The Math Forum

Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Math Forum » Discussions » Inactive » calc_reform

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: partial fractions
Replies: 0  

Advanced Search

Back to Topic List Back to Topic List  
Paul Zorn

Posts: 325
Registered: 12/6/04
partial fractions
Posted: Apr 24, 2001 12:17 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

I guess I'll take my own bait:

> Assume for the sake of argument that
> (i) all your Calc II students have TI-89s or other technology that
> plots, expands, and antidifferentiates rational functions;
> (ii) the next section in the book is on partial fractions.
> What do you do tomorrow? Why?

I might start by plotting both f1=1/(1+x^2) and f2=1/(1-x^2) ---
typographically similar but otherwise radically different, with the
first bounded and the second having two poles. Then I might have
students GRAPHICALLY antidiffi'ate both functions. The second
antiderivative wouldn't be obvious from its graph, but I'd try to
elicit or just announce that some logs are involved. and we'd
rediscover that antidifferentiating 1/linear rational functions gives
logarithmic results. Then we'd ask the TI-89 for help, see what it
says, and take it from there.

I'd stress a few main points:

(1) the divide and conquer idea ... try to *restructure* the original
fraction as a sum of objects that are simpler in a
sense appropriate to the task at hand;

(2) finding numerical coefficients that do the right things;

(3) that systems of linear equations come up naturally;

(4) that answers involve just a few predictable ingredients:
logs, arctangents, and some basic rational functions.

I'd de-stress hand manipulations, but I'd force students (by hand)
through a few simple examples. I'd also force students to
complete a square (e.g., in antidiff'ing 1/(x^2+2x+2) and
1/(x^2+2x-2) , less as a hand technique than to reveal
important but somewhat hidden structural differences.


Paul Zorn
Department of Mathematics
St Olaf College 507-646-3414 office
1520 St Olaf Avenue 507-646-3116 fax
Northfield Minnesota 55057-1098



To UNSUBSCRIBE from the calc-reform mailing list,
send mail to:

with the following in the message body:

unsubscribe calc-reform your_email_address

-Information on the subject line is disregarded.

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2018. All Rights Reserved.