One of the five recommendations put
forth by the Institute for Educational Studies in Developing Effective Fractions Instruction for Kindergarten Through 8th
Grade (2010) is, “to help students understand why procedures for
computations with fractions makes sense” (p. 5). One way to help students make
sense of procedures for computations with fractions is to leverage procedures
they may already understanding for computations with whole numbers. The
argument is that fractions represent numbers and the reasoning to solve
problems involving operations with fractions is closely related to the
reasoning to solve problems involving operations with whole numbers because both
problems involve numbers. For example, “addition is addition” whether the
problem involves whole numbers or fractions. That is, addition is a process of
defining numerically the amount of objects in two or more sets of like objects
with same unit (e.g 9 apples and 4 apples is 13 apples). The purpose of this first blog is to make a clear link with addition. The necessary
understandings of fractions as well as important terminology will be highlighted.
Depending
on the operation and the specific situation it is important to understand the
different possible meanings of a fraction. In some instances it is important to
interpret the fraction 3/4 as a scalar. For example, is
3 copies of 1/4 or 3(1/4),
in which the 1/4,
or fourth, is the iterating unit—that is 1/4 is repeated 3 times. In other instances the
fraction 3/4 is interpreted as an operation. For example,
in the abstract problem 3/4 * 16, the fraction 3/4 can be thought of through the lens of partitioning
16 into 4 equal groups {4} and choosing 3 of those equal groups {3 equal groups
of 4 = 12}.
Consider
the following addition problems.
Jenny has 4 whole apples and her friend Sue has 9 whole apples. Between the two of them how many total whole apples do they have?
Jenny has 4 whole apples and her friend Sue has 9 whole apples. Between the two of them how many total whole apples do they have?
Jenny has 1/3 of a whole pound of cheese and her friend
Sally has 3/4 of a
whole pound of cheese. Between
the two of them how much cheese do they have?
The operation of addition is about describing numerically the sum of the number of elements in two or more sets of the same objects sharing the same iterative unit. For example, to answer the question “How many total whole apples do they have?,” the iterative unit is 1 whole and apple is the object. In performing the computation the action of addition involves combining 4 copies of 1 whole apple with 9 copies of 1 whole apple for a total of 13 copies of 1 whole apple or 13 apples.
The
critical aspect of addition in contextual situations is that not only is the
object is the same, but so is the iterative unit. These ideas are lost
sometimes when discussing de-contextualized problems such as 1 + 1/2. The answer 1 and 1/2 is describing the numerically the combining
action of 1 copy of 1 whole unit with 1/2 copy
of 1 whole unit yielding 1 and 1/2 copies of 1 whole unit or 1 1/2 whole units. It seems important to mention
that 1 + 1/2 can also be thought of as the sum of 2 copies
of 1/2 whole unit and 1 copy of 1/2 whole unit which is 3 copies of 1/2 whole units or 3(1/2) or 3 halves.
While thirds and fourths are
different iterative unit preventing us from completing the combining action of addition
it is possible to re-define each of those fractions in terms of the same
iterative unit. That is, 1 third is the same as 4 twelfths or 1 copy of 1/3 is the same as 4 copies of 1/12 .
Likewise 3 fourths is the same as 9 twelfths or 3 copies of 1/4 is the same as 9 copies of 1/12.
Now that both fractions are
expressed in terms of the same iterative unit the combining action of addition showing
the sum of 4 twelfths and 9 twelfths as 13 twelfths i.e. --An
analogous solution to the whole number addition problem.
