## MSTAR Interventions

### Common Misconceptions and How to Prevent Them

#### Examples for Preventing or Correcting

Some students confuse the terms numerator and denominator.

Teach students what numerators and denominators represent.

Some students believe that fractions must be less than 1. They believe that the numerator must be less than the denominator; otherwise, there would not be a sufficient number of pieces.

• Explain to students that fractions can represent any position on a number line. Use fractions equal to and greater than 1 as examples when appropriate.
• Teach students that rational numbers include fractions. Teach students that rational numbers include whole numbers.
• Teach students that of a whole is 5 parts of the unit fraction .

Some students count the small dividing marks (tick marks) on a number line rather than the intervals.

Draw thin ovals over each interval. Another method is to have students place their finger on 0, slide their finger to the right and count aloud each fractional part as they read each tick mark.

Some students believe that fractions are equivalent only when they look identical or have the same number of selected parts.

• Teach students that equivalent fractions occupy the same position on the number line.
• Teach students that fractions are parts of a whole, so the size of the fraction depends on the size of the whole.

Some students believe that performing an operation always changes the value of the original quantity, even when multiplying or dividing by 1.

• Teach students that multiplying or dividing by 1 does not change the original quantity or its position on the number line, including when the operation is performed on a fraction. Use manipulatives as appropriate.
• Because students may be familiar with the identity property of multiplication when working with whole numbers, make the connection between using the property when working with whole numbers (a subset of rational numbers) and rational numbers (fractions).
• Teach students that the identity property applies when the original number is a fraction and when 1 is written in fractional form.

Some students perform an operation only on the numerator when working with fractions.

Teach students that performing an operation only on the numerator is dissimilar to applying the multiplicative identity property and results in changing the fractional value and its position on the number line. Use manipulatives as necessary.

Some students erroneously use additive instead of multiplicative reasoning when identifying or generating equivalent fractions. For example, given , these students add 8 to both the numerator and the denominator to get , rather than multiplying both the numerator and the denominator by 8 (multiplying the whole fraction by 1) to get .

• Teach students that performing an operation only on the numerator is dissimilar to applying the multiplicative identity property and results in changing the fractional value and its position on the number line. Use manipulatives as necessary.
• Teach students to contrast the effects of adding with the effects of multiplying. Teach students that addition can be written as multiplication. For example, 6 + 6 = 12 could also be written as 6 x 2 = 12.
• Use manipulatives as appropriate and relate the concept to when students were introduced to equivalent fractions by folding paper.