## Rational Functions and Jewelry Production

By Kristina

Math SL is a Diploma Programme (DP) course for grades 11 and 12, yet I’m unit planning in the IB Middle Years Programme (MYP) style since it’s consistent with our grades 6-10. MYP math has four criteria: A – Knowing and Understanding, B – Investigating Patterns, C – Communicating, and D – Applying Math in a Real-Life Context.

I’ve been racking my brain to come up with an authentic assessment (not a test) for my Rational Functions Unit. In this unit, we are focusing on the following objectives from the IB Math SL syllabus:

*2.5 The reciprocal function x → 1/x, x ≠ 0, its graph and self-inverse nature. The rational function x → ax+b/cx +d and its graph. Vertical and horizontal asymptotes. Applying rational functions to real-life situations.*

My statement of inquiry for the unit is “Representing change and equivalence in a variety of forms has helped humans apply our understanding of scientific principles.” I’d been looking at population growth, water flow, and light (via Until Next Stop) after checking out the MTBoS, but couldn’t quite get the right assessment.

Then I found The Math Projects Journal Optimum Bait Company task and hint cards. This approach seemed great for scaffolding problems like this one, which students had had trouble with:

When printing out the Optimum Bait Company task, I ran into a coworker in the teachers’ lounge. We got to chatting about modeling production costs. She is an English teacher, but I feel like she’s my in-person MTBoS because she brings so many interesting approaches to teaching and problem-solving. We talked about the larger question “how do you know when to enter a market?” and concepts such as barrier to entry (overall cost of running a business, competition, willingness to buy), sunk cost, marginal cost, and bringing in investors.

This conversation led me to create my own task about my friend’s company (Winter Hill Jewelry), because the marginal cost of production of earrings seemed like it would be similar to the bait problems (and the problems from the textbook). I asked her what kind of printer she used and how much filament was needed for a pair of earrings. She replied “Flash Forge and on average, 3 grams of filament per pair of dangle earrings.”

I found pricing for fish hooks and PLA filament on Amazon, then found a few prices for FlashForge 3D printers. I recalled that Vanessa had space at Artisans Asylum, so I checked out their memberships and studio rental costs. I realized that Artisans provides 3D printer usage to members (at a cost of $0.10 per gram of PLA used), so I moved the “buy your own 3D printer” option to a homework problem.

I used the slides and task document linked at the bottom of this post to introduce the task and support students’ work. I started by asking “what questions could you ask about these earrings?” and “what do you notice?” Some students went right to “how much do they cost?” Some focused on geometric patterns in the designs. Others picked up on the fact that they were 3D printed (they’d done some 3D printing in summer enrichment).

I explained the origin story of Winter Hill Jewelry and then asked them to brainstorm what goes into the cost of making 3D printed earrings. Lots of great ideas came out in the groups: ads, URL, shipping, electricity, workspace/rent, labor, taxes, investors, and more. I showed them the monthly recurring costs (workspace, 3D printer PLA, and fish hooks), then explained that we were going to use hint cards to support their perseverance through longer independent work. The hint cards didn’t work as I had intended, so I changed strategy for the second class (offering more conversation when I noticed they were off versus holding to the “only thing I can offer is a hint card”). For example, when a student asked “am I right?” when proposing a per-unit cost of $652, I told him “I own three pairs of these earrings. Do you think it makes sense on a teacher’s salary to buy something that’s $652 before profit?”

We’ll finish out the task on Monday and revisit the traditional form of IB exam-style problems–I am curious to see if the students do better as a result of exploring problems like this one!