Parts of a Part: Multiplying Two Fractions

When we multiply whole numbers, like $2 \times 3$, the answer gets bigger. But when we multiply a fraction by a fraction, we are finding a 'part of a part.' Imagine a square representing one whole. If we want to find $\frac{1}{2} \times \frac{1}{2}$, we first split the square in half vertically. Then, we split it in half horizontally. The spot where the two halves overlap is our answer! Because we split the whole square into 4 equal pieces, that one overlapping piece represents $\frac{1}{4}$ of the whole.

Drawing models is helpful, but there is a faster way! To multiply fractions, you simply follow the Straight Across Rule. You don't need a common denominator like you do for addition. You just multiply the numerators (top numbers) to get the new numerator, and multiply the denominators (bottom numbers) to get the new denominator. This works because the denominators determine the total number of pieces in our area model grid.

This is a concept called scaling. When you multiply a number by a fraction less than 1, you are taking only a portion of that number. Think of it like this: if you have $\frac{1}{2}$ of a dollar, you have 50 cents. If you take $\frac{1}{2}$ of that 50 cents, you only have 25 cents. The product $\frac{1}{4}$ is smaller than both of the original fractions we started with!