Thursday, April 26, 2012

Soft Drink Project Part 2: The Brainstorm

This post will make a lot more sense if you read the framework for the project in "Soft Drink Project Part 1: The Framework".

I left the classroom energized; I could not remember a time that I was more pleased with a lesson that I had taught. In fact, I wouldn't even call it teaching. I was observing. The process of brainstorming began organically. I had my doubts that it would continue the following Monday. Typically, students can't even remember where they sit after a weekend--let alone what task they ended on.

Monday came and, much to my astonishment, students came into the room, ran right to the board, and began to draw out configurations they were thinking of over the weekend. I must have looked surprised, because they gave me weird looks... looks that said:

"What, you think we don't do math on weekends?"

I took a minute to settle them down for attendance and then reiterated the task. As a class we brainstormed what the final goal should be. Just like the previous lesson, I opened up the floor for suggestions and began recording:

"The box should contain less cardboard"
"The box needs to be easy to ship"
"The box needs to be stackable"
"The box should waste less space"
"The box should look cool"
"The box should be original"

It was obvious that students had put thought into their goals before the class began. After some debate on the importance of each property, we settled on the fact that the box needs to improve sales. If that means taking an environmental angle, then so be it. If that means developing some kind of gimmick, then run with it. I insisted, however, that a total surface area be submitted. Just as the ground rules were laid out, a hand shot up. This was shocking for two reasons:
  1. The student who raised his hand NEVER raises his hand.
  2. I'm not sure I have ever been able to engage this student before this moment.
He asked:
"We are so worried about the boxes. What if the boxes are not the problem?"

Again, I contain my inner rejoicing and probe further...

"What do you mean by problem?"

The student went on to explain how the surface area and volume were too large, and maybe this was not the fault of the box. Maybe the cans were to blame. In his eyes, the fundamental problem was that the cans were cylinders and the boxes were rectangular prisms. He thought they should match.

I told him that he was well on his way to a zero waste model; his project is one of the most compelling of the group. he got right to work.

Students got into groups of 3-4 and began scribbling formations on paper. Some asked for cans, others needed ways to hold cans in place for measurements and estimations. One student came up with the idea to hold cans together with elastic bands. This idea was wildly popular and soon I was running to the office for more.

I assumed that triangles may figure prominently in our investigation, but that only scratched the surface. Triangles were followed by hexagons, diamonds, and trapezoids. One group designed a Tetris inspired model. Most of the models were physically built with cans on their tables.

As I circulated, I would move a can, and ask if I created a more efficient model. Usually I was quickly rebuffed for wrecking their model, but the justifications were solid. Each group presented an impromptu sales pitch for their idea. I once again played skeptic--and was amazed.
A cylindrical model takes shape
Triangular prism is constructed and analyzed
Another triangular prism gets edges curved after being traced onto paper
A diamond is designed, but soon re-arranged for stacking reasons
A 10-pack is designed. Students trying to save volume without curving their box edges
As the class progressed, I noticed one very quiet group was still standing at the board. I approached them to tell them to get to work, but before I could open my mouth, I overheard an argument:

"You can't sacrifice convenience for surface area"

"It can be the smallest package in the world, but people won't buy it if it won't fit in their fridge"

As I got closer I saw two diagrams written on the board. This group was too slow getting to the supplies table, and wasn't able to get pop cans. Instead of letting that stop them, they picked up some markers. One diagram was a trapezoidal model and the other was a parallelogram:
The trapezoid 10-pack (on the right)

The parallelogram model

I didn't interfere with the debate, but did clarify what each side was saying. One didn't want to do the trapezoid because the cans were still sitting directly on top of one another. That did not conserve space and made it too high to fit in a fridge. The other didn't think a 6-pack was enough and the triangular prism only held six.

As they continued to talk, I left. I listened to the new can design from the student who thought the shapes should match. He had designed a system where triangular prism cans would fit into a triangular prism box. I told him I was very pleased, but still wanted the cans to hold 355mL. That hurdle will be tackled in a subsequent blog post.

As I ventured my way back to the front group, they had developed a solution. They would have two 6-packs connected by a perforation. That way consumers could tear them apart and put one half into the fridge and one half in storage. Or one half in the garage and the other in the camper. They sold me on the versatility.

As class began to wind down, students got to the measurements of their designs. Diameters, heights, and radii were all jotted down.
Cylindrical design gets initial measurements
Triangular base is measured
It took a full hour of running around, but most groups finished the class with a rough idea of how they wanted to attack the problem. Up to this point, estimation and conjectures had been made, but few real measurements and calculations to back them up. The next challenge is that of accuracy. How can the formulae be applied to the box? How are the measurements going to be recorded from the cans? How much space needs to be added for tabs?

All these struggles will be documented in the next installment as students move through the design phase into fabrication.

After the brainstorm, one thing was crystal clear:

I was completely helpless. The students owned the class now--and they knew it!


Saturday, April 21, 2012

Soft Drink Project Part 1: The Framework

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:
"Anything else?"

"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:
  1. 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.
  2. 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.

Friday, April 6, 2012

Unexpected Lesson Extension

It is very hard to develop an active atmosphere in a math classroom--especially at the high school level. I believe there are two main reasons for this: 1) Students have been slowly trained throughout their schooling that a "good" math student is one that listens, absorbs, and repeats. 2) The content often reaches beyond what most teachers deem to be "constructable". Rather than fight with these two restraints, I began my implementation of Problem Based Learning in a class with manageable curriculum content filled with students who never learned to sit still in the first place.

As time progressed, the room slowly took the form that best suited the class. Skills were introduced with complex, engaging problems and worked on in a collaborative setting. As the basic skills with the content and various technologies were honed, I moved them into a more open-ended, project-based setting where they needed to mix initiative, critical thinking, and problem solving with their new found content skills to create rich mathematical experiences.

My role changed to that of a half-time tech support guy and half-time interested observer. In between battles with the unreliable wireless internet connections and antiquated laptops, I had chances to enter into discussions with students about their approaches to the tasks. More often than not, students find a familiar path and stick to it, but the best thing about the extra Independence of PBL is that students who are ready to expand have the freedom to pose new problems and search for solutions. Every so often a student will question the status quo, and it always leads to student empowerment.

There is nothing quite like the look on a student's face when they realize that they have discovered a possible caveat or extension in a project or prompt. They are far more likely to take on a challenge that they create; the project setting opens up room for student problem formation.

I have a student in my class that took a modified pathway until Grade 10. To be honest, I was weary of putting her in such an independent environment. She recently provided the greatest moment to date in one of my PBL focused classes.

We were working on a very structured series of problems entitled, "Centimetre Cube Exploration". It required the students to answer a variety of questions on surface area using ordinary centimetre cubes, GeoGebra, Microsoft Word, and Google SketchUp as their tools. A full handout can be downloaded here.

She had needed lots of support through the first few questions, and had landed on the question asking her to construct a 10-cube structure that had the maximum surface area. We talked about what caused the surface area to shrink. We developed the idea of overlapping sides, and how fewer overlaps meant a larger surface area. This was all great learning; this was the learning I expected. The great thing about the new ecology of mathematics I have developed is that I don't always get what I expect.

Shortly thereafter, the student approached me with a 2-cube design in hand. It was simply two cubes stacked on top of one-another. Sensing another mundane fix, I rolled my chair over and she asked a question that made me look like a fool.

"Mr. Banting, you said that this was the maximum surface area two connected cubes could have, right?"

She then showed me the cubes which looked like this:
"... but wouldn't there be more surface area if we did this?"

She then simply twisted the cubes to look like this:
The look on my face must have been priceless, because she started to laugh. The scene went on for quite a while. Slowly but surely, every student had approached the desk to see what was up. The student beamed as she explained how twisting the cubes made eight additional triangles to measure. After about 10 minutes, I re-routed her efforts back to her work and began to calculate the additional area myself.

I modeled the two images above on the same software the students were using, and created this cross-section using GeoGebra:

From there I used linear equations, isosceles triangles, regular polygons, area geometry, and elementary trigonometry to find the exact area of the eight new pieces of surface area (coloured purpely-blue above). I used GeoGebra to verify the results.

I used the epiphany in two ways. The students who were nearing completion and needed enrichment were commissioned to find the missing area. Those students who were still struggling through the original task were shown the conclusions when they were complete. The experience meant something different for every student. Some saw it as more work, some saw it as a chance to solve a difficult problem, and some saw it as a launching pad into more questions.

For me, it was a chance to feel like a student again. Providing a space that encourages students to be creative always ends up in unsuspected surprises, but some of my richest moments as a teacher are those in which I don't feel like the teacher at all.