Here are three gas station pumps. Which ones are trying to rip you off? Can you tell just by looking?
After your students have that debate and share their reasons (expected: “the third is a ripoff because it’s moving faster”) invite your students to collect data for each pump and enter it at Desmos. Here we’re establishing a need for a graphical representation. It may reveal patterns that our eyes can’t detect.
The third act helps clarify the underlying trends. The third pump is spinning faster, but the price and the gas still exist in a proportional relationship. The first pump, meanwhile, pumps less gas per dollar the longer it runs.
I am indebted to William G. McGowan and Sean Berg, whose NCTM 2016 session description included the words “gas pumps have been hacked,” and there went my weekend.
Their description reminded me how important it is to expose students to counter-examples of the relationships they’re studying, protecting against over-generalization. (ie. “Everything is proportional. That’s the chapter we’re in!”) I’m becoming fascinated, in general, by problems that ask students to prove that a mathematical model is broken rather than just apply a model that works.
[Download the goods.]
Iâ€™ve written before about expanding teaching to the â€œneighborhoodâ€ of the special case. If we always show the highlight reel, students never get appreciation for how special and how powerful the ideas are. So I like that this lesson is about finding the non-proportional â€œripoffâ€ as it stands out in contrast to the â€œnormal/expectedâ€ proportional relationships. (Ironically I would have said that proportionality is the special case, and nonproportionality is the â€˜normalâ€™â€“ before thinking about what we expect as consumers. â€˜normalâ€™ is all subjective!)