Polynomial Operations as Glue: Second Year Algebra

A couple years ago, I suddenly realized that my students rarely evaluated quadratic expressions. And when I thought about it, I could see why.

Create a table of values for y = x² -6x – 16. Start with -3

These are kids who aren’t too great at working with negatives, yes? And it’s a whole bunch of work for a relatively small gain. Makes it tough to guess and check, to work velocity problems, and so on. I want something simpler.

Enter the Remainder Theorem: the remainder of the division of a polynomial f(x) by a linear polynomial x-a is equal to f(a).

We usually teach synthetic substitution when introducing with the Fundamental Theorem of Algebra, which is when we give advanced students the bad news—at a certain point, factoring higher-degree polynomials becomes guess and check. Here’s the Holt book, for example: Chapter 5, Quadratics, covers evaluation by substitution (aka, plug it in). Chapter 6, Polynomials (meaning degree greater than 2), covers polynomial division, synthetic substitution/division, remainder theorem, and factor theorem, leading up to the fundamental theorem of algebra. Notice, too, that the book is a tad soulless on two of the more remarkable theorems, as I write about here.

So this is screwed up. First, quadratics are polynomials, thankyouverymuch. Second, synthetic substitution/division solves the problem I started with: it’s brutal to evaluate quadratics if you can’t do it in your head—and most of my students can’t.

Then, there’s the fact that polynomial operations in algebra 2 are like kissing a sister; students don’t really learn the purpose for these operations until math analysis and calculus. Over half my students are in their last high school course and won’t be taking anything more advanced in college, but they will need knowledge of these operations for math placement tests. The other half will be moving on to math analysis, and need the skills.

Over the past two years, I’ve played with different ways of teaching polynomial operations, and different ways of introducing synthetic substitution for quadratics.

My algebra II/intermediate algebra class is comprised of four modeling units: linear equations (and inequalities), quadratic equations, exponential functions, and probability. I intersperse polynomial operations, inverses and logarithms between these four units. Logarithms fit organically with exponential functions; polynomial operations and inverses, not so much in a world where I’m not going on to the more rigorous parts of algebra 2. But inverses work as a good review of multistep equations, so the kids get some good practice in another skillset they need. Leaving polynomial operations as just….out there.

I haven’t been terribly unhappy with this, given the purely functional nature of the lessons, but I want my kids to know synthetic sub/div, dammit, and I want an organic way of introducing it. Right now, I go from linear equations to polynomial operations, ending with multiplication, which takes me into quadratics. That works, but not as smoothly as I want.

A couple days ago, I was pondering how to explain the synthetic substitution/division problem as a blog post, when I suddenly thought of a way to better integrate polynomial operations in and around my first two modeling units. I can use function operations as a method of introducing the transitions. Normally, I just introduce the function notation so they’re familiar with it. (Composites don’t normally show up on the test, and are covered again in pre-calc.)

This is just an outline, but remember that I have all the units done. All I’m describing, broadly (without any curriculum yet) is the transitions, the points at which I introduce and then return to polynomial operations.

After Linear Equations and Inequalities,

I could start with a question like: “Part 1: Sami needs three more dollars to buy the new hoodie that he wants. Model a relationship between the money Sami has and the money he needs, and plot.”

Then, Part 2: “If Sami skips the hoodie, he needs just one more dollar to buy a ticket to the pizza feed on Friday. Model a relationship between the money he has and the money he needs, and plot.”

Part 3, starting as a discussion: “How much more money does Sami need if he wants both the hoodie and the ticket to the pizza feed?” My guess, although I’m happy to be wrong, is the kids will say that Sami needs four more dollars. And so how can they use the graphs to show otherwise?

So we can show graphically and algebraically that adding the two equations together will give us one equation that we can use to see how much more money Sami needs. At this point, I can introduce polynomial addition and subtraction in its simplest form. This will just be a couple days–one for addition, one for subtraction. But it allows me to reinforce linear graphing one more time, in addition to the new concept.

Then I can move from addition and subtraction to multiplication.

Introduction to Quadratics

I’ve always introduced quadratics with the modeling exercise above, then moved onto binomial multiplication. I really like the possibilities that come up after adding and subtracting linear functions, by asking the question (without the graph, at first):

“Okay, we’ve added two lines. What happens when we multiply two lines?”

In class discussion, I’ll point out the negative values, the positive values and the points at which one graph is positive and one negative. What’s going to happen when these are multiplied? (Hey, it never hurts to remind them about negative integer operations.) I haven’t completely thought through implementation—I definitely want them graphing this. Maybe give them the two lines at first, have them multiply the values.

I mentioned earlier that I’ve been looking for a better method of modeling quadratics. While this approach doesn’t involve situation modeling, it does organically introduce the shape of a parabola. It will also help them spot zeros.

And this leads in perfectly to my binomial multiplication unit, which I already extend to include higher degree polynomials. With the strongest kids, I can even give them three lines and have them determine what a cubic function looks like.

Factoring, Division, Remainder, Synthetic Sub/Div

Then, when I’m moving from binomial multiplication to factoring, I can show a graph like the one at right and ask:

“So we multiplied the linear equation by another linear equation to get the parabola. What are the equations you see, and what’s the missing linear equation?”

which, of course, brings up function division, and allows me to introduce factoring as a variant of division–and, a month or so after we’ve done linear equations, they get to review the concepts. As I write this, I’m trying to think if it makes more sense to introduce long division and synthetic substitution at this point, or to work on factoring for a while and then bring up division. TBD.

If you’re not familiar with synthetic sub/div, take a look at long division and synthetic division side by side:

Synthetic sub/div is far easier than substitution, even in quadratics. It’s also noticeably easier when evaluating fraction values for velocity problems.

After I’ve finished all of linear and all of quadratics, I can do a few days on polynomial operations and function notation, just to wrap up.

Again, this is very skeletal. I just had the idea because of the writing challenge. Thanks, blog! But I know it will work; I can feel it. I just have to be careful and think through the transitions thoroughly, make sure I’ve given the kids plenty of support. For example, I don’t want to overemphasize the function operations of this. I just want the kids to be comfortable with the notion of addition, subtraction, multiplication and division of equations. That will give me the entrance to teach them synthetic div/sub, as well as the reason for practicing polynomial ooperations.

Those of you who are thinking, “Hey. This is really algebra one.” well, welcome to my world. My kids learn a whole bunch of first year algebra in my algebra II and geometry classes. But I cover about 60% of the algebra II standards to kids with very weak skills, and the class is pretty conceptually interesting, I think. It’s definitely not just a rehash of algebra one.

I’ve also been thinking a lot about this post on curriculum mapping, which I found very interesting. I hope it’s okay that I borrow his image:

I was talking with Kelly Renier (@krenier), director at Viking New Tech, and we began discussing the concept of “power standards” or “enduring understandings” or “What are the Five Things you want your students to know when they leave your class?” then build out from there. However, we didn’t discuss building those Five (or whatever number) Things out into linearly progressing units, but rather concentric circles.

So this is absolutely how I teach, as regular readers may know. Teaching Algebra, or Banging Your Head with a Whiteboard covers, literally, the Five Big Ideas of algebra I. I also have them for geometry and algebra II (for my students, anyway). I thought the advantages of this approach were interesting in that I didn’t realize how many teachers don’t do this already. Again, quoting:

1. Students get to revisit a general topic every few weeks, rather than a one-and-done shot at learning a concept.
2. Students have time to “forget” algorithms and processes and when they see a scenario they have to fight their way through it accessing prior or inventing new knowledge, rather than relying on teacher led examples. Yes, I consider this a benefit.
3. Teachers may formatively assess more adeptly.
4. Students may see math as a more connected experience, rather than a bunch of arbitrary recipes to follow.
5. It probably better reflects the learning process, which happens in fits and starts, and frankly, cannot be counted upon to be contained within a specified time frame.

This is a really good explanation of what I see as the advantages to my approach. I have never taught in anything approaching a linear fashion, probably because I used CPM, which spirals as a matter of course, in my first two years and was nonetheless shocked at how much kids forgot. So once their forgetting is shoved in your face, it’s hard to go back to the linear curriculum design.

I don’t obsess with getting every single connection made from the first time I teach the class. Sometimes I’ll just acknowledge, as I’ve done with polynomial operations up to now, “Hey, this is kind of an odds and ends thing you just need to know.” There’s nothing wrong with making clean breaks between some units—it doesn’t automatically turn the curriculum linear. For example, I make a very clean break between quadratics and exponentials, because the kids have never seen exponentials before. I show the connections between linear and exponential functions, but I also don’t just lead in. NEON SIGN: NEW EQUATION is a helpful way for kids to realize they’re getting something new. (Common Core says they’ll be learning this in Algebra I. Jesus. These people are friggin’ delusional.)

Going back to who I am as a teacher, I start with explanations. Not necessarily verbal explanations every time, but making sense of a concept before doing it is an essential element of my teaching.

That doesn’t mean I start with a lecture, which I rarely do, or an explanation. I often begin a unit or a concept with an activity. But if I’m asking my students to engage in an activity with no concept or prior understanding, then they can be sure it’s going to be simple, straightforward, and illustrative.

Two Math Teachers Talk

Hand to god, I will finish my post about the reform math fuss I twittered in mid-week, but I am blocked and trying to chop back what I discuss and I want to talk about something fun.

So I will discuss Dale, a fellow math teacher who was a colleague at my last job. Dale is half my age and three days younger than my son. Yes. I have coworkers my son’s age. Shoot me now.

He and I are very different, in that he is an incredibly hot commodity as a math teacher, whose principal would offer him hookers if he’d agree to stay, and gets the AP classes because he’s a real mathematician who majored in math and everything. He turns down the hookers because he’s highly committed to his girlfriend, who is an actual working engineer who uses math every day. I am not a hot commodity, not offered hookers, and not a real mathematician. I also don’t have a girlfriend who is an actual working engineer using math every day, but there’s a lot of qualifiers in that last independent clause so don’t jump to too many conclusions.

He and I are similar in that we both were instantly comfortable with teaching and the broad requirements of working with tough low income kids who don’t want to be in school, and extremely realistic about cognitive ability. We also don’t judge our students for not liking math, or get all moral about kids these days. (Of course, he is a kid).

We are also similar in that we like beer and burgers (he has a lamentable fondness for hops, but no one’s perfect), and still meet once or twice a month at an appropriate locale to talk math. I tell him my new curricular ideas, which he is kind enough to admire although his approach is far more traditional, and ask him math questions, particularly when I was teaching precalc; he tells me that most of the department wants him to be head, despite his youth and relative inexperience. We also talk policy in general. It’s fun.

“I have some news for you,” I told him, “but you will laugh, so you should put down your beer.”

He obligingly takes a pull on his schooner of Lagunitas IPA and sets it down.

“A new study came out,” I said, “and apparently, many high school algebra and geometry courses have titles that don’t actually match the course delivered.”

Dale, who clearly thought I was going in a different direction, did a double take. “Wait. What?”

“The word used was ‘rigor’. Like, some Algebra I courses don’t actually cover algebra I. Same with geometry.”

He looks at me. Takes another pull. “Like, not all algebra teachers actually cover the work formula?”

“Like, not all algebra teachers cover integer operations and fractions for two months. Like not all algebra teachers spend two weeks explaining that 2-5 is not the same as 5-2.”

“Uh huh. Um. They did a study on this?”

“They did.”

“They could have just asked me.”

“They can’t do that. They think math teachers are morons. But there’s more.”

“Of course there is.”

“Apparently, the more blacks and Hispanics and/or low income students are in a class, the less likely the course’s rigor will match the course description.”

He sighs. “I need more beer. Ulysses!” (that’s actually the bartender’s name.) “I’m assuming that nowhere in this study did they even mention the possibility that the students didn’t know the material, that the course content depended on incoming student ability?”

“Well, not in that study. But you know what happens when we point that out.”

“Oh, yeah. ‘It’s all that crap they teach in elementary schools!’ Like that teacher in that meeting you all had the year before I got here. ‘Integer operations and fractions! Damn. Why didn’t I think of that?‘”

“Yes. Actually, the researchers blamed the textbooks, which was a pleasant change from the platitude–and-money-rich reformers who argue our standards are too low.”

“Did anyone ever tell them if it were that simple, whether textbook or teacher, then we could cover the missing material in a few weeks and it’d all be over? Wait, don’t tell me. Of course they told them. That’s the whole premise behind….”

“Algebra Support!” we chorused.

“But then there’s that hapless AP calculus teacher stuck teaching algebra support. He spent, what, a month on subtraction?”

“And the happy news was that at the end of the semester, the freshmen went from getting 40% right on a sixth grade math test to 55%.”

“The bad news being at the end of the year, they forgot it all. Net improvement, what–2 points?”

“Hell, I spend the entire Algebra II course teaching mostly Algebra I, and while they learn a lot, at the end of the course they’re still shaky on graphing lines and binomial multiplication. And I don’t even bother trying to teach negative numbers, although I do try to show them why the inequality sign flips in inequalities.”

“But it’s our fault, right?”

“Of course. But that’s not the best part.”

“There’s a best part?”

“If you like black comedy.”

“The Bill Cosby sort, or the Richard Pryor catching himself on fire sort?”

“Someone doesn’t know his literary genres.”

“Hey, we can’t all be English majors. What’s the best part?”

“The best part is that Common Core is supposed to fix all this.”

“Common Core? How?”

“By telling us teachers what we’re supposed to teach.”

I’d forgotten to warn Dale, who was mid-gulp. “WHAT???”

I handed him a napkin. “You’ve got beer coming out your nose. Yes. Checker Finn and Mike Petrilli always use this example of the shifty, devious schools that, when faced with a 3-year math requirement, just spread two years of instruction over three!”

“Wow. That’s painful.”

“Well, they don’t much care for unions, either, so I guess they think that when faced with a mandate that’s essentially a jobs program for math teachers, we teachers use it as an opportunity to kick back. But that’s when they are feeling uncharitable. Sometimes, when they’re trying to puff teachers up, they worry that teachers will need professional development in order to know the new material.”

“How to teach it?”

“No. The new material.”

“They think we don’t know the new material?”

“Remember, they think math teachers are morons. On the plus side, they think we’re the smartest of teachers. (Which we are, but that’s another subject.) There’s still other folks who complain because ed schools don’t teach teachers the material they’re supposed to be teaching.”

“But we know that material. That’s what credential tests are for. You can’t even get into a program without passing the credential test.”

“Do not get me started.”

“So when the test scores tank, they’ll say it’s because teachers don’t know the material?”

“Well, they’ve got the backup teachers don’t have the proper material to teach the standards, in case someone points out the logical flaws in the ‘teacher don’t know the material’ argument.”

“Sure. If it ain’t in the textbook, we don’t know it’s supposed to be taught!”

“Don’t depress me. Yes, either we don’t know what’s supposed to be taught or we don’t know how to teach it without textbooks telling us to.”

Dale starts to laugh in serious. “I’m sorry, Governor. I would have taught vectors in geometry, but since it wasn’t on the standards, I taught another week of the midpoint formula.”

“I’m sorry, parents, I would have dropped linear equations entirely from my algebra two class, but I didn’t know they were supposed to learn it in algebra one!”

“Damn. A whole three weeks spent teaching fraction operations in algebra when it’s fifth grade math. I could have spent that time showing them how to find a quadratic equation from points!”

“I didn’t know proofs were a geometry standard. Why didn’t someone tell me? Here I had so much free time I taught my kids multi-step equations because my only other option was showing an Adam Sandler movie!”

“Stop, you’re killing me.”

“No, there’s too many more. Who the hell went and added conics to the standards and why wasn’t I informed? Here I spent all this time teaching my algebra II kids that a system of equations is solved by finding the points of intersection? Apparently, my kids didn’t bother to tell me that they’d mastered that material in algebra I.”

“I can’t believe it! Four weeks killed teaching kids the difference between a positive and a negative slope! Little bastards could have told me they knew it but no, they just let me explain it again. No wonder they acted out–they were bored!”

My turn to snarf my beer.

“Jesus, Ed, I’ve wondered why we’re pulling this Common Core crap, but not in my deepest, most cynical moments did I think it because they thought we teachers just might not know what to teach the kids.”

“That’s not the most depressing, cynical thought. Really cynical is that everyone knows it won’t work but the feds need to push the can—the acknowledgement that achievement gaps are largely cognitive—down the road a few more years, and everyone else sees this as a way to scam government dollars.”

“New texbooks! New PD. A pretense that technology can help!”

“Exactly. I’d think maybe it was another effort to blame unions, but no.”

“Yeah, Republicans mostly oppose the standards.”

“Well, except the ‘far-seeing Republicans’ who just want what’s best for the country. Who also are in favor of ‘immigration reform’.”

“Jeb Bush.”

“Bingo. You’ll be happy to know that libertarians hate Common Core.”

“Rock on, my people!”

“Yeah, but they want also want open borders and privatized education.”

“Eh, nobody’s perfect.”

“But all that depressing cynicism is no fun, so let me just say that I would have taught sigma notation except I thought that letter was epsilon!”

“Hey, wait. You do get sigma and epsilon confused!”

“No, I don’t, or I wouldn’t call the pointy E stuff sigma notation, dammit. I just see either E shape out of context and think epsilon. Why the hell did Greeks have two Es, and why couldn’t they give them names that start with E? Besides, the only two greek letters I have to deal with are pi and theta, and really, in right triangle trig there’s no difference between theta and x.”

“Well, you’re going to have to stop making that mistake because thanks to Common Core, you’ll know that you’re supposed to teach sequences and series.”

“Damn. So I won’t be able to teach them binomial multiplication and factoring and let them kick back and mock me with their knowledge, which they have because they learned it all in algebra I.”

“Here’s to Common Core and math research. Without them, America wouldn’t be able to kid itself.”

We clinked glasses just as Maya, Dale’s girlfriend walked in, a woman who actually uses centroids, orthocenters, and piece-wise equations in her daily employment. The rest of the evening was spent discussing my search for more real-life models of quadratics that don’t involve knowing the quadratic formula first. She offered road construction and fruit ripening, which are very promising, but I still need something organic (haha), if possible, to derive the base equation. So far area and perimeter problems are my best bet, which gives me a good chance to review formulas, because until Common Core comes out I won’t know that they learned this in geometry. I wondered if velocity problems could be used to derive it. Dale warned me that it involved derivations. Maya was confused by my describing velocity problems as “-16 problems”, since gravity is either gravity is either 32 ft/sec/sec or 9.8 m/s/s. Dale interpreted. I’m like Jeez, there are people who know what gravity is off the top of their heads? This is why I don’t teach science. (edit: I KNEW I should have checked the numbers. I don’t do physics or real math, dammit. Fixed. )

But all that’s for another, happier, post.

Spring 2013: These students aren’t really prepared, either.

I’m teaching Geometry and Algebra II again, so I gave the same assessment and got these results, with the beginning scores from the previous semester:

I’m teaching two algebra II classes, but their numbers were pretty close to identical—one class had the larger range and a lower mode—so I combined them.

The geometry averages are significantly lower than the fall freshmen only class, which isn’t surprising. Kids who move onto geometry from 8th grade algebra are more likely to be stronger math students, although (key plot point) in many schools, the difference between moving on and staying back in algebra come down to behavior, not math ability. At my last school, kids who didn’t score Proficient or Advanced had to take Algebra in 9th grade. I’d have included Basic kids in the “move-on” list as well. But sophomores who not only can’t factor or graph a line, but struggle with simple substition ought not to be in second year algebra. They should repeat algebra I freshman year, go onto geometry, and then take algebra II in junior year—at which point, they’d still be very weak in algebra, of course, but some would have benefited from that second year of first year.

Wait, what was my point? Oh, yeah–this geometry class class is 10-12, so the students took one or more years of high school algebra. Some of them will have just goofed around and flunked algebra despite perfectly adequate to good skills, but a good number will also be genuinely weak at math.

On the other hand, a number of them really enjoyed my first activity: visualizing intersecting planes, graphing 3-D points. I got far more samples from this class. I’ll put those in another post, also the precalc assessment.

I don’t know if my readers (I have an audience! whoo!) understand my intent in publishing these assessment results. In no way am I complaining about my students.

My point in a huge nutshell: how can math teachers be assessed on “value-added” when the testing instrument will not measure what the students needed to learn? Last semester, my students made tremendous gains in first year algebra knowledge. They also learned geometry and second year algebra, but over half my students in both classes will test Below Basic or Far Below Basic–just as they did the year before. My evaluation will faithfully record that my students made no progress—that they tested BB or FBB the year before, and test the same (or worse) now. I will get no credit for the huge gains they made in pre-algebra and algebra competency, because educational policy doesn’t recognize the existence of kids taking second year algebra despite being barely functional in pre-algebra.

The reformers’ response:

1) These kids just had bad teachers who didn’t teach them anything, and in the Brave New World of Reform, these bad teachers won’t be able to ruin students’ lives;

2) These bad teachers just shuffled students who hadn’t learned onto the next class, and in the Brave New World of Reform, kids who can’t do the work won’t pass the class.

My response:

1) Well, truthfully, I think this response is moronic. But more politely, this answer requires willful belief in a delusional myth.

2) Fail 50-60% of kids who are forced to take math classes against their will? Seriously? This answer requires a willful refusal to think things through. Most high schools require a student to take and pass three years of math for graduation. Fail a kid just once, and the margin for error disappears. Fail twice and the kid can’t graduate. And in many states, the sequence must start with algebra—pre-algebra at best. So we are supposed to teach all students, regardless of ability, three years of increasingly abstract math and fail them if they don’t achieve basic proficiency. If, god save us, the country was ever stupid enough to go down this reformer path, the resulting bloodbath would end the policy in a year. We’re not talking the occasional malcontent, but over half of a graduating class in some schools—overwhelmingly, this policy impacts black and Hispanic students. But it’s okay. We’re just doing it for their own good, right? Await the disparate impact lawsuits—or, more likely, federal investigation and oversight.

Reformers faithfully hold out this hope: bad teachers are creating lazy students who could do the work but just don’t want to. Oh, yeah, and if we catch them in elementary school, they’ll be fine in high school.

It is to weep.

Hey, under 1000 words!

My math classes: are they prepared? Um. No. So what?

After over a month talking about policy, it’s fun to write about math classes for a change.

I’m teaching geometry, Algebra II, and Math For Kids Who Haven’t Passed The State Graduation Test Yet.

I’ve given this algebra readiness test to all my students for the past three years. I got it from a senior teacher at my last job, and it’s an excellent assessment of a students’ basic numeracy and first semester algebra skills. Can they substitute? work with negatives? multiply binomials? factor a quadratic? I don’t much care about second semester algebra (graphing parabolas, quadratic formula); my geometry students won’t need it, and my algebra II students will be reviewing the material again.

I know what some of you are thinking. “Why the heck are you giving your geometry and algebra II students a test in pre-algebra and first semester algebra? They already know that material, don’t they?”

This is me laughing at you naive folks. Ha ha!

Or, I could show you a graph of the results.

So the algebra II kids have taken a full year more of math than the geometry kids, and both groups have passed algebra. But the algebra II class is usually taken by kids who made it this far by their toenails, got low scores in both algebra and geometry. Geometry 9 is ninth graders who passed algebra the first time but chose not to take the honors course—and who aren’t in A2/Trig.

My Algebra II class is substantially stronger on average than last year’s class, which averaged around 20 wrong. I’ve got about 8 kids that got 0-2 wrong after finishing the test in ten minutes and should be taking A2/Trig, or even Honors. The Geometry 9 class is slightly stronger on average than my class from last year, which averaged around 12-13 wrong. My geometry classes last year were the strongest I’ve ever taught, but had a much weaker bottom than this class does. The strongest students in the Math Support class got 13-15 wrong, which is impressive.

For the uninitiated, there are two big pieces of info in this graph:

In all three classes but particularly the A2 and Geo, the range of scores on what should be an easy test is huge. The weakest students in both geometry and A2 got barely half right. I’m used to this—handling wide ranges in ability is probably my greatest strength as a teacher, although administrators don’t value it much. However, really think about that range and what it represents in terms of the ability gap within one classroom. And remember–this is a school that provides honors courses, so it tracks much more than my last two schools. The gap this year is considerably less than the scores from last year (which I can’t find, so you’ll have to take my word for it).

The other news is, of course, that the average score for the geometry class should be 5-6 wrong, and the algebra II students should, in a world where we worry more about what kids learn than what their transcript says, knock the test out of the park.

About half the kids in both geometry and algebra II classes should not be taking formal college prep courses, but rather an interesting math applications course, in which they continue to apply what they’ve already learned, rather than pile on new stuff.

Oh, well. That’s what we get for pretending ability doesn’t matter.

I don’t want to sound cynical or discouraged. I’m pumped. They’re a great group of kids and are stronger on average than my last year’s kids, with lots of high achievers. But what’s “normal” for me is clearly not anything that reformers understand or anticipate when they talk about high expectations or proficiency for all.