What are some examples that could be used when teaching subroutines or methods in the context of OO?
I almost always teach programming by connecting each component of programming with something in real life. For instance, when explaining variables, I ask the students to imagine them as boxes in which you put things, like food items or gifts or something similar. 

For methods, I ask them to think of their friends. For instance, there will be a 'rich' friend, and then there are 'stingy' friends, 'useful' friends and so on. So, bring up a scenario. For instance, lets say the challenge, you are in trouble and you want to ask for help. 

You have 3 friends - rich, stingy and useful. Instead of friends, think of them as methods, and you (as in the individual) is the program or the main method. Now, you need help, which friend aka method are you going to call...you will obviously call the 'useful' friend/method.

So, essentially, whenever you want something to be done, you design/code a method for that, just like how in life, you want something done, you call upon a friend. So, methods are like friends who help out get things done. 

Of course, in an actual classroom, I pick students, make them do role-play (pretend that they are functions) and make it all very active (and hopefully fun) and it seems to work for me. 

and of course, once they get the concept, I jump into code right away, building small methods that do simple print statements, then upping it to display simple variables and then complex stuff including sending and getting values