Fraction multiplication using area models - practice worksheet with visual grids.

To solve the problems using the area models, we need to follow these steps:

1. Understand the Area Model: Each rectangle is divided into smaller sections, and the shaded areas represent fractions. The product of two fractions can be visualized by shading one fraction along one axis and the other fraction along the perpendicular axis. The overlapping shaded area represents the product.

2. Calculate the Product of Fractions: Multiply the numerators together and the denominators together to find the product of the fractions.

3. Verify with the Area Model: Check that the overlapping shaded area in the model corresponds to the calculated product.

Let's solve each problem step by step:

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Example (Ex):

\[
\frac{1}{3} \times \frac{1}{2} =
\]
- The first fraction, \(\frac{1}{3}\), is represented by shading 1 out of 3 columns.
- The second fraction, \(\frac{1}{2}\), is represented by shading 1 out of 2 rows.
- The overlapping shaded area is 1 out of 6 total sections.
- Therefore, \(\frac{1}{3} \times \frac{1}{2} = \frac{1}{6}\).

---

Problem 1:

\[
\frac{1}{2} \times \frac{3}{4} =
\]
- The first fraction, \(\frac{1}{2}\), is represented by shading 1 out of 2 columns.
- The second fraction, \(\frac{3}{4}\), is represented by shading 3 out of 4 rows.
- The overlapping shaded area is 3 out of 8 total sections.
- Therefore, \(\frac{1}{2} \times \frac{3}{4} = \frac{3}{8}\).

---

Problem 2:

\[
\frac{1}{5} \times \frac{2}{4} =
\]
- Simplify \(\frac{2}{4}\) to \(\frac{1}{2}\).
- The first fraction, \(\frac{1}{5}\), is represented by shading 1 out of 5 columns.
- The second fraction, \(\frac{1}{2}\), is represented by shading 1 out of 2 rows.
- The overlapping shaded area is 1 out of 10 total sections.
- Therefore, \(\frac{1}{5} \times \frac{2}{4} = \frac{1}{10}\).

---

Problem 3:

\[
\frac{3}{4} \times \frac{2}{3} =
\]
- The first fraction, \(\frac{3}{4}\), is represented by shading 3 out of 4 columns.
- The second fraction, \(\frac{2}{3}\), is represented by shading 2 out of 3 rows.
- The overlapping shaded area is 6 out of 12 total sections, which simplifies to \(\frac{1}{2}\).
- Therefore, \(\frac{3}{4} \times \frac{2}{3} = \frac{1}{2}\).

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Problem 4:

\[
\frac{6}{8} \times \frac{1}{2} =
\]
- Simplify \(\frac{6}{8}\) to \(\frac{3}{4}\).
- The first fraction, \(\frac{3}{4}\), is represented by shading 3 out of 4 columns.
- The second fraction, \(\frac{1}{2}\), is represented by shading 1 out of 2 rows.
- The overlapping shaded area is 3 out of 8 total sections.
- Therefore, \(\frac{6}{8} \times \frac{1}{2} = \frac{3}{8}\).

---

Problem 5:

\[
\frac{1}{3} \times \frac{1}{7} =
\]
- The first fraction, \(\frac{1}{3}\), is represented by shading 1 out of 3 columns.
- The second fraction, \(\frac{1}{7}\), is represented by shading 1 out of 7 rows.
- The overlapping shaded area is 1 out of 21 total sections.
- Therefore, \(\frac{1}{3} \times \frac{1}{7} = \frac{1}{21}\).

---

Problem 6:

\[
\frac{2}{6} \times \frac{1}{2} =
\]
- Simplify \(\frac{2}{6}\) to \(\frac{1}{3}\).
- The first fraction, \(\frac{1}{3}\), is represented by shading 1 out of 3 columns.
- The second fraction, \(\frac{1}{2}\), is represented by shading 1 out of 2 rows.
- The overlapping shaded area is 1 out of 6 total sections.
- Therefore, \(\frac{2}{6} \times \frac{1}{2} = \frac{1}{6}\).

---

Problem 7:

\[
\frac{2}{9} \times \frac{1}{4} =
\]
- The first fraction, \(\frac{2}{9}\), is represented by shading 2 out of 9 columns.
- The second fraction, \(\frac{1}{4}\), is represented by shading 1 out of 4 rows.
- The overlapping shaded area is 2 out of 36 total sections, which simplifies to \(\frac{1}{18}\).
- Therefore, \(\frac{2}{9} \times \frac{1}{4} = \frac{1}{18}\).

---

Problem 8:

\[
\frac{5}{7} \times \frac{1}{2} =
\]
- The first fraction, \(\frac{5}{7}\), is represented by shading 5 out of 7 columns.
- The second fraction, \(\frac{1}{2}\), is represented by shading 1 out of 2 rows.
- The overlapping shaded area is 5 out of 14 total sections.
- Therefore, \(\frac{5}{7} \times \frac{1}{2} = \frac{5}{14}\).

---

Problem 9:

\[
\frac{1}{2} \times \frac{2}{5} =
\]
- The first fraction, \(\frac{1}{2}\), is represented by shading 1 out of 2 columns.
- The second fraction, \(\frac{2}{5}\), is represented by shading 2 out of 5 rows.
- The overlapping shaded area is 2 out of 10 total sections, which simplifies to \(\frac{1}{5}\).
- Therefore, \(\frac{1}{2} \times \frac{2}{5} = \frac{1}{5}\).

---

Problem 10:

\[
\frac{5}{6} \times \frac{3}{6} =
\]
- Simplify \(\frac{3}{6}\) to \(\frac{1}{2}\).
- The first fraction, \(\frac{5}{6}\), is represented by shading 5 out of 6 columns.
- The second fraction, \(\frac{1}{2}\), is represented by shading 1 out of 2 rows.
- The overlapping shaded area is 5 out of 12 total sections.
- Therefore, \(\frac{5}{6} \times \frac{3}{6} = \frac{5}{12}\).

---

Problem 11:

\[
\frac{1}{3} \times \frac{1}{2} =
\]
- The first fraction, \(\frac{1}{3}\), is represented by shading 1 out of 3 columns.
- The second fraction, \(\frac{1}{2}\), is represented by shading 1 out of 2 rows.
- The overlapping shaded area is 1 out of 6 total sections.
- Therefore, \(\frac{1}{3} \times \frac{1}{2} = \frac{1}{6}\).

---

Final Answers:

\[
\boxed{
\begin{aligned}
1) & \quad \frac{3}{8} \\
2) & \quad \frac{1}{10} \\
3) & \quad \frac{1}{2} \\
4) & \quad \frac{3}{8} \\
5) & \quad \frac{1}{21} \\
6) & \quad \frac{1}{6} \\
7) & \quad \frac{1}{18} \\
8) & \quad \frac{5}{14} \\
9) & \quad \frac{1}{5} \\
10) & \quad \frac{5}{12} \\
11) & \quad \frac{1}{6}
\end{aligned}
}
\]