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:
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?”
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….”
“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.
“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…”
“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.
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.
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:
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:
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.
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).
Last semester, I taught modeling to my math support class, and they really enjoyed it:
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.