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Maximum Volume: The Box Problem

This is one of my favorite lessons.  Depending on the class and the amount of time I have to dedicate to this activity, I have several versions.  But almost all versions have popcorn involved.  I love popcorn.

 You can begin by giving students a plain piece of computer paper with very little instruction (I love giving minimal instruction as it allows the students to be creative).  Their task is to make a 3-dimensional object with the paper that has the maximum volume.  They can use scissors and tape, but cannot extend the paper with the tape.  I will fill what they create with popcorn.  The more volume, the more popcorn!  And yes, I buy bulk popcorn for this activity, which I pass on to all of my students in other classes.  A lot of times, movie theaters will donate a bag, other times I just go to a local popcorn store and buy a big bag for $10.  It's worth it.  Plus, did I mention that popcorn is my favorite snack?

 Once students have openly discussed their 3D shape and why they constructed their design, we move into the official activity.  I give the groups this sheet to follow as they move forward.  I assign each group a value for x (the side lengths that will be cut from the corners of their stock paper to form the open-top box).  (I am including snips of the PowerPoint that I use with this lesson.)  Before we actually start calculating volume, I have each group hold up their boxes and we talk about which box will contain the most amount of popcorn, or if they will all encompass the same amount since all groups began with the same size of paper.  Groups do tend to know that not all boxes have the same volume.
Once the kiddos have created their box, they are to use a ruler to measure lengths to find the volume.  Some will realize that they do not need a ruler to measure because they are subtracting 2x from the length and width, with x being the height.  Hot dog!  Either way, each group goes to board and puts their volume in the appropriate cell of the table that I have constructed.

 Inevitably, there will be groups that do not calculate the correct volume.  It actually opens up a great discussion as to which x-values have volumes that do not make sense.  Students will recognize the pattern of the volume that increases, maxes out, and then decreases.  Yay.  
To summarize, students will use old-school calculations to find the volume of a box (V = lwh) and follow up with using the features of a calculator to find the maximum volume.  And I may have forgotten... we all get to eat popcorn!

~SSB

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