No, no, we’re not talking about your pet goat’s behavioral problems.
We’re talking about a certain type of math problem—a type of problem to which I used to devote a couple of classes per year when I was teaching Honors Geometry. It’s called a goat problem because… well, take a look at a couple of examples:
- “After making the hat, Ms. Hamavid seized control of the goats at the summer camp where she worked. One day she tethered her favorite goat, Goatzilla, to a post at the 60° corner of a barn that was in the shape of a 30°–60°–90° triangle with a 100-meter hypotenuse. If Goatzilla’s leash was an amazing 75 meters long, find the exact area of land over which she could graze.” (From April 13, 2017. The hat reference is to the problem that preceded it on the same test.)
- “Gaga Ball is often played on a court that has the shape of a regular octagon. Suppose that a particular octagonal Gaga Ball pit has side length 12 feet. While she plays, Katherine has tethered her guinea pig to a corner of this pit (on the outside, so she doesn’t get crushed) with a leash 18 feet long. What is the area of ground that Katherine’s guinea pig can roam?” (From May 4, 2018)
I know, I know, guinea pigs aren’t goats, but they’re mathematically equivalent, aren’t they? Anyway, if you make some sketches, you’ll see that the solution is a function of both the length of the leash and the angles of the pen/barn. The longer the leash, the more complicated it gets.
Anyhow, these are what we call “goat problems,” and I’ve always been fond of them. They can be difficult, but they are usually solvable by mathematically strong ninth-graders. So I was delighted to see an article by Patrick Honner in Quanta Magazine “about the infamous grazing goat,” as he puts it. The drawing above comes from that article, in which Honner carefully explores the various parameters that enable a simple-looking problem to evolve into a rich panoply of options. The most interesting problems—which for some reason I never wrote or assigned—are the inverse ones, where the area is given and one of the original parameter is left to be found by the student. The article presents some great examples of that, including one that was not solved until recently. That one goes beyond high-school geometry, but most the article doesn’t.
Go read it!