TRIGONOMETRY 11TH - 12TH

MARY BORNE

Overview
The students work in groups to graph groups of polar equations and answer questions concerning these classic curves. After discussing the answers, the students are then required to develop equations for given graphs of classic curves.

Expected outcome
The students should be able to recognize the equations of different classic curves by the position and value of the numbers, cosine and sine. They should also be able to determine how the graph of a polar equation will look by the value of these terms.

Background information
This activity should be used in a trigonometry class that has completed the unit on the trigonometric functions sine and cosine. The students and teacher should be able to use graphing calculators in polar mode. The students should also be able to graph a point given the polar coordinates.

Materials
Each student or student group needs a graphing calculator, worksheets, and a pencil.
The students will also need to refer to their textbook, Advanced Mathematical Concepts .

Standards
The NTCM Standards encourages a shift towards more active student involvement in the learning process and the use of technology. This lesson provides a student centered experience that extends logical reasoning skills by investigating the connection between the equations of a polar equation and its graph. Mathematics as Connections encourages students to analyze and identify important mathematical relationships using a graphing calculator. The lesson also incorporates Mathematics as Communications by having students clarify their thinking by formulating ideas in writing.

Engagement
Graphing the first polar equation on the overhead graphing calculator will show the students what they need to do and how to proceed. I will then discuss with the class what to look for and how to answer the questions. The shapes of the classic curves encourages students to continue the activity to see the different shapes they can form on their calculators.

Exploration
The student will be placed into groups and given the graphing polar equations worksheet. I will give the students instructions and graph the first polar equation to show the students how to proceed. The student groups will work together to discover the classic curves obtained when the polar equations are graphed. We will discuss the answers to the worksheet together on the following day. The students will be required to complete a second worksheet to identify the equations of graphs of classic curves.

Explanation
The first worksheet helps the students to discover how the position and value of the numbers and trig functions affect the graphs of the classic curves. The following class is used to determine the equations of the classic curves given their graphs. The second worksheet is used to help the students recognize what they have learned. 

Evaluation
The second worksheet will be collected and graded. The students will continue to work in the assigned groups and each student will hand in their own worksheet for a grade. The skills and knowledge gained in this lesson will be tested on the polar coordinates chapter test.

Extensions
Graphing the polar equations will help the students to make the connection when they are learning to change polar coordinates to rectangular coordinates and back The students can visually see the points on the polar axis and compare the point on the rectangular axis.

References
Advanced Mathematical Concepts Merrill Publishing
Trigonometry Activities: For the TI82 & TI85 Graphing Calculators
by C. R. Dennis & L. M. Neal , International Thomas Publishing
 

Section ( 9 - 2 ) Name________________________

Graphing polar equations Partner_______________________
 

Sketch a graph of each equation and complete the following:

1) r = 8 cos x                                     2) r = 6 sin x
 

If r = 2a cos x , then a determines___________________________________________________

2a determines______________________________________________________

sin and cos determine _______________________________________________

3) r = 1 + 2 cos x                             4) r = 2 + 4 sin x                         5) r = 2 + 2 cos x
 

If r = a + b cos x , then sin and cos determine__________________________________________

a determines_____________________________________________________

a + b determines__________________________________________________

a = b determines__________________________________________________

6) r  = 9 sin 2 x
 

In r  = a cos 2 x , a determines ___________________________________________________

the number of petals is determined by_______________________________

7) r = 2 cos 5x                             8) r = 3 sin 3x

9) r = 3 cos 4 x                            10) r = 5 sin 6x
 
 

If r = a cos nx and n is odd , then the number of petals is determined by _____________________

a determines_____________________________________________

If r = a cos nx and n is even, then the number of petals is determined by _____________________

 
When n is odd there are ____ petals and when n is even there are _____ petals

Using the classic curves on page 241, name each of the graphs you have sketched:

1)___________________________ 2)___________________________
 

3)___________________________ 4)___________________________
 

5)___________________________ 6)___________________________

 
7)___________________________ 8)___________________________
 

9)___________________________ 10)___________________________