Just because this is a math class, it doesn't mean that students shouldn't 1) be increasing their graphing skills and 2) learning how to analyze data to look for evidence that support a hypothesis.
So we built on yesterday's activity and collected the distance around the outside and the boxes inside every student's square. While displayed on the screen, I tasked the class with graphing "boxes inside vs. distance outside." There was a brief review of the importance of scale and which axis is which, but most students did just fine. There were some gripes that they had to plot *gasp* 25 data points, but I stressed that we needed a lot to smooth out the relationship in case anyone made mistakes with their measurements (which happened more than it should have).
In the end, I used a document camera to show the results of their hard work. The only major conclusion I was hoping they'd get to is that the relationship is NOT linear. They should have learned about parabolas and what general equation would create what we're seeing, but very few did.
I then used Excel to create my own graph of the same data. I did this so that I could easily show different potential trendlines to see which looked the best. I also displayed the correlation coefficient, but only described it as "the higher this number, the better the fit." Students quickly saw that the polynomial trendline actually fits better than the linear (it helps to have a couple of students make BIG squares to make this obvious). Then I had Excel provide the quadratic that describes the curve, and explained that our 'y' variable was really "Area" and 'x' was "Perimeter." They obviously knew these words, but seemed to be OK with my intended misdirection.
The end result is A=0.06(P^2) (the constant will vary based on the data, the three classes hovered around 0.06).
I continued on with a deeper analysis that I stressed was just FYI - students would not be responsible for doing this on their own. Essentially, I approximated 0.06 to 0.0625 so that it could be written as 1/16. Including the constant with the P^2, I rewrote the equation as (1/4 P)^2 which combines with the idea that P = 4s to create A=s^2.
Looking back, the honors classes were able to follow and at least some appreciate the derivation. This was a complete cluster in my regular class and threw everything off track for a full day. They simply don't have the attention span or the appreciation for something that's not required, as those students often struggle with connecting enhancement with greater success.