5th Grade Area of Rectangles with Fractional Sides Test Solutions and Scoring (PDF)

(a) \(5 \times 3 = 15\) tiles: each tile is \(\frac{1}{15}\) square unit.

(b) \(3 \times 2 = 6\) tiles, so \(\frac{6}{15} = \frac{2}{5}\): \(\frac{2}{5}\) square unit.

Scoring: 1 point for each correct answer.

(a)

Multiply the numerators and multiply the denominators: \(\frac{3}{5} \times \frac{1}{4} = \frac{3 \times 1}{5 \times 4} = \frac{3}{20}\).

The area is \(\frac{3}{20}\) square foot.

(b)

Multiply the numerators and multiply the denominators: \(\frac{7}{8} \times \frac{2}{3} = \frac{7 \times 2}{8 \times 3} = \frac{14}{24}\).

Simplify: \(\frac{14}{24} = \frac{7}{12}\).

The area is \(\frac{7}{12}\) square yard.

Scoring: 1 point for each correct area in simplest form with square units.

(a)

Write the mixed number as improper fraction: \(2\frac{1}{2} = \frac{5}{2}\).

Multiply the numerators and multiply the denominators: \(\frac{5}{2} \times \frac{1}{3} = \frac{5 \times 1}{2 \times 3} = \frac{5}{6}\).

The area is \(\frac{5}{6}\) square foot.

(b)

Write the mixed numbers as improper fractions: \(1\frac{1}{2} = \frac{3}{2}\) and \(1\frac{3}{4} = \frac{7}{4}\).

Multiply the numerators and multiply the denominators: \(\frac{3}{2} \times \frac{7}{4} = \frac{3 \times 7}{2 \times 4} = \frac{21}{8}\).

Write as a mixed number: \(\frac{21}{8} = 2\frac{5}{8}\).

The area is \(2\frac{5}{8}\) square inches.

Scoring: 1 point for each correct area with square units.

(a) Each unit square has \(3 \times 4 = 12\) tiles: \(\frac{1}{12}\) square foot.

(b) \(4 \times 3 = 12\): 12 tiles.

(c) \(12 \times \frac{1}{12} = 1\): 1 square foot. Check: \(\frac{4}{3} \times \frac{3}{4} = 1\).

Scoring: 1 point for each part.

(a)

Multiply the numerators and multiply the denominators: \(\frac{5}{6} \times \frac{5}{6} = \frac{5 \times 5}{6 \times 6} = \frac{25}{36}\).

The area is \(\frac{25}{36}\) square foot.

(b) \(\frac{25}{36}\) is less than \(\frac{5}{6} = \frac{30}{36}\). Multiplying by a fraction less than 1 gives a smaller number.

Scoring: 1 point for the area; 1 point for the correct comparison with a reason.

(a) \(\frac{7}{4} \times \frac{5}{2} = \frac{35}{8} = 4\frac{3}{8}\): \(4\frac{3}{8}\) square feet.

(b) The second poster has area \(2 \times 2 = 4\) square feet. The first poster is greater because \(4\frac{3}{8} > 4\).

Scoring: 1 point for each part.

(a) \(\frac{9}{4} \times \frac{4}{3} = \frac{36}{12} = 3\): 3 square feet.

(b) \(3 \times 6 = 18\): $18.

Scoring: 1 point for each correct answer.

(a) \(\frac{2}{3} \times \frac{1}{4} = \frac{2}{12} = \frac{1}{6}\): \(\frac{1}{6}\) square yard.

(b) \(2 \times \left( \frac{8}{12} + \frac{3}{12} \right) = 2 \times \frac{11}{12} = \frac{22}{12} = 1\frac{5}{6}\): \(1\frac{5}{6}\) yards.

Scoring: 1 point for each correct answer with units.

(a) \(\frac{8}{3} \times \frac{3}{4} = \frac{24}{12} = 2\): 2 square feet.

(b) \(\frac{5}{8} \times 2 = \frac{10}{8} = 1\frac{1}{4}\): \(1\frac{1}{4}\) square feet.

(c) \(2 – 1\frac{1}{4} = \frac{3}{4}\): \(\frac{3}{4}\) square foot.

Scoring: 1 point for each part.