Quick! How many triangles can you find in the following diagram?
Look daunting? Maybe—but I’ll give you the solution to this one. There are 35 triangles in this pentagon. Upper school math teacher Alton Price recently offered this puzzle as part of a weekly school-wide math challenge, open to all, and posted on walls around the middle and upper campus.
This week, Alton’s math challenge will be more formidable:
How many triangles can you find in this one?
Before posting this more problematic polygon for the weekly math challenge, Alton posed it as a dilemma to students to grapple with in his elective math class, Introduction to Problem Solving and Probability. There’s no easy answer to this one (believe me)—and that’s ok.
In fact, that’s the point. Alton’s class is designed to challenge students’ minds.
“We start off every class with a problem,” he tells me. What’s different from a traditional math class is that Alton doesn’t feed students formulas or algorithms—tricks to help them solve. Rather, students tackle the problems—sometimes successfully, and sometimes not—through reason, trial, and error. And they often discover and construct the formula themselves.
What I witnessed in a visit to his class is a problem-solving process that’s rigorous and collaborative.
Eleventh grader, Noah G., jumps in—walking up to the diagram, projected on the classroom screen. “If we can count all of the points,” he says, “then we can figure out the triangles. There are only so many ways you can connect three lines to make a triangle.”
Tenth grader, Gabe F., looks at the diagram and notices its symmetry. “The shapes on the right are just reflections of the shapes on the left, and vice versa,” he says. “Let’s count the triangles on the right side and just double them…. That will give us the number.”
Alton adds his thoughts, guiding the discussion: “I see polygons… lots of polygons. Maybe there are patterns we can identify? It could be easier than we think.”
It turns out that Alton’s attempts at problem-solving are authentic. He doesn’t have a solution to the puzzle either—a vulnerable position for some teachers but one that Alton eagerly takes on here in the spirit of learning. This polygon was offered up from a teaching colleague, music teacher Hagai Izraeli, who was inspired by last week’s math challenge and drew up this one himself. He’s been seeking his own solutions for a week. “I told him we’d give it a try,” Alton tells me. “I want to figure this one out as much as the kids.”
I get hooked on finding a solution, too, and find myself trying to sketch ideas on my copy.
Ninth grader Nora B. takes a different approach. She heads out of the room with a copy of the polygon and returns a few minutes later with several more copies. She sits down, takes out a scissors, and cuts out as many different-shaped triangles as she can. Gabe notices and admires the approach: “Nora, that is really smart.”
After more than a half-hour of grappling, the kids are getting a better sense of the problem but…not the solution. Alton okays a math-related break. Gabe, fellow 10th grader, John B., and a few others play a quick game of chess to clear their minds.
So what are these kids learning without finding a solution? “For one,” Alton shares, “they’re able to employ their reasoning skills and apply the knowledge they’ve already gained to a new situation.” What’s more, they’re internalizing that many of life’s challenges—math related or not—don’t always have easy answers.
They’re also learning that getting there is sometimes half the fun.
After the others have moved on to other work, I check back in with Nora—still cutting out and arranging triangles—to see how her work is going. “It’s complicated… very complicated,” she says, and I sense her frustration when she adds, “and not very fun.”
I ask her, “So why keep at it?” She turns her head and smiles, belying her true feelings: “Ok… it’s a little fun.” She goes back to her work, confident that she can figure out a solution.
~ By Steve Barrett, Director of Outreach, Teaching, Learning