measurement ratios tasks unit analysis

Leaky Faucet Task

This idea is not my own. The only problem is, I don’t exactly know who it belongs to. I remember tweeps talking about about a task where a leaky faucet’s effect was analysed on a water bill. When I encountered the situation at my Uncle’s house, I had to capture the modelling in action.

The best part was the conversation from intrigued (and weirded out) relatives as I ducked and dived around the tap to get a good angle. We got into a conversation about teaching, and they were happy to present any questions that came to their minds.

The premise of the task goes as follows:
Leaky Faucet Task from Nat Banting on Vimeo.
After the video is watched, I would garner a list of questions that the students may have. My relatives provided the following:
How much does the leaky faucet cost you?
Is it worth it to fix the tap?
When will the sink fill with water? 
(quickly altered to append a “if the drain was clogged” clause)
How long would it take to drown in the water leaking onto the floor? 
(followed by the fact that you could drown in an inch of water.)
My personal question:
Would you be able to tell if the faucet was leaking just by looking at your water bill?
(Again, stolen from someone, somewhere.)
The questions generally fall in two categories. Those that involve unit analysis, and those that involve some measurement of volume. Although you can certainly entertain the volume oriented questions, my goal here is to focus on rates. The task is great because its difficulty can grow depending on the information you give to students.
Students make a “wish list” of pieces of information they need to know to solve the problem. This is where you need to make a curricular decision. My province focuses on conversions within a single system of measurement (either SI or Metric) at the Grade 10 level. The Grade 11 level inserts conversions between the two.
I ask for the answer in mL because that opens the interesting conversation of mL and cubic centimeters.
I play the following informational video to grant some sort of “rate” of water leakage:
I also have the two measuring cup pictures on hand so I don’t have to search through the video to find the correct spot. 
I also give them the cost of water. If I want them to convert between systems, I give them a picture of my statement in $/cubic feet. If I don’t, I will do the conversion before class and have it written on the board when asked for it. 
A snippet from my water bill
Then I let them work. At this point, they have developed methods to solve the problem that are not based solely on multiplicative structures. Some will get the conversion from previous experiences with ratios or rates. Others will group together “chunks” and use an additive strategy. How much does 100mL cost? How long will it take for 100mL to leak? Students often fall back on tables of values and linear relations. 
Because my curricular goal is unit analysis, I take mental notes of all strategies but highlight those that use a unit analysis framework. At the end of class, I use a discussion to highlight and connect ideas. We then can move into altering the problem and getting rapid results. 
Every time my uncle walked through the kitchen, he would mumble to himself that he should really get to fixing the faucet. Depending on cost of hardware and time, it may not be such an obvious decision. 
P.S. The work and answers to this task are purposely left out. As my Math Ed. prof @MatthewMaddux would say, “I guess I lost the answer key.”

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