IB Maths

If you feel brave, an Extended Essay in maths is bound to be interesting. Check out the 4 Excellent Extended Essays below for an idea of what's involved
Growing Weed
Poker Playing
The 3n+1 conjecture
Billiard Problem

You can also read the extended essay guide here. Start reading from page 6 because the first bit is ToK blab. Then skim until page 15, which is the start of the Assessment Criteria. Then skip to page 34 for Biology, 43 for Chemistry and page 96 for Maths (yay!), which also has sample questions.

Although it is aimed at University level students, these guide to writing mathematical essays by Prof. Tom Korner contains many useful tips (if you look hard enough) on writing about mathematics:
https://www.dpmms.cam.ac.uk/~tw1k/Excess.pdf
https://www.dpmms.cam.ac.uk/~twk1/Essay.pdf

Changes to the IB HL syllabus from Summer 2014

Exam Skills

Here is a powerpoint on exam skills.
Here are some sample questions.

The Exploration

Some marked sample explorations.
The IBO samples (SL and HL).

Some past NLCS year 13 sample explorations in physics (HL), medicine (HL) and medicine (SL).

Sample explorations and markschemes from the IB to get an idea of Marking Criteria.

Over 100 suggested themes for an exploration.

Some archived articles from Significance magazine. These vary in themes from physics and politics to zombies.

You might be able to get an idea for a pure maths exploration by solving a problem. Some good problems to struggle with can be found here, as well as in past maths challenges or olympiads. You would have to go beyond the problems, but they could be interesting starting points.

If you want to do some original statistical analysis, xkcd has just released masses of data from its survey.

Ways to bribe the maths department...