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This applet was designed to help investigate the interrelationship between cut length and volume. In particular, the previous interactions with the first Box Problem applet allowed the student to recognize the relationship between cut length and position of the cut. Using this, the student can build a functional relationship, based on a single variable, relating cut length and volume. This applet is designed to explore that functional relationship and help students come to a realization that in modeling real-world phenomena, a careful examination of the domain of a the functional relationship is necessary.

How can one ask questions to help students interact with the second Box Problem applet in meaningful ways? From our experience, the first question that probably should be asked involves the relationship between the first Box Problem applet and this second Box Problem applet. In particular, we can see in this new applet that the point P varies as the point Q moves, what information drawn from the first Box Problem applet allows us to control the point P? Another question should focus on constructing the volume of the box based upon the length of cut. Essentially, a series of questions to help lead students to developing a functional relationship between volume and cut length, such as:

- How does one determine the volume of a rectangular box?
- What would be a description for the length of the box?
- What would be a description for the width of the box?
- What would be a description for the height of the box?
- How can you use these descriptions to build a functional relationship between volume and cut length?

Inherent to these questions are questions about what are the constants and what are the variables? Such a question is important when students are faced with multiple letter-based descriptions such as those expressed in the formula for \( V(l) \) :

\[ V(l) = \left( {B - 2l} \right)\left( {{1 \over 2}A - 2l} \right)\left( {2l} \right) \]

Once students have developed appropriate functional relationships between cut length and volume, a variety of questions relating to that function can be asked. In general, we typically ask students to investigate the graph of the function on a hand-held graphing calculator. And then, we start asking various questions to help the students compare and contrast the graphical representation produced by the graphing calculator and that of the applet. For instance, we might ask:

- Why does the graphing calculator seem to show you more of the graph than the applet does?
- Why does the applet truncate the graph?
- What conditions should be on the domain of the function and how do they relate to physically constructing a box?

So far, these questions have focused primarily on a static rectangular sheet of cardboard. That is, we have not asked questions that forced students to think about varying the size of the cardboard.

- For different lengths or widths, the applet's graph seems to change shape near the right-hand terminus, what mathematical reason can you provide for this change or provide an argument that it does, in fact, not change?
- Are there two (or more) non-isomorphic sheets of cardboard, so the maximal volume is the same? If so, identify them and if not, explain why not.
- Are there two (or more) non-isomorphic sheets of cardboard, so the placement of the maximal cut is the same? If so, identify them and if not, explain why not.

Here, the questions are designed to force students to think beyond the original question and attempt to distill commonality and explore generalizations. Stretching students to think beyond and to ask ''what if'' questions is an important aspect of interacting with this applet. We sought to design the applet to support both the investigation of a specific scenario as well as a broad range of extended questions related to the original problem.

Thomas Hern (Bowling Green State Univ.) and David Meel (Bowling Green State Univ.), "Thinking Outside the Box -- or Maybe Just About the Box," *Loci* (February 2010), DOI:10.4169/loci003321