This school year’s first AlgPoW, “The Custom of Customs” was a challenging problem to start the year with! In the office, we debated whether it was appropriate or not, and here is some of our thinking:

- The problem has many valid approaches, and it’s neat to compare them all. Logical reasoning, guess and check, writing an expression with one variable, writing an expression with two variables, etc.
- The problem really focuses our attention on
*Understanding the Problem,*and approaches such as “Notice and Wonder,” “Act it Out,” and “Key Words.” Which of those are you familiar with? Which do you want to learn more about? - You can revisit this problem at different times through the year to focus on how you would solve it with different approaches. What happens when you use
*Guess and Check*to solve the problem? What if you*Make a Mathematical Model*?

Plus, we liked that the problem had an international flair because some Math Forum PoW members are using the PoWs to start online conversations about math and problem solving across countries and continents. Cool, huh? They are tweeting at the #powplanet hashtag, and will be using blogs and videochats at the PoW Planet website.

What’s so important about mathematical conversations? What can people learn from sharing their math ideas and reading others’ ideas? What’s important when sharing your ideas and talking about others’? I think this problem gives us a lot of good ways to think about those questions!

For one thing, “The Custom of Customs” problem has two different answers, depending on how you think about the situation. The problem focuses on two wine merchants who are forced to pay customs duties at a border. Neither has enough cash to pay the duties, so they pay using some of their casks of wine. What’s open to interpretation is whether they should pay the customs duties on those casks of wine they give to the border agents. What do you think?

Here are some examples of student work from this problem, and some of the questions I would ask those students. What would you ask? What do these examples make you think about good ways to facilitate student learning?

let n be the duties of 20 casks

let c be the price of each casks20 casks=n

64 casks=64n/205c-64n/20=-40 x2

2c-n=40 x510c-6.4n=-80

10c-5n=200-1.4n=-280

n=200 >> this is for 20 casks, so for 1 cask= 102c-40=200

2c=240

c=120

I am really excited to learn more about how this student, from Indonesia, chose to work on the duties of 20 casks. I never would have thought of that and it seems like a neat idea! After that I have some wonderings about what some of the other numbers represent, and what the units are, to help me understan the student’s reasoning. For example, I’m not sure exactly what the terms in the 2c – n = 40 equation represents are. Maybe some units would help me follow? If I had to guess, I’d say the relationship is: “cost of 2 casks in francs, minus the total duty on twenty casks in francs = 40 francs,” but I’m not sure.

Here is another example that I’d like to know more about, this one from a student in the US.

Each cask cost 120 francs and the duty on a cask is 10 francs.

6 casks + 40 francs + 2 casks – 40 francs = 7 casks

10 casks would be 1 cask – 20 francs

I think the student had a typo in the 2nd line, and meant to write 5 casks + 40 francs + 2 casks – 40 francs = 7 casks. But more than that I’m wondering what the student did next. If they had any ways, for example, to use their unique discoveries: the duties on 84 casks was 7 casks, and the duties on 10 casks would be 1 cask – 20 francs. I wonder if there are ways to figure out the duties on other amounts of casks? I wonder if there are ways to find out the duties in pure francs from those two pieces of information? How did the student work with that data. It’s such a neat strategy to try different calculations and see what sorts of results you get, and how you can combine results to make new calculations!

So… I thought that reading students’ thinking stimulated my thinking and led me to neat wonderings and math that was new to me! When students labeled their values and said what the terms represented, it was easier for me to follow their ideas and learn more, but either way I had a good time reflecting on what their ideas could have been. I hope that the students who get involved in the PoW Planet project also have a good time sharing their math ideas and getting new “wonderings” from one another.

**Some “Custom of Customs” links in case you are interested:**

- The problem [available for free thanks to the Math Forum Financial Education project].
- Information about accessing all of our current PoWs for two weeks with a free Math Forum trial account.
- Information about becoming a Math Forum Problems of the Week Member. Compare prices – consider starting with a $25 membership giving you access to all of this year’s Current PoWs — and now you can create 36 student logins as well!