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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 2.3.1 The student will use algebraic and/or geometric properties to measure indirectly.

Assessment Limits:

  • “Measure indirectly” means to use mathematical concepts such as congruence, similarity, and ratio and proportion to calculate measurements.
  • Similarity and congruence will be directly stated or implied (scale drawings, enlargements).
  • Items may require the student to make comparisons.
  • This indicator may incorporate measuring.
  • This indicator does not include right-triangle trigonometry.

Extended Constructed Response (ECR) Item - Released in 2003

A golf ball has a diameter of 1.68 inches. Twelve golf balls will be packaged in a rectangular cardboard box. There are three choices for how the golf balls will be arranged.

Choice A:  12 golf balls in one row

Choice B:  2 rows of golf balls, each row 6 balls long

Choice C:  3 rows of golf balls, each row 4 balls long

Complete the following in the answer box below:

  • Which of the three choices requires the least amount of cardboard? Use mathematics to explain how you determined your answer. 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 incorrectly answers that "They are equal in the amount of space/cardboard they take up." The student provides some explanation and work that indicate the strategy involved is the determination of the dimensions and the area of each box. Having some of the correct dimensions is minimal, but relevant, understanding. While the strategy that considers only each of the boxes' areas is inappropriate to solve the problem, the calculation of the area of the boxes demonstrates relevant understanding that could be applied toward calculations of the surface area of each box.


Anchor Paper #2

image of student response

Score for Anchor Paper #2: Rubric Score 1

Annotation: This response demonstrates a minimal understanding and analysis of the problem. The student correctly answers "Choice C," but the inappropriate strategy leading to that decision appears to rely on comparisons of the length of two sides of each box. There is some relevant understanding in the student's explanation for finding the length of a side of the box for Choice A.


Anchor Paper #3

image of student response

Score for Anchor Paper #3: Rubric Score 2

Annotation: This response demonstrates a conceptual understanding and analysis of the problem. The student correctly answers "Choice C" and states that surface area, an appropriate strategy, is calculated. However, the student provides three incorrect surface area values of 67.68 in., 50.88 in., and 47.52 in. The work reveals that the student fails to consider the diameter of the golf ball in the calculations of the surface areas, thus demonstrating an incomplete correct strategy.


Anchor Paper #4

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 student incorrectly answers "Box B." After inappropriately calculating the volume of a golf ball, the student uses the volume value, instead of diameter, to calculate surface area. The understanding of the appropriate strategy of applying surface area to this problem conveys conceptual understanding; however, the misapplication of volume in place of diameter demonstrates an incomplete correct strategy.


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 student correctly answers "3 rows of golf balls, each 4 balls long is the best choice because it has the least surface area." While the surface area calculations for Box A and Box C are correct, calculations for the surface area of Box B reveal an error in the length dimension. The student multiplies the length by 12, rather than 6, which results in an incorrect surface area value of 214.5042 instead of the correct value of 112.90 square inches.


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 correctly answers "Choice C." Although a good explanation of the application of surface area is provided, no values for dimensions or surface areas are given. Without supporting values, this response lacks a full explanation.


Anchor Paper #7

image of student response

Score for Anchor Paper #7: Rubric Score 4

Annotation: This response demonstrates complete understanding and analysis of the problem. The student correctly answers that Choice C is the best because it has the least amount of cardboard. Math calculations provide a full explanation for the correct strategy of surface area.


Anchor Paper #8

image of student response

Score for Anchor Paper #8: Rubric Score 4

Annotation: This response demonstrates a complete understanding and analysis of the problem. The student correctly answers "Choice C is the best because it has the least surface area." Math calculations provide full explanation for the correct strategy of finding surface area.


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 2.3.1:
Skill Statements | PUBLIC RELEASE ITEMS | Lesson Plans |