Modeling Linear Equations

“I have a certain amount of nickels and dimes that add up to $2.10.”

“Sam bought a number of tacos and burritos. The tacos were $2 and the burritos were $3. Sam spent $24.”

“Janice joined a gym with a sign-up fee of $40 and a monthly rate of $25.”

I put these three statements on the board and told my Algebra II students to generate a table of values and graph the values.

Some students were instantly able to use their “real-life” math knowledge to start working. Others needed a push and were then able to start. Some needed calculators to work out the values, but all 90 students were, with minimal prompting, able to use that part of their brain that had nothing to do with school to create a list of possibilities. Only a few questioned the lack of an “answer”, and none needed the explanation more than once.

We did this for three days. Over half my students were able to determine the slope of the line and link it back to the word problem without my expressly teaching it. Some of them improved as time went on; few of them are completely incapable of linking the word model to an equation. With discussion, most students began to realize that models with two changing numbers adding up to a constant (nickels and dimes, tacos and burritos) had a negative slope, because an increase in one led to a decrease in the other.

Then, for two days, I gave them problems in equation form:

“y = 3x+2”

“2x + 5y = 45”

“3x – 2y = 6”

“y = -.5 + 50”

They had to generate a table of values and graph the line. When some of them had difficulties, I pointed out the links to word models (what if you were buying burritos for $2 and 6-packs for $5–how much money did you have to spend?) and they got it right away. The subtraction models were the most difficult (which I expected, and didn’t emphasize).

And then, for a couple days, I mixed it up–gave them word models and equations.

I began writing the answer to the most common question in huge letters on the white board: YES, YOU CAN JUST PICK ANY NUMBER.

We’ve just spent two days doing the same thing with word models that provide two points but no relationship.

“Janice joined a gym that had a monthly rate and a signup fee. After three months, she’d paid $145. After 6 months, she paid $190.”

“Brad buys grain for his livestock on a regular cycle and re-orders it when he runs out. After 3 days, he had 72 pounds left. After 7 days, he had 24 pounds left.”

So they have to create the table, find the slope from the table and graph it. Day 2, they had to do that while also answering questions (“What was the signup fee? What was the monthly rate? How much grain did Brad use daily? When would he have to reorder?)

At no point during these two weeks did I work problems algebraically. Some students did, but in all cases I encouraged them to think about other ways to work the data.

They improved dramatically. I gave them a quiz, and all but a few students were able to do the problems with a minimum of questions, although they needed lots of reassurance. “Okay, I think I know what’s going on, so there must be something wrong.” A few kids gave me the “I have no idea how do to this” and I was pretty brutal in my lack of sympathy because they were the kids who don’t pay attention.

I spent the entire first semester teaching them linear and quadratic equations–graphing, systems, solving, factoring, the works. Algebra II is a course designed for students who don’t want or aren’t ready to move to Algebra II/Trigonometry–or who failed that and need a third year. So the first semester is a rehash of Algebra I. I covered it, they all learned a lot–and yet, the semester final was dismal. Some of it was Christmas crazy, and then I wasn’t happy with the test. But nonetheless, they should have done better.

So I mulled this over Christmas break. All but a few of my students are juniors or seniors. Some will be taking a proficiency exam in a few months. Others will be taking a proxy for the exam in their state tests. I’ve always been more focused on their college tests than their knowledge of second year algebra. I want my students to test out of remedial math, or spend as little time as possible in it.

That’s why I’ve decided to spend a month helping them use their math knowledge–the knowledge they see as entirely separate from algebra and geometry, their “real-life” knowledge–to model data.

If this works properly, the strongest students will have a much deeper understanding of the equations and how they relate to the data. The weaker students will be able to work problems using their inherent math ability, rather than struggling to turn the problem into an abstract representation they don’t recognize.

I’m going to finish up linear equations with maps, to give them a better understanding of midpoint, distance, parallel and perpendicular.

For more samples and boardwork, see Modeling Linear Equations, part 3
Then it’s onto quadratics.

Teaching Geometry

I taught two geometry sections my first year at a different school, and while I didn’t do a particularly good job (the classroom management problems were horrible for a different reason, and the curriculum was CPM–ick), I came away with useful insights that have really improved my execution this year.

Geometry Then and Now

Back in the dark ages, we used to say “There are two sorts of people in the world: those who prefer algebra and those who prefer geometry.” This mindset comes from a time when advanced students took algebra in 8th grade, most of everyone else college bound took it in 9th or 10th grade, and then followed it up with geometry, Algebra II, and precalc if there were enough time. Students who were really bad at math took Basic Math or Business Math and maybe took algebra their senior year. If this sounds familiar to you, fine–but it’s not like that anymore.

“Students who were really bad at math” were not representing our nation’s racial balance, and research unsurprisingly showed that students who went beyond Algebra II in high school had higher college completion rates. Naturally, this meant that everyone should take algebra as early as possible, cognitive ability or readiness be damned. The resulting carnage of this policy did not lead to re-evaluation, but rather to the determination that pre-algebra preparation should start earlier–and, of course, kids who fail algebra need to take it again.

Consider the effect of this policy on the average, “okay at math” kid today. Starting in sixth grade, it’s All Algebra, All the Time. By the time they get to geometry, “math instruction” and “algebra” are virtually synonymous–and they don’t even realize it. Kids have spent three, four, or even five years with algebra preparation or instruction. Specifically, using processes to solve for an unknown.

And then: geometry. Good god, what fresh hell is this? Facts. Vocabulary. Relationships. And then, in some weird way, you use these facts and vocabulary and relationships to come up with more facts and vocabulary and relationships. There’s no solving. There’s not even an answer. Half the time the book gives you the answer but then expects you to explain it using, god help us all, facts and vocabulary and relationships.

This is a whole galaxy away from “I like algebra better than geometry.” First off, all but 10-15% of my students found algebra completely unmanageable, so they aren’t looking back fondly at an easier subject. They’re trying hard not to curl up in a fetal position at the realization that math gets worse than the horror of the past three years.

Geometry teachers would do well, I think, to acknowledge this confusion. I tell my students some version of what I’ve just explained above and I see the light dawn. They get it. They might not get geometry, yet, but they get why they feel so lost. And that helps them move forward.

Try, try again
Two years ago, I could see that many of my students weren’t getting it. I retaught, thought of other ways to explain things, but I didn’t understand the degree of their lostness until relatively late in the first semester. I adjusted my teaching more, but I still hadn’t figured out why they were so lost.

This year, I was teaching parallel lines and transversals in week 2 or so and I suddenly realized that most of my students didn’t get it. They weren’t complaining, they weren’t acting out, they were just lost. I recognized the look from two years ago, and was now able to distinguish a furrowed brow of mild confusion from a blank look of utter nihilistic despair.

So at the end of day 2, I did a thumb check. “Okay, guys, my sense is a lot of you are feeling lost. Thumbs up if you feel confident, sideways or down if you’re kind of or totally lost.” And sure enough, most of the class was sideways or down.

I told the class I would come up with a different way to explain it. The next day, I used Geoboards, rubber bands, and little wooden geometric shapes to create a visual image of corresponding, alternate interior, and so on. (I’ll write that up some time). The lesson was very effective in helping students understand the angle relationship. But more important, the students recognized that I had stopped everything, rethought the lesson, came up with a radically different way of explaining the concepts–and had gone through this effort because I could see they were lost. The feedback from the new lesson was very enthusiastic, kids felt much less lost–but more importantly, they felt like I understood their confusion and was willing to spend time and effort helping them out.

This created a lot of good will, and since then they’ve been very trusting of my oftentimes bizarre way of building visual images to help them grasp geometric concepts.

It’s easier to do this in geometry than algebra, since geometry is new to everyone. Even my top students appreciate the occasional visual exercise, and I always have extra challenges for them. In algebra, some kids are lost right from the beginning, and it’s impossible to reteach everyone. (Which means, now that I think about it, that if I differentiate immediately after my assessment test in algebra, I might have an easier time. Hmmm.)

But fundamentally, it’s important to understand that time spent at the beginning, pacing be damned, will really pay off in student investment. I now realize that many of my geometry students two years ago had checked out because none of it made sense, and I didn’t pick up on that early enough to intervene. I took five days to explain parallel lines and transversals rather than two, but every minute of it was well spent.

De-emphasize what they won’t use

Most college graduates think proofs, logic, and construction as quintessential geometry subjects. That’s because we never use them again. We don’t spend any time in formal logic, never do formal proofs, and as for construction, forget it.

So I mostly dump them (which is how I make up the five days on transversals). Not completely. My Holt text starts with these five chapters:

1. Foundations (Undefined Terms, Segments, Angles, area/perimeter formulas, coordinate geometry, transformations)
2. Geometry Reasoning (logic and proofs)
3. Parallel and Perpendicular Lines
4. Triangle Congruence
5. Triangle Properties

I dumped transformations entirely. I then took coordinate geometry, proofs, and logic and broke them up into tiny digestible chunks (coordinate geometry was review), rather than cover them all at one time, and covered about 20% of the material. So rather than an entire section on proofs, I introduced algebraic proofs at a natural pause point, when I had a day or two between major sections. I just introduced it; my goal was familiarity and recognition but not competence. Then, after introducing congruent triangles, I introduced two column proofs, and the students used congruence shortcuts to create two column proofs. This was much more successful than introducing a whole chapter on proofs when they were still in the WTF stage.

Yes, I know, the purists out there, assuming anyone is reading, is shocked. What? Proofs and logic introduce an invaluable way of thinking logically and methodically! Yep. But ask geometry teachers in heterogeneous classrooms if their kids understand proofs, and they will sigh. There’s just no way to get the lower ability half of the population to understand proofs and they’ll never use it again. I could spend lots of time trying, but I have better things to do with their time.

Ideally, I’d love to make my top students go through rigorous proofs, but it would take more instruction time than I can manage in differentiation. I hope to figure this out at some point, but I’m not as practiced at teaching geometry as I am at algebra.

So you’re thinking my class is too easy, right? Well, we just had our semester final and the geometry teachers agreed to start with a common assessment, built by a traditional geometry teacher who had covered far more material than I had in the first semester. I considered the test a little too easy and more picayune than I would build, so I substituted some harder questions. I didn’t dump more than two or three of the questions that we didn’t cover, because I felt pretty confident the students could figure it out–and, for the most part, they did.

Always remember where they are going
Geometry is just a brief respite. The next year, it’s back to algebra II, another course that causes a lot of carnage. Half of my class has extremely weak algebra skills, half of the rest are adequate, and the top students were rarely challenged with tough material. They need the practice. My sophomores will be taking the algebra and pre-algebra intensive state graduation test and my juniors are taking the SAT. Algebra is a big part of their testing load this year.

So I teach my geometry course as Applying Algebra with Geometry Facts. My students will never again need to prove that triangle ABC is congruent to triangle XYZ, but they will always need to know how to find the angle measures of a triangle whose angle ratio is 2:3:5. They will never use a compass again, but they will need to know what to do if Angle A and Angle B are supplementary and Angle A = 4x+ 13 and Angle B = 2x + 17 and they need to solve for x.

The problem is that the state tests tend to emphasis more traditional geometry. Aggravating, really, given that the state has clearly de-emphasized traditional geometry in its overall curriculum, but so be it.

Results

I had told all my students this year that if they showed up and worked, they’d pass with a D-. In my Algebra II course, several students did not in any way demonstrate understanding of the material we covered that year (not for lack of trying, in most cases), but I kept my promise.

But in my Geometry classes, my D students were genuinely Ds. They struggled, but got Ds or “respectable Fs” (50% or higher) on all the tests and quizzes. On the 100 question final (40 correct is a D-, 15 point grade scale instead of 10), 15 students failed. All but two had “respectable” Fs (answered 30 or more questions correctly, and those two were just below 30. The distribution was pretty close to normal, the average score and the mode were C. So far, so good.

Teaching Algebra I

This is the first year I have been completely uninvolved with “first year” algebra. I use quotes because in high school, almost all–say, 80%–have taken the class at least once, and a good 20-30% have taken it twice. A fraction have taken it three times.

I wake up each day grateful that I’m not teaching algebra I, despite the fact that I’ve spent more time thinking about how to teach algebra than any other subject. But I rejoice nonetheless. No class is designed more perfectly to slap you in the face with the insane inadequacies of our educational policy. See Tom Loveless’s study,The Misplaced Math Student, for great research on the idiocies of putting unprepared students in eighth grade algebra.

Challenges

Schools with majority URM (under-represented minorities, aka black and Hispanic) can simply pretend to teach algebra. I’m speaking here of charter schools and urban comprehensive schools with no high-achieving population to worry about. Pretending to teach algebra doesn’t do much for test scores, but it’s a lot easier for classroom management if you can give struggling students something they know how to do.

But many Title I schools are in the suburbs, which aren’t as economically homogeneous as they used to be and these schools have it much tougher. First, they can’t track, because their majority minority population sued back in the 90s. So all their classes are “heterogeneous”, progressive-speak for “put functionally illiterate/innumerate kids, struggling but not completely unskilled kids, ready-to-learn kids and highly skilled kids all in the same classroom and yammer about differentiation”. Second, because they have kids who are ready to learn algebra, they have to actually teach algebra. So they have to figure out a pace that doesn’t lose the middle and doesn’t bore the top. The bottom is largely left out of the equation. Unless you differentiate, and differentiation is a lot of work.

Last year, I taught all Algebra I: three regular courses and one Intervention, a double period course. Because the best thing to do with kids who are terrible at math is give them twice as much time to feel inadequate.

Differentiation

I slowly moved to a differentiation model. First, I started giving the top kids separate lessons, which was partially successful. It gave them more challenges and they did very well, but they felt neglected because I didn’t always get back to close up with them. By the end of the first semester, I’d seen a second group, skilled but slightly less motivated, start to break away from the pack. At the same time, I had about 20 students who simply had not demonstrated understanding of the first semester basics–and I’m very flexible in demonstrating understanding. These students, I decided, would go “on contract”. I would give them a D- if they demonstrated they could graph a line, factor a quadratic, solve a complicated multi-step equation, and make a reasonable stab at solving a system.

So I had four groups, which I called Purple, Black, Gold, and Blue. They had different objectives, different assessments, but not different standards. A Gold student could get, at best, a B-. If any gold student started doing better than that it was time to move them up. It worked beautifully, if “beautifully” can be used to describe teaching kids who still struggled with math. By setting up the groups, I was able to formalize the process of working with them–“Okay, golds, you’re clear on what you’re doing? Good, I have to check in with the Purples”. I could give myself time to close up with the top kids, who all agreed they were not neglected in the second half. I designed lessons in three day chunks and staggered them so that I could introduce a lesson to Gold while Purple, Black, and Blue (heh) were on day 2 or 3 of their lesson. The contract kids in particular did very well; I still lost nearly half of them to chronic absence and/or refusal to work, but three of them moved out of Contract entirely and the rest of that half passed with a D-. I had a low failure rate.

Most people who saw my class in action said it looked like an enormous amount of work, and it was. But you pick your stress levels. I found it far more stressful to teach one lesson to 30 kids with a four year range in ability levels than I did to design four lessons for smaller groups.

Results

Did the kids learn algebra? I was able to compare the scores of 200 students who took algebra two years in a row. I had by far the most students of any algebra teacher. My students incoming test scores were 15 points below two of the teachers, and 30 points below one of them. In other words, I got most of the low-scoring kids, and I had no students with Advanced or Proficient scores on their previous year’s test, unlike the other teachers. When I broke down the comparison by incoming ability level, I did roughly as well as all the other teachers. Specifically, my students improved at a slightly lower rate than the other three teachers, which makes sense when you consider I had more low ability kids, but my standard deviation was, along with one other teacher, much lower.

For example: Teacher A, who had a class average 30 points higher than mine, had spectacular results with a couple of the 10 Far Below Basic students she had, but the others didn’t do as well. So the standard deviation of her new average was huge, whereas mine was pretty tight. Translation: I improved more students by smaller amounts, but had few huge wins. My improvements, like to like, were on par with the best teacher in the department, an algebra specialist who I very much admire, although our teaching methods couldn’t be more different. In my opinion, my differentiation allowed me to be a better teacher. But I don’t think it works for everyone.

I had no kids with Advanced scores; I attribute this in part to my failure to really challenge my top students. On the other hand, my top students all came in with Basic scores the year before, and almost all of them moved to Proficient (a few moved to high Basic). A number of the contract kids had Below Basic rather than Far Below Basic scores, which was a good sign.

We had a very good year last year, and our year was widely considered successful because of Algebra results. I taught more algebra students than anyone else, so in addition to specific results, it’s fairly evident I did a decent job. Nonetheless, I was moved to Geometry and Algebra II this year. As I said, I give thanks daily.

Classroom Management

So if my kids learned well, why am I so thankful to be out of algebra? Because classroom management difficulty correlates directly with the percent of the class with no skin in the game, and in my classes that was 10-20%. I would regularly check my students’ grades in all their classes; all my disruptive students were failing all their classes. For the most part, they are just waiting out the clock until they can go to alternative schools of some sort. They don’t mind going to school; it’s fun. It’s social. It’s their life. Learning, behaving, and engaging–not so much.

I say this to all new teachers: getting this population under control is imperative. Start by moving kids away from each other. I have often pulled a desk (or two) way up front and made a talking kid (or kids) move to that seat. The disruptor will usually then go to sleep. Fine by me. Many teachers are philosophically opposed to sending disruptive kids out of the classroom. Get over it. If moving seats didn’t work, or multiple warnings didn’t work, give them the boot. Ignore the pressure your management will put on you to stop sending them out of the classroom (although you will, inevitably, think 8 times before doing it). It’s incredibly hard to be patient and controlled enough to send kids out of the classroom until and you won’t always win the battle, but you must fight. Sing me no sad songs about these kids and their problems. They are making it impossible to teach the others, and you owe the others every bit as much.

What is essential, though, is to always hold to the rule no harm, no foul. I kick a kid out one day, he wants to come in the next day and works, I give him my full attention. And behavior never affects the grade. I had more than one student who showed functional understanding of algebra despite being a monumental pain who was often sent out. Those kids passed.

And it’s also essential to reach out in every way you can. I told every one of these students that I wanted to pass them, that I understood this was a course they didn’t like and didn’t want. I also turned to differentiation because I hoped that it would give the unmotivated strugglers a sense of possibility–and it did work, for a lot of them. You never want to take their behavior personally. But you also don’t want to tolerate it.

So don’t bleat ineffectually at your students, and don’t teach to a room that isn’t quiet.

Curriculum Paths and Classroom Management

Survey math teachers and you will find that Algebra I classes have the most severe classroom management challenges, quite apart from being difficult to teach content-wise. We put kids in classes they don’t understand and make them take that same class two or even three times. By the time they are sophomores, these kids know they can’t graduate. They are failing all their other classes, too, but in math, a lot of the high achievers get winnowed out into geometry, many before they even get to high school. The proportion of nohopers is insanely high.

By Geometry and Algebra II, most of your kids have skin in the game. Even if they don’t like math, they have something to lose. Classroom management moves from being a nightmare to a manageable challenge.

I’ve been working with a colleague on a math sequence that will find a path for the no-hopers. Remember, from a purely pragmatic standpoint, that the more kids who fail algebra, the more kids who will leave your school for a credit-generating factory–and take their attendance dollars with them.