🎯Today's objective: Find the volume of right rectangular prisms with fractional edge lengths.
Need a hint?
Re-read the problem and underline the numbers and the question.
Pick one representation (model, table, or equation), show your
steps, and check that your answer makes sense for the situation.
The school is building a time capsule! The students voted to bury a
rectangular box that is 15 inches long,
9 inches wide, and 8 inches tall.
The principal wants to know: (1) How many cubic inches of space is
inside the box? (2) Students plan to fill the bottom of the box with
a layer of foam cubes that are each 1.5 inches on
every side. How many foam cubes fit in one layer on the bottom? How
many layers fit? How many total foam cubes fill the box? Your team
must calculate the volume and prove the cube-packing plan works.
Investigation
The Problem: (A) Find the volume of the time capsule
box (15 in x 9 in x 8 in). (B) How many 1.5-inch foam cubes fit along
the length? Along the width? Along the height? (C) How many cubes in
one layer? How many layers? How many total cubes? (D) What is the
total volume of all the foam cubes? Is it the same as the box volume?
Why or why not?
Visual Model: Layers Build Volume
Step-by-Step Investigation Guide
Identify the three dimensions. The box is 15 in
long, 9 in wide, and 8 in tall. Write l = 15, w = 9, h = 8.
What is the difference between area (2 dimensions) and volume (3
dimensions)?
Find the base area (B). The base is the bottom
face: B = l x w = 15 x 9 = 135 in². This tells you the area of
one layer.
Why is the base area measured in square inches, but volume is in
cubic inches?
Multiply by the height. V = B x h = 135 x 8 = 1,080
in³. Also check: V = l x w x h = 15 x 9 x 8 = 1,080 in³.
Why do both formulas (V = Bh and V = lwh) give the same
answer?
Figure out the cube packing. Each foam cube is 1.5
in on a side. Divide each box dimension by 1.5: Length: 15 / 1.5 =
10 cubes. Width: 9 / 1.5 = 6 cubes. Height: 8 / 1.5 = 5.33... so
only 5 full layers fit.
What happens to the 0.5 inches of leftover space at the top? Can
a partial cube fit?
Count the total cubes. One layer: 10 x 6 = 60
cubes. Total: 60 x 5 = 300 cubes. The volume of 300 cubes: 300 x
(1.5 x 1.5 x 1.5) = 300 x 3.375 = 1,012.5 in³.
Why is 1,012.5 less than 1,080? Where did the missing 67.5 cubic
inches go?
Compare box volume to cube volume. Box = 1,080
in³. Cubes = 1,012.5 in³. Difference = 67.5 in³ of
empty space at the top (the gap where a 6th layer does not fit).
Is it possible to fill a box completely with cubes that do not
evenly divide the height? Why or why not?
Language Support: Key Vocabulary
volume — the amount of space inside a 3-D
shape, measured in cubic units (in³)
cubic unit — a small cube used to measure
volume; 1 in³ = a cube 1 inch on each side
layer — one flat level of cubes that fills
the base of the box
base area (B) — the area of the bottom face
of a prism; B = length x width
prism — a 3-D shape with two identical
bases and rectangular sides
right rectangular prism — a box shape where
all angles are 90 degrees
capacity — how much a container can hold
inside
gap / leftover space — empty space where
cubes do not fit perfectly
Sentence Frames
"The volume of the box is _____ in³ because I multiplied
_____ times _____ times _____."
"One layer has _____ cubes because _____ fit along the length and
_____ fit along the width."
"Only _____ full layers fit because _____ divided by 1.5 equals
_____ with _____ left over."
Multiple Representations
Layer Diagram
Draw the bottom of the box as a rectangle. Divide it into a 10 x 6
grid of 1.5-inch squares. This is one layer (60 cubes). Then show
5 stacked layers from the side.
Best for: seeing how layers of cubes build volume
Two Formulas
V = l x w x h = 15 x 9 x 8 = 1,080 in³
V = B x h = (15 x 9) x 8 = 135 x 8 = 1,080 in³
Both give the same answer. V = Bh is useful when you already know
the base area.
Best for: connecting the formula to the layer concept
Packing Table
Table: Dimension | Box Size | Cube Size | How Many Fit |
Leftover
Length: 15 / 1.5 = 10, 0 leftover
Width: 9 / 1.5 = 6, 0 leftover
Height: 8 / 1.5 = 5, 0.5 in leftover
Best for: organizing the division and spotting the gap
Team Roles
FacilitatorKeeps the group on track, watches the timer, makes sure everyone
speaks.Mission 12 task: Make sure the team finds the box volume, the
cubes per layer, total cubes, AND explains the leftover
space.
Model BuilderDraws the layer diagram, labels dimensions, and creates the
packing table.Mission 12 task: Draw a top-view grid showing 10 x 6 cubes in one
layer, and a side view showing 5 stacked layers with the gap at
the top.
Precision CheckerChecks all multiplication and division, verifies units
(in³), compares V = lwh with V = Bh.Mission 12 task: Verify that 300 x 3.375 = 1,012.5 in³, and
that 1,080 - 1,012.5 = 67.5 in³ of gap. Check both volume
formulas match.
ReporterPrepares the defense: claim, evidence, and one mistake the team
caught.Mission 12 task: Explain why the cubes do not fill the box
completely and what the gap means. Connect layers to the volume
formula.
Timed Lab Phases
Ready
Click a phase, then press Start.
03:00
Read the scenario out loud. Assign roles.
Underline the box dimensions and the cube size.
What are the 3 dimensions of the box? What is the cube size?
What is the difference between area and volume?
Estimate: is the volume closer to 500, 1,000, or 2,000 cubic
inches?
Checkpoint: Every teammate can state the 3 dimensions and knows
the volume formula.
Calculate the box volume two ways. Then start
the cube-packing analysis.
V = l x w x h = 15 x 9 x 8 = ?
V = B x h: first find B = 15 x 9, then multiply by 8.
Do both formulas give the same answer?
Start dividing: 15 / 1.5, 9 / 1.5, 8 / 1.5.
Checkpoint: Volume = 1,080 in³ using both formulas.
Division for cubes is started.
Complete the cube packing. Find total cubes and
calculate the gap.
One layer: 10 x 6 = 60 cubes. Layers: 5. Total: 60 x 5 = 300.
Volume of all cubes: 300 x (1.5)³ = 300 x 3.375 = ?
Gap = 1,080 - 1,012.5 = ? What does this gap mean?
Draw the layer diagram and label the gap.
Checkpoint: 300 cubes, 1,012.5 in³ filled, 67.5 in³
gap explained.
Prepare your presentation. Connect layers to
volume. Explain the gap.
State the box volume and how you calculated it.
Show the layer diagram: how cubes fill the base, how layers
stack.
Explain why 300 cubes do not fill the entire box.
Share one mistake your team caught and fixed.
Checkpoint: Reporter can present volume, cube count, and gap
explanation in under 60 seconds.
Challenge Zone
Extension: If the foam cubes were 2 inches on each
side instead of 1.5 inches, how many would fit? (Hint: 15/2 = 7.5, so
only 7 fit along the length.) How does changing the cube size affect
the gap? Is there a cube size where the gap is exactly zero?
What If? What if the box were 15 in x 9 in x 12 in
instead (taller)? Calculate the new volume. Now how many layers of
1.5-inch cubes fit? Is the gap bigger, smaller, or the same?
Real-Life Connection: Think about packing a moving
box with books. Each book takes up space (volume). When you stack
them, some space is always wasted between the books. Volume tells you
the maximum capacity, but packing is never 100% efficient. This is why
real shipping companies care about both volume and packing patterns.
Defense Preparation
Questions Your Team Must Answer
What is the volume of the time capsule? Show both formulas.
How many foam cubes fit in one layer? How did you figure that out?
Why do only 5 layers fit even though 8 / 1.5 = 5.33?
What is the gap, and what does it mean for the time capsule?
How are V = lwh and V = Bh connected? When would you use each one?
Sentence Starters for Your Defense
"The volume is _____ in³ because _____ times _____ times _____
equals _____."