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Understand Volume as Packing with Unit Cubes

Students understand that the volume of a solid figure equals the number of unit cubes needed to fill it completely without gaps or overlaps. They connect the abstract concept of cubic units to a physical packing model.

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Understand Volume as Packing with Unit Cubes

Grade 5 · Understanding Volume · question 1 of 3

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20 rungs. A star at the top.

The box holds one layer of cubes. Count them.

Count the cubes.

How it is taught

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Volume is how many unit cubes pack a solid with no leftover space.

Count every cube, including ones you cannot see.

Read it: Volume: cubes that fill a box

Volume is how much space a solid takes up, counted in unit cubes. This box is packed 3 by 2 by 4: 6 cubes in a layer and 4 layers, so 24 cubes. Its volume is 24 cubic units.

  1. Count the cubes in the bottom layer: 3 rows of 2 is 6.
  2. Count the layers: the box is 4 cubes tall.
  3. Multiply: 6 cubes in a layer × 4 layers = 24 unit cubes.

A unit cube is 1 unit long, 1 unit wide and 1 unit tall: 1 cubic unit.

A box packed with no gaps and no overlaps holds the same number of cubes in every layer.

So the volume is the cubes in a layer times the layers: 6 × 4 = 24 cubic units.

Volume is the number of unit cubes that fill a solid, with no gaps and no overlaps.

Counting only the cubes on the front, and missing the ones hidden behind them.

Count a whole layer, 3 × 2 = 6, then every layer: 6 × 4 = 24, the hidden cubes too.

A box is 6 by 3 by 6. Its volume is ___ cubic units. — 108. The base is 6 × 3 = 18, and 6 layers: 18 × 6 = 108.

3 × 2 × 2 box = 12 cubic units

Volume tells us how much three-D space a shape fills. Let's find the volume of this box together!

Grade 5 · Packing a box with unit cubes

We measure volume in unit cubes — little cubes that are one unit on every side. Each one is 1 cubic unit.

Our empty box has two layers. Each layer fits two rows of three unit cubes — with no gaps and no overlaps.

Let's pack the bottom layer first. Slide in three cubes for the front row.

Now add three more cubes behind them.

The bottom layer is full — that's six cubes!

Now stack the top layer. Place three cubes in the front row.

Then three more cubes in the back row.

The top layer holds six cubes too.

Six cubes plus six cubes equals twelve cubes in all.

The volume is twelve cubic units. We packed the whole box — nice work!

Now you try. Pack the box in your mind and type the volume. How many cubic units?

What you see

While they climb, your page writes one true line to ask about tonight, and a short note each Sunday.

MayaGrade 3 · Mathematics
Did you know?

A chessboard has 64 squares — share them out between the 8 rows and each row holds exactly 8.

Today’s goal
1 star — one skill mastered, every rung of its ladder climbed
Climbing now
Divide within 100

An example student.

MayaGrade 3 · Mathematics

Maya earned 2 stars this week — Interpret multiplication as equal groups, which took 6 answers over 3 days before its star landed 3 days ago and Multiply by 6, 7, 8 and 9. Next up: Divide within 100.

An example week.

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