Believing that fractions' numerators and
denominators can be treated as separate whole numbers. Students often add or subtract the numerators and
denominators of two fractions (e.g., 2/4 + 5/4 = 7/8 or 3/5 – 1/2 = 2/3). These
students fail to recognize that denominators define the size of the fractional
part and that numerators represent the number of this part. The fact that this
approach is used for multiplication of fractions is another source of
confusion.
Failing to find a common denominator when adding
or subtracting fractions with unlike denominators. Students often fail to convert fractions to a common,
equivalent denominator before adding or subtracting them, and instead just use
the larger of the 2 denominators in the answer (e.g., 4/5 + 4/10=8/10).
Students do not understand that different denominators reflect different-sized
unit fractions and that adding and subtracting fractions requires a common unit
fraction (i.e. denominator).
Believing that only whole numbers need to be
manipulated in computations with fractions greater than one. When adding or subtracting mixed numbers, students
may ignore the fractional parts and work only with the whole numbers (e.g.,
53/5 – 21/7 = 3). These students are either ignoring the part of the problem
they do not understand, misunderstanding the meaning of mixed numbers, or
assuming that such problems simply have no solution.
Leaving the denominator unchanged in
fraction addition and multiplication problems.
Students often leave the denominator unchanged on fraction multiplication
problems that have equal denominators (e.g., 2/3 × 1/3 = 2/3). This may occur
because students usually encounter more fraction addition problems than
fraction multiplication problems. They incorrectly apply the correct procedure
for dealing with equal denominators on addition problems to multiplication.
Failing to understand the invert-and-multiply
procedure for solving fraction division problems. Students often misapply the invert-and-multiply
procedure for dividing by a fraction because they lack conceptual understanding
of the procedure. One common error is not inverting either fraction; for
example, a student may solve the problem 2/3 ÷ 4/5 by multiplying the fractions
without inverting 4/5 (e.g., writing that 2/3 ÷ 4/5 = 8/15). Other common
misapplications of the invert-and-multiply rule are inverting the wrong
fraction (e.g., 2/3 ÷ 4/5 = 3/2 × 4/5) or inverting both fractions (2/3 ÷ 4/5 =
3/2 × 5/4). Such errors generally reflect a lack of conceptual understanding of
why the invert-and-multiply procedure produces the correct quotient. The
invert-and-multiply procedure translates a multi-step calculation into a more
efficient procedure.
