OK, for this experiments, we will deal with springs and masses, so again, it shouldn't be something difficult. The task this time is simple:
You will be given a "mystery" object in which you need to determine its mass. You are given a set of springs, and a set of calibrated masses. In addition, you will also have access to a ruler and a stopwatch if you need them. Figure out how you can determine, as accurately as possible, the mass of this mystery object. You must describe explicitly how you go about doing this determination.
Now, of course, in many intro physics class, this type of experiment typically requires that they find the spring constant by looking at the extension versus force or mass applied to the spring. I'm going about this the other way. Forget about the spring constant for now. The key thing here is that the student learns about the relationship between the spring extension as different masses are added to the spring. This to me would be the most obvious technique that most of the students would do. They would find the extension of the spring with different masses. Then, when given mystery mass, they may have to do some interpolation or extrapolation to estimate the mass of that object.
Now, there's also a possibility that some students may do this differently. They could, instead, let each of the known masses oscillates one at a time and find the relationship between the mass and the period of oscillation. They won't end up with a straight line, but as in the previous suggested experiment, this is OK. While we tell them they need to do this as accurately as possible, in the end, we really don't care as long as they explain what they did and how they did it. So even if they had to extrapolate/interpolate by hand, this is perfectly fine.
Now, what we can do further is this. For the students that did the first method (hanging the mass and finding the spring extension), we can ask them this:
Now, often it is difficult to get the spring to be very still - the mass tends to oscillate up and down. So maybe it might also be a good idea to see if we can make use of this property to see if there's an additional relationship here between the mass on the spring, and the period of oscillation. Can you determine the mass of the mystery object this way? Does it give the same answer? It is always more convincing when two different methods give consistent answers.
For those who did the the second method (oscillating the mass and finding the period), you then say:
Oscillating the spring doesn't allow you to read off the mass very quickly, which is something you need quite often. So is there another way to determine the mass quicker? How about looking at how much the spring extends as you hang different masses? Can this lead you to a different way to measure the mystery mass? Does this value agrees with the one you got earlier? It is always more convincing when two different methods give consistent answers.
.. and voila, you've gotten them to do this in both ways! They also learn that in science, it is always more convincing when you can show a consistent result from two different techniques (although, to be technically accurate, these are not really two different techniques, but this is a good enough demonstration at this level). Now the fun starts if they come up with very different answers. This is where they need to figure out (with the help of an instructor) on what went wrong. To me, figuring out what went wrong is as important and what went right.
After the students have done both, you then can pose an additional question such as this:
What you have now is a graph that you always need to use whenever you want to determine a mass. Is there a way to know the mass of something without having to resort to using such a graph? Can we figure out a way in which, if we know how much the spring extends, we can simply punch that number in and out comes the mass?
I think you know where I'm going with this, don't you? Considering that the students should have a background in sufficient mathematics, they would have seen a straight line equation. If not, a bit of help and hand-holding is called for, which, at this point, should be alright.
So in essence, we have done the mass-spring experiment, but done in a different manner. Rather than giving out the necessary steps that the students have to do, we instead "coerced" them into doing them by a series of questions and tasks that we want them to accomplish by themselves. Inadvertently, they "discover" Hooke's Law by themselves.