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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 7 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 indicates little attempt to apply 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 the 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. However, the volume ratios (4/10 and 6/15) are incorrect. The student once again uses ratios of diameter to height of the same cylinders rather than the ratio of the volumes. Because there is no understanding of volume, this response demonstrates minimal understanding and analysis of the problem.


Anchor Paper #2

image of student response

Score for Anchor Paper #2: 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 adds 2 to the height 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.


Anchor Paper #3

image of student response

Score for Anchor Paper #3: Rubric Score 1

Annotation: This response indicates little attempt to apply a reasonable strategy. There is no explanation for how the student arrived at the incorrect height of 12 for the large cylinder. In the attempts at volume, work reveals a misunderstanding of B in the volume formula V=Bh, using the diameter of the base rather than the area of the base. The student gives an incorrect ratio of volumes (9:5) by using the incorrectly calculated volumes. This response demonstrates minimal understanding and analysis of the problem.


Anchor Paper #4

image of student response

Score for Anchor Paper #4: 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 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 of the strategy used to find the large cylinder's height is clearly presented and fully developed. Volume is said to be attempted. Despite the correct explanation for the finding of the volumes, the values of these volumes are not indicated. These unknown volumes were then "rounded" and used to "set up a ratio." However, the ratio of the volumes (13:28) is incorrect. The justification is undeveloped. The response demonstrates a conceptual understanding and analysis of the problem.


Anchor Paper #5

image of student response

Score for Anchor Paper #5: 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, "I found the volume for each cylinder, which is 90 and 40," supports the solution. However, the student used incorrect values for the volume. The response demonstrates a conceptual understanding and analysis of the problem.


Anchor Paper #6

image of student response

Score for Anchor Paper #6: 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 (8:27) is given. Full justification for the volume ratio is in the statement, "the ratio of the volumes of similar figures is the scale factor cubed." The response demonstrates a complete understanding and analysis of the problem.


Anchor Paper #7

image of student response

Score for Anchor Paper #7: Rubric Score 3

Annotation: This student applies a reasonable strategy that leads to a correct solution in the context of the problem. The student converts the ratio of the diameters into a factor of 1.5 and uses that factor to correctly calculate the height of the large cylinder. The correct ratio of the volumes of the cylinders (8/27) is given. Full justification is provided in the correctly calculated volumes of both cylinders that comprise the volume ratio. The response demonstrates a complete understanding and analysis of the problem.


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 |