GEOMETRY FOR SANTA

or in other words...

POLAR GEOMETRY

Instead of graphing using Cartesian coordinates (x, y), another way to approach graphing is to change the coordinate system to polar coordinates and graph using (r,). What do you think r and  should represent?

Using your knowledge of trigonometry, what should be the relationship between (x, y) and (r, )?

Consider the polar equation r = 3. What do you expect it to graph?______

Change your calculator  to POL. Look at your  and decide what you need to change, and then  to see if you are correct. Were you?______

What  values did you need for a complete graph?

min = ______max = ______step = ______

Write an equation for this graph in rectangular (Cartesian) form.

Write parametric equations for this graph.

Now change your toDEGREE, your min to 0, max to 360, andstep to 90,

And then change  to 4: ZDecimal

and sketch the resulting graph to the right.

What type of figure is it?______

Why do you think we got this graph?

We can graph an equilateral triangle by changing one number in our . What do we need to change?

How can we graph a hexagon? (Check your hypothesis by graphing.)

How can we graph a pentagon? (Check your hypothesis by graphing.)

What is the relationship between the step and the figure?

Calculator limitations: What is the maximum number of sides a figure can have before it looks like a circle?

Geometry Limitations: Change your step to 80. What happens to your figure and why?

Make a conjecture about properties of the step necessary to graph a closed figure.

Now, with your step at 80, change your max to 720. What happens and why?

Make another, more complete, conjecture about the relationship between the step and the max in order to graph a closed figure.

Geometry extention #1: Can we rotate these geometric figures?

Start with a different value formin and see what happens. Do you need to change yourmax?

How can you rotate this figure in the opposite direction?

Make a conjecture about how to rotate a figure through any number of degrees either clockwise or counterclockwise.

*** Now use your parametric equations. Does changing theTmin have the same effect as changing the min?

Write another way to rotate the parametric equations.

In general, people recognize regular polygons more readily if they appear to be "sitting on their base". Rotate regular triangles, quadrilaterals, pentagons, hexagons and more, if needed to complete the table below and thereby create a formula by which we can rotate any ngon so that its lower base is horizontal.

Number of sides / 3 / 4 / 5 / 6
Angle and direction of rotation

Conclusion:

Geometry extentions #2: A beautiful figure is the pentagram. We can graph a pentagram by changing our step (Tstep)to 144 and our max (Tmax) to 720. Graph a pentagram.

Why do these values of step (Tstep) and max (Tmax) create this graph?

Can you create a septagram (a 7 sided star)? What changes did you make?

Can you make another, different septagram? How?

Can you graph a Star of David (a 6 sided star) using these changes? Try it!!

Why doesn't this method work on an even sided star?

Find another way to graph the Star of David!!