SUBJECT: Trigonometry

TOPIC: Graphing in the polar coordinate system

INTERACTIVITY: Students will observe how to plot points in polar coordinates. Students will reveal graphs of polar equations by pulling tabs on the board and uncovering "hidden" information. Student will play tic-tac-toe with each other using their new knowledge of how to plot points in polar coordinates.

DIRECTIONS FOR USE:

3 ways to win polar tic-tac-toe

It looks like the polar bears are in love, but I think they are trying to show you something, pull the heart tab, and lift up each polar bear. (This reveals a graph of a limaçon.)

For more directions, see my worksheet.

TIME: approximately 15 minutes

SPECIAL CONSTRUCTION: The heart shaped graph (limaçon) is less complicated to construct than the flower shaped graph.

For the limaçon, I cut out a heart from red construction paper. Then I pasted a graph of a limaçon (graphed on geometer’s sketchpad) on another sheet of paper so that the graph could be seen through the cut out heart. Then I took a sheet of pink paper and put it between the cut out heart and the graph. I attached a tab to the pink paper so that students could pull on the tab and the pink paper would lift to reveal the graph of the limaçon.

For the flower shaped graph, I graphed it on geometer’s sketchpad. I cut out the same kind of shape from purple paper and placed it on green construction paper. The whole idea is that before a student pulls the tab, all they can see is the flower made from construction paper, but once they pull the tab, they can see the graph paper flower. I cut slits into the construction paper an inch wide. I then cut the graph of the flower into slits as well. Then I cut the graph paper horizontally at the top and bottom of the slits, being careful not to cut the strips of paper off completely. I reinforced the graph paper underneath with index cards. I poked the graph paper through the slits in the green construction paper. The tricky part about this construction is that you need to make sure the graph paper doesn’t get lost behind the construction paper or else it will never come out again. Therefore, I recommend putting clear tape on the edges of the graph that will be sliding through the slits. The clear tape will be the only thing "visible" when the graph is retracted into the construction paper. Attach an index card to the edge of the graph paper, so that you can pull the graph through the slits. I also attached a back piece of construction paper to the whole thing to keep it together.

I also have a lot of flaps with information on them. Simply folded sheets of construction paper with pictures on the outside have information on the inside of them. For example, I used Archimedes’s picture on the outside of a flap and placed his famous spiral on the inside of the flap.

By highlighting the axis and drawing four circles(with their center being the origin), I constructed a tic-tac-toe board with geometer’s sketchpad.

I made the "North Pole" a review of degrees to radians. I cut out two circles of the same size. By using a brass fastener in the center of the circles, the top circle can rotate around. There is an arrow pasted on the top circle and another arrow, the axis, pasted on the lower circle. The top circle revolves revealing the degrees and radians (written on the bottom circle) that correspond with the arrow’s angle from the axis arrow.

There is another set of arrows, one polar axis and one rotating "candy cane" arrow that help students understand the concept of graphing in polar coordinates. These arrows are also put together with a brass fastener. I also poked a cotton ball into the brass fastener to represent the origin (and to appear like a snowball for the Polar theme). The candy cane appearance is not merely decorative, but acts as the units for r.

CREDIT: This will be an extra credit assignment. Students will learn about radians in their everyday trigonometry work, but polar coordinates are not usually covered at that time. However, I think it is an interesting topic for students to learn about.