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?

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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?)

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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

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“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.

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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 –

Previous posts: Part 1, Part 2.

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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)

Next: **Pop Box Project: Part 3 – Trade Show and Reflections**

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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**.

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Every being cries out in silence to be read differently. Do not be indifferent to these cries. – Simone Weil

I was substitute teaching today for a group of students I had taught last semester. I noticed that one of the girls had new glasses, and I told her that they looked nice. She replied, surprised, “You’re the only teacher that’s noticed, and you don’t even teach me.”

Not pretending to be a hero. There have undoubtedly been countless occasions that I’ve been blind when I should have seen. Just tying a knot in my mind to notice, amidst the chaos, the students I’m not seeing in my own classroom.

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In order to occasion and nurture flow, Liljedahl argues, teachers should be intentional in providing hints and extensions that keep students in balance between their abilities and the challenge of the current task (2016a). Although I typically opt for this approach—that is, offer the same task to all student groups, and provide hints and extensions as necessary—, today, I found success in offering students choice of tasks (coupled with hints and extensions from me, as necessary).

The unit is trigonometry (Foundations and Pre-calculus Mathematics 10), and we are a few days away from the exam. I wanted students to engage with a variety of problems involving triangles where they could apply their understanding of trigonometric ratios and the Pythagorean theorem, and initially considered a “speed dating” review (see this post). However, I decided to change the format, wanting students to spend the lesson immersed in *doing* mathematics, rather than listening and sharing, even if it meant they engaged with a smaller number of problems. (I still love the speed dating idea, and would like to give it a go sometime in the future.)

Luckily, I had already made up a series of 12 problems for the speed dating activity, which I labeled with the astrological signs (also Nat’s idea), which is why I’m referring to it as the Zodiac review. Some of the problems were taken straight from the textbook, or adapted slightly. My tweak was to add an approximate difficulty level to each problem (1-4, with 1 being the easiest). This morning, I made a few copies of each problem and stuck them to the board with magnets. I also put up the answers (not solutions) for verification. See photo below.

I grouped students randomly (as I do every day), provided each group with a marker and a whiteboard, and gave the following instructions:

- Decide, as a group, where you would like to start; if it’s too easy, go up a level, if it’s too hard, go down. Once you’re done with a problem, put it back on the board.
- Try to work through as many of the astrological signs as you can, but it’s not a race. Keep track of which ones you have done, and take photos of your work as “notes.”

Any questions? Nope. The students got right to work; some grabbed more than one problem right off the bat. Now, when offering this kind of choice to students, there is always the fear that they will choose wrong – especially that they will take the easy way out, and stick with the simpler problems. This is not what happened. Check out which problems flew off the board first:

A few comments, noticings, and takeaways:

- Since students were working on different tasks and not encountering new concepts, there was no need for a whole-group discussion to close the lesson. I focused on facilitating communication
*within*groups, as I discuss here. - A few of the groups spent the majority of the class working on one of the level four problems. I did not hear one “this is too hard” or “let’s try an easier problem” – they wanted to,
*needed*to solve it, and*by golly*they would, as if their life depended on it. When I sensed that they were stuck, I directed groups working on the same or a similar problem to talk to each other (side note: I did not plan for this, but I now realize this is a good reason to make several copies of the same problem for this activity). All of these groups eventually found the solution, and were able to experience a nice connection to the previous unit (systems of equations).

- For the most part, students were engaged until the bell rang, which caught us all off guard. In several cases, I was surprised to see students who tend to shy away from participating taking the lead on a problem. Cell phones were being used as calculators or to take photos of the work. I hesitate to confirm (despite the title of this post) that the students were in flow because the experience is very personal, but the outward signs were there.
- One group ran up to me at the end of class to excitedly share how they had solved one of the problems using Desmos. The problem is as follows:

*You and your friend Michael are 38 m apart, both west of Big Ben. From your vantage point, the angle of elevation to the top of Big Ben is 65°. From Michael’s vantage point, the angle of elevation is 49,5°. What is the height of Big Ben?
*

They drew a diagram, and figured that the tangent ratio would be involved. They reasoned that since the tangent ratio is essentially the slope of the hypotenuse of a right triangle, they plotted the line *y* = (tan65)*x*, to represent your vantage point. Then, they plotted the line *y* = (tan49.5)*x* + *b* to represent Michael’s vantage point, adjusting *b* until the distance on the *x*-axis was 38. Zooming out, they found the point of intersection of the lines (84, 97), from which they concluded that the height of Big Ben is about 98 m. My recreation of the graph appears below.

You guys. I couldn’t even. What an awesome connection between trigonometry, linear relations, and systems of equations – three of the major topics of the course. They couldn’t have demonstrated any better their deep understanding of the concepts at hand.

Many hours later, I’m still buzzing from this class.

Because today was a shortened day, I will give students some time to continue their work tomorrow. Since none of the groups worked through even half of the problems, I will put up the same questions I offered today, but have added a few more level 3 and 4 problems into the mix. (Since I ran out of Zodiac signs, I used the names of some Harry Potter creatures… close enough.)

If you’re interested in the problems, I’ve uploaded a PDF here; it’s heavier on higher-level problems (of course, depending on your students, you may assign different levels to the problems). Adjust as you please.

Let’s see if we can keep up the mad flow for two days in a row…

**References**

Liljedahl, P. (2016a). Building thinking classrooms: Conditions for problem solving. In P. Felmer, J. Kilpatrick , & E. Pekhonen (Eds.), *Posing and solving mathematical problems: Advances and new perspectives*. New York, NY: Springer.

Liljedahl, P. (2016b). Flow: A framework for discussing teaching. In *Proceedings of the 40th Conference of the International Group for the Psychology of Mathematics Education* (Vol. 3, pp. 203-210).

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tl;dr: Sometimes, a “problem” in the classroom is really a symptom of a bigger issue.

Here’s the backstory. Lately, I’ve been struggling with a lack of engagement in whole-group discussions after small-group work. I was growing increasingly frustrated that the majority of the students – especially the students who had a good grasp on the concepts – didn’t volunteer to share their thinking, and that they had no qualms about chatting with their friends when a student (or me) did choose to put forth a question or share an idea. Increasingly, I was feeling like I was losing them, and searched for explanations ranging from summer being around the corner, to deficiencies on the part of my students (*kids these days!*), to the possibility that I am a terrible teacher for whom there is, frankly, no hope. (During a recent conversation, Jimmy Pai described teaching as a balancing game between overconfidence and despair, and these days I’ve been leaning heavily to the right.)

After a particularly rough class, I decided yesterday that resolving the issue amounted to (temporarily) giving students some type of external motivation to listen and to participate. “What if,” I conjectured, “I gave an exit slip at the end of class, asking students to recap and reflect on a strategy to a class problem that was different from their own?” I was satisfied with throwing this at the wall and seeing if it stuck, until I shared my problem with a fellow teacher. Here’s some of that conversation, which I’d like to share because it caused a substantial shift in my perspective [**emphasis** mine]:

- As much as possible, allow enough
**time**so that*every*group can finish the task (give**fewer,**but**meatier**problems), and check in often with groups that seem to be making very slow progress. **Check in**with*every*group at least once. If someone appears to be working by themselves, bring them into the conversation – “Could you explain why you did this?” “What would you do here?”- Enforce
**one marker per group**. If a student appears to be hogging the marker, ask them hand it off to someone else. - Always have
**extension and reflection questions**ready for groups that are done quickly. Some go-to questions: “Can you solve it another way? E.g., without using method X?” and “I want you to talk about how you can explain this strategy clearly to someone who did it another way.” - When you notice an interesting or efficient strategy,
**stop the action**and have the group explain to everyone what they did. In some cases, it may be more appropriate to get two groups to share and compare strategies. - During whole-group discussion, ask students to
**compare**their strategy with that of another group. “What’s similar? What’s different? Why do you think they did X? How is X represented in your strategy?”

Many of these are well-established strategies for supporting collaborative learning in the classroom, and they’re not new to me, either – but I know that I had been letting some of these slip as I focused on the norms that my *students* weren’t following.

Today, I started class by saying that our goal for the lesson was twofold: To do mathematics, and to work well together. I had two problems prepared for the entire class, which gave the students and myself a lot of room to stretch our feet (see point 1). The problems were not elaborate: pulled straight from the textbook, they asked students to find all of the missing sides and all of the missing angles for a given triangle. Here’s the first:

There was, however, a catch: Each group was given one constraint among the following:

- Use only
*sin*. - Use only
*cos*. - Use only
*tan*. - Don’t use the Pythagorean theorem.
- Don’t use the fact that the angles in a triangle add up to 180 degrees.

(Nat Banting has referred to such restrictions in similar tasks as *enabling* constraints, a notion developed by Davis, Sumara, and Luce-Kapler: “‘The common feature of enabling constraints is that they are not prescriptive. They don’t dictate what *must* be done. Rather, they are expansive, indicating what *might* be done, in part by indicating what’s not allowed’ (Davis, Sumara, & Luce-Kapler, 2015, p. 219). By restricting what can be done, action orients itself to the possible.” The whole post is definitely worth a read.)

You may have noticed that some of these are impossible. In my discussions with groups, some admitted that they “had to cheat,” which led to great discussions about which trigonometric “tools” can be used in a given situation. As soon as a group finished and I was satisfied that everyone in the group understood, I asked them to check their work by solving the triangle given another constraint (see point 4). Finally, I asked individual groups to find the “easiest” strategy possible.

I circulated around the room, making a point of checking in with every group and bringing students into the conversation when they seemed to be disengaged (see points 2 and 3). At one point, I stopped the action (see point 5) when a group told me that the task they had been given was impossible, and gave a very clear explanation for why. Even better, it turned out (and I hadn’t noticed) that they had made a mistake in calculating one of the sides, and other groups had evidently done the same, because two answers were put forth with confidence by the class. This led to a great, spontaneous whole-group troubleshooting session, which helped to reinforce the fact that the sides in a trigonometric ratio depend on the “vantage point” (angle) you take. I swear – and this makes my heart leap – I haven’t seen so many hands shoot up to share in a long time, and several students even volunteered to go up to the board. We resubmerged, and soon after I set students to work on a different problem (see below) with the same constraints.

Near the end of class, we regrouped again for a brief discussion – notably, *not* simply a show-and-tell (prompts: “Which constraint did you hate the most? Why?” “Which one do you think was easiest?” “What would you do on a test, if you were pressed for time?”). Finally, I gave students an exit slip – but not as a way (as I had originally envisioned) to test their attention; they were to solve a triangle using any strategy they wished, then to verify using another. Again, my heart leaped when I was marking (!), because 30 out of 32 students *destroyed* the problem, and the other 2 were well on their way.

Now, I know that this class may have been an anomaly; at least some regression to the mean is likely to follow. I also still have a long way to go in supporting productive collaboration in my math classroom. Nonetheless, this was an incredible learning opportunity for me, demonstrating in a powerful way – in addition to the power of teacher collaboration – that turning a problem upside down and changing your perspective can make all the difference, perhaps revealing that the “problem” in question is really a symptom of a bigger issue.

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