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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.

I updated 101questions today to include a single major new feature: a lesson editor.


Creating webpages like this soaks up too much of my time. I have to upload files in three different places. Changing a single word in the lesson means firing up an FTP client. Changing anything about an image takes ten minutes at least. None of this is creative work.

So I put together the task editor I want to use. You can add supporting materials — photos, videos, questions, teacher notes, student notes, links, and more. You can re-order them quickly, all from the browser. More fun is that other users can download them quickly. Click the "Download" button and Internet pixies will zip all the resources up and send the file to your computer.


I've been using it for a couple of weeks and I'd like you to use it also.

I've added other features some of you have asked for:

Better tagging.

You can add tags like "pizza" or "basketball" or "money." You can type a few key mathematical terms into the Common Core search bar and it will locate standards for you. Of course, all of this will make the search engine much smarter.


A smarter search engine.

People e-mail now and then telling me in kind terms how awful this spreadsheet is. I'm in total agreement. Unless you're fluent in Common Core shorthand, it's impossible to find tomorrow's topic today. So now you can head to my page on 101questions, click Search, and then click "Search this user." Type in what you're looking for. Click "Has lesson" to narrow down my material to everything that's been a little more developed. Click the grade boxes to tighten the results down even more.


Try it out. Add some tags to your old material. Leave me some comments here. I'll need as much useful criticism as you can offer. Let's make this great together.

[3ACTS] Dueling Discounts


Ask your students to write down which one they'd use. Some students will assume you should always use $20 off. Others will assume you should always use 20% off. Still others will (rightly) understand that it depends on the cost of the item you're buying.

Our goal here is to get all of those responses on paper, emptied out of the students' head. If one student in the class blurts out "It depends!" we'll lose a lot of the interesting and productive preconceptions lurking about.

Take a show of hands. Ideally you'll find some disagreement. At this point, students should try to convince each other of their position.

Offer the material from act two here: a bunch of items that will test out their hypotheses.

Once we reach the understanding that it's better to take a percentage off the large expensive items and better to use the fixed value with the small cheap items, it might seem natural to ask:

Where's the break-even point? Where do cheap items become expensive items? For what dollar cost should you use one coupon versus the other?

Then generalize some more:

If the coupons read "x% off" and "$x off", where is the break-even point? Does your answer work for every x?

BTW. There's a perplexing little pile of coupons assembling at 101questions right now. Great work, everybody.

Featured Comment

Kate Nowak:

“If you are allowed to apply one coupon, and then the other on a purchase, does it matter in which order you apply them?” is also a really nice question.

Mary Hillman

You need to be careful in your use of “small” and “large.” An iPod is small (yet expensive) compared to a large bouncy ball (inexpensive).

Bill Carey:

What’s so compelling about the three-act math project isn’t that it does a better job of teaching the body of knowledge of mathematics; it’s that it reshapes the cultural practice of mathematics in a way that more closely reflects how grown-ups engage in mathematical inquiry.

That's the goal anyway, particularly w/r/t mathematical modeling. Pick any definition of modeling you want — the IB, the Common Core, the modeling cycle, anything. They all define modeling in similar terms. Here's the Common Core. It's scary:

  1. identifying variables in the situation and selecting those that represent essential features,
  2. formulating a model by creating and selecting geometric, graphical, tabular, algebraic, or statistical representations that describe relationships between the variables,
  3. analyzing and performing operations on these relationships to draw conclusions,
  4. interpreting the results of the mathematics in terms of the original situation,
  5. validating the conclusions by comparing them with the situation, and then either improving the model or, if it is acceptable,
  6. reporting on the conclusions and the reasoning behind them.

That is a huge list of important, valuable skills. The scary part is how little our curriculum helps students develop those skills. Here's a task from Pearson's Algebra I text, which is pretty typical in this regard:

That brave little icon indicating the "Modeling" practice begs the question: Is this modeling? Who is doing the modeling? Try to locate each of the six parts of modeling in that textbook problem:

  • Who is identifying essential variables? Where?
  • Who is formulating the model for those variables? Where?
  • Etc.

Then do the same for any arbitrary three-act lesson plan.

The three-act structure isn't the only worthwhile approach to modeling and it's still a work in progress. But we should all stop pretending that including some real, physical, made-from-atoms item in a word problem does justice on its own to the complicated, exhilarating stew of skills we call "modeling."

BTW. While you're at it, feel free to compare the Common Core modeling standard against the Common Core modeling assessments. As you may know, there are two consortia developing the assessments. Here is an item from SBAC and an item from PARCC. They are more different than they are alike.

Featured Comment

Bowen Kerins offers up a useful analysis of the SBAC and PARCC tasks.

[3ACTS] Finals Week

Which of these drinks has the strongest caffeine concentration? Can you rank them from strongest to weakest? I couldn't. What information would you need to know to find out?

Two options here:

One, have students talk about the information they'd need and how they'd get it. (Two questions central to mathematical modeling.) Then just give them that information.

Two, have students talk about the information they'd need and how they'd get it. Give them the names of the drinks and have them research the information themselves.

I'm not too uptight about the difference. Option two has the students practicing their Google-fu. Option one costs less time.

Here are the goods. I'm pretty sure caffeine doesn't work the way I illustrate it in the third act, so you may want to skip that clip.

I also included the cost of each drink in case you'd like to ask students to calculate the "bang per buck" ratio as a follow-up, a cool improper fraction that reads "milligrams per ounce per dollar."

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