Hook, line, and sinker: Calculus bait

I was reading — as I think we all were — that New York Times article “Building a Better Teacher.” In that article, a number of ideas and sentences and thoughts leaped out at me, especially concerning Doug Lemov’s taxonomy. (Yes, like you, I’ve already pre-ordered the book and cannot wait for it to arrive.) One of Doug’s points is:

The J-Factor, No. 46, is a list of ways to inject a classroom with joy, from giving students nicknames to handing out vocabulary words in sealed envelopes to build suspense.

I love the idea of sealing things up and unveiling them. So in my calculus class, right after we finished anti-derivatives but before we embarked on integrals, I gave my kids 15 or 20 minutes and this picture.

I showed them a Chinese take out container which I shook (and it rattled), and I said it had very special prizes inside. I showed them a fancy envelope and gave them each a notecard that they would place in the envelope. With their name, and their area estimate.

Each kid worked individually — using anything they had on them like rulers, straightedges, calculators. One student asked if he could use a scale from the physics lab (I said no, mainly because of the time issue.) I did this in two classes. Both seemed into it, but one was definitely more into it than the other.

What was interesting to me was how hard it was for them. Not the estimating, or the making of triangles and rectangles and other smaller pieces. What was hard for them was being asked to do something that they didn’t know how to do. It happened multiple times that kids were sheepishly telling me that they didn’t know how to start (they had already drawn auxiliary lines and broke the figure up into smaller pieces — um… you DID start, darlin’), that they were doing it wrong (um, didn’t I say there was no wrong way to do this?), that they didn’t know the right way (um, see my last um). They were telling me this to assuage some part of their psyche that was telling them that they had to be right. I told them to STOP BEING CONCERNED ABOUT KNOWING THE RIGHT WAY and just TRY SOMETHING! Then they did.

I also mentioned that last year someone got the answer right to TWO decimal places — setting the bar high.[1]

At the end of the allotted time, I collected the notecards, put them in the envelope, and sealed it with a flourish.

I told them it would take a week or so before we could unveil the envelope (“but Mr. Shaaaaaaaaaaah”) and find out who came the closest to the real answer. And how would we find the real answer?


This was their hook for integrals. The next day (today) I introduced the idea of area under the curve being related to that anti-derivative thingamajig that they had been working on. I got at least 4 questions whining about needing to know who got the closest answer. I stoically responded “you’re going to find out when you figure out the true answer… soon.” The hook worked, and the bait is waiting to be won. For them, the bait is getting the surprise inside that dang Chinese take out box. For me, well, they are now curious.

[1] That was technically true, but slightly a lie. The exercise we did last year was different. I gave various pairs of students the same graph with different gridlines… and I had them estimate. So, for example, one pair of students got:

So clearly their estimation was going to be better — and it is unsurprising they could get an estimation to 2 decimal places. And last year we talked about how the more gridlines you have, the better your estimate can be.



  1. Cute! I have a friend in a biology lab and they need integrals from time to time. Their boss told them to cut it out and weigh the paper. =P

    Now *I* want to know what’s in the box. As long as it’s not actual chinese food that’ll be a week old, I’m in!

  2. I’m not impressed with Lemov’s philosophy, or at least what I could make of it from the article. Still, this sounds like a really fun idea – I may have to give it a try. (Probably not this year though – almost half my class are retaking the course for a grade upgrade, which sucks the wind out of anticipation and inquiry.)

    1. I’m just curious why you weren’t impressed with Lemov’s ideas… Personally, I thought the idea of having some concrete teacher tips/ideas to try out in the classroom, sounds like something I would be into. I guess what appealed to me is that someone who was doing ed research was offering something concrete and useful for the classroom.

      As for kids retaking the course, ARGH! That’d be so frustrating! That’s a whole new can of worms.


  3. Nice. It’s a great example of creating a need for instruction. They’re practically begging to be taught how to find the area now. Our revised curriculum talks about allowing students to construct their own meaning. Teachers are afraid they won’t be able to show students how to do anything anymore. That’s not the intent at all. I saw Phil Schlechty (Student Engagement) talk a few months ago, and he said that direct instruction is a very effective teaching method, once you create a need for it. Your students now have a need for the instruction you’re about to provide.

  4. The problem with research like Lemov’s and most ed research I find is that they tend to assume that all classes and students are the same or at least average out to be the same over time. I’ve found that one student can change a class.

    I found the article pretty laughable at times, for example:
    The whole idea that the same same class will be terrors with one teacher and angels/scholars with another because the second teacher “stands still while giving instructions”

  5. Thanks for writing this, Sam – I can already feel it helping me connect my summer camp experience with my classroom in yet another way. As the director of a summer camp I routinely spice up meetings and training sessions with mystery envelopes hidden under chairs, play-acting as different types of news reporters, etc, and yet I’ve never even considered doing these things in my calculus class. Why not?!

    I have this ingrained idea that school is siryus bisnuss, you know? wtf. Thanks again!

  6. Great Article!

    If I could write like this I would be well chuffed ;-)

    The more I read articles of such quality as this (which is rare), the more I think there might be a future for the Web. Keep it up, as it were.

  7. This looks like a great activity. I may have missed it somewhere, but can you tell me what the piecewise function is for the graph?

    1. Hi Jane,

      Sorry for the SUPER late reply. I saw you posted on a mathforum site, but I couldn’t find a way to reply. In any case, just send an email to ____ at gmail.com, where the ____ is samjshah. I can send you the fxn / solution. I don’t want to post it here in case my kids see it and it ruins their fun!


Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s