Monday, November 3, 2014

Implementing The Core: Draining A Tank - A Real-World (?) Quadratic Model Problem

From today (of course the wording of the problem will exceed 140 characters!)...
Water is flowing out of a tank. The number of gallons after t min is given by the function
V(t) = k-2t-t^2. [Assume t≥0 and other suitable restrictions]
If 153 gallons remain after 3 min, in how many additional min will the tank empty?
I'll even provide an answer: 9 min
Problems like these which *artificially* model the real world are common these days on standardized tests but let's go beyond assessment issues.
Before throwing this problem out to the class I usually began with some thought-provoking questions to deepen understanding. For example:
(1) How do we know if the water is flowing out at a constant rate or not? Explain this to your partner.
[Suggested Answer: Constant rate implies a linear model]
(2) Draw a rough sketch before determining k. How can we do this if we don't have a value for k?
(3) Why is the quadratic model given more reasonable than say t^2-2t+k?
[Suggested Answer: The coefficient of the quadratic term should be negative since the quantity of water is decreasing. Note that students most often reply "'Because we want graph to open down!" This is insufficient IMO.
(4) What is the meaning of k both graphically and in the context of the problem?
[Suggested Answer: Graphically, k is the V-intercept; in the application, k = quantity of water at start or t=0]
(5) What strategy do we typically employ when working with function problems?
[Suggested Answers: Make a t,V(t) table; sketch a graph]
(a)  Using a parameter like k makes it harder to just punch it into the graphing calculator. Common assessment technique these days. Students should be encouraged to also solve the problem with technology afterwards but that's teacher preference.
(b) Like most standardized test questions the quadratic doesn't require the quadratic formula, but for classroom discussion it certainly doesn't have to unless you're reinforcing factoring skills.
(c) Is asking for the "additional" number of minutes overkill here? A 'gotcha' ploy? Or does it discriminate as a difficult item should? If strong students, i.e , those who score high, do poorly then the question may be invalid. Serious issue here. What do you think?

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