Cube Crate Teacher Guide Teacher Guide • Grade 5 Math
Cube Crate Architects
60 Minutes
North Carolina Standards Alignment
NC.5.MD.4: Recognize volume as an attribute of solid figures and measure volume by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.
NC.5.MD.5.i: Find volume of a right rectangular prism with whole-number side lengths by packing with unit cubes; show volume equals multiplying edge lengths or \(B \times h\); represent threefold products.
NC.5.MD.5.ii: Apply \(V = l \times w \times h\) and \(V = B \times h\) to find volumes of right rectangular prisms in real-world and mathematical problems.
Learning Target
Students will measure volume by counting unit cubes, connect layered cube packing to \(V = B \times h\), and solve real-world problems using \(V = l \times w \times h\).
Materials Needed
Unit cubes (24+ per pair) & Blueprint Slides
Prism Packing Guided Practice
Cargo Volume Worksheet
Instructional Pacing & Teacher Script
1. Hook & Cube Counting (Slides 1–3) 10 min
Introduce the shipping crate scenario. Define a unit cube (\(1 \times 1 \times 1\)) and cubic units (\(\text{cm}^3, \text{in}^3, \text{ft}^3\)). Model counting one full layer first so hidden cubes are included.
2. Packing Layers to Formulas (Slides 4–7) 15 min
Have pairs build a \(4 \times 3\) base layer (12 cubes) and stack 3 layers (\(h = 3\)). Connect Base Area \(B = l \times w = 12\) to \(V = B \times h = 36\), \(V = l \times w \times h\), and \((4 \times 3) \times 3 = 4 \times (3 \times 3)\).
3. Guided Practice: Prism Packing 15 min
Complete Part A together. Have pairs work through Part B (\(B \times h\)) and Part C (Threefold Products), checking that students account for hidden cubes in 3D drawings.
4. Independent Worksheet & Debrief (Slide 8) 20 min
Students complete Cargo Volume Worksheet independently (15 min). Close with Slide 8 reflection: "When is \(V = B \times h\) more helpful than \(V = l \times w \times h\)?"
Common Misconceptions
Counting only visible faces: Prompt students to count the full bottom layer first, then multiply by height.Adding instead of multiplying: Reinforce repeated addition of equal layers rather than \(l + w + h\).
Differentiation
Support: Provide physical unit cubes to build each prism layer-by-layer before writing equations.Extension: Design two different crates that hold exactly \(48\text{ ft}^3\) and compare their dimensions.
Cube Crate Architects
Complete Answer Keys
Teacher Reference
Prism Packing Guided Practice — Answer Key
Part A: Count Unit Cubes (NC.5.MD.4)
1. \(3 \times 2 \times 2 = \mathbf{12\text{ cubic units}}\) (6 cubes on bottom layer, 2 layers).
2. \(4 \times 2 \times 3 = \mathbf{24\text{ cubic units}}\) (8 cubes on bottom layer, 3 layers).
Part B: Base Area × Height (NC.5.MD.5.i)
3. Base Area \(B = 5 \times 3 = \mathbf{15\text{ cm}^2}\); Height \(h = \mathbf{2\text{ cm}}\); Volume \(V = 15 \times 2 = \mathbf{30\text{ cm}^3}\).
4. Base Area \(B = 4 \times 4 = \mathbf{16\text{ in}^2}\); Height \(h = \mathbf{3\text{ in}}\); Volume \(V = 16 \times 3 = \mathbf{48\text{ in}^3}\).
Part C: Threefold Products & Associative Property (NC.5.MD.5.i)
5. \((5 \times 2) \times 4 = 10 \times 4 = \mathbf{40\text{ ft}^3}\) and \(5 \times (2 \times 4) = 5 \times 8 = \mathbf{40\text{ ft}^3}\). Both groupings give the same volume because multiplication is associative.
Cargo Volume Worksheet — Answer Key
Section 1: Unit Cubes & Formulas (NC.5.MD.4 & NC.5.MD.5.i)
1. \(3 \times 3 \times 2 = \mathbf{18\text{ cubic units}}\).
2. \(6 \times 2 \times 4 = \mathbf{48\text{ cm}^3}\).
3. \(V = B \times h = 24 \times 5 = \mathbf{120\text{ in}^3}\).
4. Multiple choice: B (\(4 \times (3 \times 5)\) or equivalent \(60\text{ ft}^3\)).
Section 2: Real-World Cargo Problems (NC.5.MD.5.ii)
5. \(V = 8 \times 4 \times 3 = \mathbf{96\text{ ft}^3}\).
6. \(V = B \times h = 35 \times 6 = \mathbf{210\text{ in}^3}\).
7. Crate A: \(6 \times 4 \times 5 = \mathbf{120\text{ cm}^3}\). Crate B: \(5 \times 5 \times 5 = \mathbf{125\text{ cm}^3}\). Crate B holds more by \(5\text{ cm}^3\).
8. Missing height: \(72 \div (6 \times 3) = 72 \div 18 = \mathbf{4\text{ feet}}\). Accept clear explanation showing division or \(18 \times ? = 72\).
Volume Blueprint Slides Grade 5 Math • Blueprint NC.5.MD.4 • NC.5.MD.5
Cube Crate Architects
Measuring 3D space with unit cubes, layered packing, and volume formulas.
Today's Goal: Measure & calculate volume of rectangular prisms Let's Build!
What Is Volume?
NC.5.MD.4
Volume
The amount of 3D space inside a solid figure.
Measured in Cubic Units:
\(\text{cm}^3\) • \(\text{in}^3\) • \(\text{ft}^3\) • unit cubes
1 1 1
1 Unit Cube
A cube with edge lengths of \(1 \times 1 \times 1\) equals 1 cubic unit .
Think: Area covers a flat floor (2D). Volume fills the whole crate (3D).
Counting Unit Cubes
NC.5.MD.4
3 units 2 units 2
Don't forget hidden cubes in back!
Step 1: Count 1 Layer
\(3 \times 2 = \mathbf{6\text{ cubes}}\) on the bottom
Step 2: Multiply by Layers
\(6 \text{ cubes} \times 2 \text{ layers} = \mathbf{12\text{ cubic units}}\)
Architect Tip: Always count a full layer first so no hidden cubes are left out!
Packing Layers: \(V = B \times h\)
NC.5.MD.5.i
Base B = 4 x 3 = 12 h = 3
Capital \(B\) = Area of the Base
\(B = 4 \times 3 = \mathbf{12\text{ square units}}\)
Multiply Base Area by Height (\(h\))
\(V = B \times h = 12 \times 3 = 36\)
\(36\text{ cubic units}\) fill the crate completely without gaps or overlaps.
The Formula: \(V = l \times w \times h\)
NC.5.MD.5.i
Two Equivalent Formulas
\(V = B \times h\)
Base Area × Height
\(V = l \times w \times h\)
Length × Width × Height
Example: \(l = 5\text{ cm},\; w = 3\text{ cm},\; h = 2\text{ cm}\)
\(V = 5 \times 3 \times 2\)
\(V = 15 \times 2\)
\(V = 30\text{ cm}^3\)
Notice: \(l \times w\) gives you the Base Area \(B\)!
Grouping Edge Lengths
NC.5.MD.5.i
A crate measures \(5\text{ ft}\) by \(2\text{ ft}\) by \(4\text{ ft}\) . Does grouping change the volume?
Way 1: Bottom Layer First
\((5 \times 2) \times 4\)
\(= 10 \times 4\)
\(= 40\text{ ft}^3\)
Way 2: Side Slice First
\(5 \times (2 \times 4)\)
\(= 5 \times 8\)
\(= 40\text{ ft}^3\)
Associative Property: \((l \times w) \times h = l \times (w \times h)\) always gives the same volume!
Real-World Cargo Challenge
NC.5.MD.5.ii
Crate A: Edge Lengths
A shipping crate is \(8\text{ ft}\) long, \(4\text{ ft}\) wide, and \(3\text{ ft}\) tall.
\(V = 8 \times 4 \times 3 = 96\text{ ft}^3\)
Crate B: Base Area Given
A storage box has a base area of \(35\text{ in}^2\) and a height of \(6\text{ in}\) .
\(V = 35 \times 6 = 210\text{ in}^3\)
Always check your units! Volume answers must use cubic units (\(\text{ft}^3\), \(\text{in}^3\), \(\text{cm}^3\)).
Architect's Debrief
Wrap-Up
Question 1
How do hidden unit cubes affect your count, and how does layering help?
Question 2
When is \(V = B \times h\) more helpful than \(V = l \times w \times h\)?
Great work today, Cube Crate Architects!
Prism Packing Guided Practice Name: ____________________________________ Date: __________________
Guided Practice • Cube Crate Architects
Prism Packing Guided Practice
NC.5.MD.4 • 5.MD.5.i
Unit Cube: \(1 \times 1 \times 1 = 1\text{ cubic unit}\) Layer Formula: \(V = B \times h\) Edge Formula: \(V = l \times w \times h\)
Part A Count Unit Cubes by Layers
Count the cubes in one full layer, then multiply by the number of layers to find the volume.
1. Prism A
Cubes in 1 layer:
Number of layers:
Total Volume:
2. Prism B
Cubes in 1 layer:
Number of layers:
Total Volume:
Part B Relate Cube Packing to \(V = B \times h\)
3. A rectangular crate has length \(l = 5\text{ cm}\), width \(w = 3\text{ cm}\), and height \(h = 2\text{ cm}\).
Base Area (\(B = l \times w\))
Height (\(h\))
Volume (\(V = B \times h\))
Part B Base Area × Height (Continued)
4. A square-base shipping box measures \(l = 4\text{ in}\), \(w = 4\text{ in}\), and \(h = 3\text{ in}\). Find the base area \(B\), then calculate the total volume.
Base Area (\(B\))
Height (\(h\))
Total Volume (\(V\))
Part C Threefold Products & Associative Property
5. A cargo crate measures \(5\text{ ft}\) long, \(2\text{ ft}\) wide, and \(4\text{ ft}\) tall. Solve using two different factor groupings.
Grouping 1: \((5 \times 2) \times 4\)
Grouping 2: \(5 \times (2 \times 4)\)
Explain: Why do both groupings give the exact same volume?
Cargo Volume Worksheet Name: ____________________________________ Date: __________________
Independent Practice • Cube Crate Architects
Cargo Volume Worksheet
NC.5.MD.4 • 5.MD.5
Section 1 Unit Cubes & Volume Formulas
1. Find the volume by counting unit cubes.
Volume (cubic units):
2. Find the volume of a prism with:
Length \(l = 6\text{ cm}\)
Width \(w = 2\text{ cm}\)
Height \(h = 4\text{ cm}\)
Show work & volume (\(\text{cm}^3\)):
3. A rectangular crate has a base area of \(B = 24\text{ in}^2\) and a height of \(h = 5\text{ in}\). Calculate its volume.
4. Which expression represents the volume of a crate that is \(4\text{ ft}\) long, \(3\text{ ft}\) wide, and \(5\text{ ft}\) tall?
A) \(4 + 3 + 5\)
B) \(4 \times (3 \times 5)\)
C) \((4 \times 3) + 5\)
D) \(4 \times 3 \times 3\)
Section 2 Real-World Cargo Problems (NC.5.MD.5.ii)
5. A wooden shipping crate is \(8\text{ ft}\) long, \(4\text{ ft}\) wide, and \(3\text{ ft}\) tall. What is the volume of the crate?
6. A storage bin has a rectangular base with an area of \(35\text{ in}^2\) and a height of \(6\text{ in}\). How many cubic inches of space are inside the bin?
7. Crate A measures \(6\text{ cm} \times 4\text{ cm} \times 5\text{ cm}\). Crate B measures \(5\text{ cm} \times 5\text{ cm} \times 5\text{ cm}\). Which crate holds more, and by how many cubic centimeters?
8. Architect Challenge: A cargo container has a total volume of \(72\text{ ft}^3\). Its length is \(6\text{ ft}\) and its width is \(3\text{ ft}\). What is its height? Explain how you know.
