# Day 28: What Type of Discontinuity?!

This was a red-letter day in precalculus…at least for the teacher.  As an opening question, I asked my class this simple question:

Is x = 3 the location of an infinite discontinuity or a removable discontinuity?  Explain how you know.

As I was walking into class to set up, I decided to change it up.  I asked the students to answer the question silently and individually.  Then, once everyone had an answer, I asked them to move to one side of the room or the other depending on their answer of vertical asymptote versus removable discontinuity (hole).  Then as a group, they needed to come up with one convincing mathematical reason their choice was correct.  You can see the two huddles below.

Then I was the scribe as each group gave a reason.  After each group commented, the other group needed to give a convincing statement to either support their stance OR refute the other group’s statement…thus the different colors.  After each group gave a statement, they had time to re-huddle and/or change sides.  And then we repeated the process.  I was after student voice and academic discourse targeted at each other and not me as the conduit.  They desperately wanted the answer, but I didn’t give it.  So they had to resort to the definitions and theorems to add sophistication and depth to their arguments.

Although this took longer that I had originally planned for the opener (and I had to adjust the subsequent lesson) the end result was that the students, through disciplined discourse, were able to convince themselves of the answer AND had a much deeper understanding of the underlying concepts.  It was so exhilarating to see the metamorphosis of their graph of the ideas behind vertical asymptotes and point discontinuities.  They were so proud, and somewhat surprised that they could actually reason through to the correct answer.  I never did tell them 😛