Comparing Fractions

There are many ways to compare fractions.  Some methods are quick and easy.  But a lot of methods require a lot of math.  The best way to compare a group of fractions is to figure out the best method for any given set of fractions (and, in a large set, you might employ different methods to compare different specific fractions!)

NOTE: There are many ways to think about fractions and the steps below give you some great starting points.  BUT, if a student is having a hard time thinking about fractions in many different ways, teach them to find common denominators. They will need and use that process as they start doing computation with fractions and will allow students to compare ANY fractions.  It can be more work than some of the other methods, but it's reliable and useful.  So, for some students, common denominators are the best way to go!

When comparing fraction there are a few questions you want to ask:

1. Are any of these fractions over 1 whole? (If the numerator is greater than the denominator, it's greater than 1 whole.)
2. Are the fractions greater than one half? (A fraction is equal to 1 half if the numerator is half of the denominotor.  If the numerator is greater than half of the denominator, the fraction is greater than one half.)
3. Do the fractions have the same denominator? (You can always compare fractions with the same denominator.  If the denominator is the same, just compare the numerators, $\displaystyle{\frac {3}{7}}$ will always be less than $\displaystyle{\frac{4}{7}}$.)
4. Do the fractions have the same numerator? (The larger the denominator, the smaller the fraction, so if the numerators are the same, you can compare.  Because something cut into 6ths results in smaller pieces than something cut into 5ths, $\displaystyle{\frac{4}{6}}$ will always be less than $\displaystyle{\frac{4}{5}}$.)
5. Do the fractions convert into obvious decimals? (Remember: $\displaystyle{\frac{1}{4}=.25, \frac{1}{2}=.5,\frac{3}{4}=.75}$, etc.)
6. Can you easily put convert the fractions to a common denominator? (If the denominators are multiples and/or factors of each other, this can be easy!)
7. Remember, every fraction is a division problem, and you can always divide to find a decimal and compare! ($\displaystyle{\frac{1}{6}=1\div6=.1667}$.)

Example: Order the fractions from greatest to least: $\displaystyle{\frac{3}{6}, \frac {10}{9}, \frac{1}{4}, \frac{7}{8}, \frac{1}{8}}$

Go through the questions above:

1. Are any fractions over 1 whole?  Only $\displaystyle{\frac{10}{9}}$ has a numerator greater than a denominator, so $\displaystyle{\frac{10}{9}}$ is the greatest fraction in this list.
2. Are there any fractions over one half? $\displaystyle{\frac{3}{6}}$ is equal to one half. $\displaystyle{\frac{7}{8}}$ is greater than one half.  So now we have is: $\displaystyle{\frac{10}{9}, \frac{7}{8}, \frac{3}{6}}$
3. Do any fractions have the same denominator? Yes, but based on the other questions, we already know that $\displaystyle{\frac{1}{8}}$ is lesss than $\displaystyle{\frac{7}{8}}$.
4. Do the fractions have the same numerator? Yes, the last two fractions have the same numerator! $\displaystyle{\frac{1}{4}}$ must be greater than $\displaystyle{\frac{1}{8}}$ because there is one piece of each, and something cut into fourths is cut into much bigger pieces than something cut into eights. 

The final order is: 

$\displaystyle{\frac{10}{9}, \frac{7}{8},\frac{3}{6},\frac{1}{4},\frac{1}{8}}$

Sets of fractions can be hard to compare or easy to compare -- but very often, going through the steps above will help you order at least a few of the fractions, cutting down on the work you have to do to put the rest in common denominators, or convert them into decimals.