## MSTAR Interventions

### Common Misconceptions and How to Prevent Them

#### Examples for Preventing or Correcting

Some students believe ratios always compare a part to a whole, like fractions.

Provide an example in which students compare a part to a part (e.g., 1 green apple to 3 red apples). Explain that ratios can compare part to part, not only part to whole. The ratio 1 to 3 compares part of the apples to another part of the apples.

Some students struggle to simplify ratios.

Demonstrate an example of simplification, explicitly describing the steps:
1. Check for common factors. (The easiest thing to remember is to use the prime numbers 2, 3, 5, 7, and 11.)
2. Divide each part of the ratio by the common factor.

Example:
Show students 8 red and 4 yellow counters.

Ask, “What is the ratio of red to yellow counters?” ()

Ask, “How do I know whether this ratio will simplify?” (Both numbers must divide by a common factor.)

Ask, “What divides into 8 and 4 evenly?” (2 and 4)

Say, “When 2 factors divide evenly, to simplify, choose the largest factor.” (4)

Say, “When I divide by 1 whole, the ratio stays the same. Because 4 is a common factor, I need to divide by  , which is 1 whole."

Ask, “What is 8 divided by 4? What is 4 divided by 4?” (2, 1)

Say, “So  becomes  when it is simplified.”

Ask, “What is the common factor for the ratio 10 to 15?” (5)

Some students believe that ratios and rates can always be simplified to a mixed number or a whole number.

Remind students that ratios and rates must compare 2 quantities; therefore, you often cannot simplify a ratio or a rate to a mixed number. Examples: 15 boys to 5 bats becomes 3 boys to 1 bat, not 3. 14 girls to 4 boys becomes 7 girls to 2 boys, not 3 .