One of the things that I have learned over the years is to let go of any preconceptions I have about how a problem should be solved. I have methods I prefer, but my students need to develop their own. Never has this been more obvious than it was today.
We are beginning to play around with non-linear functions and so I gave my class the following problem:
You're going to build a garden and need to build a fence around it. If you have 120' of fencing, how would you set it up in order to have the biggest garden?
...or something like that.
I didn't specify rectangle although I figured most kids would default to that. I didn't mention the barn in the back of the property that could be used as one of the sides. I just kept the prompt as simple as possible so I could see where they took it.
Many assumed a square right off the bat. And one group felt pretty ambitious and looked up some shapes in their school planner only to settle on the typical decagon. (Good luck calculating the area of that one, fellas.)
So, I'm walking around and seeing groups all proud of themselves by defining x as one side and y as the other and they're arguing about whether 2x + 2y = 120 or x + y = 60 is the better equation. I ask a few questions like, "so how will that equation help you determine the largest garden" to which they reply, "we'll find the maximum."
"Mmm-kay," I say as I mutter to myself, "I'll be back..."
Then I walk up to a group of three boys who usually push the envelope when it comes to creative problem solving and I see the equation x + y = 60. I think to myself, "oh, no not you too."
Then I look a little closer and I see this other thing they're working on.
xy > 900
I double take.
Before any of them even address my incredulous look, one of them says, "let's go put it in GeoGebra."
"So how'd you come up with that?"
"Well we figured that x + y = 60 tells us how much fencing we have. And since a square would give us 900 ft2, we want to know if there is an area out there that's greater than 900."
"Uh, yeah. That's, uh, yeah, that's exactly how I'd do it."
Not really. This is what I really said.
These kids used a system. Not in a million years would I have ever considered using a system to solve this problem and three 13 year-olds set me straight.
Man, I love this job.