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THE SCHOOL DISTRICT OF LEE COUNTY Academic Plan Pre-Calculus Honors (1202340) Adopted Instructional Materials: 1-1 Graphs of Functions 1-2 Polynomial & Rational Functions Precalculus with Limits, A Graphing Approach 2-1 Trigonometric Functions 2-2 Analytic Trigonometry Additional Course Information English Language Development ELD Standards Special Notes Section: Teachers are required to provide listening, speaking, reading and writing instruction that allows English language learners (ELL) to communicate information, ideas and concepts for academic success in the content area of Mathematics. For the given level of English language proficiency and with visual, graphic, or interactive support, students will interact with grade level words, expressions, sentences and discourse to process or produce language necessary for academic success. The ELD standard should specify a relevant content area concept or topic of study chosen by curriculum developers and teachers which maximizes an ELL’s need for communication and social skills. Page 1 of 20 3-1 Trigonometry Applications 3-2 Analytic Geometry 4-1 Limits 4-2 Exponential & Logarithmic Functions 4-3 Sequences & Series Professional Development Math Practices by Grade Level Build Relationships: Teach More Than ‘Just Math’ Sorting Equations Video: Research shows that formative assessments have a significant impact on student learning gains. This video is just one example of using formative assessment to inform instruction. CPALMS MFAS Training Research around formative assessment shows that students make greater learning gains when they are accountable for their own learning and the learning of their peers. The video, Facilitating Peer Learning, is a good example of a math classroom where students are engaged with one another. Five “Key Strategies” for Effective Formative Assessment Helpful Websites Khan Academy: Practice by Grade Level Standards https://www.khanacademy.org/commoncore/m ap Shmoop: Math videos http://www.shmoop.com/video/math-videos KUTA Software: Test and Worksheet Generator http://www.kutasoftware.com/ Teaching Channel: Videos and Best Practices https://www.teachingchannel.org/ Geogebra: Math web applets https://tube.geogebra.org Inside Mathematics: Videos and Best Practices http://www.insidemathematics.org/ Updated: June 8, 2016 THE SCHOOL DISTRICT OF LEE COUNTY Academic Plan 1-1 Pre-Calculus Honors (1202340) Adopted Instructional Materials: Precalculus with Limits, A Graphing Approach Big Idea: Graphs of Functions Standards Math Content Standards Suggested Literacy & English Language Standards MAFS.912.F-BF.1: Build a function that models a relationship between two quantities 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. 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. c. 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. MAFS.912.F-BF.2: Build new functions from existing functions MAFS.912.F-BF.2.4: 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) = 2𝑥 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. MAFS.912.N-CN.3: Use complex numbers in polynomial identities and equations. MAFS.912.N-CN.3.9: Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. LAFS.1112.SL.1.1: Initiate and participate effectively in a range of collaborative discussions (one-on-one, in groups, and teacher-led) with diverse partners on grades 1112 topics, texts, and issues, building on others ideas and expressing their own clearly and persuasively. a. Come to discussions prepared, having read and researched material under study; explicitly draw on that preparation by referring to evidence from texts and other research on the topic or issue to stimulate a thoughtful, well-reasoned exchange of ideas. b. Work with peers to promote civil, democratic discussions and decision-making, set clear goals and deadlines, and establish individual roles as needed. c. Propel conversations by posing and responding to questions that probe reasoning and evidence; ensure a hearing for a full range of positions on a topic or issue; clarify, verify, or challenge ideas and conclusions; and promote divergent and creative perspectives. d. Respond thoughtfully to diverse perspectives; synthesize comments, claims, and evidence made on all sides of an issue; resolve contradictions when possible; and determine what additional information or research is required to deepen the investigation or complete the task. Suggested Mathematical Practice Standards MAFS.K12.MP.4.1: Model with mathematics. What real-world situations could be analyzed and solved by producing inverse functions? Page 2 of 20 Updated: June 8, 2016 MAFS.K12.MP.8.1: Look for and express regularity in repeated reasoning. What are the similarities and differences among parent graphs when applying transformations? Essential Outcome Question(s) In what ways can you analyze and describe a function? Aligned Learning Goals Identify a variety of functions given a graph or equation, including linear, quadratic, cubic, rational, square root, exponential, logarithmic, trigonometric, inverse trigonometric, absolute value, and greatest integer functions Determine key features of functions, including domain, range, y-intercepts, zeros, increasing, decreasing, even, odd, and rates of change Use functions to model real world problems Identify graphs of parent functions Identify the transformations of parent functions given an equation or a graph Use vertical and horizontal shifts, reflections, and nonrigid transformations to graph functions Add, subtract, multiply, and divide functions Find the composition of functions Write functions as the composition of two functions Use function compositions to model and solve real-world problems Find inverse functions both informally and algebraically Verify whether or not functions are inverses algebraically and from graphs Page 3 of 20 District Adopted Materials Supplemental Resources Strategies for Differentiation Precalculus w/ Limits MAFS.912.F-BF.2.4: Lesson: Invertible or Not? INTERVENTION Section 1.1 MAFS.912.F-BF.1.1: Parent Functions Handout Chapter 1 Sections 2 – 6 Suggested Lesson Grouping: Lessons 1 & 2 Lesson 4 Lessons 5 & 6 MAFS.912.F-BF.1.1: Foldable Parent Polynomial Function Behavior ENRICHMENT Section 1.7 MAFS.912.F-BF.1.1: QR Code Scavenger Hunt Updated: June 8, 2016 THE SCHOOL DISTRICT OF LEE COUNTY Academic Plan 1-2 Pre-Calculus Honors (1202340) Adopted Instructional Materials: Precalculus with Limits, A Graphing Approach Big Idea: Polynomial & Rational Functions Standards Math Content Standards Suggested Literacy & English Language Standards MAFS.912.F-BF.1: Build a function that models a relationship between two quantities 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. 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. c. 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. MAFS.912.N-CN.1.: Perform arithmetic operations with complex numbers. MAFS.912.N-CN.1.3: Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. MAFS.912.A-APR.4: Rewrite rational expressions. MAFS.912.A-APR.4.6: 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. 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. MAFS.912.N-CN.3: Use complex numbers in polynomial identities and equations. MAFS.912.N-CN.3.9: Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. MAFS.912.C.1: Limits and Continuity. MAFS.912.C.1.12: Understand and use the Intermediate Value Theorem on a function over a closed interval. 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.WHST.3.9: Draw evidence from informational texts to support analysis, reflection, and research. Page 4 of 20 ELD.K12.ELL.MA.1: English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics. Suggested Mathematical Practice Standards MAFS.K12.MP.2.1: Reason abstractly and quantitatively. How did you decide how to approach this problem? MAFS.K12.MP.3.1: Construct viable arguments and critique the reasoning of others. What do you think about _____’s work? Is your answer different than ____’s? If so, how? Updated: June 8, 2016 MAFS.912.C.1.13: Understand and apply the Extreme Value Theorem: If 𝑓(𝑥) is continuous over a closed interval, then f has a maximum and a minimum on the interval. MAFS.912.A-APR.3: Use polynomial identities to solve problems. MAFS.912.A-APR.3.5: 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. Essential Outcome Question(s) What tools can help identify the zeros of a polynomial function? How are various polynomial theorems helpful in solving problems? Aligned Learning Goals District Adopted Materials Analyze quadratic functions and use key features to solve real-world problems Use transformations to sketch the graph of a polynomial function Determine the end behavior of a polynomial function Understand and define zeros of a function Identify the zeros of a factored polynomial function and use them to sketch the graph the function Divide polynomials using long division and synthetic division to find the zeros of a polynomial function Apply the Remainder Theorem when dividing polynomials Apply the Rational Zero Test and Descartes’s Rule of Signs to find zeros of a polynomial function Apply the Fundamental Theorem of Algebra to determine the number of zeros of a polynomial function Use various methods to find all the zeros, including rational and complex zeros, of a polynomial function Understand and use the Intermediate Value Theorem on a function over a closed interval Understand and use the Extreme Value Theorem Add, subtract, multiply, and divide complex numbers Find the conjugate of a complex number Page 5 of 20 Precalculus with Limits Chapter 2 Sections 2 – 5 Chapter 8 Section 4 Supplemental Resources Strategies for Differentiation MAFS.912.N-CN.1.3: Complex Numbers Maze INTERVENTION Polynomial Review Materials ENRICHMENT Lessons 6 & 7 Lesson 8 MAFS.C.1.13: Extreme Value Theorem Suggested Lesson Grouping: Ch. 2 Lessons 1 & 4 Lesson 2 Lesson 3 Lesson 5 Ch. 8 Lesson 4 Updated: June 8, 2016 Find complex solutions of quadratic equations Use the Binomial Theorem to calculate binomial coefficients Use binomial coefficients to write binomial expansions Use Pascal’s Triangle to calculate binomial coefficients Formative Assessment Options: MAFS.9.12.A-APR.3.4: Performance Task Trina’s Triangles Page 6 of 20 Updated: June 8, 2016 THE SCHOOL DISTRICT OF LEE COUNTY Academic Plan 2-1 Pre-Calculus Honors (1202340) Adopted Instructional Materials: Precalculus with Limits, A Graphing Approach Big Idea: Trigonometric Functions Standards Math Content Standards Suggested Literacy & English Language Standards MAFS.912.F-TF.1: Extend the domain of trigonometric functions using the unit circle MAFS.912.F-TF.1.1: Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle; Convert between degrees and radians. MAFS.912.F-TF.1.2: 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. MAFS.912.F-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, cosine, and tangent for 𝜋 − 𝑥, 𝜋 + 𝑥, and 2𝜋 − 𝑥 in terms of their values for 𝑥, where 𝑥 is any real number. MAFS.912.F-TF.1.4: Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. MAFS.912.F-TF.2: Model periodic phenomena with trigonometric functions. MAFS.912.F-TF.2.5: Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. MAFS.912.F-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. MAFS.912.F-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. MAFS.912.G-SRT.3: Define trigonometric ratios and solve problems involving right triangles MAFS.912.G-SRT.3.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. 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.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. ELD.K12.ELL.MA.1: English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics. Suggested Mathematical Practice Standards MAFS.K12.MP.1.1: Make sense of problems and persevere in solving them. What is this problem asking? Could someone else understand how to solve the problem based on your explanation? MAFS.K12.MP.4.1: Model with mathematics. Does your solution make sense? What do you know about the situation already? Essential Outcome Question(s) In what ways are trigonometric functions useful? In what ways can you determine the measure of a central angle? Page 7 of 20 Updated: June 8, 2016 Aligned Learning Goals Describe angles using appropriate vocabulary Understand and use both degrees and radians to measure central angles Solve real-world problems involving central angles, arcs, and sectors Model converting between degrees and radians Identify a unit circle and describe its relationship to real numbers Determine the value of basic trigonometric functions using right triangle geometry Evaluate trigonometric functions using the unit circle and a calculator Use domain and period to evaluate sine and cosine functions Solve real-world right triangle problems using trigonometric ratios Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers Use the unit circle to find trigonometric values in both mathematical and real-world contexts Identify and use trigonometric identities to solve problems Find reference angles Evaluate trigonometric functions of angles and real numbers Sketch the graphs of sine, cosine, and tangent functions, including understanding and applying transformations in both mathematical and real-world contexts Sketch the graphs of cotangent, secant, and cosecant functions, applying transformations in both mathematical and real-world contexts Explain how restricting the domain of the graph of a trigonometric function allows for the inverse to be constructed and use inverse functions to solve problems Evaluate and graph inverse trigonometric functions Evaluate the composition of trigonometric functions Solve real-world problems involving right triangles using trigonometric functions and the Pythagorean Theorem Page 8 of 20 District Adopted Materials Supplemental Resources Strategies for Differentiation Precalculus with Limits MAFS.912.F-TF.1.1: Lesson: Radians vs. Degrees INTERVENTION Right Triangle Review Material MAFS.912.F-TF.1.4: Lesson: Graphs from the Unit Circle Lesson 4.8, pp 329-31 Chapter 4 Omit Harmonic Motion Suggested Lesson Grouping: Lessons 1 & 2 Lessons 3 & 4 Lessons 5 & 6 Lesson 7 Lesson 8 EXTENSION MAFS.912.F-TF.2.5: Lesson: Ferris Wheel and Periodic Motion Updated: June 8, 2016 THE SCHOOL DISTRICT OF LEE COUNTY Academic Plan 2-2 Pre-Calculus Honors (1202340) Adopted Instructional Materials: Precalculus with Limits, A Graphing Approach Big Idea: Analytic Trigonometry Standards Math Content Standards Suggested Literacy & English Language Standards MAFS.912.F-TF.2: Model periodic phenomena with trigonometric functions. MAFS.912.F-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. MAFS.912.F-TF.3: Prove and apply trigonometric identities. MAFS.912.F-TF.3.8: Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1 and use it to calculate trigonometric ratios. MAFS.912.F-TF.3.9: Prove the addition and subtraction, half-angle, and double-angle formulas for sine, cosine, and tangent and use these formulas to solve problems. MAFS.912.G-SRT.3: Define trigonometric ratios and solve problems involving right triangles MAFS.912.G-SRT.3.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. MAFS.912.A-APR.3: Use polynomial identities to solve problems. MAFS.912.A-APR.3.4: 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. LAFS.1112.WHST.2.4: Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience. ELD.K12.ELL.MA.1: English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics. Suggested Mathematical Practice Standards MAFS.K12.MP.2.1: Reason abstractly and quantitatively. Explain how you know that a given trig equation models the given real-world situation. MAFS.K12.MP.5.1: Use appropriate tools strategically. What methods can be used to solve trigonometric equations? Essential Outcome Question(s) How are trigonometric identities used to solve problems? Aligned Learning Goals Understand and write fundamental trigonometric identities, including reciprocal identities, quotient identities, Pythagorean identities, cofunction identities, and even and odd identities District Adopted Materials Supplemental Resources Strategies for Differentiation Precalculus with Limits MAFS.912.F-TF.2.7: Shmoop: Solving Trig Equations INTERVENTION Trigonometry Review Work Chapter 5 Page 9 of 20 Updated: June 8, 2016 Use trigonometric identities to evaluate trigonometric functions and simplify/rewrite trigonometric expressions Use various methods to prove trigonometric identities by manipulating trigonometric expressions with Pythagorean identities Prove the Pythagorean identity (sin 𝜃)2 + (cos 𝜃)2 = 1 and use it to calculate trigonometric ratios Solve equations involving trigonometry in mathematical contexts Use inverse trigonometric functions to find solutions to trigonometric equations modeling real-world problems Prove the sum and difference identities for sine, cosine, and tangent Prove the double-angle and half-angle identities for sine, cosine, and tangent Use the sum and difference identities and the double-angle and half-angle identities to rewrite expressions and solve problems Page 10 of 20 MAFS.912.T-TF.3.8: Lesson Videos MAFS.912.F-TF.3.9: Shmoop: Verifying Trig Equations MAFS.912.F-TF.3.9: I Have Who Has Trig Identities MAFS.912.F-TF.3.9: Trig Function Maze Updated: June 8, 2016 THE SCHOOL DISTRICT OF LEE COUNTY Academic Plan 3-1 Pre-Calculus Honors (1202340) Adopted Instructional Materials: Precalculus with Limits, A Graphing Approach Big Idea: Trigonometry Applications Standards Math Content Standards MAFS.912.N-CN.1.: Perform arithmetic operations with complex numbers. MAFS.912.N-CN.1.3: Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. MAFS.912.N-CN.2: Represent complex numbers and their operations on the complex plane. MAFS.912.N-CN.2.4: 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. 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. For example, (1 + 3 i) = 8 because (1 + 3 i) has modulus 2 and argument 120. MAFS.912.N-VM.1: Represent and model with vector quantities. MAFS.912.N-VM.1.1: 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). 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. MAFS.912.N-VM.1.3: Solve problems involving velocity and other quantities that can be represented by vectors. MAFS.912.N-VM.2: Perform operations on vectors. MAFS.912.N-VM.2.4: 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 componentwise. Page 11 of 20 Suggested Literacy & English Language Standards LAFS.1112.SL.1.2: Integrate multiple sources of information presented in diverse formats and media (e.g., visually, quantitatively, orally) 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. Suggested Mathematical Practice Standards MAFS.K12.MP.2.1: Reason abstractly and quantitatively. What do you notice about…? MAFS.K12.MP.6.1: Attend to precision. What labels should you use to accurately represent the problem? Updated: June 8, 2016 MAFS.912.N-VM.2.5: Multiply a vector by a scalar. a. Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c = . b. 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). MAFS.912.G-SRT.4: Apply trigonometry to general triangles. MAFS.912.G-SRT.4.10: Prove the Laws of Sines and Cosines and use them to solve problems. MAFS.912.G-SRT.4.11: 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). MAFS.912.G-SRT.4.9: 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. Essential Outcome Question(s) How can vectors be used in real-world problem solving? Aligned Learning Goals Prove the Laws of Sines and Cosines Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles 1 𝑎𝑏 sin 𝐶 2 Derive the formula 𝐴 = Recognize, understand, and use symbols and vocabulary associated with vectors, including magnitude and direction Represent vectors using sketches, symbols, rectangular components, and in terms of magnitude and direction Add and subtract vectors, adding end-to-end, component-wise, and by the parallelogram rule Multiply a vector by a scalar Calculate magnitudes for vectors, sums of vectors, and scalar multiples of vectors Page 12 of 20 for the area of a triangle District Adopted Materials Supplemental Resources Strategies for Differentiation Precalculus with Limits MAFS.912.G-SRT.4.11: When To Use The Law of Sines INTERVENTION Trigonometry Review Work Chapter 6 MAFS.912.G-SRT.4.11: Law of Sines Ambiguity MAFS.912.N-VM.1.1: Lesson: Vectors: Tip to Tail MAFS.912.N-VM.1.3: Lesson: Amusement Park Physics MAFS.N-CN.1.3: Lesson: Can You Hear Me Now? Dividing Complex Numbers Updated: June 8, 2016 Solve real-world applications of velocity and other quantities that can be modeled by vectors Plot complex numbers in the complex plane Find the absolute value of complex numbers Write the trigonometric form of complex numbers Perform operations with complex numbers written in trigonometric form Convert complex numbers from rectangular form to polar form and vice versa Explain how a complex number in polar form and rectangular form represent the same number Find products and quotients of complex numbers in polar form and represent them geometrically on the complex plane Page 13 of 20 MAFS.912.N-VM.2.4: Online Game: Vector Island Updated: June 8, 2016 THE SCHOOL DISTRICT OF LEE COUNTY Academic Plan 3-2 Pre-Calculus Honors (1202340) Adopted Instructional Materials: Precalculus with Limits, A Graphing Approach Big Idea: Analytic Geometry Standards Math Content Standards MAFS.912.G-GPE.1: Translate between the geometric description and the equation for a conic section. MAFS.912.G-GPE.1.1: Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. MAFS.912.G-GPE.1.2: Derive the equation of a parabola given a focus and directrix. MAFS.912.G-GPE.1.3: Derive the equations of ellipses and hyperbolas given the foci and directrices. Suggested Literacy & English Language Standards LAFS.1112.WHST.1.1: Write arguments focused on disciplinespecific content. a. Introduce precise, knowledgeable claim(s), establish the significance of the claim(s), distinguish the claim(s) from alternate or opposing claims, and create an organization that logically sequences the claim(s), counterclaims, reasons, and evidence. b. Develop claim(s) and counterclaims fairly and thoroughly, supplying the most relevant data and evidence for each while pointing out the strengths and limitations of both claim(s) and counterclaims in a discipline-appropriate form that anticipates the audiences knowledge level, concerns, values, and possible biases. c. Use words, phrases, and clauses as well as varied syntax to link the major sections of the text, create cohesion, and clarify the relationships between claim(s) and reasons, between reasons and evidence, and between claim(s) and counterclaims. d. Establish and maintain a formal style and objective tone while attending to the norms and conventions of the discipline in which they are writing. e. Provide a concluding statement or section that follows from or supports the argument presented. Suggested Mathematical Practice Standards MAFS.K12.MP.7.1: Look for and make use of structure. How does simplifying this equation relate to…? Page 14 of 20 Updated: June 8, 2016 Essential Outcome Question(s) What are similarities and differences among the equations of conic sections and the graphs of conic sections? Aligned Learning Goals Understand and use vocabulary associated with conic sections Derive the equation of a circle of given center and radius using the Pythagorean Theorem Complete the square to find the center and radius of a circle given by an equation Derive the equation of a parabola given a focus and directrix Derive the equations of ellipses and hyperbolas given the foci and directrices Find asymptotes of and graph hyperbolas Find eccentricities of ellipses Analyze and graph equations of circles, ellipses, parabolas, identifying key features Identify conics sections from equations Solve real-world problems involving conic sections Classify conics from their general equations Page 15 of 20 District Adopted Materials Supplemental Resources Precalculus with Limits MAFS.912.G-GPE.1.2: Chapter 9 Sections 1 – 3 Lesson: Intro to Conic Sections Hands-On Strategies for Differentiation ENRICHMENT Pg. 663: Rotations Lessons 4, 5, 6, & 7 Conic Sections Extra Practice Updated: June 8, 2016 THE SCHOOL DISTRICT OF LEE COUNTY Academic Plan 4-1 Pre-Calculus Honors (1202340) Adopted Instructional Materials: Precalculus with Limits, A Graphing Approach Big Idea: Limits and an Introduction into Calculus Standards Math Content Standards MAFS.912.C.1: Limits and Continuity. MAFS.912.C.1.1: Understand the concept of limit and estimate limits from graphs and tables of values. MAFS.912.C.1.2: Find limits by substitution. MAFS.912.C.1.3: Find limits of sums, differences, products, and quotients. MAFS.912.C.1.4: Find limits of rational functions that are undefined at a point. MAFS.912.C.1.5: Find one-sided limits. MAFS.912.C.1.9: Understand continuity in terms of limits. MAFS.912.C.1.10: Decide if a function is continuous at a point. MAFS.912.C.1.11: Find the types of discontinuities of a function. Suggested Literacy & English Language Standards LAFS.1112.RST.2.4: Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in a specific scientific or technical context relevant to grades 1112 texts and topics. ELD.K12.ELL.SI.1: English language learners communicate for social and instructional purposes within the school setting. Suggested Mathematical Practice Standards MAFS.K12.MP.2.1: Reason abstractly and quantitatively. Essential Outcome Question(s) What methods can be used to find the limit of a function? Aligned Learning Goals Understand and explain the concept of a limit Estimate limits of functions at both a fixed point and at infinity Determine if limits of a function exists Use properties of limits and direct substitution to evaluate limits Evaluate one-sided limits of functions Understand and explain continuity in terms of limits Decide if a function is continuous at a point Find limits of sums, differences, products, and quotients Page 16 of 20 District Adopted Materials Supplemental Resources Precalculus with Limits MAFS.912.C.1.1: Lesson: Leap Frog Limits Review Game Chapter 11 Sections 1, 2, & 4 MAFS.912.C.1.11: Types of Discontinuity Limits Video Series Strategies for Differentiation MAFS.912.C.1.1: Shmoop: Graphing and Visualizing Limits MAFS.912.C.1.1-4: Worksheet Practice on Limits Updated: June 8, 2016 Find limits of rational functions that are undefined at a point Find limits of sequences Page 17 of 20 Updated: June 8, 2016 THE SCHOOL DISTRICT OF LEE COUNTY Academic Plan 4-2 Pre-Calculus Honors (1202340) Adopted Instructional Materials: Precalculus with Limits, A Graphing Approach Big Idea: Exponential and Logarithmic Functions Standards Math Content Standards Suggested Literacy & English Language Standards MAFS.912.F-BF.1: Build a function that models a relationship between two quantities. 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. 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. c. 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. MAFS.912.F-BF.2: Build new functions from existing functions. MAFS.912.F-BF.2.5 - Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. LAFS.910.WHST.3.9: Draw evidence from informational texts to support analysis, reflection, and research. ELD.K12.ELL.MA.1: English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics. Suggested Mathematical Practice Standards MAFS.K12.MP.4.1: Model with mathematics. What other ways could you use to model the situation mathematically? MAFS.K12.MP.7.1: Look for and make use of structure. How can you use what you know to explain why this works? Essential Outcome Question(s) How can the properties of exponential and logarithmic expressions be used to solve equations? Aligned Learning Goals Understand and graph exponential functions Understand and graph logarithmic functions Use the properties of logarithms to expand and condense logarithmic expressions Use the One-to-One Property and properties of exponentials and logarithmic expressions to solve exponential and logarithmic equations Page 18 of 20 District Adopted Materials Supplemental Resources Precalculus with Limits Log Review Jigsaw Puzzle Chapter 3 Sections 1 – 4 Worksheet Practice on Exponential and Logarithmic Functions Strategies for Differentiation Updated: June 8, 2016 THE SCHOOL DISTRICT OF LEE COUNTY Academic Plan 4-3 Pre-Calculus Honors (1202340) Adopted Instructional Materials: Precalculus with Limits, A Graphing Approach Big Idea: Sequences and Series Standards Math Content Standards Suggested Literacy & English Language Standards MAFS.912.A-APR.3: Use polynomial identities to solve problems. MAFS.912.A-APR.3.4: 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. MAFS.912.A-APR.3.5: 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. MAFS.912.F-BF.1: Build a function that models a relationship between two quantities. MAFS.912.F-BF.1.2 - Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. LAFS.910.WHST.3.9: Draw evidence from informational texts to support analysis, reflection, and research. ELD.K12.ELL.MA.1: English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics. ELD.K12.ELL.SI.1: English language learners communicate for social and instructional purposes within the school setting. Suggested Mathematical Practice Standards MAFS.K12.MP.4.1: Model with mathematics. What other ways could you use to model the situation mathematically? What connections can you make between different representations of the situation? MAFS.K12.MP.7.1: Look for and make use of structure. How can you use what you know to explain why this works? What patterns do you see? Essential Outcome Question(s) What methods can be used to find the sums of series of numbers? Page 19 of 20 Updated: June 8, 2016 Aligned Learning Goals Understand and write explicit and recursive formulas from given sequences Understand summation notation, and use it to find sums of series Find the nth term of arithmetic or geometric sequences Find the sum of finite arithmetic or geometric series Find the sum of infinite geometric series Page 20 of 20 District Adopted Materials Supplemental Resources Precalculus with Limits Worksheet Practice on Sequences & Series Strategies for Differentiation Chapter 8 Sections 1 – 3 Updated: June 8, 2016