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Q&A #1493

Teachers' Lounge Discussion: Dividing fractions

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From: Marielouise 
To: Teacher2Teacher Public Discussion
Date: 2000052415:24:51
Subject: Re: Division and Reciprocals

Hi, Jennifer,

How you define a reciprocal depends upon how old, or how
mathematically mature your student is.  A reciprocal of a number is
itself a number so thatthe product of the reciprocal and the number is
one.  Younger students will understand a reciprocal of a number # as 1
over # or  1/number.  Sometimes students view a reciprocal as the
number turned upside down.  For example if the # is 2/3, then 1/# is
1/(2/3) or 3/2.

Why do we use reciprocals?  Again this depends upon the maturity of
the
student.  Think about addition.  When we wish to "undo" addition we
subtract which is the opposite of addition.  When we multiply we are
doing repetitive addition.  That is, 3 x 4 is adding 4 three times or
4 + 4 + 4.   If we divide 4 by 3 we are looking for three number # so
that when the number is added three times we arrive at 4.  Therefore,
# + # + # = 4.  Since the numbers are all equal the # is 4/3 or one
and one-third.  Think of taking 1/3 of 4.  Is this not dividing it
into three equal parts so that the sum is 4?

I think it makes sense to teach division by teaching multiplication;
for example, what is 3 x 4 to get twelve and then asking the children
to divide 12 into 3 groups where each group has the same number in it.
Do your students know their time-tables?  Knowing them makes division
much easier.

A teaching example for dividing 4 by 3 is to have four rectangles of
equalsize.  To divide these into three piles with the same amount of
area in each requires that one rectangle goes into each of the three
piles and then the fourth rectangle be cut into thirds.

A nice example for dividing 3 by 4 is to take three rectangles. One
way is to take 1/4 off of each rectangle so that each of the four
piles has the same amount of area. A second way is to cut each of the
rectangles into 1/4 of the area.  You will then have twelve pieces,
each of which is of the size "1/4."  The twelve pieces can be placed
three in each of the four piles.   Each pile has 3 "1/4" or 3/4.

I hope that these ideas have helped you.

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