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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.

Brief Constructed Response (BCR) Item - Released in 2000

Look at the two cylinders shown below. The ratio of corresponding diameters is equal to the ratio of corresponding heights.

Complete the following in the answer box below:
  • What is the height of the large cylinder? Use mathematics to explain how you determined your answer. Use words, symbols, or both in your explanation.
     
  • Find the ratio of the volumes of the cylinders. Use mathematics to justify your answer.
     

The following 12 Sample Student Responses represent a range of score points.

Sample Student Response #1

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Score for Sample Student Response #1: Rubric Score 3

Annotation: This response demonstrates application of a reasonable strategy that leads to the correct solution in the context of the problem. The student correctly sets up a proportion and correctly solves for y, the height of the large cylinder. The explanation of the strategy used to find the large cylinder's height is clearly presented and fully developed. The correct ratio of the volumes of the cylinders (27:8) is given. Full justification for the volume ratio is in the statement "the ratio of the volumes of 2 figures is the ratio of the heights cubed. Therefore I cubed 3 and got 27, then cubed 2 and got 8." The student actually uses the cube of the radii rather than the heights. This is not a significant mathematical error since the relationship holds for all corresponding parts. The response demonstrates a complete understanding and analysis of the problem. Compare to Anchor Paper #6.
Sample Student Response #2

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Score for Sample Student Response #2: Rubric Score 1

Annotation: This response indicates little attempt to apply a reasonable strategy for the entire item. However, the student sets up a proportion that can be used to find the height of the large cylinder and correctly solves for y, the height of the large cylinder. This proportion compares the diameter of the base to the height of the same cylinder, rather than corresponding parts. The explanation is clearly presented and fully developed in mathematical symbols. There is no attempt to express the volumes as a ratio or to calculate volume. Because there is no evidence of understanding of volume, overall this response demonstrates minimal understanding and analysis of the problem. Compare to Anchor Paper #1.
Sample Student Response #3

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Score for Sample Student Response #3: Rubric Score 3

Annotation: This student applies a reasonable strategy that leads to a correct solution in the context of the problem. The student correctly sets up a proportion and correctly solves for the height of the large cylinder. The explanation of the strategy used to find the large cylinder's height is clearly presented and fully developed. The volume ratio of the cylinders (8pi/27pi), while not simplified, is correct. Full justification for the volume ratio is provided in the correctly calculated volumes of both cylinders. The response demonstrates a complete understanding and analysis of the problem. Compare to Anchor Paper #7.
Sample Student Response #4

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Score for Sample Student Response #4: Rubric Score 2

Annotation: This response indicates the incomplete application of a reasonable strategy. The student sets up a proportion that works to find the height of the large cylinder and correctly solves for that height. This proportion compares the diameter of the base to the height of the same cylinder rather than corresponding parts. The explanation of the strategy used to find the large cylinder's height is clearly presented and fully developed. Volume is attempted. However, the ratio of the volumes (9:4) is incorrect. The values for volume indicate an error in the value for B in the volume formula V=Bh. Justification for the ratio is undeveloped, as the incorrectly calculated volumes were used to support the justification. The response demonstrates a conceptual understanding and analysis of the problem. Compare to Anchor Paper #5.
Sample Student Response #5

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Score for Sample Student Response #5: Rubric Score 1

Annotation: This response indicates little attempt to apply a reasonable strategy. The student attempts to solve for the height of the large cylinder, but sets up the proportion incorrectly. There is no attempt at the second bullet. The response demonstrates a minimal understanding and analysis of the problem. Compare to Anchor Paper #3.
Sample Student Response #6

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Score for Sample Student Response #6: Rubric Score 2

Annotation: This response indicates the incomplete application of a reasonable strategy. The student sets up a proportion that can be used to find the height of the large cylinder and correctly solves for its height. This proportion compares the diameter of the base to the height of the same cylinder rather than corresponding parts. The explanation of the strategy used to find the large cylinder's height is clearly presented and fully developed. There is an attempt at volume. However, the ratio of the volumes (10:1) is incorrect. The value for the large cylinder's volume is correct, while the value for the small cylinder's volume indicates an error in the value for B in the volume formula V=Bh. Justification for the ratio is not well developed, as calculated volumes were intended to be support. The response demonstrates a conceptual understanding and analysis of the problem. Compare to Anchor Paper #4.
Sample Student Response #7

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Score for Sample Student Response #7: Rubric Score 2

Annotation: This response indicates the incomplete application of a reasonable strategy. The student correctly sets up a proportion and correctly solves for the height of the large cylinder. The explanation of the strategy used to find the large cylinder's height is clearly presented and fully developed. Volume is attempted. However, the ratio of the volumes (9/4) is incorrect. The computations for volume indicate an error in the value for B used in the volume formula V=Bh, with the student using the diameter of the cylinder instead of the area of the base. The response demonstrates a conceptual understanding and analysis of the problem. Compare to Anchor Paper #5.
Sample Student Response #8

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Score for Sample Student Response #8: Rubric Score 3

Annotation: This response demonstrates application of a reasonable strategy that leads to the correct solution in the context of the problem. The student correctly sets up a proportion and correctly solves for y, the height of the large cylinder. The explanation of the strategy used to find the large cylinder's height is clearly presented and fully developed. The correct ratio of the volumes of the cylinders (27:8) is given. Full justification for the volume ratio is in the use of the knowledge that the volume ratio is the scale factor cubed. The response demonstrates a complete understanding and analysis of the problem. Compare to Anchor Paper #6.
Sample Student Response #9

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Score for Sample Student Response #9: Rubric Score 2

Annotation: This response indicates the incomplete application of a reasonable strategy. The student correctly sets up a proportion and correctly solves for the height of the large cylinder. The explanation of the strategy used to find the large cylinder's height is clearly presented and fully developed. Volume is attempted. However, the ratio of the volumes (9:4) is incorrect. The justification is undeveloped. The values for volume indicate an error in the value for B used in the volume formula V=Bh. The response demonstrates a conceptual understanding and analysis of the problem. Compare to Anchor Paper #5.
Sample Student Response #10

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Score for Sample Student Response #10: Rubric Score 1

Annotation: This response indicates little attempt to apply a reasonable strategy. This student applies an inappropriate linear strategy to solve for the height of the large cylinder. Instead of using proportion, the student "guessed" and adds 2 to the height, perhaps because the base increased by 2. The volumes are never expressed as a ratio, but there is an attempt to calculate volume. The volume of the small cylinder is correct, and using the incorrect height value for the large cylinder, volume was calculated for the large cylinder. Because there is no understanding of proportion, this response demonstrates minimal understanding and analysis of the problem. Compare to Anchor Paper #2.
Sample Student Response #11

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Score for Sample Student Response #11: Rubric Score 3

Annotation: This student applies a reasonable strategy that leads to a correct solution in the context of the problem. The student sets up a proportion that works to find the height of the large cylinder and correctly solves for that height. This proportion compares the diameter of the base to the height of the same cylinder rather than corresponding parts. The explanation of the strategy used to find the large cylinder's height is clearly presented and fully developed. The volume ratio of the cylinders (40pi/135pi), while not simplified, is correct. Full justification for the volume ratio is provided in the correctly calculated volumes of both cylinders. The response demonstrates a complete understanding and analysis of the problem. Compare to Anchor Paper #7.
Sample Student Response #12

image of student response

Score for Sample Student Response #12: Rubric Score 1

Annotation: This response indicates the incomplete application of a reasonable strategy. The student sets up a proportion that works to find the height of the large cylinder and correctly solves for that height. This proportion compares the diameter of the base to the height of the same cylinder rather than corresponding parts. The explanation of the strategy used to find the large cylinder's height is clearly presented and fully developed. There is no attempt at finding the volumes and expressing their ratio in the response. The response demonstrates minimal understanding and analysis of the problem. Compare to Anchor Paper #1.

Additional Resources

Anchor Papers used in scoring

Brief Constructed Response (BCR) Rubric
Print: Scoring Rubric (pdf)
Score 3

The response indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. The representations are essentially 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 2

The response indicates application of a reasonable strategy that may be incomplete or undeveloped. It 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 attempt to apply a reasonable strategy or applies an inappropriate 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 |