I'm working on this project with Esther Billings and Pam Wells, who both have a great grasp on teaching this subtle concept and are amazing teachers. Pam is a wonder at working from student/participant work towards her goals, and Esther has this great integration of research understanding with what it means for teaching.

*Objective*: TPW understand variables can represent changing quantities or represent an unknown; experience doing algebra with meaning and understanding of underlying concepts.

The day before they had spent a lot of time working on and watching student work for finding the patterns from pictoral sequences and expressing the relationship symbolically. Including the very nice Modeling Middle School Math video of the Beams and Rods problem, from the Math in Context curriculum Building Formulas lesson (videos 8-13).

We wanted to keep the element of having the teachers have a chance to do mathematics instead of just talk about it, and we also wanted it to connect to the Connected Mathematics Project Bags of Gold problem from the Moving Straight Ahead unit. Swapping problems help develop the idea of equality, which is central to the idea of variable as unknown. The bags of gold gets at it by putting the same amount in each bag, but you don't know how much that is. So if 2 bags and 3 coins is the same as 15 coins...

I hate contrived situations, so my first thought about swapping was Magic cards (or Pokemon or Yu-Gi-Oh) but that seemed irrelevant to the audience. Pam had the nice Stuart Murphy book Dinosaur Deals about that sort of thing. My next inspiration was currency trading. Found the current rates, jiggled it around, and realized it was entirely proportional. Save it for later. I wanted an exchange plus. I thought about a classroom rewards situation (these were teachers) that a friend uses, and then complicated it. So I wrote this problem:

**Classroom Rewards**

For small achievements or solid whole group work, students in Ms. Smittyck’s class earn a white chip (like presenting a problem at the board). For more significant achievements or an action that benefits others (like raising your grade in the class or helping another student meet a standard), a student can earn a red chip. For very notable work or effort or action on behalf of another student (like figuring out a way to increase recycling in the classroom), students can earn a blue chip. If you can get to 6 blue chips, you get a jolly rancher on Fridays for free.

Two white chips can be traded in for a jolly rancher.

Six white chips can be traded in for three red chips and a jolly rancher.

Five red chips can be traded in for two blue chips and a jolly rancher.

What would be a fair trade involving white chips and blue chips? At the end of the year, what would be a fair trade for blue chips in terms of jolly ranchers? What other questions does this raise?

*Connections*: Does the situation make sense? Do you need more information? What does an answer to the problem look like?

*Focus*: How will you answer the question? Have you ever solved a problem like this one? Is there a representation that would be helpful?

*Activity*: Solve the problem. Try to record your thinking.

Extension: If you were going to let the class trade in chips as a whole group for a pizza party or doughnut day, what would make a reasonable goal? Why?

*Reflection*: How would you check your work? Now that you know the answer would you solve it another way?

**Teacher Work**

This was my first time with the group, so I was worried: was it too messy? Too easy? Too contrived? (I hate contrived problems - especially mine!)

But they were amazing. They dived right into the problem, asking great making sense questions. What is this? How does that work? Why wouldn't they just...?

People worked with equations (because that seemed more mathy), tables (because that seemed helpful) and making pictures with blocks (which we had put out on the table beforehand without comment). The blocks group had the most rapid progress and worked things out in multiple ways. Nobody's answer matched mine. But that wasn't the point. They did want to be told the answer, but were okay with 'later.'

The free Jolly Rancher caused the most problems. People thought you were trading the 6 blue for 1 Jolly Rancher, or that it was in addition or a one time thing. The other confusing thing was that the red and blue chips come out surprisingly (to people there) close in value. Given that, I reworked the problem a bit. This retains the messiness and elements of non-proportionality. If you want to make the problem considerably cleaner, make it 11 white chips for three red chips and a Jolly Rancher

**Classroom Rewards, v2**

For small achievements or solid whole group work, students in Ms. Smittyck’s class earn a white chip (like presenting a problem at the board). For more significant achievements or an action that benefits others (like raising your grade in the class or helping another student meet a standard), a student can earn a red chip. For very notable work or effort or action on behalf of another student (like figuring out a way to increase recycling in the classroom), students can earn a blue chip. If you get into the 6 blue chips club, you get a jolly rancher every Friday without having to trade anything in.

Two white chips can be traded in for a jolly rancher.

Twelve white chips can be traded in for three red chips and a jolly rancher.

Ten red chips can be traded in for two blue chips and a jolly rancher.

What would be a fair trade involving white chips and blue chips? At the end of the year, what would be a fair trade for blue chips in terms of jolly ranchers? What other questions does this raise?

**Questions**

How would you work on this problem? How would students? Is it too messy for students? If you're interested, I'd be curious about your comments.

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