School Improvement in Maryland
Public Release Item Scoring Information Return

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 2002

The two triangles shown below are equilateral.

Complete the following in the answer box below:
  • What is the ratio of the smaller triangle's area to the larger triangle's area?
  • Use mathematics to justify your answer.

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 correct area ratio (1:16) is given; however, no justification is provided.


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 calculations and explanation reveal that the student finds the correct 2:8 (when simplified equals 1:4) ratio of the sides. However, the incorrect area ratio of 1:4 is provided as the answer. Because the ratio of the sides to area is not a linear relationship, the strategy of applying the side ratio to area is inappropriate. The student fails to recognize that area is a squared function and, therefore, the area ratio is the square of the side ratio. There is no justification present.


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 uses an appropriate strategy of comparing the areas of the triangles. The correct area ratio of 1:16 is found. However, in using this strategy the student makes an error in the value of the height of each triangle. Rather than finding the actual height value, the student uses the side value of each triangle. An attempt at justification (the two incorrect area values) is provided.


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 attempts to solve the problem using the appropriate strategy of comparing the areas of the triangles. The correct area of each triangle is determined. However, in setting up the ratio the student rounds the area values and gives an incorrect ratio of 1/14. This error in the strategy results in an incorrect answer. Correct values for the area of each triangle provide the justification.


Anchor Paper #5

image of student response

Score for Anchor Paper #5: Rubric Score 3

Annotation: This response demonstrates complete understanding and analysis of the problem. The work reveals that the student applies an appropriate strategy. The student states the principle "in two similar figures" the "ratio of perimeters = a:b" and the "ratio of areas = a2:b2." Using the base lengths of each triangle, the student sets up the ratio of the sides (2:8) and reduces the ratio to 1:4. The correct ratio of the areas (1:16) is present, fully justified by the understanding of the stated principle.


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 work shows that the student applies an appropriate strategy of comparing the areas of the triangles. The areas of both triangles are correct, with the student supplying a correct area ratio of .0625. (The ratio .0625 is an equivalent value of 1/16.) Correct values for the area of each triangle provide the justification.


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/2002_231_geo10.xml
Resources for 2.3.1:
Skill Statements | PUBLIC RELEASE ITEMS | Lesson Plans |