Tag Archives: systems of equations

Great Moments In Teaching: The Third Dimension (part II)

In our last episode, the class was engaged in sense-making, 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 two-dimensional 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:

3dplot1112colored

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 three-D?”

“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, x-y 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:

3dambiguouspoint

“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 two-dimensional 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.”

2ddistance

“How do you find the distance of a prism?” said Francisco.

“What would the distance be?”

I drew a prism:

prismdistance

“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.

prismvertex

“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.

prismdoubletriangles

“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?”

“x2 plus y2 equals w2,” offered Patty.

“Yes, and the other one would be “w2 plus z2 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.”

prism2equations

“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 x2 plus y2 and pointed it at w2 in the second equation

prismsubstitution

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: “X2 PLUS Y2 PLUS Z2 = 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.

prismfinalformula

“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 3-dimensions 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.)

3dexampledistance


Great Moments in Teaching: The Third Dimension (part I)

“How many other dimensions are there?”

“Well, four, according to Einstein, and five according to Madeline L’Engle, if you’ve read A Wrinkle in Time.”

“I have!” Priya’s hand shot up. “It’s a tesseract!”

I was impressed. Not many girls read that classic anymore. “But we’re going to stick to three dimensions.”

“Isn’t real life three dimensions?” asked Tess.

“Yes. But if you think of it, up to now, we’ve only been working in two. We’ve spent a lot of time in the coordinate plane thinking about lines. In two dimensions, 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.”

Now, no mocking my terrible art skills” and I put up this sketch, the drawing of which occurred to me the night before, and was the impetus for the lesson.

3dclassroom

Everyone gasped, as they had in the previous two classes. My instincts about that clunky little sketch proved out, beautifully. No clue why.

“Holy sh**,” groaned Dwayne, the good ol’ country boy who offered to paint my ancient Honda if I gave him a passing grade. He doesn’t like math. He’s loud and foul and annoying and never shuts up. That last sentence is a pretty good description of me, so I’m very fond of him. “What the hell is that? Get it off the screen, it hurts my eyes.”

“That is awesome,” offered Talika, a senior I had last year for history. “How long did it take you to draw that?”

“What is that white stuff spread everywhere? Did someone get all excited?” asked Dylan, a sophomore whose mother once emailed me about his grade, giving me the pleasure of embarrassing him greatly by describing his behavior.

“Ask your mother,” I replied, to a gratifying “ooooo, BURN!” from the class, who knew very well what had happened.

“How did you draw that?” asked Teddy, curious. “It’s not ordinary graph paper, right?”

“No, it’s isometric paper, which allows you to draw three dimensional images. So…”

“This is really stupid,” said Dwayne. “I’ve taken algebra 2 three times and no one’s ever taught me this.”

“Best I can tell, no one’s ever taught you anything , and not just not in algebra 2,” I replied, earning another “Oooooo” from the class and an appreciative chuckle from Dwayne.

“It’s weird, though, because in two dimensions, you start in the middle,” offered Manual, who was consulting with Prabh, another bright kid who rarely speaks.

“That’s a good point! For example, if we were going to plot seating positions in this room in two dimensions, we’d start with Tanya,” I said, moving to the class center and indicating Tanya, who looked a bit confused. “So Tanya would be the origin, and Wendell would be (1,0), while Dylan would be (1,-1).”

“I’m not negative!” Dylan said instantly, talking over my attempt to continue. “You’re saying I’m negative. You don’t like me.”

“Hard to blame anyone for that,” said Wendell who is considerably more, er, urban than Teddy, with pants down to his knees and a pick that spends some time in his hair. Despite his occasional class naps, he maintains a solid C+, and could effortlessly manage a B if I could just keep him awake. “S’easy, dude. It’s like one of those x y things, like we’re all dots on the graph.”

“You’re one down and one to the right of me,” pointed out Tanya.

Dylan was interested in spite of himself. “So Talika’s, like, (0, 4)?”

“Yes,” several students chorused.

“Then I’m negative 8.” said Dwayne, unhappy with any conversation that doesn’t have him at the center.

“More like….(-2,2), yeah,” says Cal.

Ben speaks up, “But how come Tanya’s at the center for mapping the room’s people, but your sketch is, like, from the left?”

“Or right?” Sophie, from the back.

“Or is it….outside?” asks Manuel.

“Yes, it’s kind of like you’re standing on a desk in Ms. Chan’s room and the walls are transparent,” says Ben, more certainly. Ben is repeating Algebra 2 after having taken it with me last semester. Very bright kid who clowned incessantly, confident in his ability to learn without really trying, only to learn that Algebra 2 was different from other nights, and he wasn’t finding the afikoman. I advised him to repeat. The big sophomore not only agreed, but specifically asked to repeat with me. His attitude and behavior is much improved. I ran into him while walking across the courtyard a few weeks earlier, and he said “I just realized I was Dwayne and Dylan combined last semester, and it’s so embarrassing. I’m really sorry.”

“I’m not enough of an artist to know if I could have drawn this any other way. It just seemed intuitive to me last night, when I came up with the idea.”

“See, I knew it,” trumpeted Dwayne. “You’re making this up!”

“Yeah, I know this isn’t in any algebra book” said Wendy, a sophomore whose excellence in math is often hard to discern beneath her complaints. “This is just some weird thing you’re doing to make us think about math.”

I picked up at random one of the four algebra 2 books sitting on my desk (I’m on the textbook committee) and walked over to Wendy’s desk, opening it to the “Three Dimensional Systems” chapter. She looked, and said “Ok, maybe not.”

“So just as we can plot points in two dimensions, we can plot points in three. Take Aditya here,” my TA, who was watching the circus in amusement. “How could we represent him as a point on my graph?”

Teddy said instantly, “Yeah, I’ve been working that out. I can’t figure out which the new one is, and what do we call it? Where’s x, where’s y?”

Sanjaya said, “I think the part along the ground is x. Like if you go along the bookshelf?”

“Like this?”

3dstartoutline

“Yes,” Sanjaya said, confidently. “That has to be x. So you could count to Aditya, right?”

“Count which way?”

“The bottom!” “The bottom line!” “the bottom..axis, thing. The X!” comes a chorus of voices.

I start counting, and while I do, Sophie objected. “But hang on. I still don’t see what the new thing, direction, is. What’s the third?”

“Up! said Calvin, who rarely participates and often tunes out so far he can’t keep up. But he was watching this with interest. “You know how the class map with Tanya was going north and south and east and west. But it’s all flat, like. This picture has an up.”

“Yeah!” Ben got it. “Cal’s right.”

Dwayne has begun to grasp this. “So you can’t just draw a line? You have to follow along the…things?”

“The axes.” I finish counting along the “bottom” axis and go over to my bookshelf in the furthest corner of my room. “So the sketch starts here…QUIET! One conversation at a time, and I’m the STAR here. The origin starts at this bookshelf. I am walking along the wall, hugging it, on my way to Aditya. Does everyone see how they could track my progress on the axis?”

“YES!” from various points of the room.

“What the hell are you doing?” Dwayne is watching me carefully hug the wall.

“Everyone except Dwayne?”

“YES!” much louder.

I walked along the wall to the table where Aditya sat (fourth along the wall) and stop.

“So now what?”

“You’re there.”

“No, not yet.” countered Nadine. You have to go out….” she waved me towards her. “this way.”

“Yeah, towards Aditya,” this from Talika.

I stepped out 2 steps or so. “That all?”

Josh frowned. “Yeah. You’re there. Except…”

“UP!” Sophie shouted from the back. “That’s the third axis!” General approval reigns loudly, until I wave them all quiet, or try to.

“You go up 4!” Teddy shouted.

“OK. So Aditya is about 40 units out along the wall, 2 units out towards…the door, and 4 units up. Yes?”

“Yes!”

“So let’s draw that.”
3dfinalprism

Of course, while I’m drew this, general mayhem is ongoing with my back turned. I shouted “QUIET! or “Could someone stick a sock in Dwayne?” a few times.

“Wow, so it’s a…cube?”

“A prism, yes. So here’s what we’ve done. We’ve taken the two dimensional x-y coordinate plane and extended it.”

“We extended it up,” from Sophie.

“Yes. And now, instead of a rectangle, we have a three-dimensional rectangular prism. And we can describe things now in three dimensions. But we can do more than that. So let’s step away from my classroom sketch….”

3dcoord

“Whoa. What’s that?” Dylan.

“Man, that’s f***ed up. I just started to get this, and now you’re….” Dwayne, of course.

“No, it’s fine,” Manuel said. “It’s just like the whole thing moved to the center.”

“Oh, I see. It’s like there’s four rooms, all cornered.” Wendell.

“Yes, exactly. Except now, you want to stop thinking about it as a room and think of it as a coordinate plane. As Sophie says, the new plane is the up/down one. So the old x is now here. The old y is now here. The z is the straight up and down one. I think of it as taking the 2 dimensional plane and kind of stepping back and looking down on it.”

“That’s just….”

“DWAYNE BE QUIET. One thing to remember: when you see a 3-dimensional plane, they may be ordered differently. There’s a whole bunch of rules about it that make potentially obscene finger orientations, but I promise I won’t test you on that.”

“So let’s say we’re plotting the point (8,4,5). I’m going to show you how to do it first. Then I’ll go through why. Start by plotting the intercept along each of the planes.”

“Man, does anyone else get this?”

“YES. Shut up, Dylan,” says Natasha.

3dplot845beg

“The trick to remember when you’re graphing in 3-d is to stay parallel to the axis you’re drawing along. So never cross over the lines when plotting points. Now let’s add the yz and xz planes.

“What? This is weird. Why are you drawing so many rectangles?” Patty, frowning.

3dplot845mid

“What you have to visualize is that it’s like we’re drawing sides. So far, I’ve drawn,” I look around and grab three of my small whiteboards, “the bottom and two of the sides. Hold this, Natasha, Talika.” and I build the walls. The kids in the back stand up and look over.

“Oh, I see,” Teddy again. “You’re drawing the prism again.”

“Right. It’s just looking different because the axis is in the center.”

“You do all this just to plot one point?” Sophie, ever the skeptic.

“Yes, but remember this is more just to illustrate, to see how you can extend the dimensions. So after you draw the three sides, joining the intercepts for xy, yz, and xz intersections, you extend those out–again, along the lines.”

3dplot845

“So the point we’re graphing is going to be at the vertex, the intersection of the three planes, the furthest point from the origin–just like in two dimensions, the point is at the intersection of the two lines.”

“That’s really complicated.” Wendy sighed.

“No, it’s not” “Don’t you see the…” Ben, Manuel, Teddy, Wendell and others jump in at the same time, while Dwayne bellowed, Wendy and Tess were asking questions of the room, and, as the writer says, pandemonium ensued. It was a shouting match, yes, but they were shouting about math. The Naysayers, the Doubters, and the Apostles were all marking their territory and this was no genteel, elegant, “turn and discuss this with your partner”, no think-pair-share nonsense. This was a scrum, a brawl, a melee conducted across the room with the volume up at 11—but just like any good fight, there was order beneath the chaos, a give and a take at the group level.

And for you gentle souls wondering about the quiet kids, the introverts, the shy ones who need time to think, they were enthralled, watching the game and making up their mind. It may not look like everyone gets time to talk, but pretty much every time you read me call on a kid, it’s a quiet one. And I shush the room. Then the quiet kid sits there in shock as he or she realizes oh god, I’ve got the mike and I can’t be a spectator anymore.

Anyway, the story goes on with a second great moment, but I’m getting better at chunking and this half had too many details I didn’t want to give up. I’ll stop here for dramatic effect. Because oh, lord, I was high as a kite in this moment, watching the room, realizing I was riding a tremendous wave of energy and excitement. Yeah. ME. On Stage. Making Drama.1

Now I just had to come up with a good ending.

*********************
1I’m not congratulating myself, saying I’m proving kids with the great moment. No, the great moment is mine. I’m standing there going oh, my god, this is a great moment in teaching, in my life. For me! The kids, hey, if they liked it, that’s good.


Modeling Linear Equations, Part 3

See Part I and Part II.

The success of my linear modeling unit has completely transformed the way I teach algebra.

From Part II, which I wrote at the beginning of the second semester at my last school:

In Modeling Linear Equations, I described the first weeks of my effort to give my Algebra II students a more (lord save me) organic understanding of linear equations. These students have been through algebra I twice (8th and 9th grade), and then I taught them linear equations for the better part of a month last semester. Yet before this month, none of them could quickly generate a table of values for a linear equation in any form (slope intercept, standard form, or a verbal model). They did know how to read a slope from a graph, for the most part, but weren’t able to find an equation from a table. They didn’t understand how a graph of a line was related to a verbal model—what would the slope be, a starting price or a monthly rate? What sort of situations would have a meaningful x-intercept?

This approach was instantly successful, as I relate. Last year, I taught the entire first semester content again in two months before moving on, and still got in about 60% of the Algebra II standards (pretty normal for a low ability class).

So when I began intermediate algebra in the fall, I decided to start right off with modeling. I just toss up some problems on the board–Well, actually, I start with a stick figure cartoon based on this lesson plan:

modelingsketch

I put it on the board, and ask a student who did middling poorly on my assessment test, “So, what could Stan buy?”

Shrug. “I don’t know.”

“Oh, come on. You’re telling me you never had $45 bucks and a spending decision? Assume no sales tax.”

Tentatively. “He could just buy 9 burritos?”

“Yes, he could! See? Told you you could do it. How many tacos could he buy?”

“None.”

At this point, another student figures it out, “So if he doesn’t buy any burritos, he could buy, like,…”

“Fifteen tacos. Why is it 15?”

“Because that’s how much you can buy for $45.”

“Anyone have another possibility? You? Guy in grey?”

Long pause, as guy in grey hopes desperately I’ll move on. I wait him out.

“I don’t know.”

“Really? Not at all? Oh, come on. Pretend it’s you. It’s your money. You bought 3 burritos. How many tacos can you get?”

This is the great part, really, because whoever I call on, and it’s always a kid who doesn’t want to be in the room, his brain starts working.

“He has $30 left, right? So he can buy ten tacos.”

“Hey, now, look at that. You did know. How’d you come up with ten?”

“It costs $15 to get three burritos, and he has $30 left.”

So I start a table, with Taco and Burrito headers, entering the first three values.

“And you know it’s $15 because….”

He’s worried it’s a trick question. “…it’s five dollars for each burrito?”

I force a couple other unwilling suckers to give me the last two integer entries

“Yeah. So see how you’re doing this in your head. You are automatically figuring the total cost of the burritos how?”

“Multiplying the burritos by five dollars.”

“And, girl over there, in pink, how do you know how much money to spend on tacos?”

“It’s $3 a taco, and you see how much left you have of the $45.”

“And again with the math in your head. You are multiplying the number of tacos by 3, and the number of burritos by….”

“Five.”

“Right. So we could write it out and have an actual equation.” And so I write out the equation, first with tacos and burritos, and then substituting x and y.

“This equation describes a line. We call it the standard form: Ax + By = C. Standard form is an extremely useful way to describe lines that model purchasing decisions.”

Then I graph the table and by golly, it’s dots in the shape of a line.

datamodelmockup

“Okay, who remembers anything about lines and slopes? Is this a positive or a negative slope?”

Silence. Of course. Which is better than someone shouting out “Positive!”

“So, guy over there. Yeah, you.”

“I wasn’t paying attention.”

“I know. Now you are. So tell me what happens to tacos when you buy more burritos.”

Silence. I wait it out.

“Um. I can’t buy as many tacos?”

“Nice. So what does that mean about tacos and burritos?”

At this point, I usually get some raised hands. “Blue jersey?”

“If you buy more tacos, you can’t buy as many burritos, either.”

“So as the number of tacos goes up, the number of burritos…”

“Goes down.”

“So. This dotted line is reflecting the fact that as tacos go up, burritos go down. I ask again: is this slope a positive slope or a negative slope?” and now I get a good spattering of “Negative” responses.

From there, I remind them of how to calculate a slope, which is always great because now, instead of it just being the 8 thousandth time they’ve been given the formula, they see that it has direct relevance to a spending decision they make daily. The slope is the reduction in burritos they can buy for every increased taco. I remind them how to find the equation of a slope from both the line and the table itself.

“So I just showed you guys the standard form of a line, but does anyone remember the equation form you learned back in algebra one?”

By now they’re warming up as they realize that they do remember information from algebra one and earlier, information that they thought had no relevance to their lives but, apparently, does. Someone usually comes up with the slope-intercept form. I put y=mx+b on the board and talk the students through identifying the parameters. Then, using the taco-burrito model, we plug in the slope and y-intercept and the kids see that the buying decision, one they are extremely familiar with, can be described in math equations that they now understand.

So then, I put a bunch of situations on the board and set them to work, for the rest of that day and the next.

DataModelingstart

I’ve now kicked off three intermediate algebra classes cold with this approach, and in every case the kids start modeling the problems with no hesitation.

Remember, all but maybe ten of the students in each class are kids who scored below basic or lower in Algebra I. Many of them have already failed intermediate algebra (aka Algebra II, no trig) once. And in day one, they are modeling linear equations and genuinely getting it. Even the ones who are unhappy (more on that in a minute) are getting it.

So from this point on, when a kid sees something like 5x + 7y = 35, they are thinking “something costs $5, something costs $7, and they have $35 to spend” which helps them make concrete sense of an abstract expression. Or y = 3x-7 means that Joe has seven fewer than 3 times as many graphic novels as Tio does (and, class, who has fewer graphic novels? Yes, Tio. Trust me, it’s much easier to make the smaller value x.)

Here’s an early student sample, from my current class, done just two days in. This is a boy who traditionally struggles with math—and this is homework, which he did on his own—definitely not his usual approach.

StudentSampleModeling

Notice that he’s still having trouble figuring out the equation, which is normal. But three of the four tables are correct (he struggles with perimeter, also common), and two of the four graphs are perfect—even though he hasn’t yet figured out how to use the graph to find the equation.

So he’s doing the part he’s learned in class with purpose and accuracy, clearly demonstrating ability to pull out solutions from a word model and then graph them. Time to improve his skills at building equations from graphs and tables.

After two days of this, I break the skills up into parts, reminding the weakest students how to find the slope from a graph, and then mixing and matching equations with models, like this:

MixandMatch2

So now, I’m emphasizing stuff they’ve learned before, but never been able to integrate because it’s been too abstract. The strongest kids in the class are moving through it all much faster, and are often into linear inequalities after a couple weeks.

Then I bring in one of my favorite handouts, built the first time I did this all a year ago: ModelingDatawithPoints. Back to word models, but instead of the model describing the math, the model gives them two points. Their task is to find the equation from the points. And glory be, the kids get it every time. I’m not sure who’s happier, them or me.

At some point in the first week, I give them a quiz, in which they have to turn two different models into tables, equations, and graphs (one from points), identify an equation from a line, identify an equation from a table, and graph two points to find the equation. The last question is, “How’s it going?”

This has been consistent through three classes (two this semester, one last). Most of the kids like it a lot and specifically tell me they are learning more. The top kids often say it’s very interesting to think of linear equations in this fashion. And about 10-20% of the students this first week are very, very nervous. They want specific methods and explicit instructions.

The day after the quiz, I address these concerns by pointing out that everyone in the room has been given these procedures countless times, and fewer than 30% of them remember how to apply them. The purpose of my method, I tell them, is to give them countless ways of thinking about linear equations, come up with their own preferred methods, and increase their ability to move from one form to another all at once, rather than focusing in on one method and moving to another, and so on. I also point out that almost all the students who said they didn’t like my method did pretty well on the quiz. The weakest kids almost always like the approach, even with initially weak results.

After a week or more of this, I move onto systems. First, solving them graphically—and I use this as a reason to explitly instruct them on sketching lines quickly, using one of three methods:

SketchingLines

Then I move on to models, two at a time. Last semester, my kids struggled with this and I didn’t pick up on it until a month later. This last week, I was alert to the problems they were having creating two separate models within a problem, so I spent an extra day focusing on the methods. The kids approved, and I could see a much better understanding. We’ll see how it goes on the test.

Here’s the boardwork for a systems models.
systemsboard

So I start by having them generate solutions to each model and matching them up, as well as finding the equations. Then they graph the equations and see that the intersection, the graphing solution, is identical to the values that match up in the tables.

Which sets the stage for the two algebraic methods: substitution and combination (aka elimination, addition).

Phew.

Last semester, I taught modeling to my math support class, and they really enjoyed it:

SlopeWorkMathsupportdatamodel1Mathsupportdatamodel2MathSupportdatamodel3

Some sample work–the one on the far right is done by a Hispanic sophomore who speaks no English.

Okay, back at 2000 words. Time to wrap it up. I’ll discuss where I’m taking it next in a second post.

Some tidbits: modeling quadratics is tough to do organically, because there are so few real-life models. The velocity problems are helpful, but since they’re the only type they are a bit too canned. I usually use area questions, but they aren’t nearly as realistic. Exponentials, on the other hand, are easy to model with real-life examples. I’m adding in absolute value modeling this semester for the first time, to see how it goes.

Anyway. This works a treat. If I were going to teach algebra I again (nooooooo!) I would start with this, rather than go through integer operations and fractions for the nineteenth time.