In this clip we join Ms. Coombs as she works with the students on exploring graphs of the functions y=x² and y=x³ near x=0.  Specifically, she raises the issue of whether the graph of the functions are “flat” near the origin.  It appears that at least one student believes that several function values near the origin are equal to zero. The graphing calculator program comes into play to help in this exploration.

To further explore the “flat” behavior near the origin of the monomial functions, Ms. Coombs directs the students to consider y=x⁶.  The function of this graph, in a standard window, looks so flat near the origin that the x-axis and the function become indistinguishable.  Ms. Coombs chooses to focus on asking the students if the function is continuous near the origin. This discussion requires students to consider the continuous nature of these types of functions.

Following the classroom instruction Ms. Coombs and Pat meet to discuss the lesson.  In looking to the future, Pat talks about how students will need to be thinking about these functions in order to understand sums of functions.  We see that he envisions students thinking about sums of functions covariationally.

In a second clip from the post-class meeting, Pat explains to Ms. Coombs what he means by having the students think about moving slowly across the domain.  He uses the graphing calculator program and the functions y=-x and y=x² for a demonstration.