The original plan was to have students continue on with the same graph of their square that we've been working with for a week now and determine the relationship between the coordinates of the endpoints of the diagonals and the coordinates of the midpoint. Students being what they are (teenagers), so many of them have lost their graphs or made such a mess of them that to continue working with the same data would be a fruitless venture.
Instead, I took the classes to the computer lab to do the same activity using Geometer's Sketchpad. As much as I love GSP, I don't love trying to teach students how to use it. It's an incredibly powerful program, which means there are a TON of little things you need to know how to deal with if you want to get consistent results. Sadly, even with the most direct instructions, my students see the program as an obstacle and will often quit at the first sign of struggle.
In the end, most students got the construction created successfully, but struggled with the discovery aspect of the activity. Most of them really fight the idea of being creative in math, they are so accustomed to being spoon fed a procedure which must be learned and memorized to be regurgitated later, that the freedom to create something new is a foreign concept in this specific environment.
Some students did arrive at the idea of "middle," but couldn't make the mathematical leap to the idea of 'average' (further reinforcing my suspicion that even students who are generally "successful" in math have little idea what they're actually doing).
I will adapt the instructions to include more scaffolding, specifically the suggestion that the x coordinates should be considered separate from the y coordinates.