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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 2004

Toy blocks are used to build a tower. The surface area and volume of the tower built with these blocks is shown in the table below.

Complete the following in the Answer Book:

  • Complete the table in the Answer Book to determine the surface area and the volume for 5 and 6 blocks.
  • Write an algebraic expression to represent the relationship between the number of blocks and the surface area of the tower. Use mathematics to justify your answer.
  • If 10 blocks are used, what is the surface area and the volume of the tower? Use mathematics to explain how you determined your answers. 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 4

Annotation: This response indicates application of a reasonable strategy that leads to a correct solution within the context of the problem. The table is correctly completed. The representation, in the form of an equation (y=12x+6), is correct, and the variables are defined. The symbolic justification is clearly presented, fully developed, and supports the solution. The student replaces the x in the expression with the numbers 1 and 2, then solves for each, showing that the found solutions match the given numbers in the table. The surface area of a 10-block tower (126) is correct. The symbolic explanation is fully developed; the student replaces x in the expression with 10 and solves. The volume (40) is correct. The tabular explanation is clearly presented; the student extends the pattern to find the volume at 10 blocks and then summarizes. (I followed the pattern and figured out that the v = 4x.) This response demonstrates a complete understanding and analysis of the problem.


Sample Student Response #2

image of student response

Score for Sample Student Response #2: Rubric Score 2

Annotation: This response indicates an incomplete application of a reasonable strategy. The table is correctly completed; however, no expression is given. The surface area (126 cm²) and the volume (40 cm³) of a 10-block tower are correct. The explanations support the solutions (every time the number of blocks used increases by one, the surface area increases by 12. If there are 6 blocks and the surface area is 78, then 10-6=4 and 4x12=48 and 48+78=126 cm²...the volume increases by 4. If the volume is 24 when 6 blocks are used, then 10-6=4 and 4x4=16 and 24+16=40 cm³). This response demonstrates a conceptual understanding and analysis of the problem.


Sample Student Response #3

image of student response

Score for Sample Student Response #3: Rubric Score 1

Annotation: This response indicates little application of a reasonable strategy. The table is correctly completed, and the variables are defined. The attempted equation is incorrect, and a justification is not given. The surface area of a 10-block tower (138 cm²) is incorrect. No explanation is provided. The volume is not given. This response demonstrates a minimal understanding and analysis of the problem. Compare to Anchor Paper #2.
Sample Student Response #4

image of student response

Score for Sample Student Response #4: Rubric Score 3

Annotation: This response indicates application of a reasonable strategy that leads to some correct solutions. The table is correctly completed. The representation, in the form of an equation (y=10x+(6+2x)), is correct, and the variables are defined; however, a justification is not given. The volume of a 10-block tower (40 cm³) is correct. The explanation (the number of blocks times 4) supports the solution and is generally well developed. A correct surface area (126 cm²) is given. The symbolic explanation supports the solution; the student replaces x in the equation with 10 and solves. This response demonstrates a clear 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 |