Geometric Constructions Task: Pizza Delivery Regions

The following task is probably best suited for the a beginning unit on Geometric Constructions or use of a compass and straight edge. However, you may also wish to use it as a fun review as students reenter your classrooms in January, groggy from two weeks of sleeping late.

The task is adapted from the NCTM Illuminations task “Dividing a Town into Pizza Delivery Regions” (membership required). Students are asked to split a town into delivery regions based on proximity. We’re hoping to get students to use a compass and straight edge to construct overlapping circles.

I adapted it by making it local to my area (using a simple search of “Domino’s Pizza” on Google Maps. I then split it into a warm up in which students identify which Domino’s Pizza four specific locations would go to. From there, students are given a larger map and asked to generalize the regions (that’s where the constructions and tools come in). The other adaptation is simply a solicitation for the map and a written explanation for that map.

Here’s the task: [Pizza Delivery Regions]

And in the spirit of good assessment practices, here’s the rubric I’ll use.

Feel free to share in the comments how you’d adapt the task even further. And if you do so and implement the task, feel free to email me (gmkrall@gmail.com) and share some student work. It’s always appreciated.

As an aside, I’m starting to feel a bit odd about how often I blog about pizza….

Why don’t students feel like they’re learning? (when they actually are)

It’s something we’ve all experienced: we’ll lecture and feel like students got it. Maybe they’ll even do well on the practice problems we assign them. Then the next day or the next week we try something a bit more open ended – a problem based lesson, a 3-Act Task, an Open Middle task – and it feels … unresolved. The practice problems we assign after the task don’t look like the ones we just did, so naturally they are completed at a lower success than via our rock solid lecture.

According to research, however, it’s possible our rock solid lecure wasn’t as effective as we (or our students) thought. And it’s just as possible the more “active instruction” was more effective for student learning. In fact that’s what this recent study suggests. From Measuring actual learning versus feeling of learning in response to being actively engaged in the classroom by Deslauriers et al. (2019)

“Students in active classrooms learned more (as would be expected based on prior research), but their perception of learning, while positive, was lower than that of their peers in passive environments.”

“[W]hen students experience the increased cognitive effort associated with active learning, they initially take that effort to signify poorer learning. That disconnect may have a detrimental effect on students’ motivation, engagement, and ability to self-regulate their own learning.”

To be blunt, rote tasks are a drug for both teacher and student. With every passing lesson where students are rewarded for demonstrating low-level thinking and under- standing, it becomes more challenging to wean them off rote tasks. Making matters worse is an obsession with standardized test scores and coverage of standards, which contribute to the cycle of low-level tasks. I’ve seen the cycle countless times. It goes like this:

1. Students exhibit low performance on standardized assessment.
2. Teacher decides to use complex tasks to enhance rigor.
or students are highly reticent when it comes to tackling the task.
4. The teacher follows up the seemingly failed experiment with more rote instruction.
5. The rote instruction is seemingly well received.
6. Students exhibit low performance on standardized assessment.

A lecture, like low level tasks, can lull a class into offering false positives about student knowhow.

In many ways, this article is reassuring for those of us who promote more active learning over teacher-driven lectures. But before we pat ourselves on the back too much and say “seee?!!!!” we should eat a slice of humble pie and recognize that if students don’t feel like they’re learning, that’s almost as bad as not learning. As the authors write, “As the success of active learning crucially depends on student motivation and engagement, it is of paramount importance that students appreciate, early in the semester, the benefits of struggling with the material during active learning.” So I read the study as a call to action to make “active instruction” more public in its knowledge transfer. Here are a few ways to make sure students are learning and know that they’re learning.

1. Celebrate what students did and recap what you intended students to do. When you have a complex task that students took in another direction, first off: that’s great! That probably resulted in more permanent learning if they had done it the way you intended. Afterward, share the approach you took to solve the problem and reiterate that it’s pretty cool that they chose a different route.
2. Make your follow up problems more aligned to the task. Call an “audible,” if you will. Right after the complex task or the student centered activity, let students demonstrate their learning via a handful of practice problems that are related to the concept you just tackled. In a past blog post I called it an “audible.”
3. Ask students to rate themselves on a rubric before and after the activity. You’ll be amazed at how the simple act of self-assessing against a rubric will communicate growth for many students. Here’s an example of a Problem Solving Rubric (again, from Necessary Conditions, but modified from my employer, New Tech Network).

Over the course of a day, a week, a unit, or a grading period, ask students to rate themselves on their skill. Once at the beginning, once toward the end. They’ll be able to see their own individual growth and you’ll be able to have excellent data on how students are experiencing your class.

The research paper itself contains a few additional recommendations for making students more aware of the benefits of active learning activities and I recommend you check those out as well.

How do you communicate to students that the struggle is indeed worth it?