Precalculus, Quarter 2, Unit 2.1. Trigonometry Graphs. Overview

Transcription

1 13 Precalculus, Quarter 2, Unit 2.1 Trigonometry Graphs Overview Number of instructional days: 12 (1 day = 45 minutes) Content to be learned Convert between radian and degree measure. Determine the usefulness of radian measure and use it appropriately in solving problems. Identify angles on the unit circle in both radian and degree measure. Analyze, interpret, and model the characteristics and behavior of trigonometric functions and their operations, including finding inverses and compositions algebraically. Analyze the domain, range, and x and y intercepts of trigonometric functions, and analyze the effect of domain restrictions on the function and its properties. Determine increasing and decreasing intervals, rates of change, periodicity, end behavior, and maximum and minimum values of trigonometric functions, and determine graphically and analytically whether the functions are even, odd, or neither. Analyze informally the limiting behavior and continuity of trigonometric functions and determine limiting values of trigonometric functions pertaining to horizontal and vertical asymptotes, including equations of horizontal and vertical asymptotes. Analyze properties of functions, including injectivity (1-1), surjectivity (onto), critical points and inflection points, and recognize properties of families of functions. Mathematical practices to be integrated Use appropriate tools strategically. Use technology including, but not limited to, graphing calculators to solve-real world problems involving trigonometric functions. Use technology to visualize results from problems involving trigonometric functions. Make sense of problems and persevere in solving them. Work between different representations. Think about simpler problems to help solve more complex problems. Model with mathematics. Identify the important quantities and their relationships using mathematical models. Attend to precision. Use labels of axes and units of measure correctly. Calculate and compute accurately (including technology). Use precise mathematical vocabulary, clear and accurate definitions, and symbols to communicate efficiently and effectively. Cumberland, Lincoln, and Woonsocket Public Schools C-13

2 Precalculus, Quarter 2, Unit 2.1 Trigonometry Graphs (12 days) Essential questions How do the graphs of trigonometric functions compare to graphs of previously studied functions? Why is radian measure useful? What relationship exists between radian and degree measure, and how is this relationship utilized in trigonometry? In geometry? What real-life applications use trigonometry in degree or radian measure? What properties of trigonometric functions help characterize families of functions? How does the circumference of a circle impact the conversion between radian and degree measure? What real-life situations are modeled by trigonometric graphs? What is meant by the limiting behavior of a trigonometric function, and how is limiting behavior determined? Written Curriculum Grade Span Expectations M(F&A) AM-2 Demonstrates conceptual understanding of linear and nonlinear functions and relations from a set-theoretic perspective, and operations on functions including composition and inverse including computing inverses algebraically; analyzing characteristics of classes of functions and inverse functions (exponential, logarithmic, trigonometric) to include domain, range, intercepts, increasing and decreasing intervals and rates of change, periodicity, end behavior, maximum and minimum values, continuity, and asymptotes; analyzing properties of functions including injectivity (1-1), surjectivity (onto), critical points and inflection points. Determine graphically and analytically whether a function is even, odd or neither; analyzing informally the idea of continuity and limits; recognizing properties of families of functions including logarithmic and trigonometric, and graphs them; analyzing domain restriction and the effects of it on the function and its properties. (Local) M(G&M)-AM-7 Uses radian measure appropriately when solving problems; converts between radian measure and degree measure; and understands why radian measure is useful. (Local) Clarifying the Standards Prior Learning In grade 2, students learned how to convert units of measurements when solving problems. In grade 5, students identified acute, obtuse, and right angles. In grade 8, students developed conceptual understanding of linear and nonlinear relations. They analyzed characteristics and behavior of functions and relations to include intercepts, domain, range, maximum and minimum values, and increasing and decreasing rates. A graphical, numerical (tables, calculator), algebraic, analytical (function behavior), and verbal (writing and notation) approach was further developed. In geometry, students applied sine, cosine, and tangent ratios to solve problems. In grade 11, students interpreted, modeled, and solved problems of families of functions. Cumberland, Lincoln, and Woonsocket Public Schools C-14

3 Precalculus, Quarter 2, Unit 2.1 Trigonometry Graphs (12 days) Current Learning Students learn to convert between radians and degrees. Students learn the unit circle, both in radians and degrees. Students graph trigonometric functions and study periodicity, amplitude, and intercepts. Range restrictions on the functions and the effect of them are also incorporated into the lesson. Future Learning Students will use these concepts to solve trigonometric equations and model real-life situations. Radian measure will be used extensively in calculus. Careers in the medical, engineering, physics, and biology fields will use trigonometric graphs. Additional Research Findings Beyond Numeracy states that in determining the circumference of the earth, the Greeks used trigonometry; other areas in which trigonometry played an important role were in astrology, physics, and geography. Both right angle and non-right angle triangles are used in trigonometric analysis (pp ). Notes About Resources and Materials Cumberland, Lincoln, and Woonsocket Public Schools C-15

4 Precalculus, Quarter 2, Unit 2.1 Trigonometry Graphs (12 days) Cumberland, Lincoln, and Woonsocket Public Schools C-16

5 17 Precalculus, Quarter 2, Unit 2.2 Trigonometry Identities and Proofs Overview Number of instructional days: 13 (1 day = 45 minutes) Content to be learned Verify trigonometric identities. Simplify trigonometric expressions using identities. Use the axioms, postulates, properties, and theorems learned in algebra and geometry to create formal proofs of propositions concerning the concepts of lines, angles, circles, distance, midpoint, polygons, triangle congruence, and triangle similarity. Use the process of deductive reasoning to create formal proofs of propositions in trigonometry. Essential questions What are the different strategies for verifying trigonometric identities? Why is it important to simplify trigonometric expressions? What is the relationship between the trigonometric functions of angles and the (x, y) coordinates in a rectangular plane? Mathematical practices to be integrated Look for and make use of structure. Identify patterns and structures. Break down complicated problems into smaller, less complex problems. Construct viable arguments and critique the reasoning of others. Think logically, determine if there are errors, and explain reasoning. Use prior material and properties of mathematics to solve more complex problems involving deductive and inductive reasoning. Use proofs to make logical arguments and conclusions. Look for and express regularity in repeated reasoning. Look for general methods, patterns, repeated calculation, and shortcuts. Look for patterns to find generalizations. Evaluate the reasonableness of intermediate results. What is the relationship between the Pythagorean theorem and the Pythagorean identities in trigonometry? What important concepts and properties from algebra and geometry are used in the creation of formal proofs and propositions in circular and triangular trigonometry? What are the similarities and differences between proofs in geometry and trigonometry? Cumberland, Lincoln, and Woonsocket Public Schools C-17

6 Precalculus, Quarter 2, Unit 2.2 Trigonometry Identities and Proofs (13 days) Written Curriculum Grade Span Expectations M(F&A) AM 4 Demonstrates conceptual understanding of equality by solving equations and verifying identities involving trigonometric expressions; solving, graphing and interpreting equations involving exponential and logarithmic expressions; interpreting systems as matrix equations and solving them by computing the appropriate matrix inverse and multiplication, with or without technology; applying the intermediate value theorem to find exact or approximate solutions of equations or zeros of continuous functions. (Local) M(G&M)-12-2 Creates formal proofs of propositions (e.g. angles, lines, circles, distance, midpoint and polygons including triangle congruence and similarity). (Local) Clarifying the Standards Prior Learning In the third grade, students used models to show equivalence between two expressions. In geometry, students created formal and informal proofs of propositions utilizing geometric terminology. In algebra 1 and algebra 2, students simplified algebraic expressions. Current Learning Students use trigonometric formulas to prove identities and simplify trigonometric expressions. Future Learning The ability to simplify trigonometric expressions will be used for derivatives and integration in calculus. It will also be used in various theorems in calculus. Additional Research Findings Beyond Numeracy states that trigonometric identities indicate the unconditional equality of one complicated expression for another. Only a few of the identities are crucial, specifically sin 2 x + cos 2 x = 1 (p. 255). Cumberland, Lincoln, and Woonsocket Public Schools C-18

7 Precalculus, Quarter 2, Unit 2.3 Trigonometry Solving Equations Overview Number of instructional days: 8 (1 day = 45 minutes) Content to be learned Solve simple trigonometric equations. Solve trigonometric equations by substituting equivalent trigonometric expressions into the trigonometric equations. Solve complex trigonometric equations through the use of previously learned trigonometric identities and problem-solving strategies. Essential questions What real-life situations employ the use of trigonometric equations? Why is it necessary to incorporate basic trigonometric identities in solving complicated trigonometric equations, and which are most useful in accomplishing this? Mathematical practices to be integrated Make sense of problems and persevere in solving them. Think about simpler problems to help solve more complex problems. Check the reasonableness of the solution, check answers by using multiple methods. Use technology to solve problems. Justify and explain solutions. Persevere do not give up. Model with mathematics. Interpret results in the context of a problem. Look for and make use of structure Evaluate work and make modifications or try a new approach, if necessary. What are the similarities and differences among trigonometric equations and verifying trigonometric identities? What are the similarities and differences among solutions of trigonometric equations and other functions previously studied? Cumberland, Lincoln, and Woonsocket Public Schools C-19

8 Precalculus, Quarter 2, Unit 2.3 Trigonometry Solving Equations (8 days) Written Curriculum Grade Span Expectations M(F&A) AM 4 Demonstrates conceptual understanding of equality by solving equations and verifying identities involving trigonometric expressions; solving, graphing and interpreting equations involving exponential and logarithmic expressions; interpreting systems as matrix equations and solving them by computing the appropriate matrix inverse and multiplication, with or without technology; applying the intermediate value theorem to find exact or approximate solutions of equations or zeros of continuous functions. (Local) Clarifying the Standards Prior Learning In grades 1 and 2, students demonstrated conceptual understanding of equality by finding values that make open sentences true using models, verbal explanations, or written equations. At this time, it was limited to one operation using addition or subtraction. In grade 3, multiplication was added as an operation. In grade 4, students began to simplify numerical expressions where left to right computations were modified only by parentheses. Solving one-step linear equations of the form ax = c or x ± b = c, where a, b, and c are whole numbers was introduced. In grade 5, x = c, where a 0 type equations were a added, as well as equations of the form ax ± b = c, using selections from a replacement set. In grade 6, solving multi-step linear equations ax ± b = c, without a replacement set, were undertaken. In grade 7, equations with variables and constants on both sides were introduced, and ax ± b = cx ± d, and x b c a ± =, were solved. In addition, translating a problem-solving situation into an equation consistent with the above equations was introduced. In grade 8, solving formulas for a variable using one transformation was explored. Solving multi-step equations with integer coefficients was introduced. The commutative, associative, and distributive properties were learned; order of operations or substitution and informally solving systems of linear equations in a context were processes used to solve equations. In grades 9 and 10, symbolic, graphical, algebraic, analytic, and verbal approaches were used to solve equations. Models and representations were employed throughout. In grades 11 and 12, processes included factoring, completing the square, quadratic formula, synthetic division, and graphing to solve multi-step linear and nonlinear equations, expressions, and inequalities. Current Learning Students in advanced mathematics classes solve equations and verify identities involving trigonometric expressions utilizing graphs, tables, equations, and algebraic manipulation. (Units parallel to this have logarithmic and exponential expressions at the core of learning.) The Intermediate Value Theorem is applied to find exact or approximate solutions of equations or zeroes of continuous functions. Cumberland, Lincoln, and Woonsocket Public Schools C-20

9 Precalculus, Quarter 2, Unit 2.3 Trigonometry Solving Equations (8 days) Future Learning In calculus, the derivative concept will be applied to the trigonometric functions. A more rigorous approach of the end and asymptotic behavior of these functions will be analyzed in a variety of representations. Trigonometric applications will be investigated via the continuity and limit concept. sin(x) x will be studied as the basic premise underlying the Sandwich/Squeeze Theorem. Additional Research Findings Beyond Numeracy states that many people remember trigonometric identities indicating the unconditional equality of one complicated trigonometric expression for another. However, only a few identities are crucial; the most important is sin 2 x + cos 2 x = 1 (pp ). Notes About Resources and Materials Cumberland, Lincoln, and Woonsocket Public Schools C-21

10 Precalculus, Quarter 2, Unit 2.3 Trigonometry Solving Equations (8 days) Cumberland, Lincoln, and Woonsocket Public Schools C-22

11 Precalculus Quarter 2, Unit 2.4 Laws of Sines and Cosines and Area of Polygons Overview Number of instructional days: 7 (1 day = minutes) Content to be learned Calculate angle measures, arc lengths, area of sectors, circumference, and area of circles using trigonometric principles. Calculate angle measures and side lengths of oblique (non-right angle) triangles. Derive the law of sines and the law of cosines from previously learned formulas and principles. Calculate the area of triangles and polygons using the trigonometric law of sines and law of cosines (to include the ambiguous case). Determine the volume of polygons using partitioning with Cavalieri s Principle. Create formal and informal justification, validation, and verification proofs in the analysis of trigonometric properties and propositions involving angles, arcs, circles, polygons, and segments. Mathematical practices to be integrated Model with mathematics. Relate learning in mathematics to everyday situations involving the laws of sines and cosines or area of polygons. Identify the important quantities and their relationships using mathematical models. Attend to precision. Use labels and units of measure correctly when communicating about situations involving the laws of sines and cosines or area of polygons. Calculate and compute accurately when solving problems involving the laws of sines and cosines or area of polygons. Essential questions In what situations would it be necessary to implement Cavalieri s Principle to determine volume? In what real-life applications are the laws of sines and cosines applied in determining angle measures and side lengths? What impact do the geometry postulates for congruence and similarity have on trigonometry? How does technology validate the results of computations using the law of sines and the law of cosines? How does technology validate the results of computations using areas and volumes of polygons? What is the connection between circular and triangular trigonometry? Cumberland, Lincoln, and Woonsocket Public Schools C-23

12 Precalculus Quarter 2, Unit 2.4 Law of Sines and Cosines and Area of Polygons (7 days) Written Curriculum Grade-Span Expectations M(G&M) 12 6 Solves problems involving angles, lengths and areas of polygons by applying the trigonometric formulas (law of sines/cosines, Area = ½ ab sin C) ; and applies the appropriate unit of measure. (Local) M(G&M) Demonstrates conceptual understanding of spatial reasoning and visualization by performing and justifying constructions with compass and straightedge or dynamic geometric software. (Local) M(G&M) AM 6 Solves problems involving volume using Cavalieri s principle and derives and uses formulas for lengths of arcs and areas of sectors and segments of circles. (Local) M(G&M)-12-2 Creates formal proofs of propositions (e.g. angles, lines, circles, distance, midpoint and polygons including triangle congruence and similarity). (Local) Clarifying the Standards Prior Learning In grades 1 2, students demonstrated conceptual understanding of length, height, perimeter, and area using nonstandard units and models or manipulatives to represent polygons. In grade 3, the conceptual understanding extended to perimeter of polygons and area of rectangles on grids. Students expressed all measures using appropriate units. In grade 4, the perimeter and area extended to polygons or irregular shapes on grids or using formulas. In grade 5, perimeter and area of right triangles were studied as well as volume of rectangular prisms (cubes). In grade 6, perimeter and area of quadrilaterals and all types of triangles were studied. Volume of rectangular prisms within a problem-solving context was introduced, and segment relationships within circles (radius, diameter, circumference) were studied. In grade 7, students determined area of circles, perimeter, and area of composite figures (any quadrilateral, triangle, and parts of a circle). The surface area of rectangular prisms was studied. Volumes of rectangular prisms, triangular prisms, and cylinders were emphasized. In grade 8, surface area and volume were extended to triangular prisms, cylinders, pyramids, and cones. In grades 9 10, perimeter, circumference, area, surface area, and volume of 2- and 3- dimensional figures, including composite figures, were developed. Beginning in grade 3, students demonstrated conceptual understanding of visualization and spatial reasoning by copying, comparing, and drawing models of triangles, squares, rectangles, rhombi, trapezoids, hexagons, and circles. The study of 3-D model building of rectangular prisms began. In grade 4, octagons were studied as well as 2-D model building of rectangular prisms. In grade 5, 2-D or 3-D representations were introduced for triangular prisms, cones, cylinders, and pyramids. In grade 7, sketching 3-dimensional solids and drawing nets of rectangular and triangular prisms and nets of cylinders and pyramids were used to determine the technique for finding surface area. In grades 9 10, dynamic geometric software was used to generate 3-D objects from 2-D perspectives (or vice versa). Alternatively, this was done by solving related problems. Cumberland, Lincoln, and Woonsocket Public Schools C-24

13 Precalculus Quarter 2, Unit 2.4 Law of Sines and Cosines and Area of Polygons (7 days) Current Learning Students use appropriate units of measure to solve problems using angles, lengths, areas, and trigonometric formulas to solve polygon perimeter, circumference, area, surface area, and volume. In advanced mathematical studies, students solve problems calculating volume using arcs, sectors, and circle segments using Cavalieri s Principle. Students find area and volume of non-regular shapes and partition them into rectangles, squares, prisms, trapezoids, and cylinders. Students demonstrate conceptual understanding of spatial reasoning and visualization using compass, straightedge, and dynamic geometric software to perform, justify, and create constructions. Future Learning In calculus, these ideas will be extended into the concepts of finding area under a curve, area between curves, volumes of cross-sectional solids, net area, net accumulation, the definite integral, rectangular approximation method, trapezoidal approximation method, Simpson s Rule, and the limit definition of the integral using Riemann Sums. In many career fields including, but not limited to architecture, surveying, the arts, and computer technology (CAD) perimeter, area, surface area, and volume applications will be employed. Additional Research Findings Research on this content can be found in the following references: Beyond Numeracy (pp , , and ); Science for All Americans (pp ); Principles and Standards for School Mathematics (pp and ); and A Research Companion to Principles and Standards for School Mathematics, (pp and ). This research states that the area of a figure can be found by partitioning it into triangles or rectangles. Trigonometric ratios of similar triangles are equal. Many students understand square units as things to be counted rather than as subdivisions of the plane. Cumberland, Lincoln, and Woonsocket Public Schools C-25

14 Precalculus Quarter 2, Unit 2.4 Law of Sines and Cosines and Area of Polygons (7 days) Cumberland, Lincoln, and Woonsocket Public Schools C-26