Education Technology

Looking at Polynomials as Dynamic Mathematical Objects

Published on 06/09/2008

Activity Overview

In this activity, students use the TI InterActive!™ to construct the expression of a polynomial curve based on its dynamic properties. They explore the behavior of the curve at certain points within its domain. Students also determine the polynomial equations that meet the criteria for the design of a racetrack.

Before the Activity

  • Install the TI InterActive!™ on the computer
  • See the attached PDF file for detailed instructions for this activity
  • Print pages 1 - 7 from the attached PDF file for the class
  • During the Activity

    Distribute the pages to the class.

    Follow the Activity procedures:
    Drawing the existing track:

  • Draw the semicircle on the left side of the track with t-values ranging from π/2 to 3π/2
  • Determine the radius of the circle
  • Enter the parametric equations of the circle
  • Replace t by t - π and graph the semicircle at the right end
  • Find transformations for t-values for other sections of the track

  • Changing the Design of the Track:
  • Leave the top straight section same, and cut a portion of the bottom one
  • Reduce the size of the semi-circle
  • Determine the circumference and the parametric equations of the section

  • Finding an Expression for the Track:
  • Determine the point where the track to be constructed meets the small circle
  • List the values of (x2(t), y2(t) ) when t = π/2
  • Use a straight edge to draw a straight line joining the two points
  • Ensure that the entire track looks like a smooth closed curve
  • Understand that there is sufficient information to solve for 4 parameters
  • Realize that the polynomial expression will have degree 3 [cubic]
  • Find linear approximation of the polynomial f(x) = ax3 + bx2 + cx + d at a given point
  • Find the equation for the tangent to this polynomial
  • Graph the line and determine its slope
  • Solve for the coefficients
  • After the Activity

    Students answer questions on the activity page.

    Review student results:

  • As a class, discuss questions that appeared to be more challenging
  • Re-teach concepts as necessary