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.

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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).

2014 Dec 9. Shaun Errichiello has created a series of printable cards for students to sort:

We asked students to physically sort the cards into groups. One group contains all the cards where the 20% coupon is the better choice, the other group contains all the cards where the $20 coupon is the better choice. We changed one of the prices (the desk) to be exactly $100.

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.

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

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.”

I ran across Nora Oswald’s Super Mario Bros. first act on 101questions and was instantly charmed.

It puts its head down and does what my favorite first acts do: establish a context quickly, then leave a loose end hanging.

Nora Oswald wrote about the experience on her blog. Here are some choice bits:

The whole reason I created this 3-act task was to have students realize that not everything is linear. The students watched the first act and I encouraged them to make guesses, just a number from their gut. Many students used linear reasoning for their guess. I heard this over and over again today: “Since he’s a little lower than half the height of the pole, he must have a little less than half of the points.”

The model wasn’t what they expected. Oswald locates a similar flagpole model on a fan wiki. It wouldn’t be the worst idea at all to have students graph that relationship (step function!) and compare it to the incorrect linear model they anticipated.

We watch the third act (the answer). Once we are getting close to the third jump, the students are hooked, their eyes are glued to the screen, and one students rubs his palms together and says, “Here we go!”. When they see that the answer is 400, one student stands up, throws his pencil down, and complains that he was so close.

Love this also:

One of my students (who hardly says boo in class) threw his pencil down when he saw the third act. If that isn’t passion, I don’t know what is. They were in to it, they wanted to know why, they asked where those numbers came from, they made guesses, they tried to figure out why, they took pictures of the board before leaving class. One student even said that he was going to post about this on some gaming forum.

Christopher Danielson, recounting an imagination exercise with his son:

Me: Now … the strange thing about this elephant parade is that each elephant is a bit smaller than the one in front of it, and each one has an additional leg.

Griffin: How many legs does the last one have?

Let’s pause for a moment. This was his question, not mine. Real world be damned, this is a habit of mind thing.

Jimmy Pai, on students who dislike applied math problems:

Have I been focusing too much on applying mathematics and expanding the concept of “relevance?” Have I been expending too much energy on looking for relevance when I should play off of the interesting and awesome world that is mathematics? I had one student who became more disengaged throughout the year as everyone else was loving the relevance and exploring their own questions.

These are important posts. They remind me that it’s a flawed theory of student engagement that leads you to draw two circles, one labeled “real” and the other “fake,” and put material into either one with any kind of objective confidence.

Seven-legged elephants are real to Griffin. Numbers are real to Jimmy Pai’s student. Star Wars isn’t real, but for millions of viewers, it feels real. The techniques of storytelling can make the unreal seem real, but let’s all agree to bear in mind that what’s real to one student isn’t necessarily real to another.

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Craig:

My favorite quote [from this article] is “the quality of a task need not be judged by its relation to real life but in relation to how it engages students’”