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Q&A #3589


Multiplying fractions

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From: Pat Ballew (for Teacher2Teacher Service)
Date: Apr 04, 2000 at 18:08:52
Subject: Re: Multiplying fractions

First I want to suggest that you don't view multiplication of fractions as if
it is separate from all the other mathematical experiences of the student.
The foundation for teaching multiplication of fractions should have been a
constant process during the earliest introduction to multiplication and the
meaning of fractions.  Nothing that happens here should be a stark departure
from what they have understood about multiplication in the past, or you are,
in my opinion, doing damage rather than good.  Assuming all that is
foundation...

For a first introduction you can build a square out of a number of tiles
that has easy factors.  12x12 is usually a reasonable size and good to work
with..    With a ruler or strait edge that allows the student to partition
the square, illustrate that 2/3 or some other easy fraction can be divided
off along one edge and use the straight edge to separate out three rectangles
that are each 1/3 of the original amount (whether we think of this as 1/3 of
a unit square or 1/3 of 144 tiles, the process is the same, and that is worth
lots of talk with and between kids... multiplying fractions has no meaning
unless the "of what" has a meaning)

 Now repeat with another fraction in the perpendicular direction... 1/2 would
be an easy one.  Be sure to divide the entire square (now in three
rectangles) because we want the product of the denominators to be visible as
sixths of the square.   We have in one corner, 1/2 of 1/3 of the object we
divided.  Two of these would be 1/2 of 2/3 (which is also easy to show to be
2/2 of 1/3 making some abstract cancellation properties make sense later)
Have them draw a picture of each result after the division to lay a framework
for the figural approach and write out the results in proper fraction
notation.  We want the physical idea, the figural image, and the abstract
relationship to become associated so that in the future, the fractions will
stimulate a mental picture.

After several examples of these done by the teacher and the students and time
for students to talk and process this,,, you may want to move to a figural
model.   Now the square is just a representation and no divisions are shown.
The student draws in the division by thirds with  vertical lines, and the
same kind of conversation should occur as with the physical.  If you have
taken the time during  the physical manipulatives to have them draw a
"picture" of the physical model, they have already seen examples of what they
are doing, so we suggest that sometimes we can draw the picture without
actually having the physical model at all.
   At this early stage would be a good time to illustrate some BIG ideas that
will come up later... commutativity is easy to show  (does it matter which
fraction we use first to divide the shape??? try both ways) as is equivalent
fractions(If we take 4/6 of the shape is that different than taking 2/3 ?)
  Hopefully when you were multiplying 8x4 you used an area model as one of
the representations of multiplication so that that is already a familiar
idea, and when you introduced fractions you returned to the area model to
portion denominate pieces and enumerate the number that were required.. and
the idea of area and fractions are neither a new idea which distract from the
focus on unification of two familiar concepts, multiplication and fractions.

Good luck, I hope some of this rambling has helped give you some ideas to
work with.

 -Pat Ballew, for the Teacher2Teacher service

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