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.


About educationrealist

34 responses to “The Negative 16 Problems and Educational Romanticism

  • NewarkTFA

    So, according to you, intellectual engagement has some sort of intrinsic value? No idea could be more romantic, or a more welcome contrast to our current, gradgrindian obsession with “data.”

  • countenance

    I was 8 years old and in the 3rd grade when Challenger happened. I happened to be home from school that day for an afternoon doctor’s appointment, so I was at home watching the launch live on TV. I made it my business to watch because the teachers at my school were all abuzz over a teacher going up on the shuttle (Christina McAuliffe).

    When I watched it live, I didn’t know that something went wrong. It was only after watching Dan Rather loop it over and over again and using words indicative of mission failure and tragedy did I finally get it.

    Two weeks or so earlier, I was watching Dan Rather talking about another outer space issue: Voyager 2 visited Uranus.

  • Hattie

    “We would no more do this than write off a child with low grades as destined to stack shelves.”

    This makes me angry, and normally I’m too lazy to engage that much. Something wrong with stacking shelves? This is normally where the educational romantics I know gently, oh-so-patiently explain about higher expectations and the low wages, respect etc., accorded to shelf stackers. Like it’s akin to the laws of gravity. Nope, more to do with the educational (and immigration) policies educational romantics like to push.

    It’s like the people pushing, um, certain ethnic groups to come into the area. They then lecture me about the reality that, well, it’s just too dangerous to go to certain areas, especially after dark. Live in the real world, Hattie. Like they had nothing to do with how the real world works.

  • Roger Sweeny

    Shouldn’t the algorithm near the end read,

    h(t) = MINUS 16 t squared?

    Which is the “Negative 16” in the post title. The equation shows height to be determined by three things. Working backwards, the height you start at–s zero, a constant. The height contributed by the initial velocity of the rocket, v zero times t, something that always gets higher with time. And the height contributed by gravity pulling down, minus 16 t squared. It’s negative because the existence of gravity means the rocket goes less high that it would if gravity was somehow turned off In fact, eventually, gravity will pull the rocket down to earth; h(t) will be zero.

    As a former high school physics teacher, if there is no wind or air resistance, the rocket will come down exactly where it started. Everything on the earth has angular momentum due to the fact that the earth is rotating. That angular momentum doesn’t go away just because something is no longer connected to the earth’s surface (“Angular momentum is conserved.”). So essentially, the rocket here would continue rotating at the same speed that the earth’s surface is rotating.

  • panjoomby

    Bravo! well done. even i almost understand it! Plus, anything that makes it good for the teacher is good for the students (analogous to anything that reduces parent stress is good for the kid). seriously, a teacher who’s happy with how their instruction went does the most good for the most students …does that make utilitarianism romantic?:)

    • educationrealist

      I’m not sure that’s true. Teachers who are happy with lectures but have a class zoned out and not listening wouldn’t work. I can think of plenty of activities that wouldn’t work with this format.

      Right now, my kids are doing “problems in the round”. It’s a review technique; I just put problems on white boards around the room. They’re working and chatting. Not a lot of instruction required. Well, I find Christmas songs.

      • panjoomby

        as a stats teacher for 20 years, i figured my “value added” was how much they got from me alone – above & beyond what they would’ve got from the book alone (easy win – stat book reading results in zero knowledge – just sleep:) you’re far more active – i mainly lectured & wrote on the board – i didn’t like to yield control! & i told ’em if someone asks a question you don’t understand – DON’T listen to the answer! (they had a lot of variability in their math background, from zero all the way to professional math teachers:)

  • retired

    What kind of faith?

  • Tort

    I like this blog and I like the lesson you shared. For a guy who does not seem to be too enamored with either Common Core or John Dewey, I must admit the lesson you shared is quite Deweyian and is also a fine Common Core example.

    What do you say about that?

    • educationrealist

      I say guilty as charged. I think if you search Boaler or reform on this site, I mention that I’m anti-reform but look an awful lot like a reform teacher. The difference is that I sort by ability, which reformers say doesn’t exist.

      thanks for the kind words!

  • Tort

    I took Boaler’s Stanford online class last summer and I’ve read her main book, “What’s Maths Got To Do With It.” I read your blogs because they mostly nail the political hypocrisy behind the reform, and the fact that teachers are being targeted because big money despises the union. Not that I’m a staunch union guy (I survived 13 years teaching Catholic school without being in one), but I am hearing a lot of demagoguery by people who haven’t paid their dues in a public classroom.

    Regarding reform teaching, some of it makes sense because it’s not completely procedural and it can generate powerful thinking by our students. What I am seeing is that the lesson you invented and executed only makes good students better; but struggling students might be huddled together scratching their heads, but still, “From nothing comes nothing,” which makes a guy like me want to get right back into rows and Direct Interactive Instruction. At least I can control the clock and give them some math.

    I look forward to seeing more of your work.

    • educationrealist

      Believe me when I say that I would have considered a situation such as you describe–the top kids working, the bottom kids not–to be a complete waste of time. Every kid in the room was working.

      That’s exactly the problem with reform math and “group work”. Most of the time, my kids are just working problems, problems that I set at an appropriate difficulty level for students. Top students either know the material already so they’re doing something else, or they have harder versions of the work.

      When I taught algebra I, it was mostly the first sitution–I had kids in the class who knew a lot of algebra already, and kids who were non-functional. I taught something like four separate lessons per class (search differentiate on this site, I think it comes up).

      But look, I am stating this as a fact: the kids all did the work. There were 7 groups, I’ve included pictures of five groups. The remaining two—one top, one lower–also did the work.

  • Florida resident

    My favorite joke is about a teacher in a military school. He is deriving the formula for the horizontal distance L, which a projectile will cover after being shot from a gun at elevation angle “alpha” over the horizon with initial velocity v.
    In the process of derivation he made a typo somewhere, and instead of correct formula
    he got
    Some smart student asks:
    “Sir Coronel, Sir ! What if alpha is going closer and closer to 90 degrees ?
    Will L be going to infinity ?”
    Coronel (after two minutes of intense thinking):
    “No, in will not be going to infinity,
    since each big gun has special stopper at 88 degrees.”

  • Tort

    Ed R, I keep reading this lesson, which takes me hours because I follow the links you have embedded, and that takes me deep into connected topics. I want to try teaching this way also, but I would like to ask you if a) your students are in Algebra 2?, and b) do you think I should modify this lesson for my Algebra 1 students? If you wouldn’t mind, I would like to give to you a private email address, but I think you need to guard your privacy, and I can respect that. Thanks.

    • educationrealist

      You can email me privately at my blogname@gmail.

      I would not do this lesson for Algebra I until you’ve been doing this for a while. Common Core algebra I has exponentials, and my exponential modeling lesson is good. Best of all would probably be my linear equations modeling. Search model on this site.

    • educationrealist

      Oh, forgot. My students are in algebra II, but they are very weak. I’m basically teaching second semester algebra I.

  • Audrey

    I’m pretty good at math, but I can barely remember any of the things you describe in this lesson! It all seems almost like Greek to me. I use algebra all the time but I never use anything like this. It’s kind of hard to believe!

    • Tort

      Hello, Audrey. I majored in political science, but took a job in a Catholic high school in mathematics. My first move was to take a course in College Geometry since I recalled not doing well in my high school Geometry class. By the time I was 25 and had already finished my undergraduate degree three years earlier, the course was an easy A, a snap. For the rest of the ’80s I took evening courses in College Algebra, College Algebra and Trig, Calculus (just the first year), and Probability and Statistics and received As in all.

      Back in the 80s and 90s as a Catholic school teacher, all I needed was content knowledge and classroom management skills; the pedagogy developed with it and today I cannot be more clear when I teach. Students who don’t do well in my class say “You explain it very well,” and they like being in my class. Still, I am not executing the lesson above and may never do so if it is contingent on being in Mensa. I have found that I have to know algebra more deeply and I need to “ask more” and “tell less.” Common Core will force it.

      I don’t like having to change, but, as I heard back in the late-90s when I was going to workshops, “There has been a paradigm shift.” I chafed because things like ‘differentiating’ and ‘being the guide on the side instead of the sage on the stage’ would put me in a weakened position and would demand that I become more ‘experimental’ as a teacher–and this could only mean mistakes on my part. How could that help the kids, especially when I had to teach a section a day just to “cover” (I hate that word) the curriculum? Today, I am finding that there are not enough hours in the day to do everything that being a great teacher requires, which makes many of us ask, “Why can’t students do more?” I am not talking about truly poor, disadvantaged, or disabled kids but those that “can” but won’t. These are the kids that I am spending too much time and energy on. If that part of the ‘tumor’ shrank, then I could focus more on the truly weak. So, in “our” opinion, the general masses are lazy, spoiled, and demand steak when mac and cheese will do fine.

      I feel intimidated by having to teach in this manner, feeling like I will become extinct if I don’t adapt. Still, the above lesson illustrates power: preparing the students, getting them interested (hook), and then getting them to explore. What follows is cool too because the kids are receptive to what you aim to teach, and you are answering their questions.

      If I don’t pull it off and meet my own high standards, I just might need to transfer into the Social Studies Department where there, too, I won’t be lecturing, but instead will roll out Obamacare for the kids to discuss when we are learning about the Constitution and the principle of limited government. LOL.

  • Tort

    Dear EdR and Audrey:
    One more thing: it is extremely difficult to take somebody else’s lesson and execute it. All the more better if you have the “ability” to make your own. I don’t like reinventing the wheel, so I am going to study that which I find here, and pray that I can get familiar with all of the nuances and teach it naturally as in how I breathe.

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