1. Time: 3 – 45 minute periods

2. LESSON PLAN Unit Title: Trigonometric Functions

a. Trigonometry (or even a Gifted and Talented Geometry Class)

b. 9th or 10th grade

c. Using Exact Trigonometric Values to Graph the Sine and Cosine Functions

3. Instructional goals:

a. Content from Using the VSC Geometry, 2007

Goal 2 “The students will demonstrate the ability to solve mathematical and real-world problems using measurement and geometric models and will justify solutions and explain processes used.

Expectation 1 “The student will apply geometric properties and relationships to solve problems using tools and technology when appropriate.”

b. (technology) Students will “select and use the appropriate tools and equipment in making two-dimensional and three-dimensional representations of design solutions” (“Maryland Technology Education Voluntary State Curriculum,” 2005, p. 31).

4. Objectives: The broader context defined in the VSC states that students “will solve problems using two-dimensional figures and/or right-triangle trigonometry” (“Using the VSC Geometry,” 2007). In particular, by the end of these lessons, students will be able to:

1. find the exact value of a trigonometric function without using the calculator.

2. use the calculated exact values to graph the sine and cosine functions.

5. Rationale

Trigonometry is one of the last stops students make in the world of higher level mathematics before having access to calculus. A very important theme in the pre-calculus world is for students to gain a firm foundation in understanding functions: from an applicative, conceptual, and categorical perspective. This lesson seeks to expand the categorical perspective as students will be introduced to two more functions that exist in the mathematics world, thereby preparing them further for the world of higher level mathematics.

6. Content

• Accessing the definitions of sine and cosine ratios

• Accessing the geometry of the special-right triangles

• Graphing trigonometric functions

7. Instructional procedures

(Day 1)

I. Drill. Students will review the special-right triangle relationships by completing

a set of exercises found on the PowerPoint slide. Refer to the PowerPoint for the exercises. Continue clicking to display answers as this is discussed with students.

II. Motivation. Hold a meaningful but brief discussion to remind students about why we study functions. Use slide 4 to remind students how linear functions are used to relate the distance, rate of speed, and time for motion. Use slide 5 to remind students how quadratic functions can be used to model the trajectory of an object, particularly a baseball. Use slide 6 to remind students of how the greatest integer function is used to calculate a person’s telephone bill. Finally, use slide 7 to show students that even though their library of functions gives them access to a variety of real-world phenomenon, there are still some applications to which they do not have mathematical access, which will be gained by these series of lessons.

A much longer motivation to tap into students’ understanding of why it is that they study higher level mathematics can be found on my WebQuest. Please refer to my website for complete details on implementing this as a motivation. However, this motivational option will need approximately 3 – 5 days of implementation.

III. The teacher will have a student read the objective stated on slide 8. Suggest to students that the special angles reviewed in the drill will become the “reference” angles needed for today’s lesson. More details will follow at this time

IV. Development.

(a) The teacher use various questioning techniques to show students that they have all of the tools they need to find the trigonometric value of an angle beyond ninety degrees. The teacher will model for students how to find the trigonometric values of two hundred forty degrees on a powerpoint slide. The teacher should remind students of previously learned relationships, particularly that they should remember where the trigonometric functions are positive and negative in the coordinate plane. All of this will be done in the PowerPoint slide on a click-by-click basis. Refer to slides 9 and 10.

(b) The teacher will ask students to work in pairs to find the trigonometric values for negative five pi over four . Once students have had an opportunity to work, give go over this problem using slides 11 and 12 and/or have students display solutions on the chalkboard.

(c) Finally, the teacher will ask students to work individually to calculate the trigonometric values for three hundred thirty degrees. This solution can be shared as a class or collected.

(d) (Depending upon the speed of the class, this will probably be a good place to stop. For an advanced class, procedures d and e could be carried out on day 1. For others, this may need to be combined with tasks advised for day 2.)

(e) Suggest to students that the current approach does not work if the terminal side of the angle lies on the x or y axis. Show slide 16 where students will become exposed to the definition of a quadrantal angle.

(f) Introduce the unit circle and the associated ordered pairs using slide 16.

(g) Model for students how to find the trigonometric values of a quadrantal angle, such as using slides 17 and 18.

(h) As an assessment, allow students to work in pairs to find trigonometric values for another quadrantal angle, such as negative two hundred seventy degrees. Have students put their solutions on the board, or use slides 19 and 20 discuss the correct procedure for arriving at the solution.

(i) The teacher will assign relevant homework problems whereby students will calculate trigonometric values.

V. Summary. Students will orally recite or write a paragraph that summarizes the procedures for finding the exact trigonometric value of an obtuse or reflex angle.

(Day 2)

I. Drill. Students will review yesterday’s introductory lesson on calculating trigonometric values by the trigonometric values of various angles. A sample drill would be to find each of the following trigonometric values:

(1) cosine of one hundred thirty five degrees

(2) cosecant of three hundred degrees

II. Motivation. The teacher will reactive the discussion from yesterday regarding the need to find another function to access other real life situations.

III. Development.

(a) The teacher will ask students to open the Inspiration file whose title is “What Do The Sine and Cosine Functions Look Like?” The teacher will model for students how to use the setup of this program to calculate the requested trigonometric values. The teacher will also model for students how they will record their findings onto a sheet having the same title as the Inspiration document.

(b) The teacher will allow students to work in pairs on this activity. As students are working, the teacher will monitor student progress. At various times during the lesson, the teacher will ask students to send their work to the teacher to display for the entire class. This will allow students to see if they are on the right track.

(c) Assessment. At the end of the period, students will save their files and e-mail them to the teacher, allowing the teacher to observe student progress. Students may or may/not have finish this exercise in its entirety.

(d) Safety Valve. If anyone finishes early, they may begin the homework assignment for the day. Relevant problems will be assigned to help students to manage their skills.

IV. Summary. Students will orally reflect upon the knowledge that they needed to remember to find answers correctly. This broad question will be more fine-tuned if there is a lack of student response in the following manner:

(1) How did you know which number was placed in the numerator and the denominator? (This should facilitate a discussion centered on how students must know the definitions of the sine and cosine functions. Without knowing for instance that the sine ratio is formed by placing the length of the opposite side over the adjacent side, not much could have been accomplished.)

(2) How did you know when the unit circle was needed? (This should facilitate a discussion centered on knowing when reference angles are NOT able to be used!)

(Day 3)

I. Drill. Students will review the skill of finding exact trigonometric values through the solving two problems. (similar to day 2)

II. Motivation. The teacher will explain to students the final task that will allow them to gain access to two new functions in mathematics.

III. Development.

(a) The teacher will ask students to open the Inspiration file whose title is “Get Organized and Get It Graphed.” The teacher will model for students how they will use the information collected yesterday to categorize the angles properly and to use that information to graph points on the provided grid.

(b) The teacher will allow students to work in pairs on this activity. As students are working, the teacher will monitor student progress. At various times during the lesson, the teacher will ask students to send their work to the teacher to display for the entire class. This will allow students to see if they are on the right track.

(c) As students finish creating the graph on the screen, students will record their findings on to another handout which will be kept in their notes section. Also, the teacher will hold small discussions in groups of 4 concerning how students would graph the function beyond the domain of zero to three hundred sixty degrees.

(d) Assessment. At the end of the period, students will save their files and e-mail them to the teacher, allowing the teacher to observe student progress. Students may or may/not have finish this exercise in its entirety.

(e) Safety Valve. If anyone finishes early, students will begin the corresponding reading of the topic allowing them to gain access to needed vocabulary, particularly cycle, amplitude, and period.

IV. Summary. Students will come to the board and show the general shape of the sine

and cosine functions.

8. Evaluation procedures

Students will be questioned throughout the lesson. They will also send their two Inspiration files to receive feedback as to whether or not they are progressing in the correct direction.

9. Materials

Textbooks (for day 3), pens, pencils, 2 handouts, computers containing Microsoft PowerPoint and Inspiration

10. REFERENCES AND RESOURCES:

Using the VSC Geometry (2007). Retrieved June 25, 2008, from http://mdk12.org/instruction/hsvsc/geometry/standard2.html.

Maryland Technology Education Voluntary State Curriculum (2005). Retrieved June 29, 2008, from http://mdk12.org/instruction/curriculum/technology_education/vsc_technol....

Attachment | Size |
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Trig File 2.isf | 644.06 KB |

Trig Plan 3b.ppt | 1.09 MB |