Ordering Fractions

Parts of the same whole

Put $\frac{3}{5}$3/5, $\frac{7}{8}$7/8 and $\frac{1}{4}$1/4 in order from least to greatest. All three are parts of the same whole, so picture them as lengths of the same bar.

Tool one is the fraction wall. Each row is the same whole bar, cut into halves, quarters, fifths or eighths. Shade $\frac{1}{4}$1/4 on the quarters row, $\frac{3}{5}$3/5 on the fifths row and $\frac{7}{8}$7/8 on the eighths row. The shading of $\frac{1}{4}$1/4 is plainly the shortest, so $\frac{1}{4}$1/4 is the least of the three.

Every row is the same whole. The shading of $\frac{1}{4}$1/4 is the shortest.

Tool two: the half check

A half is a landmark to compare with. Double the top number and compare it with the bottom number. For $\frac{1}{4}$1/4, 2 × 1 = 2, which is less than 4, so $\frac{1}{4}$1/4 is less than a half.

For $\frac{3}{5}$3/5, 2 × 3 = 6, which is more than 5, so $\frac{3}{5}$3/5 is more than a half. For $\frac{7}{8}$7/8, 2 × 7 = 14, which is more than 8, so $\frac{7}{8}$7/8 is more than a half too. A fraction below a half is smaller than any fraction above it, so $\frac{1}{4}$1/4 comes first.

The bar starts at $\frac{1}{4}$1/4, and its shading stops before the half line. Cut the bar into 5 parts and shade 3 of them: $\frac{3}{5}$3/5 crosses the line.

Tool three: read the line

The half check puts $\frac{3}{5}$3/5 and $\frac{7}{8}$7/8 on the same side of a half, so it cannot tell which of the two is greater. Place all three on a number line from 0 to 1 instead.

$\frac{7}{8}$7/8 is one eighth short of 1, and $\frac{3}{5}$3/5 is two fifths short of 1. One eighth is smaller than one fifth, because a whole cut into more parts has smaller parts, so it is certainly smaller than two fifths. $\frac{7}{8}$7/8 is nearer to 1, and it is the greater. From least to greatest, the order is $\frac{1}{4}$1/4, $\frac{3}{5}$3/5, $\frac{7}{8}$7/8.

$\frac{1}{4}$1/4 falls short of a half, $\frac{3}{5}$3/5 and $\frac{7}{8}$7/8 pass it, and $\frac{7}{8}$7/8 is the nearest to 1.

Same bottom or same top

Two fractions that share a number can be ordered at once. With the same bottom number the parts are the same size, so compare the top numbers: $\frac{5}{8}$5/8 is less than $\frac{7}{8}$7/8. With the same top number there are the same number of parts, so the smaller bottom number has the bigger parts: $\frac{3}{5}$3/5 is more than $\frac{3}{8}$3/8.

Use whichever tool settles each pair, and only look further when a tool gives a tie.

Not by the bottom numbers

The usual mistake is to order fractions by their bottom numbers. For $\frac{1}{4}$1/4, $\frac{3}{5}$3/5 and $\frac{7}{8}$7/8 that happens to give the right order, but for $\frac{2}{3}$2/3, $\frac{1}{4}$1/4 and $\frac{1}{2}$1/2 it gives $\frac{1}{2}$1/2, $\frac{2}{3}$2/3, $\frac{1}{4}$1/4, and $\frac{1}{4}$1/4 is the least of all three.

The half check puts them right. $\frac{1}{4}$1/4 is less than a half, $\frac{1}{2}$1/2 is exactly a half, and $\frac{2}{3}$2/3 is more than a half, because 2 × 2 = 4 is more than 3. From least to greatest: $\frac{1}{4}$1/4, $\frac{1}{2}$1/2, $\frac{2}{3}$2/3.

On the wall, $\frac{1}{4}$1/4 has the shortest shading and $\frac{2}{3}$2/3 the longest.