Sharing the Slice: Dividing Fractions

Before we start slicing, let's look at our tools. A unit fraction is a fraction where the top number (the numerator) is always $1$. It represents exactly one equal part of a whole. Examples include $\frac{1}{2}$, $\frac{1}{4}$, or even $\frac{1}{100}$. When we divide a unit fraction by a whole number, we are taking a piece that is already small and cutting it into even smaller equal pieces. Because we are breaking a part into more parts, our answer will always have a larger denominator but represent a smaller value than what we started with.

Think of a rectangular cake. If you have $\frac{1}{3}$ of the cake left and you divide it into $2$ equal pieces, you are essentially changing the size of the pieces for the entire cake. If every third of the cake were cut into $2$ pieces, the whole cake would now have $3 \times 2 = 6$ pieces. Therefore, one of those new small pieces is $\frac{1}{6}$ of the original cake. To find your answer quickly, you can simply multiply the denominator of the fraction by the whole number.

Drawing a model helps confirm your math. Imagine a square representing '1 whole.' First, shade $\frac{1}{4}$ of it by drawing vertical lines. Next, draw a horizontal line across the middle to divide the whole square into $2$ equal rows. You will see that your original shaded $\frac{1}{4}$ section is now split into $2$ smaller boxes. If you count all the small boxes in the entire square, there are now $8$ total. This proves that $\frac{1}{4} \div 2 = \frac{1}{8}$.