Equivalent Fractions Worksheet with Pie Models for Learning Proper Fractions

Problem Description:

The task involves identifying equivalent fractions for the pie models shown in the image. Each pie model represents a fraction, and you need to write the equivalent proper fractions for each one.

Solution Approach:

1. Understand the Pie Models: Each pie is divided into equal parts, and some parts are shaded. The total number of parts represents the denominator, and the shaded parts represent the numerator.
2. Identify the Fraction: For each pie, determine the fraction represented by the shaded portion.
3. Find Equivalent Fractions: Multiply both the numerator and the denominator of the identified fraction by the same non-zero number to get an equivalent fraction.
4. Ensure Proper Fractions: A proper fraction has a numerator smaller than the denominator.

Step-by-Step Solution:

#### 1. First Row:
- Problem 1:
- Pie is divided into 5 parts, with 2 shaded.
- Fraction: \( \frac{2}{5} \)
- Equivalent fraction: Multiply numerator and denominator by 3.
\[
\frac{2 \times 3}{5 \times 3} = \frac{6}{15}
\]
- Answer: \( \frac{6}{15} \)

- Problem 2:
- Pie is divided into 4 parts, with 1 shaded.
- Fraction: \( \frac{1}{4} \)
- Equivalent fraction: Multiply numerator and denominator by 3.
\[
\frac{1 \times 3}{4 \times 3} = \frac{3}{12}
\]
- Answer: \( \frac{3}{12} \)

- Problem 3:
- Pie is divided into 3 parts, with 2 shaded.
- Fraction: \( \frac{2}{3} \)
- Equivalent fraction: Multiply numerator and denominator by 4.
\[
\frac{2 \times 4}{3 \times 4} = \frac{8}{12}
\]
- Answer: \( \frac{8}{12} \)

#### 2. Second Row:
- Problem 4:
- Pie is divided into 2 parts, with 1 shaded.
- Fraction: \( \frac{1}{2} \)
- Equivalent fraction: Multiply numerator and denominator by 4.
\[
\frac{1 \times 4}{2 \times 4} = \frac{4}{8}
\]
- Answer: \( \frac{4}{8} \)

- Problem 5:
- Pie is divided into 4 parts, with 3 shaded.
- Fraction: \( \frac{3}{4} \)
- Equivalent fraction: Multiply numerator and denominator by 2.
\[
\frac{3 \times 2}{4 \times 2} = \frac{6}{8}
\]
- Answer: \( \frac{6}{8} \)

- Problem 6:
- Pie is divided into 2 parts, with 1 shaded.
- Fraction: \( \frac{1}{2} \)
- Equivalent fraction: Multiply numerator and denominator by 4.
\[
\frac{1 \times 4}{2 \times 4} = \frac{4}{8}
\]
- Answer: \( \frac{4}{8} \)

#### 3. Third Row:
- Problem 7:
- Pie is divided into 2 parts, with 1 shaded.
- Fraction: \( \frac{1}{2} \)
- Equivalent fraction: Multiply numerator and denominator by 3.
\[
\frac{1 \times 3}{2 \times 3} = \frac{3}{6}
\]
- Answer: \( \frac{3}{6} \)

- Problem 8:
- Pie is divided into 4 parts, with 3 shaded.
- Fraction: \( \frac{3}{4} \)
- Equivalent fraction: Multiply numerator and denominator by 2.
\[
\frac{3 \times 2}{4 \times 2} = \frac{6}{8}
\]
- Answer: \( \frac{6}{8} \)

- Problem 9:
- Pie is divided into 3 parts, with 2 shaded.
- Fraction: \( \frac{2}{3} \)
- Equivalent fraction: Multiply numerator and denominator by 3.
\[
\frac{2 \times 3}{3 \times 3} = \frac{6}{9}
\]
- Answer: \( \frac{6}{9} \)

#### 4. Fourth Row:
- Problem 10:
- Pie is divided into 4 parts, with 1 shaded.
- Fraction: \( \frac{1}{4} \)
- Equivalent fraction: Multiply numerator and denominator by 2.
\[
\frac{1 \times 2}{4 \times 2} = \frac{2}{8}
\]
- Answer: \( \frac{2}{8} \)

- Problem 11:
- Pie is divided into 3 parts, with 2 shaded.
- Fraction: \( \frac{2}{3} \)
- Equivalent fraction: Multiply numerator and denominator by 2.
\[
\frac{2 \times 2}{3 \times 2} = \frac{4}{6}
\]
- Answer: \( \frac{4}{6} \)

- Problem 12:
- Pie is divided into 4 parts, with 3 shaded.
- Fraction: \( \frac{3}{4} \)
- Equivalent fraction: Multiply numerator and denominator by 2.
\[
\frac{3 \times 2}{4 \times 2} = \frac{6}{8}
\]
- Answer: \( \frac{6}{8} \)

Final Answers:

\[
\boxed{
\begin{array}{lll}
1) & \frac{6}{15} & \\
2) & \frac{3}{12} & \\
3) & \frac{8}{12} & \\
4) & \frac{4}{8} & \\
5) & \frac{6}{8} & \\
6) & \frac{4}{8} & \\
7) & \frac{3}{6} & \\
8) & \frac{6}{8} & \\
9) & \frac{6}{9} & \\
10) & \frac{2}{8} & \\
11) & \frac{4}{6} & \\
12) & \frac{6}{8} & \\
\end{array}
}
\]