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Related Rates

Related Rates. 3 Items Each item presented with conceptual troubles encountered by some students Items 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.

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Related Rates

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  1. Related Rates • 3 Items • Each item presented with conceptual troubles encountered by some students • Items given are an example of sequence of practice given to illustrate related rates concepts

  2. 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.

  3. 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

  4. 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

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