Inspired by this post, an in the vein of painless ways to improve instruction, I thought I’d share how I introduce functions. I throw up a slide of some function, usually represented by one of those circley-arrowy thingies, and tell students we’re going to play a game. I pick a domain value, and they have to tell me which range value I’m thinking of. If they guess right, I’ll give them $10, but if they guess wrong, they pay me $10. No money ever changes hands, of course, but it raises the stakes nicely. One brave student volunteers to play, and in short order, takes all my money. “Just a practice round! Just showing you the rules! Doesn’t count!” Next slide goes up, and just like that, they are hooked.

Next slide is a relation with one domain going to twelve different ranges and I try my best to wheedle some student into playing, but they know something’s up. I go through each domain value, asking “Do you know what range I’m thinking of?” I usually get someone with a relation that has a bunch of one-to-one values and a single domain that goes to two range pairs. The student often justifies playing by saying the odds are in their favor, only to have me break character and pick that same value time and time again — “oooh, this time I was thinking of the 5, not the 2! You almost got me! Let’s keep playing!” After taking all that student’s money (again: not really), everyone is more wary. Each new slide prompts a burst of hurried strategizing, followed by a decision to play with me or stay away. The first slide that has multiple domains going to a single range value always elicits some good discussion, as students initially suspect a trap, but then figure out that they will rob me blind.

After they are okay on maps, we move on to ordered pairs, tables, points on a coordinate grid, continuous graphs (where brief mention is made of the Vertical Line Test), and eventually equations. Students eventually realize that in theory, they should be able to clean house on ANY equation.

I usually save the equation part for a few weeks later, because once students realize that the game is no fun with equations, we start playing backwards. Now I give them range and they have to determine which domain I used, and all of a sudden, we’re talking about which functions are invertible and which are not. We can push it further and put up equations that have values which “break” the game like radical or rational functions, and now we’re talking about restricted domains.

The advantage to this approach is that it lets students engage functions using their already well-honed intuitions about fair vs. unfair games. Since adopting this approach, students have a much better (if at times more casual) language for what makes a function a function (not that “oh, there’s a repeated x-value, so it’s not a function” BS), and are much more able to explain why a single “problematic” value is all it takes to lose function-hood, and why it’s totally okay to have multiple domains going to the same range value.

And it sticks. First year I did the lesson like this, I had a student mis-identify a function literally MONTHS later, and another kid chimed in with: “Nooooooo, he’s gonna take your money!”