Fractions R Us

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Fractions R Us

Kirby Urner-5
I'm noticing with our novel approach to teaching
fractions, that we're discounting the importance
of LCM in connection with adding them.  The reason
for that is our use of Euclid's Algorithm for the
GCD upon instantiation, meaning when we create a
fraction object, we automatically convert to the
canonical delegate in each equivalence class,
namely that for which the numerator and denominator
are relatively prime.

The background context is "building fractions" (yes,
this is constructionism) from the ground up, in some
computer language facilitating operator overloading.
That's right, we're doing some programming as a part
of our standard mathematics curriculum, what of it?
TIs have been programmable for years, and we've heard
no one fussing ("more features!" has always been the
battle cry, no?).

As I've posted about numerous times here (math-teach),
irrespective of whether programming's involved, it's
interesting to see if a curriculum dodges Euclid's
Method for finding the GCD, escaping into "factor trees"
(which I have nothing against -- very important for
teaching the meaning of the Fundamental Theorem of
Arithmetic).  

I'd call that a kind of litmus test, to see if you
proposal meets minimum standards -- let's apply it to the
states first and foremost, see which state legislatures
are being lobbied intelligently, vs. getting fed a lot
of mumbo jumbo.  Good project for a PhD candidate.

So how *do* we add fractions then?  Of course the old
fashioned way:  a/b + c/d == (a/b)(d/d) + (c/d)(b/b)
== (ad + cb)/bd as justified by using multiplicative
identity, other axioms (I'm sure Paul could help out
here).  

Then you get gcd((ad + cb), bd)), call it thegcd (could
be 1), so your new (reduced) numerator, denominator (p,q)
become p/thegcd, q/thegcd (multiplying by x/x is to
multiply by 1, the multiplicative identity -- this came
up earlier, in justifying (a/b)==(a/b)(d/d).

How does this look in actual Python?  Here's a fraction
class (rational number class) developed to the point of
enabling its instances (p/q) to multiply and add (p,q
integers, q <> 0).  See plaintext view to restore
significant indentation.

def gcd(a, b):
   while b:
       a, b = b, a % b
   return a


class Q:

    def __init__(self, p, q):
        thegcd = gcd(p, q)
        self.p = p/thegcd
        self.q = q/thegcd

    def __repr__(self):
        return "(%s/%s)" % (self.p, self.q)
   
    def __add__(self, other):  # code for adding  (+)
        """
        (a/b + c/d) == (a*d + c*b)/b*d
        """
        return Q(self.p*other.q + other.p*self.q, self.q*other.q)
       
    def __mul__(self, other):
        return Q(self.p * other.p, self.q * other.q)

So used interactively:

>>> from whatever import Q
>>> a = Q(1,2)
>>> b = Q(1,6)
>>> a * b
(1/12)
>>> a + b
(2/3)

Kirby
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Re: Fractions R Us

Paul A. Tanner III

- --- Kirby Urner <[hidden email]> wrote:

> I'm noticing with our novel approach to teaching
> fractions, that we're discounting the importance
> of LCM in connection with adding them.  The reason
> for that is our use of Euclid's Algorithm for the
> GCD upon instantiation, meaning when we create a
> fraction object, we automatically convert to the
> canonical delegate in each equivalence class,
> namely that for which the numerator and denominator
> are relatively prime.
>
> The background context is "building fractions" (yes,
> this is constructionism) from the ground up, in some
> computer language facilitating operator overloading.
> That's right, we're doing some programming as a part
> of our standard mathematics curriculum, what of it?
> TIs have been programmable for years, and we've heard
> no one fussing ("more features!" has always been the
> battle cry, no?).
>
> As I've posted about numerous times here (math-teach),
> irrespective of whether programming's involved, it's
> interesting to see if a curriculum dodges Euclid's
> Method for finding the GCD, escaping into "factor trees"
> (which I have nothing against -- very important for
> teaching the meaning of the Fundamental Theorem of
> Arithmetic).  
>
> I'd call that a kind of litmus test, to see if you
> proposal meets minimum standards -- let's apply it to the
> states first and foremost, see which state legislatures
> are being lobbied intelligently, vs. getting fed a lot
> of mumbo jumbo.  Good project for a PhD candidate.
>
> So how *do* we add fractions then?  Of course the old
> fashioned way:  a/b + c/d == (a/b)(d/d) + (c/d)(b/b)
> == (ad + cb)/bd as justified by using multiplicative
> identity, other axioms (I'm sure Paul could help out
> here).  

The alternative proof/method I've been pushing is to multiply the whole
sum of fractions by 1 in the form of some common denominator divided by
itself, and then distribute the common denominator in the numerator
through the sum, dividing first on each fraction addend if you're not a
computer for easier computation. Here, in your example above where you
choose your common denominator to be the product of the denominators,
this alternative proof/method is:

a/b + c/d == (bd/bd)(a/b + c/d) == (ad + bc)/bd

This alternative to the "old fashioned way" is more streamlined. You do
one thing - the exact same thing - to the whole sum every single time,
no matter what common denominator you use, least common or otherwise.
That is, you never ever have to worry at all about what different forms
of 1 you have to use to multiply the different addends by.

And by the way, if you would like to see a generalization to contexts
where not all elements have multiplicative inverses, we can use this
alternative, more streamlined proof/method even in the contexts where
we don't have multiplicative inverses on all elements (including those
contexts where all elements do not have multiplicative inverses),
like the natural numbers. For here, with fraction addition, just
"rewrite" what I wrote above as:

a/b + c/d == (bd(a/b + c/d))/bd == (ad + bc)/bd

That is, where a/b and c/d are defined (and they can be in the contexts
I just mentioned), instead of "multiply the whole sum of fractions by 1
in the form of some common denominator divided by itself", we multiply
the sum by the chosen common denominator and then divide this product
by this common denominator. This application of a cancellative semiring
theorem x = (xy)/y enables us to perform the algebra (distributive
property, specifically) to get the final form (ad + bc)/bd, not only in
the contexts where all elements have multiplicative inverses, but in
those contexts where they don't as well (and this includes those
contexts where all elements do not have multiplicative inverses).

Notice that a beauty of this proof/method I recommend is that it does
not require the use of the least common denominator. It's truly a
universal method, perhaps the only universal method.

>
> Then you get gcd((ad + cb), bd)), call it thegcd (could
> be 1), so your new (reduced) numerator, denominator (p,q)
> become p/thegcd, q/thegcd (multiplying by x/x is to
> multiply by 1, the multiplicative identity -- this came
> up earlier, in justifying (a/b)==(a/b)(d/d).
>
> How does this look in actual Python?  Here's a fraction
> class (rational number class) developed to the point of
> enabling its instances (p/q) to multiply and add (p,q
> integers, q <> 0).  See plaintext view to restore
> significant indentation.
>
> def gcd(a, b):
>    while b:
>        a, b = b, a % b
>    return a
>
>
> class Q:
>
>     def __init__(self, p, q):
>         thegcd = gcd(p, q)
>         self.p = p/thegcd
>         self.q = q/thegcd
>
>     def __repr__(self):
>         return "(%s/%s)" % (self.p, self.q)
>    
>     def __add__(self, other):  # code for adding  (+)
>         """
>         (a/b + c/d) == (a*d + c*b)/b*d
>         """
>         return Q(self.p*other.q + other.p*self.q, self.q*other.q)
>        
>     def __mul__(self, other):
>         return Q(self.p * other.p, self.q * other.q)
>
> So used interactively:
>
> >>> from whatever import Q
> >>> a = Q(1,2)
> >>> b = Q(1,6)
> >>> a * b
> (1/12)
> >>> a + b
> (2/3)
>
> Kirby
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Re: Fractions R Us

Dave L. Renfro
In reply to this post by Kirby Urner-5
Paul A. Tanner III wrote (in part):

http://mathforum.org/kb/message.jspa?messageID=6135572

> Here, in your example above where you [[ Kirby ]]
> choose your common denominator to be the product
> of the denominators, this alternative proof/method is:
>
> a/b + c/d == (bd/bd)(a/b + c/d) == (ad + bc)/bd
>
> This alternative to the "old fashioned way" is
> more streamlined. You do one thing - the exact
> same thing - to the whole sum every single time,
> no matter what common denominator you use, least
> common or otherwise. That is, you never ever have
> to worry at all about what different forms of 1
> you have to use to multiply the different addends by.

I don't know how common this is with other teachers,
but it's something I've pointed out in probably every
single one of the over 100 classes I've taught since
1983 (high school and college) in which a review of
this is sometimes needed (anything at or below the
calculus 2 level), not to mention in all the tutoring
I also did in the 1970s and 1980s. I no longer teach,
but when I did, I called it the "cross-multiplying
method", and I mentioned it a few months ago in
this group (although I believe you [Paul] may have
been talking about it in this group then as well):

***********************************************

http://mathforum.org/kb/thread.jspa?messageID=5929430

{5} Give several numerical examples. Point out what
happens when you multiply the denominators -- this
always gives you a common denominator, and it leads
you to the sometimes useful "cross multiplying way
of adding fractions" a/b + c/d = (ad + bc)/bd.

***********************************************

Dave L. Renfro
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Re: Fractions R Us

Theresa Detert
In reply to this post by Kirby Urner-5
Dave,
I believe the "corss-multiplication" is referred to as "The Means-Extremes
Property" in the Algebra textbook that I use.  Theresa



At 12:40 PM 3/13/2008 EDT, you wrote:

>Paul A. Tanner III wrote (in part):
>
>http://mathforum.org/kb/message.jspa?messageID=6135572
>
>> Here, in your example above where you [[ Kirby ]]
>> choose your common denominator to be the product
>> of the denominators, this alternative proof/method is:
>>
>> a/b + c/d == (bd/bd)(a/b + c/d) == (ad + bc)/bd
>>
>> This alternative to the "old fashioned way" is
>> more streamlined. You do one thing - the exact
>> same thing - to the whole sum every single time,
>> no matter what common denominator you use, least
>> common or otherwise. That is, you never ever have
>> to worry at all about what different forms of 1
>> you have to use to multiply the different addends by.
>
>I don't know how common this is with other teachers,
>but it's something I've pointed out in probably every
>single one of the over 100 classes I've taught since
>1983 (high school and college) in which a review of
>this is sometimes needed (anything at or below the
>calculus 2 level), not to mention in all the tutoring
>I also did in the 1970s and 1980s. I no longer teach,
>but when I did, I called it the "cross-multiplying
>method", and I mentioned it a few months ago in
>this group (although I believe you [Paul] may have
>been talking about it in this group then as well):
>
>***********************************************
>
>http://mathforum.org/kb/thread.jspa?messageID=5929430
>
>{5} Give several numerical examples. Point out what
>happens when you multiply the denominators -- this
>always gives you a common denominator, and it leads
>you to the sometimes useful "cross multiplying way
>of adding fractions" a/b + c/d = (ad + bc)/bd.
>
>***********************************************
>
>Dave L. Renfro
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Re: Fractions R Us

Hypatia
In reply to this post by Kirby Urner-5
I believe the means/extremes cross multiplication refers to a proportion and would involve an equation. However, for addition and subtraction of fractions, I also have used the method Paul and Dave describe in every class to all levels of students for many years. Fractions seem to trouble them at all levels. Each time I have demonstrated this easy procedure, students ask why no one has ever shown them this? And that is a very good question!
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Re: Fractions R Us

Gary Tupper
Why? Well, I'd surmise that most text books would shun this procedure
because it is not very transparent to the student new to adding
fractions. I'd guess that most books would show that
adding fractions with common denominators can be simulated with
manipulatives - and then concentrate
on creating the common denominator. I'd further guess that teachers
often do show your method to
students who show difficulty in finding common denominators. The issue
is likely further clouded by a common desire to find the lowest common
denominator (for ease of arithmetic I suppose)

Gary Tupper
Terrace BC

vlm217 wrote:
> I believe the means/extremes cross multiplication refers to a proportion and would involve an equation. However, for addition and subtraction of fractions, I also have used the method Paul and Dave describe in every class to all levels of students for many years. Fractions seem to trouble them at all levels. Each time I have demonstrated this easy procedure, students ask why no one has ever shown them this? And that is a very good question!
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Re: Fractions R Us

Paul A. Tanner III
In reply to this post by Dave L. Renfro

- --- "Dave L. Renfro" <[hidden email]> wrote:

> Paul A. Tanner III wrote (in part):
>
> http://mathforum.org/kb/message.jspa?messageID=6135572
>
> > Here, in your example above where you [[ Kirby ]]
> > choose your common denominator to be the product
> > of the denominators, this alternative proof/method is:
> >
> > a/b + c/d == (bd/bd)(a/b + c/d) == (ad + bc)/bd
> >
> > This alternative to the "old fashioned way" is
> > more streamlined. You do one thing - the exact
> > same thing - to the whole sum every single time,
> > no matter what common denominator you use, least
> > common or otherwise. That is, you never ever have
> > to worry at all about what different forms of 1
> > you have to use to multiply the different addends by.
>
> I don't know how common this is with other teachers,
> but it's something I've pointed out in probably every
> single one of the over 100 classes I've taught since
> 1983 (high school and college) in which a review of
> this is sometimes needed (anything at or below the
> calculus 2 level), not to mention in all the tutoring
> I also did in the 1970s and 1980s. I no longer teach,
> but when I did, I called it the "cross-multiplying
> method", and I mentioned it a few months ago in
> this group (although I believe you [Paul] may have
> been talking about it in this group then as well):
>
> ***********************************************
>
> http://mathforum.org/kb/thread.jspa?messageID=5929430
>
> {5} Give several numerical examples. Point out what
> happens when you multiply the denominators -- this
> always gives you a common denominator, and it leads
> you to the sometimes useful "cross multiplying way
> of adding fractions" a/b + c/d = (ad + bc)/bd.
>
> ***********************************************
>
> Dave L. Renfro

Hi Dave,

Based on what you said and what some others have said in this thread
about the theorem a/b + c/d = (ad + bc)/bd, there might be some
confusion about what I was referring to when I wrote, "This alternative
to the "old fashioned way" is more streamlined. You do one thing - the
exact same thing - to the whole sum every single time, no matter what
common denominator you use, least common or otherwise."

What I was referring to when I said "this alternative" was not the
theorem a/b + c/d = (ad + bc)/bd, which I always share with every
student I have taught on fraction addition, but the *method* I
outlined. This theorem is just a special case of this method, which
I've never seen published or used by another teacher. This method is
identical on any number of fraction addends using any of the infinitely
many possible common denominators. Hence my reference to it being
possibly the most general method possible. Here's the method written
out abstractly: Where C is any of the infinitely many common
denominators (least common denominator, product of the denominators,
whatever) of any number of fraction addends, where the method is
written out in two alternative ways for the middle step,

a/b + . . . + y/z = (C/C)(a/b + . . . + y/z) = [C(a/b) + . . .
C(y/z)]/C

or

a/b + . . . + y/z = [C(a/b + . . . + y/z)]C = [C(a/b) + . . . C(y/z)]/C

The theorem a/b + c/d = (ad + bc)/bd is just one special case of this
general method (this special case would be where there are only two
fraction addends and where C = bd). This general method works much,
much better than the old fashioned way (as Kirby calls it) when either
there are more than two fraction addends or when we want to use a
common denominator other than the product of the denominators. This is
especially so if we are adding rational functions, when we would
normally want to to use the least common denominator to keep the
algebra simpler. In the old fashioned way, we have to multiply each
fraction addend by a different number divided by itself. With this
method I'm describing, we don't have to worry with that at all. We also
can skip the step of writing out a bunch of different fractions all
with the same denominator.

Example using the least common denominator (again writing the middle
step in two different ways):

3/4 + 5/6 + 7/8 + 11/12 = (24/24)(3/4 + 5/6 + 7/8 + 11/12) = (18 + 20 +
21 + 22)/24

or

3/4 + 5/6 + 7/8 + 11/12 = [24(3/4 + 5/6 + 7/8 + 11/12)]/24 = (18 + 20 +
21 + 22)/24

Note: When distributing through the C (in this example C = 24), divide
first on each addend for much, much easier computation - or algebra if
the context is rational functions. (For instance, with this example on
the first addend, do 24/4*3, not 24*3/4.)
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Re: Fractions R Us

Dave L. Renfro
In reply to this post by Kirby Urner-5
Paul A. Tanner III wrote (in part):

http://mathforum.org/kb/message.jspa?messageID=6136949

> Based on what you said and what some others
> have said in this thread about the theorem
> a/b + c/d = (ad + bc)/bd, there might be some
> confusion about what I was referring to when
> I wrote, "This alternative to the "old fashioned
> way" is more streamlined. You do one thing - the
> exact same thing - to the whole sum every single
> time, no matter what common denominator you use,
> least common or otherwise."

Yes, I thought you were advocating this method
essentially exclusive of anything else, although
my main point was independent of this. My main
point was that I don't think this method is as
unknown as you said. You later agree that it's
very well known and then say:

> This theorem is just a special case of this method,
> which I've never seen published or used by another
> teacher. This method is identical on any number of
> fraction addends using any of the infinitely many
> possible common denominators. Hence my reference [...]

What you described is actually what I did in class,
and I'm sure many others do as well (high school
and college). I'd point out that any common denominator
will serve, and it varies with the situation as to
how much effort you want to expend towards getting
near to (or actually equal to) the least common
denominator. Using the least common denominator
involves more work to find, but less work in the
subsequent computations ("more" and "less" being
their non-strict versions, of course), while using
the product of the denominators (or something between
the least common denominator and the product) is
the reverse. One thing about not using the least
common denominator is that you'll *always* wind
up having the result not be in lowest terms (proof
left to the math-teach reader). On the other hand,
it's not hard to come up with examples where you're
still not in lowest terms if you use the least
common denominator.

Two examples that come to mind are trig. function
manipulations (identities, etc.) and algebraic
evaluation of derivatives (of rational functions
and not-too-messy non-rational explict algebraic
functions). In the case of manipulations with trig.
functions, it's usually worth your while to spend
a little extra time finding the LEAST common
denominator. In the case of evaluating derivatives
using the difference quotient definition of a
derivative, it's usually not worth spending a
single moment worrying about a least common
denominator, since it's almost always the case
that the least common denominator is going to
be the product of the denominators.

To get back to educational matters, fractions
are a topic students have a lot of trouble with,
and this has been true since at least the early
1800s (any earlier and we'll be comparing apples
with oranges in many ways). I also think it's
gotten substantially worse in the past 30 some
years, especially the past 20 years, due to the
lack of the reinforcement of these skills, as
compared to what students used to receive in
middle school and high school, from the current
almost universal use of calculators by students
on everything numerical.

I once made a  comment in sci.math related to
this that is worth repeating. When I was in
middle school and in early high school (early
1970s, before anyone had calculators; at least,
no one at my school had a calculator before
1975), we always dreaded word problems with
decimal numbers, because those were always the
hardest to carry out the arithmetic calculations.
Naturally, most people hated word problems to
begin with, but at least when you had things
like 3/8, 2/5, or 22/7, you knew at worst
the calculations would be one and two digit
multiplications and divisions. But if you had
things like 4.52, 0.581, or 3.14, you knew
there was going to be some work involved if
you had to divide by one of these numbers
(or by a number obtained from them). This was
especially true in direct variation and inverse
variation problems.

Now we're not going to get rid of calculators,
so the issue comes down to a trade-off of how
much additional practice with fractions should
we include elsewhere in the curriculum to offset
this decreased practice, if any. I don't know
for an average high school student, but I will
say that I found lots of ways to include this
and other skills (rationalizing methods, practice
with exponents, etc.) in high school and college
precalculus and calculus classes. Instead of
lamenting over the fact that students had a lot
of trouble with these topics and then review
them in probably the same way they were taught,
I came up with assignments and worked examples
in class that required heavy use of these topics.

Here's an example from an old worksheet of mine
for calculus 1.

Show by hand (i.e. without using a calculator) that

(3/2)ln(2) + (1/2)ln(3) + (1/2)ln[(1 + 1/49)/(1 - 1/49)]

is equal to ln(5).

I certainly allowed the use of calculators (except for
certain trig. tests and for certain "calculus gateway"
tests), so students were free to check this numerically
(or check their work along the way -- in fact, I
strongly encouraged them to check their work this
way if they were unsure about what they were doing).
However, I explained that full credit always required
an appropriate amount of work, and if they wanted
to argue with me that they did some of it in their
heads, they were welcome to come by my office after
class and demonstrate this on similar problems.
[Very few people took me up on this. In fact, the
students that I was most convinced could do a lot
of the "appropriate" work in their heads were almost
always were the same students who showed the most
amount of work on their assignments.]

Dave L. Renfro
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Re: Fractions R Us

Paul A. Tanner III

- --- "Dave L. Renfro" <[hidden email]> wrote:

> Paul A. Tanner III wrote (in part):
>
> http://mathforum.org/kb/message.jspa?messageID=6136949
>
> > Based on what you said and what some others
> > have said in this thread about the theorem
> > a/b + c/d = (ad + bc)/bd, there might be some
> > confusion about what I was referring to when
> > I wrote, "This alternative to the "old fashioned
> > way" is more streamlined. You do one thing - the
> > exact same thing - to the whole sum every single
> > time, no matter what common denominator you use,
> > least common or otherwise."
>
> Yes, I thought you were advocating this method
> essentially exclusive of anything else, although
> my main point was independent of this. My main
> point was that I don't think this method is as
> unknown as you said. You later agree that it's
> very well known and then say:
>
> > This theorem is just a special case of this method,
> > which I've never seen published or used by another
> > teacher. This method is identical on any number of
> > fraction addends using any of the infinitely many
> > possible common denominators. Hence my reference [...]
>
> What you described is actually what I did in class,
> and I'm sure many others do as well (high school
> and college). I'd point out that any common denominator
> will serve, and it varies with the situation as to
> how much effort you want to expend towards getting
> near to (or actually equal to) the least common
> denominator.

I'm still not sure that we are talking about the same thing as to what
I was referring to with the phrase "this method." You said, "My main
point was that I don't think this method is as unknown as you said. You
later agree that it's very well known . . ." But what I referring to
with the phrase "this method" was not what was I referring to in terms
of "very well known." What I was referring to in terms of "very well
known" is the identity a/b + c/d = (ad + bc)/bd itself, not a method of
how to get from a/b + c/d to (ad + bc)/bd. I've never seen in print
anywhere the method I'm talking about.

The only method I've seen used in print to get from from a/b + c/d to
(ad + bc)/bd is (a/b)(d/d) + (c/d)(b/b). (This is the old fashioned
method Kirby was talking about - it's what he wrote in the original
post.) That is, getting from the former to the latter is a proof of the
identity a/b + c/d = (ad + bc)/bd, this proof used as a method. A
clumsy one, in my view.

There is another proof of a/b + c/d = (ad + bc)/bd. One that is one not
clumsy. This other proof used as a method is not a clumsy method, in my
view.

This clumsy method I just talked about involves coming up with a
different number for each fraction addend. Here, with the product of
the denominators being the chosen common multiple of the denominators,
we use two different numbers for the two different fraction addends,
these two numbers being d and b. If we have more than two different
fraction addends, say n fraction addends, this problem of having to
come up with with different numbers for different fraction addends
means we have to come up with n different numbers. And on top of all
this, this method involves rewriting the sum of n fraction addends with
different denominators with n fractions addends with the same number as
their denominators. And then we rewrite that as a single fraction.

But with the non-clumsy method I'm talking about to get from from a/b +
c/d to (ad + bc)/bd is (a/b)(d/d) + (c/d)(b/b), we come up with one and
only one number, whatever common multiple of the denominators we
choose. Instead of replacing the sum a/b + c/d with (a/b)(d/d) +
(c/d)(b/b), we replace it with the equivalent [C(a/b + c/d)]/C or the
equivalent (C/C)(a/b + c/d). Then we distribute the top C through the
sum, doing the arithmetic. And for n different fraction addends, this
means we do not rewrite the sum of n fraction addends with different
denominators with n fractions addends with the same number as their
denominators. We skip this step.)

I've never seen this non-clumsy method I'm talking about in print. If
someone can cite a source where it is in print, please do so.

I have no doubt that some teachers have realized it's there as an
alternative, and maybe you're one of them. So for C being any common
multiple of the denominators, are you saying that you taught your
students to multiply the whole sum S of fraction addends by 1 in the
form of C/C, this written as (C/C)S, or that you taught your students
to multiply the whole sum S of fraction addends by C and then divide by
C, this written as (CS)C, and then distribute the top C through S,
skipping the step of having to write out a new sum of fraction addends
all with the same number as their denominators?
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Re: Fractions R Us

Kirby Urner-5
In reply to this post by Kirby Urner-5
Dave Renfro:
 
> Now we're not going to get rid of calculators,
> so the issue comes down to a trade-off of how
> much additional practice with fractions should
> we include elsewhere in the curriculum to offset
> this decreased practice, if any.

I agree that calculators will continue to be available,
including as emulated on computer screens, however
some emerging pre-college math curricula displace them
with computers, and introduce a modicum of programming
as a way to learn the underlying concepts.

The advantage of programming over pressing buttons on
a calculator is you're focusing on the algorithms.  How
do we find the GCD?  Do we build factor trees and circle
primes in common, building up a product that way?  That's
one method for sure.  Euclid's Method is another, less
often taught, but ultimately way more effective, as
distilling to primes becomes non-trivial in a hurry,
as the numbers get big.

So whereas we look at calculators as "black boxes" that
"just know" how to add fractions, we might approach the
computer as a machine that "needs to be taught". In
teaching the computer how to add two fractions, students
also teach themselves how it's done.

In this day and age, some languages are easy enough,
sharable enough, that teaching "adding and multiplying
fractions" is actually within the range of a pre-college
student.  This approach is not exclusive of using
manipulatives, visualizations, other traditional approaches
- -- these all become mutually reinforcing, even with a
"new kid on the block" (computer programming).

Replacing the calculator with an executable math notation
(as Kenneth Iverson called APL) is not like just phasing
in a more powerful calculator.  The pedagogical
implications are potentially huge, deserving of separate
threads / conversations vs. the "appropriateness of
calculators" threads -- rather old hat and hashed over
after this many decades.

Students who build "fraction objects", gradually adding
capabilities, coming back to their evolving scripts over
a period of days, are consistently looking at the
algorithms.  Yes, they execute at superhuman speeds, but
no, it's not a black box, and comprehension of how and
why is the focus.

My purpose in this thread was to point out that if you
have a way of getting the GCD really easily (have coded
Euclid's Method, understand it as well), then it's less
important to focus on the LCM, as we can reduce to
lowest terms at the moment a "result fraction" is
obtained.  

In other words, as numerator p and denominator q enter
the "birth method" (where a fraction gets instantiated),
the gcd algorithm is automatically applied, such that
thegcd = gcd(p, q), newp = p/thegcd, newq = q/thegcd.
Put another way, fractions automatically reduce to lowest
terms:

In the interactive session below, a student is creating
Q-objects (rational numbers, members of Q).

>>> somefraction = Q(10,100)
>>> somefraction
(1/10)
>>> otherfraction = Q(12,144)
>>> otherfraction
(1/12)

If thegcd == 1, no problem, as this means the original
p,q were relatively prime, and dividing them each by 1
leaves them as is i.e. we were in lowest terms already.

We might pause here to discuss equivalence classes, the
fact that p/q with gcd(p,q)==1, is the canonical or
normalized representative of an equivalence class in Q.
This is usefully connected to a discussion of modulo
arithmetic, where we also have equivalent integers modulo
N, but a normal delegate < N.

Testing for equivalence.

Rational number class (coded by students):

>>> from whatever import Q
>>> Q(1,2) == Q(5,10)
True

Modulo integer class (coded by students):

>>> from tecc_alg2_u5 import M
>>> M.modulus = 11
>>> M(10) == M(21)
True

What's new in this picture is we replace a paper and
pencil process with machine execution, without turning
the process into a black box.  It remains apparent to
students what's happening behind the scenes.  The concepts
remain front and center, and yet we're still taking
advantage of machine execution speeds, allowing our
computations to become interestingly complex without
becoming tedious.  

Applying a rotation matrix to 12 vector objects, over
and over, making an icosahedron rotate in 1 degree
increments, isn't something you'd want to do with
paper and pencil (nor do current calculators do the
job, of actually displaying a polyhedron).  

With computers we have the best of both worlds:
attractive eye candy, a sense of crafting something
interesting, without losing touch with algebraic /
algorithmic / symbolic underpinnings.

Of course once you understand the concepts, have proved
why the algorithms are robust, you'll not be needing to
inspect the source code at every turn.  What we're
learning, versus what we're simply using, will vary with
the lesson.  By they time we're rotating polyhedra,
studying matrix multiplication, we're probably not
mystified by fraction addition.  But that's just in the
nature of mathematics itself, i.e. it's ever been thus.

Kirby
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Re: Fractions R Us

Dave L. Renfro
In reply to this post by Kirby Urner-5
Dave Renfro wrote (in part):

http://mathforum.org/kb/message.jspa?messageID=6137437

>> Now we're not going to get rid of calculators,
>> so the issue comes down to a trade-off of how
>> much additional practice with fractions should
>> we include elsewhere in the curriculum to offset
>> this decreased practice, if any.

Kirby Urner wrote (in part):

http://mathforum.org/kb/message.jspa?messageID=6138734

> I agree that calculators will continue to be available,
> including as emulated on computer screens, however
> some emerging pre-college math curricula displace them
> with computers, and introduce a modicum of programming
> as a way to learn the underlying concepts.

I thought about including something like "(or computers,
or whatever)", but decided it would distract too much
from the flow of the sentence (which is not to say that
I succeeded in doing this in other places, such as this
sentence). In 40-60 years we'll probably have voice
recognition systems that can answer questions or point
the way for finding an answer (as appropriate, such as
in a classroom) and automatic math features like how
MicroSoft Word can automatically (and usually, but not
always) correct misspelled words, with programming at
the level you're doing going the way of square root
algorithms (still useful for certain people, but not
the end user). And perhaps in 100-150 years there will
be "math chips" and other kinds as well (likely part
organic and part non-organinc) that will be implanted
in the brains of children at the appropriate time in
their development (like what we do with various
vaccinations children get today), and much of education
at this time will be learning how to use/adapt to these
implants. Also, I suspect sometime between 150 and 200
years these implants will more and more allow us to
directly link to whatever the internet evolves into.

By the way, these things are hardly much different
from where we're at today as compared to some of
the things you'll find in stories and novels by
Greg Egan (for those interested in some really
way way out there thinking).

Dave L. Renfro
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Re: Fractions R Us

Michael Paul Goldenberg
In reply to this post by Paul A. Tanner III

On Mar 14, 2008, at 7:06 PM, Paul A. Tanner III wrote:

>>
>
> I'm still not sure that we are talking about the same thing as to what
> I was referring to with the phrase "this method." You said, "My main
> point was that I don't think this method is as unknown as you said.  
> You
> later agree that it's very well known . . ." But what I referring to
> with the phrase "this method" was not what was I referring to in terms
> of "very well known." What I was referring to in terms of "very well
> known" is the identity a/b + c/d = (ad + bc)/bd itself, not a method  
> of
> how to get from a/b + c/d to (ad + bc)/bd.

> I'VE NEVER SEEN IN PRINT
> ANYWHERE THE METHOD I'M TALKING ABOUT.
[emphasis added]

Perhaps I'm VERY wrong, but there have been many posts like Dave  
Renfro's that indicate that contrary to Paul's opinion, his "theorem,"  
"method," "technique," "short-cut," or whatever his preferred  
appellation might be, is not quite unknown or original to him. I  
strongly believe that we need to collectively lie to Paul, tell him  
that indeed he's made an earth-shattering discovery for which he  
deserves the world's heartfelt appreciation, and suggest that a Fields  
Medal may very well be in his future.

Otherwise, we're going to continue to be flooded with posts on this  
dead-as-a-doornail topic every time someone raises the issue of  
fractions and their teaching. That has been the case for years and  
years. It's unbelievable that anyone could be so desperate for kudos,  
yet considering the individual in question, it's actually completely  
obvious and utterly predictable. I'm not sure that offering him  
hosannahs and world-wide acclaim would actually suffice to stem the  
flow of these periodic dissertations, but I, for one, would be happy  
to give it the old college try.

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Re: Fractions R Us

Paul A. Tanner III
In reply to this post by Gary Tupper

- --- Gary Tupper <[hidden email]> wrote:

> Why? Well, I'd surmise that most text books would shun this procedure
>
> because it is not very transparent to the student new to adding
> fractions. I'd guess that most books would show that
> adding fractions with common denominators can be simulated with
> manipulatives - and then concentrate
> on creating the common denominator. I'd further guess that teachers
> often do show your method to
> students who show difficulty in finding common denominators. The
> issue
> is likely further clouded by a common desire to find the lowest
> common
> denominator (for ease of arithmetic I suppose)
>
> Gary Tupper
> Terrace BC

Hi Gary,

As I mentioned to Dave, I'm wondering whether we're all talking about
the same method, by what vlm217 wrote below where he seemed to be
referring to Dave's "cross multiplication method" (the formula a/b +
c/d = (ad + bc)/bd) and by what you wrote above, "I'd further guess
that teachers often do show your method to students who show difficulty
in finding common denominators. The issue is likely further clouded by
a common desire to find the lowest common denominator."

Problem is, the method I'm talking about involves first finding a
common denominator, preferably the least common denominator. I just
pointed out to Kirby, when I replied to his original post, that the
method I'm talking about to work, it does not require that the common
denominator be the least common denominator. In fact, for the method
I'm talking about to work, we don't have to skip the step of writing a
new sum of fractions all having the same number as their denominators -
we just have skipping that step as an option.

As to why the method I'm talking about has, as far as I know, never in
published materials been given even merely as an alternative when in my
experience it makes things easier for those who struggle with adding
fractional expressions, including especially adding rational functions,
I don't know.

>
> vlm217 wrote:
> > I believe the means/extremes cross multiplication refers to a
> proportion and would involve an equation. However, for addition and
> subtraction of fractions, I also have used the method Paul and Dave
> describe in every class to all levels of students for many years.
> Fractions seem to trouble them at all levels. Each time I have
> demonstrated this easy procedure, students ask why no one has ever
> shown them this? And that is a very good question!
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Re: Fractions R Us

Dave L. Renfro
In reply to this post by Kirby Urner-5
Dave L. Renfro wrote (in part):

http://mathforum.org/kb/message.jspa?messageID=6139736

> In 40-60 years we'll probably have voice recognition
> systems that can answer questions or point the way
> for finding an answer (as appropriate, such as in
> a classroom) and automatic math features like how
> MicroSoft Word can automatically (and usually, but not
> always) correct misspelled words, with programming at
> the level you're doing going the way of square root
> algorithms (still useful for certain people, but not
> the end user).

Thinking about this some more, it occurs to me that
this is too awkward. This would probably sound "quaint"
to someone at that time. A better guess is that we'll
have "smart pads", something whose texture when writing
is between that of ordinary pencil/paper and a magic
slate (it's processing unit would be at the top, like
the top boarder of an 8.5 by 11 notepad with most of the
sheets gone), which interprets and corrects what you
write -- storing it in digital form that can be reworked
in many ways, such as a typed LaTeX document.

Dave L. Renfro

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Re: Fractions R Us

Kirby Urner-5
In reply to this post by Kirby Urner-5
> Dave Renfro wrote (in part):
>

> And perhaps in 100-150 years there will
> be "math chips" and other kinds as well (likely part
> organic and part non-organinc) that will be implanted
> in the brains of children at the appropriate time in
> their development (like what we do with various
> vaccinations children get today), and much of
> education at this time will be learning how to
> use/adapt to these implants.

I seriously doubt it.  I think it's a peculiar
subculture of science fiction writer that's always
thinking of ways to bypass perfectly good I/O
(fingers, eyes, ears) and wanting to wire directly
to the brain, assuming some major advances in brain
science.  Whereas prosthetic devices make sense in
some cases, in general there's no point reinventing
the wheel and short circuiting what nature has given
us.

It's also a peculiar subculture that relegates
"end users" to the category of Eloi (H.G. Wells),
helpless non-technical people who couldn't tell
a computer program from a salad, and a small elite
of digerati (Morlocks) who know how everything works.
One would hope a serious early numeracy curriculum
would work against such a distopian vision, but
the outgoing generation of math teacher on the
whole seemed content to accept it, by leaving
programming to programmers, instead of considering
it a basic skill.  I am hopeful we won't be repeating
that mistake in the future (overspecialization in
general has been a major shortcoming of 1900s style
thinking).

Kirby