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The task
Students were given the following instructions:

Design and build a scale model enlarging or shrinking an everyday object using a ratio of 1:10 (or specify a different ratio) for each dimension. Describe the process you used to complete the model, the mathematics you used, the measurements of the original and the model including appropriate units, calculations of volume of the original and the scale model, and analyze the relationship between those calculations.

Circumstances of performance
This sample of student work was produced under the following conditions:
alone - in a group
in class - as homework
with teacher feedback with peer feedback
timed opportunity for revision

This work sample illustrates a standard-setting
performance for the following parts of the standards:
a Geometry and Measurement Concepts: Be familiar with two- and three-dimensional objects.
d Geometry and Measurement Concepts: Determine and understand length, area, and volume.
h Geometry and Measurement Concepts: Choose appropriate units of measure and convert with ease between like units.
j Geometry and Measurement Concepts: Reason proportionally with measurements.
d Mathematical Skills and Tools: Measure accurately.
f Mathematical Skills and Tools: Use formulas appropriately.
c Putting Mathematics to Work: Design a physical structure.

What the work shows

 


a Geometry and Measurement Concepts: The student is familiar with assorted two- and three-dimensional objects….
d Geometry and Measurement Concepts: The student determines and understands length, area, and volume (as well as the differences among these measurements)….
h Geometry and Measurement Concepts: The student chooses appropriate units of measure and converts with ease between like units….
j Geometry and Measurement Concepts: The student reasons proportionally with measurements…to make…larger scale drawings.
The student appropriately used the 1:10 scale to increase each of the linear measurements. He converted to centimeters as appropriate, for instance, stating that two enlarged walls are 15 cm long instead of 150 mm long.
The student recognized that the half-centimeter thickness of the tagboard would be a factor in creating an accurate scale model, sixteen centimeters on each side. The two fifteen-centimeter walls are each about two extra half centimeters where they touch the adjacent walls.
The student recognized, correctly, that the volume of the enlargement is 10 x 10 x 10 = 1,000 times larger than the original. Some explanation is in order, though. Here, since the measurements of the original piece are stated, the smaller volume could have been computed and shown to be one-thousandth of the larger volume.

d Mathematical Skills and Tools: The student measures length, area, [and] volume…accurately.

f Mathematical Skills and Tools: The student uses…formulas…appropriately.

Standard formulas for volumes of rectangular prisms and of cylinders are applied accurately here. The student recognized that the volume of his creation is determined by adding the easily calculated volumes of the standard geometric shapes that are component parts.

There are two errors in the units of measure in the calculation of total volume. In the first line, “4 cm” should be “4 cm²” since the 4 represents the square of the 2 cm radius. The student computed the volume of each of the four cylindrical “bumps” as V = [r² x ] x h. On the fifth line, “256 cc” should be “256 cm²” since it is the area of the prism’s square base computed from the previous line’s “16 cm x 16 cm.” The total volume determined is given with the correct units, cubic centimeters.

c Putting Mathematics to Work: The student designs a physical structure, in which the student:

• generates a plan to build something….
The paragraph describes the process of building the enlarged LEGO®. It also describes the design issues that arose in creating the components.

• uses mathematics from Standard 2 to make the design realistic or appropriate, e.g., areas and volumes in general and of specific geometric shapes.

• summarizes the important features of the structure….
It would be helpful to the reader, and more appropriate mathematical communication, if the student had referred at least once to the “bumps” as “cylindrical bumps” or “cylinders.” The student appears to have recognized them as such when using the formula for cylindrical volume in computing the total volume later in the piece. The student observed that these cylindrical bumps could be made from the same tagboard as the block by stacking four circular half-inch thick disks to make each two-inch tall cylinder.

• prepares a presentation or report that includes the question investigated, a detailed description of how the project was carried out, and an explanation of the findings.