## Thursday, June 13, 2013

### Intro to Average Value

Last week I had an idea about how I could introduce average value in calculus next year.  When I've taught average value in the past, I felt like students just memorized a two-step procedure and several didn't see the connection to the definition of average that they've been using for years.  I know I won't be teaching this concept until...mmm...December?...but when you get excited about a lesson/idea, you just gotta follow through with it, right?

• It starts with an application to motivate the discussion and the why should we study this?
• It recalls previous knowledge.
• It applies the fundamental 3-step process of all calculus topics: (1) Start with a non-calculus idea, (2) apply a limit, (3) arrive at the calculus concept.
• It lets students practice a FRQ from a previous exam, but forces them to search through the problems to find which one would require their new tool.
• Students discover a main idea of calculus using what they already know, each other, and the text (not me).
I recently read that four classroom characteristics important for brain-compatible learning are: (1) challenge (with support), (2) relevance, (3) novelty, and (4) a positive emotional climate.  I think this packet offers all four of these.

This is new for me.  I usually never post material I haven’t actually tried on students yet.  So, feedback, please!  Like I said, I have puh-lenty of time to revise and make this better.  I’ll probably be posting a few other things for next year that I would also love feedback on before I test them out on real, live kids.

1. Rebecka,

For question #2 - Do they know the word histogram? If so, it might be clearer to use that language AND it might make them think on their own about connections to Riemann Sums

LOVE that you emphasize units of measurement in #3.

Not sure I understand what you're trying to prompt them for on #7.

For the free response question, I might refrain from telling them the exact question. I don't know how your kids are, but mine know that they can find exact AP solutions on the web for old AP FR questions. I would definitely let them know that they accomplished an old AP FR, I just might wait until after the fact for them to know that.

Nice work, thoughtful and structured. I'd love to share this with my AB colleague - I'm doing BC these days and will use some version of this as well (if it's okay with you)

1. Mr. Dardy!

Ok, I went back and forth on the histogram terminology after I read your comment. Here's my only problem with it: Histograms typically would have the score represented on the x-axis and the frequency on the y-axis. Because we're trying to find the average value of the score, I think it needs to remain what is typically the dependent variable (y) since this is will be the kids' very first intro to average value. Or can a histogram go the other way around, too? I'm no stat expert...

That also takes me into #7...I'm hoping students will realize that the two steps they found--(1) find area, (2) divide by total length of interval--is just the infinite extension to what they've always done to find an average--(1) find sum, (2) divide by the total number of items. I want them to make the connection that a definite integral is, yes, a way to find area, but also that it's an infinite sum. So, is there a way to phrase #7 better so they know where I'm going with it? Or is that something I address on a case-by-case basis? (As of now, I'm planning to give this in class so they can work together.)

Good call on not telling them it's an old AP question. If they don't finish it class, they'll have to take it home, and then they're sure to find it online! I can proctor that in class...can't do anything about it once they take it home. :)

Tell me what you think!