Posts Tagged geometry

Two Angles on Sustainable Building


8th grader, Henry C.'s, house plan

8th grader, Henry C.’s, house plan

Wildwood 8th graders have launched into a pair of projects this month, one in geometry and the other in environmental science. The discreet but linked projects illuminate the kinds of connections that Wildwood middle school teachers design to enhance their students’ math and science learning.

In Erin Hansen’s 8th grade geometry class this week, students are using their geometric knowledge and reasoning to design a house—using a variety of shapes and geometric elements. The project flows through an obvious mathematical lens: using shapes and elements as a template to construct and argue traditional geometric proofs (remember these?). All expected elements in almost any geometry class.

What I didn’t expect was the other lens through which students would view their work: Urban planning. The home design is for a future world, with limited allowances for space and requirements for energy sustainability. What’s more, students need to be able to describe the reasons and principles behind their design through a TED-type talk given to their classmates.

Matin K.'s initial sketch-up of his city plan

Matin K.’s initial sketch-up of his city plan

In conversation with students, 8th graders Matin K. and Sophie K., I realized that indeed this project is connected to Wildwood’s 8th grade environmental science curriculum.  Matin shows me a rough draft of his design on the computer app, Google Sketch-up. “Mine’s an apartment building,” he says. Looking at his plans, I notice another building—a tall tower, set within what looks like a street grid. “The tower’s part of my sustainable city project,” he tells me, “in [science teacher] Jane Kaufman’s class.” His tablemate Sophie shows me her plan, which she’s drawn on graph paper. Her home’s footprint features circular and rectangular living spaces—with an energy system powered by the sun. Sophie explains that her model home is also part of the sustainable city project unfolding in teacher Deborah Orlik’s science class.

Sophie K.'s House Plan

Sophie K.’s House Plan

Curiousity piqued, I head over to Deborah’s room to learn more about that.

“Erin knows that we’ve been doing a sustainable cities project for a couple years,” Deborah Orlik tells me. “This year, she and I made a conscious decision to put math concepts into our science project so that kids could see how they’re used in real life.”

Asher E. explains an idea to group mate Lucy O.

Asher E. explains an idea to group mate Lucy O.

Looking around Deborah’s science classroom, the scene is similar to Erin’s room. I see Deborah’s 8th graders working together in teams of 3 or 4. Some are sitting together, laptops open. Others are standing at whiteboards, drawing and talking—like Asher E. and Lucy O., who are engaged in a debate over which renewable energy source will most efficiently power their city’s public transportation system. Their classmate, Elijah D., chimes in with an idea he developed for his group—hydro-powered turbines placed in the river running through his city.

Like their project in math class, the sustainable city project will require these 8th graders to determine if their design ideas are realistic, which they’ll need to substantiate in a presentation to peers.

“They can dream big,” Deborah iterates, “but their ideas need to be plausible and supported by scientific research and mathematics.”

These two related projects are intentionally designed to allow Wildwood 8th graders to practice key skills for their academic and professional success: Creative and design thinking, research, and mathematical calculation. This kind of cross-disciplinary connection will help not only these Wildwood students in their future endeavors but will also train them in the kind of thinking that will be necessary as they work to solve the real problems that will face the world in our not-too-distant future.

~ By Steve Barrett, Director of Outreach, Teaching, and Learning



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No easy answers. And that’s okay.

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?

IMG_2875Before 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.

Noah G. offers a possible solution as Alton looks on

Noah G. offers a possible solution as Alton looks on

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.”

Alton's students consider the problem from a variety of perspectives

Alton’s students consider the problem from a variety of perspectives

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.”

Nora B. works through her geometric solution to the problem

Nora B. works through her geometric solution to the problem

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

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The Shortest Distance to Authentic Learning

The moment I walk into Erin Hansen’s middle school geometry class, a flood of math memories comes rushing back to me.  I think about Pythagoras and proofs, oblique angles and octagons.  I remember memorizing rules and postulates, and wondering how any of this would ever connect with the real world. I’ll be honest: all of my math memories are not great.

So I’m definitely not expecting it when I hear Erin tell the class “Today, we’re going to build kaleidoscopes.”

A polygon as seen in the ‘kaleidoscope’

The ‘kaleidoscope,’ as Erin demonstrates, is composed of a hinged mirror—two small glass squares taped together at one shared edge—and an orange sticky note.

Erin asks her students to look at a series of questions in their textbooks that they will eventually need to answer. “But first,” she says, making her expectations clear, “you should experiment and explore. Place the sticky note underneath and try out different angles with your mirrors to see what happens.” Erin knows that learning this new geometric content on angles and polygons (i.e., multi-sided shapes) requires her students to take this essential first step.

One student from each of the classroom’s four worktables gathers the necessary materials and distributes them.  Then, the students commence their work: really, to play, in an intellectual and developmentally appropriate way. These simple kaleidoscopes, combined with each student’s imagination, produce an array of geometric figures—from triangles, to hexagons, and heptagons (that’s a figure with seven sides, I learned).

8th graders Dani L. and Henry A. seek an up-close perspective

By encouraging learning through experimentation and play, Erin taps into her students’ natural curiosity—which drives discovery, and results in learning.  This research with a simple kaleidoscope will provide each student with valuable reference points when studying future geometric content and concepts.

Now it’s time for the class to answer the questions in the textbook, and the answers come quickly, and easily.

A handsome hexagon

“What’s the relationship between the size of the angle and number of sides it produced?” Erin asks.  The students hypothesize that the smaller the angle made with the mirror, the larger the number of sides—illustrating an inverse relationship.  Erin suggests students address another relationship, this one between the angle of the mirror and the number of sides.  Because the students could generalize from their experiments, they were able to infer that the number of sides multiplied by the mirror’s angle would always be 360—the number of degrees in a circle.

At Wildwood, classes like Erin’s use the textbook to support student learning, not drive it.

8th grader, Jacob R., stages a surreal pose with his kaleidoscope

For these middle school mathematicians, the possible extensions for learning are many.  Erin’s students can use this lesson as a foundation to learn any number of geometric concepts that the course requires—determining unknown sides and angles, understanding the Pythagorean Theorem (a2 + b2 = c2), or completing geometric proofs.

And, this lesson is a reminder that sometimes, in order to understand the sublime simplicity of geometry—you just need to play.

~ By Steve Barrett, Director of Outreach, Teaching and Learning

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