The problem is a classic in algebra and calculus, and to be honest it really did not make sense to me until I built the boxes, when I was 35 years old. Here is the problem:
- Given a standard 8 ½ inch x 11 inch piece of paper, determine a function which gives the volume of a box (without a lid) made by cutting squares from each of the corners and folding up the sides. Let x be the length of a side of the square, and write the volume as a function of x. What are the units of x and v(x)? What is the maximum volume of the box?
Granted this is an expanded version of the problem. You couldn’t ask all of these questions on a standardized test. But this problem is full of concepts like: domain, range, polynomials, graphing, roots, and volume just to name a few. My classrooms have students at varying degrees of experiences in math. For instance I will have students that are ready for calculus working alongside students who really don’t understand what volume means. In fact if you ask some of them, they will reply, “L X W X H”. If that is all students think about when they are asked to find volume, then they probably do not understand the concept of volume.
I have a bunch of 5X8 note cards, so I ask the kids to imagine cutting equal squares out of the corner of the note card and filling the resulting tray with marshmallows. How many marshmallows could we contain in this resulting tray if we don’t pack them and we do not heap them above the box? I let them fiddle around with the marshmallows. I then give them 1 minute and tell them to write their name and their guess on a piece of small yellow post it note. We then arrange them on the board in a makeshift line graph. I do this so that students buy into the problem and practice estimation. This guess cements their curiosity. They start asking themselves. “Am I right?” ” Is my answer the most?” “Is my answer even possible?” Some students will try and build the box so that it matches their answer.
As they are building these boxes some students will ask, “How big should my squares be?” I respond, “How big can your squares be?” I notice that the older I get the more I sound like a psychiatrist. But here I am trying to lead students to the domain. Eventually we will formalize the domain and the range for this problem. But first we have to see how many marshmallows we can stuff in these boxes.
What we are finding so far is that unsurprisingly the number of marshmallows divided by 8 is just under the actual volume of each box in cubic inches. Here we can talk about the gaps between mallows and ask questions like, does smashing them give us closer estimates? How about piling them? (You are on your own here. The kids ask this but I redirect them to the GeoGerba applet below:)
Finally, we talk about the function that represents the volume of the box for any square of length x cut from the corners of the 5 X 8 note card. In this case V(x) = x(5 – 2x)(8-2x) (I will let you do the distribution to find the simplified version of that polynomial;). And about the best place to explore the function is in Ted Coe’s GeoGebra worksheet at http://www.tedcoe.com( His site is incredible!!!!! BTW). Here is the html applet. I hope it works because it is awesome!!!
The power of this project comes in the form of having students tape their creations to a projected GeoGebra display of the function the class developed. Every time I do this it brings a small tear to my eyes at its beauty. To me it shows the true power of technology in our classrooms. The first time I saw this problem was Algebra II when I was 16 years old and our internet was floppy discs. Along with the fact that we only used computers when instruction was complete. I am sure my teacher at the time tried to show us the solution, but I did not register any understanding. It was one of those problems that the teacher assigns and then when no one tries, he/she solves it for them on the board while all students nod their heads that they understand. I saw it again in Calculus and I still don’t think I truly understood anything outside of (the algorithm) the fact that when I applied the derivative to the function and then set the resulting function to zero and solved I would get my maximum and minimum values. The third time I saw this problem was in my masters program at the University of Nebraska-Lincoln. Here is where I came up with the idea that maybe when we solve this problem we should actually construct the boxes. I am pretty sure that constructing the boxes actually makes this problem a Jo Boaler Low floor high ceiling problem. (YOUCUBED.ORG has changed my very perception of mathematics education). I give this problem to all of my students 9 – 12 and they all love it. Probably because they get to eat marshmallows, but I hope that at the very least some finally see the connections between stuffing marshmallows in a box and volume are the same thing. If we get to the connections between algebra, calculus, and geometry…. that would be bonus for those that are ready to make those connections. I am pretty sure I wish I lived right now. Education is awesome!!!