Rationalizing Denominators

[Martha Haehl]

I am teaching a class aimed at future algebra teachers. I posed the question last week why we rationalize denominators, such as 1/sqrt(2). Everyone's response was "because it looks prettier." These students have all taken College Algebra or higher math courses and are generally good algebra students.

I am curious why we still emphasize that subject and when you teach it (if you do), what rationale you give for the process.

Martha

Martha Haehl,
Ph.D., Education & Urban Leadership
Mathematics Instructor
816-759-4221
martha.haehl@...
Metropolitan Community College-Penn Valley
3201 S.W. Trafficway
Kansas City, MO 64111

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[Collinge, Peter (Mathematics)]

We still teach rationalizing denominators, though I have trouble
"rationalizing" that topic. Clearly, the historic reason of "it's tough
to divide by a messy decimal approximation" no longer applies. The only
real advantage I can now see to rationalizing is that if everyone
changes their result to a standard form, it's easier to compare answers
to see whether they agree.

Peter Collinge, Professor & Chair
Department of Mathematics
Monroe Community College, Rochester NY 14623

Voice: [phone]     E-mail: pcollinge@....
 
-----Original Message-----
From: owner-mathedcc@... [mailto:owner-mathedcc@...]
On Behalf Of Martha Haehl
Sent: Thursday, March 13, 2008 11:31 AM
To: mathedcc@...
Subject: Rationalizing Denominators

I am teaching a class aimed at future algebra teachers. I posed the
question last week why we rationalize denominators, such as 1/sqrt(2).
Everyone's response was "because it looks prettier." These students have
all taken College Algebra or higher math courses and are generally good
algebra students.

I am curious why we still emphasize that subject and when you teach it
(if you do), what rationale you give for the process.

Martha

Martha Haehl,
Ph.D., Education & Urban Leadership
Mathematics Instructor
816-759-4221
martha.haehl@...
Metropolitan Community College-Penn Valley
3201 S.W. Trafficway
Kansas City, MO 64111

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[Laura Bracken]

I show them the difference in long division between 1/sqrt 2 and sqrt 2/2.
Before cheap calculators, this was a big motivator for rationalizing a denominator.
Now, it still makes for good number sense - I find it easier to estimate the value of 1.414 divided by 2 than 1 divided by 1.414.

And we talk about manners and customs and that every body of knowledge has them.  

--Laura


-----Original Message-----
From: owner-mathedcc@... on behalf of Martha Haehl
Sent: Thu 3/13/2008 8:30 AM
To: mathedcc@...
Subject: Rationalizing Denominators
 
I am teaching a class aimed at future algebra teachers. I posed the question last week why we rationalize denominators, such as 1/sqrt(2). Everyone's response was "because it looks prettier." These students have all taken College Algebra or higher math courses and are generally good algebra students.

I am curious why we still emphasize that subject and when you teach it (if you do), what rationale you give for the process.

Martha

Martha Haehl,
Ph.D., Education & Urban Leadership
Mathematics Instructor
816-759-4221
martha.haehl@...
Metropolitan Community College-Penn Valley
3201 S.W. Trafficway
Kansas City, MO 64111

****************************************************************************
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* Archives at http://mathforum.org/kb/forum.jspa?forumID=184 *
****************************************************************************

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[Rick Butterworth]

I don't. I mention that my answer and book's may disagree, mention why I
had to do it 100 years when I was a student, and move on.

Rick Butterworth

-----Original Message-----
From: owner-mathedcc@... [mailto:owner-mathedcc@...]
On Behalf Of Martha Haehl
Sent: Thursday, March 13, 2008 11:31 AM
To: mathedcc@...
Subject: Rationalizing Denominators

I am teaching a class aimed at future algebra teachers. I posed the
question last week why we rationalize denominators, such as 1/sqrt(2).
Everyone's response was "because it looks prettier." These students have
all taken College Algebra or higher math courses and are generally good
algebra students.

I am curious why we still emphasize that subject and when you teach it
(if you do), what rationale you give for the process.

Martha

Martha Haehl,
Ph.D., Education & Urban Leadership
Mathematics Instructor
816-759-4221
martha.haehl@...
Metropolitan Community College-Penn Valley
3201 S.W. Trafficway
Kansas City, MO 64111

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[Lori Lampe]

I'm a high school math teacher and our old Algebra I
text (Structure and Method, Houghton Mifflin,
Dolciani, et al.) states: "it is easier to name the
decimal value of 'sqrt(35)/7' than of
'sqrt(5)/sqrt(7)'.  Of course, with a calculator it
may not matter."  

I've always explained that standardized test (e.g.
ACT, SAT) may require it, so we teach this in Algebra
I, Algebra II, and Geometry.  However, in the Calculus
class I teach, we DON'T rationalize!  Go figure....

Lori Lampe (llampe@...)


--- "Collinge, Peter (Mathematics)"
<pcollinge@...> wrote:

> We still teach rationalizing denominators, though I
> have trouble
> "rationalizing" that topic. Clearly, the historic
> reason of "it's tough
> to divide by a messy decimal approximation" no
> longer applies. The only
> real advantage I can now see to rationalizing is
> that if everyone
> changes their result to a standard form, it's easier
> to compare answers
> to see whether they agree.
>
> Peter Collinge, Professor & Chair
> Department of Mathematics
> Monroe Community College, Rochester NY 14623
>
> Voice: [phone]     E-mail:
> pcollinge@....
>  
> -----Original Message-----
> From: owner-mathedcc@...
> [mailto:owner-mathedcc@...]
> On Behalf Of Martha Haehl
> Sent: Thursday, March 13, 2008 11:31 AM
> To: mathedcc@...
> Subject: Rationalizing Denominators
>
> I am teaching a class aimed at future algebra
> teachers. I posed the
> question last week why we rationalize denominators,
> such as 1/sqrt(2).
> Everyone's response was "because it looks prettier."
> These students have
> all taken College Algebra or higher math courses and
> are generally good
> algebra students.
>
> I am curious why we still emphasize that subject and
> when you teach it
> (if you do), what rationale you give for the
> process.
>
> Martha
>
> Martha Haehl,
> Ph.D., Education & Urban Leadership
> Mathematics Instructor
> 816-759-4221
> martha.haehl@...
> Metropolitan Community College-Penn Valley
> 3201 S.W. Trafficway
> Kansas City, MO 64111
>
>

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> *
> * To unsubscribe, email the message "unsubscribe
> mathedcc" to
> majordomo@... *
> * Archives at
> http://mathforum.org/kb/forum.jspa?forumID=184 *
>
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> *
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> * Archives at
> http://mathforum.org/kb/forum.jspa?forumID=184 *
>
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>

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[DeFazio, Patrick]

Though it is not a justification for teaching rationalization as a topic
(in Intermediate Algebra) unto itself, the skill of rationalizing is
very helpful to the students later on (in Calculus) when determining
derivatives of radical functions (using the formal limit definition).

****************************************
Patrick DeFazio
Assistant Professor
Mathematics Department
Onondaga Community College
4585 West Seneca Tnpk.
Syracuse, NY 13215
315.498.2393  or   315.498.2328
defaziop@...
****************************************
 
- -----Original Message-----
From: owner-mathedcc@... [mailto:owner-mathedcc@...]
On Behalf Of Collinge, Peter (Mathematics)
Sent: Thursday, March 13, 2008 11:46 AM
To: Martha Haehl; mathedcc@...
Subject: RE: Rationalizing Denominators

We still teach rationalizing denominators, though I have trouble
"rationalizing" that topic. Clearly, the historic reason of "it's tough
to divide by a messy decimal approximation" no longer applies. The only
real advantage I can now see to rationalizing is that if everyone
changes their result to a standard form, it's easier to compare answers
to see whether they agree.

Peter Collinge, Professor & Chair
Department of Mathematics
Monroe Community College, Rochester NY 14623

Voice: [phone]     E-mail: pcollinge@....
 
- -----Original Message-----
From: owner-mathedcc@... [mailto:owner-mathedcc@...]
On Behalf Of Martha Haehl
Sent: Thursday, March 13, 2008 11:31 AM
To: mathedcc@...
Subject: Rationalizing Denominators

I am teaching a class aimed at future algebra teachers. I posed the
question last week why we rationalize denominators, such as 1/sqrt(2).
Everyone's response was "because it looks prettier." These students have
all taken College Algebra or higher math courses and are generally good
algebra students.

I am curious why we still emphasize that subject and when you teach it
(if you do), what rationale you give for the process.

Martha

Martha Haehl,
Ph.D., Education & Urban Leadership
Mathematics Instructor
816-759-4221
martha.haehl@...
Metropolitan Community College-Penn Valley
3201 S.W. Trafficway
Kansas City, MO 64111

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[coolmath2]

In a message dated 3/13/2008 8:46:37 A.M. Pacific Daylight Time, pcollinge@... writes:

We still teach rationalizing denominators, though I have trouble
"rationalizing" that topic. Clearly, the historic reason of "it's tough
to divide by a messy decimal approximation" no longer applies. The only
real advantage I can now see to rationalizing is that if everyone
changes their result to a standard form, it's easier to compare answers
to see whether they agree.

-----------------------------------

 

Yeah, what he said! 

 

I was going to write out just about the same thing...  Then, I thought I should read the responses first.

 

Yes, it's a completely outdated (pre-calculator) technique for when things were done by hand and you couldn't do long division by a radical.  Now that we have calculators, it's no longer needed -- except for the fact that it's still in all the books and on all "standardized" exams.  So, I still show it to them... and test them on two problems.   But, other than that, I happily except unrationalized answers on my tests.

 

What's MUCH worse, if you ask me, is changing sqrt(8) to 2sqrt(2).  Another outdated technique.  I ask you:  Which one takes longer to punch into a calculator?  I refuse to call the second option "simplified" because it isn't!

 

Karen

Coolmath.com

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[Crowley, Lillie F (Bluegrass)]

Well, in the calculus sequence, you need the technique to rationalize numerators. It also drives home (in the event this is brought in, which it usually isn't) the difference of two squares.
 
My pet peeve is that students show up in my calculus classes convinced that it is a sin to have a radical in a denominator.
 
Lillie R.F. Crowley, Ph.D.
138 Moloney Building, 470 Cooper Drive
Bluegrass Community and Technical College
Lexington, KY 40506-0235
[phone] (note new phone number)
lillie.crowley@...

________________________________

From: owner-mathedcc@... on behalf of Martha Haehl
Sent: Thu 3/13/2008 11:30 AM
To: mathedcc@...
Subject: Rationalizing Denominators



I am teaching a class aimed at future algebra teachers. I posed the question last week why we rationalize denominators, such as 1/sqrt(2). Everyone's response was "because it looks prettier." These students have all taken College Algebra or higher math courses and are generally good algebra students.

I am curious why we still emphasize that subject and when you teach it (if you do), what rationale you give for the process.

Martha

Martha Haehl,
Ph.D., Education & Urban Leadership
Mathematics Instructor
816-759-4221
martha.haehl@...
Metropolitan Community College-Penn Valley
3201 S.W. Trafficway
Kansas City, MO 64111

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[Bruce Yoshiwara]

Some parts of this message have been removed. Learn more about Nabble's security policy.

Yes, I agree that rationalizing denominators is not justified in developmental math classes.

But there is a reason for rationalizing denominators besides the computational.  It relates to the topic of algebraic structures (fields, rings, groups).

Adjoining sqrt(2) to the field of rational numbers gives a field of index 2.  One consequence is that any “rational” expression involving sqrt(2) with coefficients in that field can be simplified to the form a+b*sqrt(2), and that canonical form is unique within the field.  Another consequence is that sqrt(8) has a unique canonical representation in the field.  These and related consequences can be considered interesting and worth exploring in other classes. 

But in the end,  I'm basically agreeing with Peter that this is just a fancy way of saying that simplifying radical expressions allows a way to provide a unique form for correct answers in the back of the book.

Bruce Yoshiwara
Los Angeles Pierce College

---

From: CoolMath2@...
Date: Thu, 13 Mar 2008 12:25:16 -0400
Subject: Re: Rationalizing Denominators
To: pcollinge@...; Martha.Haehl@...; mathedcc@...

In a message dated 3/13/2008 8:46:37 A.M. Pacific Daylight Time, pcollinge@... writes:

We still teach rationalizing denominators, though I have trouble
"rationalizing" that topic. Clearly, the historic reason of "it's tough
to divide by a messy decimal approximation" no longer applies. The only
real advantage I can now see to rationalizing is that if everyone
changes their result to a standard form, it's easier to compare answers
to see whether they agree.

-----------------------------------

 

Yeah, what he said! 

 

I was going to write out just about the same thing...  Then, I thought I should read the responses first.

 

Yes, it's a completely outdated (pre-calculator) technique for when things were done by hand and you couldn't do long division by a radical.  Now that we have calculators, it's no longer needed -- except for the fact that it's still in all the books and on all "standardized" exams.  So, I still show it to them... and test them on two problems.   But, other than that, I happily except unrationalized answers on my tests.

 

What's MUCH worse, if you ask me, is changing sqrt(8) to 2sqrt(2).  Another outdated technique.  I ask you:  Which one takes longer to punch into a calculator?  I refuse to call the second option "simplified" because it isn't!

 

Karen

Coolmath.com

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[lynne ipina]

I had to laugh a little reading the words "very helpful".  Our calc I students just completed a midterm, and there was a fractional limit with square roots on it.  Just yesterday one of my students, a math double-major, commented that "we only rationalize a few times in our lives".  He meant to say that it was so rare and that would explain why most students missed this problem, and why this problem might have been a poor choice for the midterm:  not representative. 

Lynne Ipina
Mathematics
University of WY




-----Original Message-----
From: DeFazio, Patrick [defaziop@...]
Sent: Thu 3/13/2008 10:16 AM
To: Collinge, Peter (Mathematics); Martha Haehl; mathedcc@...
Subject: RE: Rationalizing Denominators

Though it is not a justification for teaching rationalization as a topic
(in Intermediate Algebra) unto itself, the skill of rationalizing is
very helpful to the students later on (in Calculus) when determining
derivatives of radical functions (using the formal limit definition).

****************************************
Patrick DeFazio
Assistant Professor
Mathematics Department
Onondaga Community College
4585 West Seneca Tnpk.
Syracuse, NY 13215
315.498.2393  or   315.498.2328
defaziop@...
****************************************

- -----Original Message-----
From: owner-mathedcc@... [owner-mathedcc@...]
On Behalf Of Collinge, Peter (Mathematics)
Sent: Thursday, March 13, 2008 11:46 AM
To: Martha Haehl; mathedcc@...
Subject: RE: Rationalizing Denominators

We still teach rationalizing denominators, though I have trouble
"rationalizing" that topic. Clearly, the historic reason of "it's tough
to divide by a messy decimal approximation" no longer applies. The only
real advantage I can now see to rationalizing is that if everyone
changes their result to a standard form, it's easier to compare answers
to see whether they agree.

Peter Collinge, Professor & Chair
Department of Mathematics
Monroe Community College, Rochester NY 14623

Voice: [phone]     E-mail: pcollinge@....

- -----Original Message-----
From: owner-mathedcc@... [owner-mathedcc@...]
On Behalf Of Martha Haehl
Sent: Thursday, March 13, 2008 11:31 AM
To: mathedcc@...
Subject: Rationalizing Denominators

I am teaching a class aimed at future algebra teachers. I posed the
question last week why we rationalize denominators, such as 1/sqrt(2).
Everyone's response was "because it looks prettier." These students have
all taken College Algebra or higher math courses and are generally good
algebra students.

I am curious why we still emphasize that subject and when you teach it
(if you do), what rationale you give for the process.

Martha

Martha Haehl,
Ph.D., Education & Urban Leadership
Mathematics Instructor
816-759-4221
martha.haehl@...
Metropolitan Community College-Penn Valley
3201 S.W. Trafficway
Kansas City, MO 64111

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[coolmath2]

In a message dated 3/13/2008 10:29:59 A.M. Pacific Daylight Time, lillie.crowley@... writes:

Well, in the calculus sequence, you need the technique to rationalize numerators. 

----------------------------------------------------------

 

OK, I don't want to be completely blasphemous here...  But, aren't those few "need to rationalize the numerator" limit problems just a useless textbook technique hoop to jump through too?  Does that kind of problem ever really come up anywhere in the real world?

 

Karen

Coolmath.com

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[coolmath2]

In a message dated 3/13/2008 10:34:39 A.M. Pacific Daylight Time, Ipina@... writes:

Just yesterday one of my students, a math double-major, commented that "we only rationalize a few times in our lives". 

-----------------------------------

 

Heck, I'm constantly trying to "rationalize" things in my life!!    ;-)

 

Karen

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[Prof Martin Weissman]

Some parts of this message have been removed. Learn more about Nabble's security policy.

Trouble is that when you're rationalizing the denominator
you're radicalizing the numerator. Kind of like
squeezing the toothpaste from the bottom to the top.

Professor Martin Weissman Prof Weissman's Software 246 Crafton Avenue Staten Island, NY 10314 Cell: 347-528-7837 mathprof@... www.math911.com www.MathMatchIt.com

---

From: CoolMath2@...
Date: Thu, 13 Mar 2008 13:47:21 -0400
Subject: Re: Rationalizing Denominators
To: mathedcc@...

In a message dated 3/13/2008 10:34:39 A.M. Pacific Daylight Time, Ipina@... writes:

Just yesterday one of my students, a math double-major, commented that "we only rationalize a few times in our lives". 

-----------------------------------

 

Heck, I'm constantly trying to "rationalize" things in my life!!    ;-)

 

Karen

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[Irene Duranczyk]

Some parts of this message have been removed. Learn more about Nabble's security policy.

And now for some mathematical humor. . .

I’m for any mathematical operation that maximizes radicalizing! That is the way I rationalize my activities.

 

 

 

Irene M. Duranczyk
Assistant Professor, Postsecondary Teaching and Learning
College of Education and Human Development, University of Minnesota
362 Appleby Hall
128 Pleasant Street SE
Minneapolis, MN  55455
Phone: [phone]
Fax: [phone]
duran026@...

 

From: owner-mathedcc@... [mailto:owner-mathedcc@...] On Behalf Of Martin Weissman
Sent: Thursday, March 13, 2008 1:21 PM
To: coolmath2@...; mathedcc@...
Subject: RE: Rationalizing Denominators

 

Trouble is that when you're rationalizing the denominator
you're radicalizing the numerator. Kind of like
squeezing the toothpaste from the bottom to the top.

Professor Martin Weissman Prof Weissman's Software 246 Crafton Avenue Staten Island, NY 10314 Cell: 347-528-7837 mathprof@... www.math911.com www.MathMatchIt.com