Mathematics Curriculum Document for Pre-Calculus

Transcription

1 Unit Title: Intro to Trig Time Frame: 15 blocks Six Weeks: 4 th Unit Number: 6 Curriculum Enduring Understandings (Big Ideas): Representing a situation in multiple ways leads to better understanding and more effective communication. Representing relationships mathematically helps us to make predictions and decisions in our dynamic world. Patterns help us to solve problems and understand data in the world. The student will know: Definition of an angle as a rotation Sine and cosine of special angles around the unit circle Definition of a sector Right angle definitions of trig functions Circle definitions of trig functions Graphs of all trig functions Inverse graphs of sine, cosine and tangent Laws of Sines and Cosines Definition of angles in navigation-type problems The student will be able to: Convert to and from angle measures in degrees and radians Convert to and from angle measures in decimal degrees and degrees, minutes, seconds Find coterminal angles Find reference angles Find the arc length and area of a sector Find linear and angular velocities Find the trig function values for special angles around the entire unit circle Recognize and graph transformations of sine, cosine and tangent functions Solve basic trig and inverse trig equations finding all possible solutions Model real-world situations with sinusoidal functions Solve navigation-type problems using right triangle trig as well as Law of Sines and Law of Cosines Unit Title: Intro to Trig Unit Number 6 Page 1 of 5

2 Essential Questions: What kind of situations can be represented using trig functions? How are algebraic and graphical representations connected? How do patterns help us simplify developing solutions? Student Understanding (student friendly TEKS): I can graph trigonometric and inverse trigonometric functions. (taken from PC.2.F) I can graph transformations of sine and cosine functions. (taken from PC.2.G) I can describe the restricted domains and ranges for the inverse sine and inverse cosine functions. (taken from PC.2.H) I can determine key features of trigonometric and inverse trigonometric functions. (taken from PC.2.I) I can develop and use sinusoidal functions to model problems and situations. (taken from PC.2.O) I can determine the values of trigonometric functions for special angles. (taken from PC.2.P) I can apply the unit circle to evaluate trigonometric functions and solve problems. (taken from PC.4.A) I can convert between degree and radian measure. (taken from PC.4.B) I can find and apply reference angles. (taken from PC.4.C) I can find linear and angular velocities. (taken from PC.4.D) I can determine and solve problems involving trigonometric ratios. (taken from PC.4.E) I can apply trigonometry to navigation-type problems. (taken from PC.4.F) I can use the Law of Sines and the Law of Cosines to solve problems. (taken from PC.4.G and PC.4.H) I can apply math in everyday life. (taken from PC.1.A) I can develop a problem solving model and justify my solutions. (taken from PC.1.B) I can select appropriate tools and techniques to solve problems. (taken from PC.1.C) I can communicate using appropriate mathematical language. (taken from PC.1.D) I can use various representations to communicate mathematical concepts. (taken from PC.1.E) I can analyze relationships to connect and communicate mathematical ideas. (taken from PC.1.F) I can use precise mathematical language in both written and oral communication. (taken from PC.1.G) TEKS: (PC.2) Functions. The student uses process standards in mathematics to explore, describe, and analyze the attributes of functions. The student makes connections between multiple representations of functions and algebraically constructs new functions. The student analyzes and uses functions to model real-world problems. The student is expected to: (F) graph exponential, logarithmic, rational, polynomial, power, trigonometric, inverse trigonometric, and piecewise defined Unit Title: Intro to Trig Unit Number 6 Page 2 of 5

3 functions, including step functions; (G) graph functions, including exponential, logarithmic, sine, cosine, rational, polynomial, and power functions and their transformations, including af(x), f(x) + d, f(x - c), f(bx) for specific values of a, b, c, and d, in mathematical and real-world problems; (H) graph arcsin x and arccos x and describe the limitations on the domain; (I) determine and analyze the key features of exponential, logarithmic, rational, polynomial, power, trigonometric, inverse trigonometric, and piecewise defined functions, including step functions such as domain, range, symmetry, relative maximum, relative minimum, zeros, asymptotes, and intervals over which the function is increasing or decreasing; (O) develop and use a sinusoidal function that models a situation in mathematical and real-world problems; and (P) determine the values of the trigonometric functions at the special angles and relate them in mathematical and real-world problems. (PC.4) Number and measure. The student uses process standards in mathematics to apply appropriate techniques, tools, and formulas to calculate measures in mathematical and real-world problems. The student is expected to: (A) determine the relationship between the unit circle and the definition of a periodic function to evaluate trigonometric functions in mathematical and real-world problems; (B) describe the relationship between degree and radian measure on the unit circle; (C) represent angles in radians or degrees based on the concept of rotation and find the measure of reference angles and angles in standard position; (D) represent angles in radians or degrees based on the concept of rotation in mathematical and real-world problems, including linear and angular velocity; (E) determine the value of trigonometric ratios of angles and solve problems involving trigonometric ratios in mathematical and real-world problems; (F) use trigonometry in mathematical and real-world problems, including directional bearing; (G) use the Law of Sines in mathematical and real-world problems; (H) use the Law of Cosines in mathematical and real-world problems. (PC.1) Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to: (A) apply mathematics to problems arising in everyday life, society, and the workplace; (B) use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution; (C) select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems. (D) communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate; Unit Title: Intro to Trig Unit Number 6 Page 3 of 5

4 (E) create and use representations to organize, record, and communicate mathematical ideas; (F) analyze mathematical relationships to connect and communicate mathematical ideas; and (G) display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication. Targeted College Readiness Standards: IA1, IC1, IIA1, IIB1, IID1, IIIA3, IVB1-2,VIIA2, VIIB1, VIIC2, VIIIA1-5, VIIIB1-2, VIIIC1-2, IXA1-3, IXB1-2, IXC1-3, XA1-2, XB1-3 Targeted ELPs: 1A, 1C, 1D, 1E, 1F, 1H, 2C, 2D, 2E, 2G, 2I, 3D, 3E, 3F, 3G, 3H, 3I, 3J, 4C, 4D, 4F, 4G Academic Vocabulary: Radian Sinusoidal Unit circle Amplitude Cycle Frequency Phase shift Wave length/period Bearing Heading Coterminal angle Reference angle Language of Instruction: Sine Cosine Tangent Cotangent Secant Cosecant Arc length Sector Initial ray Terminal ray Standard position angle Degrees/Minutes/Seconds Linear/Angular velocity Rotation Unwrapping Angle of elevation Angle of depression Supplementary Complementary Right angle Normal Orthogonal Unit Title: Intro to Trig Unit Number 6 Page 4 of 5

5 Harmonic/Periodic Damped Harmonic Instruction Instructional Resources: Textbook: Chapter , (will need to use additional resources for navigation problems) Technology: Career Connections/Real Life Application: Exemplar Lessons: Research Based Instructional Strategies: Assessment Student self-assessment & reflection: Acceptable evidence or artifacts: Performance Task Unit 6a Performance Task Unit 6b Performance Task Unit 6c Unit Title: Intro to Trig Unit Number 6 Page 5 of 5