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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 2002

Stephanie must complete class projects for history and mathematics. Each project is worth a maximum of 100 points if she turns it in on time. Each teacher uses a different method to calculate the maximum number of points that a student can earn for a late paper. The tables below show the two methods.

Complete the following in the Answer Book:

  • Complete the tables to show the number of possible points for projects that are 4 and 5 days late.
  • How many days late can Stephanie turn in each project and still possibly receive a score of 65 or better on each project? Use mathematics to explain how you determined your answer. Use words, symbols, or both in your explanation.
  • Stephanie can complete only one project on time. The second project will be only one day late. To receive the maximum total number of possible points, which project should she complete first? Use mathematics to justify your answer.
  • Describe the method that each teacher uses to determine the maximum possible points that a student can earn for a late paper.

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 table for the history project is completed correctly; the table for the math has not been completed. The student incorrectly identifies the math project as the project to turn in on time. The attempted justification is an invalid comparison of projects that are one day late ("Because History has more points than math if it's late"). This response demonstrates 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. The table for the history project is completed correctly. The student provides the correct solution for the number of days Stephanie can turn her history project in late; however, an explanation is not provided. The math table and number of days for the math project are not provided. Although the student correctly identifies the history project as the project to turn in on time, the student's response ("because it's a harder class") does not provide justification. This response demonstrates 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. The table for the history project is completed correctly; however, the math table is incorrect. The number of days for the history project is correct, and the number of days for the math project is consistent with the student's table. Explanations are not provided. The student correctly identifies the history project as the project to turn in on time. The justification is presented symbolically. The student shows that math would lose 4 points when it is one day late ("100 - 96 = 4 points"), but history would lose 5 points after one day ("100 - 95 = 5 points"). "Math uses 5 points each" is an invalid statement. 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 an incomplete application of a reasonable strategy. The table for the history project is completed correctly; however, the math table is incorrect. The number of days for the history project is correct. The number of days for the math project is incorrect, although consistent with the student's table. The student correctly identifies the history project as the project to turn in on time and provides a tabular justification, showing that after one day math "goes down less." The description of the history teacher's method of assigning points is well developed ("...subtract 5 points for every day late"), but the student does not identify this as the history project. 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 indicates application of a reasonable strategy that leads to a correct solution. The tables are correctly completed. The student gives the correct number of days Stephanie can turn each project in to receive a score of 65 or better, then incorrectly identifies the math project as the project to turn in on time. The justification for this choice reveals a misunderstanding ("...it's only 3 days until its late and be able to get a feasible score.") The student gives a complete description of the history teacher's method of assigning points ("...subtracts 5 points for every day...") which also provides a complete explanation for the history solution in the second part of the question. The description of the math teacher's method is not fully developed ("...subtracted in intervals of 4 {4,8,12,16...} according to day"). The student does not indicate that the points are to be subtracted from the previous day's total number of points. This 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 a correct solution. The tables are completed correctly. The number of days for the history project is correct, and the student provides a complete explanation ("...takes away 5 points each day"). The number of days for the math project is also correct; however, the explanation is not fully developed ("...takes away 4 x the number of days it's late"). The student does not indicate that the points are to be subtracted from the previous day's total number of points. The student correctly identifies the history project as the project to turn in on time, and this choice is fully justified ("...if math was late one day she would get 96 but if the history was she would get a 95"). While a complete description of the history teacher's method of assigning points is found in the student's explanation for the history solution in the second part of the question, the description of the math teacher's method is only generally developed. This 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. The tables are correctly completed. The number of days for the history project is correct, and the explanation is clearly presented ("...because each day it's late her score goes down by 5"). The number of days for the math project also is correct with a fully developed explanation ("...because each additional day late means another 4 points is lost. For example, from 100, one day late means - 4 points. Then you have 96. Another day late is - 8 points, which takes you down to 88 points..."). The student correctly identifies the history project as the one to turn in on time, and this choice is fully justified ("...she will receive a 96 in her math class for it being late only one day. In history class she would receive a 95."). The description of each teacher's method of assigning points is found in the student's complete explanations for the second part of the question. This 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. The tables are correctly completed. The number of days for the history project is correct, and the explanation is fully developed ("because each day its late 5 points are taken off, and after 7 days the maximum points she can get is 65"). The number of days for the math project also is correct with the explanation clearly presented and supportive of the solution ("...if she turns it in 4 days late she would get a 60"). The student correctly identifies the history project as the project to turn in on time. This choice is fully justified ("...because 1 day late for that and you get 95 maximum points. With the Math project you get 96 maximum points for 1 day late"). The student provides a complete description of each teacher's method of assigning points ("In History for every day late it's late, 5 points are taken off. For Math it's the # of days late times 4 taken off from the last day's maximum points."). This 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 |