A fraction times a fraction

“Of” is multiplication: $\displaystyle \frac{2}{3}$ of 24 is $\displaystyle \frac{2}{3}\times24$. A part of a part is a multiplication too: half of $\displaystyle \frac{2}{3}$ of a cake is $\displaystyle \frac{1}{2}\times\frac{2}{3}$.

A rectangle 2 units wide and 3 units long has an area of $\displaystyle {2\times3}={6}$ square units. The same works for fractions: a rectangle $\displaystyle \frac{2}{3}$ of a unit wide and $\displaystyle \frac{1}{2}$ of a unit long has an area of $\displaystyle \frac{2}{3}\times\frac{1}{2}$.

The unit square is cut into $\displaystyle {3\times2}={6}$ parts, and the rectangle covers $\displaystyle {2\times1}={2}$ of them: $\displaystyle {\frac{2}{3}\times\frac{1}{2}}={\frac{2\times1}{3\times2}}={\frac{2}{6}}={\frac{1}{3}}$. Numerator times numerator, denominator times denominator.

You can reduce even before multiplying – also a numerator of one fraction with the denominator of the other, because they are all multiplied together: $\displaystyle {\frac{2}{9}\times\frac{3}{4}}={\frac{2\times3}{9\times4}}$. Divide 2 and 4 by 2, and 3 and 9 by 3: $\displaystyle {\frac{1\times1}{3\times2}}={\frac{1}{6}}$. That keeps the numbers small.

Write them as fractions first: $\displaystyle {1\frac{1}{2}\times2\frac{1}{2}}={\frac{3}{2}\times\frac{5}{2}}={\frac{15}{4}}={3\frac{3}{4}}$.

Multiplying by a number less than 1 makes a number smaller: $\displaystyle \frac{2}{3}\times\frac{1}{2}$ is less than $\displaystyle \frac{2}{3}$ – it is only part of it. Multiplying by a number more than 1 makes it bigger. Estimate before you calculate: $\displaystyle \frac{9}{10}\times4$ is a little less than 4.

Draw $\displaystyle \frac{3}{4}\times\frac{1}{2}$ as a part of a square, and write the result.

Multiply, and reduce the result.

Choose a number (a fraction or a whole number).
Write a multiplication of it whose result is smaller than your number, and one whose result is bigger – and solve both.