Math worksheet teaching fraction multiplication using the area model, featuring examples and exercises with grid diagrams.

Problem Overview:

The task involves multiplying fractions using the area model. The area model visually represents each fraction by shading parts of a grid, and the overlap of these shaded areas represents the product of the two fractions.

Steps to Solve:

1. Understand the Area Model:
- Each fraction is represented by shading a portion of a grid.
- One fraction is shaded vertically, and the other is shaded horizontally.
- The overlapping shaded area represents the product of the two fractions.

2. Solve Each Problem:
- For each problem, determine the product of the given fractions.
- Use the area model to visualize the multiplication.

3. Answer the Final Question:
- Reflect on what the area model teaches about multiplying fractions.

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Solutions to Each Problem:

#### Example:
- Fractions: \( \frac{1}{2} \times \frac{2}{3} \)
- Grid: A 2x3 grid.
- Shading:
- Vertically shade \( \frac{1}{2} \) (1 column out of 2).
- Horizontally shade \( \frac{2}{3} \) (2 rows out of 3).
- Overlap: 2 squares are shaded in the overlap.
- Product: \( \frac{2}{6} \) (simplified to \( \frac{1}{3} \)).

#### Problem 1: \( \frac{2}{3} \times \frac{1}{4} \)
- Grid: A 3x4 grid.
- Shading:
- Vertically shade \( \frac{2}{3} \) (2 columns out of 3).
- Horizontally shade \( \frac{1}{4} \) (1 row out of 4).
- Overlap: 2 squares are shaded in the overlap.
- Product: \( \frac{2}{12} \) (simplified to \( \frac{1}{6} \)).

#### Problem 2: \( \frac{3}{4} \times \frac{1}{2} \)
- Grid: A 4x2 grid.
- Shading:
- Vertically shade \( \frac{3}{4} \) (3 columns out of 4).
- Horizontally shade \( \frac{1}{2} \) (1 row out of 2).
- Overlap: 3 squares are shaded in the overlap.
- Product: \( \frac{3}{8} \).

#### Problem 3: \( \frac{1}{2} \times \frac{2}{5} \)
- Grid: A 2x5 grid.
- Shading:
- Vertically shade \( \frac{1}{2} \) (1 column out of 2).
- Horizontally shade \( \frac{2}{5} \) (2 rows out of 5).
- Overlap: 2 squares are shaded in the overlap.
- Product: \( \frac{2}{10} \) (simplified to \( \frac{1}{5} \)).

#### Problem 4: \( \frac{2}{5} \times \frac{2}{3} \)
- Grid: A 5x3 grid.
- Shading:
- Vertically shade \( \frac{2}{5} \) (2 columns out of 5).
- Horizontally shade \( \frac{2}{3} \) (2 rows out of 3).
- Overlap: 4 squares are shaded in the overlap.
- Product: \( \frac{4}{15} \).

#### Problem 5: \( \frac{2}{3} \times \frac{3}{5} \)
- Grid: A 3x5 grid.
- Shading:
- Vertically shade \( \frac{2}{3} \) (2 columns out of 3).
- Horizontally shade \( \frac{3}{5} \) (3 rows out of 5).
- Overlap: 6 squares are shaded in the overlap.
- Product: \( \frac{6}{15} \) (simplified to \( \frac{2}{5} \)).

#### Problem 6: \( \frac{2}{3} \times \frac{5}{6} \)
- Grid: A 3x6 grid.
- Shading:
- Vertically shade \( \frac{2}{3} \) (2 columns out of 3).
- Horizontally shade \( \frac{5}{6} \) (5 rows out of 6).
- Overlap: 10 squares are shaded in the overlap.
- Product: \( \frac{10}{18} \) (simplified to \( \frac{5}{9} \)).

#### Problem 7: \( \frac{5}{6} \times \frac{3}{4} \)
- Grid: A 6x4 grid.
- Shading:
- Vertically shade \( \frac{5}{6} \) (5 columns out of 6).
- Horizontally shade \( \frac{3}{4} \) (3 rows out of 4).
- Overlap: 15 squares are shaded in the overlap.
- Product: \( \frac{15}{24} \) (simplified to \( \frac{5}{8} \)).

#### Problem 8: \( \frac{3}{4} \times \frac{4}{5} \)
- Grid: A 4x5 grid.
- Shading:
- Vertically shade \( \frac{3}{4} \) (3 columns out of 4).
- Horizontally shade \( \frac{4}{5} \) (4 rows out of 5).
- Overlap: 12 squares are shaded in the overlap.
- Product: \( \frac{12}{20} \) (simplified to \( \frac{3}{5} \)).

#### Problem 9: \( \left( \frac{3}{4} \right)^2 \)
- Fractions: \( \frac{3}{4} \times \frac{3}{4} \)
- Grid: A 4x4 grid.
- Shading:
- Vertically shade \( \frac{3}{4} \) (3 columns out of 4).
- Horizontally shade \( \frac{3}{4} \) (3 rows out of 4).
- Overlap: 9 squares are shaded in the overlap.
- Product: \( \frac{9}{16} \).

#### Problem 10: \( \left( \frac{2}{3} \right)^2 \)
- Fractions: \( \frac{2}{3} \times \frac{2}{3} \)
- Grid: A 3x3 grid.
- Shading:
- Vertically shade \( \frac{2}{3} \) (2 columns out of 3).
- Horizontally shade \( \frac{2}{3} \) (2 rows out of 3).
- Overlap: 4 squares are shaded in the overlap.
- Product: \( \frac{4}{9} \).

#### Problem 11: \( \left( \frac{4}{5} \right)^2 \)
- Fractions: \( \frac{4}{5} \times \frac{4}{5} \)
- Grid: A 5x5 grid.
- Shading:
- Vertically shade \( \frac{4}{5} \) (4 columns out of 5).
- Horizontally shade \( \frac{4}{5} \) (4 rows out of 5).
- Overlap: 16 squares are shaded in the overlap.
- Product: \( \frac{16}{25} \).

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Final Answer to the Reflection Question:

What does the area model teach you about multiplying fractions?

- The area model visually demonstrates that multiplying fractions involves finding the overlap of the shaded regions representing each fraction.
- It shows that the numerator of the product is the product of the numerators of the fractions being multiplied.
- Similarly, the denominator of the product is the product of the denominators of the fractions being multiplied.
- This method helps in understanding why we multiply the numerators and denominators separately when multiplying fractions.

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Final Answers:

\[
\boxed{
\begin{array}{ll}
1) & \frac{1}{6} \\
2) & \frac{3}{8} \\
3) & \frac{1}{5} \\
4) & \frac{4}{15} \\
5) & \frac{2}{5} \\
6) & \frac{5}{9} \\
7) & \frac{5}{8} \\
8) & \frac{3}{5} \\
9) & \frac{9}{16} \\
10) & \frac{4}{9} \\
11) & \frac{16}{25}
\end{array}
}
\]