In our last episode, the class was engaged in sensemaking, thinking aloud, arguing aloud, just plain being loud, at the math behind this sketch:
“So up to now we’ve spent a lot of time in the coordinate plane thinking about lines. In the twodimensional plane, x is an input and y is an output. A line can be formed by any two points on the coordinate plane. We’ve been working with systems of equations, which you think of as algebraic representations of the intersections of two lines. We can also define distance in the coordinate plane, using the Pythagorean theorem. All in two dimensions. So now we’re seeing how this plays out in three dimensions.”
I drew another point, showing them how the prisms were formed, how you could see a negative or positive value:

More students began to see how it worked, as I’d call on a kid at random to take me to the next step. When I finished a second point, the chaos was manageable, but still loud.
“Yo, you want me to be honest with you?” Dwayne shouted over everyone.
“NO!” I bellowed. “I want you to be QUIET!” Dwayne subsided, a little hurt, as I go on, “Look, this is a great discussion! I love watching you all argue about whether or not I’m making sense. But let’s stay on point! I got Wendy questioning whether or not I know what I’m doing, Dwayne howling every time he loses attention for a nanosecond….”
Teddy jumped in.
“Here’s what I don’t understand. How come you have to draw that whole diagram? We don’t have to do that with the usual graph….in, what, two dimensions? So why do we have to do it with threeD?”
“Great question. Here’s why. Go back to my classroom representation. According to this, the Promethean is a quarter inch from Josh, Hillary and Talika in the front row. Was anyone thinking that I’d drawn it wrong?”
“I learned this in art!” Pam, also up front and up to now watching silently, said, while comments around nearly drowned her out. I hushed everyone and told her to say it again. “It’s like…we need to draw it in a way to make our brains see it right.”
“That’s it.”
“My brain hurts to much to see anything!” Dwayne moaned.
“But we don’t have to do it with, you know, xy points.” Natasha.
“Good! Let’s go back to two dimensions. If I want to plot the point (2,3), I’m actually plotting the lines x=2 and y=3, like this:
Alex said “Oh, hey. There’s a rectangle. I never saw that before.”
“You never seen a rectangle before?” Dylan. I ignored him.
“Right. The point is actually the intersection of the two lines, forming a rectangle with the origin.”
Alex again: “Just like this one makes a cube…”
“..prism…”
“a prism with the intersection of the three points. But how come you have to draw it? I don’t have to draw a rectangle every time I plot a point.”
“But three dimensions make a single point much more ambiguous.”
Dwayne sighed loudly and held his head. “This ain’t English class. I can’t handle the words.”
“Ambiguous–unclear, able to be interpreted multiple ways. Let’s start with this point:
“What are the coordinates of this point?”
“Easy,” said Wendell. “Just count along the lines.”
“Okay. How about (1, 1, 2)?” I count along the axes to that point.
“Yeah.”
“How about (8, 10, 11)?
“They can’t be both!” protested Wendy.
“How about (2, 4, 5)?”
While I listed these points, I followed along the axes, just as Wendell suggested.
“How can you have three different descriptions of one point?” Josh asked.
“Exactly.”
“But that means there would be three different cubes…prisms?” Manuel.
“Yep. Let’s draw them.”
Something between controlled chaos and pandemonium dominanted as I drew–with class participation–three paths to that point, and the cubes. With each point, I could see again that increasing numbers were figuring out the process–start with the intercepts, create the twodimensional planes, join up the planes.


When all three were finished, I put them on screen one after the other.
Sophie, ever the skeptic, “Those can’t be the same point.”
Wendy: “I’ve been saying that.”
Arthur stood up. “No, you can see!” He came over to the Promethean and I gave him the pen. “Look. Here’s the original. Start at the origin, go two to the left, and one up. Each one of the pictures” and he shows it “you get to the point that way. So the point is the same on all the pictures.”
Sophie was convinced.
Dylan: “So how come it’s not just (2, 1)?”
Arthur looked at me. “It’s not (2,1) but it is (2, 0, 1).” I replied.
“Oh, I see it!” Wendell came up. “See, you go along the x as negative 2. Then you don’t go along the y. Then you go up 1 on z?”
“You got it.”
“Can you do it for x and y, without z?” Josh.
“Take a look. Let’s see.” And with many shouts and much pandemonium, the class decided that the point could be plotted as (3,1,0) or (3,1) on the two dimensional plane.
“Whoa, there’s lots of possible ways to get to the same point.”
“Can you figure out a way to count how many different ways there are to plot the point?” asked Manuel.
“That’s a great question, and I don’t know the answer. My gut says yes, but it’d be a matter of combinatorics. Outside the scope of this class.”
“Thank god,” said Wendy, and I shot her a look.
In other classes, I let them work independently on the handout at this point, but 4th block is crazy loud and easily distracted, so I brought them back “up front” to check their progress every 10 minutes. The whole time I was thinking, man, I’ve controlled the chaos and the skeptics this far, do I dare do the last step? Or should I keep it for tomorrow?
Here’s where the performance aspect kicked in: I wanted a Big Close. I wanted to bring it all together. Even if the risk was losing the class, losing the tenuous sense of understanding that the weaker kids had.
I wanted the win. After they’d all sketched three prisms, I started up again.
“I began this lesson with a reminder of two dimensional planes. The only thing we have left is distance.”
“How do you find the distance of a prism?” said Francisco.
“What would the distance be?”
I drew a prism:
“Oh, okay,” said Sanjana. “So the distance would be to the corners.”
“Yeah, the lower left is the origin, right? And the top right is the point,” said Sophie.
“Now, you learned this formula in geometry. Anyone remember?” I look around, and sigh. “Really, geometry is a wasted year. Does anyone see the right triangle that the distance is the hypotenuse to?”
Dwayne sighed. “God, Hypotenuse! I don’t…”
“HUSH. Tanya?”
Tanya frowned. “Would one of the legs be the….the height?”
“But where’s the other leg?” asked Jenny. “It can’t be the length or the width.”
“No. It’s across the middle,” and I drew the second hypotenuse and labeled everything.
“So now you can see that the x coordinate is the length, the y is the width, and z is the height. I just labeled the other distance w. So how do I find a hypotenuse length?”
Relative silence. I growled.
“Come on, it’s the mother ship of geometry. In fact, I mentioned it earlier.”
Still silence.
“Keerist. You all bring shame to your families.”
“Wait, do you mean the…the thing. a squared plus b squared?”
“The thing?”
“Pythagorean theorem!” half the class chorused.
“Oh, NOW you know it. So how can I use that here?”
“x^{2} plus y^{2} equals w^{2},” offered Patty.
“Yes, and the other one would be “w^{2} plus z^{2} is distance, squared,” said Dylan, who’d decided he couldn’t disrupt class, so he may as well participate.
“Great. We have two equations, right? Kind of like a system.”
“But there’s way too many variables. We only have two, right?” Sophie frowned.
“Great question again. To solve a system, we need as many equations as we have variables. But since we aren’t using any specific values at all in this discussion, we aren’t looking for a full solution.”
“So what are we looking for?”
“The EXIT!” shouted Dwayne.
I howled back, “Exit is in 15 minutes. BE QUIET!–Again, a good question. What we do in math sometimes is look for meaningful algorithms that can be formalized into useful tools. Like in this case. Right now, I’d need to go through several steps in order to find the distance of a prism. I want it to be simpler. How can I simplify or restate a system? Natasha?”
Natasha, tentatively, “We can add them up.”
“OK, like combination. Anything else? (SHUT UP DWAYNE!) Natasha? No? Fine. Dwayne?”
“Who, me? I don’t know anything!”
“Bull crap. I scrawled on a new page:
x=3y
2x + y = 14
“What do you do?”
“Wait. You mean where you…oh, ok, you put the 3y in for the x and then multiply by 2.”
“Try not to be shocked by your own comprehension. That’s correct.”
“Yes,” Dylan said. “You can substitute. How come you never have a substitute?”
“Excluding his extraneous crap, Dylan has also stumbled onto truth. Substitution. And looking at the equation, I see an opportunity. An isolated opportunity, even. We have two variables that aren’t length, width, and height, right?”
“Yeah. w and distance,” Alex offered. “But we’re trying to find the distance.”
“Exactly. What I’d like to get rid of, really, is that pesky w. And oh. Hey.”
I drew a circle around x^{2} plus y^{2} and pointed it at w^{2} in the second equation
Manuel got it first. “Holy SH**!”
Teddy, Alex, and Sophie were right behind him verbally, although Prabh and Sanjana had already figured it out and were rapidly taking notes, working ahead of me.
As I wrote down the final equation, they were shouting it along, and eventually all the class joined in: “X^{2} PLUS Y^{2} PLUS Z^{2} = DISTANCE SQUARED!” and as I finished it up, hand to god about a third of the class clapped madly (while the rest looked on in bemusement).
I bowed. I don’t, usually.
“But where’d the w go?” asked Josh in bewilderment.
“That’s the point, the system substitutes x and y so you only have to use the length and width and height!” shouted Manuel.
“So you never have to find w!” added Sophie.
“Pretty cool, huh?”
And the bell rang.
How to explain the adrenaline rush all this gave me? It took me a good hour to return to earth. They clapped! Not all of them, but so what?
So much of my time is spent slowing down math to be sure everyone gets it. I rarely can really engage and challenge the top kids “up front”. I give them challenges at other times, and they usually like my lectures, but I don’t often have the opportunity teach something in a way that makes sense to the less advanced but still captures intellects at the high end.
Here, it happened. The top kids understood that I’d piece by piece revealed that 3dimensions are just an extension of the two dimensional system they all knew, but had never thought of that way. Not only did I reveal it, but I did so while using systems, something they’d just been working with. They were admiring the artistry. They got it.
And I did it all with Dwayne and Dylan yipping at my heels.
Epic.
(Here’s an actual promethean shot from that day. The rest of them I rebuilt.)