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Goal 2 Geometry, Measurement, And Reasoning

Expectation 2.3 The student will apply concepts of measurement using tools and technology when appropriate.

Indicator 2.3.2 The student will use techniques of measurement and will estimate, calculate, and/or compare perimeter, circumference, area, volume, and/or surface area of two-and three-dimensional figures and their parts.

Assessment Limits:

  • Two-dimensional shapes include polygons, circles, and composite figures.
  • Three-dimensional shapes include cubes, prisms, pyramids, cylinders, cones, spheres, and composite figures.
  • Formulas will be provided.
  • No oblique solids will be used.
  • Items may involve applications of geometric properties and relationships.
  • Students may be required to make comparisons which do not require calculations.

Extended Constructed Response (ECR) Item - Released in 2001

The dimensions of two suitcases are shown below. Suitcase A is a trapezoidal prism, and Suitcase B is a rectangular prism.

Note: These figures are not drawn to scale.

Use the answer box to complete the following. (You will need to use additional paper to complete the answer satisfactorily.)

  • What is the volume of Suitcase A? Use mathematics to explain how you determined the volume of Suitcase A. Use words, symbols, or both in your explanation.
     
  • Both suitcases have the same volume. What is the height of Suitcase B? Use mathematics to explain how you determined the height of Suitcase B. Use words, symbols, or both in your explanation.

The following 8 Anchor Papers represent a range of score points and are used in conjunction with the rubrics to assess student responses.

Anchor Paper #1

image of student response

Score for Anchor Paper #1: Rubric Score 1

Annotation: This response demonstrates a minimal understanding and analysis of the problem. The student has correctly recognized the trapezoid shape, attempting to find the volume of Suitcase A using the volume formula for a trapezoidal prism. The student uses a slightly different way to solve the problem by using the formula ½(area B1+B2)xH (where B1 and B2 are the area of the top and bottom of the suitcase). However, the student has made an error in the formula by using an incorrect value for h. The student has made an attempt to find the height of Suitcase B by setting up the equation 3120=360t.


Anchor Paper #2

image of student response

Score for Anchor Paper #2: Rubric Score 1

Annotation: This response demonstrates minimal understanding and analysis of the problem. The student has correctly recognized the trapezoid shape. The correct formulas for volume of a general prism (V=Bh) and area of a trapezoid (A=½(b1+b2)h) are evident. However, the student has used incorrect values for b1, b2 and h. Solving for the volume of Suitcase B, the student attempts to apply the trapezoid area formula to solve for the height, an inappropriate strategy.


Anchor Paper #3

image of student response

Score for Anchor Paper #3: Rubric Score 2

Annotation: This response demonstrates conceptual understanding and analysis of the problem. The correct volume for Suitcase A (2880 in³) and correct height of Suitcase B (16 in) are clearly evident. However, there is no explanation or work shown to reveal the strategies behind these correct values.


Anchor Paper #4

image of student response
image of student response

Score for Anchor Paper #4: Rubric Score 2

Annotation: This response demonstrates a conceptual understanding and analysis of the problem. The explanations reveal that the student uses an incomplete reasonable strategy to solve the entire problem. The student attempts to find the volume of Suitcase A but with an incorrect volume formula. However, working with the incorrect volume, the student uses a reasonable strategy to solve for the height of Suitcase B.


Anchor Paper #5

image of student response

Score for Anchor Paper #5: Rubric Score 3

Annotation: This response demonstrates a clear understanding and analysis of the problem. The correct volume for Suitcase A (2880 in³) and the correct height of Suitcase B (16 in.) are clearly evident. This response lacks a full explanation for the entire problem; the student has not provided any explanation of the strategy employed for the correct volume of Suitcase A. The explanation for the height of Suitcase B has an error in expression ("180/2880" should have been expressed as 2880/180), but the operation is correctly executed and the correct strategy is evident.


Anchor Paper #6

image of student response

Score for Anchor Paper #6: Rubric Score 3

Annotation: This response demonstrates a clear understanding and analysis of the problem. The student visualizes a correct strategy to break down the trapezoidal shape into one rectangular and two triangular prisms. However, the student has mistakenly used a value of 20 instead of 24 when calculating the value of the triangular prisms, resulting in an incorrect total volume of "2640 in³'". Working with the calculated volume of 2640 cubic inches, the student explains an appropriate strategy to solve for the height of Suitcase B. The singular error of an incorrect value resulted in incorrect answers for the volume of Suitcase A and the height of Suitcase B in otherwise fully explained correct strategies.


Anchor Paper #7

image of student response
image of student response

Score for Anchor Paper #7: Rubric Score 4

Annotation: This response demonstrates a complete understanding and analysis of the problem. The student has found the correct volume of Suitcase A and the correct height of Suitcase B. The explanations of the strategies applied (using an alternative visualization of Suitcase A's volume and solving for the height of Suitcase B) are clearly presented, full developed, and supportive of the solution.


Anchor Paper #8

image of student response

Score for Anchor Paper #8: Rubric Score 4

Annotation: This response demonstrates complete understanding and analysis of the problem. The correct volume for Suitcase A ("2880 in.²") and correct height of Suitcase B ("16 in.") are clearly evident. The volume of Suitcase A lacks a correct label of cubic inches. Even though the student repeatedly uses "in.²" instead of "in.³," this was considered to be an error in labeling rather than strategy and, therefore, not a significant mathematical error. The student has provided a fully developed explanation to solve the entire problem.


Extended Constructed Response (ECR) Rubric
Print: Scoring Rubric (pdf)
Score 4

The response indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. The representations are correct. The explanation and/or justification is logically sound, clearly presented, fully developed, supports the solution, and does not contain significant mathematical errors. The response demonstrates a complete understanding and analysis of the problem.

Score 3

The response indicates application of a reasonable strategy that may or may not lead to a correct solution. The representations are essentially correct. The explanation and/or justification is generally well developed, feasible, and supports the solution. The response demonstrates a clear understanding and analysis of the problem.

Score 2

The response indicates an incomplete application of a reasonable strategy that may or may not lead to a correct solution. The representations are fundamentally correct. The explanation and/or justification supports the solution and is plausible, although it may not be well developed or complete. The response demonstrates a conceptual understanding and analysis of the problem.

Score 1

The response indicates little or no application of a reasonable strategy. It may or may not have the correct answer. The representations are incomplete or missing. The explanation and/or justification reveals serious flaws in reasoning. The explanation and/or justification may be incomplete or missing. The response demonstrates a minimal understanding and analysis of the problem.

Score 0

The response is completely incorrect or irrelevant. There may be no response, or the response may state, “I don't know.”

Explanation refers to the student using the language of mathematics to communicate how the student arrived at the solution.

Justification refers to the student using mathematical principles to support the reasoning used to solve the problem or to demonstrate that the solution is correct. This could include the appropriate definitions, postulates and theorems.

Essentially correct representations may contain a few minor errors such as missing labels, reversed axes, or scales that are not uniform.

Fundamentally correct representations may contain several minor errors such as missing labels, reversed axes, or scales that are not uniform.

Last Revised 8/16/00
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Resources for 2.3.2:
Skill Statements | PUBLIC RELEASE ITEMS | Lesson Plans |