I have to say that as a classroom orchestrator, my weakness is in setting up DRAMA in my lessons. I do, however, put on a great big show every year if there is a group of kids to whom I have to introduce functions for the first time. I do a big show of function machines (not a unique idea, obviously), by gradually feeding input numbers written on post-its into a "Function Machine" (this year, it was a black plastic bag with a "Function Machine" label on it) and flipping the post-it over inside so that the kids can be delighted by the output value that comes out. You might think the kids are too old for this, but NO! This year, one of my Grade 9 boys had to stifle a giggle when the first number transformed inside the function machine. His eyes got so big. Together, as a class, we fill out a function diagram on the board ONE post-it at a time (and the kids copy it down in their notes), so that they can see how the function diagram visually represents the mapping of the elements. At some point they start to feel very antsy to make predictions for the next outputs once they start to see a pattern between input-output pairs, so that sense of anticipation really teaches them about the predictability nature of functions. At some point, when they absolutely cannot stand how smart they are anymore, I ask, "So, if we put in some number x as the input, what will be returned?" And when they give me the answer, I both write x ----> x + 2 (for example) as a pair inside the function diagram, and underneath I write in a parallel formal notation:
f(x) = x + 2
And at that moment, with a tiny bit of clarification, it becomes crystal clear to them what each part of the function notation represents. (Obviously, it still takes practice to gain confidence/familiarity with the notation, but I think putting it directly underneath the x ----> x + 2 notation definitely helps to clarify why there is an X on the left side of the equal sign.)
Then, we repeat several other dramatic sequences (with increasingly complex patterns), each time referring firmly back to the definition on the board of a function HELPING US MAKE PREDICTIONS; each X value has exactly 1 Y value. I try to vary up the representations for the later examples, using tables or ordered pairs, so they can realize that all representations are actually equivalent. (Towards the end, to help them focus their understanding, obviously we look at cases of both shared outputs AND two outputs for same input, and we use our function definition of PREDICTABILITY to try to evaluate which case is a function and which is not.)
There are other things I do on Day 1 of functions that I think also help to clarify the definition of a function (such as a sorting activity involving cards that have maybe-functions printed on them), but the DRAMA! is what is so fun about this particular lesson introduction. I absolutely love it every year, even though it's so simple in nature and in foresight always seems potentially very cheesy. --And, trust me, I know that if my 9th-graders are glued to my every move at the board, it's a lesson to keep for the long haul. :)
Do you have your own favorite dramatic introductions of new concepts??