Sunday, September 1, 2013

Continuity and IVT

So, this isn't anything ground-breaking, but my calc kids responded so well to this one slide, that I thought maybe it's worth sharing.

Continuity has been the topic of discussion the past week.  Even though my kids learn about the Intermediate Value Theorem in PreCalculus, I wanted them to be able to do more with it than just find a couple of y-values.  They could have done that in Algebra 1.  Let's get to some more interesting questions.  So, we worked through a version of these questions.  They hated it.  I loved it.  I will use it again, no doubt.

The next day, I showed them this slide.  Again, it's nothing you can't find elsewhere, but the kids were amazingly into it.

I had the students discuss the questions with their partner before I took a class poll (thumbs up/down).  I would ask a thumbs down student to defend his position, and then a thumbs up student to defend hers.  The kids got into a couple debates, which made me super super happy.  I love it when they argue about math because then I know they're invested in the problem and they're using higher levels of critical thinking.

The most interesting one for us (I think) was the population of the earth.  The fascinating part was that even the students who said it was not everywhere continuous did not come up with the correct reasoning (or, at least the vocal ones didn't).  So, that one's a keeper for future years.

Anyway.  Some seriously good results here.


  1. Rebecka

    I really like the questions you posed here - especially as a follow up to Kate's questions. I find that students 'get' the IVT but only in the context of 'Oh, that's the one where I identify roots between consecutive integers' or some other forced context from Precalculus or Calculus. I am teaching AP Calc BC (we do it as a second year of Calculus at our school) and many of my students remembered both IVT and MVT but most confused their names. This was interesting to me as, in my mind, the names of these theorems reveal quite a bit about what the theorem says.

    1. Yes! That's exactly how I feel my students view IVT, too. They can "work" the trite textbook problems, but do they REALLY understand the theorem?

      I'd be really interested to hear more about your school's approach to calculus. So, do first year calc students take AP Calc AB, or do they take a non-AP calc course?

  2. You should also try "The height of a plane from a few minutes before take-off to a few minutes afterwards." You'd be surprised at the interesting conversations concerning whether there is a "first moment" when you are in the air or a "last moment" on the ground.

    1. Ohhhh!! Love it. Will add this for next time. Thanks!

  3. Ask the students if the IVT is true for functions in the rational plane. x, x^2, and root x on [0, 4] are good enough to really cause the students to crane their necks. This question requires the students to consider continuity and what that definition really means.


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