Unit 10 Projects — Teacher Answer Key
Volume & Surface Area | 6.G.2 & 6.G.4 | V = lwh, SA closed & open-top
For teachers only — this page shows fully worked solutions using each project's default input values. Students who enter different (valid) values will get different numbers, but the same reasoning process applies. Use these as your grading reference and model responses.
Standards: 6.G.2 — volume of rectangular prisms with whole-number and fractional edge lengths, counting unit cubes. 6.G.4 — surface area of rectangular prisms using nets.
Version A — Package Design Challenge
Phase 1 Calculate the Box Volume
Volume of a Rectangular Prism · 6.G.2
Length = 8 units
Width = 5 units
Height = 3 units
V = l × w × h
120 cubic units
V = 8 × 5 × 3
Step 1 — bottom layer: 8 × 5 = 40 unit cubes
Step 2 — stack layers: 40 × 3 = 120 cubic units
Interpretation: The box holds exactly 120 one-unit cubes arranged in 3 layers of 40.
Sample 4 / 3 / 2 Rubric Scoring — Volume (6.G.2)
4 (Expert): V = 120 cubic units with correct unit label; student explains the layer-by-layer unit-cube reasoning (40 per layer, 3 layers).3 (Proficient): V = 120 cubic units correct.
2 (Developing): Formula V = lwh used but computation error (e.g., forgot one dimension).
Phase 2 Volume with Fractional Edges
Fractional Edge Lengths · 6.G.2
Length = 2.5 units
Width = 1.5 units
Height = 3 units
V = l × w × h (same formula, fractional edges)
11.25 cubic units
V = 2.5 × 1.5 × 3
As fractions: (5/2) × (3/2) × (3/1) = 45/4 = 11.25
Fractional-cube explanation: The 0.5 edge along length produces half-cubes, and the 0.5 edge along width produces half-cubes in that direction. When every whole and partial cube is counted, they still total exactly 11.25 cubic units — the formula V = lwh handles fractional edges automatically (6.G.2).
Sample 4 / 3 / 2 Rubric Scoring — Fractional Edge Volume (6.G.2)
4 (Expert): V = 11.25 cubic units correct; student explains that fractional edges produce fractional unit cubes that still sum to the product lwh.3 (Proficient): V = 11.25 cubic units correct.
2 (Developing): Correct formula applied but arithmetic error with fractions/decimals.
Phase 3 Unfold the Net & Find Surface Area
Surface Area from the Net · 6.G.4
Length = 8 units
Width = 5 units
Height = 3 units
SA = 2(lw + lh + wh)
158 square units
| Face Pair | Dimensions | One face | Both faces (x2) |
|---|---|---|---|
| Top & Bottom | l × w = 8 × 5 | 40 sq units | 80 sq units |
| Front & Back | l × h = 8 × 3 | 24 sq units | 48 sq units |
| Left & Right sides | w × h = 5 × 3 | 15 sq units | 30 sq units |
SA = 2(40 + 24 + 15) = 2 × 79 = 158 square units
Alternatively: 80 + 48 + 30 = 158 square units ✓
Sample 4 / 3 / 2 Rubric Scoring — Surface Area / Nets (6.G.4)
4 (Expert): All three face pairs identified and labeled; SA = 158 sq units with correct formula and unit label.3 (Proficient): SA = 158 sq units correct.
2 (Developing): One face pair missing (e.g., forgot sides), or computation error in one pair.
Phase 4 Material Cost Decision
Application & Communication · 6.G.2 · 6.G.4
Surface area = 158 sq units
Cost rate = $0.05 per sq unit
$7.90 per box
158 sq units × $0.050 per sq unit = $7.90
To reduce cost: shrink the surface area by reducing at least one dimension. The smallest SA for a fixed volume uses a cube-like shape.
Quick Check — Exact Answer
A box measures 2 × 3 × 4 units. What is its volume? (V = l × w × h)
V = 2 × 3 × 4 = 24 cubic units
The calculator accepts only 24 as correct (from
checkVol()).Sample 4 / 3 / 2 Rubric Scoring — Application / Communication
4 (Expert): Cost = $7.90 correct; spec sheet includes all numbers and explains a design trade-off (e.g., smaller SA = less cost).3 (Proficient): Cost correct; spec sheet uses most numbers.
2 (Developing): Multiplication attempted; spec sheet unclear or missing cost justification.
Deliverable Package Spec Sheet — Sample Expert Response
"Our shipping box measures 8 × 5 × 3 units. Its volume is 120 cubic units — picture 40 unit cubes on the bottom layer stacked 3 layers high. The surface area is 158 square units, made up of three face pairs: top/bottom (80 sq units), front/back (48 sq units), and left/right sides (30 sq units). At $0.05 per square unit, the cardboard costs $7.90 per box. This design is a good choice because the 8-unit length provides ample room for the product while keeping costs under $10 per box."
Version B — Aquarium Build Lab
Phase 1 Tank Volume: Full Capacity
Volume of a Rectangular Prism · 6.G.2
Length = 5 units
Width = 2 units
Height = 3 units
V = l × w × h
30 cubic units
V = 5 × 2 × 3
Bottom layer: 5 × 2 = 10 unit cubes
Stack 3 layers: 10 × 3 = 30 cubic units total capacity
Sample 4 / 3 / 2 Rubric Scoring — Volume (6.G.2)
4 (Expert): V = 30 cubic units with correct unit label; unit-cube layer reasoning included (10 per layer, 3 layers).3 (Proficient): V = 30 cubic units correct.
2 (Developing): Formula used but arithmetic error.
Phase 2 Water Fill: Partial Height
Partial Fill Volume · 6.G.2
Length = 5 units
Width = 2 units
Fill height = 2 units (out of 3)
Vwater = l × w × fill height
20 cubic units of water
Vwater = 5 × 2 × 2 = 20 cubic units
The tank needs to be filled to only 2 units high to hold 20 cubic units of water.
As a fraction of full capacity: 20 / 30 ≈ 67% full.
Sample 4 / 3 / 2 Rubric Scoring — Partial Fill (6.G.2)
4 (Expert): Vwater = 20 cubic units correct; fill height (2) correctly substituted instead of full height (3); fraction of full capacity noted.3 (Proficient): Water volume = 20 cubic units correct.
2 (Developing): Used full height 3 instead of fill height 2, or arithmetic error.
Phase 3 Glass Needed: Open-Top Tank
Open-Top Surface Area from the Net · 6.G.4
Length = 5 units
Width = 2 units
Height = 3 units
SAopen-top = (l × w) + 2(l × h) + 2(w × h) [5 faces — NO top]
52 square units of glass
| Face | Dimensions | Area |
|---|---|---|
| Bottom | l × w = 5 × 2 | 10 sq units |
| 2 long sides | 2 × (l × h) = 2 × (5 × 3) | 30 sq units |
| 2 short sides | 2 × (w × h) = 2 × (2 × 3) | 12 sq units |
SAopen-top = 10 + 30 + 12 = 52 square units
Comparison: A closed box would have SA = 2(10 + 15 + 6) = 62 square units. Removing the top face saves 10 square units of glass (one l × w = 5 × 2 = 10 panel).
Sample 4 / 3 / 2 Rubric Scoring — Open-Top Surface Area (6.G.4)
4 (Expert): 5-face net correctly identified (no top); SA = 52 sq units correct using SA = lw + 2lh + 2wh; difference from closed box (62 − 52 = 10) explained.3 (Proficient): SA = 52 sq units correct; open-top formula used.
2 (Developing): Used closed-box formula (SA = 62) — did not subtract the top face — or one face missing.
Phase 4 Glass Cost & Budget Check
Cost Decision · 6.G.4
Glass area = 52 sq units
Glass price = $3.50 per sq unit
$182.00 total glass cost
52 sq units × $3.50 per sq unit = $182.00
Quick Check — Exact Answer
A tank measures 5 × 2 × 3 units (open at the top). What is its full volume? (V = l × w × h)
V = 5 × 2 × 3 = 30 cubic units
The calculator accepts only 30 as correct (from
checkTank()).Sample 4 / 3 / 2 Rubric Scoring — Application / Communication
4 (Expert): Cost = $182.00 correct; build plan uses all numbers (dimensions, full volume, water volume, glass area, cost) and justifies the design.3 (Proficient): Cost correct; build plan uses most numbers.
2 (Developing): Multiplication attempted; plan unclear or missing cost or glass area.
Deliverable Build Plan — Sample Expert Response
"Our aquarium measures 5 × 2 × 3 units. Its total volume is 30 cubic units (maximum capacity), but filled to 2 units high it holds only 20 cubic units of water — about 67% full. Because the tank is open at the top, the glass covers only 5 faces: the bottom (10 sq units) plus four side panels (30 + 12 = 42 sq units), totaling 52 square units. A closed box would require 62 square units, so the open top saves 10 square units of expensive glass. At $3.50 per square unit, the total glass cost is $182.00 — a reasonable budget for a community center tank of this size."