I was a bit distracted during the sermon in church today and started thinking about polar coordinates and how I was going to teach them this year. I mostly lectured on this topic last year (as I did for most topics), but I had a flash of insight to play battleship with my students to help build their intuition. We played battleships to introduce Cartesian coordinates when I taught 6th grade, and it was far and away my most successful lesson that year.
The Setup & Plan
You are the Commander of a small battleship (I suppose they’re called destroyers?) which has only a single gun that rotates on a pivot. You receive information from your Radarman (do they have a special name?) who operates the radar and have to let the Gunner know where to shoot. The Radarman sees his screen and reports the location of enemy ships in Cartesian coordinates (yes, I know that radar actually works essentially using polar coordinates, but hopefully my students will breeze by this detail and we can discuss that later). However, the Gunner just needs to know the angle (looking from above) to rotate the gun, and how far to shoot. Yay, we’ve found a situation, a game at that, where we need to move from (x,y) to (r,theta).
In addition to this, I did 2-3 situations with them that were really simple, and then we launched into a game of half the class vs the other half.
Game Setup for Students
I had the students sketch 10 x 10 boards (using Cartesian coordinates) onto a 3′ x 2′ whiteboard, so the x and y values went from -5 to 5. (See below for one class’s placement of their ships).
The students then placed the following 4 ships onto the board. Unlike traditional battleship, the center of the ship had to land on the Cartesian coordinates. So I sketched the following diagram for them.
The students then sketched their ships onto the whiteboards, after which I promptly took a picture with my iPad so I could ensure that there was no cheating.
Results and Student Interaction
The students really took to the game. One team even built a “fort” in the classroom so the other team couldn’t see their board!
So students had their own boards in Cartesian coordinates, but would “fire” on each other only using polar coordinates. They were then responsible for saying whether their opponents had “hit” or “missed” their ship. If they were wrong, then their opponents got another shot, so accuracy was critical.
The teams took turns, and we actually had about 10 volleys on each side with no hits! I was wondering about the probability of that happening, with both teams  when I realized that could be our warm-up tomorrow. So I had them lay the boards to the side and we’ll continue the game tomorrow.
The thing I liked best about this game was how much it had students converting from Cartesian to polar and back, and them trying to do it quickly, yet accurately! And to think they’ve done all this before I gave them the equations (actually, I gave them some of the equations, but they all forgot them in the face of a game, and resorted to problem-solving–yay!).
 Let’s see, 14 “hit-able ship spots” out of 100 possible points. The probability of a miss is 86/100, so to do that 10 times (about) would be (86/100)^10 or about 22.1%. However, neither team hit, so squaring that would be about 4.9%. Hmm, much less likely and interesting!
Edit: Ooh, as I was showing my students this probability, I realized this is wrong because the smart player wouldn’t shoot in the same place twice! Also, from -5 to 5 includes 11 points, not 10 (fence post problem/gotta include 0!). So the right math would start like this: