The purpose of models is not to fit the data but to sharpen the questions. Samuel Karlin
11th R A Fisher Memorial Lecture, Royal Society 20, April 1983.
Imagine a lesson where the teacher comes into the room and says to the class, "Evaluate f(80) for f(s) = 10(s - 65) + 15" without any further discussion. While students may understand by the end of the lesson to plug in whatever number is in the parenthesis into the function and spit out an answer by means of order of operations to get some number at the end, they may not understand what it really means. You might get a question of "May I go to the bathroom?" or "I need to go to the nurse" instead of questions that further the topic.
Fast forward to today in my Algebra 2 class where I wrote the following on the board. "This weekend I got a speeding ticket. The police officer told me the fine for speeding was $10 for every mile over the speed limit I was going plus $15 to the city government." Before I could even finish writing, I was barraged with questions like "Did you really get caught speeding?, How fast were you speeding? Where were you speeding?". They were actually anxious to solve this problem. I told them I was clocked going 82 miles per hour. What was my fine? I heard answers of $170, $185, and $320. I asked a student to explain how they arrived at $185 since it was the most popular answer. Then I built the formula around it. "So you subtracted 82 - 65? Why? Oh, because you wanted the amount I was over 65mph? Ok. Then what did you do? Why? Ok. Anything else? Oh yes, don't forget the city government fee."
As I talked I wrote:
82 - 67
10(82 - 67)
10(82 - 67) + 15
What if I was going a different speed? What would change? So if we were to put a variable somewhere, where would we put it and what could we use? Talk to a neighbor about it.
10(s - 65) + 15
Interesting. Let's make it a function since we have been talking about functions.
f(s) = 10(s - 65) + 15
What does this f(s) mean? Excellent. I love that you said it was the fine I paid depending on the speed I was going.
What if I wrote this f(92). What does that mean? Right! The fine of the speeding ticket for traveling 92 mph in a 65mph zone. Now, what would be my fine?
How did some of us get answers of $170 and $320?
And right there I went from concrete to abstract AND they had better understanding of the concept because there was meaning attached to it. Hot dog!!!! Now, function notation isn't something that is terribly hard. But every year I have a handful that seem to struggle with it. However, as I checked on their progress on a story problem they were doing on their own, they had it!
All credit by the way needs to go to Steven Leinwand. He challenged us in a professional development during the summer. Take the boring and make it real. For him, it was real. He really was pulled over in Vermont, I think. He said it was the most expensive math lesson he ever had to pay for.
I had to break the bad news to the students that I did not get a speeding a ticket, nor have I ever had a speeding ticket in my life. They were in disbelief but they did learn what function notation means. So when I threw problems at them from the Pre-calculus book about the weight of an astronaut with a more complicated formula, they had no fear of it. I'm amazed it went so well.
So for this particular speeding story problem, I created the model around the data as we progressed in class. But it wasn't to just mold numbers into some formula, we actually got deeper into what function notation meant. The students weren't just told to plug and chug. They had meaning behind it. I sharpened the questions I asked them and they had better questions to ask. They weren't bored with rote, lame math. They were involved and invested. After all, they did think their math teacher was a bad ass for speeding at 82mph in a 65mph zone.