Ms. Maria Heizel T. Salayo. Time Frame: TBD ... from those situations. Use Laws of Sines and Cosines to solve problems dealing with the .... They will give you one chance to "sell" your layout plan and budget proposal. ⢠Make notes to help ...
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Unit 3 Circular Function and Trigonometry
Unit Title: Subject / Quarter: Key Words:
Mathematics IV / Third Quarter
Year Level:
4th Year
cirles, radian, degrees, graphs, right triangle, trigonometric functions angle, law of sines, law of cosines, fundamental identities
Designed by: School District:
Ms. Maria Heizel T. Salayo Quezon City District 2
Time Frame: School:
TBD
New Era University
Unit Summary: In this unit students will learn about the Circular Function and Trigonometry. Students will have a basic understanding of right triangle and will be able to evaluate the six trigonometric functions using the "special right triangles" and the unit circle. The student will use the definitions of the six trigonometric functions to find the sine, cosine, tangent, cotangent, secant, and cosecant of an angle in standard position, given a point other than the origin on the terminal side of the angle. Circular function definitions will be connected with trigonometric function definitions. The student, given the value of one trigonometric function, will find the values of the other trigonometric functions. Properties of the unit circle and definitions of circular functions will be applied. The student will find the values of the trigonometric functions of the special angles and their related angles as found in the unit circle without the aid of a calculating utility. This will include converting radians to degrees and vice versa. The student will use a calculator to find the value of any trigonometric function and inverse trigonometric function. Students will use trigonometry to find unknown sides and angles of triangles. Students will explore and verify the trigonometric identities and be able to apply all of this in real life situations and solve practical applications. Big Idea for this Unit Understand and apply concepts, graphs, and applications of a variety of families of functions, including polynomial, exponential, logarithmic, logistic and trigonometric. Use appropriate functions to model real world situations and solve problems that arise from those situations. Use Laws of Sines and Cosines to solve problems dealing with the length of sides and measures of angles in triangles. Use trigonometric ratios to perform indirect measurements. Extend properties and concepts of right triangles to include trigonometric ratio and functions. Unit Design Status:
Completed template pages – Stages 1, 2, and 3 Completed blueprint for each performance task
Completed rubrics Suggested extensions
Directions to students and teachers
Suggested projects
Suggested accommodations
differentiated instructions
Status: Initial draft (date 01/10/10)
Revised draft (date 05/18/10)
Peer reviewed
Content reviewed
Field tested
Validated
Anchored
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STAGE 1: Desired Results Established Goals Content Standards The learner demonstrates understanding of the key concepts and principles of circles, circular and trigonometric functions, their graphs and application to real life. Performance Standards The learner: creates a variety of models to represent circles, trigonometric and circular functions. formulates accurately problem situations where circular and trigonometric functions are evident. explains thoroughly the relationship between the coordinates of a point on the unit circle and the values of the circular functions. investigates the behavior of the graphs of circular functions of real numbers. applies accurately circular and trigonometric functions in the fields of work like construction, navigation, etc. solve problems involving circular and trigonometric functions using a variety of strategies.
Essential Understandings The students understand that…
The concepts of circles, circular and trigonometric functions can be applied in different fields of work such as construction, navigation, surveying, etc.
Essential Questions
What do trigonometric functions have to do with right triangles, and what is a unit circle?
How are the concepts of circles and circular functions applied in real life?
Knowledge on trigonometric functions and identities are tools which facilitate the process of making decisions.
Why are circular and trigonometric functions and identities essential in making decision?
Trigonometry connects right triangles, circles, and wave functions.
Why do we study trigonometry?
Many values of trigonometric functions can be generated using trigonometric identities and a few special trigonometric values.
How do you apply the properties of graphs of trigonometric functions in real life?
When is it more appropriate to represent sine (cosine, tangent, etc) numerically, symbolically, or graphically?
Mathematicians model periodic behavior in the real world by using sinusoidal functions.
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STAGE 1: Desired Results (continued)
Familiarity with trigonometric identities is essential to simplify trigonometric expressions, solve trigonometric equations, or re-write other identities in simpler form. Each trigonometric function has an inverse function that is used by mathematicians to solve problems in which angle/arc measure is required. Triangular relationships can be used to solve a variety of practical problems such as areas of triangles.
How do we apply trigonometric values and formulas in solving real life problems?
Why do we need to use trigonometric equations to help us solve various reallife situations?
Why is knowledge of trigonometric functions essential in solving problems related to sports, bridges, cable system and projectiles?
How do we generate these functions?
How are triangles used to solve everyday problems?
How are triangles helpful in finding areas and in measuring distances indirectly?
How can right triangles be used to find unknown sides and angles including angles of depression and elevation?
Knowledge The students will know… A. Circular Functions and Trigonometry 1. defining the: unit circle arc lengths unit measures of an angle 2. converting from degree to radian and vice versa 3. illustrating the: angles in standard position (i.e. initial side coincident with the positive x-axis coterminal angles reference angles
Skills The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to:
illustrate and describe the terms and concepts related to trigonometry
describe and illustrate the unit circle
recognize the 45-45-90 and the 30-6090 triangles and the lengths of its sides
use reference angles to evaluate trigonometric functions in all four quadrants
evaluate trigonometric function values of special angles
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STAGE 1: Desired Results (continued) 4. visualizing rotations along the unit circle and relate these to angle measures (clockwise or counterclockwise directions) length of an arc angles beyond 360o or 2 radians 5. determining the coordinates of a point on its terminal side given an angle in standard position in a unit circle. 6. when one coordinate is given (applying the Pythagorean Theorem and the properties of special right triangles) when the angle is of the form: 1800n + 300 1800n + 60 1800n + 450 900n 7. defining the six circular functions sine cosine tangent cotangent secant cosecant 8. finding the six circular functions of angles with special values 9. give the angle, 2 2 2 or 3600 2 3600 draw the graph of: sine cosine tangent 10. describing the properties of the graphs of: sine cosine tangent 11. defining the six trigonometric functions of an angle in standard position
evaluate trigonometric function using a calculator
use right triangles and the ratio of its sides to illustrate trigonometric relationships
graph trigonometric functions and solve simple trigonometric equations
apply the trigonometric functions to "real life" problems.
find the domain, range, amplitude, period, x-intercept(s), and y-intercept of the six trigonometric functions, whenever these quantities exist
solve problems involving right triangles
apply solution of right triangles to problems on angles of inclination, elevation, and depression as well as the direction of a line
evaluate inverse trigonometric values
apply concepts related to trigonometry to solve real life problems
derive the fundamental identities in trigonometry
solve trigonometric identities
verify trigonometric identities
prove and disprove trigonometric identities
simplify trigonometric formulas and expressions
derive law of sines and law of cosines
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STAGE 1: Desired Results (continued) 12. finding the values of six trigonometric functions of an angle , given some conditions 13. solving simple trigonometric equations 14. stating the fundamental trigonometric identities B. Triangle Trigonometry 1. apply trigonometric functions, laws of sine and cosine in solving problems involving: right triangles triangles using the Law of Sines triangles using the Law of Cosines
apply the law of sines and cosines to solve problems related to oblique triangles
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STAGE 2: Assessment Evidence PERFORMANCE TASK Topic: Trigonometry Cooperative Group Project: Portfolio: Park layout and budget proposal Title: Designing a Park Special Teacher’s Directions: Most communities have parks for people to enjoy. Parks can have picnic areas. Softball or baseball fields, play areas, paths, and grassy areas. Tell them they can use the Problem-Solving guidelines to help them solve situational problem about designing their park. They will use the entire donated lot area for the park. They have to keep the cost as low as possible. Ask: How will you design your park? What will be the total cost for its development?
Project must include: 1. Research on park’s layout and design. 2. Notes maintained in group composition notebook. 3. Factors to be addressed when deciding the cost of each material. 4. Calculate the area for each activity. 5. Dimensions for each activity area. 6. Students must name their park. 7. Scale model of park’s layout and design.
Student Directions: A rectangular lot, 375 m by 500 m, has been donated to your city to be used as a park. The park is to have one tennis court, a softball field, and at least one picnic area. Volunteers will build the park, but materials must be purchased. You will play the role of city engineer and your job is to suggest a layout plan for the park, make a scale drawing of the park (poster) and propose a budget for the project. You will create a portfolio with factual information that the city has requested to back-up your result. You have to convince the city mayor and its sponsors to choose your proposal. Your plan must contain the following. 1. 2. 3. 4. 5. 6. 7.
The dimensions of a tennis court and a softball field. Decide how much space you will allow for each activity. The scale model of your drawing or layout plan. Cost of material for each picnic area. Cost of material for the tennis court. Cost of material for the softball field. Overall budget proposal for the plan.
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STAGE 2: Assessment Evidence (continued) Requirements for Presentation:
Meet frequently with group and record time. Decide who will talk about which parts of portfolio. All members must speak to the city at least once during presentation. Help group members figure what is important to tell the city mayor and its sponsor. They will give you one chance to "sell" your layout plan and budget proposal. Make notes to help you remember what you are going to say. (Note cards) Practice the presentation with your group to get timing down. Use the public speaking rubric to give each other feedback. Discuss as a group what to wear the day of your presentation
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STAGE 2: Assessment Evidence (continued) STANDARDS and CRITERIA for SUCCESS: Rubric: PUBLIC SPEAKING / ORAL PRESENTATION Trait: NONVERBAL SKILLS 4 Criteria Excellent
Body language
Holds attention of entire audience with the use of direct eye contact, seldom looking at notes Movement seems fluid and help the audience visualize
Poise
Displays relaxed, self-confidence, with no mistakes
Eye contact
Enthusiasm
Elocution
Subject knowledge
Organization
Mathematical Knowledge
Demonstrates a strong, positive feeling about topic during entire presentation Uses a clear voice and correct, precise pronunciation of terms so that all audience members can hear presentation. Demonstrates full knowledge by answering all class questions and elaboration. Presents information in logical, interesting sequence which audience can follow. Math is correctly used and cited in project with at least one other scenario.
3
2
1 Needs Improvement No eye contact with audience, as entire explanation is read from notes
Good
Satisfactory
Consistent use of direct eye contact with audience, but still returns to notes
Displays minimal eye contact with audience, while reading mostly from the notes
Make movement or gestures that enhances articulation Makes minor mistakes, but quickly recovers from them; displays little or no tension Occasionally shows positive feelings about topic
Very little movement or descriptive gestures Displays mild tension; has trouble recovering from mistakes Shows some negativity toward topic presented
Tension and nervousness is obvious; has trouble recovering from mistakes Show absolutely no interest in topic presented
Voice is clear; pronounces most words correctly, most audience members can hear presentation.
Voice is low. Incorrectly pronounces terms. Audience members have difficulty hearing presentation.
Mumbles, incorrectly pronounces terms, and speaks too quietly for a majority of students to hear.
At ease with expected answers to all questions, without elaboration.
Uncomfortable with information and is able to answer only rudimentary questions.
Presents information in logical sequence which audience can follow.
Audience has difficulty following presentation because student jumps around.
Math is correctly used and cited in project.
Math's role in project is mentioned and somewhat accurate.
Does not have grasp of information; student cannot answer questions about subject. Audience cannot understand presentation because there is no sequence of information. Math's role in project is not represented.
No movement or descriptive gestures
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STAGE 2: Assessment Evidence (continued) STANDARDS and CRITERIA for SUCCESS Rubric: Portfolio Self Evaluation Type: PROCESS Category
Format
Conventions
Quality of Information and Mechanics
Ideas
Math Scenario
Solution
Graphics
Promptness
4 You Have My Vote The portfolio has exceptionally attractive formatting and well-organized information. All of the writing is done in complete sentences. All the spelling, punctuation and grammar are correct.
All facts in the portfolio are accurate, and all spelling, grammar, and mechanics are correct. The whole portfolio communicates relevant information appropriately and effectively to the intended audience.
3
2
Pretty Convincing
Tell Me More
The portfolio has more attractive formatting and organized information.
The portfolio has some organized information with random formatting.
Most of the writing is done in complete sentences. Most of the spelling, punctuation and grammar are correct. Most facts in the portfolio are accurate, and most spelling, grammar, and mechanics are correct.
Some of the writing is done in complete sentences. Some of the spelling, punctuation and grammar are correct. Some facts in the portfolio are accurate, and some spelling, grammar, and mechanics are correct. Some parts of the portfolio communicate little relevant information to the intended audience.
Our solutions are correct.
Most parts of the portfolio communicate relevant information appropriately to the intended audience. We mostly used math to document and support all aspects of the project. We had correct math solutions.
The graphics go very well with the text, and there is a very good mix of text and graphics.
The graphics go fairly well with the text, and there is a good mix of text and graphics.
Submits the output on time.
Submits the output the following meeting after the deadline.
We used math to document and support all aspects of the project.
We sometimes used math to document and support all aspects of the project. We had a few correct math solutions. The graphics go somewhat well with the text, but there is a few mixes of text and graphics. Submits the output one three meetings after the deadline.
1 Back to the Drawing Board The portfolio has no organized information with random formatting.
Writing is not done in complete sentences. The spelling, punctuation and grammar are not correct throughout the portfolio. Minimal use of facts in the portfolio and minimal use of appropriate spelling, grammar, and mechanics. The portfolio communicates irrelevant information to the intended audience.
We forgot to include our math problems and solutions. We should go back and do this again. The graphics do not go with the accompanying text and there is no mix of text and graphics. Submits the output one week or later after the deadline.
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STAGE 2: Assessment Evidence (continued) Other Evidences Teacher observation of students working on tasks. Assessment of student work. Student pairs create problems to swap with other pairs to solve. Problem solving. Assessment of understanding through class conversation and questioning. Oral review of vocabulary words. Read Essential Question and call on several students to give their interpretations. Ask for any misunderstandings. Interpreting essential questions. Write a journal on the importance and application of circular functions and trigonometry. Selected Response/Short-answer test/quiz Quizzes, tests, portfolio checks, homework, calculation checks, table functions and graphs match. Assignments, classwork, essays – Research Unit test and Periodical test
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STAGE 3: Learning Plan Focus: Circular Function and Trigonometry The student will maintain a journal documenting his or her learning experiences. Each day the student will begin a new page of notes. The student must date each entry. The journal will document the entire student's learning. Example: key vocabulary, formulas, example problems, etc. Students will also document any questions or misunderstandings. The teacher will periodically provide written commentary to guide and reassure the student. The student will submit a weekly report detailing his or her progress toward the final product (presentation defending his or her choice). Student will research park’s layout and design. The research will be summarized in a portfolio. Student will log computer animated scale drawing of the park’s layout and design. 1. The teacher will introduce the Unit by presenting the essential questions. (Why do we study trigonometry? Why are circular and trigonometric functions and identities essential in making decision? How are the concepts of circles and circular functions applied in real life?) Students will begin their portfolios. W, H 2. Class discussion of student's experiences strolling in the park, having a picnic and spending time with their family. Describe the different activities you can do in the park. Have you ever thought about designing your own park? H, E 3. Teacher will tell the students that they are now going to learn about how to design a park. When the unit ends, students will be able to design and layout their own park with different activity areas and propose its budget. (Add if applies, PICNIC!!! Together with your students so they would be able to experience how to bond with their friends, classmates and teachers.) E, T 4. Students will learn basic information about circular functions and trigonometry through the use of mini-lessons. Mini-lesson data will be completed in composition books for each group. Students will be able to find solutions to different problems involving real-life situations. E 5. Mini-lessons will consist of practice pages from Mathematics textbooks and the internet will cover the following topics given here with a brief explanation. Explain to the students that before they can design their park and budget its expenses, there are several topics of mathematics that they must understand. A lot of examples should be given on each topic and the students should practice using a lot of examples. W, E 6. Trace the history of Trigonometry. State real-life situations that use trigonometry. Define terms related to angles; show how to generate angles with given measurements, and how to name angles. Give exercises on locating points and drawing angles in standard position, and showing the direction of rotation by an arrow. E
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STAGE 3: Learning Plan (continued) 7. Have the students show the difference between radian and degree as units of angular measurements. Do a conversion of angle measures from degrees to radians or from radians to degrees. Solve real-life problems. E 8. Summarize the concepts that were discussed in the class. R 9. Journal Entry: Let the students write and share their insights from the lessons, especially on the history of trigonometry and how this would be able to help them plan their park layout and design. Ask the students to give oral reactions on the lessons just finished. E-2, T 10. Ticket Out the Door: Write an example of radian measures and degree measures on a slips of paper and give one to each student. As students leave the room, ask them to convert radian to degree and vice versa. R, E-2 11. Begin the mini-lesson by asking these questions: What are circular functions? How do we generate these functions? What are the bases in formulating these functions? W, H 12. Review the sign of x and y in each quadrant of the rectangular coordinate plane. Define circular functions using illustrative figure. Emphasize the changes in the signs as terminal side of the angle varies. R, E 13. Show pictures of real-life applications of trigonometric functions, especially on designs. Give illustrative problems on how to find each trigonometric function of an angle whose coordinates of a point on its terminal side are given. Show the students how to find the values of the other functions given the value of one function of an angle. H, E 14. Have the students define and illustrate quadrantal angles. Explain how the values of the functions of 00 and 900 are obtained without using the calculator or table of values. Let the students work in pairs to discuss solution of the problems in real-life. E, T 15. Recall special triangles like isosceles right and equilateral triangles. In each type of triangle, what relations exist among the sides and the angles? How do we use these relations to find the values of trigonometric of 300, 450 and 600? Explain how to find the values of trigonometric functions. R, E 16. Define and illustrate a reference angle. Show how reference angles are obtained. Illustrate how to find the values of the trigonometric functions of 1200, 2100, 2250 and 3150. Have the students answer the exercises on the book. E, E-2 17. Assign groups and have the students create sketches of the six trig functions by hand using the chart and graph paper. E-2, T
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STAGE 3: Learning Plan (continued) 18. To graph the sine and cosine function, the student must know the basic graph of functions y=sin x and y=cos x. For example to sketch the graph of y=4 sin x, the students must be able to conclude that because a = 4, the amplitude is 4 times the amplitude of y=sin x. E 19. Real-World Connections: Lead the students to realize that some things and events occur at regular intervals, and thus have patterns, like the passing of day and night. Let the students work in groups to perform an activity such as solving real-life problems. E, H, T 20. Have students write a journal on what they have learned, what they want to learn more about the things that they did not understand very well. Surf the internet for the materials on the behavior of trigonometric functions other than those that were already taken up in the class. Tell them to relate it to their task of designing a park. W, T 21. If the students do not have any access to the internet, they may be asked to draw, take pictures, collect pictures of events, situations, objects, among others, that demonstrate trigonometric patterns. O, T 22. Ask the students to cut out various types of triangles with given sides. Have them find the measure of each acute angle using a protractor. Let them check the answer using trigonometric ratios. T, E 23. Let the students practice on the use of trigonometric functions when; (a) the angle is given and the values of the functions are unknown, or (b) the values of the functions are given and the angle is given. E 24. Assign the trig project. Students must create and PowerPoint, a newsletter, and a website. Allow time to work in the computer lab. Six groups are needed (one for each trigonometric function). Each group is to do the assigned function in a PowerPoint, and page in the newsletter, and a page on the website to be created. 25. Real-World Connections: Ask the students to define and illustrate by giving concrete examples of the angles of inclination, elevation, and depression. a. The angle of elevation of a mango fruit looked at by boy on the ground. b. The angle of elevation of a flying kite being viewed by a child. c. The angle of depression of a car on the street viewed from the 8 th floor of a building. d. The angle of depression of the fruits that fell on the ground as viewed by a boy from the top of the tree. e. The angle of inclination of the steps of stairs, and of a post. Have the students solve verbal problems related to these types of angles. Let them do the outdoor activity using clinometers. As much as possible relate these type of activity in planning a design of the activity areas of the park. E, T, W
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STAGE 3: Learning Plan (continued) 26. A map or a globe may be shown to determine the location of a particular point on earth. Define bearing (or direction of a line) based on the illustration. Define “bearing” further using the coordinate plane. Summarize all the concepts in solving right triangle and let the students express their insights, their fears, and their misconceptions about the lessons. E, R, E-2 27. Ticket Out the Door: Draw an example of bearing on pieces of paper and give one to each student. As students leave the room, ask them to read the direction based on the illustrations. R, E-2 28. To begin the lesson in fundamental identities, let the students answer the question “What is an individual’s identity?” Relate the real life identities with trigonometric identities. H 29. Derive the eight fundamental identities that include the reciprocal relations, Pythagorean relations and ratio relations. On proving identities, explain the guidelines. The students may work in pairs and may be asked to present and explain their proofs. E, T 30. Ticket Out the Door: Get the students to write from memory the eight fundamental identities in trigonometry. Ask them to look for identities not found in their textbook. Have them present the proof of the identity that they have chosen for the next day. R, E-2 31. Illustrate and explain how to solve trigonometric equations. Encourage students to give exact answer (rather than decimal approximations using a calculator) when solving trigonometric equations. The algebraic model is difficult for students to visualize. Encourage them to write or visualize the algebraic model, write the corresponding trigonometric equation in terms of a single trigonometric function, and then solve for x. Ask the students to write a journal on the difficulties they encountered in the study of trigonometric equation. E, E-2 32. Explain how an oblique triangle differs from the other types of triangles and the condition in which the Law of Sines and the Law of Cosines may be used. Derive the laws. When solving triangles a careful sketch is useful as a quick test for the feasibility of an answer. Remember that the longest side lies opposite the largest angle, and the shortest side is opposite the smallest angle. E 33. In cases where the Law of Cosines must be used, encourage the students to solve for the longest angle first, and then finish the problem using either the Law of Sines or Law of Cosines. Illustrate how these laws are used to solve oblique triangles and problems pertaining to oblique triangles. E 34. Ask the students to explain in their own words the meaning of the laws derived. Let them also express their insight on the process of deriving the laws. Give problem sets. Discuss the solution to the problem given. R, E-2
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STAGE 3: Learning Plan (continued) 35. At the conclusion of the unit, students review their results of their completed projects. Students’ self- and peer assess their end products using rubrics. Present them in front the class and teacher for further evaluation. Give feedback using public speaking rubrics. E-2, T Exhibit/evaluate: E-2 1. Chapter test from textbook 2. Presentation of park’s layout and design 3. Completion of tasks for portfolio 4. Peer evaluation for presentation 5. Group evaluation 6. Public speaking skills 7. Calculations, graphing, and function tables were completed correctly Teacher Responsibilities: 1. Prepare/post Essential Question 2. Help students prepare chart, table and graph 3. Gather background information and resources on designing a park. 4. Plan and conduct mini-lessons 5. Monitor daily progress through use of quizzes, journals, portfolios, and conferencing.
SUGGESTED EXTENSIONS Activities: 1. Assign groups and have the students create sketches of the six trigonometric functions by hand using the chart and graph paper. 2. Students, who feel comfortable using the GPS may select an actual odd shape parcel, use the GPS to measure the lengths of sides and angles and calculate the area. This will be compared to the county assessor’s records to determine accuracy of both methods. They write step-by-step directions for those students who struggle using the GPS. 3. Divide the students in 3-4 groups. Give these problems to each group, let them solve it and show their representation and solutions in a poster or manila paper. Present it to the class and have them explain each step. a. A man is walking along a straight road. He notices the top of a tower subtending an angle A = 60o with the ground at the point where he is standing. If the height of the tower is h = 15 m, then what is the distance (in meters) of the man from the tower? b. A little boy is flying a kite. The string of the kite makes an angle of 30 o with the ground. If the height of the kite is h = 24 m, find the length (in meters) of the string that the boy has used.
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STAGE 3: Learning Plan (continued) c. Two towers face each other separated by a distance d = 35 m. As seen from the top of the first tower, the angle of depression of the second tower's base is 60 o and that of the top is 30o. What is the height (in meters) of the second tower? d. A ship of height h = 21 m is sighted from a lighthouse. From the top of the lighthouse, the angle of depression to the top of the mast and the base of the ship equal 30o and 45o respectively. How far is the ship from the lighthouse (in meters)? e. Two men on opposite sides of a TV tower of height 20 m notice the angle of elevation of the top of this tower to be 45o and 60o respectively. Find the distance (in meters) between the two men.
Differentiated Instructions 1. For English Language Learner Using interpersonal Skills To reinforce the concepts of the lesson, place students in pairs and have the students take turns completing the Exercises. As one student works the problem, have the other student offer guidance and suggestions. Make sure students offer constructive reinforcement to each other and that each student completes at least one exercise.
2. For Struggling Students and Students with Special Needs Connecting Mathematics to Music Have students sing the song “Ode to Pythagoras” to the tune of “Hokey Pokey”, let them create their own jingle in any concept in trigonometry.
ODE TO PYTHAGORAS Sung to “hokey pokey” You take your first leg “a,” And then your next leg “b,” Take the sum of their squares, Are you following me? To use this famous theorem, For Pythagoras let’s shout, That’s what it’s all about. We’re not quite finished yet, Cause there’s a third side to see, You know this tri-angles right, Are you following me?
To use this famous theorem, For Pythagoras let’s shout, That’s what it’s all about. Now add the square of leg “a,” To the square of leg “b,” You get the square of side “c,” Are you following me? To use this famous theorem, For Pythagoras let’s shout, That’s what it’s all about. a2 + b2 = c2
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STAGE 3: Learning Plan (continued) 3. For Above Average Students Connecting Mathematics with History and Science Trigonometry, in ancient times, was often used in the measurement of heights and distances of objects which could not be otherwise measured. For example, trigonometry was used to find the distance of stars from the Earth. Even today, in spite of more accurate methods being available, trigonometry is often used for making quick and simple calculations regarding heights and distances of far-off objects. For this, the value of various trigonometric functions is needed. Students can use an Internet search engine to search about the different methods of finding the distances during the ancient times and today’s modern tools. Compare them and write an essay about the development of these tools how it makes peoples lives better.
SUGGESTED PROJECTS Triangulation and Surveying As you know, triangulation is the technique of dividing part of the plane into triangles in which almost all angles and a few sides are measured. From these measurements, all other distances are determined. Find out more about how surveyors use triangulation and other aspects of trigonometry in their work. You may want to concentrate on current applications such as how a new subdivision will be surveyed, or you might want to do some historical research on how a community near you was originally surveyed.
