Some things I'll add:
-Calculus. Whatever you've learned in methods, because it can feel like half of specialist is just extending on what you do in methods. For example, if you don't understand that dy/dx = dy/dt * dt/dx, you will be confused by all of differential equations and rates of change, and even some of kinematics. Also, most of specialist calculus is concerned with building tools for integration (kind of like how in differentiation you have the product rule, the chain rule, etc.), so if you don't understand basic integration, you'll struggle here, too.
-Ability to interpret real world situations. So much of specialist is based on modelling problems, it's super important to be able to to read a question and figure out what it would look like in real life.
-Graphing. Being able to draw graphs quickly and interpret things about them (in particular, where is the graph equal to 0, where is it equal to 1, where is there an asymptote, etc.) - not only because you can use this to answer questions, but there are even more questions in specialist where you need to be able to draw a graph quickly so that you CAN answer the question
Trigonometry (unit circle, exact values, graphs).
There is trigonometry involved in...
Area of Study 1 (Functions and Graphs)
Area of Study 2 (Algebra)
Area of Study 3 (Calculus)
Area of Study 4 (Vectors)
Area of Study 5 (Mechanics)
Area of Study 6 (Probability and Statistics)
(OK no trig in AOS 6, but I think you get the point 
)
What if I told you that circular functions are require to prove some of the things you do in AoS6, just that they're outside of the study design?
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