Teaching Math: Shortcut for Comparing Fractions

Math students are just like the rest of us...they're always looking for shortcuts. Once students understand the concept of comparing fractions, there's no harm in teaching them a math shortcut for comparing fractions. I always save the "math tricks" for the end, after comprehension of a skill has occurred. The students react with, "Why didn't you just show us this shortcut in the first place?" Sorry, kids, it's called understanding.

Prerequisites
As with any math lesson, hands-on experience is always a good place to start. Let the students actually manipulate ¾ and ½ to discover that ¾ is greater. Download printables for maneuvering and comparing fractions at sites such as Donna Young.org. Once concrete examples have been experienced, teach students to compare fractions by finding equivalent fractions with common denominators (same bottom numbers). Then compare numerators (top numbers).

Compare 1/2 and 1/5.
1/2 = 5/10 and 1/5 = 2/10
When the denominators are the same, it's easy to deduce that 5/10 is greater than 2/10 by simply comparing numerators. 5 of something is more than 2 of something. Now we can see that 1/2 > 1/5.

If you have computer access, a convenient way for students to practice comparing fractions is "virtual manipulation". The National Library of Virtual Manipulatives includes an effective, interactive segment on comparing fractions. Students can see visual representations of fractions they're comparing. They can also see where the fractions land on a number line.

Alert
Don't assume that students understand the difference between the less than< and greater than > signs, even at the middle school level. They have often been taught in elementary school to identify these signs by thinking that the "alligator's mouth eats the greater number". Thus, the number on the open end of the sign is greater. This is helpful, but doesn't always transfer well when students try to read a math problem from left to right. For example, in the comparison 2/10 < 5/10, students will say, "Five-tenths is greater than two-tenths" rather than reading it left to right, "Two-tenths is less than five-tenths".

Shortcut
The shortcut for comparing fractions is to cross multiply numerators with diagonal denominators and compare the results. I teach students to record the products next to the numerators. It's important to read the results from left to right. Emphasize the question, "Is the first fraction greater than, less than, or equal to the second fraction?"

Examples
Compare 2/3 and 4/5
Multiply the numerator 2 by the diagonal denominator 5.
The result is 10. Record the result next to the 2.
Multiply the second numerator 4 by its diagonal denominator 3.
The result is 12. Record the result next to the 4.
Read from left to right.
10 is less than 12; therefore, 2/3 < 4/5.

Compare 5/6 and 3/4
Multiply the numerator 5 by the diagonal denominator 4.
The result is 20. Record the result next to the 5.
Multiply the numerator 3 by the diagonal denominator 6.
The result is 18. Record the result next to the 3.
Read from left to right.
20 is greater than 18; therefore, 5/6 > 3/4.

Compare 4/5 and 8/10
Multiply the numerator 4 by the diagonal denominator 10.
The result is 40. Record the result next to the 4.
Multiply the numerator 8 by the diagonal denominator 5.
The result is 40. Record the result next to the 8.
Read from left to right.
40 is equal to 40; therefore, 4/5 = 8/10.

Once students learn and practice the shortcut for comparing fractions, this method will save them lots of time in future computations.

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