Comparing fractions (age 8+)

A tape diagram is a long rectangle that stands for one whole. You split it into equal pieces to show a fraction. It is a great tool for seeing which fraction is bigger.

When two fractions have the same top number, they show the same count of pieces, but the pieces can be different sizes. The bottom number tells how many equal pieces make the whole. A bigger bottom number means smaller pieces.

For example, compare 2/3 and 2/5. Draw two rectangles the same size. Split the first into 3 parts and shade 2. Split the second into 5 parts and shade 2. The shaded part of the first rectangle is clearly longer, because thirds are bigger pieces than fifths. So 2/3 is bigger than 2/5.

Try it yourself: draw tape diagrams for 3/4 and 3/8. Which shaded part is longer? The one with fourths, so 3/4 > 3/8.

Fractions with the same top number are easy to compare once you know the trick.

Take 3/5 and 3/8. Both have 3 on top, so both show 3 pieces. Now look at the bottoms. Fifths are bigger pieces than eighths, because a whole cut into 5 pieces has bigger pieces than a whole cut into 8. Three big pieces is more than three small pieces, so 3/5 > 3/8.

The rule to remember: when the top numbers are the same, the fraction with the smaller bottom number is bigger. It feels backwards, but it makes sense. More cuts means smaller pieces.

Now try comparing 4/6 and 4/9. The tops match at 4. Sixths are bigger than ninths, so 4/6 > 4/9.

Comparing fractions means figuring out which one is greater.

Step 1: Look at the bottom numbers. Are they the same? If yes, the pieces are the same size, so just compare the top numbers. The bigger top number wins.

Step 2: If the bottoms are different, check the top numbers. If the tops are the same, the smaller bottom number wins, because its pieces are bigger.

Step 3: If neither the tops nor the bottoms match, draw two same sized rectangles. Split one to match the first fraction and the other to match the second. Shade the parts and see which shaded area is bigger.

Step 4: Write your answer with a symbol. The open side of the symbol points at the bigger fraction, like a hungry mouth eating the bigger one. So 5/6 > 2/6 and 3/8 < 3/4.

Try 2/3 and 2/7, then 4/9 and 7/9.

You can compare fractions by drawing two wholes the same size and seeing which shaded part is bigger.

Here is how it works. Take 4/6 and 4/9. Draw two same sized rectangles. Split the first into 6 equal parts and shade 4. Split the second into 9 equal parts and shade 4. Now line the rectangles up. The shaded part of the first rectangle reaches farther, so 4/6 is bigger than 4/9.

Pictures help because you can see the pieces. When you compare, always start with two wholes that are exactly the same size. If one whole is bigger, the picture will trick you.

A quick check: the fraction whose shaded part covers more of its whole is the greater fraction. Use a > or < symbol to record what you see, or = if the shaded parts match exactly.