As Goalkeeper0 and Mr. Fullerton suggested, I decided to give the soap-and-water approach to understanding flux a shot. If you'd like to try this experiment but can't find one of those ancient metal coat hangers, here's a different approach:
>>>Credit to Goalkeeper0<<<
I bent a coat hanger into a solenoid with 5(ish) loops and filled the dish basin in our sink with soapy water. I'm not sure if it's because here wasn't enough soap in the water or a different factor, but the darn soap layer kept popping before I could pull the hanger completely out of the basin. Here's low quality evidence of my findings;
...Again, you really can't see much in the picture. I recommend doing the experiment yourself. To give you a better idea of what the soap spiral looked like, it resembled the shape of fusilli pasta. EX:
Yum. Love the stuff
Again, this shape is meant to demonstrate how the number of N turns in a solenoid effect the flux. The more loops in the coat hanger, the more bubble-surface there is in our solenoid. This helps reinforce the equation Mag. Flux = BANcos, with B= Mag. field, A= area and N= number of turns in the solenoid.