Curriculum Map

Transcription

1 Analysis of Functions/Trigonometry Curriculum Map Mathematics Florida s Volusia County Curriculum Maps are revised annually and updated throughout the year. The learning goals are a work in progress and may be modified as needed.

2 Florida s s for Mathematical Practice 1. Make sense of problems and persevere in solving them. (MAFS.K12.MP.1) Solving a mathematical problem involves making sense of what is known and applying a thoughtful and logical process which sometimes requires perseverance, flexibility, and a bit of ingenuity. 2. Reason abstractly and quantitatively. (MAFS.K12.MP.2) The concrete and the abstract can complement each other in the development of mathematical understanding: representing a concrete situation with symbols can make the solution process more efficient, while reverting to a concrete context can help make sense of abstract symbols. 3. Construct viable arguments and critique the reasoning of others. (MAFS.K12.MP.3) A well-crafted argument/critique requires a thoughtful and logical progression of mathematically sound statements and supporting evidence. 4. Model with mathematics. (MAFS.K12.MP.4) Many everyday problems can be solved by modeling the situation with mathematics. 5. Use appropriate tools strategically. (MAFS.K12.MP.5) Strategic choice and use of tools can increase reliability and precision of results, enhance arguments, and deepen mathematical understanding. 6. Attend to precision. (MAFS.K12.MP.6) Attending to precise detail increases reliability of mathematical results and minimizes miscommunication of mathematical explanations. 7. Look for and make use of structure. (MAFS.K12.MP.7) Recognizing a structure or pattern can be the key to solving a problem or making sense of a mathematical idea. 8. Look for and express regularity in repeated reasoning. (MAFS.K12.MP.8) Recognizing repetition or regularity in the course of solving a problem (or series of similar problems) can lead to results more quickly and efficiently.

3 Analysis of Functions/Trigonometry: Common Core State s At A Glance First Quarter-AOF Second Quarter-AOF Third Quarter-Trig Fourth Quarter-Trig Unit 1-Functions and Graphs MAFS.912.F-IF.3.7 MAFS.912.F-BF.1.1 MAFS.912.F-BF.2.4 Unit 2- Quadratic Functions and Complex Numbers MAFS.912.F-IF.3.8a MAFS.912.N-CN.1.3 MAFS.912.N-CN.2.5 MAFS.912.F-IF.3.7a MAFS.912.N-CN.3.9 Unit 3- Polynomial Functions MAFS.912.F-IF.3.7c MAFS.912.F-IF.3.8a MAFS.912.N-CN.3.9 MAFS.912.A-APR.2.2 Unit 4-Rational Expressions and Functions MAFS.912.A-APR.4.6 MAFS.912.A-APR.4.7 MAFS.912.F-IF.3.7d MAFS.912.F-IF.3.8a Unit 5-Exponential and Logarithmic Functions MAFS.912.F-BF.2.5 MAFS.912.F-IF.3.7e MAFS.912.F-IF.3.8b MAFS.912.F-LE.1.4 Unit 6- Trigonometric Functions MAFS.912.F-TF.1.1 MAFS.912.F-TF.1.2 MAFS.912.F-TF.1.3 MAFS.912.G-SRT.3.8 Unit 7- Trigonometric Applications MAFS.912.G-SRT.4.9 MAFS.912.G-SRT.4.10 MAFS.912.F-TF.2.7 MAFS.912.F-TF.3.8 MAFS.912.F-TF.3.9 Unit 8- Graphing Trigonometric Functions and their Inverses MAFS.912.F-TF.1.4 MAFS.912.F-TF.2.5 MAFS.912.F-TF.2.6 Unit 9- Vectors MAFS.912.N-VM.1.1 MAFS.912.N-VM.1.2 MAFS.912.N-VM.1.3 MAFS.912.N-VM.2.4 MAFS.912.N-VM.2.5 MAFS.912.N-CN.2.4 MAFS.912.N-CN.2.6

4 The following English Language Arts CCSS should be taught throughout the course: LAFS.1112.RST.1.3: Follow precisely a complex multistep procedure when carrying out experiments, taking measurements, or performing technical tasks; analyze the specific results based on explanations in the text. LAFS.1112.RST.2.4: Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in specific scientific or technical context relevant to grades texts and topics. LAFS.1112.RST.3.7: Integrate and evaluate multiple sources of information presented in diverse formats and media (e.g., quantitative data, video, multimedia) in order to address a question or solve a problem. LAFS.1112.SL.1.1: Initiate and participate effectively in a range of collaborative discussions with diverse partners on grades topics, texts, and issues, building on others ideas and expressing their own clearly and persuasively. LAFS.1112.SL.1.2: Integrate multiple sources of information presented in diverse media or formats in order to make informed decisions and solve problems, evaluating the credibility and accuracy of each source and noting any discrepancies among the data. LAFS.1112.SL.1.3: Evaluate a speaker s point of view, reasoning, and use of evidence and rhetoric, assessing the stance, premises, links among ideas, word choice, points of emphasis, and tone used. LAFS.1112.SL.2.4: Present information, findings and supporting evidence, conveying a clear and distinct perspective, such that listeners can follow the line of reasoning, alternative or opposing perspectives are addressed, and the organization, development, substance, and style are appropriate to purpose, audience, and a range of formal and informal tasks. LAFS.1112.WHST.1.1: Write arguments focused on discipline-specific content. LAFS.1112.WHST.2.4: Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience. LAFS.1112.WHST.3.9: Draw evidence from informational texts to support analysis, reflection, and research.

5 MAFS.912.F-IF.3.7a 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. SMP #7, #8 MAFS.912.F-BF.1.1 Write a function that describes a relationship between two quantities. a. Determine an explicit expression, a recursive process, or steps for calculation from a context. b. Combine standard function types using arithmetic operations. c. Compose functions. SMP #4, #7 Course: Analysis of Functions/Trigonometry Unit 1-Functions and Graphs In what ways can function be built? identify that the parent function for lines is the line f(x)=x. identify the point-slope form of a linear functions as graph a line in point-slope form and use the graph to show where the starting point (x 1, y 1 ) and the slope (m) are represented on the graph. identify the slope-intercept form of a linear function as f(x)=mx+b graph a line in slope-intercept form and use the graph to show where the y-intercept (b) and the slope (m) are represented on the graph. identify the standard form of a linear function as Ax+By=C use the definition of x-intercept and y-intercept to find the intercepts of a standard form line and graph the line. relate the constants A, B, and C to the values of the x-intercept, y- intercept, and slope. define explicit and recursive expressions of a function. identify the quantities being compared in a real-world problem. write a explicit and/or recursive expressions of a function to describe a real-world problem. recall the parent function apply transformations to equations of parent functions. combine different parent functions (adding, subtracting, multiplying, and/or dividing) to write a function that describes a real-world problem. compose two or more functions. explain a multi-step real world problem in terms of function composition and write an equation to describe the composition. Focus on the linear functions section of this standard in this unit. A general overview of functions would fit well here.

6 Unit 1-Functions and Graphs (cont) MAFS.912.F-BF.2.4 Find inverse functions. a. 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. b. Verify by composition that one function is the inverse of another. c. Read values of an inverse function from a graph or a table, given that the function has an inverse. d. Produce an invertible function from a non-invertible function by restricting the domain. In what ways can function be built? define inverse of a function. write the inverse of a function by solving f(x)=c for x. explain that after solving f(x)=c for x, c can be considered the input and x can be considered the output. write the inverse of a function in standard notation by replacing the x in my inverse equation with y and replacing c in my inverse equation with x. use the composition of functions to verify that g(x) and f(x) are inverses by showing that g(f(x))=f(g(x))=1. recall the definition of one-to-one function. decide if a function has an inverse using the horizontal line test. use the definition of function, inverse function, and one-to-one function to explain why the horizontal line test works. list values of an inverse given a table or graph of a function that has an inverse. identify and eliminate the part(s) of a graph that cause it to fail the vertical line test. state the domain of a relation that has been altered. write the inverse of the invertible function in function notation.

7 Unit 2-Quadratic Functions and Complex Numbers MAFS.912.F-IF.3.8a 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. SMP #2, #7 MAFS.912.N-CN.1.3 Find the conjugate of complex number; use conjugates to find moduli and quotients of complex numbers. SMP #7 MAFS.912.N-CN.2.5 Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. SMP #7 What are the defining characteristics of quadratic functions? explain that there are three forms of quadratic functions (standard, vertex, and factored form) and that the graph of all three forms is a parabola. find the x-intercepts of a quadratic written in factored form and use them to find the axis of symmetry. use the axis of symmetry of a quadratic to find the vertex of a parabola. identify the line of symmetry and the vertex of a quadratic written in vertex form. sketch a graph of a parabola written in vertex form. tell if a quadratic written in vertex form has x-intercepts by looking at the equation. use algebra to find the x-intercepts of a quadratic written in vertex form. convert a standard form quadratic to factored form by factoring and to vertex from by completing the square. write the function that describes a parabola in all three forms when I am given a graph with the x-intercepts, y-intercept, and vertex labeled. determine the conjugate of a complex number (a+bi and a-bi). explain why multiplying a complex number by its conjugate results in a real number. multiply complex numbers determine the quotient of two complex numbers using the conjugate of the denominator. calculate the modulus, r, of a complex number (r = a 2 + b 2 ) add, subtract, multiply, and divide (using conjugation) complex numbers. calculate the modulus and argument of a complex number. represent addition, subtraction, multiplication, and division of complex numbers by graphing on the complex plane. use geometric representation on the complex plane to perform operations with complex numbers. explain what is happening geometrically when a complex number is squared, cubed, etc. relate conjugation to a simple geometric transformation and explain the relationship. justify why it is easier to add, subtract, multiply, and divide in polar or rectangular form.

8 Unit 2-Quadratic Functions and Complex Numbers (cont) MAFS.912.F-IF.3.7a 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 quadratic functions and show intercepts, maxima, and minima. SMP #7, #8 MAFS.912.N-CN.3.9 Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. SMP #7 What are the defining characteristics of quadratic functions? explain that the parent function for quadratic functions is the parabola f(x)=x 2 explain that the minimum or maximum of a quadratic is called the vertex. identify whether the vertex of a quadratic will be a minimum or maximum by looking at the equation find the y-intercept of a quadratic by substituting 0 for x and evaluating. estimate the vertex of a quadratic by evaluating different values for x. use calculated values while looking for a minimum or maximum to decide if the quadratic has x-intercepts. estimate the x-intercepts of a quadratic by evaluating different values of x. graph a quadratic using evaluated points. use technology to graph a quadratic and to find precise values for the x-intercept(s) and the maximum or minimum. explain the Fundamental Theorem of Algebra in my own words by using simple polynomials and their graphs. use the Linear Factorization Theorem to demonstrate that a quadratic polynomial, f(x)=a 2 (x c 1 )(x c 2 ) has two linear factors under the set of complex numbers. solve a quadratic equation in factored form for its zeroes even if the zeroes are complex. Include quadratic functions with complex roots.

9 Unit 3- Polynomial Functions What are the defining characteristics of polynomial functions? Learning Targets MAFS.912.F-IF.3.7c define a polynomial and identify the degree of a polynomial. Graph functions expressed symbolically and use pictures to explain why x 3, x 4, and higher degree polynomials do not have show key features of the graph, by hand in parent functions as x and x 2. simple cases and using technology for more determine the x-intercepts of a polynomial when looking at a graph of the complicated cases. function. c. Graph polynomial functiosn, identifying determine the multiplicity of the x-intercepts of a polynomial when looking at zeros when suitable factorizations are a graph of the function. available, and showing end behavior. approximate the factored equation of a polynomial function when looking at a SMP #2, #3 graph of the function. determine the end behavior of a polynomial by looking at the degree and leading coefficient of the equation. use technology to graph a polynomial and to find precise values for the x- intercept(s) and the maximums and minimum (turn points). MAFS.912.F-IF.3.8a 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. use algebra to find the x-intercepts of a quadratic written in vertex form. convert a standard form quadratic to factored form by factoring and to vertex from by completing the square. write the function that describes a parabola in all three forms when I am given a graph with the x-intercepts, y-intercept, and vertex labeled. Apply the process of factoring quadratics to polynomial functions. SMP #2, #7 MAFS.912.N-CN.3.9 Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. SMP #7 explain the Fundamental Theorem of Algebra in my own words by using simple polynomials and their graphs. use the Linear Factorization Theorem to demonstrate that a quadratic polynomial, f(x)=a 2 (x c 1 )(x c 2 ) has two linear factors under the set of complex numbers. solve a quadratic equation in factored form for its zeroes even if the zeroes are complex. Apply the Fundamental Theorem of Algebra to polynomial functions.

10 Unit 3- Polynomial Functions (cont) What are the defining characteristics of polynomial functions? Learning Targets MAFS.912.A-APR.2.2 divide polynomials using long division and synthetic division and apply the Know and apply the Remainder Theorem: Remainder Theorem (when appropriate) to check the answer. For a polynomial p(x) and a number a, the apply the Remainder Theorem to determine is a divisor (x-a) is a factor of the remainder on division by x-a is p(a), so polynomial p(x). p(a)=0 if and only if (x-a) is a factor of p(x). SMP #8

11 MAFS.912.A-APR.4.6 Rewrite simple rational expressions in different forms; write a(x) in the form q(x)+r(x), b(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. SMP #5, #7 MAFS.912.A-APR.4.7 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. SMP #2, #7 MAFS.912.F-IF.3.7d 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. SMP #2, #3 Course: Analysis of Functions/Trigonometry Unit 4- Rational Expressions and Functions What are the defining characteristics of rational functions? define rational expression determine the best method of simplifying the given rational expression (inspection, long division, computer algebra system). simplify rational expressions by inspection or by using long division. simplify complicated rational expressions using a computer algebra system. write a rational expression a(x) where a(x) is the dividend and b(x) is the divisor in b(x) the form q(x)+ r(x) where q(x) is the quotient and r(x) is the remainder. b(x) apply the definition of a rational number to explain why adding, subtracting, multiplying, or dividing two rational numbers always produce a rational number. apply the definition of a rational expression to explain why adding, subtracting, multiplying, or dividing two rational expressions always produce a rational expression. add and subtract rational expressions. multiply and divide rational expressions. define rational function as the ratio of two polynomials. state the end behavior of a rational function when looking at a graph of the function. find the y-intercept and the x-intercepts of a rational function. describe the end behavior of a rational function by interpreting the graph. determine if a rational function has a horizontal asymptote and find the asymptote. sketch a graph of a rational function based on its domain, x-intercepts, y-intercept, and end behavior. use technology to graph rational functions and to find precise values for the x- intercept(s) and the maximums and minimum (turning points).

12 Unit 4- Rational Expressions and Functions (cont) MAFS.912.F-IF.3.8a 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. SMP #2, #7 What are the defining characteristics of rational functions? use algebra to find the x-intercepts of a quadratic written in vertex form. convert a standard form quadratic to factored form by factoring and to vertex from by completing the square. write the function that describes a parabola in all three forms when I am given a graph with the x-intercepts, y-intercept, and vertex labeled. Use factoring to simplify rational functions, as well as identify key features of the graph of the rational function.

13 Unit 5- Exponential and Logarithmic Functions What is the relationships exist between exponential and logarithmic functions? MAFS.912.F-BF.2.5 state and explain the inverse relationship of an exponential function is a logarithmic Understand the inverse relationship between function (and vice versa) exponents and logarithms and use this estimate or solve for the values of logarithms by evaluating powers of the base. relationship to solve problems involving create the inverse of an exponential equation when given its table or graph and write logarithms and exponents. the inverse in logarithmic form. SMP #3, #6 solve problems with variables in an exponent or logarithm by applying the inverse relationship to logarithms and exponents. MAFS.912.F-IF.3.7e 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. SMP #2, #3 MAFS.912.F-IF.3.8b Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. b. Use the properties of exponents to interpret expressions for exponential functions. SMP #2, #7 explain that the parent function for exponentials is f(x)=b x where b is a positive number. determine the domain, range, and end behavior (horizontal asymptote) of an exponential function when looking at its graph. classify exponential functions in function notation as growth or decay. substitute convenient values for x to generate a table and graph of an exponential function. explain how a simple geometric transformation changes a growth graph to a decay graph. distinguish between exponential functions that model exponential growth and exponential decay. interpret the components of an exponential function in the context of a problem (e.g. y = t 3 describes the quantity that was initially 5 and increases 22.5% every three years.) use the properties of exponents to rewrite an exponential function to emphasize one of its properties (e.g. y = t 3 is approximately y = t, which means that increasing 22.5% every three years is about the same as increasing 7% per year.) Logarithms are challenging for students. Review the definition and properties of logarithms. Only cover exponential and logarithmic functions in this unit.

14 Unit 5- Exponential and Logarithmic Functions (cont) MAFS.912.F-LE.1.4 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. SMP #5, #7 What is the relationships exist between exponential and logarithmic functions? define exponential function and logarithmic function write an exponential equationa b ct d = d and log b = ct are equivalent. use powers of 2 or 10 to estimate the value of log 2 (x)or log 10 (x). use a calculator to evaluate a logarithm with a base of 10 or e. apply the change of base formula to evaluate the logarithm with a base of 2 using a calculator. a

15 Unit 6- Trigonometric Functions MAFS.912.T-TF.1.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. SMP #6 MAFS.912.T-TF.1.2 Explain how the unit circle in the coordinate plane enables the extensions of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. SMP #2, #3 When does a function best model a situation? define unit circle, central angle, and intercepted arc. define the radian measure of an angle. extend the definition of radian measure to show that an angle measure of one radian occurs when the length of the arc and the raidus of the circle are the same. use a similarity approach to find the radian measure of central angles in circles that are not unit circles. define a radian and unit circle. label the unit circle in radians since it is known that one revolution of the unit circle is equal to 2pi radians. draw central angles of given radian measures on the unit circle with the vertex at the origin and the initial ray on the positive x-axis. recall that on the unit circle, the cosine of an angle is defined to be the x- coordinate where the terminal ray of the angle crosses the unit circle and the sine of an angle is defined to be the y-coordinate where the terminal ray of the angle crosses the unit. identify the cosine and sine of an angle when given a graph of the unit circle with the coordinates labeled. explain why the right triangle definitions of cosine and sine does not allow cosines and sines to have negative values, while the unit circle definitions of cosine and sine allow cosines and sines to have negative values define co-terminal angles. identify many co-terminal angles when given a radian measure explain why co-terminal angles will all produce the same output when evaluated as the inputs of a trigonometric function. -Elements of standards T-TF.1.1, T- TF.1.2, and T-TF.1.3 should be taught in conjunction. -Include secant, cosecant, and cotangent and their reciprocal relationships with sine, cosine, and tangent; include their measures in the unit circle.

16 Unit 6- Trigonometric Functions (cont) MAFS.912.T-TF.1.3 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 π-x, π+x, and 2 π-x in terms of their values of x, where x is any real number. SMP #3, #7 MAFS.912.G-SRT.3.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. SMP #1, #4 When does a function best model a situation? convert between the radian and degree measures of angles using the relationship that one revolution of the unit circle is equal to 2 π radians and 360 degrees. recall the sine, cosine, and tangent ratios for right triangles. draw a right isosceles triangle ( ) in the first quadrant and use it to find the values of cosine, sine, and tangent of π/4 draw an equilateral triangle symmetric about the positive x-axis of the unit circle and use it to find the values of the cosine, sine, and tangent of π/6. use a geometric reflection and the values of the sine, cosine, and tangent of π/6 to find the values of the cosine, sine, and tangent of π/3. explain that each of the triangles drawn in the first quadrant can be reflected to create a congruent triangle in the second, third, and fourth quadrants. define a reference angle. use reference angles and arguments using reflections and rotations to find the values of sine, cosine, and tangent for π-x, π+x, and 2 π-x in terms of their values for x. use angle measures to estimate side lengths and vice versa. solve right triangles by finding the measures of all sides and angles in the triangles using Pythagorean Theorem and/or trigonometric ratios and their inverses. draw right triangles that describe real world problems and label the sides and angles with their given measures. solve application problems involving right triangles, including angle of elevation and depression, navigation, and surveying. Creating a unit circle on a paper plate (or a circle on a coordinate plane) helps students visualize the unit circle and see the patterns in the values of sine, cosine, and tangent.

17 Unit 7- Trigonometric Applications What relationships exist among angle measures and side lengths in triangles? MAFS.912.G-SRT.4.9 understand that two right triangles are created when an altitude is drawn from a Derive the formula A = 1 ab sin C vertex. 2 for the area of a triangle by find the length of a triangle s altitude by using the sine function. drawing an auxiliary line from a use the traditional area formula of a triangle A=1/2*base*height and the sine function vertex perpendicular to the to generate an equivalent area formula A = 1 ab sin C. 2 opposite side. calculate the area of a triangle using the formula A = 1 ab sin C, using any angle of the SMP #2. #7 2 triangle. MAFS.912.G-SRT.4.10 Prove the Law of Sines and Cosines and use them to solve problems. SMP #2, #7 MAFS.912.T-TF.2.7 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. SMP #1, #4 derive the Law of Sines by drawing an altitude in a triangle, using the sine function to find two expressions for the length of the altitude, and simplifying the equation that results from setting these expressions equal. draw an altitude to create two right triangles and can establish the relationships of the sides in each right triangle using the sine and cosine functions of a single angle in the original triangle. derive the Law of Cosines using the Pythagorean Theorem, two right triangles formed by drawing an altitude, and substitution. generalize the Law of Cosines to apply to each included angle (a 2 =b 2 +c 2-2bccosA) use the Law of Sines and Law of Cosines to solve real world problems. solve a trigonometric equation using an inverse trigonometric function use the period of the function to identify multiple solutions of a trigonometric equation. evaluate solutions using a calculator interpret solutions in the context of the problem. compose an original problem situation, construct a trigonometric function to model it, and solve the equation for the inputs that produce specific outputs.

18 Unit 7- Trigonometric Applications (cont) What relationships exist among angle measures and side lengths in triangles? MAFS.912.F-TF.3.8 derive the Pythagorean identity sin 2 θ + cos 2 θ = 1 by using the unit circle definitions of Prove the Pythagorean identity cosine and sine and applying the Pythagorean Theorem. sin 2 θ + cos 2 θ = 1 and use it to use the Pythagorean identity sin 2 θ + cos 2 θ = 1 to calculate the value of sin θ or cos θ find sin θ, cos θ, and tan θ given when I am given sin θ or cos θ and the quadrant of θ. sin θ, cos θ, or tan θ and the sin θ use the quotient identity tan θ = to calculate tan θ. quadrant of the angle. coseθ SMP #7, #8 MAFS.912.F-TF.3.9 Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. SMP #3, #7 use right triangles to prove the sum of angles identities for cosine and sine. convert the sum of angles identities to the difference of angles identities. use the sum and difference formulas for sine and cosine to prove the sum and difference formulas for tangent. recognize angles that can be written using sums of π/3, π/4, and π/6. use the sum and difference formulas for sine, cosine, and tangent to solve for the exact values of the trigonometric functions of other angles. Include double angle formula.

19 Unit 8-Graphing Trigonometric Functions and Their Inverses MAFS.912.T-TF.1.4 Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. SMP #7, #8 MAFS.912.T-TF.2.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. SMP #4, #7 MAFS.912.T-TF.2.6 Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed. SMP #2, #8 How can periodicity be displayed on a graph? label the radian measures on the unit circle with their corresponding ordered pairs (cosine and sine values) identify the quadrants for while cosine, sine, and tangent are positive. use points and their corresponding angle measures on the unit circle to demonstrate that the cosine function is even and the sine and tangent functions are odd. use arguments based on reflections to show that the cosine function is even and the sine and tangent functions are odd. use the unit circle to explain that the periods for cosine and sine are 2pi and the period for tangent is pi. define amplitude, frequency, and midline of a trigonometric function. explain the connection between frequency and period. recognize real-world situations that can be modeled with a periodic function by identifying the amplitude, frequency (or period), and midline. write a function notation for the trigonometric function that models a problem situation, given the amplitude, frequency (or period), and midline of a periodic situation. define inverse function, one-to-one function, and horizontal line test. explain why the functions f(x)=sinx, g(x)=cosx, and h(x)=tanx do not have inverses using the graphs of the functions from 0 to 2pi. eliminate the parts of the graphs of f(x)=sinx, g(x)=cosx, and h(x)=tanx that cause them to fail the horizontal line test and write the new domains after the eliminations. explain that restricting the domain is a mathematical way of saying, I m not allowing values of x that cause the function to fail the horizontal line test. restrict the domains of functions so that they have inverses when given a picture of the graphs. defend the statement, Restricting the domain of a trigonometric function so that the function is always increasing or always decreasing, allows the function to have an inverse. state the restricted domains of f(x)=sinx, g(x)=cosx, and h(x)=tanx.

20 Unit 9- Vectors and Complex Numbers How does knowledge of real numbers and geometry help when working with vectors and matrices? MAFS.912.N-VM.1.1 explain that a vector represents two quantities, magnitude (length), and Recognize vector quantities as having direction (angle). both magnitude and direction. Represent recognize that a vector can be expressed as a directed line segment, written vector quantities by directed line v = v =, PQ where P is the initial point and Q is the terminal point. segments, and use appropriate symbols represent a vector as a directed line segment. for vectors and their magnitudes (e.g. v, use appropriate notation for the magnitude of a vector: v, v, v, or v. v, v, v) SMP #6 MAFS.912.N-VM.1.2 Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point. SMP #7 explain the component from a vector as v =< v 1, v 2 >= (v 1, v 2 ) in terms of horizontal and vertical travel from a starting location. find the component form of a vector with initial point (p 1, p 2 ) and terminal point (q 1, q 2 ) by applying the formula v =< v 1, v 2 >= (q 1 p 1, q 2 p 2 ). MAFS.912.N-VM.1.3 Solve problems involving velocity and other quantities that can be represented by vectors. SMP #1, #4 identify the quantities described in a problem that should be represented as vectors. decide if the information given in a problem allows me to write the vector using components or using direction and magnitude. convert between component form and magnitude and direction form as appropriate. solve real world problems that can be represented by vectors (velocity, force, navigation, etc.) by drawing them to scale. perform operations (addition, subtraction, scalar multiplication) with vectors. solve real world problems that can be represented by vectors (velocity, force, navigation, etc.) by using vector arithmetic.

21 Unit 9- Vectors and Complex Numbers (cont) How does knowledge of real numbers and geometry help when working with vectors and matrices? MAFS.912.N-VM.2.4 Part A Add and subtract vectors. use the triangle inequality to specify a range of possible magnitudes for the sum of two vectors. a. Add vectors end-to-end, calculate the sum of two vectors in component form. component-wise, and by the add vectors by adding them end-to-end (position one vector so its initial point coincides with the parallelogram rule. Understand terminal point of the other). that the magnitude of a sum of add vectors by applying the Parallelogram Rule (position the vectors so both have their initial two vectors is typically not the point at the origin). sum of magnitudes. use pictures to show that the vectors for end-to-end vector addition are present on the b. Given two vectors in magnitude parallelogram method diagram. and direction form, determine explain when the magnitude of the sum of two vectors does equal the sum of their magnitudes. the magnitude and direction of explain when the magnitude of the sum of two vectors is the difference of their magnitudes. their sum. explain why the magnitude of the sum of two vectors does not have to equal the sum of the c. Understand vector subtraction magnitudes of each vector v + w v + w v-w as v+(-w) where w is the Part B additive inverse of w, with the convert from magnitude and direction form to component form to add vectors. same magnitude as w and pointing in the opposite convert from component form to magnitude and direction form to determine the magnitude and direction. Represent vector direction of the sum of vectors. subtraction graphically by explain why it is easier to calculate the sum of two vectors in magnitude and direction form by connecting the tips in the converting them to component form, adding them, and converting the components back to appropriate order, and perform magnitude and direction. vector subtraction componentwise. explain that w the additive inverse of w (w + -w = 0), has the same magnitude as w but points Part C SMP #1, #7 in the opposite direction. apply vector addition methods to solve vector subtraction problems.

22 Unit 9- Vectors and Complex Numbers (cont) How does knowledge of real numbers and geometry help when working with vectors and matrices? MAFS.912.N-VM.2.5 Part A Multiply a vector by a scalar. explain the difference between a vector and a scalar. a. Represent scalar multiplication graphically represent the scalar multiple of a vector on the coordinate plane. by scaling vectors and possibly reversing calculate the product of a scalar and a vector in component form. their direction; perform scalar multiplication Part B component-wise, e.g. as v x, v y = calculate the magnitude of a scalar multiple by multiplying the absolute value of the (cv x, cv y ). scalar by the magnitude of v ( cv = c v ). b. Compute the magnitude of a scalar compute the direction of a scalar multiple using inverse tangent. multiple cv using cv = c v. Compute explain when scalar multiplication changes the direction of the original vector and the direction of cv knowing that when when it does not. c v 0, the direction of cv is either along explain the connection between scalar multiplication of vectors and geometric v (for c>0) or against v (for c<0). reflections and scale changes. SMP #7 MAFS.912.N-CN.2.4 Represent complex numbers on the complex plane in rectangular plane in polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number. SMP #2, #7 MAFS.912.N-CN.2.6 Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints. SMP #7 define the complex plane. plot complex numbers using rectangular coordinates (x, y) on the complex plane. plot points using polar coordinates on the coordinate plane. plot complex numbers using polar form (r,θ) on the complex plane. represent the complex point (r,θ) as the complex number z=rcos θ+risin θ use right triangles to derive the conversion formulas and explain why the rectangular and polar forms of a complex number represent the same number. calculate the difference between two complex numbers. calculate the modulus of this difference, which represents the distance between the two complex numbers. calculate the midpoint of a line segment between two complex numbers by averaging the real and imaginary parts of the endpoints.