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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 1.1.1 The student will recognize, describe, and/or extend patterns and functional relationships that are expressed numerically, algebraically, and/or geometrically.

Assessment Limits:

  • The given pattern must represent a relationship of the form y = mx + b (linear), y = x2 + c (simple quadratic), y = x3 + c (simple cubic), simple arithmetic progression, or simple geometric progression with all exponents being positive.
  • The student will not be asked to draw three-dimensional figures.
  • Algebraic description of patterns is in indicator 1.1.2

Extended Constructed Response (ECR) Item - Released in 2005

Aisha used squares to make the pattern of figures below.

Complete the following in the Answer Book:

  • Using the pattern, draw Figure 5 and Figure 6 in the Answer Book.
  • Complete the table in the Answer Book to determine the perimeter of each figure.
  • Write an expression that can be used to determine the perimeter of the nth figure in this pattern.
  • If this pattern continues, which figure will have a perimeter of 140 centimeters? Use mathematics to explain how you determined your answer. 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.

Anchor Paper #1

image of student response

Score for Anchor Paper #1: Rubric Score 1

Annotation: This response indicates little application of a reasonable strategy. The figures are incorrectly drawn. While the table is correctly completed, no expression is given, and the figure with a perimeter of 140 units is not provided. 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 little application of a reasonable strategy. Figures 5 and 6 are correctly drawn. Although the table is correctly completed, no expression is given, and the figure with a perimeter of 140 units is not provided. 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 an incomplete application of a reasonable strategy. Figures 5 and 6 are accurately drawn, and the table is correctly completed. However, no expression is given. The student correctly determines that figure 35 will have a perimeter of 140 units, but an explanation is not provided. The response demonstrates a conceptual understanding and analysis of the problem.


Anchor Paper #4

image of student response
image of student response

Score for Anchor Paper #4: Rubric Score 2

Annotation: This response indicates an incomplete application of a reasonable strategy. Figure 6 is accurately drawn; however, Figure 5 is not provided. While the table is correctly completed, no expression is given. The student correctly determines that figure 35 will have a perimeter of 140 units, and the explanation supports the solution. The student extends the pattern until the perimeter of 140 units is reached at figure 35. The 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 indicates application of a reasonable strategy that leads to some correct solutions. The student does not address the first part of the problem. The table is correctly completed, and the representation, in the form of an equation (P=4n) rather than an expression, is correct. The variable p is not defined. The student correctly determines that figure 35 will have a perimeter of 140 units. The explanation supports the solution. (I found this because for every figure times 4, gave me the perimeter {P=4n}; therefore, if I plug in 140 for the perimeter, I will get n=35.) The response demonstrates a clear understanding and analysis of the problem.


Anchor Paper #6

image of student response

Score for Anchor Paper #6: Rubric Score 3

Annotation: This response indicates application of a reasonable strategy that leads to some correct solutions. Figures 5 and 6 are accurately drawn, and the table is correctly completed. The representation, in the form of an equation (F(4)=P) rather than an expression, is correct. The variables are defined. The student correctly determines that figure 35 will have a perimeter of 140 units; however, an explanation is not given. The response demonstrates a clear understanding and analysis of the problem.


Anchor Paper #7

image of student response

Score for Anchor Paper #7: Rubric Score 4

Annotation: This response indicates application of a reasonable strategy that leads to a correct solution within the context of the problem. Figures 5 and 6 are accurately drawn, and the table is correctly completed. Although in the form of an equation set equal to "p," defined as the perimeter, the representation is correct. The student correctly determines that figure 35 will have a perimeter of 140 units. The fully developed explanation supports the solution. (I found this answer simply by placing 140 in the place of p in my expression, 140=4n. Then I divided both sides by four.) The response demonstrates a complete understanding and analysis of the problem.


Anchor Paper #8

image of student response

Score for Anchor Paper #8: Rubric Score 4

Annotation: This response indicates application of a reasonable strategy that leads to a correct solution within the context of the problem. Figures 5 and 6 are accurately drawn, and the table is correctly completed. The correct expression (4n) is given. The student correctly determines that figure 35 will have a perimeter of 140 units. The fully developed explanation supports the solution. (I placed what numbers I had to where they belong like 140=4n. Then I divided 140 by 4 to get 35.) The response demonstrates a complete understanding and analysis of the problem.


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 1.1.1:
Skill Statements | PUBLIC RELEASE ITEMS | Lesson Plans |