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Q&A #6232


Demonstrating division of fractions with pictures or manipulatives

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From: Jeanne (for Teacher2Teacher Service)
Date: May 01, 2001 at 17:14:25
Subject: Re: Demonstrating division of fractions with pictures or manipulatives

Hi Beverly,

I think its great that you are looking for models to help you show "why
dividing a negative fraction by a negative fraction will give a positive
number."  And I think that manipulatives and drawings are excellent tools.
They are effective for use when teaching some of the concepts leading up to
the example you cited in the subject line.  However, tools have
limitations.   And so, I also believe this specific problem and others of
this degree of difficulty should come at a time when these tools are no
longer necessary.  Prior to reaching this level of complexity, students
should already have had experience using manipulatives, drawings, and
pattern analysis in multiplying and dividing integers, and with multiplying
and dividing positive fractions.

Here are some lesson ideas I use with my students...

Multiplication/Division of integers.
Idea #1:  My favorite manipulative for teaching the arithmetic of integers
is tile spacers.  (Tile spacers are used to keep the distance between tiles
equal when someone is laying tiles for a floor, or a countertop, or a wall.
You can buy them in bulk very inexpensively from hardware stores.)  I like
them because they are shaped like plus signs.  One can trim a plus sign with
a pair of scissors to make a negative.  The spacers are soft plastic.

I start my kids with the concept of making zero.  There a lots of
ways to make zero:  one "+" and one "-" ,  two "+" and two "-", three "+"
and three "-", ...  I have them start with several zeroes on their working
area to begin with.

Example:  3 times -2

"3 times -2" can be interpreted to mean "put in 3 groups of -2."  So their
work area could look like

     + + + + + + + + + +
     - - - - - - - - - -  (--) (--)  (--)  <----Put in
So, 3 times -2 equals -6.


Example:  -3 times -2.

If "3 time -2" means "put in 3 groups of -2,"  then "-3 times -2" can be
interpreted to mean "take out 3 groups of -2."

     + + + + + + + + + +
     - - - -                     (- -) (- -)(- -)  ---->Take away
So, -3 times -2 equals +6.



Idea #2:  Pattern analysis.  The "negative times a negative equals a
positive" is a bit strange for many students, so I like to have more than
one way of having kids look at this concept.

Start with:        Then multiply each
                  integer by 2:

     .                  .        .
     .                  .        .
     .                  .        .    I ask them to look at the sequence
     4                  4 * 2 =  8    of numbers generated by the products.
     3                  3 * 2 =  6    Without too much trouble they say
     2                  2 * 2 =  4    numbers are even and the sequence is
     1                  1 * 2 =  2    decreasing.
     0                  0 * 2 =  ?     I ask what comes next?  0
    -1                 -1 * 2 =  ?       ...what comes next?   -2
    -2                 -2 * 2 =  ?       ...what comes next?   -4
    -3                 -3 * 2 =  ?
    -4                 -4 * 2 =  ?    With these patterns we support the idea
     .                  .        .    that a negative times a positive is
     .                  .        .    negative.
     .                  .        .


Now start with:       And multiply each
                    integer by -2:

     .                  .        .
     .                  .        .
     .                  .        .
     3                  3 * -2 =  -6      I ask them to look at the sequence
     2                  2 * -2 =  -4    of numbers generated by the products.
     1                  1 * -2 =  -2    With a little help they say the
     0                  0 * -2 =  ?    numbers are even and the sequence is
    -1                 -1 * -2 =  ?    increasing.
    -2                 -2 * -2 =  ?       I ask what comes next? 0
    -3                 -3 * -2 =  ?        ...what comes next?   2
    -4                 -4 * -2 =  ?        ...what comes next?   4
    -5                 -5 * -2 =  ?        ...what comes next?   6
     .                  .        .
     .                  .        .    With these patterns we support the idea
     .                  .        .    that a negative times a negative is
                                      positive.


Division by fractions:
Idea #3:  Pictures or fraction squares.
Example:  3 divided by 1/2

"3 divided by 1/2" can be interpreted to mean "how many 1/2 are in 3
wholes?"

Start with 3 wholes.
     -------     -------     -------
    |       |   |       |   |       |
    |       |   |       |   |       |
    |       |   |       |   |       |
     -------     -------     -------
Divide each whole in half.
     -------     -------     -------
    |       |   |       |   |       |
    |-------|   |-------|   |-------|
    |       |   |       |   |       |
     -------     -------     -------
There are 6 halves in 3.  So 3 divided by 1/2 equals 6.


Example:  3/4 divided by 1/2.

"3/4 divided by 1/2" can be interpreted to mean "how many 1/2 are in 3/4?"

If this is a whole,.........then this is a picture of 3/4.
     -------                  -------         ---
    |       |                |   |   |       |   |
    |       |                |---|---|  -->  |---|---
    |       |                |   |   |       |   |   |
     -------                  -------         -------

How many half's are in the 3/4?
     -------
    |       |
     -------
Answer:  1 complete half and 1/2 of a half.  or 1 1/2.


Idea #4:  Unit analysis.

Before we go on, we need to take a side trip into the analysis of the words,
numerator and denominator.  The word, numerator, comes from a Latin word
which means to count.  The word, denominator, comes from a Latin word which
means to name.  So, 3/4 tells us that we have 3 of a part called "fourth"
and can be written as 3 fourths.


Example:  3/4 divided by 1/2

     3 fourths
    ----------  Since these fractions don't have the same name, we need
     1 half     to change the name of at least one of them.


     3 fourths             3 fourths
    ----------  ----->    -----------   The units (or names) divide out...
     1 half                2 fourths

So, 3/4 divided by 1/2 equals 3/2  (or 1 1/2).


Example:  2/3 divided by 3/4.

     2 thirds           2 (4 twelfths)           8 twelfths
    ----------  -----> ----------------  -----> ------------
     3 fourths          3 (3 twelfths)           9 twelfths


So, 2/3 divided by 3/4 equals 8/9.

This method, of course, requires that the students have a great deal of
experience in equivalent fractions


Now back to your original problem... -2/3 divided by -3/4.  When my students
and I do a problem of this complexity, I ask them to break this problem down
into subproblems:  signs and division.  Then we combine the results of
solving the subproblems to get a final answer.

Hope this helps.

 -Jeanne, for the T2T service


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