Multiplying fractions using models worksheet - Free Printable

Let's solve each of these fraction multiplication problems using the area models provided. The area model helps visualize multiplying two fractions by showing how much of the whole is shaded in both directions.

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🔷 How Area Models Work:

- The rectangle is divided into parts based on the denominators.
- One dimension represents the first fraction (e.g., 1/3), and the other represents the second (e.g., 1/2).
- The overlapping shaded region represents the product.

We’ll go through each problem step by step.

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✔ Example:

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

- Rectangle divided into 3 columns (for thirds) and 2 rows (for halves).
- Shaded area = $1$ column × $1$ row = $1$ out of $6$ total parts.
- So, $\frac{1}{3} \times \frac{1}{2} = \frac{1}{6}$

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Now let’s solve each one:

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1) $\frac{1}{2} \times \frac{3}{4}$

- Divide rectangle into 2 rows (for 1/2) and 4 columns (for 3/4).
- Shade 1 row (half) and 3 columns (three-fourths).
- Overlap = $1 \times 3 = 3$ parts out of $2 \times 4 = 8$
- Answer: $\boxed{\frac{3}{8}}$

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2) $\frac{1}{5} \times \frac{2}{4}$

- Simplify $\frac{2}{4} = \frac{1}{2}$, but keep as $\frac{2}{4}$ for model.
- Rectangle: 5 columns (for fifths), 4 rows (for fourths).
- Shade 1 column (1/5), 2 rows (2/4).
- Overlap = $1 \times 2 = 2$ parts out of $5 \times 4 = 20$
- $\frac{2}{20} = \boxed{\frac{1}{10}}$

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3) $\frac{3}{4} \times \frac{2}{3}$

- 4 columns, 3 rows → total 12 parts.
- Shade 3 columns (3/4), 2 rows (2/3).
- Overlap = $3 \times 2 = 6$ parts → $\frac{6}{12} = \boxed{\frac{1}{2}}$

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4) $\frac{6}{8} \times \frac{1}{2}$

- Simplify $\frac{6}{8} = \frac{3}{4}$, but use model.
- Rectangle: 8 columns (for eighths), 2 rows (for halves).
- Shade 6 columns (6/8), 1 row (1/2).
- Overlap = $6 \times 1 = 6$ parts → total $8 \times 2 = 16$
- $\frac{6}{16} = \boxed{\frac{3}{8}}$

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5) $\frac{1}{3} \times \frac{1}{7}$

- 3 columns, 7 rows → total 21 parts.
- Shade 1 column, 1 row → overlap = $1 \times 1 = 1$
- Answer: $\boxed{\frac{1}{21}}$

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6) $\frac{2}{6} \times \frac{1}{2}$

- Simplify $\frac{2}{6} = \frac{1}{3}$, but model uses 6 columns.
- 6 columns, 2 rows → total 12 parts.
- Shade 2 columns (2/6), 1 row (1/2).
- Overlap = $2 \times 1 = 2$ parts → $\frac{2}{12} = \boxed{\frac{1}{6}}$

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7) $\frac{2}{9} \times \frac{1}{4}$

- 9 columns, 4 rows → total 36 parts.
- Shade 2 columns, 1 row → overlap = $2 \times 1 = 2$
- $\frac{2}{36} = \boxed{\frac{1}{18}}$

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8) $\frac{5}{7} \times \frac{1}{2}$

- 7 columns, 2 rows → total 14 parts.
- Shade 5 columns, 1 row → overlap = $5 \times 1 = 5$
- $\frac{5}{14} = \boxed{\frac{5}{14}}$

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9) $\frac{1}{2} \times \frac{2}{5}$

- 2 columns, 5 rows → total 10 parts.
- Shade 1 column (1/2), 2 rows (2/5).
- Overlap = $1 \times 2 = 2$ → $\frac{2}{10} = \boxed{\frac{1}{5}}$

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10) $\frac{5}{6} \times \frac{3}{6}$

- Note: $\frac{3}{6} = \frac{1}{2}$, but model uses 6.
- 6 columns, 6 rows → total 36 parts.
- Shade 5 columns, 3 rows → overlap = $5 \times 3 = 15$
- $\frac{15}{36} = \boxed{\frac{5}{12}}$

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

- Same as example!
- 3 columns, 2 rows → 6 total parts.
- Shade 1 column, 1 row → overlap = 1
- $\boxed{\frac{1}{6}}$

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✔ Final Answers:

| Problem | Answer |
|--------|--------|
| Ex) $\frac{1}{3} \times \frac{1}{2}$ | $\frac{1}{6}$ |
| 1) $\frac{1}{2} \times \frac{3}{4}$ | $\frac{3}{8}$ |
| 2) $\frac{1}{5} \times \frac{2}{4}$ | $\frac{1}{10}$ |
| 3) $\frac{3}{4} \times \frac{2}{3}$ | $\frac{1}{2}$ |
| 4) $\frac{6}{8} \times \frac{1}{2}$ | $\frac{3}{8}$ |
| 5) $\frac{1}{3} \times \frac{1}{7}$ | $\frac{1}{21}$ |
| 6) $\frac{2}{6} \times \frac{1}{2}$ | $\frac{1}{6}$ |
| 7) $\frac{2}{9} \times \frac{1}{4}$ | $\frac{1}{18}$ |
| 8) $\frac{5}{7} \times \frac{1}{2}$ | $\frac{5}{14}$ |
| 9) $\frac{1}{2} \times \frac{2}{5}$ | $\frac{1}{5}$ |
| 10) $\frac{5}{6} \times \frac{3}{6}$ | $\frac{5}{12}$ |
| 11) $\frac{1}{3} \times \frac{1}{2}$ | $\frac{1}{6}$ |

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🧠 Summary:

To multiply fractions using an area model:
1. Draw a rectangle.
2. Divide it vertically for the first fraction.
3. Divide it horizontally for the second fraction.
4. Shade the overlapping region.
5. Count the shaded parts over total parts.

This visual method reinforces that:
$$
\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}
$$

Let me know if you'd like a diagram explanation or printable version!