Try this - the best I can do off the top of my head and without the aid of a whiteboard:
Consider two circular portions of the earth's surface, one centred at the pole and the other centred at the equator. The circle at the pole, when viewed from above in free space will be rotating (at about 15 degrees per hour) whilst the circle at the equator will have no rotation. A similar circle anywhere on the earth's surface away from the equator will exhibit a degree of rotation (proportional to the cosine of the latitude).
Now, apply your normal explanation for coriolis, starting from the centre of an imaginary circle (which you have established exhibits a degree of rotation) and you will find that it works equally well irrespective of the geographic direction of the pressure gradient.
A mathematical solution, whilst undoubtedly more accurate, is neither necessary nor desirable for the average student.