Teach 180: I Took the Noticing and Wondering Back (Day 92)

Today it was like my PreCalculus students were reading my mind.  We just finished reviewing domain, vertical asymptotes and horizontal asymptotes for rational functions using some individual practice with deltamath.  A.K. raises her hand and asks, "Can there be asymptotes other than vertical and horizontal, like slanty ones?"  She asked this about 10 seconds before I was going to introduce the concept!  I replied, "Let's look at one of those in Desmos."

The students could see the slant asymptote in the graph, but how did I know what it was?  I told students to rewrite f(x) by doing the division.  At that point the students could easily see the equation for the slant asymptote was the quotient. In previous years, most students would be ok with that and not think anything of the remainder.  But G.L. raises his hand and asks, "But what about the remainder?"  More music to my ears!  Noticing something and wondering about it!  Unfortunately, I erred at that point and took the wondering out of the students hands. (Face-palm.)  I explained that as x got bigger the remainder divided by the divisor would get smaller, meaning it was approaching 0. Why didn't I turn that back on the class?  I could have had them enter 6/(x-1) in Desmos in a table to investigate what happens as x gets larger (or smaller).  It's great that they are noticing and wondering, but now I need to get out of the way and give them a chance to figure it out.