I have been in such a program when I was 12-14 (run by the William Stern foudnation in Hamburg, Germany) and the curriculum consisted mostly of very diverse 'math' problems prepared in a way to make them accessible to us in a func way without introducing too much up-front terminology or notation. Examples I remember of the spot:

• turing machines (dresses as short-sighted busy beavers)

• generalized Nim really with lots of matches

• tilings of the plane

• conveys game of life (easy on paper)

More I just looked up in an older folder:

• distance metrics on a graph

• multi-way balances

• continuous fractions (cool for approximations; I still use this)

• logical derivations about beliefs of people whose dream are indistinuishable from reality

• generalized magical squares

• Fibinacci sequences and http://en.wikipedia.org/wiki/Missing_square_puzzle

• Drawing fractals (the iterated function ones; with printouts of some)

In general only an exposition was given and no task to solve. Or some introductory initial questions, The patterns to be detected were the primary reward.

We were not introduced to really practical applications but I'm unsure whether that had been helpful or rather whether it had been interesting. My interest at that time stemmed from the material being systematic patterns that I could approach abstractly and symbolically and 'solve'. I'm not clear whether the Sequences would have been interesting in that way. Their patterns are clear only in hindsight.

What should work is Bayes rule - at least in the form that can be visualized (tiling of the 1/1 grid) or symbolcally derived easily.

Also guessing and calibration games should work. You can also take standard games and add some layer of complexity on them (but please not arbitrary but helpful ones; a minimum example is: Play Uno but cards don't have to match color+number but some number theoretic identity e.g. +(2,5) modulo (4,10)).