Students must understand fraction concepts deeply in order to compare them with understanding. First we compare fractions concretely, using the same physical models we used in other fraction lessons.
Students should also have a chance to practice drawing pictures to compare fractions. Again, I recommend that students draw rectangles or number lines since it is difficult to draw accurate fractions of a circle. *Note: It’s important that students realize that the whole, the entire rectangle, must be the same size for each fraction.
Finally, students compare fractions abstractly. After having a lot of practice with fractions, students should be able to reason out many of them mentally:
Fractions with the same denominator. For example, comparing ⅖ and ⅘ is easy. Both fractions are comparing equal parts, fifths. 4 parts must be greater than 2 parts. Therefore, ⅘ is greater than ⅖.
Fractions with the same numerator. For example, comparing ⅓ and ⅕ is easy. To make the first fraction, the whole was cut into three equal pieces. Since the wholes of both fractions are the same size, the thirds must be bigger than the fifths. Therefore, since you are only comparing one of the thirds and one of the fifths, ⅓ must be greater than ⅕.
Using benchmark numbers. Once students can easily change fractions into benchmark fractions such as ½, comparing many fractions becomes easier. For example students can reason that 2/7 must be less than a half because 2 is less than half of 7. Similarly, students can reason that 5/7 must be greater than a half because 5 is more than half of 7. It is also possible to use this reasoning strategy to compare ⅗ and 4/9. Students can see that ⅗ is more than a half because 3 is more than half of 5. They should also be able to tell that 4/9 is less than a half because 4 is less than half of 9. Therefore, ⅗ is greater than 4/9.
Fourth graders will not be able to mentally compare all fractions, of course! In these cases, I encourage students to use physical materials, draw pictures, and/or find common denominators or common numerators: