Developmental Progression for Volume Measurement
Volume Quantity Recognizer
- Filling and Packing. Recognizes capacity as an attribute, beyond implicit acting on materials.
- “This box holds a lot of toys.”
- Building. Builds with blocks, associating more blocks with terms like “big” and fewer blocks with terms like “small”
- Comparing. Initially, recognizes volume as an attribute, describes objects with words such as big, small, and tiny. Eventually, may compare volume recognizing only one dimension.
- Filling. Fills a container using another (smaller container) and counts the number needed to completely fill the larger container.
- Packing. Places cubes into a rectangular box to fill it; eventually, may pack entire box with a focus on leaving no gaps.
- Building. Given a 3-D object constructed of cubes, recognizes and counts cubes (the child may be counting “blocks” or even “squares”) on multiple faces.
- Comparing. Compares objects by physically or mentally aligning; refers to a least two dimensions of objects.
- May begin to associate number of scoops/cubes in comparison, simply using a bigger number for larger container.
- Filling. Able to estimate number of scoops needed to fill. Able to attend to both the portion of container filled and the portion remaining unfilled. Recognizes when container is half full.
- Packing. Exhibits initial spatial structuring. Packs box neatly and completely with cubes; may count one cube at a time, while packing, to determine total.
- Building. Exhibits initial understanding of cubes as filling a space (completely, without gaps). Counts on all faces of a 3-D object constructed of cubes, has a developing sense of the cube as a unit, begins to recognize that squares on adjacent faces of a rectangular prism and that share a side are faces of the same cube.
- Comparing. Compares objects by physically or mentally aligning and explicitly recognizing three dimensions (Note: Although cylindrical containers have only two dimensions to vary, height and radius, children at this level demonstrate understanding of 3-D space.)
- Compares the volume of objects by counting the number of cubes, showing initial understanding of cubes as filling space (as described above); may break a larger object into smaller pieces in order to “see” all the cubes. Recognizes that objects can look different, but still contain the same number of cubes.
Volume Unit Relater and Repeater
- Relates size and number of units explicitly; understands that fewer larger than smaller units will be needed to fill or pack a given container. Can accurately convert units in 1:2 ratio.
- Filling. Uses simple units to fill containers with accurate counting, completely filling the scoop each time. After one unit has been poured into the container can anticipate the volume of the container by iterating the height filled by the unit exhaustively along the height of the container.
- Packing. Uses discrete units to pack a container without gaps and with accurate counting. Able to iterate unit throughout volume, maintaining equal unit size and spacing.
- Building. Exhibits developing understanding of cubes as filling a space. Counts cubes, not faces (or “faces-as-cubes”).
- Comparing. When comparing two 3-D objects in cases such as, congruent objects containing different numbers of units, non-congruent objects containing the same number of units, or non-congruent objects containing different numbers of units, describes correctly the relative volumes of objects by reasoning about unit size.
Initial Composite 3-D Structurer
- Understands cubes as filling a space. Explicitly relates size and number of units to volume. Uses additive reasoning (e.g., skip-counting to obtain total). Conversion of units develops to include ratios other than 1:2.
- Filling. Relates number of cubes to cubic units as measured by capacity. Given a graduated cylinder marked in cubic-inch units, child understands that sand filled to the 10 in the cylinder would fill a box that holds ten, 1-inch cubes.
- Packing. Begins to visualize and operate on composite units such as rows or columns (what we call a 1 x 1 x n core). Iterates to pack the space completely, accounting for “internal/hidden” cubes. Decomposes space, allowing for accurate use of units and subunits. Recognizes when a box is half full, visualizes remaining rows or columns.
- Building. Develops more accurate counting strategies. Counts systematically, accounting for internal/hidden cubes, and moves to operating on composites, including rows and columns.
- Comparing. Connects volume as number and volume as space. Develops sense of conservation for the cases in which transformation is involved.
3-D Row and Column Structurer
- Able to coordinate flexibly filling, packing, building aspects of volume. Shows a propensity for additive comparisons (e.g., “this one has 12 more”) but may show some nascent multiplicative comparisons (e.g., “this one is four times as big”).
- Initially counts or computes (e.g., number of rows times number of columns) the number of cubes in one layer (1xmxn), and then uses addition or skip counting by layers to determine the total volume. Eventually moves to multiplication (e.g., number of cubes in a layer times number of layers).
- Operates fluidly and flexibly on units (cubes), units of units (rows or columns), and units of units of units (layers). Composes and decomposes array ←→ layers ←→ rows/columns ←→ units. With perceptual support, can decompose 3-D arrays into other, complex 3-D arrays (not only layers, rows, or columns) and calculate the number of these smaller arrays in the larger array.
3-D Array Structurer
- Has an abstract understanding of the rectangular prism volume formula. Shows a propensity for multiplicative comparisons, coordinates multiplicative and additive comparisons flexibly.
- With linear measures or other similar indications of the three dimensions, multiplicatively iterates cubes in a row, column, and/or layer to determine volume. In multiple contexts, can compute the volume of rectangular prisms from their dimensions and explain how that multiplication creates a measure of volume.
- Develops the ability to visualize and operate on both horizontal and vertical layers, even without perceptual support.
- Can decompose 3-D arrays into other, complex 3-D arrays (not only layers, rows, or columns) and calculate the number of these smaller arrays in the larger array (by using repeated addition, multiplication, or division). Coordinates the spatial and symbolic decompositions.