The Algebraic Nature of Fractions: Developing Relational Thinking ...

Relational Thinking and Fractions 15
these classrooms are rare – and provide a study of the possibility of integrating fractions
and algebra in the upper elementary grades.
Each problem involved division with a remainder to be taken into account in the
quotient. For each case, we describe the strategy and then note how children used
fundamental properties of operations and equality in their solutions.
Case 1: Measurement division.
The first case comes from a combination fourth- and fifth-grade class, working on
the following measurement division problem:
“It takes ____ of a cup of sugar to make a batch of cookies. I have 5
sugar. How many batches of cookies can I make?”
The students were given a variety of number choices for the divisor. In order of
difficulty, these choices were
work with
3
8 of a cup of sugar.
1
2 ,
1
4 ,
3
4 , and
1
2
cups of
3
8 . Several students, including Jill, chose to
Jill began her strategy by drawing upon the basic multiplicative relationship
described above to generate familiar groupings of three eighths that would simplify the
calculation (Table 1). She said she knew that 8 three-eighths would be 3, which meant
that 4 three-eighths would be half that much, or 1
be 4
1 , and 12 three-eighths would therefore
2
1
1
(Fig. 4). At this point, she knew that she needed only 1 more cup to use up all 5
2 2
cups. Again Jill used the relationship between
3
8
and 3 as a reference point. She said that
because 8 three-eighths was 3, a third as many three-eighths would be a third as much, or
1. That is, ( 1
3
3 × 8) × 8
12 + 8
3
1 is 1, and 3 × 8 is
batches, which would be equal to 14
8
3 . She concluded that she could make a total of
2
3 batches.