A Title III

Learning Unit

____________________

Model Schools Learning Unit

Displayed on the State Education Department Learning

Experience Form

DESCRIPTION OF LEARNING UNIT

Title: Trigonometric Functions for Precalculus

Written by Gerald F. Peters for Mathematics grades 11-12

Gouverneur High School, 113 E. Barney St., Gouverneur, NY 13642

School: Phone 315-287-1900 Fax 315-287-7963

Home: Phone 315-287-1687 Fax 315-287-7965

Typed using Microsoft Word in Windows 98

LEARNING CONTEXT

Learning Standards Addressed: NYS Math B Curriculum--Key Idea 7: Patterns/Functions

• Represent and analyze functions, using verbal descriptions, tables, equations, and graphs (7B)
• Model real-world situations with the appropriate functions (7G)
• Analyze the effect of parametric changes on the graphs of functions (7D)
• Apply and interpret transformations to functions (7F)

Connections to other curriculum areas include the study of electricity and harmonic motion in physics. Also the writing done by the students connects to the writing standards in the English Language Arts curriculum.

Prerequisites include a knowledge of how to draw the basic graphs of Y=Sin X and Y=Cos X and the understanding of amplitude and frequency as they apply to trig graphs. Students should also have a good command of the use of written English as they will be expected to produce reports that are "ready to publish".

ESSENTIAL QUESTION

How can trig functions be used to model real life phenomenon?

PROCEDURE

Initiating Activity (Launch): Students view the video "Sines and Cosines-Part I (Cal Tech tape) to link prior knowledge to this unit. When the tape is done, students pair up and take 5 minutes to produce a summary of the key ideas shown on the tape. Students can use drawings, words, or both to create a summary. Several pairs then report their summaries to the whole class until most of the key ideas have surfaced. The teacher's role is to show the video and facilitate the summarizing reports.

Acquiring and organizing new knowledge: In groups of three or four, students work through Activities #34-38 on the Sine function taken from "Graphing Calculator Activities" from Dale Seymore Publications. These five sequential activities will introduce students to the transformation procedures needed to graph functions in the form of y=A[Sin B(x-C)] + D beginning with the graph of y=Sin x. The teacher only has to monitor the groups, provide hints as needed, and provide feedback (answers) as the students finish the activities.

Further organize and store knowledge: Students complete a sequential step graphic organizer to show and explain the steps needed to graph y=3[Sin 2(x-60)] - 2 starting with y=Sin x and drawing a new graph following the order of operations for each transformation encountered. Students work together in collaborative groups with the teacher assisting only when a whole group is stuck. Feedback is provided by having students write possible solutions on the board. Each of these group members then share in the job of explaining the group's work to the whole class. Corrections are made through class discussion if needed.

Refining and Extending the Knowledge:

Students complete a compare and contrast graphic organizer for the functions y=2[sin(x-30)] + 1 and y=2[Cos(x-60)] - 3. The teacher provides support for groups needing it. Feedback can be accomplished as in the last activity.

In the next activity students (in groups) work through Activity 39 ("Graphing Calculator Activities"-Dale Seymore). Students are asked to develop a trig function to model ocean tides and determine when it is safe to sail a boat with a given draft. Each group constructs a full-page graph showing its function that models the tide and a written explanation that constructs support for its choice of safe sailing times. Each group gives a brief report of its work with each group member being responsible for a significant part of the report.

The final activity to extend and refine student knowledge is a lab activity called "Music to My Ears", a public domain activity designed for use with a Calculator Based Laboratory (CBL) sold by Texas Instruments. The CBL is attached to a TI graphing calculator (we use TI-82's) and a special microphone probe. The students work in groups and measure the pressure waves created by a vibrating tuning fork. The calculator displays a trig function that represents the sound waves emitted by the tuning fork. The students use the trace function on the calculator to determine the period and then the frequency of the function. The calculated frequency is compared to the actual frequency stamped on the tuning fork and the percent error is calculated by the same process used in science classes. The activity then leads them through the steps needed to find the periods, amplitude, phase shift, and finally the complete equation of the function. Students also enjoy experimenting with the graphs of tones from some of the musical instruments that they play. With an electronic keyboard, students can easily discover that frequency doubles with each octave on the keyboard. This connects to the physics curriculum.

Applying the knowledge (Meaningful use task):

Students (in groups) work through the development of George Brett's biorhythms in the activity "Biorhythms" from the spring 1994 issue of "Eightysomething!" published free by Texas Instrument Corporation (www.ti.com/calc/docs/80xthing.htm).

Each student then develops the trig functions that model his/her own physical, emotional, and intellectual biorhythms and graphs those functions for the upcoming month. Students then use the process of experimental inquiry to examine the possible validity of the theory of biorhythms. Specific steps in the inquiry process are:

• Students (individually) make personal predictions for the upcoming month based on their acquired knowledge of the biorhythm theory and the graphs of their own biorhythms. These predictions are recorded and set aside for the month.
• Students design and keep a personal daily journal for the nest month recording all significant events in their lives that might possibly be affected by their biorhythms. Also recorded were their reactions to the events.
• At the end of the month students compare their predictions with their journals, and examine the results to determine possible connections with the theory of biorhythms.
• Each student then writes a formal report for the project that meets the criteria shown in the special rubric for this project.

INSTRUCTIONAL/ENVIRONMENTAL MODIFICATIONS

No instructional modifications were needed for this group of average and above average students. Individual learning differences were addressed by extra individual help to needy students while working in small groups. Student desks were moved into and out of small group arrangements as needed.

MATERIALS & SUPPLIES

The video used for the launch activity can be any video that provides an overview of the basic of Y=Sin x and Y=Cos x. Carolina Biological Supply Company (1-800-334-5551) sells such a video for $19.95 that would fulfill the objective.

Each student should have access to a graphing calculator (about $85 each). TI-82's were used in this class.

For the "music lab", each group needs a CBL ($170 per group) and a microphone probe ($30 per probe). The CBL's and microphone probes are sold by many educational supply companies that sell graphing calculators. One such company is D & H (1-800-340-1006).

The teacher needs to acquire the book "Graphing Calculator Activities" written by Charles Lund and Edwin Andersen ($12.95) and published by Dale Seymore Publications. Also, the teacher needs to acquire the spring 1994 issue of "Eightysomething" from Texas Instruments (see Website above) or any equivalent information about biorhythms that explain the theory and how the functions are developed for an individual. The Texas Instruments Website can also provide the software program and the "Music to My Ears" activity sheet.

ASSESSMENT TOOLS & TECHNIQUES

Students were given a traditional test following the activity on ocean tides. The test consisted of traditional graphs and an applied problem similar to the tide activity.

Informal observation of individual and student progress was ongoing throughout the unit as the students worked through the activities. Errors in student thinking and confusion over concepts were addressed by assisting students as needed. Students were able to detect and correct most errors themselves as they worked collaboratively. On an acreage day, students worked in small groups about eighty percent of the time.

Each class ended with a 5-minute journal entry related to the main objective of the class. These journal entries served as a closure activity for each class and provided assessment information that was used to plan the next class.

The biorhythm project (MUT) was assessed using the following rubric:

1. You clearly but briefly explained the task in the first paragraph so that a reader who has never seen this task before would understand the basic theory of biorhythms and the task you performed. E(10)---VG(9)---G(8)---M(7)------------P(4)------(0) Comments:
2. You explained in depth how you developed the equations and graphs that model your own biorhythms. Your explanation shows your knowledge of trig functions, their graphs, and the vocabulary associated with them. E(20)---VG(19)---G(16)------M(14)-----P(8)---(0) Comments:
3. You stated the predictions you made for the month and explained how they were obtained using the graphs of your biorhythms. E(10)---VG(9)---G(8)---M(7)----P(4)----(0) Comments:
4. You7 explained what criteria were used to design your journal and how you used the journal to record data for a month. E(10)---VG(9)---G(8)---M(7)----P(4)-------(0) Comments:
5. You evaluated how well your predictions compared to the data collected in your journal. You provided a complete and accurate explanation of the outcomes in terms of relevant facts. Your explanation show that you can apply your knowledge of trig functions/graphs to a real life phenomenon. E(20)---VG(19)---G(16)---M(14)----P(8)--------(0) Comments:
6. The graph of your biorhythms (all 3 functions on the same axes) is done neatly, is completely labeled, and is identified as Appendix A. A copy of your completed daily journal is included as Appendix B. Appropriate tools were used to give these parts a professional appearance. E(10)---VG(9)---G(8)---M(7)-----P(4)-------(0) Comments:
7. Your work is "ready to publish" by being free of errors in English or typing, cross outs, or other flaws that give the work less than a professional appearance. E(20)---VG(19)---G(16)---M(14)---------P(8)----------(0) Comments:

OVERALL SCORE OF THE TASK:______________

Comments:

TIME REQUIRED

Planning Time: about 15 hours

Implementation: Launch and acquiring/storing activities: 3 class periods (86 minutes each) Refine/extend activities: 2 class periods (86 minutes each)

Applying the knowledge (MUT): 3 classes (86 minutes) to complete the graphs and predictions, students complete their diaries outside of class for a month, then about 20-30 minutes of each class for a week is used (in groups) for developing the conclusions needed to do the reports. The students write their reports outside of class and spend 30 minutes of one class to peer assess each other's rough "all-but-polished" reports. They finish the reports outside of class.

Assessment: 2 class periods (86 minutes) for the traditional tests plus the 30 minutes to peer assess the biorhythm reports in class 2 or 3 days before the final reports are due.

It takes about 30 minutes to properly grade each student's biorhythm report. Students will benefit from 30 minutes in class devoted to reading several graded reports submitted by classmates.

REFLECTION

As of this writing, this unit has been taught three times and revised twice.

• The four performance indicators addressed in this unit (listed on page 2) are all assessed within the unit. The MUT provides evidence of attainment of indicators 7B, 7G, and 7D. The "sequential step" graphic organizer assesses 7F, and the music and tide problems assess 7G, and 7B. The traditional test and the "compare and contrast" graphic organizer provide evidence on indicators 7B, 7D, and 7F.
• This unit utilizes student centered learning to a high degree. The materials are designed for students to collaboratively learn and experiment with the trig concepts. Students actively construct most of their own knowledge with the teacher acting as the guide on the side to provide assistance and feedback as needed.
• The unit supports both the learning standards for both Math English. The report rubric was designed collaboratively with an English teacher.
• One could further expand the unit to the English standards by developing a rubric for the oral reports.
• The constructivist approach of this unit reflects best practices that are outlined in the NCTM math standards as well as those described in the classroom activities published in the NYS Core Curriculum for Mathematics. The most recent cognitive theories of learning are also reflected in the student-centered approach to the activities in this unit. The writing component requires each student to reflect deeply on what he or she knows in order to express that knowledge clearly in writing.