Category Archives: math

Tales from Zombieland, Calculus Edition, Part I

A couple weeks ago, I met with a charming math zombie who I coach for the SAT. “Could you help me study for a pre-calc test instead?”

She brought out her book, a hefty volume, and turned to chapter 4, page 320

I took one look and skidded to a stop.

“What the hell…heck. This is calculus.”

The mother sighed. “Yes, they cover calculus in pre-calculus so that everyone is ready for AP Calc next year.”

Huh. Remember that, folks, the next time you hear of a school with a 100% AP pass rate. They are teaching the kids some of the calculus the year before.

“OK, I can maybe help you with this but before we start: I don’t usually work in calculus. I’m pretty good conceptually, and my algebra is awesome, but at a certain point I’m going to have to send you back to the teacher.”

“That’s fine; I really need any help I can get.”

First up. “Use the limit process to find the derivative of f(x) = x2 – x + 4.”

“What on earth is the limit process?” I turn back in the book, leafing through the pages.

“I have no idea.”

“Well, you must have worked the problem before.”

“I don’t know how.”

“Maybe they mean the definition of a limit, the slope thingy.” I look at the next problem, which also focuses on slope, and decide that must be it.

“So you know the definition of a limit, right?”

“No, not really. I know the derivative of this is 2x-1.”

“Yes, but what is the derivative?”

“I don’t know. I don’t understand this at all.”

“Um, okay. The derivative of any function is another function, that returns the slope of the tangent line for any given point on the original function. The tangent line represents…um, .not just the average rate of change between two points, but the instantaneous rate of change at that point.” (I am not using math terms; whenever mathies get together and talk about the “intuitive” definition of a derivative I want to slap them. I checked a few places later, like this one, and I think I’m on solid ground.)

“Yeah, but why do we care about the rate of change?”

I should mention here that her teacher and I went to ed school together, and I’m certain she (the teacher) explained this multiple times from various perspectives.

“You say you know the derivative is 2x-1, yes?”

“Right. You’re saying that’s the slope of the line?”

“Almost. The derivative is the means of finding the slope of a tangent line to any point on the function, with various caveats I’m going to skip right now. Remember, most functions do not change at constant rates. You can find the average rate by finding the distance between any two points, and finetune that average by picking two points closer and closer together. The slope of the tangent line, which means the line is intersecting only at one point, is the….” I can see she doesn’t care, and her understanding is definitely ahead of where it was just five minutes earlier, so I stopped for the moment.

She sighed hopelessly. “Look, can’t I just find the derivative?”

I scrawled something like this:

“Oh, I remember that. Okay.” And she plugged it all in and calculated rapidly. “How come I have an h left over?”

I was a tad flummoxed, but then remember. “Oh, h approaches 0, so it’s basically negligible. I think that’s right, but check with your teacher. Now, what does this represent?”

“I have no idea.”

“Suppose I ask you to find the derivative when x=1, or at the point, um, (1,4).”

“I plug 1 in for x in 2x-1, which is 1. Then I write the equation y-4=1(x-1).”

“So graph that.”

“I don’t know how. It’s a line, right?” She thinks a bit, then converts the equation to slope intercept. “Okay, so it’s y=x+3.”

“Now, graph the parabola.”

“Um…” I sketched it for her, and marked (1,4). “Now sketch the line.”


“See how it just intersects at the point, perfectly tangent? That’s what a derivative does–it returns the slope of the line through that point that will intersect at just one point.”

“Yeah, I saw this before.”

“And it made quite an impression. Stop waving this off. You want to feel less hopeless about math? This is why you have no idea what’s going on. So gut it up and focus.” She nodded, somewhat chagrined.

“The slope of the line at that point indicates the slope of the original function at that point, which is the instantaneous rate of change. Remember: most functions don’t change at a constant rate. Finding the rate of change at a single point is an essential purpose of calculus. So pick another point and try it.”

“OK, I’ll try -1. What do I do first?”

“What do you need to know?”

She looked at the graph. “I need to know the slope of the line….which I get from plugging in -1 to the derivative 2x-1, which is….-3. And then I—”

“Stop for a minute. Say it. What did you just find out?”

“The derivative for x=-1 is -3, which means…the slope of the line where it meets the graph is -3?”

“Slope of the tangent line. And what does that represent?”

She frowned in concentration and looked at the sketch I’d drawn. “That’s the rate of change at that point. But where is that tangent line intersecting? Oh, I need the plug that in…” She did some work. “So the point is (-1,6), and the slope is -3, and that’s why I use point slope, because I have a point and a slope.”

“And remember, you don’t have to convert from point slope to slope intercept. I just do it because I find it easier to sketch roughly in y-intercept form.”


“But how does this work in problem 2? They don’t give me an equation but they want me to find a derivative.”


“You can find the equation from the graph.”

“Oh, that’s right. But I checked the answer on this, and it’s just -1, which makes no sense.”

“Sure it does. Graph the line y=-1.”

She thinks for a minute. “It’s just a horizontal line.”

“And the slope of a horizontal line is…”

Pause. “Zero. But does that mean the derivative is 0?”

“Which would mean what?”

“The rate of change is zero?”

“How much does a line’s slope change?”

“It doesn’t.” I wait. “You mean a line has a zero change in its rate of change?”

“There you go. And doesn’t that make sense?”

“So….because a line has a slope, which is the same between every point, its derivative is zero. So the derivative is….oh, that’s what you mean when you say other functions don’t change at a constant rate. OK. So lines are the only functions whose derivative is zero?”

“Um, yes, I think. But a derivative can return zero even if the function isn’t a line. ”

She sighed. “It’s much easier to just do the problem.”

I’m going to stop here, because I want to go through several of the conversations in detail so I’ll do a Part 2.

In my last post, I pointed out that Garelick and Beals and other traditionalists are, flatly, wrong in their assertions that procedural competence can’t advance well in front of conceptual understanding.

At the risk of stating the obvious, here is a nice, charming, perfectly “normal” calculus student who understands how to find a derivative, how to work the algebra to find a derivative, and yet has absolutely no idea or caring about what a derivative is—and complains in almost identical words to the middle school girl in G&B’s article. She just wants to “do the problem.”

Our entire math sequencing and timing policy is based on the belief that kids who can do the math understand the math. Yet increasingly, what I see in certain high-achieving populations is procedural fluency without any understanding.

In case anyone wonders, I’m not engaging in pointed hints about East Asians (I tend to come right out and say these things), although they are a big chunk of the zombie population. The other major zombie source I’ve noticed is upper income white girls. I have never met a white boy zombie, or a black or Hispanic zombie of any gender, although perhaps they are found in large numbers elsewhere. But the demographics of my experience leads me to wonder if culture and expectations play a big part in whether a student is willing to put the time and energy into faking it. Or maybe it’s easier for people with certain intellectual attributes (a really good memory, for example) to fake it.

Anyway, I’ll do a part 2, and not solely to reveal zombie thinking. I was planning on writing about this session before the G&B piece appeared. Not only did I enjoy the chance to work with calculus, but I also have really started to understand how unrealistic it is to teach calculus in high school. I’m moving towards the opinion that most kids in AP Calc don’t understand what the hell’s going on, thanks to the unrealistic but required pacing.

Oh and yes, I don’t know much calculus. Forgive me if my wording isn’t correct, and feel free to offer better in the comments.

The Test that Made Them Go Hmmmm

So school has begun and despite my palpitations about the boredom of only two familiar preps, I’m pleasantly busy. Last year was a hell of a lot of work, and given the nosedive that my writing time took, I should maybe not be so eager for a less…familiar schedule. So instead of demanding new classes, I accepted the first semester, threw a minor temper tantrum when no one listened about second semester and all is well. Algebra 2 in particular is proving a delightful challenge, given my new emphasis on functions.

In no small part because of this planning breathing room (is anyone noticing I’m saying my panic was a total overreaction?), the senior Water Park Day registered in my awareness ahead of time. In prior years, I didn’t heed the warnings that half my class would disappear, and so would be forced to dump my lesson plan on the Day itself, when the smaller classes would just have a day to practice. But thanks to this old, familiar schedule that gives me more time, I anticipated the impact.

So for the first time, I was able to give serious thought to having a day to pursue math without regard to subject matter or schedule. I could have a “math day”! Then I remembered Grant Wiggins’ challenge to math teachers everywhere in the form of a conceptual knowledge quiz.


Grant proposed this as an actual test: I will make a friendly wager: I predict that no student will get all the questions correct. Prove me wrong and I’ll give the teacher and student(s) a big shout-out.

What math teachers think their kids would know the answers? I certainly didn’t. In some cases, they probably were taught, but in others, I doubt an elementary school teacher would ever think to bring them up. But even if all the concepts were taught by fifth grade, how many kids of that age could really appreciate the questions?

Most of the questions tease at the paradox….wrong word? tension? between the functional day-to-day applications of arithmetic, and the amazing truths that underlie them. John Derbyshire wrote, in Prime Obsession, that “arithmetic has the peculiar characteristic that it easy to state problems in it that are ferociously difficult to solve.” (I was rereading Prime Obsession last night; there’s tons of useful thought material for math teachers. I need to go get his book on algebra.)

Arithmetic looks easy. (And certainly in the last twenty years, the rush to shove everyone into calculus has led to a certain contempt for “basic arithmetic” classes.) But even if elementary school age children are capable of understanding its ideas fully (and most of them aren’t), they haven’t experienced several years’ utility of arithmetic. They haven’t had time to get bored of the routine rules that they are expected to remember (mind you, many don’t, but leave that for another day.) Yeah, yeah, invert and multiply. Yeah, yeah, you can’t divide by zero. Wait, what the hell do you mean multiplication isn’t repeated addition?

To really enjoy this test, to be fascinated by the underlying truths–or misconceptions–behind certain everyday math tools, requires familiarity with “the rules”. Time spent in the trenches of doing math just because.

That’s when a teacher can spend an enjoyable hour taking the kids back through a re-examination of the basics and what they really know. I’d much rather discuss these concepts with adolescents who have survived two or three years of high school math than try to force sixth graders to “demonstrate conceptual understanding” of dividing by zero.

I had no real expectations—no, that’s wrong. I had hopes. My sense was the students would be interested in the exploration, if I didn’t take on too much or dive in to the wrong end of the pool. But which end was the wrong end?

So for each of my four classes–two Algebra 2, two Trigonometry–I gave them the test and 20 plus minutes to write down their thoughts. I was alert to the possibility that kids would use five minutes to doodle and fifteen to giggle, but in each class the bulk of students asked for and got an additional five minutes to finish up. I collected their answers and will share some of them in later posts; they were often detailed and thoughtful.

After the writing time, the students had a few minutes to “share out” in their groups, so they could learn what questions puzzled their classmates—and also as reassurance that they weren’t alone in their befuddlement. Again, this seems different from Grant’s intent; he considered it a real test that the students would either answer correctly or leave blank in confusion. I listened in on many conversations; they were rich with exchange as the students realized they weren’t alone in their uncertainty.

But certain questions also sparked genuine debate and interest. More than a few students offered up multiplying negatives as an example of multiplication being something other than repeated addition. In every case I witnessed, their group members, who had written something to the effect of “isn’t it always repeated addition?” instantly recognized the roadblock that negative numbers posed to their definition. I came across more than one group arguing whether multiplying by zero counted as repeated addition (“yes, it does. If I have zero groups of five, I have zero!”). Interestingly, no one came up with the roadblock I was interested in, and I’d never once considered negative numbers until my students brought it up.

Their discussion time was about ten minutes. My goal wasn’t to have them determine the answers; rather, I wanted them all to have a shared experience before we discussed them as a class, and I gave them the “answers” (to the extent I knew them). That way, there’d be more of a sense of “we”–yeah, we thought of zero, too! yeah, we all have 3F=Y–that’s not the answer? yeah, we think dividing by zero gives you zero–it doesn’t?

So then we went through the answers as a group.

I had taken a subset of Grant’s list, ignoring the last three items. Doing it again, I would have swapped out question 2 for question 11 “appropriately precise”), because while question #2 is good, it really requires its own day. The rest of them are easily covered and discussed in at most 15-20 minutes each.

The questions I really wanted to spend time on, to explain in at least introductory depth, were 1, 3, and 5. From a practical standpoint, I wanted to be sure everyone understood why they got questions 4, 6, and 8 wrong, assuming most missed at least one of them. I was genuinely interested to see what they had to say about 7 and 9 but was going to take most of my lead from them. Question 10, I wanted to know if the trig students knew it; obviously, my algebra 2 students learn about imaginary numbers for the first time.

My trig classes are quite different in nature. Both are small, just 25 in each. Both are doing quite well; I have no kids who simply shouldn’t be there, as I did last year. My first block class is stronger, on average, but has more surly kids who mouth off. It’s very irritating, frankly, since the five or six kids giving me quite nasty sass are seniors who are doing relatively well (Bs and Cs), and who openly acknowledge that they think I’m a hell of a teacher. Two of the surlies had me last year for algebra 2, when they were much less trouble, and had been switched into my class because they were failing with another teacher. But these other teachers, who they didn’t like (and often failed, forcing them to retake a fake summer school course if they couldn’t switch to my class), didn’t get nearly the lip. I’m a tad flummoxed. My second block class has more kids who are amiable and interested but not taking the class as seriously as they should, so several more low scores on the first test. First block has a stupendous top tier, but it’s just three or four kids. Second block has a top tier of close to eight, but they aren’t quite as strong.

Anyway, I was expecting more interesting conversation from second block, and I had it backwards. First block was on point, even the cranky ones. They loved the test, wrote detailed responses, discussed it thoroughly in group, and were wildly participatory in the open discussion. Easily 90% of them came up with the correct response to imaginary numbers (and the ones from my algebra 2 class identified multiplying by i as 90 degree rotations in the complex plane, which was quite gratifying, thanks so much). Second block, the amiable, mildly uninterested ones pulled things down slightly, goofing around and making jokes while the stronger kids would have preferred more time to explore things. The conversation was still great, the students learned a lot and enjoyed the discussion, but I had the enthusiasm levels backwards.

My algebra 2 classes, I nailed in terms of expectations. Block three is a fairly typical profile, except I have a lot more sophomores than usual (which is due to our school successfully pushing more kids through geometry as freshmen). But still a good number of seniors who barely understood algebra I, a lot of whom are just hoping to mark time til graduation without ending up in summer school. (One of my specialty demographics.) And in between, juniors and seniors who are often thrilled to find themselves actually understanding math and succeeding beyond anything they’d ever hoped (another specialty of mine). Typically, many of the seniors were in class, as they lacked the the behavior or grade profile (and sadly, in some cases, the money) to go to the water park. So I expected conversation here to be a bit lower level, with less interest. Happily, everyone engaged to the best of their ability and many told me later how much they loved just “talking about math”. I spent much more time on questions 4, 6, and 8, and could see them all really registering why they’d made the mistakes they did. But they still were enthralled by questions 1, 3, and 5, which is great because it’s going to give them some memories when we review percentages in preparation for exponential functions.

Last up was block 4 algebra 2, a ridiculously strong class; only five students are of the usual caliber I expect. The seniors are all well above average ability level. Two of the kids are so skilled that I’ve already introduced three dimensional planes and the matrix, while still forcing them and the other really strong kids to deal with complex linear word problems (mixture questions! I usually skip them, so it’s a trip). They stomped all over the test, writing at great length, discussing it with their teams and then shouting out to other groups to see what they’d answered for multiplication. The class discussion took so long that I actually allowed it to continue for 20 minutes into the next day, when I invited one of my mentees to watch. He came away determined to try the test in his honors geometry class.

Look, the whole day was teacher crack. Take a day. Try the test. I’ll be discussing individual questions and my explanations in future posts, but this introduction is offered up as invitation. High school teachers working in algebra 2 or higher would be a good starting point. Honors classes in algebra and geometry would also benefit. Every math teacher can find links from this test to their math class—but then, that’s not the point.

As for me, I started out the day with hope, but also a determination to see it through as part of a way to honor Grant Wiggins, who felt very strongly that students needed to do more than just march through curriculum. I promised myself I wouldn’t abandon the effort even if it went wrong. It didn’t go wrong. Quite the contrary, the test sparked delighted interest and intellectual curiosity among students who are often hard to push into exploring mathematics in depth. So hey, Grant, thanks for the idea–and the inspiration.

Functions vs. Equations: f(x) is y and more

I wanted to talk about function algebra, which naturally would include a reference to function notation.

So here’s the frustrating thing about writing this blog. I try to include links to other sites that explain a concept, so that I don’t have to reinvent the wheel for my reading audience. But a google gives me these results: useless links that do little more than say “f(x) is the same as y”. That’s not math. That’s test prep. And there’s nothing wrong with test prep, but every one of these sites purport to be math teaching sites, and hey, I’m not a mathematician, but shouldn’t we be explaining what f(x) means?

Someone somewhere is saying “See, this is why we need teachers to be math majors, instead of English majors who get 800 on the GRE quant section. You can’t substitute math understanding that comes with the study of these important principles.” That someone somewhere is wrong. I used to think that in my early days, until I had too many conversations like this:

Me, to AP Calculus teacher WHO MAJORED IN MATH: Hey, what do you tell your kids about function notation?

AP Calculus teacher WHO MAJORED IN MATH: f(x) is the same as y.

Me, nonplused: Well. Yeah. But I mean about why we developed function notation, what it serves that can’t be served by….

AP Calculus teacher WHO MAJORED IN MATH: It’s just notation. Don’t be confused.

Me: I’m not confused. But they serve different purposes, and I’m just trying to be sure I accurately capture…

AP Calculus teacher WHO MAJORED IN MATH: They don’t serve different purposes. It’s just notation f(x) is the same as y.

Me: Ok.

In my experience, very few math teachers WHO ACTUALLY MAJORED IN MATH care about these things either. My beer drinking buddy is an exception (and he’s now department head), and he’s the only math teacher I’ve found so far who was interested in my work on this subject.

Textbooks? McDougall Litell, CPM has a lot of those function machines. But no explanation. Holt does a little better but I didn’t understand that until I understood what I was looking for.

So I spend more time looking for a good link. Otherwise, I have to spend a lot of time figuring out how to explain function notation accurately, or at least inoffensively, so that people reading this blog don’t make me remind them that, for chrissakes, I’m an English major not a mathematician! That takes time. It’s not time I wanted to spend. I don’t want to tell you what function notation is, in a way that will pass expert muster. I want to tell how I build on function notation to teach function algebra. But I can’t do that well without explaining function notation, which I didn’t set out to do. This leads to many blog entries taking much more time than they should. The original intent for my function algebra post was to be just a quick little throwaway.

I began writing this post nearly a month ago, and got stalled looking for a way to characterize the explanation. You may be wondering why I would explain something I don’t understand—but that’s not it, really. I just don’t know what to call it. And that’s fine for teaching, not so much for writing, and so I spend hours trying to figure out the correct query. Which took me, literally, up until today.

Just fifteen minutes ago (as I write this sentence) I finally found the kernel in this discussion on function notation before Euler, in which someone writes:

but [Newton] refers to these as equations, not functions, and admittedly (written the way they are) that is exactly what they are. It seems anything that we would today write as a function, Newton described in words, such as:

HA. I learned something I hadn’t quite understood completely before–a function and an equation are not the same thing. Googling “what is the difference between an equation and a function” led me to the right websites. I realize now that I wasn’t just looking for an explanation of function notation, but rather why and when we use functions vs. equations.

Here’s an explanation that covers what I was trying to say.

So my research paid off. In practice, what I’ve been doing in this lesson is introducing function operations and function notation as a way to overcome a constraint in using equations.


Sami needs $15 more to buy the new hoodie that he wants. But if Sami skips the hoodie, he needs just three more dollars to buy a ticket to the pizza feed on Friday. If Sami has x dollars, how much money, in terms of x, does Sami need if he wants both the hoodie and the ticket to the pizza feed?

The first thing the kids think is that Sami needs $18 more.

I say okay, Sami has $20. How much does the hoodie cost? $35. How much does the pizza feed cost? $23. How much ….oh. Huh, say the kids. He needs a lot more than $18.

Depending on how goofy I feel, I might get out some fake money. I count out $20, give it to a quiet student. How much more for the hoodie? Count out another $15. Now how about the…Right about then, a student gets it: you need the $20 twice.

So then we go to the board and model the two different equations for each purchase.


So if we are getting both things, what are we doing? Adding, the class choruses.

Ah, now there’s a new wrinkle. The kids have been adding equations for a while now, in systems. So I say, let’s try to add these equations.


Is that right? We test it with $20 and the kids realize that the right side “works” (that is, we get $68) but the left side says we still need to divide by 2, which would be…wrong.

“So what’s happening is that we are running into the limits of an equation. An equation tells us that two expressions occupy the same point on a number line–that is, after all, what “equal” means.”

“But when we use multiple variables in equations, then the equation becomes a relationship between two variables, an if-then. If y=x + 15, then the point (3, 18) is a solution because setting x=3 and y=18 creates an equation that has both sides occupying the same point on the number line. If 3x + 2y=12, then (2,3) is a solution because setting x=2 and y=3, etc.”

But in an equation, the variables are values. So in the Sami case, we can’t treat y as a collection point. We can’t keep track of the dependent variable because it varies, obviously. The y in the first equation has a different value from the y in the second equation. If we wanted to keep them separate, we could use two different variables, like z = x + 15 and y = x + 3. Or we could number the ys: y1 = x+15, y2 = x+3.

“Using the language of functions makes a lot of these constraints disappear.”

“First, logically. Functions are different in a key way from equations: a function is an output. An equation is a relationship between variables. Yes, y=x+3 and f(x)= x+3 yield the same results, which is why we teachers always tell you to remember that ‘y and f(x) are the same thing’. However f(x) isn’t a variable, but an output. So when we add two functions, we’re adding outputs. Remember, too, that a function doesn’t even have to be an equation, like in the cell phone code example.

Then there’s function notation, invented by Euler. Function notation enables unique names, usually a single letter. But it doesn’t have to be. You can get creative with the letter names and the input values.”

“Function notation is just more elegant and efficient, too. Instead of saying ‘if x=7’ you can just say f(3). Once you define the function named ‘f’, anything can be input, even another expression, like f(a+7). And then, instead of saying ‘y=’ and solving for x, write f(x)= 3.”

“So let’s call Sammy’s cash on hand c, and then create a function h for hoodie, and p for pizza feed.

h(c) = c+15
p(c) = c+3

In both cases, c represents the money Sami has, so the input value is the same. But the output value varies based on the function used.”

“Now, this is a small difference. But how many have you been told that f(x) is the same as y?” Bunch of hands raised.

“Yep. And in a lot of ways, it is. But you have to be wondering why, if they’re the same thing, we bother teaching you about function notation.” Lots of nods.

“So as you move on into advanced math, you’ll start to learn other reasons why we sometimes use functions and other times use equations. For now, it’s enough to know that function notation allows us to keep track of our different outcomes.

“Once we can do this, we can actually create an entire math with functions. They can be added, subtracted, multiplied. They have inverse operations.”

“But then why do we use equations?”

“Well, for one thing, functions don’t do systems well. Remember, when we solve systems, we are expecting both the x and the y (and any other variables) to be equal. Functions don’t handle that well. So you’ll see that we switch back and forth between equations and functions as needed.”

When you need to add expressions, functions are great. So now we can add h(c) and g(c).

h(c) + p(c) = (x + 15) + (x + 3) = 2x + 18

“Because we are adding outcomes, and have a unique way of tracking each outcome, we can add them properly. Remember, too, that since a function doesn’t need to be an equation, I can add or subtract outcomes without even having an equation. If a(x) = 9 and b(y) = 17, then b(y) – a(x) is 8, and I don’t have to care if a(x) and b(y) are generated by an expression or a rule or a code or a random happenstance—provided, of course, that random happenstance is only one per input.”


I know. You’re wondering why I don’t just follow the AP Calculus teacher’s “f(x) is the same as y”. Well, it turns out that function operations are a big part of pre-calc, so they’ll use this later.

In the meantime, I give them some practice with function notation (I stole this at random). Not enough. Kids don’t really know it later. But at least they’re exposed to it.

Then I go on to linear function addition and subtraction. I usually just put problems on the board.

Sample quiz:


Here’s a test question:


And from here I go on to linear function multiplication (aka quadratics) and, eventually, rational expressions (linear function division).

Like teaching congruence with isometries, I can’t argue that using functions to further our work in linear and quadratic equations is better. I find it more…elegant, maybe?

But the execution isn’t quite there. This is the first year I’ve really taught this whole sequence: introducing functions, function addition/subtraction/notation, function multiplication, inverse functions, rational expressions. Writing it up has revealed an obvious improvement. Up to now, my function illustration has been a quick standalone lesson. Then later I introduce the notion of function addition and in doing so, bring up function notation.

This is goofy, now that I look at it. In the future, I’ll introduce functions and then go into function notation. I can spend a day or two on that, quiz that early. Then I can go back into linear equations or inequalities (the placement is flexible) and then bring up function addition and subtraction, with function notation already covered.

You know what’s irritating? The huge effort described at the beginning of this post to figure out how to describe what I was teaching led me to this. The huge effort underwent solely in order to write this post. Which I was griping about. In learning how to describe function notation for my readers, I learned that the proper way to characterize my work is as a difference between functions and equations, and that led to an idea for better sequencing.

This is kind of a placeholder post. Obviously, I’m in flux about this right now. My linear equations unit has been in good shape for a while. This gives me plenty of room to add flourishes, introduce more complicated topics onto a subject the students know well. Meanwhile, linear function multiplication has proven to be a great introduction to quadratics. So now I’m involved in putting it all together.

Next up in this sequence: the post that I really wanted to write, on my quadratics introduction.

Sorry for the slow rate of posts lately. I did five in April, then got lazy.

Illustrating Functions

Function definitions aren’t usually tested on either the SAT or the ACT and since I never worked professionally with math, functions were something I’d barely considered in algebra a billion years ago. So for the first few years of teaching, I kind of went through the motions on functions: unique output for each input, vertical line test, blah blah. I didn’t ignore them or rush through them. But I taught them in straight lecture mode.

Once I got out of the algebra I ghetto (which really does warp your brain if that’s all you do), I accepted that a lot of the concepts I originally thought boring or unimportant show up later. So it’s worth my time to come up with the same third way activities and lessons for things like functions or absolute value or inverses that I do for binomial multiplication and modeling linear equations and inequalities.

So every year I pick concepts to transfer from pure lecture/explanation to illustration. Sometimes it’s spur of the moment, other times I plan a formal curriculum change. In the case of functions, the former.

Last year I was teaching algebra II/trig and–entirely in passing–noted a problem in the Holt book that looked something like this:

and had two simultaneous thoughts: what a boring question and hey, I could really do something with that.

So the next day, I tossed this up on the board without comment.


I’ve given these instructions three times now–a2/trig, trigonometry, algetbra 2–and the kids are always mystified, but what the heck, the activity seems simple enough. No student ever reads through the entire list of instructions first. They spend a lot of time picking the message, with many snickers, then have fun translating the code twice.

But then, as they all try to translate someone else’s message using the cell phone code, bam. They realize intuitively that translating the whole-alphabet code would be an easy task. And with a few moments of thought, they realize why the cell phone code doesn’t offer the same simple path. They don’t know what it means, exactly. But the students all realize that I’ve demonstrated a difference that they’d never considered.

From there, I graph the processes, which is usually a surprise as well. The translation process can be graphed?



At this point, I can usually convince kids to remember the Vertical Line Test, which they were taught in algebra I. At that point, I go through the definitions for relation, function, and one-to-one function, using a Venn diagram (something like this with an addition inner circle for one to ones). Then I go back through what the students vaguely remember about functions and link it to the correct code example.

Thus the students realize that translating the message into code is a function in either code key. I hammer this point home, because the most common misconception kids get from this is that all functions must be one to one. Both are functions. Each letter has one and only one number assigned, and the fact that one translation key puts several letters to the same number is irrelevant for its determination as a function. Reversing the process, going from numbers to letters, only one of them is a function.

Then I sketch parabolas and circles. Are they both functions? Are either of them one-to-one functions?

In Algebra 2, I do this long before the inverse unit. In Trig, I introduce it right before graphing the individual functions as part of an overview. In both classes, the early intro gives them time to recognize the significance of the difference between a function and the more limited case of the one-to-one function–particularly in trig, since the inverse functions are very limited graphs for exactly the reason. In algebra II, the graphs reinforce the meaning of the Horizontal Line Test.

I haven’t taught algebra I recently, but I’d change the lesson by giving them a coded message and ask them to translate with the cell phone code first.

This leads right into function and not-function, which is all they need in algebra I.

I have periodically mentioned my mixed feelings about CPM. Here’s a classic example. The CPM book introduces functions with the following example.

Okay. This is a terrible example. And really boring. Worst of all, as far as this non-mathie can tell, towards the end it’s flat out wrong. A relation can be predictable without being a function (isn’t that what a circle is?). But just looking at it, I got an idea for a great test question (click to enlarge):


And I could certainly see some great Algebra I activities using the same concept. But CPM just sucks the joy and interest out of the great starting ideas it has.

Anyway. I wanted to finish up with a push for illustrations. What, exactly, do the students understand at the moment of discovery in this little activity? I’ve never seen anyone make the intuitive leap to functions. However, they do all grasp that two tasks that until that moment seemed identical…aren’t. They all realize that translating the message in the whole-alphabet code would be a simple task. They all understand why the cell phone code translation doesn’t lend itself to the same easy translation.

I look for illustrative tasks that convince kids to think about concepts. As I’ve said before, the tasks might kick off a unit, or they might show up in the middle. They may demonstrate a phenomenon in math, or they might be problems designed to lead the students to the next step.

The most common pushback I get from math teachers when I talk about this method is “I love the idea, but I don’t have enough time.” To which I respond that pushing on through just means they’ll forget. Well, they’ll probably forget my lessons, too, but–maybe not so much. Maybe they’ll have more of a memory of the experience, a recollection of the “aha” that got them there. That’s my theory, anyway.

There’s no question that telling is quicker than illustrating or letting them figure it out for themselves. Certainly, the illustration should be followed by a clear explanation with much telling. I love explaining. But I’ve stopped kidding myself that a clear explanation is sufficient for most kids.

That said, let me restate what I said in my retrospective: The tasks must either be quick or achievable. They must illustrate something important. And they must be designed to lead the student directly to the observations or principles you want them to learn. It’s not a do it yourself walk in the park. Compare my lesson on exploring triangles with this more typical reform math example. I resist structure in many aspects of my life, but not curriculum.

In researching this piece, I stumbled across this really excellent essay Why Illustrations Aid Understanding by David Kirsh. I strongly recommend giving it a read. He is only discussing the importance of visual illustrations, whereas I’m using the word more broadly. Kirsh comes up with so many wonderful examples (math and otherwise) that categorize many different purposes of these illustrations. Truly great mind food. In the appendix, he discusses the limitations of visually representing uncertainty.


On reading this, I felt like my students did when they realized the cell phone message was much harder to translate: I have observed something important, something that I realize immediately is true and relevant to my work–even if I don’t yet know why or how.

Assessments with Multiple Answers

Multiple Answer Math tests are my new new thing, and I’m very pleased with how it’s going so far. I thought I’d talk about some of the problems in depth, see if anyone has suggestions.

Most of these questions come from an A2/Trig test I wrote this weekend, focusing on systems of equations, but my tests are always cumulative.


One of the things I really like about this format: I can combine free response and selected responses very easily. So here they had to graph the plane, then answer questions which may or may not have to do with the graph. So I could both test their ability to graph a plane see if they understand how distance works in three dimensions, check out their attention to detail, and see if they remember what a trace is. Query: is “slope of a trace line” acceptable? I’ve never taught 3-dimensional systems before, and the book only said “trace”. But when I was teaching it, I kept forgetting and say “trace line”. I wanted them to demonstrate they could visualize the plane in three dimensions and see the slopes of the lines forming the plane, and I couldn’t find any sample questions. Probably an oddball question.

“a” and “e” contain typos. I originally had a different line, until I realized it’d be too hard to graph on the coordinates I provided. So I changed everything, or tried to. Missed two things. First, I intended “a” to be correct, but forgot to change the constant. That’s okay, it will allow for attention to detail. But “e” is just a kluge question, since I changed the points but forgot to change the distance. Before, it was a test of evaluation; now it’s a more obvious wrong answer.


This question makes me very happy. Transformations, function operations, evaluation, and then a transfer of knowledge test! We’d never done any problems like “e” before. No one squawked, and I even saw some kids solving it graphically.


(I stole this graph from online, but can’t find it any more. If it’s yours, let me know and I’ll provide a link.)

I tiptoed conceptually into linear programming, but we did a lot with feasibility regions and of course, systems of inequalities. I describe my approach for Algebra II, but I step it up a bunch of paces for A2/Trig. I expect them to be able to graph lines and inequalities. They get review during the modeling section, but that’s all.


Another one I just think is elegant because it approaches the absolute value from so many different angles: algebraically, graphically, and then a function conceptual question for good measure.


I use this on both Algebra II and Algebra II/Trig. We math teachers try to beat into the kids’ heads the idea that a function can be defined or expressed in four ways: verbal, algebraic, graphic, and tabular. If this were a multiple choice question, students would just test one value and see what happens. But it’s multiple answer, and plugging in numbers takes a long time. Plotting the points and sketching the lines, on the other hand, works very nicely and very quickly—if you know how to graph those lines.

Every so often you can really mess with the kids’ minds, like this:


None of the “obvious” answers are right. The kids really have to trust their abilities.


This is almost pure concept. I introduced the algebra of rational expressions; we’ll do the graphs later. Well over half the kids correctly selected e, but a lot missed b. Ack.

Here’s a couple that work for either pre-calc or algebra II. The quadratic runs the gamut from conceptual to technical. The circle question is more purely technical, but that’s because there’s a lot to test.



I’m having a much easier time grading these now, once I realized I was actually creating True/False tests.

Still to be resolved, however: I have to distinguish between “left the problem blank because I didn’t know” and “not true”. Right now, I evaluate the test to determine what the student is doing, but in the future I think I’m going to have a field they can mark “T” or “F”. If it’s blank, it’s wrong.

So, for example, take a look at this question again:


Answers A, D, and E are true. The others are false. I give this question 14 points, 2 for each letter.

Almost all my students correctly select A as true, because they’ve built the equation themselves as an exercise and understand the parameters. They likewise know that B is false. Some of them read “maximum” as “initial” and wrongly select C, but many otherwise weak students with good attention to detail get it correct. So even my weak students are likely to get 6 points on these three letters.

Then we get to the tougher ones (they aren’t always in order of difficulty). Students have to understand what elements of the parabola equate to max height, time to max height, and zero height. Obviously, I cover these extensively, but kids have a harder time with this. I don’t just teach them a method. I expect them to know that max height is the parabola’s vertex, so that the x value is time to max height, and the y value is the height.

I had at least 12 students who correctly factored the problem, thus correctly NOT selecting E, but also NOT selecting D. Strong technically, weak on the concept of a “zero”. I gave them partial credit (a point) and yelled at them on the paper: things like “Noooooooooo!” and “Arggggghhhh.” and arrows and question marks and “Yo! What do you think (2t-3) means, exactly!?!”

The vertex questions E and G give students the most trouble, but that seems to be less about concept and more about a reluctance to work with fractions. My algebra II students actually do better than my precalc students because we spend a whole unit on this, as opposed to a few days in precalc.

So an average weak student will get 8-10 points out of 14. Very few students get all 14 points, maybe 8 out of 60. Most get 10-12. If they show their work and I can see they were on the right track with just an algebra error, I give partial credit. Other times, I can clearly see their math was terrible, even if they got the right answer. In those cases, I mark the question correct and then dock them 2 points for bad math.

While I don’t normally review tests, I always go through these and give the correct answers and discuss grading decisions.

I strongly recommend giving these a try. They’re lots of fun to make and again, typos are a lot easier to hide.

Assessing “Upper Level” Math Students on Algebra I


I am teaching Algebra II/TRIG! Not Algebra II. First time ever. Last December, I gave the kids a packet with the following letter:

Hi! I’m looking forward to our course.

Attached is a packet of Algebra I review work to prepare you for our class. If you are comfortable with linear and quadratic equations, then you’re in good shape. If you’re not, it’s time to study up!
Our course will be challenging and fast-paced, and I will not be teaching linear equations and quadratics in their entirety—that is, I expect you to know and demonstrate mastery of Algebra I concepts. We will be modeling equations and working with applied knowledge (the dreaded word problems) almost constantly. I don’t just expect you to regurgitate solutions. You’ll need to know what they mean.
I’m not trying to scare you off—just put you on your toes! But you should put in some time on this, because we will be having a test when you come to class the first full day. That test will go in the gradebook, but more importantly, it will serve as notice. You’ll know if you’re prepared for the class.

Have a great holiday.

Reminder: My school is on a full-block schedule, which means we teach a year’s content in a semester, then repeat the whole cycle with another group of students. A usual teacher schedule is three daily 90-minute classes, with a fourth period prep. I taught algebra II, pre-calc, and a state-test prep course (kids killed) last semester, and this semester I have A2/Trig and two precalcs.

(Notice that I am getting more advanced math classes? Me, too. It’s not a seniority thing. It’s not at my request. It’s possible, and tempting, to think they noticed the kids are doing well. I know the first decision to put me in pre-calc last year was deliberate, a decision to give me more advanced classes because they wanted a higher pass rate. But I honestly don’t know why it’s happening. Maybe they cycle round at this school, moving teachers from high to low and back again.)

So I said the first full day, and today was a half day, but the kids had a whole packet to work on and I wanted to understand I wasn’t screwing around. If they’d done the work, they’d do fine on the test. If they were planning on cramming, too bad so sad.

I was originally going to do a formal test, but decided to just throw a progression of problems on the board. Then I typed it up for next time, if I teach the class again.


How’d they do? About a third of them did well, given the oddball nature of the test. A couple got everything right. Most of them stumbled with graphing the parabola, which is fine. Some of them knew the forms (standard, point slope), but weren’t sure how to convert them.

Another three passed–that is, answered questions, showed they’d worked some of the packet. The rest failed.

Of the ones who failed, easily half of them had just blown off the packet but have the chops. The other half of that third I’m not sure of.

If you are thinking that kids in Algebra II/Trig should know more, well, they were demonstrably a step ahead of my usual algebra 2 classes. And I think some of them just didn’t know I was serious. Wait until that F gets entered, puppies. Like I told them today: “There’s a lower level option here. Take it if you can’t keep up.” Whoo and hoo.


I’ve now taught pre-calc twice. The first time, last spring, I was stunned at the low abilities of the bottom third, which I didn’t really understand fully for two or three weeks, leaving some of them hopelessly behind. I slowed it down and caught the bulk of the class, with only four to five students losing out on the slower pace (that is, they could have done more, but not all that much more). So when I taught it again in the fall, I gave them this assessment to see how many students could graph a line, identify a parabola from its graph, factor, and use function notation. If you’re thinking that’s pretty much the same thing I do with the A2/Trig classes, well, yeah. Generally, non-honors version of course is equivalent of honors version of previous year.

I don’t formally grade this; the assessment happens while they’re working. I can see who stumbles on lines, who stumbles on parabolas, who needs noodging, who works confidently, and so on. I was able to keep more kids moving forward in the semester/year just ended using this assessment and a slightly slower pace. One of the two classes is noticeably stronger; half the kids made it through to the function operations before asking for assistance.

This assessment also serves as a confidence booster for the weaker kids. Convinced they don’t understand a single bit of it, they slowly realize that by golly, they do know how to graph a line and multiply binomials. They can even figure out where the vertex should be, and they might have forgotten about the relationship between factors and zeros, but the memory wasn’t that far away.


While I just threw together the A2/Trig course, I put a huge amount of thought into this precalc assessment last fall. I think it’s elegant, and introduces them to a lot of the ideas I’ll be covering in class, while using familiar models.

Part II is just a way of seeing how many of them remember trig and right triangle basics:



If you’re interested in assessing kids entering Algebra (I or II) or Geometry, check out this one–multiple choice, easy to grade, and easy to evaluate progress.

Multiple Answer Math Tests

As previously explicated in considerable detail, I’m deeply disgusted with the Common Core math standards—they are too hard, shovel way too much math into middle school. If I see one more reporter obediently, mindlessly repeat that [s]tudents will learn less content, but more in-depth, coherent and demanding content my head will explode.

Reporters, take heed: you can’t remove math standards. The next time some CC drone tells you that the standards are fewer, but deeper, ask for specifics. What specific math standard has been removed? Do students no longer have to know the quadratic formula? Will they not need to know conics? No, not colonics. That’s what you all should be forced to endure, for your sins. In all likelihood, the drone has no more idea than you reporters do about high school math, so go ask Jason Zimba, who reiterates several times in this interview that the standards are fewer, but go deeper. (He also confirms what I said about algebra, that much of it is moved to middle school). Ask him. Please. What’s left off?

Pause, and deep breath. Where was I?

Oh. Tests.

So the new CC tests are not multiple choice, a form that gets a bad rap. I give my kids in algebra one, geometry, and algebra two lots of multiple choice tests—not because I prefer them, and they aren’t easier (building tests is hard, and I make my own), but because my top students aren’t precise enough and they need the practice. They fall for too many traps because they’re used to teachers (like me) giving them partial or most of the credit if all they did is lose a negative sign. Remember, these are the top kids in the mid-level or lower math classes, not the top kids at the school. These are the kids who often can get an A in the easier class, and aren’t terribly motivated. My multiple choice tests attempt to smack them upside the head and take tests more seriously. It works, generally. I have to watch the lower ability kids to be sure they don’t cheat.

We’ve been in a fair amount of PD (pretty good PD, at that) on Common Core; last fall, we spent time as a department looking at the online tests. The instructors made much of the fact that the students couldn’t just “pick C”, although that gave us a chuckle. Kids who don’t care about their results will find the CC equivalent of picking C. Trust them. And of course, the technology is whizbang, and enables test questions that have more than one correct answer.

But I started thinking about preparing my students for Common Core assessments and suddenly realized I didn’t need technology to create tests questions that have more than one answer. And that struck me as both interesting and irritating, because if it worked I’d have to give the CC credit for my innovation.

On the first test, I didn’t do a full cutover, but converted or added new questions. Page 1 had 2 or 3 multiple answer questions and 3 was free-response, but on that first test, the second page was almost all multiple response:


I had been telling the kids about the test format change for a week or two beforehand, and on the day of the test I told them to circle the questions that were multiple answer.

It went so well that the second test was all multiple answer and free response. I was using a “short” 70-minute class for the test, so I experimented with the free-response. I drew in the lines, they had to identify the inequalities.



I like it so much I’m not going back. Note that the questions themselves aren’t always “common core” like, nor is the format anything like Common Core. But this format will familiarize the kids with multiple answer tests, as well as serve my own purposes.


  • Best of all, from my perspective, is that I am protected from my typos. I am notorious, particularly in algebra, for test typos. For example, there are FIVE equations on that inequality word problem, not four. See the five lines? Why did I put four? Because I’m an idiot. But in the multiple answer questions, a typo is just a wrong answer. Bliss, baby.
  • I can test multiple skills and concepts on one question. It saves a huge amount of space and allows the kids to consider multiple issues while all the information is in RAM, without having to go back to the hard drive.
  • I can approach a single issue from multiple conceptual angles, forcing them to think outside one approach.
  • It takes my goal of “making kids pay attention to detail” and doubles down.
  • Easier, even, than multiple choice tests to make multiple versions manually.
  • Cheating is difficult, even with one version.


Really, only one: I struggle with grading them. How much should I weight answers? Should I weight them equally, or give more points for the obvious answer (the basic understanding) and then give fewer points for the rest? What about omitting right answers or selecting wrong ones?

Here’s one of my stronger students with a pretty good performance:


You can see that I’m tracking “right, wrong, and omit”, like the SAT. I’m not planning on grading it that way, I just want to collect some data and see how it’s working.

There were 20 correct selections on nine questions. I haven’t quite finished grading them, but I’ve graded two of the three strongest students and one got 15, the other 14. That is about right for the second time through a test format. Since I began the test format two thirds of the way through the year, I haven’t begun to “norm” them to check scrupulously for every possible answer. Nor have I completely identified all the misunderstandings. For example, on question 5, almost all the students said that the “slope” of the two functions’ product would be 2—even the ones who correctly picked the vertex answer, which shows they knew it was a parabola. They’re probably confusing “slope” with “stretch”, when I was trying to ascertain if they understood the product would be a parabola. Back to the drawing board on that.

Added on March 7: I’ve figured out how to grade them! Each answer is an individual True/False question. That works really well. So if you have a six-option question, you can get 6/6, 5/6, 4/6 etc. Then you assign point totals for each option.

I’ll get better at these tests as I move forward, but here, at least, is one thing Common Core has done: given me the impetus and idea for a more flexible test format that allows me to more thoroughly assess students without extending the length of the test. Yes, it’s irritating. But I’ll endure and soldier on. If anyone’s interested, I’m happy to send on the word doc.

Note: Just noticed that the student said y>= -2/3x + 10, instead of y<=. It didn't cost her anything in points (free response I'm looking for the big picture, not little errors), but I went back and updated her test to show the error.

The Negative 16 Problems and Educational Romanticism

I came up with a good activity that allowed me to wrap up quadratics with a negative 16s application. (Note: I’m pretty sure that deriving the algorithm involves calculus and anyway, was way beyond the scope of what I wanted to do, which was reinforce their understanding of quadratics with an interesting application.) As you read, keep in mind: many worksheets with lots of practice on binomial multiplication, factoring, simpler models, function operations, converting quadratics from one form to another, completing the square (argghh) preceded this activity. We drilled, baby.

I told the kids to get out their primary quadratics handout:


Then I showed two model rocket launches with onboard camera (chosen at random from youtube).

After the video, I toss a whiteboard marker straight up and caught it. Then I raised my hand and drop the marker.

“So the same basic equation affects the paths of this marker and those rockets–and it’s quadratic. What properties might affect—or be affected by—a projectile being launched into the air?”

The kids generated a list quickly; I restated a couple of them.


Alexandra: “What about distance?”

I pretended to throw the marker directly at Josh, who ducked. Then I aimed it again, but this time angling towards the ceiling. “Why didn’t Josh duck the second time?”

“You wouldn’t have hit him.”

“How do you know?”

“Um. Your arm changed…angles?”

“Excellent. Distance calculations require horizontal angles, which involves trigonometry, which happens next year. So distance isn’t part of this model, which assumes the projectile is launched straight….”


“What about wind and weather?” from Mark.

“We’re ignoring them for now.”

“So they’re not important?”

“Not at all. Any of you watch The Challenger Disaster on the Science Channel?”

Brad snickered. “Yeah, I’m a big fan of the Science Channel.”

“Well, about 27 years ago, the space shuttle Challenger exploded 70 some seconds after launch, killing everyone on board when it crashed back to earth.” Silence.

“The one that killed the teacher?”

“Yes. The movie—which is very good—shows how one man, Richard Feynman, made sure the cause was made public. A piece of plastic tubing was supposed to squeeze open and closed—except, it turns out, the tubing didn’t operate well when it was really cold. The launch took place in Florida. Not a place for cold. Except it was January, and very cold that day. The tubing, called O-ring, compressed—but didn’t reopen. It stayed closed. That, coupled with really intense winds, led to the explosion.”

“A tube caused the crash?”

“Pretty much, yes. Now, that story tells us to sweat the small stuff in rocket launches, but we’re not going to sweat the small stuff with this equation for rocket launches! We don’t have to worry about wind factors or weather.”

“Then how can it be a good model?” from Mark, again.

“Think of it like a stick figure modeling a human being but leaving out a lot. It’sstill a useful model, particularly if you’re me and can’t draw anything but stick figures.”

So then we went through parameters vs. variables: Parameters like (h,k) that are specific to each equation, constant for that model. Variables–the x and y–change within the equation.

“So Initial Height is a parameter,” Mark is way ahead.

Nikhil: “But rocket height will change all the time, so it’s a variable.”

Alissa: “Velocity would change throughout, wouldn’t it?”

“But velocity changes because of gravity. So how do you calculate that?” said Brad.

“I’m not an expert on this; I just play one for math class. What we calculate with is the initial velocity, as it begins the journey. So it’s a parameter, not a variable.”

“But how do you find the initial velocity? Can you use a radar gun?”

“Great question, and I have no idea. So let’s look at a situation where you’ll have to find the velocity without a radar gun. Here’s an actual—well, a pretend actual—situation.”neg16question

“Use the information here to create the quadratic equation that models the rocket’s height. In your notes, you have all the different equation formats we’ve worked with. But you don’t have all the information for any one form. Identify what information you’ve been given, and start building three equations by adding in your known parameters. Then see what you can add based on your knowledge of the parabola. There are a number of different ways to solve this problem, but I’m going to give you one hint: you might want to start with a. Off you go.”

And by golly, off they went.

As releases go, this day was epic. The kids worked around the room, in groups of four, on whiteboards. And they just attacked the problem. With determination and resolve. With varying levels of skill.

In an hour of awesomeness here is the best part, from the weakest group, about 10 minutes after I let them go. Look. No, really LOOK!


See negative 2.5 over 2? They are trying to find the vertex. They’ve taken the time to the ground (5 seconds) and taken half of it and then stopped. They were going to use the equation to find a, but got stuck. They also identified a zero, which they’ve got backwards (0,5), and are clearly wondering if (0,4) is a zero, too.

But Ed, you’re saying, they’ve got it all wrong. They’ve taken half of the wrong number, and plugged that—what they think is the vertex—into the wrong parameter in the vertex algorithm.. That’s totally wrong. And not only do they have a zero backwards, but what the hell is (0,4) doing in there?

And I say you are missing the point. I never once mentioned the vertex algorithm (negative b over 2a). I never once mentioned zeros. I didn’t even describe the task as creating an equation from points. Yet my weakest group has figured out that c is the initial height, that they can find the vertex and maybe the zeroes. They are applying their knowledge of parabolas in an entirely different form, trying to make sense of physical data with their existing knowledge. Never mind the second half—they have knowledge of parabolas! They are applying that knowledge! And they are on the right track!

Even better was the conversation when I came by:

“Hey, great start. Where’d the -2.5 come from?”

“It’s part of the vertex. But we have to find a, and we don’t know the other value.”

“But where’d you get 2.5 from?”

“It’s halfway from 5.”

Suddenly Janice got it.

“Omigod–this IS the vertex! 144 is y! 2.5 is x! We can use the vertex form and (h,k)!!”

The football player: “Does it matter if it doesn’t start from the ground?”

Me: “Good question. You might want to think about any other point I gave you.”

I went away and let them chew on that; a few minutes later the football player came running up to me: “It’s 2!” and damned if they hadn’t solved for a the next time I came by.

Here’s one of the two top groups, at about the same time. (Blurry because they were in the deep background of another picture). They’d figured out the vertex and were discussing the best way to find b.


Mark was staring at the board. “How come, if we’re ignoring all the small stuff, the rocket won’t come straight back down? Why are you sure it’s not coming back to the roof?”

“Oh, it could, I suppose. Let me see if I can find you a better answer.” He moved away, when I was struck by a thought. “Hey….doesn’t the earth move? I mean yes, the earth moves. Wouldn’t that put the rocket down in a different place?”

“Is that it?”
“Aren’t you taking physics? Go ask your teacher. Great questions.”

I suggested taking a look at the factored form to find b but they did me one better by using “negative b over 2a” again and solving for b (which I hadn’t thought of), leading to Mark’s insight “Wait–the velocity is always 32 times the seconds to max height!”

The other kids had all figured out the significance of the vertex form, and were all debating whether it was 2.5 or 2 seconds, generally calling me over to referee.

One group of four boys, two Hispanics, one black, one Asian (Indian), all excellent students, took forever to get started, arguing ferociously over the vertex question for 10 minutes before I checked on them to see why they were calling each other “racist” (they were kidding, mostly). I had to chastise the winners for unseemly gloating. Hysterical, really, to see alpha males in action over a math problem. Their nearly-blank board, which I photographed as a rebuke:


The weaker group made even more progress (see the corrections) and the group to their left, middling ability, in red, was using standard equation with a and c to find b:

My other top group used the same method, and had the best writeup:

Best artwork had the model wrong, but the math mostly right:

  • All but one group had figured out they wanted to use vertex form for the starting point.
  • All but one group had kids in it that realized the significance of the 80 foot mark (the mirror point of the initial height)
  • All the groups figured out the significance of five seconds.
  • All the groups were able to solve for both a and b of the standard form equation.
  • The top three groups worked backwards to find the “fake” zero.
  • Two groups used the vertex algorithm to find b.
  • All the groups figured out that b had to be the velocity.

So then, after they figured it all out, I gave them the algorithm:

h(t)=-16t2 + v0t + s0.

Then I gave them Felix Baumgartner, the ultimate in a negative 16 problem.

And….AND!!!! The next day they remembered it all, jumping into this problem without complaint:projmotfollowup

Charles Murray retweeted my why not that essay, saying that I was the opposite of an educational romantic, and I don’t disagree. But he’s also tweeted that I’m a masochist for sticking it out—implying, I think, that working with kids who can’t genuinely understand the material must be a sad and hopeless task. (and if he’s not making that point, others have.) I noticed a similar line of thought in this nature/nurture essay by Tom Bennett says teachers would not write off a child with low grades as destined to stack shelves –implication that stacking shelves is a destiny unworthy of education.

The flip side of that reasoning looks like this: Why should only some students have access to a rich, demanding curriculum and this twitter conversation predicated on the assumption that low income kids get boring curricula with no rigor and low expectations.

Both mindsets have the same premise: education’s purpose is to improve kids’ academic ability, that education without improvement is soulless drudgery, whether cause or effect. One group says if you know kids can’t improve, what a dreary life teaching is. The other group says dreary teaching with low expectations is what causes the low scores—engage kids, better achievement. Both mindsets rely on the assumption that education is improvement.

Is it?

Suppose that in six months my weakest kids’ test scores are identical to the kids who doodled or slept through a boring lecture on the same material. Assume this lesson does nothing to increase their intrinsic motivation to learn math. Assume that some of the kids end up working the night shift at 7-11. Understand that I do make these assumptions.

Are the kids in my class better off for the experience? Was there value in the lesson itself, in the culmination of all those worksheets that gave them the basis to take on the challenge, in the success of their math in that moment? Is it worth educating kids if they don’t increase their abilities?

I believe the answer is yes.

Mine is not in any way a dreary task but an intellectual challenge: convince unmotivated students to take on advanced math—ideally, to internalize the knowledge for later recall. If not, I want them to have a memory of success, of achievement—not a false belief, not one that says “I’m great at math” but one that says “It’s worth a try”. Not miracles. Just better.

I would prefer an educational policy that set more realistic goals, gave kids more hope of actual mastery. But this will do in the meantime.

I have no evidence that my approach is superior, that lowering expectations but increasing engagement and effort is a better approach. I rely on faith. And so, I’m not entirely sure that I’m not an educational romantic.

Besides. It’s fun.

Isometries and Coordinate Geometry

Michael Pershan’s post on teaching congruence reminded me that way back in the beginning of summer, I’d been meaning to write up some of my geometry work, which I think is pretty unusual. Still on the list is the lesson sequencing, but here is some thoughts and sample problems on integrating Isometries and coordinate geometry.

To summarize my earlier work, explicated in Teaching Congruence, or Are You Happy, Professor Wu?, I was unhappy with the circular reasoning that geometry books present in congruence sections. Triangle ABC is congruent with Triangle DEF because all their sides and angles are congruent, and congruence is when the shapes have congruent sides and angles. Professor Wu’s writing taught me the link between congruence, similarities, and isometries (aka, transformations, or translations, rotations, and reflections). I’d previously skipped isometries, since the kids don’t need them much and they’re easy to figure out, but this discovery led me to use isometries as an introduction to congruence and similarity.

But all book chapters on isometries are very thin, or they rely on non-coordinate shapes, which is largely a waste of time. Was there any way I could bring back some other concept while working with isometries, particularly with my top students?

Which leads me straight to coordinate geometry. The most immediate tie-in is helping students figure out rotation, the most difficult of the transformations. A 90 degree rotation around a point involves perpendicular lines (“..and class, what is the relationship of perpendicular slopes, again? Class? Waiting!”). Moreover, the kids learn that the slope of the line connecting a point and its reflection must be perpendicular to the line of reflection. Finally, dilations involve all sorts of work with parallel lines. All of these reinforcements are excellent for weaker students, and are yet another reason to introduce transformations, even if only as a prelude to congruence.

But I wanted a meaningful connection for my top students, who usually grasp the basics quickly. What could I give them that would integrate algebra, coordinate geometry, and a better understanding of transformations?

Over the summer, I taught an enrichment geometry class to seventh graders whose parents got mad because I wasn’t assigning enough homework. My boss backed me–thanks, boss!—and the kids did, too—thanks, kids!—and not for the usual reasons (these are not kids who celebrate a lack of homework). The kids all told the boss that they were surprised that they weren’t able to just follow the pattern and churn out 50 problems of increasing difficulty in the same vein. “I have to really think about the problem,” said more than one, in some astonishment.

So, for example:

Homework: Reflect Triangle LMN [L: (-1,4), M: (0,7), N: (-4, 10) over line y=x+2. Prove it.

So we discussed the steps before they left. I actually posed it as a couple of questions.

  1. If you sketched this and just estimated points the reflection, what would be the key information you’d need to pin down to go from “estimation” to “actual answer”?
  2. Can you think of any coordinate geometry algorithms that might help you find these points?

And working with me, they came up with this procedure for each point:

  1. Find the equation of the line perpendicular to the reflection line.
  2. Find the solution to the reflection line and the perpendicular line. This solution is also the midpoint between the original point and its reflection.
  3. Using the original point and the midpoint, find the reflection point.
  4. Prove the reflection is accurate by establishing that the sides of the original triangle and the reflection are congruent.

And here it is, mapped out in Desmos—but honestly, it was much easier to do on graph paper. I just wanted to increase my own Desmos capability.


This is the cleaned up version. Maybe I should put the actual work product here. But I’m not very neat. Next time I’ll take pictures of some of the kids’ work; it’s gorgeous.

When we came in the next day, the kids excitedly told me they’d not only done the work, but “figured out how to do it without the work!” Sure, I said, and we then predicted what would happen with the reflection of y=x+3, y=-x + 4, and so on.

But what about reflecting it over the line y=-2x?


Gleesh. I didn’t have time during summer to investigate why the numbers are so ugly. The kids got tired after doing two points, and I told them to use calculators. But we did get it to work. We could see the fractions begin in the perpendicular line solutions, since we’re always adding .5x to 2x. But would it always be like that?

However, I’ve got one great activity for strong kids done–it reinforces knowledge of reflection, coordinate geometry, systems of equations, and some fairly messy algebra. Whoo and hoo.

Down side–for the first time in two years, I’m not teaching geometry this year!

All the more reason to document. Next up in this sequence is my teaching sequence. But if anyone has ideas about the translation that makes the second reflection have such unfriendly numbers, let me know.

Hey, under 1000!

Algebra 1 Growth in Geometry and Algebra II, Spring 2013

This is part of an ongoing series on my Algebra II and Geometry classes. By definition, students in these classes should have some level of competence in Algebra I. I’ve been tracking their progress on an algebra I pre-assessment test. The test assesses student ability to evaluate and substitute, use PEMDAS, solve simple equations, operate with negative integers, combine like terms. It tiptoes into first semester algebra—linear equations, simple systems, basic quadratic factoring—but the bulk of the 50 questions involve pre-algebra. While I used the test at my last school, I only thought of tracking student progress this year. My school is on a full-block schedule, which means we teach a year’s content in a semester, then repeat the whole cycle with another group of students. A usual teacher schedule is three daily 90-minute classes, with a fourth period prep. I taught one algebra II and one geometry class first semester (the third class prepared low ability students for a math graduation test), their results are here.

So in round two, I taught two Algebra 2 courses and one Geometry 10-12 (as well as a precalc class not part of this analysis). My first geometry class was freshmen only. In my last school, only freshmen who scored advanced or proficient on their 8th grade algebra test were put into geometry, while the rest take another year of algebra. In this school, all a kid has to do is pass algebra to be put into geometry, but we offer both honors and regular geometry. So my first semester class, Geometry 9, was filled with well-behaved kids with extremely poor algebra skills, as well as a quarter or so kids who had stronger skills but weren’t interested in taking honors.

I was originally expecting my Geometry 10-12 class to be extremely low ability and so wasn’t surprised to see they had a lower average incoming score. However, the class contained 6 kids who had taken Honors Geometry as freshmen—and failed. Why? They didn’t do their homework. “Plus, proofs. Hated proofs. Boring,” said one. These kids knew the entire geometry fact base, whether or not they grokked proofs, which they will never use again. I can’t figure out how to look up their state test scores yet, but I’m betting they got basic or higher in geometry last year. But because they were put into Honors, they have to take geometry twice. Couldn’t they have been given a C in regular geometry and moved on?

But I digress. Remember that I focus on number wrong, not number right, so a decrease is good.


Again, I offer up as evidence that my students may or may not have learned geometry and second year algebra, but they know a whole lot more basic algebra than they did when they entered my class. Fortunately, my test scores weren’t obliterated this semester, so I have individual student progress to offer.

I wasn’t sure the best way to do this, so I did a scatter plot with data labels to easily show student before/after scores. The data labels aren’t reliably above or below the point, but you shouldn’t have to guess which label belongs to which point.

So in case you’re like me and have a horrible time reading these graphs, scores far over to the right on the x-axis are those who did poorly the first time. Scores low on the y-axis are those who did well the second time. So high right corner are the weak students at both beginning and end. The low left corner are the strong students who did well on both.

Geometry first. Thirty one students took both tests.


Four students saw no improvement, another four actually got more wrong, although just 1 or 2 more. Another 3 students saw just one point improvement. But notice that through the middle range, almost all the students saw enormous improvement: twelve students, over a third, got from five to sixteen more correct answers, that is, improved from 10% to over 30%.

Now Algebra 2. Forty eight students took both tests; I had more testers at the end than the beginning; about ten students started a few days late.


Seven got exactly the same score both times, but only three declined (one of them a surprising 5 points—she was a good student. Must not have been feeling well). Eighteen (also a third) saw improvements of 5 to 16 points.

The average improvement was larger for the Algebra 2 classes than the Geometry classes, but not by much. Odd, considering that I’m actually teaching algebra, directly covering some of the topics in the test. In another sense, not so surprising, given that I am actually tasked to teach an entirely different topic in both cases. I ain’t teaching to this test. Still, I am puzzled that my algebra II students consistently show similar progress to my geometry students, even though they are soaked in the subject and my geometry students aren’t (although they are taught far more algebra than is usual for a geometry class).

I have two possible answers. Algebra 2 is insanely complex compared to geometry, particularly given I teach a very slimmed-down version of geometry. The kids have more to keep track of. This may lead to greater confusion and difficulty retaining what they’ve learned.

The other possibility is one I am reminded of by a beer-drinking buddy, a serious mathematician who is also teaches math: namely, that I’m a kickass geometry teacher. He bases this assertion on a few short observations of my classes and extensive discussions, fueled by many tankards of ale, of my methods and conceptual approaches (eg: Real-life coordinate Geometry, Geometry: Starting Off, Teaching Geometry,Teaching Congruence or Are You Happy, Professor Wu?, Kicking Off Triangles, Teaching Trig).

This possibility is a tad painful to contemplate. Fully half the classes I’ve taught in my four years of teaching—twelve out of twenty four—have been some form of Algebra, either actual Algebra I or Algebra I pretending to be Algebra II. I spend hours thinking about teaching algebra, about making it more understandable, and I believe I’ve had some success (see my various posts on modeling).

Six of those 24 classes have been geometry. Now, I spend time thinking about geometry, too, but not nearly as much, and here’s the terrible truth: when I come up with a new method to teach geometry, whether it be an explanation or a model, it works for a whole lot longer than my methods in algebra.

For example, I have used all the old standbys for identifying slope direction, as well as devising a few of my own, and the kids are STILL doing the mental equivalent of tossing a coin to determine if it’s positive or negative. But when I teach my kids how to find the opposite and adjacent legs of an angle (see “teaching Trig” above), the kids are still remembering it months later.

It is to weep.

I comfort myself with a few thoughts. First, it’s kind of cool being a kickass geometry teacher, if that is my fate. It’s a fun class that I can sculpt to my own design, unlike algebra, which has a billion moving parts everyone needs again.

Second, my algebra II kids say without exception that they understand more algebra than they ever did in the past, that they are willing to try when before they just gave up. Even the top kids who should be in a different class tell me they’ve learned more concepts than before, when they tended to just plug and play. My algebra 2 kids are often taking math placement tests as they go off to college, and I track their results. Few of them are ending up in more than one class out of the hunt, which would be my goal for them, and the best are placing out of remediation altogether. So I am doing something right.

And suddenly, I am reminded of my year teaching all algebra, all the time, and the results. My results look mediocre, yet the school has a stunningly successful year based on algebra growth in Hispanic and ELL students—and I taught the most algebra students and the most of those particular categories.

Maybe what I get is what growth looks like for the bottom 75% of the ability/incentive curve.

Eh. I’ll keep mulling that one. And, as always, spend countless hours trying to think up conceptual and procedural explanations that sticks.

I almost titled this post “Why Merit Pay and Value Added Assessment Won’t Work, Part IA” because if you are paying attention, that conclusion is obvious. But after starting a rant, I decided to leave it for another post.

Also glaringly on display to anyone not ignorant, willfully obtuse, or deliberately lying: Common Core standards are irrelevant. I’d be cynically neutral on them because hell, I’m not going to change what I do, except the tests will cost a fortune, so go forth ye Tea Partiers, ye anti-test progressives, and kill them standards daid.


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