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I'm teaching a college algebra course and I'm trying to design a few projects that involve modeling with quadratic functions. So far I have two ideas that involve downward-facing parabolas (projectile motion and the profit function given a linear demand curve).

I'd like to design a third project that involves some real-world application of an upward-facing parabola, preferably something where the minimum has some interesting physical interpretation.

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One example that comes to mind is modelling the position of a diver (or of a diver's head).

Let $t$ be the time elapsed since jump. Let $d(t)$ be the diver's distance from the water and define $d(t)>0$ to be "above water" and $d(t)<0$ to be below water. (It should be obvious why $d(t)=0$ represents being "at the surface"). If we use a quadratic model, then:

$$d(t)=a(t-h)^2+k$$

where $a$ is a positive parameter (to get an upward parabola), $h$ is the time at which the diver reaches the lowest point and $k$ indicates the furthest distance reached below the surface.

An even better model would be to have 2 parabolas (and make a function defined by parts), one that is downside for the initial jump and one that is upside for the rest of the movement (falling, going through the water and resurfacing).

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Answer: Air friction often modeled as a square (comes from compression of air in front of you). I think there are some hyper complicated formulas that add even higher terms, but square is really common. If you want to get fancy, can add a linear term for sliding and/or rolling friction. You can look at terminal velocity for a falling object or at automobile travel as common real world examples. I guess it depends on your point of view with downward facing, but to me, the force is increasing (e.g. if in a car, takes more gas pedal ("accelerator pedal") to match the force.

Long comment: I worry when teachers push "projects". Think it appeals to them as interesting content but is not necessarily best approach pedagogically. "Drill" (much reviled) is often more effective pedagogically. When I say interesting, I mean interesting TO THE INSTRUCTOR, not the students. Recall that for the students the material is new anyways (or if repeating course, is challenging).

In particular if you are teaching college algebra in college, you are likely dealing with weaker kids (by selection process). I would think of experimenting in pedagogy or just make your own internal challenge to increase class performance versus last year on some semi-objective reference to give yourself motivation.

Watch the movie Stand and Deliver (note the different drills as well as the progressive improvement in results by year) and read the Jay Matthews biography. Read the Escalante article in JNE: https://files.eric.ed.gov/fulltext/ED345942.pdf And it doesn't just need to be him (and he evolved and changed and improved over time, so not a static reference). But look at other people, flipped classroom ideas, old salts, whatever. You could even run some little study or publish something. Do an A/B trial across two sections for instance. (Sample size may not be statistically great, but still it really gets you thinking.)

[This is a general concern I have with the content uber pedagogy attitude. But, in your case, I actually found a thread on MSE where you specifically said you were bored with teaching.]

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    $\begingroup$ Not sure what Jared meant by "project", but would you agree that giving context to what the math students are studying is a factor that could increase their motivation, thus making the content interesting to them? Also, you claim that college students are weaker and that flipped classrooms would be worth looking at. But if the reason they are weaker is their socioeconomic background and perhaps lack of a sense of responsibility towards studies, are flipped classrooms really a good idea? (Not American, not 100% sure what age/social group we are dealing with here with College). $\endgroup$ – orion2112 Dec 24 '18 at 2:00
  • $\begingroup$ The thing I like about flipped is the emphasis on doing versus watching. $\endgroup$ – guest Dec 24 '18 at 6:49
  • $\begingroup$ I am not anti application per se. And sure I get the factor you mentioned. My point is that college teachers tend to overemphasize complicated project assignments and underemphasize familiarization and drill. $\endgroup$ – guest Dec 24 '18 at 6:53
  • $\begingroup$ I see your point very well and I agree with you, doing is pedagogically better than watching. Relative to flipped, I was asking because from what I'm hearing, it sounds like a very polarized subject and I like to get people's opions. Some people love it, others hate it, so I was curious. $\endgroup$ – orion2112 Dec 24 '18 at 7:20
  • $\begingroup$ I don't have a strong feeling on flipped. Sounds sort of touchie feelie to me. But I like that some aspect of practice versus listening is involved. Could do same thing with old school recitation, practice problems and change papers. Bigger point was to look for innovation/improvement in the pedagogy, not just the content. After all, it is a richer and less well known topic. $\endgroup$ – guest Dec 24 '18 at 13:28

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