# Thinking Outside the Box -- or Maybe Just About the Box

Author(s):
Thomas Hern (Bowling Green State Univ.) and David Meel (Bowling Green State Univ.)

#### The first Box Problem applet

One element in the applets that we have designed are the multiple representations and alternative ways of conveying information. For instance, the image shown below is a screen shot from the ClosedBox applet, which allows students to explore various scenarios. The student can manipulate the lengths A and B by just pulling on the yellow points A' and B'. The yellow points P and Q also move. In doing so, the other components of the applet change in accordance with these manipulations. The power here is that students can interact with a wide variety of examples and see if the conjectures they make hold up to empirical investigation. In addition, the student can move the corners of the box, in the lower right-hand corner of the applet, and see if the box will actually close or not, an important aspect if you want the box to hold something. The last elements in this applet are the two different graphical indicators of maximal volume. The one graph shows the volume with respect to P or Q while the other is held constant and the bar graph next to it displays the percentage of maximal volume obtained by the current configuration. If the volume is too large, one can resize the vertical unit, a yellow point denoted by U, to get the graphs comfortably into the grey viewing window.

As students play with these to attempt to improve their maximal value score, they will be led to ask a variety of questions, such as:

• Is there a relationship between the cut length and the position of the cut that maximizes the volume?
• Under what conditions does the volume drop to zero?
• Is there a best way to orient P and Q so that the maximum is achieved?
• How do the two graphics on the bottom left-hand side interact with each other?

Figure 8: The first Box Problem applet
Click Here to open the first Box Problem applet

Warning: The first Box Problem applet page, entitled ClosedBox, is best viewed in 1024 x 768 resolution or greater and may take up to a minute to load.

The goal of this exploration is to help students recognize that even though this problem may at first appear to be a problem involving two variables, the cut length and the position of the cut, focusing on maximizing the volume allows one to turn the problem into a problem of one variable. By exploring the applet, students should find that if the cut length is held constant and the position of the cut is allowed to vary, then the volume of the corresponding box is described by a constrained quadratic equation (expressed by the blue graph), and if the position of the cut is held constant while the length of the cut is allowed to vary, the volume of the box is described by a constrained linear equation (expressed by the red graph). The constraints result from the physical constraints on the box as well as whether, when the box is constructed, it will hold its contents, i.e. the flaps meet or overlap so there are no holes in the box. All it takes is some imagination after working through multiple examples and noticing the relationship between these two variables that maximizes the volume for any particular condition. From our perspective, being able to explore, quantify, and utilize is an important aspect of learning mathematics.

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