The first thing I noticed when I walked into Caitlin’s third grade class was that every student was, with bowed head, focused on writing in their math notebooks — even Elliot, who is prone to distraction and generally nervous about math. These students were invested.
“We’re starting with Pilar’s Yard,” Caitlin told me. “100 foot perimeter. But they’re allowed to make any polygon they want. Like triangles.”
Many homes in our school’s neighborhood are squeezed onto tiny but inordinately expensive plots of land, some of which are shaped irregularly. Puzzle pieces. Why wouldn’t Pilar have a triangular yard.
I glanced at some of the work in the class. Sadie had drawn a star-shaped yard, with ten equal sides of a length of 10 feet. Unorthodox? Sure, but it fit the constraints. Meanwhile, a few seats down, Marc had drawn a series of rectangles: 40 by 10, 30 by 20, 25 by 25.
Elliot’s Triangles
“I’m going to make triangles,” Elliot told me eagerly. “Then I only have to add three numbers to get 100.” Honestly? Clever.
Elliot continued: “but I’m feeling a little stuck.”
I waited a few beats before suggesting that we draw a triangle to help us keep track. Elliot drew a roughly equilateral triangle, and then looked back at me, expectantly. A few more beats.
“Okay, what if one of the sides is… I don’t know… 20 feet.”
Elliot labeled one side of the triangle “20,” and then immediately said, “okay, so there’s 80 left. That’s 40 and 40!”
“Beautiful!” I smiled and admired his work. “Is that the only possible triangle if one of the sides is 20? Or could we make another one?”
Elliot took this very seriously. “Forty and forty is the way to make 80,” he said sternly.
“Yes! Do the other two sides have to be the same length?”
It was like a lightbulb went off over Elliot’s head, and he drew another triangle. “What about 20 and forty-ONE. Then the last side is 39. They can be all different!”
I started to take a picture of what he was doing, but his brain was a locomotive with no brake. He started drawing more and more triangles.
“Okay, but also what if we had a 10 and a 10. That makes 20, and then we need 80 more, so it’s 10 and 10 and 80.”
“Yes! Love it!” It was then that I realized that… wait. That triangle wouldn’t work. A side of 10 feet and a side of 10 feet would need a third side of less than 20 feet to be joined into a triangle. A final side length of something like 5 or 14 would make a lot more sense than 80.
But Elliot kept going.
“And 30 + 30 is 60, and so 40 more.” That one works! “And 5 + 5 is 10 and 90 more!” That one doesn’t. Hmm.
I considered whether it was even worth it to tell Elliot that these triangles were impossible. The point of the task was to generate perimeters of 100, and, if we could suspend the physical nature of a triangle, Elliot was able to come up with many different ways to sum to 100. Like 60 + 39 + 1 does equal 100! Wasn’t that enough?
And the kids that focused on rectangular yards didn’t have this problem at all. As long as they had two pairs of congruent sides (like 20 + 30 + 20 + 30, or 2 + 48 + 2 + 48) their rectangle was possible.
Impossible Triangles
This thinking is actually addressed in 7th grade, and then implicitly in high school geometry.
7.G.A.2
Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
Standards are benchmarks that help us craft a mathematical story of a grade, and, more importantly, across grades. Most of the time spent in math class should be spent on grade level content, even if that means thinking about how to push that content deeper without necessarily advancing to future grade level content. And sometimes we brush up against future content, without expecting students to develop a deep understanding of it. We’re just making connections. Building neural pathways.
This is from a Grade 7 lesson (Desmos 6-A1 Math: Grade 7, Unit 7, Lesson 5)
The green dots represent possible values for the green side length (shown as 6 in the diagram). It looks like some of the values that work are 6, 7.1, 8 1/2, etc. Five is too small, and 11 is too large. Why is that? What do we notice about those numbers in relation to the two given sides of 3 and 8?
When the sum of two sides is equivalent to the measurement of the third side — 3 + 5 = 8, or 8 + 3 = 11 — a triangle cannot be formed because the sides would lie next to one another in a line. If the third side is longer than the sum of the other two sides — 3, 2, and 8 — they won’t join to form a triangle, either. There would be an extra length hanging off. (Assuming that all lengths are given in the same unit of measurement. Yes, Devil’s Advocate, that one is for you.)
Sharing Elliot’s Triangle Dilemma
Classroom teacher Caitlin and I decided it would make sense to amplify Elliot’s thinking around generating triangle answers, with a focus on how he determined missing addends rather than on the impossible nature of some of his solutions. I facilitated the discussion.
“So Elliot wanted to create a triangle with a perimeter of 100 feet,” I started. “I think some of you were also trying to make triangles. Do a ‘me too’ sign if you were also working on that,” I said.
Brayden’s hand shot up. “I was, but I tried 33 and 33 + 33 + 33 = 99, which isn’t 100.”
“Oh, that’s a perfect idea for us to built from!” I grinned. “So there’s this question… do they all need to be the same side length? Can they have different lengths?”
I saw some students nodding, and some shaking their heads no.
“Okay, so here’s how Elliot started.” I picked up a dry erase marker to write on the board.
20 + ___ = 100
Elliot immediately chimed in. “I know that 2 + 8 = 10, so 20 + 80 = 100,” he stated proudly. I recorded this on the board.
“Okay, so 2 tens + 8 tens = 10 tens, which is 100,” I restated. “Love that. But that seems like that’s only two sides… a 20 and an 80…”
“Oh, I made the 80 into a 40 and a 40,” Elliot clarified.
“Right, right, so then you end up with 20 + 40 + 40 = 100.” I drew the triangle on the board next to it, with the sides labeled. “Here we go! A perimeter of 100 feet.”
“And then I did 20 + 39 + 41,” Elliot continued.
I added this to the board. “So it sounds like you’re okay having all three sides being different lengths.”
“Yeah!”
Sadie raised her hand. “But couldn’t you break down the other number? What if you used 20 + 80, and instead of breaking down the 80, you broke down the 20 into 10 + 10.”
The impossible triangles strike again!
Should I mention it? Most of the triangles I had drawn were not to scale in the slightest. My 20-40-40 triangle looked downright equilateral.
In teaching, we have to make split-second decisions, and, in that moment, I decided I’d go for it. If I could go back in time, I might make a different decision. But here’s what happened:
“I wonder what this triangle would look like!” I mused, as I drew two (roughly) congruent lengths to represent the side lengths of 10. Then I added an outstretched 80.
Sadie laughed. “That’s not a triangle!”
“It’s not?!” Classic teacher move: feigning comic surprise. “Why not?”
Sadie continued: “It’s not closed.”
“Okay, so, technically, it’s not a triangle,” I said. “Sadie’s right that it isn’t a closed shape. But also if we could put together a triangle with the side lengths 10 + 10 + 80, it would have a perimeter of 100. And that’s what the learning was about today. Let’s come up with two other triangles — real or imaginary — that work.”
Students suggested 30, 30, and 40, and 20, 50, and 30.
The next day, we would share some shortcuts for coming up with the perimeter of rectangles.
The Main Road, and Detours
When we’re planning for a lesson, we identify the essential learning for students. That doesn’t mean that every student has the same takeaway. The standards set benchmarks (to be met by the end of the year), and then every day we continue to support students in building understanding, making connections, and consolidating or practicing knowledge.
The lesson today was about understanding that the perimeter of a polygon is the sum of all of its sides, and from there
How big a detour can students handle? …and how many detours? The responses to these questions feels like it depends entirely on the students in front of you. Maybe Sadie benefited from the Impossible Triangle detour, and Jack tuned it out, but Lia feels confused. We work to achieve a balance between the main road and the detours we choose. These decisions are complex. That’s something I love about teaching.
Jenna Laib // Embrace the Challenge 
I love everything about this. First, you & Caitlin honored the student’s thinking about finding 3 numbers whose sum is 100. Second, you avoided the pitfall similar to “rules that expire” where learning in earlier grades gets eclipsed by later concepts that contradict previous ideas; in this case that just because 3 lengths sum to a particular perimeter doesn’t mean they will work as side for a triangle (which students will encounter later as the triangle inequality theorem).
I also noticed that students seem to have a preconceived notion that triangles should have equal sides, but hopefully this learning episode expanded their conception of what triangles can be. [Chris Danielson has promoted using non-regular triangles and other shapes in children’s toys, books, & learning experiences.]
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Oh, I hadn’t thought about how “Elliot” and others may have been working off the idea that triangles (and other shapes) should have equal sides! Like he may have been simplifying because doubling and halving are easier, and kids often gravitate towards those relationships, but I think you’re right — it may go deeper and into his geometric ideas. He certainly wasn’t the only one!
Thanks! 💛
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I love this, Jenna! Your writing always reminds me how important it is for a teacher to know not just his/her grade level math but where students are coming from and what they’re going to learn later down the road of learning. I wonder how we can make that knowledge more accessible to more teachers.
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YES! Absolutely! And most curriculum include information about this, and I’d hazard a guess that these pages would win the prize for “most skipped” in the Teacher’s Guide Awards.
I think there are lots of opportunities to do this through content-based coaching, but that’s a labor intensive model that many schools and districts can’t afford right now. I also think there’s a lot of benefit to switching positions every now and then! I feel very fortunate that I’ve been able to work K-8.
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This blog post about impossible triangles is intriguing and makes me want to try and solve the puzzle. I appreciate how the author breaks down the problem and provides helpful tips to solve it. Overall, a fun and challenging read for any math enthusiast.
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