In his instant-classic lecture, “Mathematics for Human Flourishing” (2017), Francis Su lists play as the first fundamental human desire that mathematics can help cultivate: “Everyone can play. Everyone enjoys play… [And] mathematics makes the mind its playground.” Looking back, the possibility of extending my time in the sandbox is probably what attracted me to pursuing a math degree, and later, to becoming a math teacher. Years later, I’m playing the game with my own students, but as I found out quickly, teaching play is hard work. As Su writes, “It’s actually harder than lecturing, because you have to be ready for almost anything to happen in the classroom.” Brent Davis (1996, p. 222) echoes this sentiment:

“[The teacher] is assigned the task of presenting possibilities and, through attending to students’ responses to these possibilities, opening spaces for play. Such play-fullness is only feasible when one allows for departure from the anticipated … fluidity in the structured … and uncertainty in the known.”

I try hard to open such spaces for my students, choosing tasks that yield themselves to multiple solution paths, encouraging students to pose problems, and exercising a certain freedom in lesson structure that allows us to follow interesting paths that we stumble upon along the way. Of course, there are limits to this freedom, given that we must work within the structure of the curriculum and time constraints. But while structure and constraints may seem antithesis to play, both Su and Davis agree that on the contrary, structure can support playfulness:

There is usually some structure: even babies know ‘peekaboo’ follows a certain pattern, but there is lots of freedom within that structure.

– Su, 2017Play is not so much an activity as it is an acceptance of uncertainty and a willingness to move … But it is not an abandonment of our quest for structure, order, pattern, and comprehensibility. Quite the opposite, these are the ends of play.

– Davis, 1996, p. 222

Play, then, isn’t about purposelessness or disorder; play is having freedom of movement within a framework of constraints. And sometimes, opportunities for play can arise unexpectedly, even within the most structured of tasks.

These days, we’re playing radicals.

After spending a few days working on evaluating, estimating, and ordering radical numbers, I taught my students how to play Radical War (credit to Andrew Stadel for sharing the idea). The game is played in partners using a deck that includes a variety of rational and irrational numbers, with values ranging from 1 to 5 (available here). The deck is split between the players. On each turn, both players turn up a card; whoever has the higher card takes both cards into his or her hand. (Special rules apply when both players turn up cards of the same value; full rules available here.) The game is played until one player has won the deck.

As I circulated and listened in on the conversations, I witnessed some great reasoning. The game offered a great, low-risk context to practice estimating and ordering rational and irrational numbers, one that my students seemed to really enjoy. Somewhat ironic, though, that the game itself does not seem to offer a lot of space for play, as Su or Davis would define it. There is plenty of structure given by the rules, but where’s the freedom for movement? With the constraints that are provided, there’s no room for strategy, for deliberating and choosing between possible actions; only luck (or cheating) will determine the winner. Below the surface, then, not much different from a set of drills, with the added twist of competition.

Except that, as anyone who has ever played Monopoly knows, sometimes play is about throwing out the rule book and inventing your own game.

As I circulated, I came upon a group that had abandoned Radical War after a quick victory by one of the players. Each now held their cards fanned out in front of them:

When I asked them if they were going to start round two of Radical War to play best out of three, they sheepishly admitted: “Well… we kind of invented our own game.” And so they told me about Radical Cheat. It’s based on a well-known card game called Cheat (or Bull****, in some circles):

Cards are distributed evenly. On the first turn, a player puts down an Ace card and announces, “one Ace.” If the player does not have an Ace card or would like to get rid of more than one card, he may bluff and play non-Ace cards. If any of the other players do not believe the announcement, they can call “Cheat!”. The person who played the cards must turn them over and show the challenger whether he is bluffing or not. A player who is caught bluffing must pick up the entire pile and add it to his hand. If a challenged player is not bluffing, then the challenger must pick up the discard pile. When the rank to play reaches Kings, it then goes back to Aces and the numbers start again.

Of course, because the cards are non-standard, composed of rational and irrational numbers ranging from 1 to 5, the rules needed to be modified. First and most obviously, the students decided that play would start with 1s rather than Aces and end at 5s rather than Kings. Then, and less obviously, to account for the non-integer numbers in the deck, the students decided that the values on the cards would be rounded to the nearest integer. This way, 1.2 would count as 1, √3 would count as 2, √8 would count as 3, and so on. Simple, and brilliant in its simplicity.

I observed the game play for a few minutes. The element of competition was still present, and students needed to use just as much reasoning as they did in the previous game, because they were required to estimate the values of their cards in order to round them to the nearest integer. In fact, the stakes for accurate estimates were even higher during Radical Cheat, because players couldn’t rely on their partner to estimate the value of their cards for them—if they made an error, they risked taking the entire discard pile.

I was impressed. Instead of abandoning the game and putting an end to their reasoning about radicals once they got tired of the task, the students demonstrated their ability to play with mathematics, in the sense that Davis describes: “a willingness to move,” without abandoning structure, order, pattern, and comprehensibility. While discarding some of the original constraints (the rules for Radical War), they used the tools they had at their disposal (the cards) to create a new game to play. Note that this form of play is different from playing the game of school. In the school game, which tends to be about learning how to stay within explicit and implicit boundaries, rules are often rigid and not up for negotiation.

But isn’t play what mathematics is all about? As Su (2017) writes, in mathematics,

We play with patterns, and within the structure of certain axioms, we exercise freedom in exploring their consequences, joyful at any truths we find.

Not to mention that some of the most interesting advancements in mathematics came about when mathematicians pushed against boundaries and discarded seemingly inflexible “rules” to create new games to play.

I used to think that this kind of playful, high-level mathematical activity was next to impossible to attain within the rigid constraints of curriculum. These days, I have a new perspective. As this experience reminded me, playfulness in mathematics is not about abandoning constraints, which can actually serve to spark creative thought by offering something to think about in the first place. Rather, playfulness involves knowing which rules to keep, and which rules can be broken—say, the rule that exponents must only be positive whole numbers, or that a set of cards must only be used to play Radical War—and giving students the space to break them, and to explore the consequences.

“Alright, so that’s the game! Let’s play.”

*References*

Davis, B. (1996). *Teaching mathematics: Toward a sound alternative*. New York, NY: Garland.

Su, F. (2017). Mathematics for human flourishing [Blog post]. Retrieved from mathyawp.wordpress.com/2017/01/08/mathematics-for-human-flourishing/. Audio recording available at youtu.be/xEtDvc1SWm8

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To backtrack a little, a key principle underpinning the discussions in the book is that of *performance*:

In all social encounters we play roles that we desire to perform competently. Embarrassment typically involves discrediting information that undermines our performances. (Newkirk, 2017, p. 9)

A second underpinning principle is what Newkirk calls the *awkwardness principle*:

Any act of learning requires us to suspend a natural tendency to want to appear fully competent. We need to accept the fact that we will be awkward, that our first attempts at a new skill will, at best, be only partial successes. Moreover we need to allow this awkwardness to be viewed by some mentor who can offer feedback as we open ourselves up for instruction. (p. 10)

It is already easy to see how these principles can work against one another. While, in Newkirk’s words, “there is a great deal of happy talk these days about welcoming failure, the need for failure, and learning from failure, even the gift of failure—this tribute comes primarily from people who generally succeed” (p. 11). For learners who haven’t yet experienced success, the fear of misperforming the roles that underpin our very sense of self can seriously hinder or arrest the learning process.

As a result of this tension, even the seemingly simple act of asking for help can become fraught with anxiety or avoided entirely. Admittedly, I have always been a student who has sought help unabashedly, from primary school through university and beyond. Making this a habit early on, and without having to be coached to do so, I often asked my teachers and professors to clarify ideas during class or after the lecture; I made note of contact information and, as needed, sought advice on making progress with challenging problems during office hours or over email. In university, I worked on assignments (“lived,” we used to say) in the math help centre and formed study groups with friends. Somehow, my desire to learn almost always outweighed any risk of “looking dumb” (or, equally offensive in some circles, looking like a “suck-up”) and was rewarded and reinforced by understanding (and, yes, good grades). And finally, when I became a high school teacher, I assumed that my students would naturally and without delay ask for help when they needed it, too. If they didn’t come at lunch when I asked them to “*just* see me if you have any questions, my door is always open,” it was because they were doing fine… or lazy, or unmotivated, or otherwise beyond my help. A classic case of false consensus.

Of course, things are almost always more complicated than we first assume when it comes to human behaviour, and Newkirk makes a compelling list of reasons why students who *are* interested in learning, who are, in fact, *not* lazy *nor* unmotivated, may fail to ask for help. Newkirk’s students are in college, but many of the reasons may apply just as well to middle-school and high-school learners:

Seeking help in this way makes the student appear too invested in school work. It feels like grade grubbing, even bordering on cheating. You need to keep your distance.

If suggestions for improvements are offered, this will create an expectation or obligation for the student. That is, the teacher may look to see if the advice has been taken—and be disappointed if it is not.

Seeking help is perceived as an admission of inadequacy or failure, which would be better to hide. Students want to think of themselves as good students, competent students, and asking for help would undermine that identity. So they vote with their feet—better to risk a poor grade than seek help. […]

No matter how open the professor’s [or teacher’s] door, there is often the perception of interruption or intrusion—the student is disrupting important work.

The issue of social class may also play into this reluctance. A professor [and, to some extent, a teacher] is seen as a member of an elite group […]. And students may feel intimidated by what they perceive as a cultural divide.

Students don’t know how to seek help. Though faculty might view help seeking as a self-evident skill, it clearly is not. It requires the capacity to define a problem that you want help on—and often the student has only a global feeling of confusion. In effect, seeking help can compound the embarrassment of the student. (1) It can be embarrassing to seek help in the first place, and (2) it can be embarrassing to have difficulty describing the help you want. (p. 56)

Newkirk goes on to explain that for some students, the difficulty of help seeking is compounded by racism and stereotype threat. He shares the story of Jonathan Gonzalez, a black student who found himself in the following predicament in college:

Jonathan went to a tough high school in the Bronx, but earned a scholarship to Wheaton College where he slowly sank from view and, like many students in his situation, failed to graduate. […] But it wasn’t purely or even primarily a lack of preparation that did him in; it was a sense of embarrassment, a feeling that he stood out, did not belong. […] If he reaches out for help […] he makes himself “vulnerable”; […] he confirms the stereotype that he doesn’t really belong, that he’s not ready. […] And if he fails to get help, and does poorly, he also confirms the stereotype. It’s a catch-22. (pp. 58-59)

For other students still, seeking help “aligns the learner with the institution of school—it announces that doing well matters to the student” (p. 59). For some students, such public displays of caring can be socially and even physically threatening.

And suddenly, I feel—yes—embarrassed to have thought so little of my students, to have assumed that they could all have “just” come for help when they needed it, to not have realized that asking for help is highly complicated by valid social and personal concerns, nor that help seeking is a skill like any other. A learnable and teachable one, Newkirk assures. Aside from explicit coaching, he suggests that one way to eliminate help-avoidance is to simply make it expected:

We apply this principle in health care when we schedule annual physicals or semiannual visits to the dentist. We make it normal. I have much different luck if I schedule regular conferences with students—this practice suggests that getting help is not something you do in a time of crisis, or when you have a “problem.” (p. 72-73)

Quoting Peter Johnston, Newkirk also encourages the use of prompts like, “Who came up with an interesting problem?” “Failure or disappointment is less scary,” suggests Newkirk, “if we can name it, share it, and see it as a normal and expected feature of thinking and working.” (p. 73)

A counteracting tension that I recognize for teachers, and maybe especially for teachers of mathematics, is that we are increasingly encouraged to move away from the image of “helpers” and to promote “productive struggle,” to encourage persistence in times of difficulty, to advocate using one’s own mental and physical resources to solve a problem and resisting the temptation to immediately ask for help. Of course, this is not in conflict with Newkirk’s message. We *can* promote productive struggle but still recognize that there are many situations where perseverance alone will not lead to success, where struggle ceases to be productive—for example, when students don’t know what to do when they are stuck, or consistently have trouble identifying relevant information in problem situations. But I do wonder if simplistic directives by teachers to “persevere” and “persist” complicate matters even further for students who are already wary of seeking help. How do I know when my struggle is becoming unproductive? If I ask for help, am I revealing a lack of perseverance? Does asking for help mean giving up, a personal failure? Unless our championing of “productive struggle” is accompanied by coaching on when and how one *should* seek help, as well as by work on normalizing help seeking, maybe we are doing more harm than good.

All this to say: In teaching and learning (and maybe even in spiritual matters, too), the idea that if you just “ask, and it will be given to you” is much less straightforward than I had assumed. Newkirk’s *Embarrassment *leaves me humbled, challenged, and eager to keep exploring the emotional underside of learning (and teaching, too).

**In this class, I felt like I, my ideas, and my questions mattered.**(scale of 1-5, from Never to Always)**In this class, I felt challenged.**(scale of 1-5, from Never to Always)**In this class, I often felt stressed out.**(scale of 1-5, from Never to Always)**The best aspect of this course was…****The worst aspect of this course was…****One surprising or interesting thing that I learned was…****One thing I’d like to tell Mlle V, in all honesty, is…**

And here are some highlights from my Grade 9 class.

**The good**

- Many students commented on the organization of the class and the “teaching style,” with most comments being positive. Students enjoyed that we often worked on problems in groups and on whiteboards, practiced through games or other interactive activities, and didn’t see much of the textbook in class. This was heartening to hear, because many of these same students were very concerned at the beginning of the semester that we weren’t relying on the textbook and were openly distrustful of the learning activities (“Oh god, we’re going to fail the final,” one student exclaimed during the second class). The shift in routine was unexpected and maybe even scary for some students, who seemed to need something tangible to hold on to to prove that they were learning something. I’m glad that most of them came to find value in our learning activities.
- At the beginning of our unit on polynomials, we spent a few days exploring binary and other base arithmetic. Several students pointed to this as the most interesting thing they learned during the semester. I had enjoyed these lessons too, but I am reconsidering their value as an introduction to polynomials. As engaging as these classes were, the connection was tenuous to say the least; next year, I will spend much more time building on students’ understanding of the base 10 system and its representations (e.g., base 10 blocks). Seeing how much students enjoyed exploring arithmetic in different bases, though, I’ll have to find another place for them in the semester, maybe as an extension of our unit on rational numbers.
- “Thank you for helping me with my homework, I couldn’t have done it without you!” Appreciation doesn’t fuel our work as teachers, but it’s still nice to hear once in a while that your efforts are valued by your students.
- “You are amazing and I am very happy that you were my teacher.” Alright, now I’m just showing off.
- “I talked too much and disrupted the class.” “Thank you for helping me and I’m sorry for being distracting.” The kind of notes that remind me I that love these kids… yes,
*all*of them, even the one who took to crawling under the desks last week while others were trying to focus on reviewing for the final exam.

**The bad**

- A few students commented on my relative lenience with regard to classroom behaviour, finding it difficult at times to concentrate because of students who–let’s say things as they are–wouldn’t shut up. I guess many teachers simply show students the door when they’re being disruptive, which my students sometimes went so far as to suggest that I do; not wanting my students to miss out on learning, though, I tried my best to redirect them towards their work. This isn’t to say that I always succeeded, and did ask students to leave for a period of time several times during the semester. Of course, I’m aware that students often disengage from a task not because they don’t want to learn or because they want to distract others, but because they don’t know how to get a foothold into the problem or how to use the resources within their group (i.e., themselves and the other students). I am still working on the larger goal of helping students become persistent problem solvers, and I can’t say that I have consistently done this well. Next semester, I am going to try to be more explicit in teaching good problem solving practices, including what to do when you get stuck.
- A few students suggested that more time should be spent in the classroom working on problems from the textbook. I take this as a call for more individual work and practice in the classroom, which is definitely something to take into account. This is something I will try to consistently try to make time for next semester as I work on developing instructional routines in my classroom.

**The ugly**

At the beginning of this class period, the last one before the final exam, I had students complete an entrance slip on solving inequalities. The first question asked:

I was discouraged to find many, if not most of my students unable to solve this problem, even if they could now virtually solve inequalities in their sleep. I tried to guide them by asking, “What does it mean to solve an inequality?” and “When you solve an inequality, what are you looking for?” To which many responded, “isolate the variable” and “*w,*” and not something along the lines of “find all of the values of the variable that make the inequality true.” Although I felt that we had laid the groundwork for making sense of solving linear equations by working on a variety of balance scale problems early on during the unit, representing them with inequality statements, and connecting the process of determining the unknown value to that of solving inequalities, clearly the connection was lost on or discarded by many of my students once they latched on to key aspects of the procedure (“isolate the variable,” “what you do to one side, you do to the other,” etc.). Task propensity seems to be a relevant idea here. Evidently, they had mastered the procedure, but lost the *meaning* of the process along the way. Interpreting solutions and connecting these to big mathematical ideas is something we clearly need to focus more on going forward.

A humbling experience, and a lot to reflect on as I prepare for the next semester.

]]>As advertised, the survey (click here if you’d like to respond) consisted of one question: “In your opinion, what’s the perfect banana ripeness? (For eating, NOT for making banana bread.)” The options were “More green than 1,” “1,” “2,” “3,” … and so on until “15,” and “More brown than 15.”

Here’s what I learned:

**Y’ALL LOVE TALKING ABOUT BANANAS.**

You wrestled with the choices, shamed your fellow MTBoS-ers for their preferences, and pondered why some might lean towards one end of the scale or the other. So far, I’ve received 1609 responses… and counting. Clearly, I hit a strong nerve.

Now, some of you may have assumed that I was collecting the data for a brilliant lesson in my classroom… well, prepare to be disappointed (although I do offer some suggestions for lessons below). In fact, the question emerged in my Grade 9 classroom when I saw a student eating a very green banana. Given that my Banana Number is 7 (I expect this metric to become no less iconic than the Erdős number) and that I find any banana less ripe than this to be basically inedible, I was perplexed. I brought up the following image on Google and did a quick poll:

Like you, my students were eager to debate the issue, which made me think it might offer an opportunity to take a brief detour into some data analysis. That night, I found the 1-15 photo (by Rebecca Wright; click here for original; I’m not sure who added the numbers) and made the poll, which I tweeted out and also had my students take in class. We briefly discussed what they expected the results to look like, and we moved on to that day’s lesson on fraction multiplication, as I was hoping to get a few more responses before analyzing them with my students. I expected a few dozen, maybe a hundred responses… so it would be an understatement to say that I was bowled over by the overwhelming reaction.

So, what now?

First, here are the (preliminary, given that responses are still rolling in) results (click here to see the data in Google sheets):

The mean ripeness preference in this sample is 7.34 and the standard deviation is 1.96. This means that about 74.8% (mean +/- 1 standard deviation) of people prefer bananas that are between 6 and 9 on the ripeness scale, and about 95.8% (mean +/- 2 standard deviations) prefer bananas between 4 and 11 on the ripeness scale.

What I find interesting is that the frequency decreases on both sides of the mean, but begins to tick up again both at 1 and 14. This gives the distribution “heavy tails,” with more extreme values than a normal distribution would predict. (For example, the normal distribution would predict that essentially 0 people in the same sample would choose 15 as their most preferred banana to eat, as compared to 14 in the actual sample.) I wonder if this may be because there are no values past 15 or lower than 1, so the frequencies “pool” there (i.e., if the photo included bananas that were even more ripe than 15 and even more green than 1, this would distribute the frequencies currently at 1 and 15 to neighbouring numbers, because these groups would now be able to fine-tune their preference).

More importantly: What might you do with this in your classroom?

This is where I hope you will chime in with your ideas. Like I said, I don’t expect to spend a lot of time on this with my own students, because it’s outside the scope of our Grade 9 curriculum. But here are some directions I might take it if I had the time and/or if I was lucky enough to teach statistics:

- Have students predict the shape of the distribution (after having taken the survey themselves). After analyzing the results and noting the main features of the distribution (how far you take this depends on your students and their background knowledge), ask students to design and carry out a survey where they think they might get a similar distribution. For example, Nat suggested that a similar distribution might be observed for milk & coffee preferences, with extreme left being black – no milk – and extreme right being milk – no coffee – (although I think this might be skewed towards the left if polling adults, and right if polling teenagers). (
*N.b. for SK readers: This may be an appropriate task for outcome SP9.2 – Demonstrate an understanding of the collection, display, and analysis of data through a project, or an introductory task for outcome FM20.6 – Demonstrate an understanding of normal distribution.*) - Connect to the topic of probability by asking students to make estimates for various samples. For example, in a sample of 500 people, how many would you expect to have a banana number of 5? 10?
- Have students refine the experiment to consider the effect of different factors on banana ripeness preference, such as age. (Others suggested nationality, but that would be more difficult to test.) Are there other ways to improve the experiment? (Of this, I am sure – I spent not more than 3 minutes designing the survey, certainly not expecting it to go as far as it did.)
- As Marla Goldberg suggested, reverse the task by presenting the results first and asking students, “What’s Going on in This Graph”? For instance, you might show students the following sequence, asking them to reflect after the first two and predict / explain the axis labels:

- Statistics students can test the data for normality. As I noted earlier, the distribution has heavy tails, and although the normal distribution predicts that 99.7% of the data would fall within 3 standard deviations of the mean, only 97.3% of the data does so in this sample (the difference seems small, but the data sample is quite large). I haven’t performed the necessary analysis, so I leave it to you. Might the distribution be multimodal?

I’d love to hear your ideas. (Here’s the link to the data again.)

Until then…

What’s your toast number?

]]>As a way of reviewing integer operations, we spent some time playing Integer Bingo, which I wrote about here. A task involving Fraction Talks images served as a re-introduction to fractions, and was followed by a clothesline number line task to elicit reasoning about comparing and ordering fractions. When we moved on to fraction operations, I used Nat’s (Min + max)imize structure (see also 16 Boxes) and the Open Middle idea to design (I use the term loosely, because the general idea is not mine) the following task:

Students draw four boxes and an operation to create a fraction expression (determined prior to the game), as above (image credit to Open Middle). Before I roll four dice, we decide on whether we are aiming for the largest sum, the smallest sum, or the sum that is closest to 0. After each roll, students must make a decision about where to put the number, and cannot change their minds once the number has been written. (With my students, I use two dice that have the numbers 1, 2, 3, -4, -5, -6 and two that have the numbers -1, -2, -3, 4, 5, 6.) After four rolls, they determine the sum. I ask volunteers to share their expressions and their thinking, and we work together to determine the answer that is largest/smallest/closest to 0. Repeat, changing the goal and/or the operation after one or several rounds.

Although I had planned the task to last only about 20 minutes, it ended up sustaining productive discussion about fraction operations for an entire class period, with both of my Grade 9 classes (and for some time the next day, too). Both misconceptions and strong reasoning emerged, especially when the task was modified so that a subtraction sign separated the fractions and when a box was added before the first fraction to create a mixed number. The next day, I extended the task by rolling the 4 dice *simultaneously* and asking students to find (and provide justification for) the biggest/smallest/closest to 0 sum or difference (again, determined prior to rolling) that was possible by forming two fractions with the 4 numbers.

Because the previous activity took longer than I had expected, I didn’t get a chance to try out a new task that I had planned, also for reviewing fraction operations, inspired both by (Min + Max)imize and… **The Price is Right**. I am sharing it both to document it for the future and in the hopes that you will try it with your own students and report back with feedback!

Again, at each round, four dice are rolled (and again, I use the integer dice described above), giving students four numbers to create two fractions with (modify the number of dice and the numbers on the dice as desired). Each student uses these dice to make two fractions, which they secretly add; mini-whiteboards would work well here, so that students can hold them up for all to see at the end. Then, a final die is rolled (say, with 0,1,1,1,2,2 on the sides). This die is the target number (“price”). The students now compare their sums. Whoever is closest to the target number, *without going over*, is the winner. Points might be allocated based on whether or not the target number is reached exactly (e.g., 10 points for hitting the target number exactly; 5 points for being the closest, without going over).

I like the idea because it invites students to reason about fractions in a way that goes beyond drill (“& kill”), because they must think not only about how to perform the operations involved, but also about how to form fractions that will meet the constraints of the game (the sum must be less than or equal to 2) and about the *risk* of forming certain fractions. E.g., a larger sum is more likely to win if 2 is called, but this outcome is less likely than 1. 0 is an even less likely target number, but a student who chooses the strategy of always making a fraction below 0 (if possible) is guaranteed to never overshoot the target. What other strategies might emerge? And do they change as the game goes on? I anticipate the task to elicit interesting discussions about risk and probability, in addition to fraction operations. Once students have played a few rounds, the game could be played in partners.

I’m not sure that I’ll have time to try out this task during the current unit, but I do hope to try it sometime this semester. I’d love to hear about it, along with any feedback or extensions, if you give it a go with your own students! Finally, if you’re interested in more great fraction tasks for students, you will find a wealth of ideas here. (And have I mentioned www.fractiontalks.com?)

]]>I’d like to share one task that, I think, had something to offer to all of my students: Integer Bingo, which is based on a task developed by and discussed in Serradó (2016). I chose this task for several reasons:

- It gives students an opportunity to review integer operations, and me an opportunity to assess their understanding and note any misconceptions in this area.
- The review is embedded within a task that challenges students to wrestle with the less-familiar concepts of randomness, relative frequency, theoretical probability, and elementary outcomes.
- It’s easily adaptable (to suit a variety of grade levels and to target a variety of concepts) and extendible. Not only does it offer practice with a targeted concept, it can serve as a launch into the study of probability.
*Who doesn’t like bingo?*

Here’s how the task was enacted in my class.

*Preparation:* In preparation for the task, I marked two sets (one yellow and one white) of nine ping-pong balls with the integers from -4 to +4 and placed each set into a paper bag. (A random number generator such as this one can also be used, but I elected to use physical random generators for a particular reason, which I’ll discuss below). I also generated a 4×4 bingo card for each student in my class using Excel, with each cell containing a number that is the sum of two random integers between -4 and +4. Numbers may be repeated. (Click here to download a set of 38 unique cards.)

*Game play – Round 1:* Unlike in most bingo games, where the goal is to get a line, students’ goal in Integer Bingo is to get a “blackout” (to mark off all of the numbers on their card). During the game, I repeatedly draw, with replacement, a number from each bag. If the sum of those numbers is on a student’s card, they can cross it off; if it appears more than once, they cross it off only once per draw. Draw, add, repeat, until someone crosses off all of the numbers on their card and calls “bingo!”

The review, obviously, is in determining the sums (another skill can easily be targeted by changing the operation, or changing the numbers – of course, this changes the possibilities for numbers on the cards). At several points during the first round, I asked students to explain how they reasoned through the computation. The numbers are small, but the task still allowed misconceptions to surface – e.g., “Two negatives make a positive, right?” Important conversations to have at the start of the year.

*Design: *After the first round and the first “bingo!,” which took somewhere between 5 and 10 minutes (next time, I’d like to time it for comparison), I presented students with the following challenge: “In your groups, design a card that you think will be most likely to win. You must be able to justify your choices of numbers.” I handed out blank cards, whiteboards, and markers.

Circulating, I noticed that some groups filled out their cards in a matter of several minutes or less. Here’s an example of one of these:

When I asked them about their choices, many students stated that “these were the numbers that came up most often” – in other words, their arguments were based purely on the observed frequencies, rather than on an underlying theoretical probability distribution. This might explain why, for example, -1 appears more often on the above card than 0 and 1. Other students seemed to be looking *beyond* the observed frequencies to the underlying distribution, referring to “more possibilities” or “more ways to get” certain numbers, but couldn’t clearly articulate their reasoning beyond this when prodded.

I brought the class together again to discuss these issues. Through the conversation, we were able to establish that one way to reason about more likely outcomes is to list all of the ways that the integers -8 to 8 can be produced as a sum of integers between -4 and 4. This discussion served as a theoretical foothold, allowing students to articulate their intuitions through the mathematical process of determining possibilities. I offered students time to explore these possibilities, probing and challenging their thinking as I interacted with the groups. Because many of the groups had already hastily filled in the card I handed out at the outset of the construction phase, I eventually handed out a second card for them to fill out using what they had discovered. (Next time, I will hand out the blank cards near the end of this phase.)

*Game play – Round 2:* Once all of the groups had finalized their cards, we played the game again. I allowed student groups to play with both of their cards (mainly because I was worried about time; otherwise, I would have asked them to choose the one they thought would be more likely to win).

As most groups had filled their cards with 0, ±1, ±2, and ±3, there were audible groans around the room when sums such as 8 and -7 came up – sometimes, several times in a row. Again, we played until a group crossed off all of the numbers on their card. This round was noticeably shorter than the first (but I would like to time both rounds next time to compare).

*Discussion:* During the post-game discussion, I asked students to explain why, if 0 and ±1 were so likely and ±7 and ±8 were so rare, we still observed them several times during the drawing process, which led us to discussing the distinction between *possibility* and *probability*: “It’s rare, but it can still happen.” We also grappled with the question of whether -7 and -8 were equally likely outcomes, which hinged on the question of whether (-3,-4) was the same outcome as (-4,-3) (a question I had posed to several of the groups during the design phase, which sparked some debates among the students). “Yes, because it’s the same sum,” argued several students; “no, because you can get it in two ways,” argued others. Having physical random generators was helpful during this particular discussion, because the drawing process was transparent. As one student eventually explained, “to get -7, you can either draw a yellow -3 or -4 from the first bag, and then you need to draw a white -4 or a -3 from the second bag. But to get -8, you need a -4 from both bags. So -7 is more likely.”

Shortly after this point, the class drew to a close.

Next week, as we continue our review, I am going to extend this task by having students design cards most likely to win if the numbers drawn are multiplied. Since students will be familiar with game play, and because I would like them to consider not only the probabilities, but also the possibilities themselves, we won’t start with a practice round.

To reiterate, this task fulfilled my criteria for a back-to-school review activity, but also has other bonus features that made it stand out to me as a Good Task:

- It gave me an excuse to briefly delve into the topic of probability with my students, which is unfortunately not part of the Math A90 curriculum. Those who
*are*lucky enough to be teaching probability may find this to be a fruitful space to explore several important concepts related to the topic; I could see this task being stretched over several days, as I suggest below. - It gave my students an opportunity to review a basic concept (integer operations), and gave me an opportunity to assess their understanding and note any misconceptions in this area.
- The review was embedded within a task that challenged students to wrestle with less-familiar concepts (in this case, randomness, relative frequency, theoretical probability, and elementary outcomes). As a result, it was appropriate and engaging for most students in the room – and additionally so, because there were stakes attached to designing a card well: namely, winning the game. (Although I’m not into bribing students into learning, I will note that when suckers are on the line, students tend to pay attention.)
- It’s readily adaptable and extendible. Next week, we will consider possibilities when integers between -4 and 4 are multiplied; the task can be made even more interesting when the operation is randomized as well (e.g., use a die with multiplication, addition, and subtraction symbols). Imposing the winning arrangement to be a line, an X, or an L (for example) adds yet another layer to the task. If, on the other hand, your students are just starting to explore the concept of probability, you might wish to (as in Serradó, 2016) introduce another round between the introductory round and the construction phase, where students must choose between two pre-made bingo cards, play the game with their selection, and reflect on the outcome before designing their own card. In this context, I think the task could be easily stretched over several days, with several iterations of choose-play-reflect and/or construct-play-reflect that give students opportunities to develop increasingly sophisticated reasoning about randomness, uncertainty, and probability.

If you have other ideas for how this task can be adapted or extended, I’d love to hear them. I’m looking forward to trying out different variations with my students as we warm up for the new school year.

**References**

Serradó, A. (2016). Enhancing reasoning on risk management through a decision-making process on a game of chance task. Paper presented at the 13th International Congress on Mathematics Education, Hamburg, Germany, July 2016. Available at https://iase-web.org/documents/papers/icme13/ICME13_S13_Serrado.pdf

]]>“The one real goal of education is to leave a person asking questions.” –Max Beerhohm

Undoubtedly, many of these are spoken in a hyperbolic manner, and should be interpreted as such. (Myself, I would be wary of anyone who believed that there are precisely *N *goals of education, true for all places and at all times.) Nevertheless, this sentiment came to my mind last week, the first week of school.

This semester, I am teaching one Grade 10 and two Grade 9 math classes, and I am happy to be teaching many of my Grade 9 students from last year in Grade 10 this year. I knew we would have another good semester when, on the first day of class (only 15 minutes long), the students skipped formalities and picked up right where we had left off in January. Although I had planned a team-building activity for the brief period, a student asked, almost immediately after class had started: “So anyway, is 0.999… = 1?” The other students jumped right in, arguing vehemently *yes! *or *no!*, trying to articulate some deep concepts that are, technically speaking, beyond the Grade 10 curriculum; this led us to other questions, such as the meaning of 0^{0}, 0, 0/*a,* and *a*/0. I noticed that some students had changed their mind about certain issues from last year, especially regarding the question of whether 0 is, in fact, a number (in particular, more were leaning towards *yes*, and were able to articulate why). I am sure they know by now that these kinds of questions are a great way to get me off track from my lesson plan, but there also seemed to be genuine curiosity in their questions, and genuine emotion in their responses.

“Have patience with everything that remains unsolved in your heart. Try to love the questions themselves, like locked rooms and like books written in a foreign language.” –Rainer Maria Rilke

The students’ responses during this discussion were interesting in and of themselves, but what struck me most was simply that they had remembered these questions and debates from our time together last year, which were not part of the curriculum and had certainly not been on any homework assignment, quiz, or unit exam. Presumably, they had learned some mathematical content in my class, but what seemed to stick with them was the *questions – *and, in particular, questions that can’t be answered using a rule or algorithm, questions to which the answer may not be black or white. They had also remembered that my classroom was a space for mathematical curiosity, where they could, and would, wrestle with such tough questions – even if they didn’t immediately get answered, and even if they technically didn’t “count.”

I left school that day with my heart full. Fostering and supporting (mathematical) curiosity is one of my main goals as a mathematics teacher, and it was heartening to see that I was able to at least nourish, if not spark, an inquiring attitude about mathematics among this group of students. It is my greatest hope that when I see my current Grade 9 students in Grade 10 next year, they, too, remember the questions, and are eager to tackle them anew.

And so the work begins again. Next week, my Grade 9 students will be tackling the following problem, sparked by an interesting question a student posed last week regarding the possibility of “stacked” exponents: “Which is bigger: 2^{710} or 2^{710}?” I can’t wait to see what other questions we wrestle with this term.

At last, students’ prototypes were coming to life. The penultimate lesson was a (productive) mess of paper, tape, and running to other classrooms to borrow meter sticks.

I wanted to give students an opportunity to share their hard work with other teachers and students, so earlier in the week, I began organizing a trade show. In terms of physical set-up, preparation was minimal; the biggest hurdle was finding volunteers for students to interact with. I sent out an email inviting other staff members to join, and ended up with about 6 staff participants; another math teacher also very kindly volunteered his Grade 11 class to join in.

On the morning of the show, I taped large numbers on several of the tables in the cafeteria, corresponding to name tags I handed out to the student groups. After about 30 minutes of last-minute preparations (we were running a little late), the class headed down. The staff and student volunteers joined us shortly thereafter.

Here are the instructions that I provided prior to the trade show:

*Students’ goal for this project was to design and build a prototype of a more effective and/or efficient pop box. Some groups have chosen to focus on efficiency—less cardboard, less empty space, or both—while others have chosen to focus on aesthetics, opting for a unique and interesting design.*

*Your role is that of a soda company executive who is looking to increase soda sales. When you meet with a group, they will present their brief sales pitch to you. You are encouraged to ask any or all of the following questions, if they haven’t been answered during the sales pitch, to learn more about their design and to push their thinking further:*

*Is your design more environmentally friendly than the regular pop box design?*

How so?*Are these boxes easy to ship? to stack on store shelves?**How will this design encourage more people to buy pop?**Who do you think this design will appeal to most: kids, teenagers, adults, or another category of people?**What other ideas did you entertain before deciding on this particular design? Why did you choose this design as opposed to the others?**You have improved the design of the box, but have you thought about improving the design of the can? How might you do so?*

*After interacting with a group, please give a grade out of 3 (3 is the highest, 1 is the lowest) for each of the following criteria:*

**Innovation
**The pop box design is unique, interesting, and/or eye-catching in comparison to the standard rectangular box.

**Precision
**The prototype is functional and has been built with care and precision.

**Strength of Argument
**The designers present a convincing argument with regard to the efficiency and/or effectiveness of their design.

*Note that these grades will *not *contribute to students’ marks for the project. *

[Note that the informal evaluation was not meant to be the highlight of the trade show and, in fact, I didn’t let my students know that participants were going to be evaluating them until the morning of the show; this didn’t seem to stress them out (I did explain that it would not influence their grades), and maybe turned their enthusiasm up a notch. Tomorrow, I will award small prizes (obviously, a can of pop) to groups with top marks in each category.]

And, once everyone found their table, they were off!

I joined the participants in speaking with the student groups and heard some fantastic pitches. The students were enthusiastic and proud of their designs, and they sold them well. I summarize just a few of them:

- One group surprised me by making a rectangular box that was nevertheless quite innovative. The box held 20 cans in total, but instead of the same kind of pop, there were 5 different flavors, and 5 flaps that opened to reveal them. Genius! Perfect for a get-together when you want to offer a variety of options, but don’t want to buy 3 boxes of pop. The design was practical, unique, and very appealing (the group ended up earning top marks in the “Strength of Argument” category). Among all of the designs, this one made me wonder most, “Why isn’t this on store shelves already?!”

- Another group played up the environmental angle of their Pepsi Flower design. “There is very little empty space. Also, the box is so cute, you can reuse it when you’ve drank the pop! Can you imagine storing baby clothes in there?! Adorable.” (Side note: This group also engaged in some great mathematical thinking to find the surface area of their box, which was full of rounded edges.)

- The group of students who created a triangular box pitched theirs as efficient
*and*fun: “When you’re done drinking the pop, put the cans back in the box, and you get bowling pins!” (Side note: This group refused to let go of the mathematics and*insisted*on finding the exact surface area of their design – despite my suggestion to estimate part of it – , which turned out to be a very interesting problem. See Part 2.)

- One group designed their cylindrical box to contain 38 cans, 19 per layer, for a very specific reason: 19 circles packed in the smallest possible circle that contains them results in the most dense packing up to 20. (See this Wikipedia article.) This was based on their own research, and was completely unprompted by me. (The abundance and variety of this group’s ideas was quite astounding – they spent two days just in the brainstorming phase, and a significant portion of this time arguing about the relative efficiency of installing a Pepsi pipeline in cities where they generated the most profit. Talk about out-of-the-box thinking – literally!)

This group’s pitch was, in fact, a rap, and its over-the-top cheesiness makes me laugh every time I watch it:

To sum it all up, I was incredibly proud of my students, and I hope they are proud of their hard work as well. Throughout the experience, I have been blown away by students’ creativity, collaboration, persistence in solving unfamiliar and challenging problems, not to mention by the variety of curricular and extra-curricular mathematics that emerged. Nat, whose Soft Drink Project this project was based on, sums it up nicely:

The sheer volume of work that went into the different designs was several times more than I could have ever gotten out of the same students with a dictated assignment. It showed me that an interesting starting point, a little bit of student control, and a willingness to learn alongside can create unbelievably powerful learning.

A willingness to learn alongside was certainly critical, as I was often pushed to exercise my problem-solving skills alongside students; as such, my role shifted even further from *knowledge authority* to *co-participant *in the problem-solving process. I found this to be tremendously exciting and fun, and I hope the experience helped to validate students’ feelings of being competent, creative mathematicians.

Finally, now that I’ve dipped my feet into project-based learning and survived, I am itching to explore its affordances further in other courses, with more curricular concepts.

Stay tuned…

Some resources related to the project, in case you are interested:

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

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

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

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

Pop Box Project –

Pop Box Project –

Last day, after a few lessons of stirring up ideas related to packaging design and strengthening understanding of the concepts of surface area and volume, students were introduced to the unit project: Design a more effective pop box. Although we had previously focused on comparing the efficiency of different packages in terms of amount of material used and percentage of wasted space, for their design projects students could choose to focus instead on creating a more unique and interesting package if they felt it would increase sales. Students left the classroom buzzing as ideas already started to emerge.

**Day 5**

I opened the lesson with a brief review of the task. The students were anxious to start brainstorming and paid little attention to this introduction; some groups already had a variety of ideas to mull over and debate, and wasted no time in doing so. A collection of empty cans, kindly lent to the class by the recycling team, was made available for the students to manipulate as they debated various ideas. Large elastic bands were provided to hold groups of cans in place, if necessary; students scribbled and sketched their emerging ideas on whiteboards, taking photos of viable options as they went along.

Little prodding was needed on my part, except with two groups. One claimed that they were done within two minutes, pitching a 4-can design that was otherwise identical to the regular box. I felt that they could do better, and after some discussion (Could you make it more efficient? What if I move these cans like this?), they began considering other options. Another group sat stumped for quite some time, finding that all of the designs that they thought were interesting were not particularly efficient. I reassured them that they could choose to focus on one of these factors, but this suggestion was ignored – fortunately, because they ended up settling on an interesting flower-shaped design that also minimized empty space.

Otherwise, my role was mainly to probe thinking about the reasoning behind the proposed ideas and to provide supplies as necessary (e.g., measuring tape, scrap paper). I was amazed with the variety of designs that emerged and with the engagement in the task – I happily observed students who are typically less engaged in class coming alive, even taking the lead in their groups in the brainstorming process. Some of the designs that emerged (not all of which were adopted) included a dumbell-shaped box, a fidget spinner box, a triangular box, a hexagonal box, a tube, a C-shape (for Coke), a Christmas wreath, a Coca-Cola Cake, an inuksuk, a circular 19-can design that minimized empty space, Tetris-inspired boxes (interestingly, this also came up in Nat’s class), and likely many other short-lived designs that were discarded for reasons of practicality or feasability.

Students debated ease of storage and shipping, efficiency, aesthetics, marketing, number of cans… e.g., “Nobody would buy an odd number of cans.” This led to an interesting discussion. Is it really about sharing (in which case, even numbers *are* better because they tend to have a larger number of factors), or is it a matter of convention? (Isn’t it just as likely that there are an odd number of friends sharing a box of pop? Moreover, has anyone ever opened a box of pop at a party and proceeded to distribute cans evenly among all the party-goers?)

By the end of the period, many of the groups had a rough idea of a design that they would go forward with. Their next challenge would be to take accurate measurements and begin the task of bringing their idea to life.

**Days 6-8**

The room buzzed with productive activity as students made measurements, sketches, and computations.

A variety of materials were made available (some borrowed from other classrooms as needed), including elastic bands, masking tape, glue, paper, rulers, meter sticks, compasses, protractors, and centimeter paper. I observed two groups independently make rough scale models of their design out of paper and tape, which helped them visualize where they needed to add flaps and what the box would look when it was flattened into a net. I briefly stopped the action to highlight this idea.

The students were progressing at various rates, but no time was wasted. However, very quickly I began to lament not having saved quite enough time for this project, as I observed several groups discard an interesting idea for a more traditional design because of time restrictions (e.g., the Tetris and the wreath designs). If some groups rushed through certain phases of the project, it was my fault for not having provided enough time; this will be one of the changes next time around. Nevertheless, the designs were steadily taking shape.

As students turned to the mathematics of surface area, volume, and empty space, I met with groups who were dealing with more complex shapes. More often than not, the students were able to independently simplify complex shapes into more familiar ones by dividing them into polygons; other groups, especially those whose designs had rounded corners, found the mathematics to be more involved.

One particularly interesting problem was finding the perimeter and area of a 10-can triangular design in which the cardboard wrapped around the cans. Initially, the group conjectured that the cardboard covered 1/2 of the circumference of a can in the rounded corners. However, when they compared their calculations with their actual measurements, they found that the numbers did not match (even taking measurement error into account). In my conversation with the group, they realized that their initial estimate was too large. “The cardboard would cover 50% of the circumference of an “end can” only if they were in a line,” they reasoned, and that if the cans were arranged in a square, 25% of the circumference would be covered by cardboard. So, for a triangle, they conjectured that the coverage would be somewhere in between – about one-third. Eventually, we proved that this was indeed true, which allowed us to find the total perimeter around the cans.

This authentic process of conjecture-verification-revised conjecture-proof was really great to witness. I noted more than once that the students were accountable to their physical models in a way that they were not towards, for instance, answers in their textbooks – if the calculations did not reflect real-world measurements, it was back to the drawing board to check and revise their work. After all, the groups wanted to build a working prototype, and it wouldn’t do to have unsightly gaps or too much empty space. Determining the area of the rounded triangle also proved to be an interesting problem, involving finding the area of a circular segment created by a chord that was smaller than the diameter. (I had initially suggested to this group that they could estimate using grid paper, but they would not have it)

When it came to finding the volume of the cans, again, students were not satisfied to (over-)estimate it by assuming that the cans were cylinders. They suggested using water; I had some Play-Doh on hand. Can you guess how we found a more accurate estimate?

Many other interesting problems were encountered and solved over the course of the week. All this to say, the mathematics did not get lost in the design and construction of the boxes, which I had worried about at the outset of the project. On the contrary, students were doing more and more interesting mathematics than I had anticipated (curricular concepts included surface area, volume, trigonometry, proportional reasoning, and factoring, among other mathematical concepts), and were more concerned about the accuracy of their calculations than ever. Moreover, the mathematics slowly materialized into a tangible, working prototype that, I would hope, provided more satisfaction than finding that your answer matched with that given in the back of the textbook. Nat wrote a great post about the mathematics involved in the pop box project, so I won’t say more about it except to quote him on the following point:

The lesson for me (and all teachers interested) is that

we can still tailor a student-driven class around a content-driven curricula.

At last, the 3D models were coming to life. Students were meticulous in their measurements and cutting and would not be rushed. However, the trade show was around the corner, and prototypes and sales pitches would need to be ready for the show; this necessitated adding an extra day afterwards to complete final calculations. (Again, more time will be one of the biggest changes when I do this project next time around.)

In my next and final post, I will discuss the trade show, gush about my students’ final products, and share some final reflections about the project. For now, I leave you with a rubric that I provided students to refer to over the course of the project, which helped them keep track of the work done and the work still to be completed. It is also the rubric I will be using to grade their work.

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

Pop Box Project – **Rubric** (French)

Previously:

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

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

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

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

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**.