I would like them to realize it's non-linear first I would think contemplate then calculate for this. What activity could we design that would help students like this one develop their thinking?

I agree: it seems as if this student is trying to fit a U-shaped function into a line-shaped paradigm. Knowing that this sort of equation produces a U will make it more likely that they will test negative x-values, or at least more reliably guess the rest of the shape. While there were a lot of great observations, the one that stood out to me was that this student could probably learn to recognize that this sort of equation will produce a U shape. I wonder how come the student did not use any negative values in her table.” Jonathan: “I notice that there is a disconnect in the student’s knowledge of linear vs. Lane: “Does he have an eraser? Does he get confused calculating with zero? Does he know the shape of a parabola? Does he know that a function cannot possible have one point on top of another? Does he sometimes get confused with x^2 and 2x? Could he have analyzed his own mistakes with a calculator? By not checking with a calculator will some of his errors snowball and cause further confusion? Is the student feeling frustrated? I think it is good this student understands the choice of input does not have to be in a particular order.” Kevin: “These don’t seem to be in any particular order.” They do not recognize that this is a quadratic function and try to get a straight line.” Guest: “This is a common error for my students as well. The written answers people offered were also really interesting.

I’m not sure what to make of all that overlaid, but I’m definitely interested. Here is the overlay showing everything that everyone circled. I asked people to circle something they noticed in this student’s work. I made a little Desmos activity to see if it’s possible to use their activity builder to share and comment on student work. What do you notice about this student’s thinking? What do you wonder about it?

Desmos graphing activity