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Curves, quadratics, and cubics


This thread introduces curves as a variant of lines in which the inclination is not constant but instead varies - first linearly, and then quadratically.  In this context, the curves are generated by summing their inclination (step size) - which is essentially the slope of tangent lines.  In order to provide intuitions related to the algebraic and graphical properties of quadratic and cubic functions, programs plot both the generated curves and their inclination.   Exercises focus student attention towards the relationship between the inclination graph's zero-crossings and the generated graph's turning points such as maxima and minima.  The effect of biasing the inclination upon the generated graph's size and zero-crossings is also examined.  Finally, a graphical construction using manipulatives is used to provide students with intuitions relating the graphs examined in this thread to second and third order polynomials.

Lesson 1: Quadratics

This lesson has not been posted yet.

Lesson 2: Cubics

This lesson has not been posted yet, but a draft powerpoint prepared for it is attached to this page. 

This lesson was developed by Eric Freudenthal and Art Duval.  It extends lesson from iMPaCT related to generating curves as second-order sums, but focuses more directly on building intuitions related to the properties of polynomials.

© 2011 Eric Freudenthal and collaborators.  All rights reserved.
Eric Freudenthal,
Feb 13, 2011, 4:12 AM