Mathematics is about the reasoning that can be made precise. The different branches of mathematics reason about different objects: numbers, shapes (rigid or stretchy), games, arrangements and relations, and other things for which the words do not exist in the everyday language. There are some branches of mathematics exploring the reasoning itself. Pretty much any set of rules of reasoning one can normally think of is equally powerful (we say equiconsistent). The large cardinals is a name of for various rules that are stronger.
To explore reasoning about different things, give a puzzle to the class. Many people like them, and do not think of these as maths. I would consider "hats" puzzles, or some topological puzzle (e.g. is this picture an unknot? are there two antipodal points on the surface of the Earth with the same temperature? etc). There are good puzzles books to get ideas from, such as those by Martin Gardner and Raymond Smullyan.