IB recently changed Internal Assessments from having investigations of Type I and Type II to having a portfolio.
Even though IB would no longer require students to do this type of investigation, I kept this mathematical investigation around anyway, as I think the process students go through to arrive at a formula is very valuable. I basically treat it as a project.
The investigation has them look into the problem of determining the maximum number of "slices" a pizza may be cut into with a random assortment of n cuts. For example, with 3 random cuts, a pizza could be sliced into a maximum of 7 parts. (The cuts should not all emanate from the center of the pizza.)
Please download the pdf preview file first, so you can see exactly what's included; the product file is a word document, which you may personalize for your students.