This post is the first in a series describing a set of classes in my Grade 11 Workplace and Apprenticeship class. I have designed the course around the ideals of Project-Based Learning (PBL); students encounter a series of tasks, problems, and prompts that necessitate three crucial qualities: Collaboration, Critical Thinking, and Communication. Each unit leaves ample room for student extensions and mathematical forays into more elaborate pursuits. This unit was no different. Students studied the topics of Surface Area and Volume through a series of tasks, problems, and prompts–one of which ballooned into the subject of this blog series.
I had designed a set of lessons inspired by Timon Piccini. If you are in need of a similar inspiration read @MrPicc112’s blog post. Basically, students are asked to analyze the differing approaches to packaging employed by the soft drink powerhouses Coke and Pepsi. The first lesson was designed to calculate the differing surface areas of the two packaging options. The second was designed to increase the number of cans to 24, explore all possible arrangements, and find the most efficient packaging options. Both were fairly structured, with a low floor for all students to begin, but a high ceiling for the precocious few to pose deeper problems.
I began the first lesson with as much ambiguity as possible. I placed a box from each company on the front table and said five words:
“Do you have any questions?”
A short silence was followed by a response:
“Can I have a pop?”
A quick smile and revision of the question:
“Do you have any mathematical questions?”
“How many cans are there total?”
I can work with that. I asked the class for an answer. Very quickly a boy in the front told the class that there were 24. I asked him for an explanation. He said that there were 12 in each box, and 12 times two was twenty-four. I turned to the class and asked if they bought it. I got a obvious, and resounding, “yes”.
So I continued:
“How many boxes fit in the room?”
I write it on the board.
“How expensive was each brand?”
“Which box is bigger?”
I write all suggestions on the board, but dwell on this one. I ask them to explain themselves. What do they mean by “bigger”?
“Which takes up more space?”
“Which has the greater volume?”
“Which uses less cardboard?”
“Which has the smaller surface area?”
Notice, these students had been exposed to the mathematics of surface area and volume previously, so they were able to translate their human queries into mathematics language. The questions began to pour in as the class became more comfortable.
“What is the volume of each can?”
“How much space is there between the cans inside the boxes?”
“Which company is easier on the environment?”
“Is it cheaper to buy your pop in bottles?”
“How many other ways can the cans be arranged?”
Each suggestion was written on the board and explained fully to the class. When the class began to exhaust its thinking, I began by congratulating them on doing real mathematics. I explained how mathematics has evolved throughout the years on questions much like the ones they had just posed. It was now our job to answer these questions one by one.
I asked them to choose an easy one to begin. We looked at the volume of each can. The conversion between millilitres and cubic centimetres was looked up and explained. When the class deemed that was all the information they needed, we answered the question. Interestingly enough, this conversion came up when they began to answer their second question: “Which has the greater volume?”
We answered with relative ease. We even calculated the empty space with relative ease with the conversion. I asked why we didn’t just use the volume of a cylinder and got a few responses:
“The cans aren’t quite cylinders”
“That would require more measurement”
“Bigger chance to make a mistake”
All great thoughts, so we moved on to the surface area problem. After some initial clarification on the differences between surface area and volume, we had our answer. I used Timon’s “reveal” video as a validation.
That was supposed to be the end of day (and lesson) one. But a student asked a very interesting question:
“Can’t we do better?”
I tried to contain my excitement while soberly questioning him:
“How do you mean?”
“Like wouldn’t there be less waste if the cans were in a triangular prism?”
That question sparked a firestorm in the class. Students were running to the board and drawing schematics and stating their rationale. In fact, the students were so engaged in the new problem that I took a vote to either pursue this answer or continue with the next lesson. The result was unanimous: they wanted to pursue their own task.
A couple things need to be highlighted here:
The PBL framework of the class provided the freedom for students to pose problems. Their previous experiences with the various tasks aided in their development of this new one.
The previous tasks had built up skills in the students. This enabled them to pose expert problems. A solid foundational set of skills increased the inquiry power of the students. The attributes of a triangular prism were part of their mathematical arsenal; that enabled them to elicit the prism as a possible solution.
The class ended in a buzz. The next day I opened up the floor for brainstorming options. That process is the topic of the next post.