The students take a class, the teacher teaches the class. If you say that someone "takes a class," that means they are in the role of a student.
This is true for math, science, art, and sports (in the U.S., "sport" classes are often called gym class, or, more formally, physical education, which is often shortened to phys ed).
Therefore, I would say:
John teaches phys ed.
Also, if you don't like the verb teaches (presumably because it implies more formal instruction), you could use a generic verb like runs or manages or leads – but don't use takes:
John runs the gym classes.
Dana manages the SL&LP.
That said, some phys ed teachers might take exception to the notion that they aren't really "teaching." As one website says, the core goal of physical education teachers has always been to "help students of all grade levels improve their health and understand the importance of physical education and health."