"Predicting Functions" Worksheets
Preliminary Exercise designed by Dr. Therese Shelton
for Calculus I  Fall 1998
Instructional Notes
 
 

Introduction
My colleagues and I at Southwestern University used this exercise in Calculus I on the first day of class, Fall 1998.  It is a simple exercise in which every student can participate.  It is active, collaborative, discovery-based learning that can build intuition.  It also serves as a review of crucial concepts.

If you wish to use this exercise, here are instructions.  I'd appreciate feedback on your use.

Set-up
Create a grid and photocopy it.  On page 1, draw a piece of a function.  On page 2 redraw that piece and add another piece.  Continue this process.

I made three functions, labeled A, B, and C.  The first two used the same small size of grid; the third used a larger grid size.  I chose a continuous, smooth curve for A, a function with a jump discontinuity for B, and a function with a corner for C.  On B, I gave non-consecutive pieces and let them fill in.

Student Activity

  • Form the students into groups. I used 7 groups of 4 students.  Inform the students that each sheet has a function that is bounded, meaning that it fits on the page.  Ask them to think about what a function is.
  • Pass out Page 1 of Function A.  The group is to make an educated guess about the next piece of the function.  Each group will sketch the next piece and record a reason for the choice.
  • Pass out Page 2 of Function A.  In addition to the previous instructions, each group will record any reactions to the revealed piece, assumptions the group made in their choice that made it differ from the revealed piece, and assumptions in the choice of the next piece.
  • Repeat as necessary.
  • When the exercise is complete, each student or group will record some observations.  The instructor may wish to give hints for this.

    • Rationale for Exercise/Discussion Questions

  • Did students assume the function was continuous?  Smooth? This exercise emphasizes that some functions are more predictable; that is one way that continuity and smoothness are "nice" attributes.  Were students aware of their assumptions?
  • Was it easier to predict a piece when "connecting" pieces?  What assumptions were made?
  • Were predictions based on continuing the slope? (This gets at the idea of "local linearity", a buzzword in the Calc Reform.)  the curvature?  something else?
  • It’s good for them to see there is not just one "correct" answer.  I found most students saying, "Oh, good.  We were close," as though mine were the "correct" graph.
  • We can refer back to this exercise in speaking of partitions, non uniform partitions, refinements, and the resolution of graphing calculators.

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     Student Reactions   - a collection of quotes from my section of students.