Skill: Use area models for fraction multiplication

To multiply fractions, use a rectangle as an area model. Each fraction describes a length of the rectangle:

• The denominator tells how many equal parts to divide a side into.
• The numerator tells how many of those parts to use.
• The overlapping shaded parts show the product.

Example: Find $\frac{2}{3}\times\frac{3}{4}$

1. Draw a rectangle.

This rectangle represents $1$ whole.

2. Show $\frac{3}{4}$.

Divide the rectangle into $4$ equal columns. Shade $3$ of the $4$ columns.

3. Show $\frac{2}{3}$.

Divide the same rectangle into $3$ equal rows. Shade $2$ of the $3$ rows in a different direction.

4. Count the overlapping parts.

The rectangle now has

$$3\times 4=12$$

equal small parts.

The two shaded regions overlap in

$$2\times 3=6$$

of those parts.

So,

$$\frac{2}{3}\times\frac{3}{4}=\frac{6}{12}=\frac{1}{2}.$$

The answer is $\boxed{\frac{1}{2}}$.

This also means “find $\frac{2}{3}$ of $\frac{3}{4}$.” Multiplication finds the overlapping area; it is different from adding the fractions.