For several years I have students explore polar equations with a little puzzle solving. Eventually, students stumble upon some observations about how to build the graphs. As an example:

For something like this, students will point out that there is something awfully coincidental about the presence of 3 and 5 in the equation and the values of the various intercepts on the graph. That leads to a discussion where I have ignorantly offered them the shorthand that "the two parameters add together or subtract" to get you various places. It doesn't tell the entire truth. The presence of the + in the equation confuses a few. Why would you subtract these? Same thing happens if it was r = 3 - 5 cos theta. It still reaches out to 8, and 3 + 5 still equals 8 but that's weird mister, there's no + sign! And in the end you're memorizing a lot of rules about one integer being larger than the other, etc.

What I should have done is given a proper explanation, something I didn't figure out until a week too late last year. Now I was prepared.

Started with the same activity, students made the same observation about the weird additive property thing going on. But rather than let them hold onto a shortcut, we looked at why.

After the initial analysis, I gave them a new set of graphs with the same goal: determine the equation. After they succeeded using whatever patterns they had internalized, I had them do a breakdown on the corners (0, 90, 180, and 270). If a loop is present, there will be a negative value at one of the corners. The largest dimension happens even in the presence of a - because the value of the sin or cos is -1 there. This had an added benefit of helping students understand loops better. What does it mean to go -3 in the direction of 90? It helps sell the idea that polar coordinates have a different concept of positive and negative. It also gives them a more universal idea about how to graph a polar equation. The corners will tell me what I need to know.

I feel much better about this unit now.