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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 2003

Dion is standing next to the Washington Monument. He is 5 feet 10 inches tall. His shadow is 16 inches long. The shadow of the Washington Monument is 127 feet long, as shown in the diagram below.

Note: The figures are not drawn to scale.

Complete the following in the answer box below:
  • Find the approximate height of the Washington Monument. Use mathematics to explain how you determined your answer. Use words, symbols, or both in your explanation.

The following 6 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 fails to provide a height for the monument but correctly finds the scale factor (4.375) of Dion's height to his shadow. "I took the monuments height & divided it by 4.375 & got how long the shadow of the monument is" explains a strategy that will never lead to a correct answer. The measurement of the monument's shadow (127 ft.) is given. For a correct strategy, the student needs to find the height of the monument rather than the length of the shadow.


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 calculates an incorrect height of 29.03 ft. Some understanding of similarity is demonstrated in the use of proportion. However, the proportion is incorrectly set up: the ratio of Dion's height to his shadow is set equal to the ratio of the monument's shadow to its height (rather than the monument's height to its shadow). This strategy will never lead to a correct answer because the ratios are not equal to each other.


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 fully explains a correct strategy (the ratio of Dion's height to his shadow set equal to the ratio of the monument's height to its shadow). However, an error in the conversion of 5 feet 10 inches into 5.10 (rather than the correct 5.83) leads to an incorrect solution of 40.5 feet. This response indicates an incomplete application of a reasonable 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 explains a correct strategy by setting up an appropriate proportion (the ratio of Dion's height to his shadow set equal to the ratio of the monument's height to its shadow). The response gives the incorrect ratio of 30:8, not 35:8. This error results in the incorrect answer of 476 feet.


Anchor Paper #5

image of student response

Score for Anchor Paper #5: Rubric Score 3

Annotation: This response demonstrates a complete understanding and analysis of the problem. The student provides a full explanation of a correct strategy (the ratio of Dion's height to his shadow, both expressed in feet, set equal to the ratio of the monument's height to its shadow), which leads to a correct answer "x=~556...x=555.625 ft."


Anchor Paper #6

image of student response

Score for Anchor Paper #6: Rubric Score 3

Annotation: This response demonstrates a complete understanding and analysis of the problem. The student correctly answers "555.6 ft." and provides a full explanation of a correct strategy for finding the monument's height. The student gives the scale factor of Dion's height to his shadow (35/8) and multiplies that scale factor by the monument's shadow (127).


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
/share/clg/xml/public_release/mathematics/2003_231_geo06.xml
Resources for 2.3.1:
Skill Statements | PUBLIC RELEASE ITEMS | Lesson Plans |