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Public Release Item Scoring Information Return

Goal 1 Functions and Algebra

Expectation 1.1 The student will analyze a wide variety of patterns and functional relationships using the language of mathematics and appropriate technology.

Indicator 1.1.2 The student will represent patterns and/or functional relationships in a table, as a graph, and/or by mathematical expression.

Assessment Limits:

  • The given pattern must represent a relationship of the form mx + b (linear), x2 (simple quadratic), simple arithmetic progression, or simple geometric progression with all exponents being positive.

Extended Constructed Response (ECR) Item - Released in 2001

Certain ice crystals have six sides. Flakes are formed when these six-sided ice crystals connect. One pattern is shown below.

Complete the following in the Answer Book:

  • Complete the table to show the total number of crystals in each of the first 6 stages of flake growth.

  • Based on the drawing at Stage 3, draw the pattern for the same flake at Stage 4.
  • Draw a scatter plot using the data in the completed table.
  • Write an equation that represents the total number of crystals (c), at each stage number (n). Use mathematics to explain how you determined your equation. Use words, symbols, or both in your explanation.

The following 4 Sample Student Responses represent a range of score points.

Sample Student Response #1

image of student response
image of student response

Score for Sample Student Response #1: Rubric Score 1

Annotation: This response indicates little application of a reasonable strategy. The representations are incomplete. The student has extended the pattern correctly in the table, and the representation of the snowflake at Stage 4 is accurate. The graph, however, is incorrect. The student has copied the numbers from the right column of the table onto the x-axis, creating a non-uniform scale and reversing the axes with respect to the independent and dependent variables. Although the student indicates an understanding of the constant ("Each stage the crystals add six new crystals onto them."), this is not expressed as an equation. This response demonstrates a minimal understanding and analysis of the problem.


Sample Student Response #2

image of student response
image of student response

Score for Sample Student Response #2: Rubric Score 4

Annotation: This response indicates application of a reasonable strategy that leads to a correct solution within the context of the problem. The pattern is correctly extended in the table, and the snowflake structure for Stage 4 is accurately represented. While the graph lacks a title and contains a minor scale error on the y-axis, the coordinates are clearly labeled, and the points are correctly placed. The student determines a correct equation that represents the total number of crystals at each stage. The explanation ("...each stage number if you multiplied it by 6 and added one then that gives you the crystals.") is logically sound, fully developed, and supports the solution. This response demonstrates a complete understanding and analysis of the problem. Compare to Anchor Paper #8.
Sample Student Response #3

image of student response
image of student response

Score for Sample Student Response #3: Rubric Score 3

Annotation: This response indicates application of a reasonable strategy. The representations are essentially correct. The representation of the snowflake structure at Stage 4 is missing, but the table, graph, and equation are correct and complete. The explanation, although it supports the solution, is not fully developed. The student does not indicate that the "+1" represents the initial crystal, instead stating that "the stages numbers increased by 1." This response demonstrates a clear understanding and analysis of the problem.


Sample Student Response #4

image of student response
image of student response

Score for Sample Student Response #4: Rubric Score 2

Annotation: This response indicates an incomplete application of a reasonable strategy. The representations are fundamentally correct. The pattern is correctly extended in the table, and the representation of the snowflake structure for Stage 4 is correct. The graph correctly represents the data, including labels and uniform scales and omitting only a title. The student indicates an understanding of the constant ("...at each stage 6 crystals form."), but does not express this data as an equation. This response demonstrates a conceptual understanding and analysis of the problem. Compare to Anchor Paper #5.

Additional Resources

Anchor Papers used in scoring

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 1.1.2:
Skill Statements | PUBLIC RELEASE ITEMS | Lesson Plans |