## Thursday, August 16, 2012

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.