Well, my understanding is that you can either preserve distance or shape but not both... And in general, the poles are the regions suffering most, as it was the ones have tended to be the least interested in...

There are a large number of projection types beyond the classical Mercator and each has its own advantage and drawback:

http://en.wikipedia.org/wiki/Map_projection.


Personally, I think that you could stick a compass on the first picture you posted and simply use that as your handout map. It does look pretty awesome!