Kent just wrote a great post about his 8th Grade functions unit. He and I share a lot of the same activities and priorities, but I have an observation and a few related ideas to share related to “What’s My Rule?,” a game both he and I play with kids.*
* I learned about it from the CME Algebra 1 text.
The function concept is famously difficult to pin down. The trickiness of the definition follows its history. There’s even a fun article to read that traces the concept’s historical development (thanks Christopher!). Here’s a preview:
Tricky stuff, right?
One conceptual issue I’ve seen again and again in my teaching has to do with the relationship between rules, machines, formulas and functions. Not all rules that we commonly talk about are functions; not all functions are rules; not all formulas have rules; not all rules have machines…pick two: not all of one is like the other.
A major goal of my functions unit to help kids separate these ideas. So the very first thing I do is poke at it.
Three Rounds of What’s My Rule?
For the first round of “What’s My Rule?” I started with a function that has a rule that you could write a formula for: minus 1 from your input and then square it. This is to make sure we all know what’s going on, and Kent does this with his class as well (see his Day 6).
(I also prompted kids from the very start to notice that different inputs might have the same output.)
Next up, though, I want to move them towards rules that they don’t know how to write formulas for. I showed them a piecewise rule: double if it’s odd, halve it’s even.
Notice the “No”? That’s a domain limitation. My rule refuses to handle numbers that don’t make sense for it.
Now, this is a very similar second move to Kent’s. Kent’s second move is to use the rule “less than the input.” It’s a good one! But I’m doing something slightly different. Kent is showing them a non-function, but I’m trying to show them functions that don’t have formulas. I’m also establishing that functions don’t have to accept any inputs.
That brings us to Round 3. Now I want to show a non-function. I want to emphasize that non-functions are mean, inconsistent places where what’s true one moment is not true the next. See if you can figure out what this rule is:
Stumped? My rule is that I get to decide whether to halve the input or multiply it by 10. I choose based on whim, my mood, whatever I feel like. I do what I want.
Perhaps — if you’ve studied too much math or have been teaching for a while — you would suggest that this is not a function because there is no rule. But very clearly there is a rule! The rule is I get to choose one of two procedures, whichever I want at the moment. This is a rule, but it has no formula and it certainly feels different than the other rules.
A rule is not a function is not a formula. I give a very rough definition on the board of function: a function is a rule that’s fair. This is not the definition they’ll have by the end of the unit.
What about machines?
By the way, I’ve left out machines.
Kent brings up machines early, but I think that machines emphasize consistency in effect. If I were talking about “What’s My Rule” in terms of machines, I might get stuck. Could a machine have a “Do whatever you want?” rule? I’d rather emphasize the human choice in these early examples before bringing in mechanical metaphors.
Three Good Questions
At this point, I made a lucky call: I asked kids to take out their notebooks and write a question they could ask that would test their understanding of “function.” Each question that we talked about was golden.
Question 1: What about a rule that says, “the first time an input is called, do _____ to it; the second time an input is called, do ____ to it,” and so on? Would that be a function?
Question 2: Does a function need to have a formula?
Question 3: Does every formula have a function?
These kids were thinking about the right stuff!
Make Your Own Rule
I asked kids to play a few rounds of this game with each other, making up their own rules. Two kids came up with interesting rules that seemed worth sharing with the class. I typed them up and handed them out as the warm up for our next session.
The first really pushes on the relationship between rules and functions. His hint is to think of this table in chunks of 4 rows:
The second rule uses a different formula then what kid are used to seeing:
Who cares what a function is?
We’ve all seen those questions that show you one indiscriminate blob with arrows to some other blob and ask if it’s a function or a relation. These questions are dumb and shouldn’t be what we’re aiming for with our function unit.
Instead, we should be worried about trigonometry. sin(x), cos(x) and tan(x) are very, very different feeling for kids than the functions they’ve seen before in algebra. They hardly seem like they have formulas at all, they’re just rules.
And we should be worried about the idea that in computer programming you can define functions that lookup information and don’t have anything to do with numbers at all. And you can use cases to define those functions, and they can even involve randomish parameters.
The function vs. non-function perspective is interesting, but there is a lot of depth in the function vs. rule vs. formula (vs. machine) comparisons. That’s where I like to make sure my kids spend a lot of time.
A Few Other What’s My Rule? Rules That Are Fun
- Take a book up to the board. Have kids call out numbers. Turn to those pages and then call out the first word from the top of the page as an output.
- Kids call out colors, and you look at your watch and call out the time.
- Kids call out numbers and you just repeatedly say “NO” unless they say “5,” in which case you say “5.”
- Kids call out years and you look up the world population for that year.
- Kids call out numbers and you respond with words that are that many letters long. (And you say “NO” if they say numbers above 10 or so.)
And, as always, the question is whether this is a rule, does it have a formula, and is it a function.
What are some other fun prompts for this game that could press on the relationship of these related ideas? Would love to hear your thoughts.