Area Model Multiplication Worksheets - Free Printable

Problem Description:

The task involves solving fraction multiplication problems and using area models to represent the solutions. The goal is to match each multiplication problem with its corresponding area model.

Steps to Solve:

#### 1. Understanding Fraction Multiplication:
- When multiplying two fractions, you multiply the numerators together and the denominators together.
- For example, if you have \(\frac{a}{b} \times \frac{c}{d}\), the result is \(\frac{a \times c}{b \times d}\).

#### 2. Using Area Models:
- An area model represents the multiplication of fractions by dividing a rectangle into smaller parts.
- The total area of the rectangle represents the product of the two fractions.
- Each fraction is represented by shading a portion of the rectangle.

#### 3. Solving Each Problem:
Let's solve each problem step by step and match it with the correct area model.

---

Problem 1: \(\frac{1}{2} \times \frac{1}{3}\)

- Multiplication:
\[
\frac{1}{2} \times \frac{1}{3} = \frac{1 \times 1}{2 \times 3} = \frac{1}{6}
\]
- Area Model:
- Divide the rectangle into 2 equal vertical parts (for \(\frac{1}{2}\)).
- Divide the rectangle into 3 equal horizontal parts (for \(\frac{1}{3}\)).
- The intersection of these divisions creates 6 smaller rectangles.
- Shade 1 out of the 6 smaller rectangles.
- Match: This corresponds to the area model where 1 out of 6 parts is shaded.

---

Problem 2: \(\frac{2}{3} \times \frac{1}{4}\)

- Multiplication:
\[
\frac{2}{3} \times \frac{1}{4} = \frac{2 \times 1}{3 \times 4} = \frac{2}{12} = \frac{1}{6}
\]
- Area Model:
- Divide the rectangle into 3 equal vertical parts (for \(\frac{2}{3}\)).
- Divide the rectangle into 4 equal horizontal parts (for \(\frac{1}{4}\)).
- The intersection of these divisions creates 12 smaller rectangles.
- Shade 2 out of the 12 smaller rectangles (or simplify to 1 out of 6).
- Match: This corresponds to the area model where 2 out of 12 parts are shaded.

---

Problem 3: \(\frac{3}{4} \times \frac{1}{2}\)

- Multiplication:
\[
\frac{3}{4} \times \frac{1}{2} = \frac{3 \times 1}{4 \times 2} = \frac{3}{8}
\]
- Area Model:
- Divide the rectangle into 4 equal vertical parts (for \(\frac{3}{4}\)).
- Divide the rectangle into 2 equal horizontal parts (for \(\frac{1}{2}\)).
- The intersection of these divisions creates 8 smaller rectangles.
- Shade 3 out of the 8 smaller rectangles.
- Match: This corresponds to the area model where 3 out of 8 parts are shaded.

---

Problem 4: \(\frac{1}{3} \times \frac{1}{2}\)

- Multiplication:
\[
\frac{1}{3} \times \frac{1}{2} = \frac{1 \times 1}{3 \times 2} = \frac{1}{6}
\]
- Area Model:
- Divide the rectangle into 3 equal vertical parts (for \(\frac{1}{3}\)).
- Divide the rectangle into 2 equal horizontal parts (for \(\frac{1}{2}\)).
- The intersection of these divisions creates 6 smaller rectangles.
- Shade 1 out of the 6 smaller rectangles.
- Match: This corresponds to the area model where 1 out of 6 parts is shaded.

---

Problem 5: \(\frac{2}{5} \times \frac{1}{3}\)

- Multiplication:
\[
\frac{2}{5} \times \frac{1}{3} = \frac{2 \times 1}{5 \times 3} = \frac{2}{15}
\]
- Area Model:
- Divide the rectangle into 5 equal vertical parts (for \(\frac{2}{5}\)).
- Divide the rectangle into 3 equal horizontal parts (for \(\frac{1}{3}\)).
- The intersection of these divisions creates 15 smaller rectangles.
- Shade 2 out of the 15 smaller rectangles.
- Match: This corresponds to the area model where 2 out of 15 parts are shaded.

---

Problem 6: \(\frac{3}{4} \times \frac{2}{3}\)

- Multiplication:
\[
\frac{3}{4} \times \frac{2}{3} = \frac{3 \times 2}{4 \times 3} = \frac{6}{12} = \frac{1}{2}
\]
- Area Model:
- Divide the rectangle into 4 equal vertical parts (for \(\frac{3}{4}\)).
- Divide the rectangle into 3 equal horizontal parts (for \(\frac{2}{3}\)).
- The intersection of these divisions creates 12 smaller rectangles.
- Shade 6 out of the 12 smaller rectangles (or simplify to 1 out of 2).
- Match: This corresponds to the area model where 6 out of 12 parts are shaded.

---

Final Matches:

1. \(\frac{1}{2} \times \frac{1}{3} = \frac{1}{6}\) → Match with the area model where 1 out of 6 parts is shaded.
2. \(\frac{2}{3} \times \frac{1}{4} = \frac{1}{6}\) → Match with the area model where 2 out of 12 parts are shaded.
3. \(\frac{3}{4} \times \frac{1}{2} = \frac{3}{8}\) → Match with the area model where 3 out of 8 parts are shaded.
4. \(\frac{1}{3} \times \frac{1}{2} = \frac{1}{6}\) → Match with the area model where 1 out of 6 parts is shaded.
5. \(\frac{2}{5} \times \frac{1}{3} = \frac{2}{15}\) → Match with the area model where 2 out of 15 parts are shaded.
6. \(\frac{3}{4} \times \frac{2}{3} = \frac{1}{2}\) → Match with the area model where 6 out of 12 parts are shaded.

---

Final Answer:

\[
\boxed{
\begin{array}{c|c}
\text{Problem} & \text{Matched Area Model} \\
\hline
1. \frac{1}{2} \times \frac{1}{3} & \text{(1 out of 6 shaded)} \\
2. \frac{2}{3} \times \frac{1}{4} & \text{(2 out of 12 shaded)} \\
3. \frac{3}{4} \times \frac{1}{2} & \text{(3 out of 8 shaded)} \\
4. \frac{1}{3} \times \frac{1}{2} & \text{(1 out of 6 shaded)} \\
5. \frac{2}{5} \times \frac{1}{3} & \text{(2 out of 15 shaded)} \\
6. \frac{3}{4} \times \frac{2}{3} & \text{(6 out of 12 shaded)} \\
\end{array}
}
\]