The Theory of Measurement-Volume (ToMV) construct describes how children come to

understand foundations of volume measure. These foundations include the nature of

units, dynamic generation of volume as a product of area and length, and development of

formulas to find the volume measure of right and non-right prisms, and of cylinders. Each

of six levels reflects a cluster or network of related understandings. Performances

described by the six levels of the construct range from informal, direct comparison of

volumes to using Cavalieri’s principle to find the volume of different non-right prisms

and cylinders. Instruction designed to support learning should not proceed as if all

understandings at an earlier level must be stable and widespread (i.e., “mastered”) before

students can learn anything represented in later levels. Instead, the construct should be

interpreted as a map of understandings worthy of assessment, with more challenging and

generally later developing ideas and procedures represented in higher levels, and with the

lower levels representing ideas and procedures that serve as building blocks for forms of

reasoning constituting higher levels. Depending on the context and content of reasoning,

the same student may perform at different levels, because each level represents an ideal

kernel of reasoning that learners may or may not see as productive for particular purposes

and contexts. When learners are faced with novel situations, they may “revert” to the

forms of reasoning expressed by earlier, lower levels. But we expect that, in general, if

they have received prior experiences with forms of reasoning described by lower levels,

children will revisit them only briefly when they are relevant to new contexts. For

example, if length is the initial context of measure, understanding the limitations of

failing to tile units of length measure (e.g., leaving gaps or overlaps) may develop over a

considerable period of time (e.g., weeks or months). But when properties of units of

measure are once again considered for establishing measure of volume, we anticipate that

children will more quickly, perhaps within a day, realize the consequences of failing to

tile units of volume measure

ToMV1 ToMV2 ToMV3 ToMV4 ToMV5 ToMV6

ToMV1 and ToMV2

At the initial point of learning, level ToMV1, the student informally describes volumes

as space inside or space contained by an object. Level ToMV2 builds on these informal

conceptions of volume as space occupied to compare volumes of specific objects,

perhaps by nesting one within another, or by filling containers with popcorn or sand and

comparing the amounts needed to do so.

ToMV1 ToMV2 ToMV3 ToMV4 ToMV5 ToMV6

ToMV3

At level ToMV3, students accumulate units to compare volumes indirectly. For example,

students find and compare the volumes of right rectangular prisms by filling them with

unit cubes and counting the cubes. Properties of units are recognized as important for

establishing accurate comparisons of volumes. Students recognize that a volume measure

is a ratio: a volume of 36u3 means that the volume is 36 times that of 1u3. Students

initially find volume by counting unit cubes in structures where all the cubes are visible;

subsequently, they develop strategies to visualize and account for “hidden” cubes that

cannot be seen. Counting hidden cubes is enabled by mentally imposing a lattice of unit

cubes on the space contained by the structure.

Figure 1. Rectangular prism with no hidden cubes

Figure 2. Rectangular prism with hidden cubes

ToMV1 ToMV2 ToMV3 ToMV4 ToMV5 ToMV6

ToMV4 and ToMV5

Structuring the space as a 3-D lattice continues at level ToMV4, where students find and

compare whole number volumes if only parts of the volume are structured. For example,

they may use cubes to find the height of a hollow right rectangular prism and then cover

its bottom layer. Using this information, they conclude that the volume measure can be

obtained as a product of length, width (of the base) and the height.

Figure 3.

ToMV5 extends these strategies to find and compare volumes that involve fractional

units.

Figure 4. Rectangular prism that contains fractional height units.

Figure 5. Rectangular prism that contains fractional width unit.

ToMV1 ToMV2 ToMV3 ToMV4 ToMV5 ToMV6

ToMV6

At ToMV6, students generate and interpret volume dynamically as sweeping an area

through a length. This entails thinking of volume as a continuous quantity. For example,

the volume measure of a right rectangular prism with a base of area 54 cm2 and height 10

cm is conceived of as a sweep of the base-area through the height, with the result of 540

cm3.

Figure 6. Students start with a 54cm2 base and a height of 10cm.

Figure 7. Base is swept throught the height, resulting in a measure of 540cm3.

ToMV1 ToMV2 ToMV3 ToMV4 ToMV5 ToMV6

Based on sweeping areas through lengths, students generate, use, and explain formulas

for the volume of right rectangular prisms, including those with fractional dimensions. As

their experience with sweeping area through height continues, students find and compare

volumes of other right prisms (e.g., those with triangular and hexagonal bases), cylinders,

and composite structures. Using sweeping of area of a base through height, students

generate, use, and explain formulas for the measure of the volume of cylinders and right

prisms. For example, the volume of a triangular prism can be thought of the product of

the area of triangle and the height of the prism.

Figure 8.

Finally, students use Cavalieri’s principle to explain why the volume measure of non-

right prisms and cylinders is equal to that of their corresponding right prisms and

cylinders.

Figure 9.

ToMV1 ToMV2 ToMV3 ToMV4 ToMV5 ToMV6

Level Performances Examples

6

Find and

compare

volumes

dynamically

as area

swept

through

length

and/or as

product of

area and

length.

ToMV6D Explain and use

Cavalieri’s principle to

find the volume of

different non-right prisms

and cylinders.

“This stack of cards has the same volume even if it isn’t

stacked up neatly because you are still taking the same

area of the bottom card, and pulling it to the same height,

so it would still be 2 x 3 x 4 = 24 cubic inches.”

“If the stack of cards is pushed outward for each card, it is

also pushed inward by the same amount on the opposite

side, so nothing has changed. The volume must be the

same.”

ToMV6C Find and compare volume

of a variety of prisms.

“If the area of the base is about 3 and 1

2 square inches and

the height is 4 inches, then I would say that the volume of

the hexagonal prism is base area x height, or 3.5 times 4,

which is about 14 cubic units.”

ToMV6B Find and compare

volumes of right cylinders

using sweeping of area

through height.

Generate, use, and explain

formulas for volume of

right cylinders.

Find and compare surface

areas of right cylinders

using sweeping.

“If the area of the base is about 3 squares and I move that

through the height of the side, 11 cubes, then I would get

about 33 cubic units.”

“So if the area of the base is about 3 squares and the height

is 11, then you can just say that the base area times the

height equals the volume, so 3 x 11 = 36 cubic units.”

“If I wrap the grid paper around the cylinder, the distance

around the base [circumference] is about 9 units, and then

if I pull that through the height of 16, I get 9 x 16 = about

144 square units for the surface area of the side of the

cylinder.”

ToMV6A Find and compare

volumes of right

rectangular prisms

(including some with

fractional dimensions)

using sweeping of area

through height.

Generate, use, and explain

formulas for volume of

right rectangular prisms.

Justify sweeping as a

strategy for finding

volume.

(For a 3 x 4 x 4 hollow prism without markings) “If the

area of the base is 12 squares, and I move that all the way

through the height of the box, 4, then I would get 48 cubic

units.”

“So if the area of the base is 12 and the height is 4, then

you can just say that the base area times the height equals

the volume, so 12 x 4 = 48 cubic units.”

“If the area of the base is length x width, or 3 x 4, and I

layer that 4 times for the height, I would get 48 cubic

units, or 3 x 4 x 4 = 48.”

5

Find and

compare

volumes

with

fractional

units using

partial

structuring

strategies.

ToMV5

Find and compare

volumes of right

rectangular prisms

(including some with

fractional dimensions)

using units with partial

structuring strategies.

(For a 3 x 4 x 4.5 hollow prism without markings) “I can

use the cubes to make a bottom layer of 12 and a height of

4 and looks like half a cube, then multiply those to get 54

cubic units.”

4

Find and

compare

whole

number

volumes by

partially

structuring

volumes.

ToMV4

Find and compare

volumes of right

rectangular prisms using

partial structuring

strategies.

(For a 3 x 4 x 3 hollow prism without markings) “I can use

the cubes to make a bottom layer of 12 and a height of 3,

then multiply those numbers together to get 36 cubic

units.”

“This box has a base of 12 and height of 3 = 36 cubic

units, so it is smaller than the box that has a base of 10 and

height of 4 = 40 cubic units.”

3 Find and

compare

volumes using

units.

ToMV3C Find and compare

volumes of right

rectangular prisms by

counting unit cubes,

including “hidden

cubes.”

Create different right

rectangular prisms of

equal volume.

Explain volume measure

as a ratio.

(For a 3 x 3 x 4 with hidden unit) “I can count the cubes to

find that there are 34 cubes on the outside, and then there

would be 2 cubes you can’t see on the inside, so the whole

thing is 36 cubic units.”

“These two take up the same amount of space, even

though they do not look the same, because they each

contain 36 cubic units.”

“45 in3 means that the volume of this structure is 45 times

that of the unit cube—the cube with length, width and

height of 1 inch or 1 in3 .”

ToMV3B Find and compare

volumes of right

rectangular prisms by

counting unit cubes (no

“hidden cubes”).

(For a 2 x 2 x 9 lattice) “I can count the cubes to find that

there are 36 cubes all together.”

ToMV3A

Recognize and explain

properties of units (e.g.,

identity, tiling).

“Cubes are better than beans for filling the box, because

they are all the same size with no gaps in between.”

“You have to use cubes because they take up the space

inside. If you just drew squares to cover the box, the

inside would still be empty.”

2

Find and

compare

volumes using

direct

comparison

strategies

and/or indirect

comparison

without units.

ToMV2B

Compare volumes of

different solids by filling

them.

“The popcorn filled this first box, but the same amount of

popcorn wasn’t enough to fill the second box. The second

box can hold more popcorn inside it than the first box

can.”

“When we filled the first one with water, we used all of

the water. When we filled the second one, some water was

left over, so it doesn’t hold as much as the first one.”

ToMV2A Compare volumes of

different solids using

nesting.

“The red one fits inside of the blue one, so its volume is

smaller.”

1

Recognize and

compare three-

dimensional

shapes by

informally

attributibuting

space inside.

ToMV1C Differentiate surface

area from volume.

“The buildings have different numbers of windows (faces

of cubes composing the structure), but it looks like they

have about the same amount of space inside.”

“The outside of the cylinder is like a wrapped rectangle,

but the cylinder has stuff inside.”

ToMV1B Compare space inside or

space contained by two

or more objects.

“It looks like this one is bigger inside.”

“This one holds more than that one.”

ToMV1A Recognize space inside

or space contained by

object.

“You can put something inside it.”