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

Relational Thinking and Fractions 19
she consistently constructed and transformed equations in ways that brought her closer to
the solution of the basic problem. Again, that is essentially what solving algebra
equations is all about.
Case 2: Partitive division
problem:
Our second case involves a sixth-grade boy, Keenan, who solved the following
“Two thirds of a bag of coffee weighs 2.7 pounds. How much would a whole
bag of coffee weigh?”
This problem involves partitive division and differs from the previous division
problem in that the goal is to find out how much per group rather than to find out how
many groups. Keenan’s strategy included the transformation of quantities for the purpose
of simplifying calculations as well as the flexible use of several fundamental properties of
operations and equality (Fig. 5).
--- Insert Fig. 5 about here ---
To start, Keenan recognized that the problem was a division problem and wrote
2.7 ÷ 2
3 . He remarked, “Two divided by
worry about is the seven tenths. Seven tenths divided by
2
3
is going to be really easy, all I really need to
2
3
isn’t easy to think about so
[long pause] if I make them both thirtieths, it would be easier.” He notated his thinking so
that it read:
21 20
÷
30 30