Related RatesRelated Rates

3 Items Each item presented with conceptual

troubles encountered by some studentsItems given are an example of sequence of

practice given to illustrate related rates concepts

Item 1:

TanA=x/15 vs. Tan(A(t))=x(t)/15

Given dx/dt, find dA/dt at t=1. Students may be more comfortable with algebra

skills and first representation invokes that compartment (language).

Symbolic usage of single letter as “hidden” function is unfamiliar

More successful if reminded of implicit differentiation with appropriate notation

Sometimes time is fixed, other times varying. Invites students to “plug in” at the wrong stage.

Item 2:

Ladder problem Many have the wrong intuition-

1ft horizontally=1 ft vertically, infinite v Students can get good at setting up and computing

in context of section, more difficult to identify when to use and interpret the results

One of first multivariate problems they see Multivariable makes multi-representations more

difficult Difficult to connect algebra and calculus Tension with skill and understanding

Item 3:

Lollipop problem Routine method does not apply Generalizing method to this problem involves

connecting with the conceptual understanding of derivative, which may still be new and weak

Requires connections to many compartments of mathematics knowledge (tables, algebra, geometry, rate of change, related rates)

Again, multivaribles are new and hard to visualize Must make decision on where to compute RC and

whether to use ARC