Does My Fraction 1/1 Story Work?

Date: 11/04/2002 at 01:31:54
From: Brent Wiley
Subject: Does my fraction 1/1 story work?

Analogies and stories have always helped me in understanding practical 
applications for math. Here's a story I came up with to explain the 
conversion of any fraction n/n being simplified to one (if n is 
greater than one and an integer). Tell me if you think the story 
properly relates the concept. If not, where does it break down? Your 
help is greatly appreciated. Thank you for your time.


Any integer over itself is a one-to-one ratio that can be simplified 
to one. The equation would be n/n = 1/1 = 1. 64/64 = 1/1 = 1 works 
because 64/64 is a one-to-one ratio. Each person in the group of 64 
gets one piece out of the 64-piece pie. One piece for one person. One 
to one simplifies to one because one goes into one 1 time. Every whole 
number has a hidden fraction. For example 78 can be written as 78/1, 
84 can be written as 84/1 etc. These numbers are all in terms of one.  
1/1 is like saying "one in terms of one." We can just as easily say 
"one."  78 in terms of one is 78, 62/1 in terms of ones is 62, etc.  
In support of this, 62/1 would mean that one person gets 62 pieces of 
pie. 2/1 means one person gets 2 pieces of pie. In a restaurant, if a 
cook in the kitchen asks a waiter "how many?"  The waiter can say 
either "Give two to every one person" or he may just say "two," 
implying that quantity for each person alone. This is equivalent to 
why whole numbers are not written as fractions in terms of one. It is 
assumed the numbers are in terms of one. As a last point, how would 
you visually draw 1/1? You would just draw one pie (one of one).  

The one problem I have is explaining why 1/1 = 1 using the pie/person 
analogy. There is one pie (in the numerator) and one person (in the 
denominator). So I always think there are two "things" present, the 
person and the pie. How could they ever become one?  Thus the reason 
for the kitchen story towards the end of the paragraph. The story is 
supposed to explain the disappearance of the person when 1/1 is 
simplified to one.

Date: 11/04/2002 at 10:24:52
From: Doctor Ian
Subject: Re: Does my fraction 1/1 story work?

Hi Brent, 

>Analogies and stories have always helped me in understanding 
>practical applications for math. 

Then use them, by all means. However, you can get into trouble
whenever you try to stretch an analogy too far, which may be what's
happening here. 

In the end, a fraction is just a division that you haven't bothered to
do yet. So the fraction 5/7 just means '5 divided by 7', and that's
_all_ it means. You can try to model the division by thinking about
pies, but thinking about the model instead of the operation can lead
you into confusion. 

>Any integer over itself is a one to one ratio which can be simplified 
>to one. The equation would be n/n = 1/1 = 1.  
>64/64 = 1/1 = 1 works because 64/64 is one to one ratio. Each person 
>in the group of 64 gets one piece out of the 64-piece pie. One piece 
>for one person. One to one simplifies to one because one goes into 
>one 1 time.  

So, you can think of this as 'simplifying to one', or you can think of
the equation 

  1/1 = 1 

as saying "1 divided by 1 equals 1".  In the long run, this latter
approach is more useful, because it extends readily to equations with
variables that can take on any real values, while the pie-based
approach does not.  

(Consider a fraction like 1/x. What does that mean, in terms of
cutting up pies, if the value of x turns out to be something like the
square root of 2?)

>Every whole number has a hidden fraction. For example 78 
>can be written as 78/1, 84 can be written as 84/1 etc. These numbers 
>are all in terms of one. 1/1 is like saying "one in terms of one." 
>We can just as easily say "one."  

Or again, instead of "one in terms of one" (a phrase that may not be
meaningful to anyone but you), we can say "one divided by one." 

>The one problem I have is explaining why 1/1 = 1 using the pie/person 
>analogy. There is one pie (in the numerator) and one person (in the 
>denominator). So I always think there are two "things" present, the 
>person and the pie. How could they ever become one? Thus, the 
>reason for the kitchen story towards the end of the paragraph. The 
>story is supposed to explain the disappearance of the person when 1/1 
>is simplified to one.

In fact, once you introduce units into the situation, they only
disappear when they appear in both the numerator and the denominator.  

Let's look at a couple of examples.  Suppose I have $100, and you have
$200. What is the ratio of what I have to what you have?

  what I have     100 dollars
  ------------- = -----------
  what you have   200 dollars


                  100 * (1 dollar)
                = ----------------
                  200 * (1 dollar)


                  100   (1 dollar)
                = --- * ----------
                  200   (1 dollar)

Now, as we've noted, anything divided by itself is just 1. So the
factor on the right cancels to 1, leaving us with

                  100   
                = --- * 1
                  200   

In other words, we have no units. This is just a ratio of two numbers, 
and there's no way to tell by looking at it what it's supposed to 
mean, or what kind of process might have led to it. This could be a 
ratio of angles, or a ratio of temperatures, or a ratio of weights.  

But let's say we want to divide 6 pies evenly among 3 people. Now we 
have

  pies      6 pies      6   pies
  ------- = --------- = - * -------  = 2 (pies/person)
  persons   3 persons   3   persons

That is, the units stay around instead of cancelling. We end up, not
with '2', but with '2 pies per person', in much the same way that if
we travel 45 miles in 30 minutes, we end up with

  45 miles    45   miles
  -------- = --- * ----- = 90 miles/hour
  1/2 hour   1/2   hours

or 90 miles per hour, and not just '90'.  

That is, when you introduce units, you end up with two results: a
number, and some combination of units that tells you how to interpret
the number. Note that in the case of the speed, I could have done 
this:

  45 miles     45   miles
  ---------- = -- * ------- = 1.5 miles/minutes
  30 minutes   30   minutes

The numbers look different, but 90 miles per hour has the same
_meaning_ as 1.5 miles per minute. 

In the same way, note that

  60 miles
  ---------- = 1 mile/minute
  60 minutes

  60 miles
  -------- = 60 miles/hour
   1 hour

So in one case, you get '1', and in the other case, you get '60'. But
that leaves out the most meaningful part of the computation, i.e., we
get 1 of _what_?  Or we get 60 of _what_? 

Of course, in many contexts people will assume that everyone is using
the same units, in which case they may not bother to state them
explicitly. For example, if someone says "A cop caught me going 75 
last week on the way home from work," we assume that me means 75 miles
per hour.

Or the desired units might be included in a request for information:
"How many inches long is the counter?" In a case like this, an answer
like '64' is unambiguous.  

But suppose someone asked you to measure the length of a thing, and
you responded with '119'.  Would that answer be complete?  No, you'd
have to say '119 cm', or '119 inches', or '119 hand widths', or
something like that. 

All of which is to say that the short answer to your question is that 

  1 zig
  ----- = 1 zig per zag
  1 zag

which is only '1' if a zig is the same as a zag. You may choose not to 
state the units, if they're unambiguous. But ignoring something
doesn't make it go away.  

Does this make sense? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 11/04/2002 at 20:54:16
From: Brent Wiley
Subject: Does my fraction 1/1 story work?

Dr. Ian, 

Thank you so much for your reply. Plain and simple - it made sense.