Continuity at a Point

This summer I attended an AP Summer Institute with Sergio Stadler at the Marist School in Atlanta, GA. I would highly recommend this APSI to anyone teaching AP Calculus for the first time. I taught AP Calculus AB last year, so some was repetitive but I still gained a lot from going. One of the specific things I brought home was an activity on continuity at a point. I was excited to teach this lesson since many students struggled with this last year. They struggled because I just presented the definition and then we tried to work some problems. They didn’t have an understanding of where it came from or why. Try understanding this as a high school student…

I posed the following scenario to my students:

Post 2  Post 3

I then showed them 8 different graphs.

Post 4Post 5Post 6Post 11Post 10Post 9Post 8Post 7

Here is a link to all of the graphs. (These were all colleges my students from last year ended up at.)

I then asked them to work with a partner and do the following:

Post 12

I immediately had students asking what continuous meant. My response was that “I wasn’t quite sure.” After a few minutes students had no problem using roads and bridges in their explanation.

I then asked them to take their non formal definition of continuous and make it formal.

Post 13

Students realized they had to use limits to deal with the roads and function values to deal with the bridge.  The final step was to write it formally with the three conditions as seen at the top of this post.

Post 14

This lesson went well because the students did the heavy lifting. I just clicked next on some slides and let them talk it out. We talked about if the roads don’t meet does it even matter with the bridge is? If there is not a bridge, does it even matter if the roads meet?

Students responded well with this lesson and continued to use the non formal notation (bridges, roads, etc) to help them write the formal notation on homework and tests. Here is an example on the test.

photo“not continuous, the roads meet but there is no bridge” (Then translated into formal notation)

Again, this was not something I created but I felt I had to share since it went so well.

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