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Goal 2 Geometry, Measurement, And Reasoning

Expectation 2.2 The student will apply geometric properties and relationships to solve problems using tools and technology when appropriate.

Indicator 2.2.1 The student will identify and/or verify congruent and similar figures and/or apply equality or proportionality of their corresponding parts.

Assessment Limits:

  • Students will demonstrate geometric reasoning and justify conclusions. Although the focus is on geometric theory, answers to some items may include a numeric answer.
  • Corresponding measurements include length, angle measure, perimeter, circumference, area, volume, surface area and lateral area.

Extended Constructed Response (ECR) Item - Released in 2004

In the figure below, ΔABD and ΔCDB are isosceles triangles. The vertex angles, angleABD and angleCDB and, are congruent as shown below.

Complete the following on a piece of paper and/or in the answer box below:
  • Prove that quadrilateral ABCD is a parallelogram.
     

The following 8 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. While there is evidence of a proof process, no indication of a strategy to prove a parallelogram is given. The relevant statement (BD is congruent to BD) is correctly justified by the reflexive property. The statements (ΔABD is congruent to ΔCDB) and (angleA is congruent to angleC) could be proved true; however, the justifications given for both are incorrect. The student makes an attempt to prove congruent triangles, but the triangle proof is incomplete. No indication of the conditions that would substantiate a parallelogram 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. Although there is evidence of a SAS (side-angle-side) strategy, no indication of a strategy to prove a parallelogram is given. The statement (BD = BD) is essentially correct with an error in notation; however, the justification (perpendicular bisector) is incorrect. The statement (AB = DC ), with the same notation error, could be proved true, but is incompletely justified (isosceles). The SAS (side-angle-side) triangle proof is incomplete, and the student provides no indication of the conditions that would prove a parallelogram.


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. In attempting to find appropriate angles to prove parallel lines, the student reveals an understanding that opposite parallel sides will prove a parallelogram. However, the student does not relate the triangle's base angles to each other to establish congruence and does not supply the justification for why the angles would establish parallel lines; thereby, demonstrating 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 tries a strategy that reveals the understanding that congruent opposite angles will prove a parallelogram. Because all four base angles in the triangles are not proved congruent, and the student does not state that angle addition is required, this response demonstrates an incomplete application of a reasonable strategy. The student incorrectly states (ABCD is a quadrilateral), rather than a parallelogram.


Anchor Paper #5

image of student response

Score for Anchor Paper #5: Rubric Score 3

Annotation: This response demonstrates a clear understanding and analysis of the problem. The student fails to state the given congruent angles in a SAS (side-angle-side) proof and also neglects the statement (BD is congruent to BD by reflexive property). Using corresponding parts, the student proves BC is congruent to AD. The response gives congruent opposite sides to prove a parallelogram, but provides the justification (definition of a parallelogram).


Anchor Paper #6

image of student response

Score for Anchor Paper #6: Rubric Score 3

Annotation: This response demonstrates a clear understanding and analysis of the problem. The student supplies a complete SAS (side-angle-side) triangle proof, but does not provide the congruent angles necessary for the stated AAS (angle-angle-side) proof. Using corresponding parts, the student's statement (both pairs of opposite sides is congruent to ) establishes a parallelogram.


Anchor Paper #7

image of student response

Score for Anchor Paper #7: Rubric Score 4

Annotation: This response demonstrates a complete understanding and analysis of the problem. The student establishes (AB // DC) because (alternate interior angles angleABD and angleBDC are congruent). After establishing (AB is congruent to DC), using the isosceles triangles and the reflexive property, the student proves a parallelogram (since one pair of sides is congruent and parallel).


Anchor Paper #8

image of student response

Score for Anchor Paper #8: Rubric Score 4

Annotation: This response demonstrates a complete understanding and analysis of the problem. The student gives a correct SAS (side-angle-side) triangle proof. Claiming congruent angles through corresponding parts and using angle addition, the student proves a parallelogram by congruent opposite angles are congruent.


Extended Constructed Response (ECR) Rubric
Print: Scoring Rubric (pdf)
Score 4

The response indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. The representations are 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 3

The response indicates application of a reasonable strategy that may or may not lead to a correct solution. The representations are essentially correct. The explanation and/or justification is generally well developed, feasible, and supports the solution. The response demonstrates a clear understanding and analysis of the problem.

Score 2

The response indicates an incomplete application of a reasonable strategy that 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 application of a reasonable 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 2.2.1:
Skill Statements | PUBLIC RELEASE ITEMS | Lesson Plans |