Reflections and Extensions
This paper has attempted to illustrate how applets can broaden the investigation of mathematical relationships and reach beyond mere algebraic manipulations. The ability to investigate, conjecture, test, and pose new questions in an electronic environment is important to the development of understanding. Beyond this, such investigations are enjoyable for students. For instance, students mentioned the box problem in their weekly Calculus 1 journals and two examples are provided below:
''This last week we learned of Indeterminate terms and L' Hopital's rule. Did quite a few examples on how to perform the rule, contionued [sic] this for four days, then I think we made boxes. I believe that I learned how to do L' Hop's rule fairly well, hopefully I could do well on a test, and the box making was fun and surprised me in that you fit it into the lesson. I enjoyed the interactivity.'' - Charles
''This week we learned about maximum volume of boxes and solved a pretty cool problem involving James Bond and Bombs. I love these sorts of problem-solving and I hope that we do more of it in the future.'' - Dan
Our goal has always been to get students to think that problem-solving is fun and the interaction between building physical boxes and the computer applets helps students draw essential mathematical connections. The lessons that can be drawn from these activities bolster students' perceptions that mathematics is eminently useful and applicable to real-world phenomena much more than the typical box problem shown in many of the calculus texts available on the market.
But is that as far as one can go into the world of boxes? We should also mention that this problem of optimal box building could be further extended by including the tab to affix the sides of the box to each other without the use of a taped joint. At one level, our discussion has focused its attention on construction of a minimal shipping container of the Regular Slotted Container type; however, if we were to introduce the requirement of a tab it does impact the problem. It should be noted that there are general governmental guidelines for the width of such a tab but 1 1/4 - 1 1/2 inches is typical. For specific guidelines about tab regulations, one should check the Fiber Box Handbook (Fiber Box Association, 2005), the National Motor Freight Classification (NMFC) Item 222 (American Trucking Association, 1970) for common carrier motor freight shipment, or the Uniform Freight Classification (UFC) Rule 41 (Dolan, 1991) for rail shipment. In any case, we think that after exploring the box problem through these applets, students might be better prepared to think how to construct a RSC containing tabs (as shown in figure 5) with maximum volume?
Beyond even looking at a tabbed RSC, the world of boxes is much more encompassing. In this paper, we have focused our attention on the most popular box, a regular slotted container, but there are a host of other types of boxes fabricated from a single piece of corrugated cardboard that students can investigate, including:
A Full Overlap Container (FOL), see figure 10a, that is resistant to rough handling since all flaps are of the same length (the width of the box) and when closed, the outer flaps come within one inch of complete overlap,
A Five Panel Folder (FPF), see figure 10b, that features a fifth panel used as a closing flap completely covering a side panel and includes end pieces composed of multiple layers that provide stacking strength and protection of long articles with small diameter, or
A Once Piece Folder (OPF), see figure 10c, typically used books and printed materials since it has a flat bottom with flaps forming the sides and ends along with extended flaps that meet toform the top (GoPackaging.com, 2006).
Figure 11: Other types of boxes manufactured from a single piece of corrugated cardboard
Each of these boxes would provide an opportunity to extend investigation and gain a better sense of the issues behind box construction. We certainly hope that this paper hasn't boxed you in but rather opened your eyes to the ways boxes can be used to help students see the ability of calculus to play a meaningful role in examining real-world problems.