There are lots of great reasons to use this task from NCTM’s Illuminations site, which asks students to derive an algebraic function from a problem situation. But one of those reasons *isn’t* “to show students why they should derive algebraic functions.”

It’s a *real world* problem, by most definitions of the term. But let’s not let that fact satisfy us. It’s possible for math to be real-world, but also unnecessary. For example, I can ask students to use trigonometry to calculate the height of a file cabinet. But that math isn’t *necessary* when a measuring tape would suffice.

The same is true here. I can find Stages 1 through 5 by multiplying by three successively. So why *did* we invent algebraic representations? Life would be so much easier for both the student *and* the teacher if we relaxed that condition.

But if we added the question, “How long would it take the entire world to experience a good deed?” we will have both a) identified the need for algebraic functions —Â to calculate outputs given *any* input, even *distant* inputs —Â and b) put students in a position to *experience* that need.

That’s a two-step process. With the line, “Describe a function that would model the Pay It Forward process at *any* stage,” the author satisfies the first step. He understands the value of algebraic functions, himself. But without our added question, that’s privileged knowledge and we’re hoping students infer it. Instead, let’s put them directly in the path of that knowledge.

Real world, and also necessary.

## 10 Comments

## Rene Grothmann

March 29, 2016 - 10:04 pm -Does everyone find three NEW persons? Or select the three at random and from which community? Real world problems are really difficult.

## Xavier

March 30, 2016 - 2:10 am -Completely agree with you. Actually I use this problem in my class for 1) introducing power notation (3^step) and 2) guess how many steps are necessary to benefit whole world (alÃ 3-acts operas) And it works.

For experience, I recommend you that you change “world” for “your town”:

Act 1

https://www.youtube.com/watch?v=4x_0NsuldGQ (spanish)

(supposing Trevor was here, in our town)

How many steps are necessary to benefit all the people in our town?

Act 2

– Population of the town (Wikipedia source)

Act 3

Just put the calculator in desktop and count how many multiplication you make

Then, remake the question for “World”. Here [http://somenxavier.github.io/operes-en-tres-actes/Pay-it-forward.html] it’s the old 3-act opera in catalan.

See that 3^0 = 1 makes sense here: in the step 0 it’s just Trevor. Trough my years of experience this is the only way I achieve my students remember this “rule”.

I surprised when I see NCTM uses Trevor process for other purposes.

## Xavier

March 30, 2016 - 2:40 am -Why don’t we simulate?

Take a cardboard, make mini cards with the name of persons in your school. Take your friends and decide three friends to help….

How many steps we need to help whole school?

## Howard Phillips

March 30, 2016 - 2:52 am -I would be bolder, and leave out the last blue sentence. Let them figure out what has to be done. It does say “describe”, so a perfectly good description is “the number at any stage is three times the number at the previous stage”, which in math form is f(n+1)=3f(n). Then ask for an explicit solution.

## Laurie Hailer

March 30, 2016 - 6:57 am -I can’t wait to hear more about this at NCTM! Thanks for thinking about this, researching it and writing and sharing. You’re the best, Dan Meyer! ;)

## Jamalee Stone

March 30, 2016 - 7:48 am -Thanks for the improvement on the problem Dan! And thank you for sharing your remarkable insight with the rest of us!

## Thom Gibson

March 30, 2016 - 5:46 pm -I was thinking about this recently. Did Robert Kaplinky’s ‘Which Ticket Is The Best Deal’ (http://robertkaplinsky.com/work/ticket-option/).

I was hoping students would use unit rate but many of them didn’t. A lot of them used pesky logic! Some of them found out how much it would cost for 120 tickets at each value. I think I should have had a question like ‘order these from cheapest price per ticket to most expensive price per ticket’ but that question isn’t as meaningful as ‘which is the best deal.’

Tried to create the headache but they went for the organic tea instead of the aspirin.

## Karim

March 31, 2016 - 3:46 am -+1

“When will everyone experience a good deed” is definitely a better and more motivated question. Double-counting’s a bummer, though. (We ran into this in our ancestry lesson, “Family Tree,” when asking whether everyone on earth is related.)

## l hodge

March 31, 2016 - 3:29 pm -You wrote of students proving that a model is broken a few posts ago. This is a good problem for that — double counting, non-participation, etc. Of course, we are probably no longer dealing with a “warm-up problem” at that point.

This MARS Ponzi scheme task.is a good start for a more complicated exponential growth task. The context requires quite a bit more thought. Also the student is asked to think about what might go wrong.

## Annie Forest

April 2, 2016 - 1:28 pm -Isn’t it amazing that one or two simple (but intentional) additional questions changes the whole feel of this? It’s a great example of how a problem can become more relevant to students without having to completely change the context to something that we *think* students will find more interesting (sports, pop stars, ice cream, or whatever).

Also, loving Thom’s comment… students used “pesky logic” and “they went for the organic tea instead of the aspirin.” ha!