Curriculum Design Template Content Area: Mathematics Course Title: Pre Calculus Grade Level: 11 Functions and Graphs Polynomials and Rational Functions Exponential and Logarithmic Functions Marking Period 1 Introduction to Trigonometry Trigonometric Graphs Marking Period 2 Solving Trigonometric Equations Trigonometric Identities and Proofs Trigonometric Applications Marking Period 3 Analytic Geometry Limits and Continuity Marking Period 4 Date Created: May 2012 Board Approved on: August 27, 2012
Course Title: Pre-Calculus Grade Level: 11 th & 12 th Overarching What are Functions? What types of functions exist? How can we solve all polynomial equations? How can we use symmetry and transformations to graph polynomial functions? How can we graph rational functions? What are exponential and logarithmic functions? How can exponential and logarithmic functions be used in real-world applications? What is trigonometry and how is it used? What do the graphs of the trigonometric functions look like? How can we solve trigonometric equations? What are the trigonometric identities and how can we verify them? How can we use trigonometry to solve triangles? How does trigonometry relate to vectors? What are the conic sections? How can we write equations to describe the conic sections? How can we convert from rectangular coordinates to polar coordinates? What is a limit and what does it tell us about the behavior of a function? Overarching Enduring Understandings Students in Pre-Calculus will learn about various types of functions: polynomial, inverse, piece-wise, exponential, logarithmic, and trigonometric. Emphasis will be placed on graphical as well as algebraic knowledge of each type of function. Real-world applications will be infused throughout the curriculum as appropriate. Additionally, students will learn about the conic sections and vectors as well as the concept of limits. Course Description This course is designed for students who have successfully completed algebra 2 and/or who wish to pursue higher level mathematics. In addition to those topics normally covered in a pre-calculus course, concepts such as conic sections and limits will be explored. Graphing calculators are used on a daily basis and it would be beneficial to have one's own so that homework assignments may be completed more easily. This course is designed for the college bound student who intends to attend a 4-year college and/or a STEM career. Non-seniors who take and pass pre-calculus are eligible to take calculus as a senior if they choose to do.
Technology and 21 st -Century Life and Careers Standards 8.1.12 A. Basic Computer Skills and Tools 3. Construct a spreadsheet, enter data, use mathematical or logical functions to manipulate and process data, generate charts and graphs, and interpret the results. 8.1.12 A. Basic Computer Skills and Tools 5. Produce a multimedia project using text, graphics, moving images, and sound. 8.1.12 B. Application of Productivity Tools 9. Create and manipulate information, independently and/or collaboratively, to solve problems and design and develop products. 8.1.12 B. Application of Productivity Tools 11. Identify a problem in a content area and formulate a strategy to solve the problem using brainstorming, flowcharting, and appropriate resources. 8.2.12 B. Design Process and Impact Assessment 3. Develop methods for creating possible solutions, modeling and testing solutions, and modifying proposed design in the solution of a technological problem using hands-on activities. 9.1.12.C.5 Assume a leadership position by guiding the thinking of peers in a direction that leads to successful completion of a challenging task or project 9.1.4.F.2 Establish and follow performance goals to guide progress in assigned areas of responsibility and accountability during classroom projects and extra-curricular activities. 9.2.12.A.1 Analyze the relationship between various careers and personal earning goals. 9.2.12.A.2 Identify a career goal and develop a plan and timetable for achieving it, including educational/training requirements, costs, and possible debt. 9.2.12.A.5 Evaluate current advances in technology that apply to a selected occupational career cluster. 9.3.12.C.2 Characterize education and skills needed to achieve career goals, and take steps to prepare for postsecondary options, including making course selections, preparing for and taking assessments, and participating in extra-curricular activities.
Functions and Graphs What is a function? How can we determine the domain and range of a function? How can we apply transformations to parent graphs to graph more complicated functions? What are the characteristics of certain functions? Can we create new functions from old functions? What are inverses and are they necessarily also functions? Function, domain, range, piece-wise function, vertical line test, increasing, decreasing, maxima, minima, interval notation, concavity, point of inflection, parent graphs, transformation, symmetry, composite function, inverse function, horizontal line test, oneto-one, domain restriction Students will be able to: Determine if a relation or graph is a function. Find the domain and range of a given function. Evaluate piece-wise defined functions and greatest integer functions. Define and recognize parent functions. Graph transformations of the parent graphs. Determine if a graph has x-axis, y-axis or origin symmetry. Determine if a function is even, odd or neither. Form sum, difference, product, quotient and composite functions and find their domains. Define and find inverse functions from tables, graphs and equations. Determine if an inverse is a function using composition of functions. Standards associated with objectives F-IF.1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). F-IF.2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. F-IF.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. F-IF.5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. F-IF.7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. F-IF.8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. F-BF.1. Write a function that describes a relationship between two quantities Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time. F-BF.3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. F-BF.4. Find inverse functions. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x 3 or f(x) = (x+1)/(x 1) for x 1. (+) Verify by composition that one function is the inverse of another. (+) Read values of an inverse function from a graph or a table, given that the function has an inverse. (+) Produce an invertible function from a non-invertible function by restricting the domain. Give students characteristics of a graph: when it is increasing, decreasing, concave up, concave down, where the extrema are located, where points of inflection are located and have them sketch the graph. Have students collect data from the most recent census. Have them build a scatter
plot of the data, find its inverse, find the domain and range of both functions. Have students create a composite function on an index card. Students must them swap cards and figure out the component functions of the composite functions. Have students (in a group) develop a lesson on piecewise functions. They may present their lesson in any way they choose: blackboard, Smartboard, video, Jing, etc. At the end of each chapter in the Holt Pre-Calculus: A Graphing Approach there is a can do calculus section. Have students who are ahead of the curve in the main lesson work on these projects to gain a stronger foundation for Calculus. Polynomial and Rational Functions How can we simplify rational expressions? Is there a limit to the number of roots a function has? Can a polynomial be simplified and what is the best way to do so? Can we sketch a graph of a polynomial function of degree greater than 2 without a graphing calculator? What are asymptotes and what do they tell us about the behavior of a graph? What is the Fundamental Theorem of Algebra If given the roots of a polynomial function, can we determine the shape of the graph and a general equation for the function? What do polynomial functions looks like if they have complex roots? Polynomial, constant term, degree, linear, quadratic, cubic, quartic, synthetic division, zeros, factors, Remainder Theorem, Factor Theorem, irreducible, continuous, end behavior, multiplicity, vertical asymptote, horizontal asymptote, oblique asymptote, discontinuity, holes, Big-Little Concept, complex roots, complex conjugate, Fundamental Theorem of Algebra Students will be able to: Divide polynomials using synthetic division and apply both the Remainder Theorem and the Factor Theorem. Determine the maximum number of zeros a polynomial can have. Factor a polynomial completely. Recognize the shape of basic polynomial functions, including behavior at the x- intercepts of specific multiplicities, local extrema, and end behavior as well as write the general equation of the polynomial function if given the graph. Find the domain, intercepts, and asymptotes of rational functions and graph them. Apply the Fundamental Theorem of Algebra.
Find complex conjugates, the number of zeros of a polynomial and give complete factorizations of a polynomial expression. Standards associated with objectives F-IF.7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. d. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. A-APR.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). A-APR.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. A-APR.4. Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x 2 + y 2 ) 2 = (x 2 y 2 ) 2 + (2xy) 2 can be used to generate Pythagorean triples. A-APR.5. (+) Know and apply the Binomial Theorem for the expansion of (x + y) n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal s Triangle. 1 N-CN.9. (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. Give students the roots of a polynomial function. Have them sketch the general shape of the function and then determine a polynomial equation that describes the graph. Have students determine the size square that should be cut from each corner of a rectangular-shaped piece of paper to create an open-top box with a given value. Students will need to determine a polynomial to determine the solution. Have students do a complete graph of a rational function, complete with intercepts, asymptotes, and extrema. Any student wishing to explore complex numbers further may want to explore the Mandelbrot Set, which is the set of complex numbers c such that the orbit of 0 under the function does not approach infinity. Examine each diagonal of Pascal s Triangle. Explain why the nth term of each diagonal represented by a polynomial. Find examples of patterns in the triangle that can be described by linear, quadratic, cubic, quartic, and quintic polynomials. Present findings in a variety of ways individual to each student/group. At the end of each chapter in the Holt Pre-Calculus: A Graphing Approach there is a can do calculus section. Have students who are ahead of the curve in the main lesson work on these projects to gain a stronger foundation for Calculus.
Exponential and Logarithmic Functions What are rational exponents? How do we simplify radical and rational exponents? What are exponential and logarithmic functions? Can we transform exponential and logarithmic functions as we do polynomial functions? Is there a real-world purpose to exponential and logarithmic functions? How can we solve exponential and logarithmic equations? Rational exponent, radical exponent, rationalizing, irrational, Exponential function, exponential growth and decay, e, compounded interest, common logarithm, regular logarithm, natural logarithm, natural exponential function, properties of logarithms Students will be able to: Define and apply both rational and irrational exponents. Simplify expressions containing radicals and rational exponents. Graph and identify transformations of exponential functions. Use exponential functions to solve real-world applications. Evaluate common, regular and natural logarithmic functions both with and without a calculator. Convert flexibly between exponential and equation equivalents. Graph and identify transformations of logarithmic functions. Apply the properties of logarithms to simplify expressions involving logarithms. Solve exponential and logarithmic equations as they apply to real-world problems. Standards associated with objectives N-RN.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 5 1/3 to be the cube root of 5 because we want (5 1/3 ) 3 = 5 (1/3)3 to hold, so (5 1/3 ) 3 must equal 5. N-RN.2. Rewrite expressions involving radicals and rational exponents using the properties of exponents. F-IF.7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. F-BF.3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific
values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. F-LE.1. Distinguish between situations that can be modeled with linear functions and with exponential functions. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. F-LE.2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). F-LE.3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. F-LE.4. For exponential models, express as a logarithm the solution to ab ct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. F-LE.5. Interpret the parameters in a linear or exponential function in terms of a context. A-APR.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. A-APR.7. (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. Have students build a chart that compares the interest gained on a principal amount at a fixed interest rate for a fixed number of years but with varying compounding periods. Have a discussion on what compounding period seems better and would it be best to compound continuously? Would the money grow without bound? Have students graph both and. Have a discussion on the comparison of these two graphs (domain, range, increasing, decreasing, concavity, symmetry, etc.) Have students practice converting from exponential form to logarithmic form and vice versa. This is an important skill when solving both types of equations. Have students collect data from the most recent census or from another population source (biology data, death rates, etc.). Have them build a scatter plot of the data and determine what type of model best fits the population growth or decay: linear, exponential, logarithmic, etc. then, have them determine a regression equation describing the data.
At the end of each chapter in the Holt Pre-Calculus: A Graphing Approach there is a can do calculus section. Have students who are ahead of the curve in the main lesson work on these projects to gain a stronger foundation for Calculus. Introduction to Trigonometry What is a trigonometric ratio? What is a reference triangle? How can we use the special right triangles from geometry to find trigonometric values? Is there another way to measure angles other than degrees? How are degrees and radians related? Can angles have a negative measure? What does this mean? What is a unit circle? What are trigonometric identities and how are they used? Angle, vertex, sides, degrees, minutes, seconds, hypotenuse, adjacent, opposite, trigonometric ratios, special angles, SOHCAHTOA, angle of elevation, angle of depression, initial side, terminal side, coterminal, unit circle, radian, linear speed, angular speed, cosine, sine, tangent, secant, cosecant, cotangent, exact value, quotient identities, reciprocal identities, Pythagorean identities, period, periodicity identities, negative angle identities Students will be able to: Define the six trigonometric ratios of an acute angle in terms of a right triangle. Evaluate trigonometric ratios using reference triangles, calculator, and special right triangles. Solve right triangles using trigonometric ratios and apply trigonometric ratios to real-world problems. Define radian measure and convert flexibly between radians and degrees. Extend the definition of angle measure to negative angles and angles greater than 180 degrees. Define the trigonometric ratios in the coordinate plane and in terms of the unit circle. Develop and apply basic trigonometric identities. Standards associated with objectives F-TF.1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
F-TF.2. Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. F-TF.3. (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosines, and tangent for x, π + x, and 2π x in terms of their values for x, where x is any real number. F-TF.4. (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. G-SRT.6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. G-SRT.7. Explain and use the relationship between the sine and cosine of complementary angles. G-SRT.8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Using geometer sketchpad 5.0, show examples of graphs of sine functions and show visually their number of cycles Use geometer sketchpad to measure arc length, radian measure, and radius of any given circle. Show using the unit circle where tangent functions are undefined, and find its asymptotes Build a unit circle using reference triangles and the special right triangle ratios. Allow students to work with pictures of graphs of non whole coefficients and determine the properties of a trigonometric function Have students review the unit circle and see if they can interpolate values for non special angles Work with real-world word problems to show how trigonometric functions are used and how they will have more than one answer Students who are struggling should review the unit circle and learn to use a graphing calculator to aid in their knowledge of trigonometry Reinforce oscillations using geometer sketchpad and show the period and amplitude of basic trigonometric functions. At the end of each chapter in the Holt Pre-Calculus: A Graphing Approach there is a can do calculus section. Have students who are ahead of the curve in the main lesson work on these projects to gain a stronger foundation for Calculus.
Trigonometric Graphs What do the graphs of the trigonometric functions look like? What are the characteristics of the graphs of the trigonometric functions? What are period, amplitude and phase shift? Can we graph trigonometric transformations in the same manner as polynomial, exponential, and logarithmic functions? Even function, odd function, Even-Odd identities, period, amplitude, phase shift Students will be able to: Graph sine, cosine and tangent functions as well as their transformations. Recognize the graphs of the secant, cosecant and cotangent functions. State the period, amplitude, and phase shift of the trigonometric functions. Standards associated with objectives F-BF.3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. F-TF.5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. Given a trigonometric function with transformations, have students list what the transformations are, graph the parent trigonometric function, and then transform the parent function for the final graph. Have students graph the trigonometric functions by hand (making an x-y table) and then compare to the graphs on the graphing calculator. Have students list as many characteristics as they can abut each graph (intercepts, roots, maximums, minimums, asymptotes, etc,) Have students look up what a sinusoid graph is and report back, using whatever method they choose, as to what one is and how they can be applied. At the end of each chapter in the Holt Pre-Calculus: A Graphing Approach there is a can do calculus section. Have students who are ahead of the curve in the main lesson work on these projects to gain a stronger foundation for Calculus.
Solving Trigonometric Equations Can we solve equations that involve trigonometric functions? How do we recognize that we have all solutions to a trigonometric equation? What are inverse trigonometric functions? Basic trigonometric equation, inverse trigonometric function, arcsine, arcosine, arctangent, arcsecant, arccosecant, arccotangent, Students will be able to Solve trigonometric equations graphically and algebraically. Recognize and find all solutions to a trigonometric equation. Standards associated with objectives F-TF.6. (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed. F-TF.7. (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context. Have students sketch the trigonometric functions and determine if they are one-toone. If they are not, have them determine the domain restriction on each function that will make it one-to-one. Review solving simple algebraic equations and relate them to solving trigonometric equations, stressing that one must use the inverse of a trigonometric function when solving for an angle. Stress the importance of changing the mode of the calculator to degrees or radians as appropriate. Do an activity with them to show how not being in the correct mode CAN affect the outcome of the solution of the trigonometric equation. Have students research simple harmonic motion and sound waves and report on how trigonometry applies in these areas. At the end of each chapter in the Holt Pre-Calculus: A Graphing Approach there is a can do calculus section. Have students who are ahead of the curve in the main lesson work on these projects to gain a stronger foundation for Calculus.
Trigonometric Identities and Proofs Can we use the basic identities to prove more complicated identities? What are the cofunction identities? What are the double-angle and half-angle identities? How can we use identities to help solve complicated trigonometric equations? Cofunction identities, double-angle and half-angle identities Students will be able to: Identify possible identities by using graphs. Apply algebraic strategies to prove identities. Use the cofunction identities. Use the double-angle and half angle identities. Use trigonometric identities to solve trigonometric equations. Standards associated with objectives F-TF.8. Prove the Pythagorean identity sin 2 (θ) + cos 2 (θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle. F-TF.9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. Have students graph both sides of a given identity on the graphing calculators to see visually that the left side does indeed equal the right side. Review the following algebraic maneuvers BEFORE trying to prove trigonometric identities: Dividing fraction, adding and subtracting fractions, working with conjugates, multiplying binomials. Have students work in partners, at first, to solve some of the more basic trigonometric identities. Tell students that there are three sets of two parent functions each whose product results in another parent function. There is also a set of three parent functions whose product is another parent functions. Ask them to see how many of these four sets they can find, then prove the identity. At the end of each chapter in the Holt Pre-Calculus: A Graphing Approach there is a can do calculus section. Have students who are ahead of the curve in the main lesson work on these projects to gain a stronger foundation for Calculus.
Trigonometric Applications How can we solve non-right triangles? Is there a way to find the area of a triangle if we do not know its height but only the lengths of the tree sides? What is a vector? Can we perform operations on vectors? Oblique triangles, Law of Cosines, Law of Sines, Heron s Formula, vector, initial point, terminal point, length, magnitude, equivalent vectors, components, scalar multiplication, vector addition, vector subtraction, zero vector, unit vector, resultant Students will be able to: Solve oblique triangles by using the Law of Cosines. Solve oblique triangles using the Law of Sines. Use area formulas to find the areas of triangles. Find the components and magnitude of a vector. Perform scalar multiplication of vectors, vector addition, and vector subtraction. Perform operations with vectors, determine direction angle of a vector, and determine resultant forces in real-world applications. Standards associated with objectives G-SRT.9. (+) Derive the formula A = 1/2 ab sin(c) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. G-SRT.10. (+) Prove the Laws of Sines and Cosines and use them to solve problems. G-SRT.11. (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). N-CN.1. Know there is a complex number i such that i 2 = 1, and every complex number has the form a + bi with a and b real. N-CN.2. Use the relation i 2 = 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. N-CN.3. (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. N-CN.4. (+) Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number. N-CN.5. (+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (-1 + 3 i) 3 = 8 because (-1 + 3 i) has modulus 2 and argument 120.
N-CN.7. Solve quadratic equations with real coefficients that have complex solutions. N-CN.8. (+) Extend polynomial identities to the complex numbers. For example, rewrite x 2 + 4 as (x + 2i)(x 2i). N-VM.1. (+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, v, v, v). N-VM.2. (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point. N-VM.3. (+) Solve problems involving velocity and other quantities that can be represented by vectors. N-VM.4. (+) Add and subtract vectors. Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes. Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. Understand vector subtraction v w as v + ( w), where w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise. N-VM.5. (+) Multiply a vector by a scalar. Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(v x, v y) = (cv x, cv y). Compute the magnitude of a scalar multiple cv using cv = c v. Compute the direction of cv knowing that when c v 0, the direction of cv is either along v (for c > 0) or against v (for c < 0). Use real-world applications for the Law of Cosines ( the heading of 2 planes and the resultant distance between them, and finding the distance across a lake from a fixed point with known distances and included angle) and the Law of Sines (an airplane flying at a fixed altitude and finding its distance from a landing area). Review the distance between two points formula and the slope between two points formula and relate them to the equivalent vectors formula. Have a discussion about these first two formulas help us when we reposition vectors at the origin. There are 9 properties of vector addition and scalar multiplication. Have pairs of students take one property an explain it to the class. Research the Dot Product of two vectors and present findings. Determine how the Mandelbrot Set relates to fractal images. At the end of each chapter in the Holt Pre-Calculus: A Graphing Approach there is a can do calculus section. Have students who are ahead of the curve in the main lesson work on these projects to gain a stronger foundation for Calculus.
Analytic Geometry What are the conic sections? How can we represent an ellipse, a hyperbola, and a circle? What are the characteristics of the conic sections? Circle, center, radius, conic sections, ellipse and ellipse terms: center, major axis, minor axis, foci, vertices; hyperbola and hyperbola terms: distance difference, foci, asymptotes, center, focal axis, vertices, auxiliary rectangle; parabola and parabola terms: focus, directrix, axis, vertex; polar coordinates, origin, pole, polar axis, polar equatin, cardioids, eccentricity, Students will be able to: Define a circle and write its equation. Define and graph an ellipse. Write the equation of an ellipse. Identify characteristics of an ellipse. Define and graph a hyperbola. Write the equation of a hyperbola. Identify characteristics of a hyperbola. Standards associate with objectives G-GMD.4. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Using actual physical models as well as images on Geometer s Sketchpad, show how to generate each of the conic sections and how they are related. Have students research how conic sections are used in the development of telescopes. Review solving systems briefly in order that students will be better able to understand this concept when finding the general form of each of the conic sections. There are some REALLY interesting projects at the end of Chapter 12 in the old, green University of Chicago School Mathematics Project book Functions, Statistics, and Trigonometry. These would be really good to use for differentiated instruction. At the end of each chapter in the Holt Pre-Calculus: A Graphing Approach there is a can do calculus section. Have students who are ahead of the curve in the main lesson work on these projects to gain a stronger foundation for Calculus.
Limits and Continuity What is a limit? What techniques can be used to calculate a limit? What is the difference between a one-sided limit and a two-sided limit? How is calculating a limit related to the continuity of a function? What do limits tell us about the asymptotes of the graph of a function? Limit, vertical asymptote, horizontal asymptote, one-sided limit, two-sided limit, end behavior, continuity Students will be able to: Have a basic understanding of the intuitive idea of a limit. Understand the difference between a one-sided limit and a two-sided limit. Calculate simple limits using the properties of limits. Calculate limits at infinity and describe the end behavior of a graph. Determine if a given function is continuous at a point or on an interval. Standards associated with objectives No standards specifically relate to these topics. Have students sketch graphs with given specific characteristics. Use the graphing calculator to analyze tables and graphs of difficult functions to determine a function s limit. Complete the continuity worksheet from the University of Delaware AP Calculus course binder. Use released College Board AP Exam questions. It is very difficult to differentiate in a course of this nature since the material is brand new to everyone. One possible way to differentiate within the course is to allow students to sign out reference material such as Five Steps to a Five or AP Calculus AB Flash Cards in order for them to work at their own individual pace.