This post is the third in a series of posts detailing the happenings of a math project. To better understand the whole story, please start reading at the beginning:
The next few classes after the brainstorming class were a blur. Students would come into class, grab their previous work, and get down to business. It was the best I could do to have supplies waiting for them. I learned quickly that students can become pretty demanding when it came to their learning.
I didn’t have any problems granting their requests; none of them seemed unreasonable.
“Mr. Banting, we need some tape”
“Mr Banting, do you have any string?”
“Mr. Banting, what are we going to build these out of?”
I ran down to the main office for supplies two or three times. Every time, I would have a line of students waiting for my arrival. They were all anxious to show me their new developments. A trip to the mall produced sheets of poster board, the SRC room provided large sheets of paper for nets, the Home Economics lab had extra shoelaces…
Students had made it very clear–I was not to stand between them and their vision. Luckily for me, that vision was well steeped in mathematical knowhow. The entire process was summed up nicely by one student. In the midst of designing his zerowaste pop can/box design, he looked up at me as I was looking for a set of protractors. He smirked:
“This whole thing has kinda taken off, hey?”
Bingo.
I bought the caretaker donuts because my room was looking more and more like a garbage dump. Cut cardboard boxes, crumpled papers filled with scratched ideas, and empty pop cans filled the place.
The growing mess behind my desk. 
Slowly, but surely, the designs came to life. Each group had a rationale as to why their box provided a solution to a problem.
One group began with the idea of the 6×2 box produced by CocaCola. Instead of leaving the cans stacked directly on top of each other, they eliminated a single can and moved the top five cans into the awaiting gaps created by the bottom row. The purpose was to sacrifice one can of product to greatly diminish the amount of wasted space within the box. Also, that move reduced the height of the box, thus reducing the surface area as well. All great thoughts, and an impeccable design that used the principles of both rectangular and triangular prisms.
An 11can prototype 
A couple groups chose a triangular option, but each with their own distinctive changes. One used a base of 4×2 cans on their sides. The triangle was constructed on this base layer. The result was a 20pack of cans that used far less volume than an inefficient rectangular variety. They also assured me that their model afforded for easy fridge dispensation.
Student tests out their 20pack 
The problem of convenience was addressed by another triangular prism group. This group, the group that was initially arguing at the board, arrived at a compromise where their box would be composed of 4 parts– all triangular prism 6packs. They would then fuse them together to make a 24pack. This model still cut down on volume, but actually increased surface area marginally. They rationalized this trade off. The customer would be able to rip apart the 4 packages and store them as they saw fit. Fridge, closet, bar, camper, cabin, you name it… They were combining math skills with the 21st century skills of marketing, economics, etc.
Another group was not satisfied with the wasted volume in the corners of a triangular prism. They wanted to round the edges to follow the curved contours of the cylindrical cans. They took 3 cans and traced them as the were lying tangent to each other. From there they got their design. This group spent the majority of fabrication time in the computer lab next door working with a computer drafting program. It was the first time the student saw connections between his high school classes. This is a pretty sad, yet common, occurrence.
The modified triangular prism takes shape 
A group designed a hexagonal model that was designed to look like a large pop can itself. This was the first design that took the practicality of the task and married it to the aesthetics of product design. To me, it had the best of both worlds. They figured that a hexagon would better match the natural shape of the can–thus reducing volume waste. They were also confident that the six smaller panels would use far less material than the four larger, rectangular ones.
Student estimate a loss in surface area 
The hexagonal shape is mapped out 
On the more creative side, one group decided to keep the same rectangular box, but fill it with Tetris inspired mini packs. They were more than upfront about the drastic increase in surface area, but thought the idea was cool enough to compensate. They went into designing the various orientations and nets for the pieces. The results were quite cool.
A Tetris piece is netted and constructed 
The design process was highlighted by two things:

A visit from my administrators encouraged my kids. They could not wait to show the math off to them. One student followed my viceprincipal around asking for advise and assistance. Another student talked how they shared their math work with their family for the first time. A significant audience is crucial for PBL; seeing other people take significant interest in their work was extremely uplifting.

A student went to work one evening and asked his boss about the design. He then tweeted me the results. This doesn’t sound that amazing–unless you knew the student. He is a 50% attender who has repeated math multiple times. He just laughed it off when I highlighted the fact that he was actually doing math outside of class.
Once the initial designs were in place, the challenging task of mathematical accuracy began. I designed the task to cover the topics of surface area and volume, but it ballooned into combinatorics, geometry, circle geometry, regular polygons, estimation, triangular numbers, etc.
The important thing to note is I did not force any specific design on a group. Some groups embraced the “go green” approach, while others were engaged in their novel ideas. The group who doubled the surface area, but designed a box they felt was exciting learned just as much as the practicalminded groups who designed a box to reduce both surface area and volume. I simply sat back, provided the opportunity and freedom, and watched as the math fell from the experiment.
And fall it did…in more ways and forms than I could have possibly predicted.
Continue Reading —> Soft Drink Project Part 4: The Math
NatBanting