For our last unit of Foundations of Mathematics 10, I decided to tackle something that scared me: a project. The outcomes – determining surface area and volume – were well-suited to practical application, and I had been eyeing Nat Banting‘s soft drink project for a while now; moreover, I felt that a strong collaborative culture was finally taking root among this group of students, and that they were prepared to work on a larger, more complex problem together. (Actually, I am sure that they were prepared for this much earlier – I am still in the process of learning to trust my students, and this was a major test of the water for me). In short, it was time to take the plunge. This post is the first in a series of three detailing the experience. (Click here for Part 2.)

I decided that the first few lessons, which would serve as a launchpad into the project, would be centered around the following key ideas:

*Minimizing surface area and empty space can make for more environmentally-friendly packaging.**We can reduce a three-dimensional object to a two-dimensional object by creating a net.**We can find the surface area of complex shapes by breaking them down into simpler shapes we know.**Environmental impact is not the only factor that influences design of packaging.*

I also needed to keep in mind that the curriculum requires students to determine the volume and surface area of, among other objects, spheres, cones, and pyramids, in addition to cylinders and right prisms. While the latter two are accessible through analysis of pop boxes, the former were less likely to naturally come up, even in the design phase.

As such, I needed to front-load these particular curricular concepts at the beginning of the unit. This led me on a hunt for suitable packaging to analyze. Ping-pong balls checked off the spheres and cylinders boxes:

Cones and pyramids were more difficult to find. Luckily, it was just after Easter, and I was able to snag these boxes of chocolates at a discount:

In fact, Lindt offers quite a wide variety of mathematically interesting packages, so I also picked up a hexagonal prism, parallelepiped, and a good-old rectangular prism:

(Yes, this ended up being on the expensive side, but I bought these with the intention to keep them for years to come.)

**Day 1**

The first day of the unit was devoted to activating prior knowledge about measurement and surface area and eliciting the key ideas listed above. I began the lesson by pulling out two of the Lindt boxes (hexagonal prism and parallelepiped) and asking students: “What questions can we ask about these?”

As you can imagine, the first question was: “Can I have some chocolate?” But then, other students chimed in:

- “How many chocolates are in each one?” We opened them up and counted.
- “What’s the weight of each chocolate?” The weight was given on the box, so we were able to determine this based on the number of chocolates in each box.
- “How much do they cost? What’s the best deal?” I didn’t have the price, but the students conjectured, from experience, that the bigger box was more expensive but a better deal.
- “What’s the surface area?” “What’s the volume?” BINGO.

After some time, I chimed in with my own question. “Which one’s more environmentally friendly? How can we find out?” The students recognized the connection to the previous two questions, and decided to start with surface area. I followed up with, “What do you predict?” and “What do you need to know?”

Dimensions were measured and noted on the board. Then, students got to work (as usual, they were randomly grouped into groups of 3 and were working on large whiteboards at their tables). Some groups were prompted to draw a net; others were challenged to go further by being more precise (“What about the flaps?” “What about this open bit?”) When we came together after some time to compare results, a great discussion emerged regarding the area of the hexagonal parts of the large box.

As you’ll notice on the (messy) board above, all of the groups decided to find the area of the hexagonal part by dividing it into a rectangle and two triangles. However, while most groups determined the area of these triangles by finding their height using the Pythagorean theorem and the formula for the area of a triangle (giving an area of 10.4cm^{2 }for two triangles), one group reasoned that when you put two of the triangles together, you get a square. Therefore, you can simply multiply the two sides together, which gives an area of 16 cm^{2}. This temporarily stumped the class – after all, the sides *are* all the same length! Why would the two strategies give different answers?

Finally, someone suggested: “The angles aren’t the same.” It’s a parallelogram! “How can we check?” After some discussion, we reasoned that we could use the Pythagorean theorem to check whether the squares of the sides were equal to the square of the diagonal, and determined that they were not. Great error analysis! Proof that we can sometimes learn more from making mistakes than from doing everything right the first time around.

The lesson was coming to a close, so I engaged the students in a discussion about efficiency and effectiveness.

- “Which box is more environmentally friendly?” Based on the number of cardboard it uses per chocolate, the hexagonal prism box is (much) more efficient. But are there other factors to take into consideration?
- “Which box is more effective? What does this mean?” This led to a discussion about other factors that influence packaging design, and in particular about the balance between aesthetics and cost-efficiency. Other factors that were brought up were price, ease of shipping and storage (we noted that both packages would tile, but would leave empty space around the edges when stacked in a large box), price, uniqueness, and more.

I ended the lesson by alluding to a problem they would all soon be tackling: “How might these boxes be made more efficient?” The conversation was, unfortunately, soon brought to an end by the bell.

**Days 2-3**

The next two days of the unit proceeded in a similar fashion, so I will spare the details. On Day 2, students compared the relative efficiency and effectiveness of the ping-pong boxes. During this lesson, we also returned to the question of volume, as the students reasoned that they could also compare the efficiency of the boxes by determining the amount of empty/wasted space in the boxes, which would involve determining the volume of the contents and the volume of the box. (We compared boxes with different numbers of items by computing the *percentage* of wasted space in each.) We noted with interest that packages that used less material per item did not necessarily waste less space.

On Day 3, we returned to the chocolate boxes, with a few new ones thrown into the mix; students were challenged to determine which one, among all, was most efficient in terms of the least amount of material used and the least amount of wasted space. I introduced new formulas as the need arose, and eventually offered students a formula sheet to refer to as needed.

Finally, as foreshadowing/practice for the pop box design task, I asked students to write a brief pitch about what they viewed as the most effective box, focusing on the factors (e.g., efficiency, aesthetics, price) that they felt were most important. (In retrospect, I realize should have given students fewer boxes to compare – three instead of four -, or offered students the opportunity to choose the level of difficulty by selecting three boxes among the ones available. There was very little time for discussion and not enough to have students to make their pitches to the class.)

**Day 4**

At last: pop boxes. Actually, before this, students worked on an entrance slip that involved finding the surface area and amount of wasted space in a new chocolate box:

I was happy to observe that students had almost no difficulty with this task; little discussion was needed beyond the sharing of results. It was time to introduce the pop boxes.

By this point, students could anticipate the question: “Which one is more efficient?” After a brief debate about Pepsi *vs* Coke, we started measuring. I encouraged students to be more precise, taking into account the flaps on the boxes. Once the students were satisfied that they had the information they needed, including the height and circumference of the cans, they quickly got to work on determining which one uses less cardboard and which one wastes less space.

When a group finished early – or abandoned the task midway because they found it too easy – I posed the question that I had been waiting to ask all week: “Can you design a better pop box?” They took the bait. Markers were picked up again immediately as the students began brainstorming. (The first idea that emerged was a meter-long “Pepsi Tube,” which had very little empty space but was also, as the group admitted, rather impractical.)

Finally, after comparing results as a group and ironing out any disagreements, I posed the question again, but now to all of the students: “We can continue argue about the efficiency and effectiveness of these particular boxes… But can you design a *better* one?” A murmur arose. I offered more details about the task (the following is taken from the project overview that was given to the students the following day, and is heavily based on Nat’s original writeup):

**Your task is to design a more effective pop box.** You will validate your design with calculations of surface area and volume. **Your box does not need to contain 12 cans**: In fact, as designers, you can make any design decisions that you want as long as you can justify why the design is more effective.

*Your group will brainstorm, take precise measurements, make a net, construct a prototype, and give a sales pitch about your design.*

**Remember that commerce relies on more than mathematical efficiency**. If you come up with a sales pitch to sell more pop, design your box accordingly.

More murmurs as ideas started to emerge. I invited students to wisely choose their group members (3 students per group), and handed out group contracts for them to read together and sign (see below). (I should note that I was a little wary about students choosing their own groups, as I have relied on random grouping throughout the semester; however, I decided to trust their judgment.)

The students had only a few minutes to brainstorm together before the bell rang, but as one of the groups was walking out of the classroom, I heard them saying excitedly: “We already have about 7 ideas to choose from!”

I couldn’t wait for next week.

In upcoming posts, I will describe the brainstorming, calculating, and design phases, as well as the trade show that put a bow on the unit and gave students the opportunity to pitch their designs to an audience of teachers and other students. As a teaser, behold the following rap for the Pepsi Party Puck (“or Coke Celebration Cylinder,” depending on who signs us”):

For now, I leave you with PDFs of the project overview and group contracts which, I stress, are heavily based on Nat Banting’s work. His project binders, which offer resources not only for specific projects but also a framework for developing your own, were incredibly helpful as I organized and facilitated this project. It’s a fantastic set of resources for anyone starting out with project-based learning, and you should definitely check it out.

Pop Box Project – **Project overview** (English)

Pop Box Project – **Project overview** (French)

Pop Box Project – **Group contract** (English)

Pop Box Project – **Group contract** (French)

*Update: ***Click here for Part 2**.