## May 12, 2016

### Authentic Math

We all know the math in textbooks frequently doesn't feel authentic. Much like my painting post, the interesting math in this story isn't in the predictable computation parts. In this case the most interesting aspect was knowing what math needed to be done!

As you may know, TMC is offering the option for people to stay in dorm rooms this year. Coordinating this has been a daunting task. People have a variety of options including single or double, AC or no AC, and 2, 3 or 4 nights. Do you know how many categories that is? There's a combinatorics problem for you. Each of the room types costs a different amount, multiplied by the number of nights, plus a one time linens fee. The challenge of organizing this information, in a way that all of the members of the team could understand and easily reference is another mathematical task. Our spreadsheet is clearly labeled. You learn fast that when you have too much information that a table is the way to organize it. Do I ever give my kids too much information? Or not provide the table? Skills they need to learn... But still, we haven't done any exciting math. We've categorized and labeled and calculated a few linear functions. All things students can figure out or type in the calculator.

The challenge came when we found out that PayPal charges a fee. That fee is 2.9% plus \$0.30. We don't have cash on hand to be able to cover the fee, so we needed to charge people for it. But what do we charge? The nicest numbers are for the non A/C dorm double room, that's \$25/per person/night + \$15 linen fee. So if someone is staying all four nights we need to pay the college \$115. That means we need to get \$115 after the fee. If you're reading this with your mathematical mind engaged, you may have realized that the amount we need to charge isn't just \$115 + 2.9% of \$115 + \$0.30. This moment is where the interesting math is: this point of recognizing that percents are relative. The idea that not all operations are undone as easily as addition and subtraction. Once I had this realization the act of writing an equation wasn't particularly challenging, nor was solving for the inverse equation. I started with a specific example to check my work but rapidly generalized so I didn't have to repeat the process nine times (nine isn't the answer to the number of possible combinations, but it is the number of actual combinations, and therefore the number of invoice templates I created!) Applying a generalized function to each different category was somewhat annoying because I don't know how to tell google sheets to reference "the cell above" without having to retype the cell name each time (I could drag across to auto-fill for each row but I had four different rows for the four room types).

The authentic math I'd love to see more students working with is things like Would You Rather problems comparing 70% off to 40%, then 20% then 10% off. Anything that challenges their intuition and builds this understanding of relativity; of non-commutativity. Experimenting with inverses and learning just how much order matters. These are great opportunities to practice important skills (you'll calculate a percent four times in that one would you rather example!) while simultaneously pushing students to question their assumptions. In this case that assumption doesn't change much (we'd only be off by \$0.10 in the example) but that adds up after 99 payments (in our case) or more!