[Makeover] Meatballs

This is from Discovering Geometry.

What I Did

Basically, I three-acted the heck out of it. Which means:

• Reduce the literacy demand. Let's encode as much of the text as we can in a visual.
• Add perplexity. That visual will attempt to leave students hanging with the question, "What's going to happen next?"
• Lower the floor on the task. The problem as written jumps straight to the task of calculation. We can scaffold our way to the calculation with some interesting concrete tasks.
• Add intuition. Guessing is one of those lower-floor tasks and this problem is ready for it.
• Add modeling. We'll ask students "what information would be useful here?" before we give them that information. That's because the first job of modeling (as it's defined by the CCSS) is "identifying variables in the situation and selecting those that represent essential features." The task as written does that job for students.
• Create a better answer key. Once we've committed to a visual representation of the task, it'll satisfy nobody to read the answer in the back of the book. They'll want to watch the answer.

Here's the three-act page. Leave a response to see the entire lesson.

Show this video to students.

Ask them to write down a guess: will the sauce overflow? Ask them to guess how many meatballs it'll take. Guess guess guess. It's the cheapest, easiest thing I can do to get students interested in an answer and also bring them into the world of the task.

Ask them what information would be useful to know and how they would get it. Have them chat in groups about what's important.

If they come back at you telling you they want the radius of the pot and the radius of the meatballs, push on that. Ask them how they'd get the radius. That's tough. Is there an easier dimension to get?

Someone here may ask if the lip of the pot matters. It isn't a perfect cylinder. Give that kid a lot of status for checking those kinds of assumptions. Tell her, "It may matter. It isn't a perfect cylinder but modeling means asking, 'Is it good enough?'"

Give them the information you have.

Let them struggle with it enough to realize what kind of help they'll need. Then help them with the formula for cylinder and sphere volume. Do some worked examples.

Once they have their mathematical answer, have them recontextualize it. What are the units? If that lip matters, how many meatballs will it matter? Should you adjust your answer up or down?

Then show them the answer.

Surprisingly close. The student who decided to add a couple of meatballs to her total on account of that lip is now looking really sharp.

Let's not assume students are now fluent with these volume operations. Give them a pile of practice tasks next. Your textbook probably has a large set of them already written.

Help I Need

• Raise the ceiling on the task. My usual strategy of swapping the knowns and unknowns to create an extension task is failing me here. Watch what that looks like: "The chef adds 50 meatballs to a different pot and it overflows. Tell me about that pot and its sauce level." I'm not proud of myself. Can you find me a better extension? I'll give highest marks to extensions that build on the context we've already worked to set up (ie. don't go running off to bowling balls and swimming pools) and that further develop the concept of volume of spheres and cylinders (ie. don't go running off to cubes or frustums).

What You Did

Over on the blogs:

Over on the Twitter:

• Max Ray, Michelle Parker, and Terry Johanson are all inside my head.
• Ignacio Mancera poses a similar situation but suggests doing it live in the classroom. I don't accept the premise that "real" always beats "digital" – there are costs and benefits to consider – but I think Ignacio and Beth have the right plan here. If you have the materials, do it their way.
• Scott McDaniel suggests changing the context from meatballs in sauce to ice cubes in an iced mocha because kids drink iced coffee but don't make spaghetti. This introduces a pile of complications (like the non-spherical shape of the ice cubes and the non-cylindrical shape of the cup and the fact that the ice will float at the top of the cup) for unclear benefits. Time and again in this series I've tried to convince you that changing the context of a task does very little compared to the changes we can make to the task's DNA. Does someone (Scott?) want to make the case that the following task is a significant improvement over the original?

Call for Submissions

You should play along. I post Monday's task on Twitter the previous Thursday and collect your thoughts. (Follow me on Twitter.)

If you have a textbook task you'd like us to consider, you can feel free to e-mail it. Include the name of the textbook it came from. Or, if you have a blog, post your own makeover and send me a link. I'll feature it in my own weekly installment. I'm at dan@mrmeyer.com

2013 Jul 16. A makeover from Chris Hunter in the comments. (I had forgotten how weird Orbeez look in water.)

Featured Comment

We just talked about this problem and your makeover at Math for America. One idea was to up the stakes: I’m putting this jar of water on top of a student’s phone. How many balls can I put it before it spills over? If you are sure you are right, put your phone under the jar…

Teaching With Three-Act Tasks: Act Three & Sequel

1. Teaching With Three-Act Tasks: Act One
2. Teaching With Three-Act Tasks: Act Two
3. 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 Three & Sequel

• [18:36] "This guy wants to make a pyramid out of a billion pennies. And I'm curious how big that would be. Help me with that if you're completely finished here. Or tackle some of the other questions we had up there earlier."
• [20:20] "Is that number in between your high and low from earlier? Does it fit in the range of possible numbers for you? If it didn't we should go back and ask ourselves 'do we trust the mathematics here?'"
• [20:45] "I'm going to show you the answer here."
• [21:00] "Who guessed closest to that? Margaret or Eddie. Let's all give one clap to Eddie."
• [21:16] "Who got the closest guess overall? Who is closer? 250,000 or 300,000? One clap for these two."
• [21:50] "Let's look at other questions we had back here."
• [23:00] "How could we figure out how long it would take?"

Post-Game Analysis

Show the answer. There's the bombastic, visual element, the part that results in students cheering the answer to their math problem. It's hard for me to overvalue that reaction.

But there's another reason why students ought to see the answer to modeling tasks. (I'm not picky about answers to other tasks.) The Common Core's modeling framework asks students to "validate the conclusions" of their models. Showing the answer acknowledges the messiness inherent to mathematical modeling and allows students to discuss possible sources of error and then account for them with newer, better models.

Make good on the promises from act one. Earlier I asked students for numbers they knew were too high and too low so I asked them here to check their answer against those numbers. I said I was curious who had the closest guess so I had to find out who did and show them some appreciation. I said I hoped we would get to everybody's questions by the end of the day so I returned to those questions. If I fail to make good on any of those promises, I know they'll seem awfully insincere the next time I try to make them.

Good sequels are hard to come by. The goals of the sequel task are to a) challenge students who finished quickly so b) I can help students who need my help. It can't feel like punishment for good work. It can't seem like drudgery. It has to entice and activate the imagination.

I have one strategy I'll try on instinct: I flip the known and the unknown of the problem and see if the resulting question is at all interesting. In this case, I originally gave students the dimensions of the pyramid and asked for the number of pennies. So now I'll give them the number of pennies (one billion) and ask for the dimensions. Then I try to activate their imagination around the sequel, asking "Would you be able to build it in this room? Would it punch through the ceiling?" Etc.

In some cases, the initial task just serves to set an imaginative hook for the sequel, which is much more demanding and interesting. Once students have a strong mental image of the pyramid of pennies, I can ask them to manipulate it in some flexible and interesting ways. (Nathan Kraft has written about this recently.)

What's Missing

Formalize the math. Because I'm working with adults, I gave the math a brief treatment here. In general, act three is where the math is formalized and consolidated. Conflicting ideas are brought together and reconciled. Formal mathematical vocabulary is introduced.

Title the lesson. Lately, taking inspiration from this Japanese classroom, I ask students to provide a title that will summarize the entire lesson. Then I offer my own.

All of this happens at the end of the lesson, not the start. I'm not defining vocabulary at the start of the lesson and I'm not greeting students at the start of class with an objective on the board. Those moves make it harder for students to access the lesson, lofting interesting mathematics high up on the ladder of abstraction.

Homework

Here's my best guess how this kind of task would look in a print-based textbook. How does it differ from the task I did in Cambridge? Try to resist easy qualifiers like, "It's more boring," etc. How is it more boring? How is the math different? What are the downsides? What are the upsides? (I can think of at least one.)

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?

As soon as I know I have all the data, the exploring side of my brain just checks out. I go straight to my brains list of formulas and start looking for ones that will fit together to solve the problem. When I don’t have the numbers yet, I can almost feel synapses firing all over my brain.

That first sentence is sure a doozy. “A pyramid is made out of layers of stacks of pennies.” If you have a picture of what that means, then sure, it makes sense, but if you don’t, it doesn’t exactly give you a lot of clarification about what it means.

2013 May 25. James Key has created a nice visual proof of the formula for the sum of squares.

Teaching With Three-Act Tasks: Act Two

1. Teaching With Three-Act Tasks: Act One
2. Teaching With Three-Act Tasks: Act Two
3. 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 – 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.

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.

Teaching With Three-Act Tasks: Act One

1. Teaching With Three-Act Tasks: Act One
2. Teaching With Three-Act Tasks: Act Two
3. Teaching With Three-Act Tasks: Act Three & Sequel

I get nervous when I see long-time blog readers in my workshops on mathematical modeling with three-act tasks. I tend to assume they'll be bored. I assume that the pedagogy around these tasks has been self-evident or overly blogged-about these last few years. I should know better. It's one thing to read about these kinds of tasks. It's another to do one as a student. After a Saskatoon session last week, for instance, Nat Banting said that the process seemed tighter, and more engineered than he assumed from reading about it.

More than a few people have approached me with the impression that you simply show a photo or a video and then pursue student questions in any direction they take you. Sean Geraghty just asked me to script one of these tasks out with every question I'd ask. I'll seize that opportunity to post some video of a session I facilitated with teachers this winter around Penny Pyramid in Cambridge and clarify what I think are the important teacher moves in a three-act math task, starting today with act one.

Act One

• [00:43] "Here it is. First, I just want you to watch this very brief video."
• [01:27] "Would you go ahead and write down the first question that comes to your mind, if any? No question? That's perfectly fine."
• [01:45] "Would you introduce yourself to your neighbor and share your question? See if it's the same question, or a different question."
• [02:28] "I'm really curious what questions are out there. Just toss one out. Who else finds that question interesting?"
• [03:04] "I like that you coined a vocabulary term there for us. 'Layers.'"
• [04:24] "I would love to get to all these questions but given limited time we'll start with these ones up here."
• [04:43] "I want you to write down on a piece of paper your best, gut-level guess for how many coins there are. I'm curious who can guess the closest."
• [05:32] "Would you also write down a number you know is too high – there couldn't possibly be that many pennies – and a number you know is too low – there couldn't possibly be that few pennies. Share them with your neighbor."
• [06:09] "I'm very curious in here who has our highest guess. "
• [06:53] "What's our lowest guess in here?"

Post-Game Analysis

Act one attempts to lower barriers to entry. It's visual. It requires very little literacy from the student. (Notice that I'm using very little formal mathematical vocabulary.) It's perplexing.

Now look at the student tasks. Students are asked to to watch a video. Students are asked to pose a question. (But if you don't have one, that's okay!) Students are asked to decide if they find someone else's question interesting. Students are asked to guess at a correct answer. Students are asked to decide what an incorrect answer would look like. No one is throwing a hand up saying, "I don't know where to start." I don't know how to make it easier to start a modeling task than this.

I make three promises during act one.

1. I tell students I'm very curious who guessed closest to the answer.
2. I tell students I hope we'll get around to answering all the questions on their list.
3. I ask students to set an error check on their answer.

I'll need to make good on each of those promises by the end of act three.

I ask for student questions, but that doesn't mean you have to. (You don't have to do any of this of course.)

I have two competing goals in my head in act one. One, I want students to answer the question, "How many pennies are there?" Two, I want to know what questions students have when they see that stupid-huge pile of pennies.

I want to know their questions because students are interesting creatures and, while they spend a lot of time answering questions, they don't get a lot of opportunities to pose their own. Asking for student questions orients our community around curiosity as a shared value.

But those goals are in conflict. How do you ask students for their questions while knowing, in the back of your head, the question you're going to pursue. I know some teachers will ask for student questions and then "wait for" or "nudge students towards" the question they want to ask. I suspect this drives students crazy. It drives me crazy, this sense that there's some question the teacher wants me to ask even while she's insincerely asking me for my questions.

The quick way around this is to say, "Great. Love these questions. I hope we get to all of them. Here's one I'll need your help with first."

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 yourself to the first act of the task. We'll pick up tomorrow where I say, "What information do you need here?"

2013 May 9. As usual, a pile of great follow-ups in the comments. Kate Nowak points out a few details that I missed in my discussion. James Cleveland suggests asking for a high and low range before the more precise guess. Great call! Lots of commenters struggle to balance asking for student questions with their curriculum objectives and I respond. So does Math Forum Max. Elaine Watson maps this task to the Standards of Mathematical Practice.

2013 Jul 15. Kevin H:

One thing I do when I ask students to guess some of the given information (like the fact that each stack is 13 pennies) is to have each student write their guess on the whiteboard and then have everyone simultaneously show one student, “Bryan.” Then Bryan is tries to ball-park an average of the numbers everyone showed him. It takes about 45 s., but they seem to enjoy the process.

Cleaning The Windows Of The Luxor Hotel

Nathan Garnett, via e-mail:

I showed students a few pictures of the Luxor Hotel in Las Vegas, and then I asked them how long they it would take to wash all of those windows. We did lots of math. I had a student clean a 2ft by 5ft section of the board (with windex sound effects) so we could get a cleaning time per 10 square feet. It was a blast.

Great work. We can do something similar with Pyramid of Pennies. Students are often curious how long it took to make the pyramid. So we give students enough pennies to make the top two layers. We time them. Then they use proportions to answer how long it would take them to make the entire pyramid.

And then we list all the reasons that answer is wrong.

It doesn't account for bathroom breaks. For sleeping. For eating. For the fact that the top two layers are small and easy but the bottom ones require scaffolding and much, much more care.

Those exceptions aren't reasons to not ask the question. Those exceptions make the question more messy, more meaningful, more like actual modeling, and less like textbook modeling where air resistance is neglected, the rates are constant, the men are strong, and the women lithesome.

We need more messy modeling tasks like Nathan Garnett's.

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