Developmental Progression for Length Measurement
Length Quantity Recognizer
- Identifies length and distance as attributes.
Length Direct Comparer
- Physically aligns two objects to determine which is longer or if they are the same length.
- May use a ruler (as a stick rather than a measuring tool) to directly compare it and another object. Uses terms: long, longer, longest.
Indirect Length Comparer
- Note: Non-numerical comparison is a sub-trajectory within this LT. This level begins to develop in parallel with the subsequent measurement levels over time.
- Compares the length of two objects by representing them with a third object.
- Uses terms: long, longer, longest, short, shorter, shortest.
End-to-End Length Measurer
- Lays units end-to-end to create measures for comparison.
- Although the child can produce a meaningful measure when given all the objects, children may use fewer objects or a single object to measure.
- Uses rulers with minimal guidance.
Length Unit Relater and Repeater
- Iterates a single unit to measure.
- Relates size and number of units, at least intuitively:
- Can add two lengths to obtain the length of the whole.
- Uses rulers with minimal guidance and more success than at the prior level. Measures a length accurately with a ruler if aligned to the zero point.
Consistent Length Measurer
- Measures straight paths consistently, knowing the need for identical units, relationship between different units, partitions of unit, zero point on rulers, and accumulation of distance. Can resolve broken ruler tasks effectively.
- Begins to relate and coordinate units and subunits. For example, may use both meters and centimeters when measuring:
- Can compare two bent paths by relating the parts in a 1:1 relation of successive parts.
- Begins to estimate reasonably.
- Iterates a single length unit without gaps or overlaps along a straight path.
Conceptual Ruler Measurer
- Has an "internal" measurement tool.
- Mentally moves along an object, segmenting it and counting the segments.
- Operates arithmetically on measures. Projects or translates given lengths to determine missing lengths..
- Notices and/or is perturbed by geometric inconsistencies.
- Estimates the length of an object that is not partitioned with accuracy and without any available image of the standard unit.
- Employs explicit strategies to estimate lengths, including developing benchmarks for units (e.g., an inch-long piece of gum) and composite units (e.g., a 6-inch dollar bill) and mentally iterating those units.
Integrated Conceptual Path Measurer
- Computes perimeter of a polygon, including complex cases.
- Can find several related cases of polygons with the same perimeter, or the same area, and relate those cases to one another by logical comparison to provide evidence of an underlying pattern.
- Can re-arrange portions of a path A to B without number measures for the parts involved, arguing that both paths are the same length based on geometric properties.
- Can make changes in one part of a figure and adjust other sides to compensate for changes of length to maintain the fixed overall path length.
- Makes claims about sets of complex paths based on perimeter and other linear aspects of the objects. Can analyze length within two- and three-dimensional contexts.
- In selection of units, children show well-developed ideas of precision and accuracy.
Abstract Length Measurer
- Organizes and synthesizes sets of objects based on perimeter or collections of complex, bent paths based on overall length in two- or three-dimensional contexts to formulate and justify a valid argument.
- Can construct derived units with linear measures, such as miles per hour, and make appropriate unit conversions.
- Computes perimeter or path length, including units and divisions of units including measures of non-integer values. Can explain that this subdivision process is potentially unlimited.
- Notices and is perturbed by geometric inconsistencies within polygons (e.g., the interaction between angles and sides).
- Children use measures as evidence of the existence of patterns and relationships between quantities and construct models that describe these patterns and relationships (e.g., can use lower and upper bounds of linear estimates to order several curves by length). Measures to the degree of precision allowed by a tool by estimating to a fraction of the smallest calibration mark provided on the instrument.