Comparing Fractions

Post date: Oct 5, 2011 2:06:40 AM

At last week's MNM (say it, it's kind of funny), we discussed methods to compare fractions besides finding a common denominator. Why? Finding a common denominator is often unnecessary, too much work, and becomes procedural for students. Good mathematicians can solve problems multiple ways, so we have been exploring different ways to determine which fractions are larger than other fractions. For example:

1. Find a common numerator. Example: 3/7 is bigger than 3/8 since 7ths are bigger than 8ths. Imagine two equal-sized pizzas; one cut into 7 pieces, one into 8. The one cut into 7 pieces would have larger pieces.

2. Compare to 1/2 or other benchmark. Example: 3/7 is .5/7 from 1/2 since 1/2 = 3.5/7, 5/11 is .5/11 from 1/2 since 1/2 = 5.5/11. Since .5/11 is smaller than .5/7, 5/11 is bigger than 3/7 since it is closer to 1/2 and they are both less than 1/2.

3. Compare number of pieces and size of pieces. Example: 23/51 is bigger than 22/53 because it has more "pieces" (23 is bigger than 22) and it has bigger "pieces" (51sts are bigger than 53rds).

Many students have been using a "fishhook" or "bowtie" method they were taught in elementary school in which they "cross-multiply"--the numerator of one fraction is multiplied by the other's denominator and vice-versa. This method actually utilizes a common denominator, and I have stressed that my students understand that fact, and discouraged them from using it in general, because they do not understand why it works.

Other students have been drawing pictures of the fractions to compare them. I have been discouraging pictures because they are really hard to draw accurately, especially for unusual denominators like 7ths or 11ths. Please encourage your child to use numerical methods to compare fractions.