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Goal 2 Geometry, Measurement, And Reasoning

Expectation 2.1 The student will represent and analyze two- and three-dimensional figures using tools and technology when appropriate.

Indicator 2.1.4 The student will construct and/or draw and/or validate properties of geometric figures using appropriate tools and technology.

Assessment Limits:

  • “Validate properties” in this indicator, means justifying solutions using definitions, mathematical principles and/or measurement.
  • Students may use a compass, straightedge, patty paper, a MiraTM, and/or a mirror as construction tools. Using a ruler or protractor cannot be part of the strategy.
  • Students may use a compass, ruler, patty paper, a MiraTM, a mirror and/or a protractor as drawing tools.
  • It is acceptable to do a construction when the item asks for a drawing.
  • Paper folding and the use of MirasTM and mirrors are appropriate methods for representing, constructing, and/or analyzing figures, and their use must be referenced.
  • Constructions and drawings are limited to the two-dimensional relationships listed in 2.1.1.

Brief Constructed Response (BCR) Item - Released in 2000

Elizabeth draws a right triangle with angles of 52° and 38°.

Complete the following in the answer box below:
  • Draw a right triangle using these angle measurements. Label the measure of each angle.
     
  • Will any right triangle with angles of 52° and 38° be congruent to Elizabeth's triangle? 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 representation of the triangle is correct. However, the written portion addressing congruence does not reveal the understanding that having two triangles with the same angle measurements does not insure that the triangles are congruent because the side lengths could be different. Reflecting the triangle still gives a congruent triangle. There is no understanding of the difference between similarity and congruence.


Anchor Paper #2

image of student response

Score for Anchor Paper #2: Rubric Score 1

Annotation: The response demonstrates a minimal understanding and analysis of the problem. The representation of the triangle is correct although the angles are reversed. However, the written portion addressing congruence does not reveal the understanding that being given 3 angle measurements does not insure that triangles are congruent, as the side lengths could still be different. The knowledge that the three angles of a triangle add up to 180 degrees does not address side length. The statement "If we only had two angles known, yes, then Carls triangle could have been different" indicates further confusion. There is no understanding of the difference between similarity and congruence.


Anchor Paper #3

image of student response

Score for Anchor Paper #3: Rubric Score 2

Annotation: The response demonstrates a conceptual understanding and analysis of the problem. The representation of the triangle is fundamentally correct. The 38-degree and the 52-degree angles are not drawn within an acceptable range. The written portion addresses the difference between similarity and congruence. The lengths of the sides are addressed: "...the lengths of each side can be different from Elizabeth's triangle, but the angles can remain the same." The student has drawn a smaller similar triangle. There is understanding of the difference between similarity and congruence.


Anchor Paper #4

image of student response

Score for Anchor Paper #4: Rubric Score 2

Annotation: The response demonstrates a conceptual understanding and analysis of the problem. The representation of the triangle is correct. The written portion does not clearly address the difference between similarity and congruence. The lengths of the sides are not addressed, and changing the orientation ("placed differently") of the angles still gives congruence. However, the student has drawn a smaller similar triangle, indicating some understanding about different sizes. There is incomplete understanding of the difference between similarity and congruence.


Anchor Paper #5

image of student response

Score for Anchor Paper #5: Rubric Score 3

Annotation: The response demonstrates a complete understanding and analysis of the problem. The representation of the triangle is correct. The student clearly understands the difference between similarity and congruence with the justification "the base...maybe longer...(and) the side lengths would not be the same, so the figures would be similar not congruent."


Anchor Paper #6

image of student response

Score for Anchor Paper #6: Rubric Score 3

Annotation: The response demonstrates a complete understanding and analysis of the problem. The representation of the triangle is correct. The student clearly understands the difference between similarity and congruence, addressing different side lengths with the justification "Carl's triangle can be larger than Beth's and still have the same angle measurements or it could be smaller and still have the same angle measurements."


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 2.1.4:
Skill Statements | PUBLIC RELEASE ITEMS | Lesson Plans |