Comparing Fractions worksheet for 4.NF.2, featuring fraction comparison problems with visual aids.

Problem: Comparing Fractions

The task involves comparing fractions using the symbols `>`, `<`, or `=`. Let's solve each part step by step.

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#### Part 1: Solve each problem using >, <, or =

##### a) $\frac{3}{6} \; \square \; \frac{4}{5}$
- Step 1: Simplify $\frac{3}{6}$.
$$
\frac{3}{6} = \frac{1}{2}
$$
- Step 2: Compare $\frac{1}{2}$ and $\frac{4}{5}$.
- Convert both fractions to have a common denominator. The least common denominator (LCD) of 2 and 5 is 10.
$$
\frac{1}{2} = \frac{1 \times 5}{2 \times 5} = \frac{5}{10}
$$
$$
\frac{4}{5} = \frac{4 \times 2}{5 \times 2} = \frac{8}{10}
$$
- Now compare $\frac{5}{10}$ and $\frac{8}{10}$.
$$
\frac{5}{10} < \frac{8}{10}
$$
- Conclusion:
$$
\frac{3}{6} < \frac{4}{5}
$$

##### b) $\frac{1}{4} \; \square \; \frac{1}{6}$
- Step 1: Compare $\frac{1}{4}$ and $\frac{1}{6}$.
- Since the numerators are the same (both are 1), the fraction with the smaller denominator is larger.
- Here, $4 < 6$, so $\frac{1}{4} > \frac{1}{6}$.
- Conclusion:
$$
\frac{1}{4} > \frac{1}{6}
$$

##### c) $\frac{5}{8} \; \square \; \frac{6}{10}$
- Step 1: Simplify $\frac{6}{10}$.
$$
\frac{6}{10} = \frac{3}{5}
$$
- Step 2: Compare $\frac{5}{8}$ and $\frac{3}{5}$.
- Find the least common denominator (LCD) of 8 and 5, which is 40.
$$
\frac{5}{8} = \frac{5 \times 5}{8 \times 5} = \frac{25}{40}
$$
$$
\frac{3}{5} = \frac{3 \times 8}{5 \times 8} = \frac{24}{40}
$$
- Now compare $\frac{25}{40}$ and $\frac{24}{40}$.
$$
\frac{25}{40} > \frac{24}{40}
$$
- Conclusion:
$$
\frac{5}{8} > \frac{6}{10}
$$

##### d) $\frac{2}{5} \; \square \; \frac{4}{10}$
- Step 1: Simplify $\frac{4}{10}$.
$$
\frac{4}{10} = \frac{2}{5}
$$
- Step 2: Compare $\frac{2}{5}$ and $\frac{2}{5}$.
- Both fractions are identical.
- Conclusion:
$$
\frac{2}{5} = \frac{4}{10}
$$

##### e) $\frac{1}{3} \; \square \; \frac{2}{10}$
- Step 1: Simplify $\frac{2}{10}$.
$$
\frac{2}{10} = \frac{1}{5}
$$
- Step 2: Compare $\frac{1}{3}$ and $\frac{1}{5}$.
- Since the numerators are the same (both are 1), the fraction with the smaller denominator is larger.
- Here, $3 < 5$, so $\frac{1}{3} > \frac{1}{5}$.
- Conclusion:
$$
\frac{1}{3} > \frac{2}{10}
$$

##### f) $\frac{1}{2} \; \square \; \frac{5}{12}$
- Step 1: Compare $\frac{1}{2}$ and $\frac{5}{12}$.
- Find the least common denominator (LCD) of 2 and 12, which is 12.
$$
\frac{1}{2} = \frac{1 \times 6}{2 \times 6} = \frac{6}{12}
$$
$$
\frac{5}{12} = \frac{5}{12}
$$
- Now compare $\frac{6}{12}$ and $\frac{5}{12}$.
$$
\frac{6}{12} > \frac{5}{12}
$$
- Conclusion:
$$
\frac{1}{2} > \frac{5}{12}
$$

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#### Part 2: Solve each problem using >, <, or =

##### a)
- The first bar is divided into 4 equal parts, and 3 parts are shaded. This represents $\frac{3}{4}$.
- The second bar is divided into 6 equal parts, and 4 parts are shaded. This represents $\frac{4}{6}$, which simplifies to $\frac{2}{3}$.
- Compare $\frac{3}{4}$ and $\frac{2}{3}$.
- Find the least common denominator (LCD) of 4 and 3, which is 12.
$$
\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}
$$
$$
\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}
$$
- Now compare $\frac{9}{12}$ and $\frac{8}{12}$.
$$
\frac{9}{12} > \frac{8}{12}
$$
- Conclusion:
$$
\frac{3}{4} > \frac{2}{3}
$$

##### b)
- The circle is divided into 4 equal parts, and 3 parts are shaded. This represents $\frac{3}{4}$.
- The circle is divided into 8 equal parts, and 5 parts are shaded. This represents $\frac{5}{8}$.
- Compare $\frac{3}{4}$ and $\frac{5}{8}$.
- Find the least common denominator (LCD) of 4 and 8, which is 8.
$$
\frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8}
$$
$$
\frac{5}{8} = \frac{5}{8}
$$
- Now compare $\frac{6}{8}$ and $\frac{5}{8}$.
$$
\frac{6}{8} > \frac{5}{8}
$$
- Conclusion:
$$
\frac{3}{4} > \frac{5}{8}
$$

##### c)
- The bar is divided into 6 equal parts, and 4 parts are shaded. This represents $\frac{4}{6}$, which simplifies to $\frac{2}{3}$.
- The circle is divided into 4 equal parts, and 3 parts are shaded. This represents $\frac{3}{4}$.
- Compare $\frac{2}{3}$ and $\frac{3}{4}$.
- Find the least common denominator (LCD) of 3 and 4, which is 12.
$$
\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}
$$
$$
\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}
$$
- Now compare $\frac{8}{12}$ and $\frac{9}{12}$.
$$
\frac{8}{12} < \frac{9}{12}
$$
- Conclusion:
$$
\frac{2}{3} < \frac{3}{4}
$$

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Final Answers:

1.
- a) $\frac{3}{6} < \frac{4}{5}$
- b) $\frac{1}{4} > \frac{1}{6}$
- c) $\frac{5}{8} > \frac{6}{10}$
- d) $\frac{2}{5} = \frac{4}{10}$
- e) $\frac{1}{3} > \frac{2}{10}$
- f) $\frac{1}{2} > \frac{5}{12}$

2.
- a) $\frac{3}{4} > \frac{2}{3}$
- b) $\frac{3}{4} > \frac{5}{8}$
- c) $\frac{2}{3} < \frac{3}{4}$

Boxed Final Answer:
$$
\boxed{
\begin{array}{ll}
1. & \text{a) } <, \text{ b) } >, \text{ c) } >, \text{ d) } =, \text{ e) } >, \text{ f) } > \\
2. & \text{a) } >, \text{ b) } >, \text{ c) } <
\end{array}
}
$$