Initialization stuff

Needs["Graphics`Graphics`"]
Needs["Graphics`Colors`"]

POLAR  GRAPHING

The object of this lesson is to teach students to analyze
basic polar graphs through experimentation with 
computer graphing.

This lesson will be looking at polar equations
in the form        Sin[a theta].   Place the mouse on 
the cell , click once, and push enter to see a display of five graphs of equations in this form.

--

Clear[a,theta,lplot,pplot,t]
Do[

lplot := Graphics[Text[Sin[a theta],
{Cos[((.5 a a-5.5 a +17) (Pi/32)+.72)],
Sin[(.5 a a-5.5 a +17) Pi/32]}]];
pplot := PolarPlot[Sin[a theta],{theta,0,2Pi},
             PlotRange->{{-1,1},{-1,1}},
             PlotStyle->RGBColor[a/5.01,1-a/5.01,0],
             DisplayFunction->Identity];
Show[pplot,lplot,DisplayFunction->$DisplayFunction]
             
             
            ,{a,1,5}]

When you reach this spot, return to the first graph displayed,
group the cells--one click on second cell grouping --
and push    command Y    to see an animated display of all
five graphs.  Notice simiarities and differences in the graphs 
as the value of  a  changes. You will want to see more 
graphs of this type before you sketch the graphs on your worksheet;use the computer to graph equations by typing in the following: 
   
      PolarPlot [Sin[(insert number) theta],{theta,0,nPi}]
      
      IMPORTANT: n must be chosen so that the graph
      displayed runs from -1 to 1 on the horizontal axis.

Now, I hope that you were curious enough to try many different  
numbers -- the world does not live by whole numbers alone!
If you didn't try a variety of numbers, do so now -- I did 
while working late one night on the worksheet. Remember that
with the aid of the computer, you can explore more than ever 
before.

Now that you can quickly sketch equations in the form of 
               Sin[ a theta],
   let's change the form to   Cos[a theta].
       Get started!  You know what to do.