Two release notes here:
This task isn’t worth much if you don’t start with intuition. You should point to this image and ask your students to intuit the location of a fair horizontal cut. At the moment, I think my best option is to print out that frame and pass it out to students so they can each draw their own lines. What I need, though, is a digital system where students can adjust that line precisely to their liking and then tap submit.
After that, the students see a composite of their classmates’ guesses.
This does two things. One, it ratchets up engagement. We want to know what the answer is and who guessed closest. Two, the mathematical model gets a lot of credibility when its solution falls right in the middle of our field of guesses.
This task isn’t worth much if you don’t end with generalization. The initial task sets the hook, but it resolves quickly into computation. Where this task (and others like it) light up the board is when we say, “Okay, now tell me where to make the cut for any size wedge of cheese. Any angle. Any radius.”
The ideal outcome on a digital device is that the student comes up with an abstract function with respect to theta and r, enters it into the device, and then that abstraction gets concretized right on the original image. The student sees the result of her model on a dynamic cheese wedge. She adjusts the theta slider and sees both the wedge and the cut adjust dynamically according to her function.
That’s the ladder of abstraction right there –Â from intuition to generalization.
There’s an interesting back-and-forth in the comments with one side claiming that the obviousness of the vertical cut makes the horizontal cut kind of contrived and another side saying it doesn’t matter.