Developmental Progression for Area Measurement
Area Quantity Recognizer
- Identifies area as an attribute.
- Comparing. May use side-matching strategies in comparing areas.
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Physical Coverer and Counter
- Attends to some aspects of the structure.
- Tiling. May completely cover a rectangular space with physical tiles.
- Comparing. May make intuitive comparisons of 2D regions based on simple, direct comparisons (e.g., a child places one sheet of paper over the another piece of paper to select the sheet that covers more space).
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Complete Coverer and Counter
- Drawing. Draws a complete covering of a specific region without gaps or overlaps and in approximations of rows.
- Producing. When provided with more than the total number of physical tiles needed, can build a region of specified area (e.g., build a rectangle with an area of 12 from a pile of 20 tiles).
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Area Unit Relater and Repeater
- Quantifying. Counts individual units, guided by rows.
- Drawing. Draws a complete covering based on an intuitive row or column structure. Children attend to drawing equal sized units, one at a time.
- Comparing. Relates size and number of units (few larger, more smaller). Recognizes that different area units will result in different measures. Also recognizes that identical units should be used, at least intuitively and/ or in some situations. May choose to use simple doubling strategies. May compare areas by counting units. For example, when asked if Shape A and Shape B take up the same amount of space on the paper, a child states that they take up the same amount of space “because they both have 4.”
- Iterating. When provided with fewer than the total number of physical tiles needed, iterates individual tiles to measure and eventually cover a given region.
- Producing. Builds a region of area from an insufficient number of unit tiles through individual unit iteration (e.g., “leap frogging,” when given a set of n tiles, the child may translate the first tile to represent both tile 1 and tile n + 1, the second tile to represent both tile 2 and tile n + 2…).
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Initial Composite Structurer
- Identifies a square unit as both a unit and a component of a unit of units (a row, column, or group); however, the child needs figural support to structure the space (this may include physical motions of some of the tiles or drawing some collections of units rather than from using the dimensions.) The following are two components of this level.
- Can find reasonable estimates of regions. When asked to measure areas of regions, the child may apply an upper (counting all wholes and partials as wholes) or lower bound (counting only wholes as wholes) strategy.
- Tacitly recognizes usefulness of dimension displays as an indicator of the number of units along one dimension (row or column). When asked to draw a rectangle of specified area, may succeed in identifying dimensions of a region without correctly drawing the array of units.
- Organizes counting, drawing, or moving of objects in composites units (unit of units).
- Sublevel A. Early on, may be unaware of the congruence of rows, yet uses a unit of units (not necessarily a column, or a row) to determine a measure of a rectangular region. Structuring is very limited in scope; uses a mixture of individual units and units of units.
- Sublevel B. Later, structures a rectangular region as a set of rows (more comprehensively than before), attending to the collinearity of rows (expects rows to have the same number of units). Does not yet coordinate rows and columns of units with the linear extent of both length and width along the measured object.
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Area Row and Column Structurer
- Can decompose and recompose partial units to make whole units.
- Drawing. Uses the dimensions to constrain the placement of parallel row and column line segments. Measures both dimensions to determine the size of the iterated squares or rows of squares, as well as determine the number of rows needed in drawing. May not need to complete the drawing to determine the area by counting systematically while attending to rows (most younger children) or computation (repeated addition or multiplication).
- Comparing. Explicitly relates size and number of area units (conversion).
- Producing. When asked to draw a rectangle of specified area, may connect the measure of area to a count of area units without connecting the count of area units to the dimensions of the region.
Array Structurer
- Has an abstract understanding of the rectangular area formula.
- Quantifying. Solves for area without making a drawing. In multiple contexts, children compute the area from the length and width of rectangles and explain how that multiplication creates a measure of area. The child operates arithmetically on area measures (e.g. comparing a 2x3 rectangular unit to an 8x10 rectangular region).
- Comparing. When comparing non-congruent regions with equal areas, realizes that the regions have the same area. May use transitive reasoning.
Conceptual Area Measurer
- Has an abstract and generalizable understanding of the rectangular area formula.
- Quantifying. Restructures regions to determine how to use known area measures to find the areas of triangles, kites, trapezoids, and parallelograms. Recognizes that formulas for areas of these shapes are related to the formula for the area of a rectangle. Provides arguments and justifications as evidence of physical or figural de/recomposition and rigid transformations of these shapes. Uses geometric properties of these shapes to support reasoning.
- Comparing. When comparing non-congruent regions with equal areas, integrates and operates on qualitative and quantitative aspects of the regions.