Teach 180: Would It Be Possible To... (Day 60)

I love it when my students challenge me.  My favorite types of questions from students are the "What if..." or "Would it be possible to..." questions.  And then if the answer is yes, the natural follow-up is "Why does that happen?"

This line of questioning happened today in Calculus class.  The actual question was "Would it be possible for a function to have both a sharp point and a smooth part?" I should have asked the student to be more precise and reword his question.  But I knew the question he was really asking was: "Is it possible for a function to have both a relative max. or min. with an undefined derivative and a relative max. or min. with a defined derivative?"  Although I could have (and maybe should have) thrown the question back to my entire class, in a matter of about 15 seconds, I produced the following graph in Desmos and I asked "Why do we have cusps on this graph?"  The student recognized that it was due to the absolute value being used in the function.

Next, the student asked if it would be possible to have a cusp in the middle of the graph.  Within another few seconds, I produced the following graph.  To which I heard a student whisper, "that's cool".  Thanks again to Desmos for making it easier for me to keep me and my students curious.

(After class ended, I spent about 15 more minutes playing around with functions in desmos and have the beginnings of some pretty cool ideas for Christmas designs!  I can't wait to share my finished creation with my students.)

Teach 180: Starting to See Connections (Day 59)

One of the things I love about Desmos (besides the fact that it has better resolution than a graphing calculator and it is simple easy to use) is that it makes it easier for kids to see connections.  Today in Calculus we considered at the function f(x) = (x - 2)^2/3 + 1.  Prior to actually graphing the function in Desmos, we calculated the derivative and determined that there would be a critical point at x = 1, because the derivative was undefined at that point.   After we did a quick sketch of the graph by hand, we looked at both the function and its derivative in desmos.  The two graphs are shown on the same axes below.

We could easily see when the derivative was negative and when it was positive and how that corresponded to the left and right sides of the graph.  We could also see that the derivative had a vertical asymptote at x = 1 and that made sense since the derivative of the function was undefined at x = 1.   What was more interesting however was that the derivative had large positive values immediately to the right of x = 1, but then the derivative had smaller positive values as x got larger.  It was right around that moment that I could see the synapses in some students' brains firing as they were getting a better understanding - a visual understanding - of the derivative and its relationship to the graph of a function.  Thank you, Desmos!

Teach 180: Informal before Formal (Day 58)

Two years ago I created my very first Desmos Activity Builder lesson.  It was called "What is the Derivative, Anyway?"  This year I am teaching Calculus again and I was able to pull out this Desmos AB lesson to use in class today.  As I was reviewing it last night, I noticed several things that I think make this a good introductory AB lesson.

First, students are set up to be successful.  They have the foundation needed to do the lesson based on what they learned in our previous chapter.  And since we just had Thanksgiving break, it was a gentle way to get them thinking about math again.  Plus, I could easily use the teacher dashboard to identify student pairs who had "off" answers.  (One pair said "No" to a question that was not a "Yes/No" question and I was able to visit them for a quick discussion.)

Second, the screens build off of each other.  For example,  students are asked about intervals where the tangent lines have positive slopes and then are asked about intervals where the tangent lines have negative slopes.   

Third, it introduces vocabulary after students have had a chance to informally describe a concept for themselves.  On the screen below, students are asked "or is something else going on"?  Some students simply answered this with one word "constant" or horizontal".  Jack and Grayson said "there is a horizontal tangent, therefore the function is at a peak or valley".  To encourage more of a description of what is going on, I'm going to modify the directions on this slide slightly for future use. The informal idea of a peak or valley lead to formally talking about relative maximums and relative minimums. 

After we get through much of this current unit, I am hoping to create a Talkers and Drawers activity, like what is described by Job Orr in his blog post "Three New Desmos Activities: Talkers and Drawers".   I'll probably use it as a way to review concepts before our test in a few weeks, but it might also be a good activity to do as we ease back into things after Christmas break.