5th Grade Multiplication as Scaling Test Solutions and Scoring (PDF)

(a) \(\frac{5}{6} < 1\), so \(7 \times \frac{5}{6} < 7\).

(b) \(\frac{11}{8} > 1\), so \(7 \times \frac{11}{8} > 7\).

Scoring: 1 point for each correct symbol.

\(20 \times \frac{1}{4} = 5\), \(20 \times 1 = 20\), \(20 \times \frac{5}{4} = 25\), \(20 \times \frac{3}{2} = 30\).

Least to greatest: \(20 \times \frac{1}{4}\), \(20 \times 1\), \(20 \times \frac{5}{4}\), \(20 \times \frac{3}{2}\).

Scoring: 2 points for the correct order. Give 1 point if only one expression is out of place.

(a) Each part is 4. The bottom bar is 3 parts out of 4: \(\frac{3}{4}\). Check: \(16 \times \frac{3}{4} = 12\).

(b) Less than 1, because the bottom bar is shorter than the top bar (12 is less than 16).

Scoring: 1 point for 3/4; 1 point for a correct explanation.

(a) \(\frac{5}{6} \times 18 = \frac{90}{6} = 15\): 15.

(b) \(\frac{7}{2} \times \frac{10}{7} = \frac{70}{14} = 5\): 5.

(c) \(15 < 18\) because \(\frac{5}{6} < 1\). \(5 > 3\frac{1}{2}\) because \(1\frac{3}{7} > 1\).

Scoring: 1 point for each part (part c needs both comparisons).

(a) The scale factor \(\frac{2}{3}\) is less than 1, so the product is less than 9. Multiplying by a number less than 1 makes a number smaller.

(b) \(9 \times \frac{2}{3} = \frac{18}{3} = 6\): 6.

Scoring: 1 point for a correct explanation; 1 point for 6.

(a) Width: \(6 \times \frac{3}{2} = 9\) inches. Height: \(4 \times \frac{3}{2} = 6\) inches: 9 inches by 6 inches.

(b) \(6 \times \frac{2}{3} = 4\): 4 inches.

Scoring: (a) 1 point for each dimension. (b) 1 point.

(a) \(\frac{9}{4} \times \frac{2}{3} = \frac{18}{12} = 1\frac{1}{2}\): \(1\frac{1}{2}\) cups.

(b) \(\frac{9}{4} \times \frac{4}{3} = \frac{36}{12} = 3\): 3 cups.

Scoring: 1 point for each correct amount with units.

(a) Left. The factor \(\frac{2}{3}\) is less than 1, so the product is less than \(\frac{3}{4}\).

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

Scoring: 1 point for left with a correct reason; 1 point for 1/2.

(a) \(\frac{3}{5} \times 20 = 12\): 12 feet.

(b) \(\frac{9}{4} \times 20 = 45\): 45 feet.

Scoring: 1 point for each correct length with units.