I've been using a new approach to introduce functions to my Algebra 2 students (who presumably aren't completely new to functions). As much as I can, I'm trying to let

*need*dictate the math. At this point our definition is somewhat incomplete, but working well so far. Today we see a need for more explicit notation,*because it's not always clear to us what exactly the function rule is:*

I was amazed how quickly they got comfortable with function notation, given this set-up, but I am willing to accept that it may be partly due to previous teachers and previous foundations.

From here we jumped right into the idea of function

**compositions**(but not composition notation, because we don't

*need*it yet):

It was entirely logical to follow these with a discussion of inverses that are a bit more complex:

Here are two samples of their student work at this point. There still seems to be some ambiguity around the word

*opposite*(a great entry point for an upcoming lesson). Beyond that, however, their understanding appears to be rock solid. Everybody did a nice job of explaining the function. Only one got mixed up on defining the inverse. A teacher's dream.

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