This answer key gives a full worked solution and a scoring guide for each of the eight questions. The test is worth 20 points. Award partial credit as described, and check that every volume has cubic units, since square units show a confusion between area and volume.
Question 1: Whole-number edges (2 points)
- \(V = 9 \times 4 \times 6 = 216\). The volume is \(216\text{ in}^3\).
- \(V = Bh = 35 \times 4 = 140\). The volume is \(140\text{ ft}^3\).
Scoring: 1 point for each part. Award the point for a correct value with cubic units; give no credit if the units are square.
Question 2: Fractional edges (2 points)
\(V = \frac{3}{4} \times \frac{1}{2} \times 4 = \frac{3}{8} \times 4 = \frac{12}{8} = \frac{3}{2} = 1\frac{1}{2}\)
The volume is \(1\frac{1}{2}\text{ ft}^3\).
Scoring: 1 point for a correct product setup; 1 point for the correct simplified volume with cubic units.
Question 3: Cube packing (3 points)
- Length \(3 \times \frac{1}{2} = \frac{3}{2}\text{ in}\), width \(2 \times \frac{1}{2} = 1\text{ in}\), height \(\frac{1}{2}\text{ in}\).
- There are \(3 \times 2 \times 1 = 6\) cubes, each \(\frac{1}{8}\text{ in}^3\). \(6 \times \frac{1}{8} = \frac{3}{4}\). The volume is \(\frac{3}{4}\text{ in}^3\).
- \(\frac{3}{2} \times 1 \times \frac{1}{2} = \frac{3}{4}\text{ in}^3\), which matches.
Scoring: 1 point for each part. In part (b), award the point only if the cube volume \(\frac{1}{8}\text{ in}^3\) is used.
Question 4: Missing height (2 points)
\(84 = 7 \times 4 \times h = 28h\), so \(h = 84 : 28 = 3\).
The height is \(3\text{ ft}\).
Scoring: 1 point for a correct equation or base area of 28; 1 point for the correct height with units.
Question 5: Fish tank (3 points)
- \(30 \times 12 \times 15 = 5{,}400\text{ in}^3\).
- \(30 \times 12 \times 10 = 3{,}600\text{ in}^3\).
- \(5{,}400 – 3{,}600 = 1{,}800\text{ in}^3\).
Answers: \(5{,}400\text{ in}^3\), \(3{,}600\text{ in}^3\), \(1{,}800\text{ in}^3\).
Scoring: 1 point for each part with correct cubic units.
Question 6: Packing crates (3 points)
- \(6 \times 3 \times 3 = 54\text{ ft}^3\).
- \(\frac{3}{2} \times \frac{3}{2} \times \frac{3}{2} = \frac{27}{8} = 3\frac{3}{8}\text{ ft}^3\).
- \(54 : \frac{27}{8} = 54 \times \frac{8}{27} = 16\). Check by rows: \(6 : \frac{3}{2} = 4\), \(3 : \frac{3}{2} = 2\), \(3 : \frac{3}{2} = 2\), and \(4 \times 2 \times 2 = 16\).
Answers: \(54\text{ ft}^3\), \(3\frac{3}{8}\text{ ft}^3\), 16 boxes.
Scoring: 1 point for each part.
Question 7: Compare prisms (2 points)
A: \(5 \times 4 \times 3 = 60\text{ ft}^3\). B: \(6 \times \frac{5}{2} \times 5 = 15 \times 5 = 75\text{ ft}^3\).
\(75 – 60 = 15\). Prism B is greater by \(15\text{ ft}^3\).
Scoring: 1 point for both volumes; 1 point for the comparison and difference with units.
Question 8: Sandbox sand (3 points)
- \(8 \times 5 \times \frac{3}{4} = 40 \times \frac{3}{4} = 30\text{ ft}^3\).
- \(30 : 1\frac{1}{2} = 30 : \frac{3}{2} = 20\) bags.
- \(20 \times \$7.25 = \$145\).
Answers: \(30\text{ ft}^3\), 20 bags, \(\$145\).
Scoring: 1 point for each part.
20 points
See every chapter on the Grade 6 math page.