Multiplying fractions answers the question "what is a fraction of a fraction?" 2/3 x 3/4 means "2/3 of 3/4," and the result is 6/12 = 1/2. The algorithm is straightforward: multiply numerators and multiply denominators. But the conceptual meaning is crucial -- multiplying by a proper fraction makes the result smaller, which is counterintuitive for students used to multiplication making things bigger. The area model (a rectangle with fractional side lengths) makes this visual: 2/3 x 3/4 is the doubly-shaded region in a unit square, covering 6 of the 12 equal parts.
Use area models extensively: draw a rectangle, shade 3/4 horizontally, then shade 2/3 vertically. The overlap region shows the product. Practice with unit fractions first (1/2 x 1/3), then with non-unit fractions. Discuss why the product is smaller than either factor. Introduce simplifying before multiplying (cross-cancellation) as an efficiency tool after conceptual understanding is solid.
When you learned to multiply whole numbers, multiplication always made things bigger: 3 × 4 = 12, bigger than both 3 and 4. Fraction multiplication breaks this rule, and understanding *why* is the most important thing about this topic. Multiplying by a proper fraction (a fraction less than 1) shrinks the result. That is not a bug — it is what fraction multiplication means.
The phrase "a fraction of a fraction" captures it perfectly. When you compute 2/3 × 3/4, you are asking: what is 2/3 *of* 3/4? You already know how to find a fraction of a whole number (1/2 of 10 is 5). This is the same idea, extended: 2/3 of 3/4 is a piece of a piece. The answer, 6/12 = 1/2, is smaller than either 2/3 or 3/4 — which makes sense, because you took only part of something that was already less than one.
The area model makes this concrete. Draw a unit square (a square with side length 1). Shade 3/4 of it horizontally with one color. Now shade 2/3 of it vertically with another color. The region covered by *both* shadings is the product. Count the doubly-shaded pieces: 6 out of a total of 12 equal parts — so the product is 6/12. The algorithm — multiply numerators, multiply denominators — is just a shortcut for what the area model counts. The denominator multiplies because you are dividing the square into finer pieces; the numerator multiplies because you are taking a portion of those finer pieces.
The algorithm itself is simpler than the one for adding fractions: no common denominators needed. Multiply straight across. 2/3 × 5/7 = (2 × 5)/(3 × 7) = 10/21. That is it. After you get the product, check whether it simplifies. One efficiency trick is to simplify *before* multiplying — if a numerator and any denominator share a factor, you can cancel it early (called cross-cancellation) to keep the numbers small.
Finally, do a sanity check on your answer. If you multiplied two proper fractions and got a result *larger* than one of the originals, something went wrong. The product of two proper fractions is always between 0 and the smaller of the two factors. Using this as a quick check will catch most errors before you commit to a wrong answer.