Students’ common errors in working with fractions

iStock_000013133930XSmallWhen teaching fractions, teachers need to be on the lookout for students’ common misconceptions that lead to errors in computation.

According to the panel of 8 experts that developed the recent What Works Clearinghouse Institute of Education Sciences practice guide, “Developing Effective Fractions Instruction for Kindergarten Through 8th Grade,” these are some of the most common misconceptions and some recommendations for how to best address them.

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 the relationship between the denominator, i.e. that the denominator is the number of equal parts into which one whole is divided and that the numerator signifies the number of those parts. The fact that numerators and denominators are essentially treated as whole numbers in  multiplication only adds to the confusion.

To overcome this misconception, present a real-world problem. Ask students a question like this: “If you have 3/4 of an orange and give 1/3 of it to a friend, what fraction of the original orange do you have left?” Subtracting the numerators and denominators separately would result in an answer of 2/1 or 2.  Students should immediately recognize that it is impossible to start with 3/4 of an orange, give some of it away, and end up with 2 oranges. Such examples help students see why treating numerators and denominators as separate whole numbers is inappropriate and will make them more receptive to appropriate procedures.

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).

The same underlying misconception can lead students to make a similar error: Changing the denominator of a fraction without making a corresponding change to the numerator—for example, converting the problem 2/3 + 2/6 to 2/6 + 2/6. Number lines and other visual representations that show equivalent fractions are very helpful.

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.

A related misconception is thinking that whole numbers have the same denominator as a fraction in the problem. This misconception might lead students to translate the problem 4 – 3/8 into 4/8 – 3/8 and find an answer of 1/8. When presented with a mixed number, students with such a misconception might add the whole number to the numerator, as in 31/3 × 6/7 = (3/3 + 1/3) × 6/7 = 4/3 × 6/7 = 24/21.

Helping students understand the relation between mixed numbers and improper fractions, and how to translate each into the other, is crucial for working with fractions.

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.

Teachers can address this misconception by explaining the conceptual basis of fraction multiplication using unit fractions (e.g., 1/2 × 1/2 = half of a half = 1/4). In particular, teachers can show that the problem 1/2 × 1/2 is actually asking what 1/2 of 1/2 is, which implies that the product must be smaller than either fraction being multiplied.

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.

Reciprocal vs. invert-and-multiply. The panel suggests that teachers help students understand the multi-step calculation that is the basis for the invert-and multiply procedure. Teachers can begin by noting that multiplying any number by its reciprocal produces a product of 1, and that dividing any number by 1 leaves the number unchanged. Then teachers can show students that multiplying both fractions by the reciprocal of the divisor is equivalent to using the invert-and-multiply procedure.

“Developing Effective Fractions Instruction for Kindergarten Through 8th Grade,” Sept 2010, by Robert Siegler et al. National Center for Education Statistics. Download report on effective fraction instruction.

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