Congruent squares are going to be cut from the corners of an 8.5 inch by 11 inch piece of standard office paper, and the paper is then going to be folded to form a tray.

1. We will use s to represent the edge length of the squares that are to be removed from the paper. You will be assigned to a small work team. Each team is to make a physical model of a tray for an assigned value of s. On the bottom of the tray (or a piece of paper affixed to the tray), clearly show the value of s, the Surface Area, and the Volume of your team's tray.
   
   
2. Create formulas to find the Surface Area and Volume of the folded tray in terms of s.
   
   
3. Based on your physical model as a sample, and on the formulas as an abstract representation of the relationships, make as many reasonable conjectures as you can about how changes in the size of s will affect changes in the surface area and volume.
   
   
4. As a group, let's use the models as "points" to create a wall sized scatterplot that compares the length s to the Volume. How do you explain the way the "points" are distributed?
   
   
5. If we wanted to re-do the scatterplot to show the relationship between the length s and the Surface Area, we would have to first re-scale the vertical axis to go conveniently from minimum to maximum values. After rescaling the axis, how many of the models would remain in relativly the same location?

How could you use the collection of models to help answer the questions?



How could you use a graphing calculator to help answer the questions?



What questions can you imagine that might be helpful in guiding the discourse when using this problem with students?