I get a bit tired of the trope that students today are subjected to boring math with boring procedures and boring problems. (Although I must say, I find the real-world problems that are supposedly interesting to be quite tedious and boring). Essays abound with links to something called Lockhart’s Lament which was written by a mathematician named Lockhart and is a lamentable whine about how he found math in K-12 boring.
In his essay, he laments about why students can’t be made to be curious over the relationship between perimeter and area, and the question of what perimeters of various polygons yield the greatest area.
The problem has become a poster child of what math classes are supposed to be about, and of course Dan Meyer (of dy/Dan fame) is no exception to this. Yes, it is a good question, but one that can be attacked with the tools of mathematics rather than by guess and check or other more sophisticated iterative and inductive procedures augmented with TI calculators and Desmos-generated tools.
The principle involved is the subject of something called the Isoperimetric Theorem which can be proven with Euclidean-based geometric theorems and leads to the relationship between perimeter and area of regular n-gons. If you really wanted to teach it with “understanding” and mathematical principles, teach it in a proof-based high school geometry class. “Geometric Inequalities” by Nicholas D. Kazarinoff (from whom I had the pleasure of taking a differential equations course at U of Michigan). You can find the book on the internet. It was published by Random House as part of the New Mathematical Library and is one of the better products of the 60’s new math era.
If you insist on guess and check and other inefficient ways, then you can stick with dy/Dan, but it doesn’t take a mathematical genius to guess how I feel about this particular “lament”.