I'm pretty certain the answer is "no", but is there a "correct" way to check if a student's parametrization for a curve is appropriate? That is, there are many (many) ways to parametrize a curve, but it's not easy to check if a given set of parametric equations parametrize a given curve. Right?

Is there perhaps some nice way to handle "most" solutions a student is likely to enter? For example, given a set of parametric equations, we could plug in several points and make sure (1) they are not all the same point, (2) they all lie on the "correct" parametric curve (which seems to me only works if the parametric curve is the solution to some implicit equation).

Or to write a question that asks students to parametrize a curve, do we need to provide more guidance/structure in the answer? Like "Parametrize the unit circle with using only sin(t) and cos(t)", instead of "Enter any parametrization for the unit circle"?

One solution, of course, is to make the problem a multiple choice question, and give many incorrect parametrizations, and one correct one. But that's no fun.

Does anybody have a nice way to write problems that are asking for a parametrization of a curve (or worse, surface)?

In reply to Nick Hamblet
Tuesday, 22 June 2010, 12:39 PM

### Re: Checking Parametrizations

by Dick Lane -
A matching problem could be a bit more challenging than a plain multiple choice task --- for unit circle, offer variants of

a) [ sin(1 - 2 t) , cos(1 - 2 t) ]

b) [ cos(pi sin(t)) , sin(pi sin(t)) ]

c) [ sin(t^2) , cos(t^2) ] or [ sin(e^t) , cos(e^t) ]

d) [ (1-t^2)/(1+t^2) , 2t/(1+t^2) ] for punctured circle

e) convert r = 2 sin(theta) to rectangular, then shift

Severe ad hoc constraints seem necessary to enable Webwork to handle this as a free response task (even if adaptive parameters are used in the .PG template). E.g., requiring the parameterization to be directly proportional to arc length would be hard to assess even with access to a CAS.

I have used a similar task in Calc II as a mini-project: find several non-equivalent parameterizations for a [simple curve], show each "works", and discuss ways in which they differ.

a) [ sin(1 - 2 t) , cos(1 - 2 t) ]

b) [ cos(pi sin(t)) , sin(pi sin(t)) ]

c) [ sin(t^2) , cos(t^2) ] or [ sin(e^t) , cos(e^t) ]

d) [ (1-t^2)/(1+t^2) , 2t/(1+t^2) ] for punctured circle

e) convert r = 2 sin(theta) to rectangular, then shift

Severe ad hoc constraints seem necessary to enable Webwork to handle this as a free response task (even if adaptive parameters are used in the .PG template). E.g., requiring the parameterization to be directly proportional to arc length would be hard to assess even with access to a CAS.

I have used a similar task in Calc II as a mini-project: find several non-equivalent parameterizations for a [simple curve], show each "works", and discuss ways in which they differ.