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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 1.1.2 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 2007

A triangle, a quadrilateral, a pentagon, and a hexagon are shown below. By drawing a diagonal from 1 vertex, the quadrilateral is divided into 2 non-overlapping triangles. Since the sum of the angle measures of a triangle is 180°, the sum of the measures of the quadrilateral is 360°. By drawing the diagonals from 1 vertex, the pentagon is divided into 3 non-overlapping triangles.

Complete the following in the Answer Book:

  • In the Answer Book, draw the diagonals from 1 vertex of the hexagon so that the hexagon is divided into non-overlapping triangles.
  • Use the polygons above to complete the table in the Answer Book.
    Polygon Number of Sides Number of Non-Overlapping Triangles Sum of Angle Measures
    Triangle 3 1 180°
    Quadrilateral 4 2 360°
    Pentagon 5 3 ?
    Hexagon ? ? ?
  • Describe how the number of sides of each polygon is related to the number of non-overlapping triangles. Use mathematics to justify your answer.
  • Describe how the number of non-overlapping triangles in a polygon is related to the sum of its angle measures. Use mathematics to justify your answer.

The following 10 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 demonstrates little application of a reasonable strategy. The table is correctly completed. The student has not responded to the first, third and fourth parts of the question. This 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 demonstrates little application of a reasonable strategy. The hexagon is incorrectly divided into non-overlapping triangles but the table is correctly completed. A general relationship between the number of sides of each polygon and non-overlapping triangles is given (As the number of sides went up so did the number of non-overlapping triangles). The student does not describe the relationship between the number of sides of any polygon and its number of non-overlapping triangles and does not provide a justification. The student provides a relationship (Also every time the number of non overlapping triangles goes up so does the sum of the angles), but does not describe the relationship between the number of non-overlapping triangles of any polygon and the sum of its angle measures. A justification is not provided. This 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 demonstrates an incomplete application of a reasonable strategy. The hexagon is incorrectly divided into non-overlapping triangles. The table is correctly completed. The student provides a relationship (The more sides you have, the more non-overlapping triangles you going have), but does not describe the relationship between the number of sides of any polygon and its number of non-overlapping triangles. A justification is not provided. The relationship between the number of non-overlapping triangles in a polygon and the sum of its angle measures is correct (To get the sum of angle measures, you have to multiply the number of non overlapping triangles times 180°) but a justification is not given. 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 hexagon is correctly divided into non-overlapping triangles. The number of sides and overlapping triangles for a hexagon are correct, but the column for the sum of angle measures is incorrect. The relationship between the number of sides and non-overlapping triangles is correct (If you take the number of sides and subtract 2 you get the # of non-overlapping triangles) and the justification supports the solution (3-2=1 triangle; 4-2=2 Quad; 5-2=3 pentagon). The relationship between the number of non-overlapping triangles in a polygon and the sum of its angle measures is not given and a justification is not provided. This response demonstrates a conceptual understanding of the problem.


Anchor Paper #5

image of student response

Score for Anchor Paper #5: Rubric Score 2

Annotation: This response demonstrates an incomplete application of a reasonable strategy. The hexagon is incorrectly divided into non-overlapping triangles. The table is correctly completed. The student provides a relationship (the numbers in both go up one number each time), but does not describe the relationship between the number of sides of any polygon and its number of non-overlapping triangles. A justification is not provided. The relationship between the number of non-overlapping triangles in a polygon and the sum of its angle measures is correct (you take the number of non-overlapping triangles and you multiply 180 by that number) but a justification is not given. This response demonstrates a conceptual understanding and analysis of the problem.


Anchor Paper #6

image of student response

Score for Anchor Paper #6: Rubric Score 3

Annotation: This response demonstrates application of a reasonable strategy that leads to some correct solutions within the context of the problem. The hexagon is correctly divided into non-overlapping triangles and the table is correctly completed. The relationship between the number of sides and non-overlapping triangles is correct (how many sides there were you would subtract 2 away from it to get the number of over lapping triangles) but a justification is not provided. The omission of the prefix "non" in "over lapping triangles" constitutes a minor error. The relationship between the number of non-overlapping triangles in a polygon and the sum of its angle measures is correct (You could multiply the number of non overlapping triangles times 180°). The justification is given (because 180° is one triangle, so you would multiply 180° by the # of triangles). 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 3

Annotation: This response demonstrates application of a reasonable strategy that leads to some correct solutions within the context of the problem. The hexagon is not divided but the table is correctly completed. The relationship between the number of sides and non-overlapping triangles is correct (if you subtract 2 to the number of sides, you get the number of Non-Overlapping triangles) and the justification supports the solution (3-2=1. 4-2=2. 5-2=3. 6-2=4.). The relationship between the number of non-overlapping triangles in a polygon and the sum of its angle measures is correct (multiplied by 180 to get the sum of its angle measures) but a justification is not provided. This response demonstrates a clear understanding and analysis of the problem.


Anchor Paper #8

image of student response

Score for Anchor Paper #8: Rubric Score 3

Annotation: This response demonstrates application of a reasonable strategy that leads to some correct solutions within the context of the problem. The hexagon is incorrectly divided but the table is correctly completed. The relationship between the number of sides of each polygon and non-overlapping triangles is correct (you subract 2 from the number of sides on that shape) and the justification supports the solution (On a triangle it has three sides - two=1 overlapping triangle. In the quadrilateral it had 4 sides - two= 2 overlapping sides). The use of "sides" rather than triangles constitutes a minor error. The relationship between the number of non-overlapping triangles in a polygon and the sum of its angle measures is correct (multiplying the number of triangles times 180°) and the justification supports the solution (Hexagon 4 triangles x 180°=720°). This response demonstrates a clear understanding and analysis of the problem.


Anchor Paper #9

image of student response

Score for Anchor Paper #9: Rubric Score 4

Annotation: This response demonstrates application of a reasonable strategy that leads to correct solutions within the context of the problem. The hexagon is correctly divided into non-overlapping triangles and the table is correctly completed. The relationship between the number of sides of each polygon and the number of non-overlapping triangles is correct (There are 2 less overlapping triangles than there are # of sides) and the justification is fully developed and clearly presented (3-1=2,4-2=2,5-3=2,6-4=2). The relationship between the number of non-overlapping triangles in a polygon and the sum of its angle measures is correct (The sum of angle measures 180° times the # of non overlapping triangles), and the justification is fully developed (1x180°=180°, 2x180°=360°, 3x180°=540°, 4x180°=720°). This response demonstrates a complete understanding and analysis of the problem.


Anchor Paper #10

image of student response

Score for Anchor Paper #10: Rubric Score 4

Annotation: This response demonstrates application of a reasonable strategy that leads to correct solutions within the context of the problem. The hexagon is correctly divided into non-overlapping triangles and the table is correctly completed. The relationship between the number of sides of each polygon and the number of non-overlapping triangles is correct (To find the # of triangles using the # of sides you simply minus 2 to the # of sides) and the justification is fully developed (3-2=1; 4-2=2; 5-2=3; 6-2=4). The relationship between the number of non-overlapping triangles in a polygon and the sum of its angle measures is correct (To find the sum of angle measurements yo times # of triangle times 180). The justification is fully developed and clearly presented (1·180=180; 2·180=360; 3·180=540; 4·180=720). 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 1.1.2:
Skill Statements | PUBLIC RELEASE ITEMS | Lesson Plans |