#MTBoSBlaugust Day 6: Continuity, Limits, & L'Hopitals, Oh My!

PLT PICs at NASA

I was extremely lucky to have an out of this world (get it, get it???) PD experience this summer- going to Houston to attend the Advanced Placement Annual Conference. On top of that, I got to attend with my Calculus PLT PIC (yup....professional learning team partner in crime- because there are never enough acronyms in a school setting). She just finished her first year of teaching AP and I finally have enough under my belt to apply to be a reader, so we were coming from slightly different perspectives. Both of us knew one thing for sure, though....we could not stand to hear another word about L'Hopital's Rule by the time we left.

Anti-L'H's sentiment on conference notes

The big issue was this- there's been a lot of debate online all year about the way L'Hopital's rule is justified. Understandably, the College Board has asserted that it is mathematically incorrect to say that something equals 0/0 since 0/0 is indeterminate. This caused some unrest in the Calculus community, since many teacher have allowed students to write this for years. Easy enough adjustment for my PLT to make...make sure students evaluate each limit separately. But we got a new curve ball this year that brought some amazing scoring statistics with it. Question #5 from the AB exam was a relatively straight forward question- average rate of change, evaluating derivative, candidate test for absolute extrema, L'Hopital's Rule. 

So why oh why did only 0.013% of students (under 30 worldwide) get full credit on this question? 

The answer came from the scoring guidelines, which allocated 3 points to part d instead of a more typical 2 points:

If a student didn't state that g was continuous, they missed a point. Any while logically we know that this must be true, I think very few of us as Calculus teachers would have expected our students to state this explicitly. This was discussed ad nauseam in many sessions and left me thinking how I could better prepare my students for this type of more rigorous justification. I knew it needed to start during my limits unit and then continue throughout the year. Most importantly, it needed to have my students analyzing why a limit can be evaluated and when it does not exist vs cannot be evaluated because we are missing necessary information. 

Here's the activity I designed to start my students thinking about this during the first unit. 

Limits & Continuity Justifications (PDF)

Limits & Continuity Justifications (Word)

I've attempted to use multiple representations, lots of notation to build fluency, and to scaffold up to 2 questions more like the part d on the AP. I also am trying to help kids distinguish between when we need more information, when we have enough information, and how to justify all of that. I will be emphasizing that it doesn't just saw "evaluate," but it also says justify! Please feel free to send feedback! This is still rough and a work in progress!