The Box Problem
You have an 8.5" x 11" piece of paper. You are going to form a box by cutting square pieces out of each corner, as shown. What size square should you cut to make the box's volume as large as possible?
Here are outlines of two approaches to teaching this
problem. Compare and contrast them in terms of advantages and disadvantages.
Approach 1 
Approach 2 
Provide students with sheets of paper. Have them actually
make some boxes. 
Provide students with sheets of paper. Have them actually
make some boxes. 
Draw the diagram as shown above. Ask how to calculate the
volume of the box made from this paper. 
Draw the diagram as shown above. Ask how the endpoint of "Length of cut" will move were we to pull on it (see GSP sketch). 
Ask for a formula for the box's volume. 
Ask what the smallest cut length could be? The largest? What is the volume of the box made from the smallest cut length? From the largest cut length? What does this tell you about the volume of boxes made with cuts inbetween smallest and largest? 
Graph the volume as a function of the cut length and
examine the graph for maxima and minima 
Ask how the volume will vary as you vary the cut length,
beginning at 0 and ending at 4.25 

Ask for an estimate of the graphs's shape based on their image of the volume's variation with respect to cutlength 

Ask for a precise graph of volume with respect to
cutlength. What do we need to figure out to get a precise graph? 

Etc. 