Initially I was going to move from rectangular equation conics to polar equation conics. After staring at the tiny number of days I have remaining, it had to get bumped. It will probably get wrapped into an end of the year project once we finish our Capstone Video. For now, we skip to the polar coordinate system and the easier to digest cardiods, limaçons, and rose curves.

The lesson is preceeded by an introduction to polar coordinates, how to convert between the two systems, and a reminder that this is all just magnitude and direction from vectors. It's easy. A lot of them even remember how to compensate for coordinates that are in Quadrant II, III, and IV.

### Step 1: Wait, what?

First I pass out a pair of Polar Grids. Without any introduction, I put something like r = 4+4cos theta on the board. I use it as a demonstration for creating a list of polar points and how to plot them on a grid. We find values of r at 15 degree intervals and plot the result. After giving them a few minutes I will show them the list of points and the expected figure.

This prompts a lot of "what is this?" We set about assigning words to describe what we see: bean, heart, butt, lima bean, deformed ellipse, etc. Eventually the word "cardiod" enters the discussion. I give little hints about the possibilities of polar equations, teasing them with rose curves.

### Step 2: Playtime

The next day I give them a brief introduction. We agree that the graph from yesterday was odd. Well now they get to explore. They pair up and are given this set of instructions:

There are also some tips taped to the table to help them find the "r" and "theta" on the desmos keyboard.

And off they go. I introduced Desmos not long ago when we were doing conics, so the concept of creating sliders wasn't that big of a deal. There are a few hiccups here and there with equation entry but after a 10 minute adjustment phase, everything hums along. I have them jot answers to the discussion questions in their notebook. I didn't specify much here, just telling them to be prepared to have answers for the discussion. When groups are done I do a little pop quizzing to see that they understood what they did, specifically seeing if they figured out the pattern to rose curves and petal count.

### Step 3: Debrief

As a group we go through the answers to their questions. We hash out the theory of maximum and minimum radius. We all step back and realize that dissecting these things is pretty simple. After we're satisfied with the answers, they get a problem set of polar coordinate conversions and a series of equations where they identify the properties without looking at the graphs, knowing what we know now about the parameters a and b. There area few rose curves where they examine size and petal count.

### Step 4: Presentation

I had out a list of 12 polar equations, six cardiods and six rose curves. Each is repsonsible for two of them. They create a reference drawing in their notebook of the graphs. Intercepts, maximum and minimum radius need to be marked. These are the graphs they will reproduce in sidewalk chalk a couple days later. It falls on an early release day this year, so there was no time to figure it out as they went along. I was very impressed with the way they figured out how to access desmos in their spare time, on their phones, in the library, etc. Reminding them to start the problem with "r =" was the only technical hurdle we had to jump.

### Step 5: Working Backwards

The day following sidewalk chalk, we discuss the enormity of their art installation, a few of them commenting on how neat it was to see the whole thing at the end of the day. We pick up with the problem set I mentioned earlier. We're in pure math mode here, just spending time understanding how a and b affect the shape of a polar equation. Our last activity in polar equations is to take some figures and determine the equation responsible. I reference the desmos activity as we discuss how to determine the difference between a sin and cos graph, and the meaning of the + and - in the middle. They get 8 figures to analyze. I could've had the figures pre-loaded on the iPad, but that idea failed the paper test.

After covering the answers, we have a Pre-Calculus graduation ceremony. Limits and Derivatives here we come!

### Conclusion

This lesson is primarily analog: notebooks, problem sets, tape, glue, staples, paper grids and handmade tables. The digital aspect enhanced the experience greatly. Rather than playing with sub-par pixelated renditions, we got big, color, zoomable images. We didn't need to know how a trace function worked. We weren't retyping ten different versions of an equation to determine a trend. We could see the wonderland that was 5cos(2.5*theta). We didn't need to know how to square a window. We weren't confusing Algebra kids later in the day who innocently wanted to use "Y=" and giving me the "OMG, what happened mister!?" as they blanky contemplate what the heck "r1 =" is supposed to mean.

Then consider the 800 foot long public art installation we made.