My students and I are currently in the middle of a unit on linear systems, which follows a unit on linear relations. We had worked a lot with linear patterns during the previous unit, which offer a great foundation for developing understanding of multiple representations of linear relations. Turning to linear systems, a natural transition was to consider two or more patterns side by side, and to consider when, if ever, they would have the same number of objects. However, the model is limited in that it’s not continuous. Consider, for example, the following pair of patterns:
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 = 4x, y = 2x + 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.
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.
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.?”
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.
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?”)
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: