Taking a Moment… in Calculus

In calculus, I’ve historically asked kids to take the derivative of:

f(x)=\frac{2x^2+\sqrt{x}}{\sqrt{x}}

and students will immediately go to the quotient rule. OBVIOUSLY! There’s a numerator and denominator. Duh. So go at it!

Unfortunately, this is VERY UNWISE because it leads to a lot more work. And I was sick of my kids not taking a moment to think: what are my options, and what might be the best option available? Also, kids generally found it hard to deal when we started mixing the derivative rules up!

So I came up with a sheet to address this and paired kids to work on it.

(I’ve also had kids think they can do some crazy algebra with g(x)=\frac{x^2+1}{x+1}. This sheet also helped me talk with kids individually about that.)

For a little context, my kids have only learned the power rule, the product rule, the quotient rule, and that the derivative of e^x is e^x. They have not yet been formally exposed to the chain rule.

Without further ado…

[.pdf, .doc]

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16 thoughts on “Taking a Moment… in Calculus

  1. This is a really good exercise. I’ve done something like this with my Calc 2 students this semester with series tests. I’d give the students an infinite series, then ask them to determine whether it converges or diverges using the method of their choice. Then I’d ask them to do it again, using a different method. On the exam covering series they were given a choice of 2 out of 8 series to determine convergence/divergence and then asked to choose one of those two and do it differently. Eventually they got to the point where the first thing they’d do when they see a series is think through all the options first, THEN decide which option is best and go with it.

    I’d also add that this is a great kind of question to do with clickers (or little whiteboards). Example: Given f(x) = (x^2 + x)/(sqrt(x)). The BEST way to take the derivative of this function is
    (a) Take derivative of the top and the derivative of the bottom
    (b) Simplify the algebra and use the Power Rule
    (c) Quotient Rule
    (d) Product Rule
    (e) None of the above

    Gets some interesting discussion going as to what we might mean by “best”.

    • Great stuff, Sam! I’m at just the same moment in my calculus class where most all of the derivative rules are on the table and the synthesizing needs to start happening. I used your second through fourth pages *today*, untouched, preceded by this warm-up that was inspired by Robert’s comment:

      PDF version
      DOCX version

      I figure that offering those back is the best way for me to say thanks and hooray!

      • And you know once I introduce the chain rule I will be using your warm up!!! It’s pretty fantastic! Not pretty fantastic — TOTALLY fantastic! Thank you!

        Hey, do you do anything special/interesting/investigative/intuitive to build up to the chain rule? Or introduce it?

        SAM

  2. Love this exercise Sam. I have been having multiple conversations with my Calc class this year about the benefits of cleverness and diligence/persistence. This is the first year I have taught non-AP Calculus and the difference in their cleverness is striking. My honors kids are willing and (mostly) able to go through procedures accurately, but they do not see where they can save energy and think rather than do very well. An example from a recent quiz was to find the derivative of (x^5 + 1)/ (x + 1) To me this just screams out to divide and simplify first, none of my 23 kids saw that…

    How do we instill this instinct to analyze before diving in and just DOING the work in front of them?

    • Well, I know none of my kids would see that either (though my first instinct would also be to divide it out). I don’t know the full answer to your question, but I suppose the answer would include (a) exercises like this worksheet, (b) modeling multiple approaches and discussing the pros and cons, and (c) allowing the messiness of math in the classroom (and finding ways to encourage messiness). But yeah, easier said than done. (I don’t do it.)

  3. One of my strategies which works just a little bit is to make an encouraging scene in class whenever anyone comes up with these clever time saving strategies whether it is on an assessment or in class convos. I guess that after spending as much time as we do with limits and looking for ways to rewrite expressions such as the one I offered as an example, I WANT them to carry those instincts over to other problems. I know how difficult this type of transfer of knowledge is, but my hope springs eternal that I can structure our class conversations in such a way as to open this kind of thinking for my students. So often they take the hard way and get hopelessly lost in the maze of Algebra that this approach creates.

  4. Not being a math teacher, I’ve always wanted to know… why does anyone teach the quotient rule? I never learned it myself – it was obviously a special case of the product rule, and is more complicated to remember, so I ignored it and never ran into any case where that was an issue. Is there a deeper truth that I’m missing?

    • The Quotient Rule can always be replaced by the Product Rule, but there is a catch — you also have to do the Chain Rule. So it’s a question of whether it’s better to employ one rule that is sort of complicated (Quotient) or two rules, one of which is not that complicated and the second of which often gives students fits. Kind of gets back to that notion I mentioned about the “best” way of doing something in calculus.

      • You can’t really do much in calculus (or physics) without understanding the chain rule, so maybe the extra practice is a plus, then!

  5. jg

    I am kind of with you on this. I tend to avoid it and just use a combination of the chain rule and the product rule. The algebra of the quotient rule is often hideous

  6. Just a quick note about Example #2 on your worksheet. Since the denominator is a very simple linear expression, what about actually dividing the numerator by the denominator, and getting a polynomial (easy) with a simpler “remainder term” (quotient rule, but simpler than doing it on the original)? It may not be anyone’s preferred method, but I think it might be worth mentioning. (It also might be nice to have an example with three options!)

    • Good point – similar to what Jim said above with (x^5+1)/(x+1). Totes. We teach them polynomial division in the curriculum and then they never really see it in action except for here and there…

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