Current Digital Pythagorean Theorem Problems
I first looked at a few commonly used pythagorean problems to have a point of comparison. Here are a couple examples from IXL and Khan Academy:
IXL - (More Examples)
Khan Academy - (More Examples)
These questions can be effective for a quick check of student understanding, but do little to meet the objectives we established in part one:
Prepare students to use multiple strategies to solve problems and justify their solutions.
Situate mathematics problem-solving within a real world context.
First Prototype Attempt
The cell tower unit presented by TACIB already established a strong context by asking students to determine whether cell towers should be built near their school. Because students apply the theorem to inform a decision, students are better able to connect how solving math problems helps them make more informed decisions in the real world.
With the TACIB unit as a reference, I started with the following prototype.
I liked the following about the prototype:
It makes a clear connection between solving a problem and gaining a piece of useful information. By applying the Pythagorean theorem, students can determine whether a household can receive cellular service. The wording of the question places the emphasis on whether people can get cell coverage and not the distance between two points in a grid. Solving the math problem is merely a means to help make an important decision.
In one problem, we offer three opportunities for students to apply the Pythagorean theorem. This gives us some flexibility in how we use this problem. A teacher could for example, follow an “I do, we do, you do” approach in which they model with one household, get student volunteers to work together on computing another problem, and ask students to determine the last problem.
Another more collaborative approach could be to ask a group of students to split up the households so that they could each calculate the distance of one household, compare their work, and as a group make a decision about whether all households can receive cell coverage.
There is some opportunity for students to solve this problem in different ways. For example, students certainly can calculate the distance between every household and the cell tower, but some students might first estimate the distance of each household to the cell tower. They can then determine that certain households that are well within range may not require an additional calculation.
Here’s what I still wanted to improve:
The work here still boils down to applying a formula and at most reaches the “apply” level of Bloom’s taxonomy. While it’s important teachers offer problems that assess these lower levels of remembering, understanding, and applying, I wanted to see whether I could also offer opportunities for higher levels of critical thinking.
There is still only one right answer to the question. There’s nothing wrong with this, but I think there’s real value to showing students that there can be multiple reasonable answers to a problem. This naturally unlocks opportunities to compare and contrast answers, communicate tradeoffs, and attempt to justify the better approach for a specific context.
One of the most common ways I upgraded my tasks to the higher tiers of Bloom’s taxonomy while teaching was to ask students to optimize for a certain condition. In the second prototype, I ask students to optimize for the minimum number of cell towers that can be placed. Students now need to actually design and lay out the location of each cell tower. This requires them to iterate over a few options, compare and contrast these options, and ultimately justify which option makes the most sense.
In addition, there are multiple reasonable answers to the question, which can provide better opportunities for students to justify their answers to peers. In more standard problems, when students are asked to check their answers, they often just check if their answers match and then move on. In a problem like this, however, even if both students agree on the minimum number of cell towers, they likely have placed their cell towers in different locations. As a result, they’ll need to more critically analyze their peers’ work to make it’s reasonable despite not matching their own answer.
Final Touch Up
There are two small updates here.
The question has been changed from “What is the minimum number of cell towers that need to be built so that all households get cell coverage?” to “How many cell towers should be built and where?”
I added one house at coordinates (13, 14).
The more open ended question allows students to justify a greater variety of answers. While I imagine most students will still attempt to minimize the number of cell towers that covers all households, by adding one outlier household, a few students might question whether it’s optimal to cover all households or just the majority of them.
When such questions are asked, we can begin exploring real issues in our society. For example, is it ok to leave certain households outside of cell service range? In the real world, decision makers need to weight the tradeoffs of different approaches and rarely does a single decision benefit everyone. By providing more open ended questions, we provide an opportunity for student to start grappling with these challenging questions.
I hope this problem also communicates that even if we correctly solve the math problem, that in itself does not completely dictate the right answer. Instead, solving the math problem enables us to make a more informed decision. Depending on our priorities (minimizing costs vs ensuring equitable access), we may arrive at different solutions.
Through the current design, there are a few intentional trade-offs that are being made.
As problems become more open ended, it becomes harder to guarantee that students will solve the problem using one exact method. This makes it difficult to assess whether students have actually mastered the Pythagorean theorem since this particular problem can also be solved with other techniques.
Students also need to spend more time on other aspects of the problem that are unrelated to applying the Pythagorean theorem. For example, in the final iteration of the prototype, they need to analyze a few places where the tower can be placed before they can even do any calculations.
Despite not taking the 1-2 weeks the original TACIB project would take, this problem still requires more class time to solve than the traditional problems first presented in this post.
The problem alone does not force students to engage with social justice issues. A teacher may need to prompt this discussion by providing an example to the class or ask guiding questions to help students think outside of the box.
These are all valid concerns. Every problem in the curriculum likely can’t be this open ended given the emphasis on state testing and pressure to cover many standards. I do think, however, there’s significant value in having at least one problem like this every couple of weeks to help students appreciate math’s relevance to the real world.
In part three, I’ll try to justify that these trade-offs are worth it by incorporating this problem into a complete lesson plan.