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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Here’s the context: Last week, our school held a Box Lunch Auction, which involved homerooms preparing lunches of various sizes (e.g., for 6 people, for 10 people, for 30 people) and groups of students bidding on said lunches. The money raised was donated to various local charities.

Reflecting on the experience, G. wondered if the sizes of the lunches might be determined in a way that better suited the needs of the student body, rather than more or less randomly (truthfully, based on how much work a homeroom was prepared to do). This is the question he posed:

Let’s imagine that any two people in the school have a 50% chance of being friends. With

npeople, what’s the most likely size friend group?

I was hooked immediately.

To start, we lay down some assumptions. In particular, we assumed that if, for example, A and B were friends, and B and C were friends, then A, B, and C formed a friend group. In other words, in a friend group, everyone has at least one friend in the group, and all groups are disjoint (a student only belongs to one friend group). Then, I suggested that *n* students was a bigger problem than we could handle at the moment, and that we should simplify it to start, at least to get to know the problem a bit before trying to generalize. So, we asked: What if there were just 3 friends?

We drew a diagram:

Here’s what we found for this particular case (spoiler alert!):

If all friend groups are size 1, no students are friends with each other. The probability of this occurring is (0.5)(0.5)(0.5) = 0.125 (i.e., A and C aren’t friends, A and B aren’t friends, and B and C aren’t friends). Groups of size 1 may also occur when two students are friends with each other, but not with a third student. The possibilities are A and B being friends, B and C being friends, and A and C being friends, with the remaining student not friends with either. For the first case, A and B are friends and C isn’t friends with either A or B, which has a probability of (0.5)(0.5)(0.5). The total probability of groups of size 1, then, is (0.5)(0.5)(0.5) [none of the students are friends] + (0.5)(0.5)(0.5) [only A and B are friends] + (0.5)(0.5)(0.5) [only A and C are friends] + (0.5)(0.5)(0.5) [only B and C are friends] = 0.5. The probability of a size 2 friend group is the same as the latter three cases: (0.5)(0.5)(0.5) [only A and B are friends] + (0.5)(0.5)(0.5) [only A and C are friends] + (0.5)(0.5)(0.5) [only B and C are friends] = 0.375. Finally, the probability of a size 3 friend group is 4(0.5)(0.5)(0.5) = 0.5 (this includes the situation that all students are friends, or that, e.g., A and B are friends and B and C are friends, but B and C are not friends). (These probabilities do not sum to 1 because the probability of groups of size 1 includes the possibility of a group of size 2.) So, it turns out that size 3 friend groups and size 1 friend groups are the most likely to occur.

Interestingly, the probability that *you* (e.g., Student A) will end up in a size 1 friend group (i.e., with no friends!) is *not* 0.5. This will occur only if you are not friends with B or C, which has probability P = (0.5)(0.5) = 0.25. The probability that you end up in a size 2 friend group is equally likely: P = (0.5)(0.5)(0.5) + (0.5)(0.5)(0.5) = 0.25 (either you are friends with B and C is friends with neither you or B, or you are friends with C and B is neither friends with you or C). Finally, the probability that you end up in a size 3 friend group is P = 4(0.5)(0.5)(0.5) = 0.5, which occurs either when there are exactly two unique friend pairs (e.g., you are friends with B and B is friends with C) or when all pairs are friends with each other. From this perspective, you are pretty unlikely to end up with no friends – more likely, you will end up in a group of 2 or 3 friends. (I’m now imagining this as part of the “Welcome to high school!” presentation to Grade 9 students that’s meant to calm their nerves. I have a feeling it wouldn’t help…)

What happens when we add more students to the school? Is it always true that the largest possible friend group is the most likely? Initially, G. and I drew the following diagram and tried to do a similar analysis:

But soon, we realized that it was incomplete (can you see why?). The problem seems to get very complex quite quickly (what if there were 8 students? 100? 1000?), so we may have to get a bit more creative in our approach when we come back to it next week.

I love this problem, which is proving to be much deeper than initially expected. (But, after all: “Problems worthy of attack prove their worth by fighting back.”) I love even more that a student posed it. If you have any suggestions for a good strategy of attack, or if you recognize this as analogous to a similar problem, let me know. I’m looking forward to seeing where this one takes us.

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I should have trusted them.

Nat Banting has written about balance scales here. Although the metaphor of balance scales is referred to often in math classrooms – during the linear equations unit, students must hear the mantra “whatever you do to one side, you do to the other” at least 1000 times a class – the model really becomes powerful when students actually work with balance scale problems, and that they do so right from the start. Nat writes:

My Grade 9 students don’t see an equation for the first two weeks of their unit of solving linear equations. That is because I think students get all bogged down in the notation, and lose their problem solving intuition.

I decided to follow Nat’s advice. For a few days, my students worked through sequences of problems such as these, increasing in difficulty, where the goal is to find the “weight” of the square:

A.

I grouped students randomly and set them to work on whiteboards. I didn’t mention variables or equations. Beyond a few clarifications here and there, all students were quickly able to get to work. If they wanted to use guess-and-check, great – once the problems got harder, they would move on to more efficient strategies. After a few of days, all students eventually came to use strategies that looked very much like solving equations, and soon moved fluidly to using algebra to solving the problems. When they explained their work to the class, they would say things that sounded very much like the teacher mantra, “whatever you do to one side, you do to the other.” It’s not that the metaphor changed – it’s that it came from the *students*, and was grounded in their own reasoning and understanding of equality.

As for my worries about negative and non-integer values? For a problem like the second one above, students noticed that (given that circles were worth 1) the squares must be “lifting” the scale up, and came to realize that they could assign them negative weights. Once we moved to solving equations without the balance scale context, students were able to solve problems such as -3x – 5 = 6, because although they had now moved beyond the model, their understanding of equality was firm enough for them to adapt. Eventually, we moved on to solving equations with variables on both sides, which warranted a brief return to the balance scale context. “Wait, can you also take away variables from both sides?” “Why not?” They got it.

It’s amazing what a good concrete or visual model does to support student thinking and to smooth the transition to more abstract problems. But when I discuss this with other teachers, almost invariably I hear: “Yes, but I know that *my* kids need to *see* examples of how to solve it first, *then* try it themselves.” I can’t speak for all students’ needs, but I wish that as mathematics educators, we had more faith in their intelligence and ability to reason in sophisticated ways, if they are provided the opportunity to do so. As Nat suggests,

Give them to students. Ask them to explain their thinking. All involved may be surprised by dormant algebraic thinking that just needs an intuitive trigger.

Speaking of not trusting your students: I was really apprehensive of giving balance scale problems to my Grade 10 students today, who are nearing the end of a unit on systems of equations. I wanted them to give solving systems by elimination a try, but I wasn’t sure they were ready – most have stuck faithfully to one particular algebraic strategy and have not shown interest in moving to more efficient methods. Also, I knew that they hadn’t worked extensively with balance scales the year before. I should have trusted them.

I presented the class with the following problem:

Happily, many students recognized, and shared during the whole-group discussion, that setting up a system is *possible*, but not necessary: if you recognize that removing 2 squares is equivalent to subtracting 6, you can easily deduce that a square is worth 3 and a circle is (consequently) worth 10. First, this signaled to me that students were developing flexibility with linear systems, recognizing both what a solution *is* and that there are multiple ways of finding it. In other words, they weren’t simply following an algebraic procedure blindly. Second, it made the following suggestion natural, because the students could already imagine manipulating the weights on the scales: “What if, instead, we imagine removing 2 squares and 6 circles from the first equation? How much would we have to remove from the right side?” They got it. We translated this to equations.

I set students to work on the following sequence of problems, suggesting they try a similar strategy for each. I didn’t tell them what to do when the equations didn’t have a variable with a common coefficient. Instead, I circulated, asking questions where necessary to trigger thinking.

B.

C.

D.

E.

I found the following questions helpful in getting students to generalize the elimination method to systems where there aren’t similar terms with the same coefficient (e.g., 4x + 6y = 72 and 2x + 6y = 66, as in problem A):

- “What made this strategy work for the other problems?”
- “If this same strategy
*were*to work, what would you want to have on this scale?” - “Am I allowed to add weight to one side? How can I make sure the balance is maintained?”

For some groups, all it took was the first question for a student to say “OOOOOOOHHHHHHH! You can just multiply this one by 3,” and they were on their way. For others, it took a bit more questioning, but they got there. (When students asked “Is this right?” – they should know better – I suggested they check to see if their answer worked for both scales. My students are still working on developing strategies to analyze the reasonableness of results.) The visual model helped immensely, and when we moved to algebraic representations, students would make references to adding or subtracting weight. Several students commented on the efficiency of this strategy in comparison to others; I’m curious to see if elimination will become their new method of choice.

If you’re looking for more balance problems, Nat has a variety in his balance scales post. They are also very easy to make on your own – I whip them up in a matter of minutes using Paintbrush (the Paint equivalent for Mac), making liberal use of cut and paste. You may wish to grab the blank balance scale image below. The problems themselves are typically pulled from the textbook.

Most importantly, trust your students. If they depend, and maybe even insist, on being shown procedures before applying them to problems, maybe it’s because they’ve had too few opportunities to try problem solving on their own, and to develop confidence in their reasoning skills. All involved may be surprised with the thinking that emerges, as I was happy to find today.

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I paused the Act 2 video after the first weighing:

How much *might* each bottle and glue stick weigh?

A bit slow to start, the students nevertheless went on find a variety of possible answers. To some groups who were still stuck after a few minutes, I suggested a weight for a bottle and asked if they could go on to find the weight of a glue stick. Other groups decided to make some assumptions on their own: e.g., that the weight was distributed evenly between the 4 bottles and 5 glue sticks, or that all of the objects “by some miracle” weighed the same (“there’s lead in the glue sticks”). A few groups wrote an equation and made a table of possible values by assuming a weight for one of the objects and finding the corresponding weight of the other; one group solved the equation for one bottle, and used it to argue that it could have a wide range of weights, depending on the weight of a glue stick.

Once we had established that more information was needed, I played the next part of the video:

Although a small number of groups wrote equations right away, recognizing a system and solving it to find the weights, many had trouble getting started. I suggested writing equations as a start to several. A few groups did something I didn’t anticipate: they assumed that the two weights, 679 g and 680 g, were basically the same, which led to some really great reasoning. I was blown away by the following, for example:

The two equations are 5x + 4y = 679 and 12x + 3y = 680. We noticed that when you add 7 glue sticks and take away 1 bottle, you get virtually the same weight. So, 1 bottle is approximately equal to 7 glue sticks. By the second equation, 33 glue sticks = 680 g, so 1 glue stick is about 20.6 g and 1 bottle is 20.6(7) = 144.2 g.

How awesome is that? The two groups who made this assumption used an algebraic approach (5x + 4y = 12x + 3y) to find that 1 bottle ≈ 7 glue sticks (y = 7x) and used this information to find the respective weights. The difference between the weights found using the different strategies was essentially negligible, which served to validate the responses.

I absolutely love moments like these, when students surprise me with reasoning that I did not anticipate, and I’m reminded of the importance of giving students space to use their own strategies to solve problems. Yes, there is an argument to be made for efficiency, or precision, depending on the situation, but this is a discussion that can occur once there are several strategies to compare. (This is also what happened during last week’s trashketball task, where students slowly came on the algebra train once they were tired of making tables of values. We all eventually agreed that algebra was more efficient and gave a more precise answer.) It also gives me the confidence that students don’t need to be spoon-fed procedures before they can “apply” them to “challenge” problems at the end of a homework set. On the contrary, a good problem can function as a sandbox where procedures are developed and refined.

On the other hand, during the same period, I was reminded of how a lesson can fall flat on its face when students are forced to use procedures that don’t arise from their own work and reasoning. Some backstory: for the past few days, my students have become quite confident in solving systems of equations by solving both for the same variable, then setting them equal to each other: e.g.,

5x + 4y = 679 → y = 679/4 – (5/4)x

12x + 3y = 680 → y = 680/3 – 4x

→ 679/4 – (5/4)x = 680/3 – 4x, etc.

This strategy arose from the trashketball activity. They’re looking for when *x*, *y* are the same on both lines, so solving for *y* (or *x*) and setting the two equations equal to one another makes sense to them. It also has the advantage of making it easy to compare slopes and determine how many solutions the system has. The only “disadvantage” I see is that it often results in equations that have fractions (which many of my students struggle with – something we need to work on), and I wonder if this is why the methods of substitution and elimination are privileged in textbooks.

However, I felt obligated to teach substitution because of its prominence as a strategy for solving linear systems. I anticipated the conceptual difficulties students would have, which is why I held off until today, but nevertheless I persisted in demonstrating the procedure for the above problem, after students had shared their own solution strategies, and having the students practice it for a set of problems. A few students were able to catch on, but the majority – *especially* the strong students – were lost. *Why* can you just substitute the expression into the other one, without solving for the same variable?, they wondered. I floundered in my attempts to help students understand, because I didn’t have a good model to refer to. Near the end of class, I told students to just use a method that made most sense to them.

I think elimination will be easier to tackle because balance scales provide a good model for building conceptual understanding of the method, but I’m left wondering whether substitution is something I should bother with going forward, when students are comfortable with a strategy that is arguably no less efficient and conceptually easier to work with. Consulting the curriculum, I notice that it only states that students should learn how to solve systems “algebraically,” which suggests that I should follow my intuition – to privilege understanding over rote procedure. But am I doing a disservice to my students?

This episode brings to my mind an article I recently read by Love and Pimm (1996), titled “‘This is So’: A Text on Texts'” (it has one of the best opening lines I’ve ever read: “In the beginning was the *logos* and the *logos* was the vehicle for mathematical expression.”) In the article, Love and Pimm describe how, and by what means, the textbook has come to dominate both perceptions and practices of school mathematics. In particular, they argue that the “immutability and perceived authority” of a textbook, which inherits some of its status from the quality of being “designed by experts,” tends to limit experimentation and encourage subordination to at least some extent, sometimes even in spite of the author’s intentions (p. 400-402). Although I don’t feel particularly bound to the textbook, which I rarely refer to during lesson design, I recognize this as being part of the tension I’m feeling: a tension between the perceived need to conform to a mandated resource and what I feel, based on my observations and interactions in the classrooms, is a more appropriate strategy for my students (at least, for the time being). In this particular tug of war, I think the latter will win (but please, if you have very strong feelings about teaching substitution, comment below).

Of course, despite a rhetoric of “inquiry” and “discovery,” conformity is not something we can escape within an education system that mandates certain outcomes for all students. This is the reality teachers must learn to operate in, but I wager that most (me among them) wouldn’t favor anarchy in its place. However, it’s important to recognize when we *do* have freedom to move within a set of constraints (which can be productive, as Davis & Sumara’s notion of “enabling constraints” suggests). And as I’ve found, sometimes when you reach out and expect a boundary, you only find more space to explore.

**References**

Davis, B., Sumara, D., & Luce-Kapler, R. (2008). *Engaging minds: Changing teaching in complex times* (2nd ed.). Mahweh, NJ: Lawrence Erlbaum Associates, Inc.

Love, E., & Pimm, D. (1996). “This is so”: A text on texts. In A. J. Bishop, K. Clements, C. Keitel, J. Kilpatrick, & C. Laborde (Eds.),* International handbook of mathematics, Part 1* (pp. 371–409). Boston: Kluwer Academic Publishing.

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As several of my students noticed during their work on the task, there will always be an odd number of squares in the first pattern and always an even number of squares in the second pattern. Therefore, there would never be a step where the two patterns had the same number of squares (i.e., the system didn’t have a solution). Of course, this is true for the discrete model, and an astute observation. However, the system *y* = 4*x*, *y* = 2*x* *+ *9 *does *have a solution: namely, (4.5, 18). We could, theoretically, conceive of a step 4.5, or even continuous growth, with strange steps like 4.9143, but at this point, the model may be being stretched beyond its usefulness.

With the motivation of having students move their understanding beyond discrete systems and to motivate a need for solving systems algebraically, the next day we launched into the Trashketball task, a fantastic series of lessons developed by Jon Orr. I also borrowed heavily from Alex Overwijk’s Card Tossing task; the task is very similar, but doesn’t rely on Desmos for finding solutions. Since I wanted students to move towards an algebraic solution of systems, I decided to forgo Desmos for this activity as well.

**Day 0**

Located buckets and boxes, one per group of three students (11 of each in total). Crumpled a giant-garbage-bag’s worth of paper. Still nursing a few paper cuts.

**Day 1**

The previous day, I invited students to meet me in the cafeteria at the start of class, in front which there is a big, open space. Knowing that I would embarrass myself, I invited a phys ed teacher (Mr. M.) to embarrass himself for me. He graciously agreed. At the start of class, I introduce Mr. M. as the best trashketball player in the province. “He’s humble, so you probably haven’t heard about it, but he’s won provincials, what, six years in a row?” “Six.” The students are intrigued.

“What’s trashketball? It’s a bit underground; you probably haven’t heard of it. I’ll let Mr. M. demonstrate.” I bring out a bucket and a box full of crumpled paper balls. I ask a student to set a timer for one minute, and another to count how many balls go in. “Standard distance is four floor tiles, right?” The timer starts, and so does Mr. M. One minute later, there are 37 balls in the bucket.

I write this number on the board. The students don’t look impressed. We find Mr. M.’s shooting rate: 0.616 balls / second.

“Do you think you can do better than Mr. M.?”

“YES!!!”

And just like that, they’re hooked.

I hand out a sheet where students will record their shots and, later, make predictions and determine their shooting equation. Each will do four 1-minute trials in their group. I give the students about 5 minutes to practice, after which we decide on a set of rules:

- You must be 4 floor tiles away from the bucket.
- Your chest must be in line with your feet.
- You can only throw one ball at a time.
- You can only throw with one hand at a time.

Then, I blow the whistle to start the first trial while I time on my phone. All groups start and stop at the same time. I repeat until 12 trials in total have been completed.

It goes without saying that the students love this part of the lesson. The video below captures the mood perfectly:

Once all of the trials are completed, the students return to their whiteboards and work on finding their average shooting rate, predicting how many shots they would make in a given number of time, and determining their shooting equation. (Link to PDF / Word document.)

“Who thinks they have the best shooting rate?” A good number of students raise their hand, and shout out their rates. One girl, P., emerges as the clear winner: 0.825 shots per second. We hold a competition between P. and Mr. M. She wins easily.

Now, the key question that turns the lesson from one about linear relations (which, it easily could be, if that’s the space you want students to inhabit) to one centered on linear systems: “If P. played against Mr. M., she would win every time, because she’s better. How could we make the game a bit more interesting?” Students suggest having P. stand further back, or letting Mr. M. use both hands. “The problem is that we didn’t collect our data from different distances, or with two hands. **What if we gave Mr. M. a 10 ball head-start****?**” The students agree that this would help. “**What’s the longest Mr. M. could play against P. and still win? Or, in other words, when would they tie?**”

Note: Next time, I would save this part of the lesson for the following day because it deserves more time than I allotted, but I decided to squeeze it in because I knew Mr. M. wouldn’t be available. The students have about 10 minutes to work on the problem, and many use tables of values to find an approximate time (48 seconds). A couple of groups use algebra, which is a great sign. In the last few minutes of class, we squeeze in a test – unfortunately, the bell rings very shortly after, and even though the number of balls in both buckets is exactly the same (!!!), the effect is diminished as a tide of students pours into the hallways, taking my students away with them.

**Day 2**

On Day 2, we math. We’re back in the classroom, and I have a student volunteer their shooting rate (about 0.54). We hold a competition between this student and P.; again, P. wins easily. Again, I suggest an advantage, based on the following table that I created the day before (calibrated against P.’s shooting rate, so that the games wouldn’t be too long or too short):

I ask students to determine, given their starting advantage, how long they could play against P. and still win. They work on whiteboards in randomly-selected groups of three.

As I circulate, I notice that many of the groups have written equations to represent the number of shots made per second, and using these to create tables of values. Many start with time intervals that increase by 1 second, then notice that the increases are too small; they adjust to intervals of 10 seconds. I question some students about the values in their tables; some aren’t sure how to determine the next value after 0 seconds for the student who has the advantage. It takes a question or two from me for them to figure it out on their own. Once the students reach a point in their tables where P. starts winning again, they backtrack and decrease their intervals once more, narrowing down to an approximate time. Today, I see a few more students using an algebraic approach, recognizing that they can find the time by setting the two equations equal to one another.

Once all students have an answer, I have several groups share their strategies with the class. Below are my (messy) transcriptions of two strategies:

We test with a few pairs of students; the results are remarkably consistent with the predictions. We consider why the actual results differ slightly from the predicted values, and discuss the difference between models and reality. (“Can P. really have 24.75 balls in her bucket after 30 seconds?”)

**Day 3**

As a wrap-up, we review with another pair of shooting rates and a given advantage. Today, when I walk around the classroom, I hear students saying, “Yeah, you could do a table of values, but it takes longer. Here’s how I do it.” In other words, most groups – even without prodding on my part – have moved on to an algebraic approach, recognizing it as being the most efficient for the task. This is one of the advantages of white boards, group work, and random seating – students share their efficient strategies, and knowledge travels fast (n.b., my students use large horizontal whiteboards on their desks, rather than vertical ones). I can see that it will be a smooth transition to solving general linear systems.

We discuss what would happen if the students continued to play. Would they ever tie again? Why not? How could we make them tie sooner? later? Today, we also verify our solutions on Desmos, with graphing emerging as a third strategy to find the solution of a system.

Although it wasn’t perfect (student engagement wasn’t particularly great on the second day, but as I mentioned earlier, I would shuffle some parts of the lesson around next time), I think this was a pretty successful launch into the unit. It was particularly validating to watch the need for a procedure emerge organically, in response to a problem and the desire for efficiency (Dan Meyer would use the words ‘aspirin’ and ‘headache’). I wager that in a textbook, this task would be relegated to the end of the unit, a “real-world application” of a procedure that has been drilled and practiced a sufficient number of times – but at that point, does anything problematic about the task remain? In my view, these end-of-unit “application” tasks are just another drill in “real-world” disguise.

Post-trashketball, I have referred back to the task often, which became a useful model in the same way that linear patterns were in the previous unit. Of course, all models break down eventually, but I hope that this conceptual foundation will be a benefit to students as we move forward into greater abstraction.

Thanks again to Jon Orr and Alex Overwijk for sharing their Trashketball and Card Tossing activities; here are the links to their original posts again:

http://mrorr-isageek.com/trashketball-a-spiralled-lesson/

http://slamdunkmath.blogspot.ca/2014/05/card-tossing.html

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