Most students saw this growing the same way:
We discussed the work they had done yesterday and they seemed to be on the right track. So we returned to this question:
We reviewed the equation relating the number of cups to the height for the Styrofoam cups, emphasizing what each variable represented (I actually wrote out the words and did not use x and y). I then gave each group 5 red cups and asked them to come up with a model. It took some work but they started measuring and figuring out the rate of change. Once most groups had an equation I took one group's data (they had a constant rate of change) and we analyzed it together.
We then set up the system to find out when we would have the same number of cups and they would reach the same height. The question they were supposed to be answering had the red cups starting on the desk, but I didn't think their models would good enough for that. We got a result of 10 cups that should give the same height and this is what it looked like:
Hmmm. Clearly not the same height. What could have caused this? The algebra we did to solve the system of equations was correct, so the model for the red cups must be the culprit (we were consistent with our model for the Styrofoam cups on Monday). I asked for other groups' models for the red cups(written in brown, above):
h = 0.3n + 10.5
h = 1.2n + 2
h = 0.5n + 10.5
So many issues of values that were simply not reasonable. Could the red cup really be 2 cm without the lip? Was the lip only 0.3 cm? Back to the drawing board, as they say. With only 3 minutes left in class I asked them to try collecting data again. I am not giving up on this so there will be more tomorrow!