Improper Fractions

Keep counting past one whole

Cut a bar into 4 equal pieces. Each piece is a quarter of the bar, $\frac{1}{4}$1/4. Color the pieces in one at a time and count them: $\frac{1}{4}$1/4, $\frac{2}{4}$2/4, $\frac{3}{4}$3/4, $\frac{4}{4}$4/4. Four quarters fill the whole bar, so $\frac{4}{4}=1$4/4 = 1.

Now take a second bar, cut into quarters the same way, and keep counting: $\frac{5}{4}$5/4, $\frac{6}{4}$6/4, $\frac{7}{4}$7/4. The pieces are still quarters, so the denominator is still 4. The numerator counts how many quarters there are, and there are 7.

The count goes $\frac{1}{4}$1/4, $\frac{2}{4}$2/4, $\frac{3}{4}$3/4, $\frac{4}{4}$4/4 across the first bar, then carries on into the second bar: $\frac{5}{4}$5/4, $\frac{6}{4}$6/4, $\frac{7}{4}$7/4.

Each press adds one quarter and writes its count. After $\frac{4}{4}$4/4 the first bar is full, and the count carries on in the second: $\frac{5}{4}$5/4, $\frac{6}{4}$6/4, $\frac{7}{4}$7/4.

Improper fractions

One whole bar and 3 more quarters: $\frac{4}{4}+\frac{3}{4}=\frac{7}{4}$4/4 + 3/4 = 7/4. The numerator, 7, is bigger than the denominator, 4, and $\frac{7}{4}$7/4 is more than one whole.

A fraction whose numerator is equal to or bigger than its denominator is called an improper fraction. It is worth one whole or more. $\frac{4}{4}$4/4, $\frac{7}{4}$7/4 and $\frac{9}{4}$9/4 are improper fractions. A fraction whose numerator is smaller than its denominator, such as $\frac{3}{4}$3/4, is called a proper fraction, and it is less than one whole.

An improper fraction is an ordinary amount. $\frac{7}{4}$7/4 is seven quarters, and it has its own place on the number line, between 1 and 2.

On a number line cut into quarters, $\frac{3}{4}$3/4 is before 1, $\frac{4}{4}$4/4 is exactly 1, and $\frac{7}{4}$7/4 is past 1, three quarters of the way to 2.

Count the pieces in the whole too

How many fifths are there in one whole and $\frac{2}{5}$2/5 more? The whole, cut into fifths, is 5 fifths. The $\frac{2}{5}$2/5 adds 2 more. That is 5 + 2 = 7 fifths, so one whole and $\frac{2}{5}$2/5 is $\frac{7}{5}$7/5.

The usual mistake is to count only the 2 fifths that are drawn as pieces, or only the 5 in the whole. Both bars are part of the amount, so every piece in both of them is counted.

The whole bar holds 5 fifths and the second bar adds 2 more. Numbered in order, they come to 7: $\frac{7}{5}$7/5.