As posted on twitter.com/dmarain today...
Let's get the "answer" out of the way first.
x can = -1,1.5 or 4. Not much more to say about this, right?
If this were an SAT-type question, it might be a "grid-in" asking for a possible value of x.
So what is needed to be successful with this type of problem? A basic understanding of mean and median for sure but there are the intangibles of problem-solving here. This question requires clear thinking/reasoning. Confident risk-taking is very important also. When one seems blocked, not knowing how to start, some students just jump in anywhere and see where it goes. Insight enables a student to move in the right direction more quickly.
Many students intuitively suspect that the median could be 1 or 2 or something in between. Even if they can't precisely justify this, they should be encouraged to run with their ideas. "Guessing" the median first seems easier than guessing a mean! One can always test conjectures.
Recognizing that there are THREE cases to consider is critical here. In retrospect, this will make sense for most but they have to make that sense of it for themselves!
So why not just give a nice clean efficient solution here? Because problem-solving for most of us is not clean at all! When the student is GIVEN the solution it may help them to grasp the essence of the problem but more often it shuts down thinking and doesn't help the student learn to overcome frustration. Yes, we can provide a model solution but how will that lead to solving a similar but different problem. We learn when we construct a solution for ourselves or reconstruct other's solutions in our own way.
Annoyed yet? If you solved it, you're fine. If not, frustration sets in quickly for some. If everyone in the class is stumped we can always give a hint. I think I already did!