Re: [tinspire] Why rationalize?
Sean Bird
Rationalizing the denominator appears to be a relic from the pre-calculator days.
Interesting problem. I would venture to guess it's because
rationalizing the denominator when x is unknown may not result in the
most simplified answer when the x is all of a sudden known.
In example of 1/sqrt(x), you get a result of 1/sqrt(x) and expect
sqrt(x)/x. If you have x=3, then the expected result would be the
nicest (sqrt(3)/3), whereas, if x=4, then the result given would be
nicest (1/2). Now, I know that really, for x=4, they both simplify to
the same thing, but I would guess that the way it was programmed was to
wait until all variables are known, then simplify in the most efficient
way possible.
I also know that this isn't ideal, as the students can't use the
calculator to rationalize expressions with variables (or is that a good
thing?), but you could take it as a teaching moment.
As for "forcing" it to rationalize, I would say that it's because the
calculator simplifies each term first, so your 1/sqrt(x) *
sqrt(x)/sqrt(x) is really 1/sqrt(x) * 1.
For I am convinced that neither death nor life, neither angels nor
demons, neither the present nor the future, nor any powers, neither
height nor depth, nor anything else in all creation, will be able to
separate us from the love of God that is in Christ Jesus our Lord.
- Romans 8:38-39 (NIV)
--
Eric Findlay
AKA Eagle-Man
On 30/12/2009 8:07 PM, Joe wrote:
> Good point Nelson. Next time I will go to education.ti.com but let me
> ask you a question. This is something I have been wondering about and
> perhaps you have the answer. ( I am using the nspire cas software but
> the same thing should occur using the nspire cas handheld.) When I
> enter 1/sqrt(7) where sqrt is actually the square root sign from the
> keypad, then when I press enter, the auto simplify routine returns sqrt
> (7)/7 as expected. It has automatically rationalized the
> denominator. But when I enter 1/sqrt(7x) and press enter, in this
> case the sqrt(7)/7sqrt(x) is returned where the sqrt of 7 in the
> denominator has been rationalized but the square root of x has not.
> Why is that? Also, when I try to force the rationalization by
> entering 1/sqrt(x) * sqrt(x)/sqrt(x) the program returns 1/sqrt(x) and
> I can't find a way to force the rationalization of sqrt(x) in the
> denominator. The solution would be a "rationalize" command. How
> else can one cause the program to rationalize the sqrt(x) in the
> denominator? It's also interesting that the auto simplify routine
> won't rationalize things like 1/(sqrt(x)+1) but it will rationalize 1/
> (sqrt(7)+1). Weird huh?
Jessica Kachur
Mathematics Teacher
T3 Regional Instructor
Muka, CGC, TDI, CL2, CL3-F, CL3-S, CL3-H, TN-O, WV-N
and
Jibay, Sandy Acres lil' Phantom, CGC, CL3, CL4-F
From: Sean Bird <covena...@gmail.com>
To: tins...@googlegroups.com
Sent: Thu, December 31, 2009 9:51:54 AM
Subject: Re: [tinspire] Why rationalize?
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lee kucera
Steve Phelps
http://www.physicsforums.com/showthread.php?t=130776
There was an academic article written on this very subject by Paul H.
Schuette titled RATIONALIZING THE DENOMINATOR: WHY BOTHER? which can
be found in Mathematics and Computer Education v32(1) from 1998.
Some highlights from this article, in which Schuette surveyed some
older Algebra texts:
1865 edition of Robinson [6, p. 208], we have: "It is sometimes useful
to transform a fraction whose denominator is a surd, in such a manner
that the denominator shall become rational. The fraction is thus
simplified, because, in general, its numerical value can be more
readily calculated."
According to Wentworth in 1882 [7, p. 249]: The explicit reason given
for rationalizing the denominator in the previous three quotations is
to simplify a numerical calculation, namely a long division.
Schuette then goes on to discuss accuracy issues with respect to
"machine computations" making use of "A graphical exploration readily
carried out using ... a TI-82 graphing calculator and TI-GRAPH LINK
software (a TI-83 may be used to obtain similar results)"
Schuette's conclusion: "Rationalizing the denominator is a common
technique that has been handed down from the past. Under special
circumstances, it can be a valuable tool. Applied indiscriminately,
this technique is often a source of frustration to students, and
reinforces a belief that mathematics is mysterious, arcane, and an
ultimately useless subject. Technology has advanced to the point where
routine application of rationalizing the denominator is no longer
appropriate. It should be de-emphasized in the mathematics
curriculum."
On Dec 31, 10:51 am, Sean Bird <covenantb...@gmail.com> wrote:
> There is an interesting discussion of this topic on the math forumhttp://mathforum.org/library/drmath/view/68610.html
>
> Rationalizing the denominator appears to be a relic from the pre-calculator
> days.
>
> On Thu, Dec 31, 2009 at 12:53 AM, Eric Findlay
> <eagle-...@duetsoftware.net>wrote:
Ross
finding the square root with pencil and paper was out of the curiculum
by the time i got to high school math, cause it doesn't sound too
fun :)
Joe
intermediate calculus level which shows how to integrate 1/(sqrt(x)
+1).
To integrate 1/(sqrt(x)+1), we start by rationalizing the denominator.
The result of course is (sqrt(x)-1)/(x-1) which is then expanded into
sqrt(x)/((x-1)-1/(x-1).
Then we use the substitution u=sqrt(x) in which case x=u^2, dx/du=2u
and therefore dx=2udu.
With that substitution, the task becomes one of integrating 2u^2du/
(u^2-1)-2udu/(u^2-1), but first we need to use division to change the
improper fraction 2u^2/(u^2-1) into 1+1/(u^2-1).
Then we expand 1/(u^2-1) into 1/(2(u-1))- 1/(2(u+1)) and the rest is
easy.
We are in a position to integrate 2du+du/(u-1)-du/(u+1)-2udu/(u^2-1)
which trivial and the answer of course is: 2u+ln(u-1)-ln(u+1)-ln
(u^2-1).
Using the laws of logarithms on that result simplifies to 2u-ln(u+1)
^2.
Finally, reversing the previous substitution by using x=u^2 which
amounts to u=sqrt(x), we have the final answer which is: 2sqrt(x)-ln
(sqrt(x)+1)^2. Again using the laws of logarithms, that answer is
easily be changed into the form of the answer that the TI cas
provides.
Now here is the point. This is not a textbook problem (although
perhaps it should be), it is a problem that I worked out to
demonstrate the not so obvious but very important value of being able
to rationalize a denominator. Notice that "we start by rationalizing
the denominator." Without that ability, I honestly don't know how to
work the problem at this time. Perhaps there is another way, and one
that you will provide that doesn't require rationalizing the
denominator, but this is the way that I would work the problem for a
class or when tutoring a student, so please, don't stop teaching your
students how to rationalize the denominator or they will be lost when
they get to college and some unthinking instructor breezes thru a
problem like this without realizing that someone in the audience got
lost at step one because they were never taught how to rationalize a
denominator. Thank you.
On Dec 31, 9:44 am, Jessica Kachur <jessicakac...@yahoo.com> wrote:
> Great Point Sean regarding rationalizing being a relic! I wonder how long it will be before we stop teaching this task sort of like interpulation?
> Jess Kachur
> Mathematics Teacher
> T3 Regional Instructor
> Muka, CGC, TDI, CL2, CL3-F, CL3-S, CL3-H, TN-O, WV-N
> and
> Jibay, Sandy Acres lil' Phantom, CGC, CL3, CL4-F
>
> ________________________________
> From: Sean Bird <covenantb...@gmail.com>
> To: tins...@googlegroups.com
> Sent: Thu, December 31, 2009 9:51:54 AM
> Subject: Re: [tinspire] Why rationalize?
>
> There is an interesting discussion of this topic on the math forum http://mathforum.org/library/drmath/view/68610.html
>
> Rationalizing the denominator appears to be a relic from the pre-calculator days.
>
> For more options, visit this group athttp://groups.google.com.au/group/tinspire?hl=en-GB?hl=en-GB
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Joe
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Wayne
problem might be a more appealing approach and one that does not
require rationalizing the denominator. I will leave the details to
you to work out.
Happy new year.
Wayne
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Wayne
without rationalizing.
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Joe
If you worked the problem with the approches you suggest, you might
find that the answers differ from the rationalization/nspire cas
answer by a constant which is not convenient. Also, the integration
may be more difficult and here I'm assuming that an integration table
or cas isn't allowed. In any case, I would still use the approach I
presented and expect the student to understand rationalization. It is
a ligitimate tool to use in solving problems so I see no reason for
avoiding it.
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Wayne
Nspire CAS and is of approximately the same level of difficulty. But
that isn't the point really - I just wanted to show that
rationalization in not required. I agree that rationalization is
worthy of being taught in the classroom if for no other reason than
that a standard method of presenting results is preferable in my
opinion. I think that many of us would agree with that point.
Wayne
Nelson Sousa
irrational numbers are displayed as decimal approximations. So if CAS
isn't allowed, rationalization problems are a non-issue.
Nelson
John Benson
pretty easy put and take. I think rationalizing the denominator makes this
problem lots harder. Now, there are calculus problems where rationalizing
the numerator or denominator is useful. Taking the derivative of 1/ root(x)
using the limit definition, for example. But there is a distinction to be
made here between the process of rationalizing and the specific act of
rationalizing the denominator.
Joe
non-issue" for the calculator, and the responsibility is shifted to
the student. That requires the student to gain some basic math skills
or there will be a gazillion problems for which he or she can not come
up with the exact answer. A simple example would be to ask the
student to add 1/sqrt(2) to 1/sqrt(3) and provide the exact answer by
rationalizing and using a common denominator. I like that Nelson, as
I said, it requires the student to gain some basic math skills but we
are still stuck with the cas autosimplification not doing a complete
job of rationalizing denominators.
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Joe
Thank you for your post. It demonstrates that when there are several
ways to solve a problem, the person doing the work is not necessarily
going to use the same approach that someone else might use. Certainly
"there are calculus problems where rationalizing the numerator or
concept. Perhaps students don't need "rationalize when ever
possible," but it is still important for them to understand the
concept and be able to apply it as they will no doubt see it later in
other math courses.
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lee kucera
lk
Dr. Dan I
like the example presented above by hand anyway????
Yes, maybe a good high level thinking problem for an abstract calculus
class in college, but at any level below a college level course it
would just demonstrate to students that math is manipulating symbols
to get a "nice" answer or that there is more than one way to solve a
problem. If the integration was necessary as a result of the student
doing an application problem, then the calculation should be done by
technology anyway. These sorts of problems as well as the idea of
archaic concepts,such as rationalization, need to be removed from the
high school curriculum so that the typical high school math student
(not the math geniuses) can spend more time on reasoning and concepts
than manipulation.
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Joe
There is a saying among horse shoe'ers that "understanding how
something is done doesn't mean that you can do it." What they are
referring to is the customer who is always asking pesky questions that
wastes the shoe'ers time because after the questions are answered, the
customer still can't shoe the horse. Perhaps the same rule apply to
math? What is the point of teaching math concepts if the students
can't put them to use? All of which means that in order to learn how
to shoe a horse you must get lots of practice, and in order to learn
how to do math you also need practice. So to answer your question
then, that is a good practice problem because it demonstrates a
variety of algebra skills along with some integration. On the other
hand, if there was a machine that could shoe a horse there would be no
need to learn how to shoe a horse, so because there are math machines
(calculators) there is no point in learning math. Right? All you
need to do is push the correct button, and because the buttons are
labeled, that takes care of that and we can all quit teaching math and
do something more practical like learning how to shoe a horse.
More seriously, over the years upper level math concepts keep getting
pushed down to lower level classes with the excuse that it provides a
superior education, but that requires future carpenters and auto
mechanics to learn math that they will never ever use. So I agree
with Lee Kucera. I think that while rationalization is a ligimate
operation and as a tool it needs to be taught, it is not productive to
teach it to algebra one students who will forget it before they ever
need to use it. So who decides these things in the K-12 school
system? Textbook authors?
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Lana Golembeski
Lana