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The task
Students were given a three-part question about volumes and surface areas of cubes.

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

This is a Released Task from the New Standards reference examination. In Sample 1 and Sample 3 (completed in Spanish), the students were timed. Sample 2 was completed in Russian in a bilingual classroom. In Sample 4, the student had the opportunity for revision after peer feedback.

Sample 4 was completed in Chinese and was not timed.


These work samples illustrate standard-setting
performances for the following parts of the standards:
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.
f Mathematical Skills and Tools: Use formulas appropriately.
b Mathematical Communication: Organize work, explain a solution orally and in writing, and use other techniques to make meaning clear to the audience.
d Mathematical Communication: Exhibit developing reasoning abilities by justifying statements and defending work.

What the work shows

d Geometry and Measurement Concepts: The student determines and understands length, area, and volume (as well as the differences among these measurements), including perimeter and surface area; uses units, square units, and cubic units of measure correctly; computes areas of rectangles, triangles, and circles; computes volumes of prisms.
Work on the first question illustrates understanding of volume.
In Sample 4, the student is demonstrated understanding and determining length, area, and volume, as well as square units, and cube units of measure correctly.
Work on the second question illustrates understanding of surface area.
In Sample 1, though, equals signs (=) are used a little carelessly. Colons (:) or arrows
(-->) would be more appropriate links. Equals signs could have been retained while clarifying just what is equal or the same, for example, “Area of 1 side = 9 sq. cm.”
In Sample 4, the equals sign is used correctly. 3

In Sample 3, the labeling of only one side of the cube demonstrates an understanding of the concept of a cube and its relationship with its measurements.
In Sample 3, equals signs are used appropriately although the student missed labeling the equations for area.

h Geometry and Measurement Concepts: The student chooses appropriate units of measure with ease.
In Sample 4 (a) and (b), the student correctly chose and as the unit of measure for volume and area respectively.

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

Sample 2

Sample 2

Sample 2 Translation

Sample 3

Sample 3 Translation


b Mathematical Communication: The student organizes work, explains facets of a solution orally and in writing, labels drawings, and uses other techniques to make meaning clear to the audience.
In Sample 1, the explanation of the answer “27 cubic centimeters” is very clear and concise. A minor observation is that the student labeled three edges of the cube “3,” but did so in only two of its three dimensions.
In Sample 3, the explanation of the answer “27 ” is also very clear and concise.
The student’s understanding of the concept of the properties of a cube is clear when you note that only one side of each cube is labeled with a measurement.
In Samples 1, 2, and 3 the students gave several examples to counter Eddie’s claim. Not only is Eddie’s claim not true for all cubes, it is false for most cubes.

d Mathematical Communication: The student exhibits developing reasoning abilities by justifying statements and defending work.

In Sample 3, the small error (“sierto” instead of “cierto”) does not detract from the quality of the work.

Sample 4

Sample 4 Translation