One such topic is having student determine the function that models some form of quadratic data. I find that many students get lost in the wording of such questions and get bogged down by the details. I thought I would introduce a video to see if it helped with their understanding. I showed my class the video below and asked them to determine the function that models the height of the ball at any given time.
I played the video on the interactive white board and had a student come up to the board and put dots on the board to show the location of the ball. Once the video was finished he drew a smooth curve through the points. My question to the class was how do we determine the function that models the situation? I was amazed how engaged the students were. I received a ton of answers: We need to find the vertex. We need a set of axes. We need the 'a' value.
The video and the one question I asked produced the buy-in I was after. It was great. The nice thing about this lesson is that later in the unit when students struggled and asked for help they would say things like "Oh right, this is what we did with the video isn't it?".
As an added bonus we were able to practice a couple of times just by shifting axes around, which also reinforced the idea that the 'a' value was the same because we were talking about the same curve.
I realize that the problem is still somewhat contrived. Ideally I'd like to have a situation where students feel compelled to solve the problem and end up using the math to do it. Nobody really wants to know the function that models the path of the ball, but it was a situation they could relate to and was a step in the right direction.