The first Box Problem activity
This applet was designed to help students come to a realization that for any positioning of the fold, the maximum volume can be best achieved when the cut length is set to onehalf the smallest width of the box. Not only can students disassemble the box by pulling on the corner points, the applet helps students investigate how constraints affect the box problem. For instance, if the cut length exceeds the minimum of the length and width of the box, then the box will not close because the flaps created by the cuts will interfere with each other. Alternately, if the cut length is less than half of the minimum of the length and width of the box, then the box will not close because the flaps created will leave a hole in the box, thereby allowing the contents to fall out.
So, how does one accomplish this? In the applet, the points denoted with a yellow dot are those that are moveable. Perhaps, the first question that students should be asked concerns the midpoint fold. In particular, why does one of the folds need to occur at the midpoint? Another question should focus on the symmetry. In fact, asking ''what happens if the two folds aren't symmetric about the midpoint fold?'' can yield interesting conversations amongst the students. Getting back to the movable points, a student can set a position for where the cut is to occur and then vary the length of the cut. We suggest students carefully record the data from their various trials where the cut position, \( m( \overline{AP}) \), is set and the cut length, \( m( \overline{BQ}) \), is changed to maximize the volume for that particular cut position. The last row of each Trial is for the student to search for the maximal cut length and the corresponding cut position. The final trial is for an open exploration of a piece of cardboard of the student's choosing.

\( m( \overline {BQ}) \) 
\( m( \overline {AP}) \) 
\( m( \overline {PM}) \) 
\( m( \overline {QQ'}) \) 
\( m( \overline {BB'}) \) 
\( m( \overline {AA'}) \) 
% of max volume 
Trial 1.1 
1.5 
3.0 
4.0 
5.5 
8.5 
14.0 
97.75% 
Trial 1.2 

3.5 
3.5 

8.5 
14.0 

Trial 1.3 

4.5 
2.5 

8.5 
14.0 

Trial 1.4 




8.5 
14.0 

Trial 2.1 

2.5 
2.5 

10.0 
10.0 

Trial 2.2 

1.5 
3.5 

10.0 
10.0 

Trial 2.3 

4.25 
0.75 

10.0 
10.0 

Trial 2.4 




10.0 
10.0 

Trial 3.1 

3.0 
2.5 

8.5 
11.0 

Trial 3.2 

2.75 
2.75 

8.5 
11.0 

Trial 3.3 

4.0 
1.5 

8.5 
11.0 

Trial 3.4 




8.5 
11.0 

Trial 4.1 







Trial 4.2 







Trial 4.3 







Trial 4.4 







Exploring various positions of a cut, should lead the students' to a conjecture that for any position of a cut, the maximum volume of the corresponding box is obtained when the cut length is half the length of the shorter length between \( m( \overline {AP}) \) and \( m( \overline {PM}) \).
There are so many different questions that can be asked when students are interacting with the first Box Problem applet. In addition to those already mentioned, one can ask:

Under what conditions is the blue graph connected above the xaxis?

Under those conditions and for a particular length and width, what are the dimensions of the box with maximum volume?

Why does the red graph always appear as a slanted portion? What mathematical meaning does this red graph have? Does this slanted portion ever change slope? If so, why and if not, why not?
It should be mentioned that we have found that if students have experienced the open box problem, there is a tendency of students to either focus on the length of the cut or the position of the cut but not typically both. Some students want to make the controlling variable the position of the cut and the resulting calculus computations are more difficult but not impossible. However, if we were to focus on the computations and not a deep exploration of the problem, students would miss a perfect opportunity to look beyond the numbers and see relationships that are intrinsically interwoven in this problem.