My students and I have been brainstorming topics appropriate for the new 2014 IB internal assessments. Our department created a timeline that would allow kids to submit topic choices by early November, write an outline and then some drafts in the winter / spring, followed by final submission in May. One of my colleagues recently went to an IB Category 3 training, which focused on the new internal assessments. After we sat down as a grade-level team to discuss the issues that came up at this conference, I now feel fairly confident that I understand what is expected (as much as is possible at this early stage of implementation of the new format), so here is what I gather. I hope it is helpful to you and/or your students.
The new format entails two parts: research and your own application. The math topic selected should be one that goes beyond the IB curriculum (quadratics is too easy, for example). The student is supposed to research and read literature on the math content beyond the course related to their topic, and then break down the math literature in their own words. Then, they need to basically show "engagement" by applying the learned knowledge to a situation of their own creation.
For example, one student of mine wishes to study a certain card game. I said that is OK, and that the research would entail this student explaining step-by-step how to calculate all of the probabilities in the standard rules of the game. And then, as part of their project, they could create a second game similar to the one that they have studied, but with varied rules, and then they would re-do all calculations to show that they had made the understanding uniquely their own. This kid can also run actual game simulations to compare with their theoretical results, in order to further reflect upon their understanding.
If your kids are lost (as mine were) about exploration topics to pursue, I find that the Numbers Guy from WSJ is a good place to start to think about the connection between math and real-world issues. If they are familiar with GeoGebra, there are also possibilities of studying various GeoGebra animation-based problems that are comfortably within the realm of the SL topics. Or, any extension from 2-D to 3-D is always interesting. Besides that, Vi Hart has some unusual connections between math and various art forms, that might also jog some ideas. The kids can basically study anything, or even model data, so long as the data is sufficiently complicated. For me, a good rule of thumb is that if I can already imagine without any research what the data / result will likely look like, then the topic chosen is already too simple and not sufficiently "juicy" to get all the upper marks. In that case, the kid would need to add a twist to it to make it more interesting / complex, or abandon that idea and look for a new one.
Happy hunting for good topics! What an exciting time! Let me know if you find other sources of inspiration for you and your students. I find in my own class that if a kid picks a topic as a starting point, and they just take the time to start talking to me about it, usually we can brainstorm together until we find something that is sufficiently interesting/complex. The resulting direction may have nothing to do with what we started with, but just the process of talking a kid through a topic can jog my mind through related ideas that I have come across/read before.