I recently had students do a project where they apply the distance formula to finding the shortest path for a traveling salesperson to travel between 8 cities. The basic idea is, the students use the longitude and latitude coordinates as substitutes for the x and y coordinates of the city, then they can use the distance formula to find a pseudo-distance between the cities. Of course, on any kind of largish scale, this makes no sense, but on a small enough geographic scale the error in the distances is small, and I made sure the kids were aware of this deliberate error. This project was intended to be a chance for the students to get lots of practice using the distance formula.
If all of our students work was so neatly arranged and so carefully done, I think very soon we’d soon have much different jobs. Instead of ‘instructing our students’ we would be learning from them as equal partners. This what I strive for in my teaching.