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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 2.1.3 The student will use transformations to move figures, create designs, and/or demonstrate geometric properties.

Assessment Limits:

  • Transformations include reflections, rotations, translations, and dilations.
  • Items should go beyond the identification of transformations.
  • Essential properties and relationships include the following: congruence, similarity, and symmetry.
  • The student's explanation of a transformation must include the following:
    • translation – distance and direction
    • reflection – line of reflection
    • rotation – center of rotation, angle measure, direction (clockwise or counterclockwise)
    • dilation – center and scale factor
  • Paper folding and the use of MirasTM and mirrors are appropriate methods for performing transformations, and their use must be referenced.

Brief Constructed Response (BCR) Item - Released in 2000

A triangle with vertices at (0, 0), (0, 6), and (4, 0) is shown on the grid below.

Complete the following in the answer box below:
  • Draw the dilation image of the triangle using a scale factor of 3 and using (0, 0) as the center of the dilation on the grid provided in the Answer Book.
     
  • What is the difference between a dilation using a scale factor greater than 1 and a dilation using a scale factor between 0 and 1? Use examples 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 indicates application of an inappropriate strategy. The representation is incorrect. The student merely translates the image to the right three units and up three units--an incorrect strategy for dilation. The explanation is only partially correct. "If you draw a dilation with a scale factor of 0 or 1 your image does not get much bigger" is incorrect. Although the student's next sentence states correctly, "...if you draw a dilation with a scale factor greater than 1 your image will be much bigger than your preimage," there is no justification. The response demonstrates a 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 application of an inappropriate strategy. The representation is incorrect, but has some relevance as the student has drawn a larger triangle. The student merely adds three units to two legs of the triangle rather than multiplying by the scale factor of three--an inappropriate strategy. The explanation is incomplete and there is no justification. The student addresses a dilation with a scale factor greater than one, but not a scale factor between 0 and 1. The response demonstrates a minimal understanding and analysis of the problem.


Anchor Paper #3

image of student response

Score for Anchor Paper #3: Rubric Score 2

Annotation: This response indicates application of a reasonable strategy that leads to a correct representation that conveys an overall conceptual understanding of the problem. The written explanation only explains how the actual dilation was done and does not address the difference between a dilation with a scale factor between 0 and 1 and a dilation with a scale factor greater than 1. This response demonstrates a conceptual 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 application of a reasonable strategy that leads to a correct representation that conveys an overall conceptual understanding of the problem. The student does not address dilation with the required scale factor between 0 and 1. Although the student explains, "if the scale factor is 1 the shape would be the same, but if it was greater than 1 it would get bigger," the student does not justify this statement. This response demonstrates a conceptual understanding and analysis of the problem.


Anchor Paper #5

image of student response

Score for Anchor Paper #5: Rubric Score 3

Annotation: This response demonstrates application of a reasonable strategy that leads to the correct representation. The justification is clearly presented, fully developed and supports the solution, revealing a correct strategy. The student explains the differences in dilation by scale factors between 0 and 1 and scale factors greater than 1 in terms of multiples. ("...increases the figure by that many times"; "shrinks the figure.") The response demonstrates a complete 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 representation. The justification is clearly presented, fully developed and supports the solution, revealing a correct strategy. The student explains the differences in dilation by scale factors between 0 and 1 and scale factors greater than 1: "...a dilation with a scale factor between 0 and 1 makes the figure smaller...But when a dilation has a scale factor of greater than one is becomes bigger." He/She justifies these statements by giving specific examples: "An example is if I multiplied the point (0,6) by .5. The point would then be at (0,3)," and "...if I multiplied the point (0,7) but a scale factor of 2. The point would then be at (0,14)." 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 2.1.3:
Skill Statements | PUBLIC RELEASE ITEMS | Lesson Plans |