Steven, as far as a motivation for factoring polynomials, there may not be one good opener question that would spark interest. But what about motivation for doing bicep curls? I coach basketball , so I have some cred when it comes to bringing athletics into the classroom. My analogy is you will never be asked to flex your bicep, or do a bicep curl during a (insert athletic event here). However, the training aspect (that may on it’s own appear tedious and unrelated) allows one to perform better at other more elaborate athletic tasks. Learn how to do the exercise well, practice it, and know when it is necessary, and it then becomes useful, part of the bigger picture, even if on its surface it may appear irrelevant.

That usually is a response to the generic “when are we ever going to use this” comment, and of course it is lost on some students…

With regards to the general tone of what “style” of questioning/problem solving/curriculum choices is best, I am sure that if there was a one-size-fits-all program that worked for everyone, some Prentice Hall/Wiley/Key Curriculum/Disney/GE/Microsoft conglomerate would be marketing it to all. One of the most frustrating, yet I find interesting things about teaching is the mixing together all of these differing techniques, philosophies, activities, styles and curriculum design all come together to form a classroom environment. The ability to adapt is I feel one of the strongest traits teachers have. If I ever find that holy grail of teaching that works for all students, in all disciplines, on all of the standardized tests, hitting all of the various learning styles, accomodating all student needs, I think I would get bored.

Speaking of parabolas, I think the best in-class demonstration I’ve ever seen was in a class on Experimental Design and Measurements in undergrad. The setup was a catapult that had several variable settings, such as using 1 or 3 rubber bands, pulling the arm back 90 or 135 degrees, and something else, I don’t remember exactly.

Anyway, suppose he just varies those two settings between two values, giving 4 different settings to test. He puts down a tape measure in class then launches the ball with those 4 settings, maybe multiple times to get an average. After launching the ball and recording the results in this magical spreadsheet, he says, ok now let’s try something different. He puts on two rubber bands, chooses a different angle that we didn’t test and punches that into the spreadsheet, which linearly interpolates an estimated distance for the ball to travel. We put a bucket at the predicted distance, and this ball lands in the bucket, based on some crazy spreadsheet.

Since I can’t remember the exact contents of the spreadsheet, I must not have been scaffolded properly or something, but it blew me away. I still talk about it to this day, though in the comments section of my friend’s blog.

Why is it that however you fold a piece of paper, the fold is a perfectly straight line?

I’m ready to open my Kimono now at When Galaxies Collide which is about just that question.

What is it that prevents so many smart, dedicated, energetic people from effecting change in our profession and what might we do about it?

But if you’re meaning justifying the actual act of drawing out a parabola by hand (admittedly the rocket thing can be done just with algebra on the quadratic) you need your students to make a parabolic hot dog cooker.

http://sci-toys.com/scitoys/scitoys/light/solar_hotdog_cooker.html

They’ll have to lay out actual graph paper and sketch out the parabola first before they bend their plastic to match.