**Table of Contents**

- Teaching With Three-Act Tasks: Act One
- Teaching With Three-Act Tasks: Act Two
- Teaching With Three-Act Tasks: Act Three & Sequel

I taught using a three-act math task in Cambridge last winter. The good folks at NRich posted the video so I’m highlighting some of the pedagogy behind this kind of mathematical modeling. Ask questions and share suggestions.

**Act Two**

- [07:36] “What information do you need from me? What information will be necessary here?”
- [08:36] “I want to go ahead and capitalize ‘stack’ here. Does everybody know what stack means? Tell me how stacks and layers are related.”
- [10:10] “Are all the stacks the same?”
- [10:30] “Did you use all the same coins?”
- [11:00] “What is your estimate of how many coins are in the stack?”
- [11:45] “I’m gonna add a question to the list here: ‘Why 13?'”
- [12:15] “How many on the base layer do you think?”
- [12:47] “So what’s on the next level up? 38 by 38? 39 by 39? What am I looking for if it’s 38 by 38?”
- [13:52] “That’s everything you said you needed. You asked for this info because you had some kind of fuzzy plan in your head. Might not have been a perfect plan. But you had some need for this information. So I want to see you put that information into play somehow.”

**Post-Game Analysis**

**This is the guts of modeling right here**. Try to find a framework for modeling in mathematics that doesn’t include a line like, students need to “identify variables that represent essential features.” If students aren’t grappling with the question, “What’s important here and how would I get it?” they may be doing lots of valuable mathematics, but they aren’t modeling.

**We’re attending to precision**. When students ask me for information, I press them on units or I press them to clarify what they’re after, exactly. We coin vocabulary terms like “stack” and “layer” and emphasize that we need those terms to communicate about the task.

**Lots of different students get status in these tasks.** We’ve done a great job convincing students that they’re good in math class if and only if they’re able to memorize operations and perform them quickly and accurately. That’s it. That’s the sum of mathematical proficiency as we’ve defined it in the US.

So I love moments when I get to compliment a student for coming up with a useful vocabulary word like “stack.” Or for asking an interesting question about the pyramid. And, for totally personal, subjective reasons, my favorite moment of the whole task comes at 10:10 when a student asks, “Are all the stacks the same?” (I explain why here.)

That is a kid who is *totally unwelcome* under traditional modeling curriculum. With traditional modeling curriculum all the information is given already. The problem is stretched tight. And then along comes this bored kid who amuses herself by *poking at the problem*, by asking about exceptions and corner cases. That kid has *low* status, generally. She irritates teachers.

But with *actual* mathematical modeling, when there isn’t *any* information given, we *need* that student’s input. Her questions about exceptions and corner cases are *invaluable*. And I get the chance to turn a classroom loser into a classroom hero, to compliment that student on her sharp eye, and to turn my reproachful stare on the *other* students and say, “Did the rest of you just *assume* all the stacks were the same size? You can’t just assume that stuff!”

Moments like that. What a job, teaching.

**Look to the primary sources for answers and ask for guesses first.** The students ask me “how many pennies are in each stack?” and “how many stacks are on the base of the bottom layer?” In both cases I could have just *said* the answer (“Forty stacks along the base. Thirteen pennies per stack.”) but instead I direct their attention *back* to the raw media, taking me out of their relationship to math and the world. I also ask for guesses on both questions. Because guesses are cheap and easy and motivating for a lot of students.

**This is where I’d lecture.** Because these are teachers and not students, I don’t have to do a lot of explanation. But I begin something of a lecture here, as the teachers get blocked up. They’ve done the creative work of conceptualizing the pyramid as a sum of forty squares. No one wants to crunch those numbers by hand, though.

In the last post, Yaacov asked when these kinds of problems are useful —Â before or after learning skills. I said they’re most valuable to me *before* learning skills, or rather as the *motivation* for learning skills. I don’t expect that students will just figure everything out on their own, though. Act one helps generate the need for the tools I can offer them here in act two.

**Your Analysis**

What did you see in that clip that I didn’t talk about here? What was missing? What would you add? What would you have done differently? Go ahead and constrain your analysis to the second act of the task.

## 21 Comments

## Jason Dyer

May 13, 2013 - 10:07 am -@Kaci (continuing from the other thread):

It is quite possible to derive the sum of squares in a way students can discover. [See here.]

It requires supplies and would add about a half hour to the proceedings; plus it needs to be done after students have done geometric derivation of easier sequences first.

## hillby

May 13, 2013 - 12:09 pm -First off, thank you so much for posting these. Watching you walk through showed a ton of teaching moves that I didn’t catch in the blog posts.

Second, you said mathS. Hee hee.

## James Cleveland

May 13, 2013 - 7:11 pm -You “coined” vocabulary. I see what you did there.

What I think is most interesting, really, is how you embrace the messiness of the problem with the messiness of your computer laptop. If I use a Keynote/Powerpoint for a 3Act, it makes it hard to jump around, leave things out. It makes it hard to zoom in and out around the document where you are writing the questions, or writing the info they need. You move windows about, literally sifting through all the primary sources to find the ones you need and the students request. I think there’s something more to that than simply hitting the next slide.

(Of course, my computer [a desktop] is at the back of the room, not the front, so I do have a hard time with using it more actively. Perhaps I should move it.)

## Karl Fisch

May 14, 2013 - 7:42 am -In both cases I could have just said the answer (“Forty stacks along the base. Thirteen pennies per stack.”) but instead I direct their attention back to the raw media, taking me out of their relationship to math and the world. I also ask for guesses on both questions. Because guesses are cheap and easy and motivating for a lot of students.That’s frequently where I have issues. I’m so worried about time that I setup the problem decently, but then I give them the answers so we can move on to the (procedural) math. Hard habit to break.

What are your thoughts on having them look at the raw media to identify those answers, as opposed to you going back to the raw media and pulling out the answers? Is that a nod to time on your part, or is there a reason to do it that way?

## Jason Dyer

May 14, 2013 - 9:11 am -Ok, hard thoughts here:

I’ve arranged Acts One and Three in all sorts of ways but they are so naturally interesting to the students that just about anything seems to work. Act Two is definitely where I lose them.

I was eagerly anticipating this series just because I thought “oh, finally I can see how Dan does it and maybe I can find out how to handle things differently” — and then I find out from the video we run things roughly the same. (Also, I notice we have a similar presentation style with lots-o-windows rather than a Powerpoint. Powerpoints make me itch.)

Part of the issue with slipping this in as a problem immediately after leaning sequences is it will allow students to short-circuit thinking: “oh, I know Mr. Dyer just presented that bit on sums of squares, so that’s what we use here.” Tossing the math lecture in the middle slows down the momentum and I can see the eyes glazing.

Maybe if everything could be total-discovery? No time for that, though. Recently in Algebra I did a total-discovery lesson with arithmetic sequences — just arithmetic — and it took a full 100 minute period.

It seems like Act One and Three are attracting most of the comments, but here’s where I feel the meat is, and here is where I think I fail.

## Karl Fisch

May 14, 2013 - 9:19 am -Jason– 100 minute period? No fair.Yeah, I think Act 2 is the most difficult for me as well.

## James Key

May 14, 2013 - 9:47 am -@Jason: WOW. That picture you linked blew my mind. I was curious about the formula for the sum of squares, and I tried the approach outlined on the wikipedia page for pyramidal numbers. The picture you linked to is, like, way, way better!!! Times a million!

## James Key

May 14, 2013 - 10:01 am -I watched the full video with a colleague last Friday. The question we wrestled with was, “Into which curriculum does this fit? Which course? Which unit?” Some commenters have suggested that it could fit into a group of lessons on sequences, i.e. it’s fundamentally an application of the “sum of squares” concept.

And this is my main criticism of what I saw: Dan, you do a great job of showing how you run a 3-act math lesson. That part was awesome. The part that left me feeling a bit disappointed was how you just dropped the sum of squares formula on them: bam. Formula.

If the lesson fits into a set of lessons on formulas for special sequences, then that would bother me. Because that’s where the “real mathematics is” — not to take away from the modeling aspects, which are real math, too — but simply *applying* the formula is easy-peasy. Developing it is difficult and important work.

And if the lesson is not meant to fit there, then where does it fit?

## Eric Benzel

May 14, 2013 - 12:05 pm -“Tossing the math lecture in the middle slows down the momentum and I can see the eyes glazing.” @Jason I totally agree!

I’ve found that one method of not just “dropping” the formula or method has been to have hint cards ready…

We get to work in groups after act 1… (arithmetic solutions abounding). As each group gets to the place where they’ve found a representation, done some initial calculations, and is reaching that awesome place of “stuckness” (before they give up), *wham* I can come in with a hint card. Something that helps move them out of arithmetic world by providing a suggestions then having some reflective questions they have to answer (for example suggests them to graph, provides the scaffold of a table, or reminds of a formula).

This has helped keep the momentum of students working on answering the question while allowing me to wander around and help guide them to ways of solving that push their thinking. Also, not all groups end up needing the hint cards (and solve in efficient ways I haven’t thought of or that don’t fit the cards).

## l hodge

May 14, 2013 - 4:35 pm -Here is a very nice two dimensional visual development for the sum of squares formula.

I would say this lesson fits in as a modeling lesson. Perhaps it would be better to do a variety of modeling problems during the year rather than doing quadratic modeling problems in the quadratic unit, linear models in the lines unit, this lesson during the sum & sequence unit, etc.

I don’t think the problem lends itself to the development of the sum of the square formula. It requires a sum of squares to be evaluated, but I do not see how the problem provides insight on how to get that formula. For that reason, I would simply go to technology or some sort of estimation method if the students were not already seen the sum of squares formula at some point in the past. You don’t need a formula to get the job done here, and you likely do not have time to develop the formula in an interesting way.

I would say this lesson is more about motivating the fact that math can be used to figure things out, and perhaps that a fact or idea you learned in the past (sum of squares formula, pyramid volume) can be useful, not so much about the need to develop a formula.

## Sam Olderbak

May 14, 2013 - 9:02 pm -In the past, after asking students what information they want,I have simply given students the raw media and not done a lot of clarifying. I make part of the problem figuring out the information. For example, instead of telling them that the offset is half a penny and thus the dimensions of the second square would be 39×39 pennies, I would have just shown them both of the pictures and said something like, “All of the information you asked for is important and would help you solve this problem, however, these pictures are all that I have to help you” After watching you I feel like this may discourage students in the future from asking for different types of information. However, I still feel there is a lot of value in students figuring out the information on their own. Thoughts?

## Dan Meyer

May 16, 2013 - 6:52 am -Sam Olderbak:It depends on your instructional goals, really. You’re adding in some extra skills there —Â one of which is interpreting media —Â that are external to some of the mathematical goals. For my part, I want students to think about a) what information they need, b) how they would get it, and after that I’m happy to tell them (or show them) that information. That’s my instructional goal which dictates my route through act two.

## Matt Vaudrey

June 1, 2013 - 6:35 am -Man, I can totally vouch for the importance of driving the focus back to primary sources and media. My Algebra classes did Timon Piccini’s Megalodon 3-Act lesson toward the end of the school year and motivation to climb the “ladder of abstraction” was low. They wanted me to

give themthe very things that I wanted them to ask for and earn. In hindsight, those are media I should have prepared ahead of time; Googling them would have likely turned up the answer to the main question: “How big is that friggin’ shark?”## David Srebnick

July 4, 2013 - 5:55 am -When you first asked what information you might need in order to figure out how many coins there are, I imagine one of my students saying, the guy who built it knows how many, just ask him.

I’d like to see some motivation for figuring this out built into the problem.

Example: he knows how many coins he has, he wants to know how many levels he can make.

In all other ways, it was great to see this technique modeled… It helps me figure out how I can use it on my own.

## Lucy

July 4, 2013 - 12:03 pm -Act Two is where I would like to see more detail also. It seems like Act Two is what most of my “traditional” lessons. But I also agree with Jason that it would totally slow down the awesomeness set up by Act One.

Now I teach PreAlg and Alg 1 so a more applicable concept might be the Pythagoeran Theorem. I spend many days on the origin of the Theorem to hopefully get students to understand why it works. We draw squares in different contexts and find areas, look for patterns, etc. I even read them “What’s Your Angle, Pythagoras?” (I don’t know how much it helps, but they love being read to! And we write children’s books about math so it’s a good model.) Can this be fit in a 3 Act Task? Or do you just say “Here’s the formula. Use it.” How does that work?

I’m also just curious in general of how often this type of task is used, say, in a unit. Can the days of just doing math (practice, practice, practice) be totally gone? Or should that be a goal of mine? I love this idea, and I’m confident I can run with it, but am scared to let go of my other lessons. I’d really love to know what happens in the few days before and after this task. Thanks!

On a side note, to people who think it’s awesome to have summers off…I’ve done more learning and planning in the 2 weeks I’ve been off than I get to do during the year! I love my job.

Lucy

## Dan Meyer

July 7, 2013 - 9:06 pm -Hi Lucy, thanks for the questions. Act two is certainly where the work happens. It’s less carefree than act one. This struggle is common to all math classes, though, and the methods for working with it work in a three-act structure also. (Perhaps this blog should concern itself more with those methods.)

All things being equal, though, I’d rather help students with that struggle after I’ve perplexed them with a question-rich act one and they know the answer will be validated in act three.

Here’s a three-act task that uses the Pythagorean Theorem.

I try to use these tasks (or the principles underneath these tasks) whenever I’m introducing a new concept. I try not to say things like, “Today we’re going to learn about standard x, y, or z.” Instead I try to put students in a position where they need standard x, y, or z.

And we still practice. Lots. Fluency doesn’t come for free with these tasks. The day after a three-act tasks I’ll tell my students, “We need to get fast and good at the kind of work we did yesterday so we can do it without thinking too much about it.”

Any other questions, please let me know.

## Scott Wiens

September 14, 2013 - 8:31 am -A suggestion for all, inspired by James Cleveland’s statement: “(Of course, my computer [a desktop] is at the back of the room, not the front, so I do have a hard time with using it more actively. Perhaps I should move it.)”

I use a wifi keyboard which allows me to stand near the front of the room and type, even though my desk is at the side. It just can’t be on a metal cart as that interferes.