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 box below: 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 8 Anchor Papers represent a range of score points and are used in conjunction with the rubrics to assess student responses.

Score for Anchor Paper #1: Rubric Score 1

Annotation: This response indicates little application of a reasonable strategy. The representations are incomplete. The pattern is correctly extended in the table; however, there is no attempt to apply a reasonable strategy to draw Stage 4 or a scatter point. The student indicates an understanding of the constant ("...add six to each new answer") but does not express this in the form of an equation. This response demonstrates a minimal understanding and analysis of the problem.

Score for Anchor Paper #2: 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 structure for Stage 4 is correct. The graph is fundamentally correct. The scales are consistent, and the points are plotted correctly; however, the axes are not labeled and are reversed with respect to the independent and dependent variable, and the title is missing. There is neither an equation nor an explanation provided. This response demonstrates a minimal understanding and analysis of the problem.

Score for Anchor Paper #3: Rubric Score 2

Annotation: This response indicates an incomplete application of a reasonable strategy. The representations are fundamentally correct, demonstrating a conceptual understanding and analysis of the problem. The table, snowflake structure, and graph are correct and complete. The data is not expressed as an equation; therefore, the response does not demonstrate a clear understanding.

Score for Anchor Paper #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 graph is correct except for a missing title and an inconsistency along the x-axis. However, the student has represented six stages of flake growth in the snowflake structure rather than only Stage 4. The student indicates an understanding of the constant ("...so that adds 6 crystals each stage..."), but does not express this data as an equation. This response demonstrates a conceptual understanding and analysis of the problem.

Score for Anchor Paper #5: Rubric Score 3

Annotation: This response indicates application of a reasonable strategy. The representations are essentially correct. The snowflake structure for Stage 4 is accurately represented, and the pattern is correctly extended in the table. The graph correctly represents the data, including labels and uniform scales and omitting only a title. The student's attempt at an equation is incorrect, using alternate variables (x) and (y) that have been defined. The explanation supports the solution; however, the student does not account for the original crystal in the equation. This response demonstrates a clear understanding and analysis of the problem.

Score for Anchor Paper #6: Rubric Score 3

Annotation: This response indicates application of a reasonable strategy that leads to a correct solution. The representations are essentially 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 determines an appropriate equation, C = 7 + [(S - 1)6], that represents the total number of crystals at each stage, redefining the variable (n) as (s). However, an explanation is not provided. This response demonstrates a clear understanding and analysis of the problem.

Score for Anchor Paper #7: Rubric Score 4

Annotation: This response indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. The pattern is correctly extended in the table, and the snowflake structure for Stage 4 is correctly represented. The graph is essentially correct. The data points are plotted correctly, and the axes are labeled, but the scales are not consistent near the origin. The equation is correct with redefined variables. The explanation ("...I add one crystal (+1) for the base and I found the flake grows 6 crystals per stage.") is logically sound, fully developed, and supports the solution. This response demonstrates a complete understanding and analysis of the problem.

Score for Anchor Paper #8: Rubric Score 4

Annotation: This response indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. The representations are correct. This student correctly extends the pattern of crystals in the table and gives a correct representation of the snowflake structure at Stage 4. The graph correctly represents the data; the points are correctly plotted, and the scales are uniform. The student determines a correct equation that represents the total number of crystals at each stage. The explanation ("The pattern added 1 crystal on each arm of the snowflake at each stage after the very first one which started with 1.") is logically sound, fully developed, and supports the solution. This response demonstrates a complete understanding and analysis of the problem.

Additional Scored Student Responses