Reading a Ruler III

Date: 03/03/2005 at 23:19:37
From: Mary
Subject: reading a ruler--instructions to explain to a child

My daughter came home with a worksheet that had a ruler on it and it 
has letters on the ruler in which she is supposed to say the place 
(value) the letter represents on the ruler.  Now it's been a while 
since I have done this.  I am looking for simple, basic instructions
to teach a child how to read a ruler.
 
Of course I can instantly tell you where a 1/2 is; however, the rest 
is very vague to me.  I somehow remember that we used to count the 
lines--where the line falls on the ruler is the top number of the 
fraction and how many lines in between 0 to 1 inch represents the 
bottom number (the whole).  Is this correct?  

I like your examples on how to read a ruler.  However, if you can just
count where the line is and put that in the numerator and count how
many lines make the whole and put that in the denominator that would
be helpful to make it so that everyone truly understands how to read
the ruler.

Date: 03/04/2005 at 08:44:52
From: Doctor Ian
Subject: Re: reading a ruler--instructions to explain to a child

Hi Mary -

It sounds like you've already read these items from our archive:

  Reading a Ruler
    http://mathforum.org/library/drmath/view/58333.html 

  Reading a Ruler II
    http://mathforum.org/library/drmath/view/58439.html 

Is that correct?  If not, you might want to take a look at them.

In any case, let's look at an example.  Suppose I'm measuring 
something that's something and 7/16 of an inch long.  

                                
   |                               |
   |               |               |
   |       |       |       |       |
   |   |   |   |   |   |   |   |   |
   | | | | | | | | | | | | | | | | |  
  -----------------------------------
  xxxxxxxxxxxxxxxx

In practice, I'd look at the 1/2 mark, and say "That's too big." So
I'd look at the 1/4 mark, and say "That's too small."  So now I know
it's between 1/4 and 1/2, i.e., between 1/4 and 2/4, which is between
2/8 and 4/8.  

What's halfway between them?  3/8.  So I look at 3/8, and say "That's
too small."  So now I know it's halfway between 3/8 and 1/2, i.e.,
between 3/8 and 4/8, which is between 6/16 and 8/16.

What's halfway between them?  7/16.  And that works.  So I'm done. 

The whole system is based on successively breaking things into halves,
so I just go with the flow.  That is, you can imagine a ruler where
the markings only go down to 1/2 an inch:

   |                               |
   |               |               |
   |               |               |
   |               |               |
   |               |               |  
  -----------------------------------
   0              1/2             2/2 = 1

If we decide that's not fine enough, we double the denominators, 

   |                               |
   |               |               |
   |               |               |
   |               |               |
   |               |               |  
  -----------------------------------
   0              2/4             4/4 = 1

and that gives us room for more numerators:

   |                               |
   |               |               |
   |       |       |       |       |
   |       |       |       |       |
   |       |       |       |       |  
  -----------------------------------
   0      1/4     2/4     3/4     4/4

And so on, through 8ths and 16ths and even 32nds, if we can make our
markings finely enough.  

Of course, this is all but impossible to deal with unless you're
pretty facile at converting between equivalent fractions with powers
of 2 in the denominator, e.g., 

  1/2 = 2/4 = 4/8 = 8/16 = 16/32

That's the basic skill you need to make the system work.  

The idea of counting all the marks to figure out what the denominator
should be would work in theory, but in practice it would be pretty
slow.  But with some practice, you could do something sort of
equivalent but quicker.  

That is, you could start by identifying what size mark is next to the
end of the thing you're measuring:

                                
   |                               |
   |               |               |
   |       |       |       |       |
   |   |   |   |   |   |   |   |   |
   | | | | | | | | | | | | | | | | |  
  -----------------------------------
  xxxxxxxxxxxxx^
               |_________ This is the size I want.

Next, you could look at the 1/2 mark, and back up halfway to the 1/4
mark.  If that's the right size, you can now just count quarters.  If
it's still too small, back up halfway to the 1/8 mark.  If that's the
right size (in the example above, it is), you can just count eighths.
And so on.  

Note that when you're 'counting quarters' or 'counting eighths', you
have to count any marks that are at least as large as the one you're
using; but you can ignore any that are smaller.  So that's pretty
straightforward, but again it require at least a little facility with
powers of 2 in the denominator. 

Is this helpful? 

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