The strategy I employ in undergraduate dynamics typically follows this sequence: I provide a qualitative introduction to a particular topic, attempting to highlight a curiosity or apparent contradiction if one exists as a means of sparking interest in the subject. Then, the equations are derived, motivating the approximations and assumptions. Following this, I extract physical insight from the derivation, making use of real-world examples and appeals to their intuition and experience. Finally, I return to the equations, to show how they facilitate the application of this new or enhanced understanding to real world problems. The expectation is that by this time, most if not all students are more comfortable with the equations, and their uses (and potential misuses).

Two specific examples I will employ are gradient wind balance and the thermal wind equation. The former can start with a statement, such as: "Flow around cyclones is subgeostrophic, while it is supergeostrophic around anticyclones." This challenges their experience, as they already expect cyclones to possess faster wind speeds (because their pressure gradients are typically larger). The usual derivation is straightforward but leads to an unattractive quadratic equation possessing eight possibilities, depending on the signs of the geopotential gradient and the radius of curvature. Most texts stop there, but I go on to demonstrate why it makes sense for the flow to decelerate as it curves cyclonically, utilizing their physical intuition. With the underlying concept now understood, we can return to the equations to compute just how large departures from geostrophy can be. The development of the thermal wind concept proceeds in a similar fashion.