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Curriculum Map for: Time Frame / Content MRS21/22 M2 Essential Question(s) and ENGAGENY Links Final Draft Skills Module 2: Trigonometry Common Core Standards Sample Lesson Link, Projects and Assessments Topic A 25 days 1 1 1 1 In this topic, students develop an understanding of the six basic trigonometric functions Lesson 1: Ferris Wheels—Tracking the Height of a Passenger Car [Periodic Functions] 2 2 2 2 2 Lesson 2: The Height and Co-Height Functions of a Ferris Wheel [Periodic Functions and Graphs] 3 3 3 3 3 Lesson 3: The Motion of the Moon, Sun, and Stars—Motivating Mathematics [Unit Circle, Sine/Cosine] Sample Lessons Module 2 Objectives: Students apply geometric concepts in modeling situations. Specifically, they find distances between points of a circle and a given line to represent the height above the ground of a passenger car on a Ferris wheel as it is rotated a number of degrees about the origin from an initial reference point. Students sketch the graph of a nonlinear relationship between variables F-TF.A.1-F-TF.A.9 F-IF.C.7e F-TF.B.5 S-ID.B.6a F-TF.C.8 All Modules Assessments Link Differentiation/ Resources/ Strategies Objectives: Students model and graph two functions given by the location of a passenger car on a Ferris wheel as it is rotated a number of degrees about the origin from an initial reference position. Objectives: Students explore the historical context of trigonometry as motion of celestial bodies in a presumed circular arc. Students describe the position of an object along a line of sight in the context of circular motion. Students understand the naming of the quadrants and why counterclockwise motion is deemed the positive direction of turning in mathematics. Regents Mixed Reviews 4 4 4 4 4 Lesson 4: From Circleometry to Trigonometry [Sine and Cosine Function Values] Objectives: Students will define sine and cosine as functions for degrees of rotation of the ray formed by the positive -axis up to one full turn. Students use special triangles to geometrically determine the values of sine and cosine for 30°, 45°, 60°, and 90° degrees. 5 5 5 5 5 Lesson 5: Extending the Domain of Sine and Cosine to All Real Numbers [Reference Angles] Objectives: Students will define sine and cosine as functions for all real numbers measured in degrees. Students will evaluate the sine and cosine functions at multiples of 30° and 45°. F-TF.A.1-F-TF.A.9 F-IF.C.7e F-TF.B.5 S-ID.B.6a F-TF.C.8 Sample Lessons Module 2 All Modules Assessments Link fhhs_1 fhhs_2 fhhs_3 fhhs_4 6 6 6 7 7 7 Lesson 6: Why Call It Tangent? [Tangent Function Values] Lesson 7: Secant and the Co-Functions [Reciprocal Trig Functions] Objectives: Students define the tangent function and understand the historic reason for its name. Students use special triangles to determine geometrically the values of the tangent function for 30°, 45°, and 60°. Objectives: Students define the secant function and the co-functions in terms of points on the unit circle. They relate these names for these functions to the geometric relationships among lines, angles, and right triangles in a unit circle diagram. Students use reciprocal relationships to relate the trigonometric functions and use these relationships to evaluate trigonometric functions for multiples of 30°, 45°, and 60° degrees. fhhs_5 fhhs_6 Regents Mixed Reviews fhhs_7 fhhs_8 8 8 8 8 Lesson 8: Graphing the Sine and Cosine Functions Objectives: Students graph the sine and cosine functions and analyze the shape of these curves. For the sine and cosine functions, students sketch graphs showing key features, which include intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maxima and minima; symmetries; end behavior; and periodicity. fhhs_9 fhhs_10 9 9 9 9 Lesson 9: Awkward! Who Chose the Number 360, Anyway? [Radians] Objectives: Students explore horizontal scalings of the graph of y =sin(x). Students convert between degrees and radians. 10 10 10 10 Lesson 10: Basic Trigonometric Identities from Graphs [Translation Identities] Objectives: Students observe identities from graphs of sine and cosine basic trigonometric identities and relate those identities to periodicity, even and odd properties, intercepts, end behavior, and the fact that cosine is a horizontal translation of sine. use trigonometric functions to model periodic phenomena Lesson 11: Transforming the Graph of the Sine Function [Trig Graphs] Objectives: Students formalize the periodicity, frequency, phase shift, midline, and amplitude of a general sinusoidal function by understanding how the parameters A, w, h, and k in the formula f(x) = Asin(w(x-h))+k are used to transform the graph of the sine function, and how variations in these constants change the shape and position of the graph of the sine function. Students learn the relationship among the constant A, w, h, and k in the formula f(x) = Asin(w(x-h))+k and the properties of the sine graph. 12 12 12 12 12 12 12 Lesson 12: Ferris Wheels—Using Trigonometric Functions to Model Cyclical Behavior [Graphs, including word problems related to the graph] Sample Lessons Module 2 All Modules Assessments Link Regents Mixed Reviews Topic B: Students will 11 11 11 F-TF.A.1-F-TF.A.9 F-IF.C.7e F-TF.B.5 S-ID.B.6a F-TF.C.8 Objectives: Students review how changing the parameters A, ω, h, and k in f(x) = A sin(ω(x - h)) + k affects the graph of the sine function. Students examine the example of the Ferris wheel, using height, distance from the ground, period, and so on, to write a function of the height of the passenger cars in terms of the sine function: f(x) = A sin(ω(x - h)) + k. fhhs_11 fhhs_12 fhhs_13 fhhs_14 fhhs_15 fhhs_16 13 13 13 13 13 Lesson 13: Tides, Sound Waves, and Stock Markets [Creating a graph/equation to model a set of data] 14 14 Lesson 14: Graphing the Tangent Function Objectives: Students model cyclical phenomena from biological and physical science using trigonometric functions. Students understand that some periodic behavior is too complicated to be modeled by simple trigonometric functions. Objectives: Students graph the tangent function. Students use the unit circle to express the values of the tangent function for π - x, π + x, and 2π - x in terms of tan(x), where x is any real number in the domain of the tangent function. F-TF.A.1-F-TF.A.9 F-IF.C.7e F-TF.B.5 S-ID.B.6a F-TF.C.8 Sample Lessons Module 2 All Modules Assessments Link fhhs_17 fhhs_18 fhhs_19 15 15 15 Lesson 15: What Is a Trigonometric Identity? [Pythagorean Identities] Objectives: Students prove the Pythagorean identity sin2(x) + cos2(x) = 1. Students extend trigonometric identities to the real line, with attention to domain and range. Students use the Pythagorean identity to find sin(θ), cos(θ), or tan(θ), given sin(θ), cos(θ), or tan(θ) and the quadrant of the terminal ray of the rotation. fhhs_20 fhhs_21 fhhs_22 16 16 16 Lesson 16: Proving Trigonometric Identities Objectives: Students prove simple identities involving the sine function, cosine function, and secant function. Students recognize features of proofs of identities 17 17 17 17 Lesson 17: Trigonometric Identity Proofs [Sin/Cos/Tan of (a – b) and (a + b)] Objectives: Students see derivations and proofs of the addition and subtraction formulas for sine and cosine.Students prove some simple trigonometric identities. fhhs_23 fhhs_24 Regents Mixed Reviews