Chapter 8 — Practice examination

A museum sculpture stand contains 2 fully packed layers of 4 by 2 unit cubes. State its volume in cubic units, including cubes you cannot see.

Application-and-critique task: An exhibition packing crate contains 2 fully packed layers of 3 by 2 unit cubes. State its volume in cubic units, including cubes you cannot see. Independently prepare a packed base layer and number of identical cube layers; do not simply repeat the first application question’s computation. A student wrote this solution: “I can count one face 3×2=6 and call it all 2 layers of cubes.” Is that reasoning valid? Name the first incorrect step; demonstrate a corrected computation and independently check it.

A rectangular storage compartment measures 10 cm, 2 cm, and 9 cm along mutually perpendicular edges. What is its internal volume?

Application-and-critique task: A modular shipping container measures 4 cm, 5 cm, and 8 cm along mutually perpendicular edges. What is its internal volume? Independently prepare a labeled three-dimensional cuboid layer model; do not simply repeat the first application question’s computation. A student wrote this solution: “Volume = length×width=20 cm²; I omitted the 8 cm height.” Is that reasoning valid? Name the first incorrect step; demonstrate a corrected computation and independently check it.

A cuboid has a volume of 198 cm³ and a base of 11 by 6 cm. Recover its missing height using the base AREA.

Application-and-critique task: A rectangular aquarium model holds a volume of 210 cm³ and a base of 3 by 7 cm. Recover its missing height using the base AREA. Independently prepare an inverse multiplication diagram for the unknown edge; do not simply repeat the first application question’s computation. A student wrote this solution: “Missing height=210÷3=70 cm. I ignored width 7 cm.” Is that reasoning valid? Name the first incorrect step; demonstrate a corrected computation and independently check it.

Two NONOVERLAPPING blocks fill a display: one 9×8×4 cm and the other 4×4×5 cm. Find their total packed volume and justify why addition is valid.

Application-and-critique task: A maker constructs a display from two NONOVERLAPPING blocks: one 9×5×4 cm and the other 2×3×5 cm. Find their total packed volume and justify why addition is valid. Independently prepare two separate nonoverlapping cuboids and additive volumes; do not simply repeat the first application question’s computation. A student wrote this solution: “Combined volume = 180×30=5400 cm³; I multiplied disjoint cuboid volumes.” Is that reasoning valid? Name the first incorrect step; demonstrate a corrected computation and independently check it.

A water reservoir reading gives 2,000 cm³. Rewrite its capacity as both milliliters and liters, labeling the units.

Application-and-critique task: A flood-preparation measuring tank reports 5,375 cm³. Rewrite its capacity as both milliliters and liters, labeling the units. Independently prepare a cubic-centimeter to milliliter and liter conversion ladder; do not simply repeat the first application question’s computation. A student wrote this solution: “5375 cm³ = 5375 L; I treated a milliliter as a liter.” Is that reasoning valid? Name the first incorrect step; demonstrate a corrected computation and independently check it.

A 25 cm by 18 cm rectangular tank has a full internal height of 20 cm. The water is only 10 cm deep. Find BOTH water and unused capacity in milliliters and explain why using full height for the water is wrong.

Application-and-critique task: A water-quality technician examines a 35 cm by 15 cm rectangular tank has a full internal height of 23 cm. The water is only 19 cm deep. Find BOTH water and unused capacity in milliliters and explain why using full height for the water is wrong. Independently prepare a side-elevation diagram distinguishing water height from full height; do not simply repeat the first application question’s computation. A student wrote this solution: “Water volume=35×15×23=12075 mL, even though water height is only 19 cm.” Is that reasoning valid? Name the first incorrect step; demonstrate a corrected computation and independently check it.

A block has 5 by 2 cubes per layer and 3 layers. Count all cubes.