For full context, see these two documents. Its at page 1, bottom of the page exercise 2, number 1: https://www.docdroid.net/gMfeMMP/img-20170823-0002-new.pdf
先生に水泳をやってみたらって勧められたんです。 => the problem is みたら. I expect って to be a quotation particle/marker. I would translate the whole sentence as follows: “I was advised/encouraged by my teacher wether/if I had tried swimming.”
This interpretation is highly speculative in my eyes, but I can’t come up with something else right now. I think “みたら“ , which expresses this „trying X“ here, is in context of an unreal expression. „…たら…sentence with た-form predicative“. However, I feel very unsure about this. First, I have no idea what the omitted second part of this construction should look like. Second, たら already perfectly expresses „if“, which also works perfectly well in my translation. Therefore, it feels wrong to make things more complicated than they are. That said, the problem is that, at least in my eyes, this is not the archetypical conditional „if“. It is more like “wether” which I also displayed in my translation. I don’t know how this kind of clause would be called, or if it is it’s own class of clause at all. However, I’ve never learnt about this kind of use being applied to たら in japanese. If it is the same with たら in japanese as with „if“ in english as I lined out before, then please tell me ^^ If I’m utterly at fault though with the solution above, please point me in the right direction xD