Multiply Fractions with a Triangle Area Model

Problem

Use a triangle model to multiply fractions and find:

$$\frac{1}{2}\times \frac{2}{3}$$

Step 1: Start with One Whole

The complete triangle represents $1$ whole. Every fraction in the model will be compared with this original whole triangle.

Step 2: Shade the First Fraction

Divide the triangle into $3$ equal parts. Shade $2$ of those parts.

The shaded region represents:

$$\frac{2}{3}$$

of the triangle.

Step 3: Take a Fraction of the Shaded Part

Now take $\frac{1}{2}$ of the shaded region.

Since the shaded region is $\frac{2}{3}$ of the whole triangle, taking half of it leaves:

$$\frac{1}{3}$$

of the original whole triangle.

Step 4: Connect the Drawing to Multiplication

Taking $\frac{1}{2}$ of $\frac{2}{3}$ means multiplying:

$$\frac{1}{2}\times \frac{2}{3}$$

The triangle model shows that the result is:

$$\frac{1}{3}$$

Step 5: Use the Fraction Rule

To multiply fractions, multiply the numerators and multiply the denominators:

$$\frac{1}{2}\times \frac{2}{3} = \frac{1\times 2}{2\times 3} = \frac{2}{6}$$

Then simplify:

$$\frac{2}{6}=\frac{1}{3}$$

Step 6: State the Final Answer

Both the triangle model and the fraction multiplication rule give the same result:

$$\frac{1}{2}\times \frac{2}{3}=\frac{1}{3}$$