Rates of Change (Goals 1, 2, 5, 6, 7) A. Find the derivative using the Constant Rule B. Find the derivative using the Power Rule C. Find the derivative using the Constant Multiple Rule D. Find the derivative using Sum and Difference Rules E. Find the derivative of sine and cosine F. Understand the derivative as a rate of change and using the derivative to find rates of change G. Compare average rate of change and instantaneous rate of change and understanding the relationship H. Compare the magnitude of a function with its rate of change I. Understand and work with vertical motion problems as an application of derivatives J. Recognize f and f when given only the graph. III. The Product and Quotient Rules and Higher-Order Derivatives (Goals 1, 2, 5, 6, 7) A. Find the derivative using the Product Rule B. Find the derivative using the Quotient Rule C. Find the derivative of trigonometric functions D. Find higher-order derivatives of functions E. Use derivative rules in abstract formats

8 F. Use position, velocity and acceleration as applications of derivatives G. Recognize f, f, and f when given only the graph IV. The Chain Rule (Goals 1, 2, 5, 7) A. Find the derivative of a composite function using the Chain Rule B. Find the derivative of a function using the General Power Rule C. Simplify derivatives of functions using algebraic techniques D. Find the derivative of trigonometric functions using the Chain Rule E. Recognize f, f, and f when given only the graph F. Use derivative rules in abstract formats V. Implicit Differentiation (Goals 1, 2, 5) A. Distinguish between functions in explicit and implicit form B. Find the derivative using implicit differentiation C. Find second derivatives in terms of x and y VI. Related Rates (Goals 1, 2, 5, 6, 8) A. Find a related rate using implicit differentiation with respect to time B. Develop and understand guidelines for related-rate problems C. Use related rates to solve real-life problems D. Determine the reasonableness of solutions including sign, size, relative accuracy, and units of measurement. Applications of Differentiation (about 8 weeks) Narrative: We open this chapter with an activity involving designing the optimal package for a product (a right circular cylinder) that minimizes surface area for a constant volume. By creating a table and a graph relating radius to surface area, students theorize what the optimal radius would be. Students then examine actual containers and discuss why the actual container does (does not) minimize surface area. The results of this activity are later revisited using calculus and students compare the processes. I use the graphical perspective to examine a variety of functions and begin the discussion of extrema (relative and absolute) on intervals (open and closed) with an emphasis on whether or not the endpoints of the interval are candidates for extrema. Students first see Rolle s Theorem and the Mean Value Theorem in animations. Then they look for characteristics of graphs that would indicate the theorems apply. Finally, they take an analytic approach to the theorems. The discussion of increasing and decreasing functions has developed naturally from all of our work with graphs, so the First Derivative Test or the Second Derivative Test, as an analytic approach seems so natural and sensible to students who have come to expect that anything we can see on a graph or a table can be dealt with concretely with an analytic approach. Likewise, the students have already developed an intuitive understanding of concavity, so I have them choose their own words to describe that characteristic of a graph before I introduce the actual term. After all of our work with graphs, students make the connection between real-life problems calling for extrema and the appearance of the function when graphed. The First Derivative Test or the Second Derivative Test serves as analytic justification.

9 Using differentials as the final application of derivatives serves as a bridge to our next unit of study, integration. The culminating activity for this chapter is revisiting our initial optimization problem and complementing the graphical and numeric work with analytic computations the real answer! Topic Outline: The students should be able to I. Extrema on an Interval (Goals 1, 5, 7) A. Use the definition of extrema on an interval for various functions B. Understand the implications of the Extreme Value Theorem C. Understand and use the definition of relative extrema of a function on an open interval D. Understand and use the definition of a critical number E. Demonstrate the relationship between critical numbers and relative extrema F. Find extrema on closed intervals G. Develop and use guidelines for finding extrema on a closed interval H. Compare relative and absolute extrema II. Rolle s Theorem and the Mean Value Theorem (Goals 1, 5, 7, 9) A. Use Rolle s Theorem to justify the existence of an extreme value (and critical number) on a closed interval B. Understand and use the Mean Value Theorem (in the context of tangent lines with the same slope as secant lines) III. Increasing and Decreasing Functions and Tests for Relative Extrema (Goals 1, 7) A. Determine intervals where functions are increasing or decreasing B. Understand and recognize the implications of strictly monotonic functions C. Apply the First Derivative Test to find relative extrema analytically D. Apply the Second Derivative Test to find relative extrema analytically E. Apply the concept of increasing/decreasing to functions given in abstract formats F. Use graphing and numerical comparisons of function behaviors in conjunction with an analytic approach G. Sketch f given the graph of f IV. Concavity (Goals 1, 5, 7) A. Determine intervals on which a function is concave up and concave down B. Develop a working definition of what concave up and concave down mean when examining a graph C. Find points of inflection D. Test for concavity analytically E. Sketch graphs of functions based on verbal descriptions F. Sketch f and f given the graph of f V. Optimization Problems (Goals 1, 2, 5, 6, 7, 8) A. Solve applied maximum and minimum problems

10 B. Develop guidelines for solving applied maximum and minimum problems (including recognizing the primary and secondary equations) C. Determine the reasonableness of solutions, including sign, size, relative accuracy, and units of measurement VI. Differentials (Goals 1, 5, 6, 7) A. Demonstrate an understanding the concept of the tangent line approximation B. Compare the differential with actual change C. Estimate propagated error D. Find the differential of a function Integration (about 6 weeks) Narrative: We begin our study of integration with an exploration where I give students the derivative of a function and they tell me what the original function (the antiderivative) would be. Students discuss the strategies they used and come to a consensus. Usually the consensus will actually be an informal list of rules for finding antiderivatives! The area problem, like the tangent line problem before it, is basic to the study of calculus. We begin our examination of areas with the usual graphical and numerical approaches. Students draw curves, find upper sums, lower sums, and strategize how to make these rudimentary estimates of area more accurate. In those musings, we come right back to limits most of the time! As we move on to the analytic approach for finding the definite integral, we continue to look at graphs so that we do not lose track of the area idea in the midst of the analytic process. Also, I like to point out that the definite integral is a process for determining accumulated change. Real-life assignment problems give students a chance to use the definite integral in that light. Our closing activity is a general discussion of area. Students choose from a list of given functions. Drawing four rectangles, they use left, right, and midpoint formulas to estimate the area under their curve. Then they draw four trapezoids to estimate the area. Finally they use analytic methods to actually find the area. The activity concludes with a general discussion of the accuracy of each method, and circumstances that would dictate the use of the various methods. Topic Outline: The students should be able to I. Antiderivatives and the Indefinite Integral (Goals 1, 3, 4, 5, 6, 7, 8, 9) A. Find general solutions for a differential equation B. Use indefinite integral notation C. Use basic integration rules D. Understand the graphical significance of the constant of integration E. Find (and draw) solutions (general and specific) to a differential equation (slope fields) F. Solve vertical motion problems

11 G. Solve rectilinear motion problems H. Use the graph of f to answer questions about f and f. I. Determine the reasonableness of solutions, including sign, size, relative accuracy, and units of measurement II. Area (Goals 1, 3, 4, 5, 6, 7, 8) A. Use sigma notation to write and evaluate a sum B. Understand the concept of area C. Approximate the area of a region numerically (from a graph on a grid as well as from inscribed and circumscribed rectangles) D. Find the area of a region using limits (upper sums/lower sums) E. Find the area of regions bounded by the y-axis III. Riemann Sums and Definite Integrals (Goals 1, 3, 4, 5, 6, 7, 8) A. Understand the definition of a Riemann sum (left, right and midpoint) B. Evaluate a definite integral using limits C. Evaluate a definite integral using the properties of definite integrals D. Find definite integrals using the areas of common geometric figures and properties of integrals E. Approximate definite integrals using graphs on a grid with shaded area F. Determine the reasonableness of solutions, including sign and size IV. The Fundamental Theorem of Calculus (Goals 3, 4, 5, 6, 7, 8, 9) A. Evaluate the definite integral using the Fundamental Theorem of Calculus B. Understand and use the Mean Value Theorem for Integrals C. Find the average value of a function over a closed interval D. Understand and use the Second Fundamental Theorem of Calculus E. Evaluate a definite integral using a function graph and properties of definite integrals F. Interpret the definite integral as an accumulation function G. Use the Fundamental Theorem of Calculus in rectilinear motion problems H. Determine the reasonableness of solutions, including sign and size V. Integration by Substitution (Goals 3, 4, 5, 6, 7, 8) A. Use pattern recognition to evaluate an indefinite integral B. Find the antiderivative of a composite function C. Use a change of variables to evaluate the definite integral D. Develop guidelines for the change of variables technique E. Use the General Power Rule for Integration to evaluate an indefinite integral F. Evaluate a definite integral using even or odd functions G. Use a slope field to graphically model the general and particular solution to a differential equation H. Use a graphing utility to find the area of a region I. Determine the reasonableness of solutions, including sign and size VI. Numerical Integration (Goals 1, 3, 4, 5, 6, 7, 8) A. Approximate a definite integral using the Trapezoidal Rule B. Compare the Midpoint Rule and the Trapezoidal Rule C. Compare the results of the left end point rule, the mid point rule, the right end point rule and the trapezoidal rule for accuracy and agreement

12 D. Use tabular data or a graph to find area using the Trapezoidal Rule E. Determine the reasonableness of solutions, including sign and size Inverse Functions and Differential Equations (about 2 weeks) Narrative: Inverse functions are a review topic for most students, so that material goes quickly. The focus of this work falls mainly on slope fields. For this work, I supplement the text. My students like working together to create slope fields. Time devoted to differentiating and integrating inverse trigonometric functions varies from year to year. Topic Outline: The students should be able to I. Inverse Functions (Goals 1, 2) A. Verify that one function is the inverse of another function (graphically, numerically and analytically) B. Determin3 whether functions have an inverse C. Understand the characteristics of functions and inverses D. Understand continuity and differentiability as applied to inverse functions E. Find the derivative of inverse functions (implicit differentiation) II. Differential Equations (Goals 1, 3, 4, 6, 7, 8) A. Use separation of variables to solve a simple differential equation B. Use exponential functions to model growth and decay C. Use slope fields to visually provide solutions to differential equations D. Use initial conditions to find particular solutions to differential equations E. Recognize and solve differential equations that can be solved by separation of variables F. Use differential equations to model and solve applied problems G. Determine the reasonableness of solutions, including sign, size, relative accuracy, and units of measurement III. Inverse Trigonometric Functions (Goals 1, 2, 3, 4) A. Develop properties of the six inverse trigonometric functions B. Differentiate inverse trigonometric functions C. Remember and demonstrate the basic differentiation formulas for elementary functions D. Integrate inverse trigonometric functions E. Remember and demonstrate basic integration formulas for elementary functions Applications of Integration (about 2 weeks) Narrative:

13 I rely heavily on animations to help my students see the area and volume problems. In particular, I find that the animations help with the volume of a solid with a known cross section. The culminating activity for this work involves choosing a set of functions from a given list. The students find the area of the region between the given functions. They then find the volume of revolution of that area with two different horizontal and two different vertical axes of revolution. Finally, they imagine the region as the base of a solid with two different known cross sections. Their narrative documents the processes they used and discusses the issues pertinent to the different types of problems. Topic Outline: The students should be able to I. Area of a Region Between Two Curves (Goals 1, 3, 4, 5, 6, 7, 8, 9) A. Find the area of a region between two curves using integration B. Find the area of a region between intersecting curves using integration (vertical representative rectangles) C. Use horizontal representative rectangles to find the area of a region between two curves D. Describe integration as an accumulating process in real-life applications E. Determine the reasonableness of solutions, including sign, size, relative accuracy, and units of measurement II. Volume: The Disk Method (Goals 1, 3, 4, 5, 6, 7, 8, 9) A. Find the volume of a solid region of revolution using the disk method B. Find the volume of a solid of revolution using the washer method C. Approximate the volume of a solid of revolution from the sketch of a graph on a grid D. Find the volume of a solid with known cross sections III. Volume: The Shell Method (Goals 1, 3, 4, 5, 6, 7, 8) A. Find the volume of a solid of revolution using the shell method B. Compare the uses of the disk method and the shell method Review for AP Calculus AB Exam (about 4 weeks) (Goal 9) Final note: When time permits, we end our year with a narrative discussion of the history of calculus and major figures in the discipline. Students cite examples of the pervasive use of calculus in describing and analyzing our world. Hopefully, this assignment leads students to an appreciation of calculus as a coherent body of knowledge and as a human accomplishment.

14 South Dakota Mathematics Content Standards Alignment Grades 9-12 Course: AP Calculus AB Since Advanced standards are intended to apply to students having achieved the core mathematics standards and are more advanced than first-year algebra and basic geometry only the advanced standards apply for this course. Also, since AP Calculus AB is a college level course, much of the work only loosely aligns with high school standards. For the most part, the skills are embedded in the analytical processes of calculus itself. Advanced High School Algebra Grade Standards Indicator 1: Use procedures to transform algebraic expressions. Standard Supporting Skill: Textbook Section 9-12.A.1.1A. Students are able to write equivalent forms of rational algebraic expressions using properties of real numbers. (Application) 9-12.A.1.2A. Students are able to extend the use of real number properties to expressions involving complex numbers. (Application) Indicator 2: Use a variety of algebraic concepts and methods to solve equations and inequalities A.2.1A. Students are able to Use the quadratic formula: determine solutions of quadratic occasional practice equations. (Analysis) Use the descriminant to describe the nature of the roots: occasional 9-12.A.2.2A. Students are able to determine the solution of systems of equations and systems of inequalities. (Application) 9-12.A.2.3A. Students are able to determine solutions to absolute value statements. (Application) practice limited practice limited practice Indicator 3: Interpret and develop mathematical models A.3.1A. Students are able to distinguish between linear, quadratic, inverse variation, and exponential models. (Analysis)

15 9-12.A.3.2A. Students are able to create formulas to model relationships that are algebraic, geometric, trigonometric, and exponential. (Synthesis) 9-12.A.3.3A. Students are able to use sequences and series to model relationships. (Analysis) used in the context of approximating areas Indicator 4: Describe and use properties and behaviors of relations, functions, and inverses A.4.1A. Students are able to determine the domain, range, and intercepts of a function. (Analysis) 9-12.A.4.2A. Students are able to describe the behavior of a polynomial, given the leading coefficient, roots, and degree. (Analysis) 9-12.A.4.3A. Students are able to apply transformations to graphs and describe the results. (Analysis) 9-12.A.4.4A. Students are able to apply properties and definitions of trigonometric, exponential, and logarithmic expressions. (Application) 9-12.A.4.5A. Students are able to describe characteristics of nonlinear functions and relations. (Analysis) Change coefficients or constants: limited practice Graph the inverse of a function: limited practice Conic sections: Trigonometric Functions: Exponential and logarithmic functions: continually practicing this skill Advanced High School Geometry Grade Standards Indicator 1: Use deductive and inductive reasoning to recognize and apply properties of geometric figures G.1.2A. Students are able to Write a direct proof: determine the values of the sine, cosine, Make conjectures: limited practice and tangent ratios of right triangles. in this context (Evaluation) 9-12.G.1.2A. Students are able to determine the values of the sine, cosine,

16 and tangent ratios of right triangles. (Application) 9-12.G.1.3A. Students are able to apply properties associated with circles. (Application) 9-12.G.1.4A. Students are able to use formulas for surface area and volume to solve problems involving threedimensional figures. (Analysis) limited practice in the context of problems Related Rates Problems, Optimization Problems, Various Area Problems Indicator 2: Use properties of geometric figures to solve problems from a variety of perspectives G.2.1A. Students are able to use using graphical representations to Cartesian coordinates to verify approximate functions and areas geometric properties. (Synthesis) Advanced High School Measurement Grade Standards Indicator 1: Apply measurement concepts in practical applications M.1.1A. Students are able to use dimensional analysis to check answers and determine units of a problem solution. (Application) 9-12.M.1.2A. Students are able to use indirect measurement in problem situations that defy direct measurement. (Analysis) Advanced High School Number Sense Grade Standards Indicator 1: Analyze the structural characteristics of the real number system and its various subsystems. Analyze the concept of value, magnitude, and relative magnitude of real numbers N.1.1A. Students are able to limited practice describe the relationship of the real number system to the complex number system. (Comprehension) 9-12.N.1.2A. Students are able to apply properties and axioms of the real number system to various subsets, e.g., axioms of order, closure. (Application) limited practice

17 Indicator 2: Apply number operations with real numbers and other number systems N.2.1A. Students are able to add, subtract, multiply, and divide real numbers including rational exponents. (Application) Advanced High School Statistics & Probability Grade Standards Indicator 1: Use statistical models to gather, analyze, and display data to draw conclusions S.1.2A. Students are able to analyze limited practice and evaluate graphical displays of data. (Evaluation) 9-12.S.1.5A. Students are able to use scatterplots, best-fit lines, and correlation coefficients to model data and support conclusions. (Application) limited practice NCTM Principles and Standards for School Mathematics (2000) Grades 9-12 Number and Operations Instructional programs from prekindergarten through grade 12 should enable all students to Understand numbers, ways of representing numbers, relationships among numbers, and number systems. Indicator: Textbook Sections Develop a deeper understanding of very large and very small numbers and of carious representations of them: 1.2, 1.3, 1.4, 1.5, 3.5 Compare and contrast the properties of numbers and number systems, including the rational and real numbers, and understand complex numbers as solutions to quadratic equations that do not have real solutions: Understand vectors and matrices as systems that have some of the properties of the real-number system: Use number-theory arguments to justify relationships involving

18 Understand meanings of operations and how they relate to one another. whole numbers: Judge the effects of such operations as multiplication, division, and computing powers and roots on the magnitude of quantities: Develop an understanding of properties of and representations for, the addition and multiplication of vectors and matrices: Develop an understanding of permutations and combinations as counting techniques:

19 Compute fluently and make reasonable estimates. Develop fluency in operations with real numbers, vectors, and matrices, using mental computations or paper-and-pencil calculations for simple cases and technology for more-complicated cases: Judge the reasonableness of numerical computations and their results: continually practicing this skill Algebra Understand patterns, relations, and functions. Generalize patterns using explicitly defined and recursively defined functions: Understand relations and functions and select, convert flexibly among, and use various representations for them: continually practicing this skill Analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior: Understand and perform transformations such as arithmetically combining, composing, and inverting commonly used functions, using technology to perform such operations on more-complicated symbolic expressions: continually practicing this skill Understand and compare the properties of classes of functions, including exponential, polynomial, rational, logarithmic, and periodic functions: continually practicing this skill Interpret representations of functions of two variables:

20 Represent and analyze mathematical situations and structures using algebraic symbols. Use mathematical models to represent and understand quantitative relationships. Understand the meaning of equivalent forms of expressions, equations, inequalities, and relations: continually practicing this skill Write equivalent forms of equations, inequalities, and systems of equations and solve them with fluency mentally or with paper and pencil in simple cases and using technology in all cases: Use symbolic algebra to represent and explain mathematical relationships: continually practicing this skill Use a variety of symbolic representation, including recursive and parametric equations, for functions and relations: Judge the meaning, utility, and reasonableness of the results of symbolic manipulations, including those carried out by technology: Identify essential quantitative relationships in a situation and determine the class or classes of functions that might model the relationship: Use symbolic expressions, including iterative and recursive forms, to represent relationships arising from various contexts: Draw reasonable conclusions about a situation being modeled:

21 Analyze change in various contexts. Approximate and interpret rates of change from graphical and numerical data: continually practicing this skill Geometry Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships. Specify locations and describe spatial relationships using coordinate systems. Apply transformations and use symmetry to analyze mathematical situations. Analyze properties and determine attributes of two- and threedimensional objects: Explore relationships (including congruence and similarity) among classes of two- and threedimensional geometric objects, make and test conjectures about them, and solve problems involving them: Establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others: Use trigonometric relationships to determine lengths and angle measure: Use Cartesian coordinates and other coordinate systems, such as navigational, polar, or spherical systems, to analyze geometric situations: Investigate conjectures and solve problems involving two-and threedimensional objects represented with Cartesian coordinates: Understand and represent translations, reflections, rotations, and dilations of objects in the plane by using sketches, coordinates, vectors, function notation, and matrices: Use various representations to help understand the effects of simple transformations and their compositions:

22 Use visualizations, spatial reasoning, and geometric modeling to solve problems. Draw and construct representations of two-and three-dimensional geometric objects using a variety of tools: Visualize three-dimensional objects from different perspectives and analyze their cross sections: 6.2 Use vertex-edge graphs to model and solve problems: Use geometric models to gain insight into, and answer questions in, other areas of mathematics: Use geometric ideas to solve problems in, and gain insight into, other disciplines and other areas of interest such as art and architecture: Measurement Understand measurable attributes of objects and the units, systems, and processes of measurement. Apply appropriate techniques, tools, and formulas to determine measurement. Makes decisions about units and scales that are appropriate for problems involving measurement: Analyze precision, accuracy, and approximate error in measurement situations: Understand and use formulas for the area, surface area, and volume of geometric figures, including cones, spheres, and cylinders: 6.2 Apply informal concepts of successive approximation, upper and lower bounds, and limit in measurement situations: 3.2, 3.8 Use unit analysis to check measurement computations:

23 Data Analysis and Probability Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them. Select and use appropriate statistical methods to analyze data. Understands the difference among various kinds of studies and which types of inferences can legitimately be drawn from each: know the characteristics of welldesigned studies, including role of randomization in surveys and experiments: understand the meaning of measurement data and categorical data, of univariate and bivariate data, and of the term variable: understand histograms, parallel box plots, and scatterplots and use them to display data: compute basic statistics and understand the distinction between statistic and parameter: For univariate measurement of data, be able to display the distribution, describe its shape, and select and calculate summary statistics: For bivariate measurement data, be able to display a scatterplot, describe its shape, and determine regression coefficients, regression equations, and correlation coefficients using technological tools: Display and discuss bivariate data where at least one variable is categorical: Recognize how linear transformations of univariate data affect shape, center and spread: Identify trends in bivariate data and find functions that model the data or transform the data so that they can be modeled: occasional practice of this skill

24 Develop and evaluate inferences and predictions that are based on data. Understand and apply basic concepts of probability. Use simulations to explore the variability of sample statistics from a known population and to construct sampling distributions: Understand how sample statistics reflect the values of population parameters and use sampling distributions as the basis for informal inference: Evaluate published reports that are based on data by examining the design of the study, the appropriateness of the data analysis, and the validity of conclusions: Understand how basic statistical techniques are used to monitor process characteristics in the workplace: Understand the concepts of sample space and probability distribution and construct sample spaces and distributions in simple cases: Use simulations to construct empirical probability distributions: Compute and interpret the expected value of random variables in simple cases: Understand the concepts of conditional probability and independent events: Understand how to compute the probability of a compound event: