In my capstone class for future secondary math teachers, I ask my students to come up with ideas for *engaging* their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Perla Perez. Her topic, from Precalculus: graphing with polar coordinates.

**How does this topic extend what your students should have learned in previous courses?**

Graphing polar coordinates is usually taught in a Pre-Calculus class. Students have learned about the Cartesian Coordinates and extend their knowledge to polar coordinates. Unlike Cartesian Coordinates, which represent how to get from a specific point to the point of origin (or vice versa), the polar coordinate tells us the direction by the angle, and the distance from that point to the origin. Students will need to know how to take the measure of an angle and how to use the Pythagorean Theorem to solve for the distance which is considered the radius. Most students who are enrolled in a Pre-Calculus class have taken geometry where they have learned about the Pythagorean Theorem and what a radius is. This alongside their algebra 1 and geometry classes means they also know how to graph and plot points.

References:

http://ritter.tea.state.tx.us/rules/tac/chapter111/ch111c.htm

**How could you as a teacher create an activity or project that involves your topic?**

Polar coordinates use a different type of graph, rather than just an x and y coordinates plane. The polar coordinate plane includes symmetrical circles surrounding the center and is given a radius creating a graph that looks like a dart board. At this point students should know what a polar coordinate is. The next step is actually graphing it.

As an activity to get students excited for the wonderful world of polar coordinates, I have created a dart board game. Using an appropriate dart board, such as a magnetic one, have the students create groups of three to four student each. The point of the game is to have students create polar coordinates. The board must be properly labeled with the angles. There will be four rounds, depending on the number of members in a group. When a member throws a dart at the board it must land on a point. Wherever it lands the students must figure out the radius and the angle of the dart to the origin. This game enables students to practice finding the radius and the angle of the dart with only their previous knowledge, labels, and each other.

**What interesting things can you say about the people who contributed to the discovery and/or the development of this topic? **

Throughout centuries and in all parts of the world, mathematicians and astronomers have come to shape our understanding of the polar coordinate system. Two Greek astronomers Hipparchus and Archimedes used polar coordinates in much of their work. Though they didn’t commit to the full coordinate plane, Hipparchus first begins by writing a table of chords where he was able to define stellar positions. Archimedes focused on a lot on spirals and developed what now known as the Archimedes spiral, in which the radius depends on the angle. Descartes also used a simpler concept of coordinates, but relating more to the x-axis. In 1671, Sir Isaac Newton was one of the biggest contributors to the elements used in analytic geometry. The idea of polar coordinates, however, comes from a man named Gregorio Fontana (1735-1803), centuries later. Astronomers now use his polar coordinates to measure the distance of the sky and stars.

References:

**History of Mathematics, Vol. II**:

https://books.google.com/books?id=uTytJGnTf1kC&pg=PA324&hl=en#v=onepage&q&f=false

https://en.wikipedia.org/wiki/Polar_coordinate_system