How do math teachers kick off a new unit? When possible, I like to do something with manipulatives, or some sort of activity that introduces interesting questions. Last year, I came up with a triangle activity that I’d originally conceived of for congruent triangles, and then realized it wouldn’t work. But something at the heart of the idea struck me as fascinating, and the single day activity was extremely successful. This year, I used it to kick off triangles.
As you read this (if you read this) ask yourself: is this a constructivist lesson, in which kids discover their own meaning, and the teacher is the “guide on the side”? Or an instructivist lesson, with the teacher as “sage on stage”, telling the students the facts?
I think describing key aspects of the lesson through my interactions with the class will help clarify the lesson. Or not. Maybe it’s just a goofy delusion.
Prep: I have several questions written on the white board:
- What constraints exist in triangle construction?
- Can a triangle be made out of any three lengths?
- How many triangles can be made out of any specific three lengths?
- How can we classify (group) triangles?
- How many degrees in a triangle?
I also have some vocab words written out: constraints, properties, and room for more. I created seven of the bags, the Triangle Activity sheet, the Classifications graphic organizer. Planning this took a while until I realized I could leave the logic lesson (see below) for homework; then it all fell into place.
So first, I hand out the Triangle Activity Sheet and a bag for each group of kids (my kids sit in groups of 4, roughly arranged by ability, with strongest kids in the back):
So today we’re kicking off our triangle unit. I could lecture and give you an introduction, but I thought it might be fun to give you some specific memories about triangles to introduce the shape and help you understand that it’s a little more than ‘just take any three sides and put them together.’ So start off by emptying out the bag of everything EXCEPT the three identical paper triangles, okay?”
The kids obediently dump out the contents. They see paper fasteners (you know, those little gold things?), and strips of paper in various lengths and colors. The strips all have holes on each end, put there by a paper punch.
“Okay, first, I want you to know that this activity is going to create some really cheesy looking triangles, but it’s precisely because they are so cheesy that I like this exercise, because it proves an important point about triangles. So no giggling or mocking my strips of paper, okay? I spent hours making them.”
Naturally, a wiseass in the back of the room says “Oh, man, these are so flimsy and cheap! Who made this crap?” all to get the Killer Stare from his teacher.
“But before you start making triangles, I want you to NOT make triangles. Take a second to read the instructions in Part I.”
(Brief pause. But I do not make the rookie’s mistake, dear reader, of assuming that the kids are all obediently RTFM just because I provide them time and orders to do so.)
“Okay. Eddie, what do the instructions say?”
“Um, what part again?”
“Craig, wanna go for a trifecta?”
“You just said it. We have to find strip combinations that DON’T make triangles. But we can’t bend them.” (Craig, btw, is the wiseass. He has redeeming qualities.)
“Exactly. See the holes on the end?” Wait for yeses, don’t get enough. “SEE THE HOLES ON THE END?”
Line those up. Some of the combinations will not extend sufficiently. Remember that you can change the angle measures, and try different lengths. Now. Turn over the instructions.” I listen and look for the half sheets to turn over. “Does everyone see the smaller of the two tables? When you find a combination of strips that can NOT make a triangle, you put the three lengths in this table. Got it? No, that wasn’t loud enough. One more time: You will put the strip lengths that do NOT form a triangle in this little table. All of you should keep your notes updated, but I’m going to collect one handout from each group. Ready? Louder?” YES! “Okay, go.”
I wander around the room, periodically reminding students that they can alter the angle measure, and within 5 minutes all student groups, even the weaker ones, have at least three combinations. I call on each group to provide an example, and we soon have 8 or 9 combinations on the board.
“Excellent! Now, I’m going to say a combination of strips that’s not here on the board, and I want to close your eyes and visualize those strips. Don’t say anything, just visualize. Close your eyes!”
“Okay, visualize the strips 2, 3, and 9. Don’t answer this question, just think: can you make a triangle with strips of 2 inches, 3 inches, and 9 inches? Think for a minute, don’t say anything. Okay, now: Put your thumb UP if you think you can, thumb DOWN if you think you can’t.” The bulk of the students have their thumbs down (this is a very strong class), and even many of the struggling students are visualizing correctly and have thumbs down. A few kids, no thumbs.
“Ike, no thumb? Do you think you can make a triangle?” (Ike is in the front of the room and didn’t think to look around and get a thumbcount. He is chagrined.)
“Um, sure.” I hold up my hand to forestall the kids who want to correct him.
“Try it.” Ike tries to arrange the strips.
“No. You can’t.”
I add 2, 3, 9 on to the board. “Okay, Ike, how about 2, 3, and 3?”
“Yeah,” Ike says instantly.
“Yes, you can. That time you could see it, right?”
“Yeah. Two strips have to be longer than the other one or you can’t make a triangle. ”
“Very good! Anyone else see that?” A bunch of hands go up. “We’re going to formally learn this later, but Ike has articulated half of what’s called the Triangle Inequality Theorem. No need to write it down right now.” (I’m writing it on the board.) “In a triangle, the sum of two sides MUST be greater than the third side. There’s more to the theorem than that, but we’ll stop on this point, because I want you all to consider something. Before right now, how many of you thought you could just take any three lengths and make a triangle?” Most of the hands in the room go up, and most importantly, I see the top kids going “Hmm.” They realize that yes, indeed, they had thought that, which meant that this little exercise with no clear agenda had, in fact, taught them something. “Here’s what’s even stranger: before today, if I had asked you to make a triangle with a toothpick, a pencil, and a pogo stick, what would you have told me?”
“Wow,” said Mary. “You can’t.”
“You can’t. And you all would have instantly realized that, had you been given specifics, right?” Many nods, I own the class right now. “So this brings up something important about triangles: they have constraints. They have constraints that you have never considered, and yet if you had considered a triangle with specific questions, you would have instantly realized some of the constraints. Alan?”
“What’s a constraint?” I’m walking over to the vocabulary section while several students explain that it’s a restriction, or “something you can’t do”, and I write their definition on the board.
“So now, we’re going to build triangles. See Part two of the handout? Create triangles that meet the criteria. There are more than one combination of lengths for most, but not all, of the triangles listed. I recommend getting them all identified and lined up BEFORE you use the paper fasteners. And remember, if a hole breaks I’ve got more of the support doohickeys.”
So off they went, creating triangles with enthusiasm and precision. After 15 minutes, they all had completed tables that looked something like this:
Then I passed out this simple graphic organizer. Amazingly, many triangle classification graphic organizers (yes, there are a variety out there on the internet) miss the opportunity to visually emphasize the simplicity of triangle classification. For example, this one completely mucks it up, and this one buries the lede. This one is perfect, but it gives out all the information up front, when I wanted the kids to pull it together in a class discussion. I had already created mine which looks like a simpler version of the last, but really, how many ways are there to show it visually?
Now, some of you are saying, “WTF???? You need a GRAPHIC ORGANIZER to tell kids how triangles are classified?” Well, yes. I’m not a big fan of taking “math notes” for all but the very top kids. If a kid wants to take notes, fine. But
I hand out graphic organizers for most of the information I want kids to keep, so I can tell them what handouts to review for the test. If I want them to take down a page of notes (for example, coordinate geometry), I tell them specifically to copy down my board notes as is—and then, a few weeks later, I ask about those notes and check to see if kids have them (compliance rate this year close to 90%).
It takes about 8 minutes to go through the organizer; I look sternly at the top kids and stop them from blurting out the top-level categorizations, giving the mid-levels time to think and suggest. Then we complete the organizer working together; I tell the kids to put an example in each.
I hand out a triangle classification worksheet from KUTA (I used pages 2 and 3), and the kids work through it busily. The top kids rip through it the quickest; it’s not that hard. But this section here gave them opportunity to think:
Two of the triangle classifications aren’t possible, so when the top kids finished this, I made them write out why two classifications weren’t possible. This gave the less adept kids time to finish.
“So Jasmine, did you find a triangle type that wasn’t possible?” Jasmine stays silent, desperately hoping that someone else will shout out the answer, but I have poleaxed the usual suspects with a stare, and wait her out.
“Very good! Why isn’t that possible?” Another long wait—an eternity to Jasmine, no doubt.
“They were in different circles?”
“You..said that a triangle could be in only one circle?”
“I did! Clark, can you add to that?”
“A triangle can only be in one angle group?”
“Exactly. A triangle can only be classified by one angle type. Actually, if you put some information together, you may be able to figure out why a triangle can’t have both a right angle and an obtuse angle, but don’t worry about that now—we’ll work on that more tomorrow. Ellie, how about the other impossible triangle?”
“Good! Now that one is a bit tougher. Why isn’t it possible? Kevin?”
“Well, wouldn’t all the angles have to be equal, if all the sides are? And if they were all equal, the total degrees would be 270.”
“Very nice! We haven’t established all of the facts you used, but your reasoning is good. There’s another way, too. Maya?”
“The hypotenuse of a right triangle is the longest side, right?”
“Nice. Candy, can a triangle with three equal sides have a longest side?”
“Does a right triangle have a longest side, class?” “YES!” “So can a right triangle have three EQUAL sides?” “NO!”
(okay, I know some of the weaker students don’t quite grok this, but I’m aiming for the top students now, so I’ll pick up the pieces later.)
“Okay. Very good. The big idea, again: every triangle fits into an angle classification and a side classification. All of them. Turn over your handout.”
“This is your homework. I’m going to go through it during our advisory, so don’t sweat it now. But this organizer reveals something critical about triangle classifications that I want you to think about tonight. Also, more logic. Don’t groan!”
[Groan. They do not like logic.]
“Okay, now put the graphic organizer away. Did everyone make a parallelogram? Everyone take the parallelogram they built, and any one triangle. Apart from the number of sides, what differences do you notice?”
It takes the kids a while to figure it out, even as the parallelograms collapse the minute they pick them up. I let them mull it for a while, and they do come up with some creative offerings. I hint at it by convincing them to move them around.
But I finally tell them to hold up each figure.
“Hey! Look at that!”
“That’s not a real difference! They’re just made differently!”
“Made differently? You made them! What did you do differently?”
“Okay! They were both made of index card stock and paper fasteners. So what’s keeping the triangle up?”
“So take a look at the board, question 3. How many different triangles can be made from three specific lengths?”
“Ooookay, how many different parallelograms can be made from two pairs of strips?”
“Lots,” says Angie promptly.
“Show me.” Angie takes her parallelogram and moves it from a rectangle to a steeply slanted parallelogram.
“Perfect! Now, which has the biggest area?” Silence. “Angie, stand up and show everyone your parallelogram in the super slanted position. Okay, now push it up a little bit. Class, what just happened to the area of the parallelogram?”
“Bigger,” says Ron.
“Angie, push it up more.”
“Oh, I get it,” Karinna says. “It always gets bigger if you make the….um. I don’t know how to explain it.”
“Think in terms of height. Angie, put it back to super slant.”
“Oh!” Ike says, “If it’s taller, it has a bigger area!”
“And where does that end? Angie, keep moving it..moving it….Craig?”
“When it’s a rectangle. Because after that, it’s going to start going down again.”
“Nice. So you see, guys, a parallelogram has an infinite number of areas, although the largest area is going to be when the sides are perpendicular to each other. No need to write that down; I am just making a point really about triangles. Back to our question. How many different triangles can be made from three specific lengths?”
Pause, then “Just one?” asks Ron.
“Everyone made a 5,5,9 triangle, right? Hold it up. Look around, everyone. See any variety? Or do they all look the same? ”
“But why?” asks Effie.
“It all goes back to Euclid. How many non-collinear points in a plane?”
I kid you not, the class gasped as they realized the connection. “THREE!”
“That’s right. Any three points define a unique plane. Only one possible plane. So the triangle has structural integrity and rigidity. The parallelogram does not. If I tell you the three lengths of a triangle, it’s mathematically possible to determine the area. Not so with a parallelogram. And there you have the reason I love this little exercise. You made these shapes. You can see the triangle hold up, the parallelogram collapse. You know there’s no trick. It’s just the triangle. So if you’re going to build a bridge, or a skyscraper, what shape is going to ensure your structure won’t fall down?”
“But look around this room. See any triangles?
God love ’em, they really do look around the room.
“How about squares, rectangles and parallelograms in this room?”
“They’re everywhere,” says Zeus.
“Isn’t that weird? I don’t know if it’s human or just cultural, but we have a thing visually for rectangles. When was the last time you saw a triangular table of figures, or even a triangular dining room table? I was thinking of making the table on the back of this handout triangular, but it would distort the information! But even given our fondness for rectangles on the outside, we know to build and support our rectangles with triangles. And this goes back to the earliest times. Look at Roman architecture, and you can see triangles everywhere. When you leave the class, look outside at the classroom wall and the overhanging roof. You’ll notice struts holding it up, and what will the shape be?”
Bert: “But can’t you just add something to the rectangle to make it stay the same?”
“What, you mean like this?” I hold up two parallelograms that keep their shape:
“What’d I do?”
“Made triangles!” the class chorused.
“Indeed! Okay, for my last trick, take out the three identical triangles in the paper bag. See that each triangle has a different angle colored in? The Angle Addition Postulate says that if we align these angles so that each of them share a side with one of the others, the sum of their angles is equal to the sum of the larger angle that they form together.”
(I forgot to take a picture of this, but it’s a well known demonstration and I just stole an image off the web. Normally, you have one triangle and tear it into three parts, but I want to keep these bags as kits for future use.)
I wander round the room and check; all groups have their triangles arranged so that they can answer the key question.
“Okay. Your triangles are of all different shapes and sizes. But when you align their angles together, what do they form?”
“A straight line!”
“And how many degrees are in a straight line?”
(I am getting very loud full-class responses here, not just the top kids in the back.)
“So the angles in a triangle add up to….”
I wrap it up by going through the questions I posted on the white board, ensuring general understanding of the key concepts. And then.
“Okay, before we go through the homework, I want you to realize something. All this work you did today wasn’t proof of the facts you learned. They were demonstrations. You have demonstrated visually that a triangle has 180 degrees, that a triangle has rigidity, that the sum of two sides of a triangle must be greater than the third. And I like demonstrations! They are not to be sneered at. But don’t confuse a demonstration with proof. We’ll be proving some of these facts during the unit; in other cases, the proofs are more complicated than I want to go into. But don’t ever confuse a picture with a proof. A proof is an argument built with logic, facts, and definitions. But a picture or an experience is much easier to remember.”
So, what do you think? Was this a “discovery” lesson, the classic progressive ideal that traditionalists sneer at? Was it constructivist or instructivist? Teacher-centered or student-centered?
Instructivist, teacher-centered…..and discovery-based.
Most progressive, constructivist “discovery” lessons are insufficiently sculpted. Check out the pictures of this lesson, which is clearly covering much of the same material that I did. But the kids are spending a lot of time creating triangles with a protractor and ruler. I don’t want them spending time on that right now, so I give them the strips. Moreover, the kids are spending a lot of time putting together a presentation, which they are then going to, well, present—and everyone is going to hear the same basic information 8 or 9 times, as each group presents their “findings”—which, in the happy talk, differs for each group but in reality is the same thing over and over again. Tons of wasted time, and if you ever see a classic group-work class, at least a third of the kids aren’t doing a thing (cf Boaler’s bias).
While I wouldn’t do such an open-ended lesson, my point isn’t to argue for my lesson’s merits. I want people to notice the difference between a classic constructivist, discovery-oriented lecture and one in which kids put things together or complete discrete tasks that lead to immediate, unambiguous findings that we then translate into facts. I’m still the sage at center stage, but the kids aren’t just listening. They have sense memories and are participating in long class discussion interspersed with tasks that they can all do.
A classic instructivist approach would be a lecture, in which the teacher gives notes and the students listen and take notes. I could, of course, have covered this material in about 15 minutes. The top kids would remember it. The bottom kids would not. Plus: boring.
I am not a fan of straight lecture every day—well, actually, I rarely but ever do a straight lecture. Even when I’m talking, I’m engaging in a class discussion to move things along. But for the teacheres who do lecture daily, why not vary the routine? Find some good activities that demonstrate essential principles, with some handouts. It takes a little work the first time, and you often have to modify the activity a few times until you find the right balance of tasks and fact delivery. But the end result is almost always an enjoyable activity that gets the same information across.
But the larger point is this: many people sneer at constructivist teaching. I am not a fan, either. But so long as we are teaching kids who don’t want to learn math, we need to accept that the lecture is just zooming right over the heads of 75% of most classes. I’d rather reach a larger audience.