Cedric is making a scale model of his school. The dimensions of his school and of his model are shown below.

Step A

What is the length, in feet, of side x of the model?

Step B

• Use what you know about similar polygons to justify why your value of side x is correct. Use words, numbers, and/or symbols in your justification.
• If Cedric changes the length of his model from 3 feet to 6 feet, explain how this change will affect the value of side x. Use words, numbers, and/or symbols in your explanation.

Step A is scored 0 (Incorrect) or 1 (Correct) and assesses 2.D.1.a.
Step B is scored with a 4 point (0, 1, 2, 3) rubric and assesses Processes of Mathematics.

Note: Twenty "Sample Student Responses" follow below. Each response appears on its own separate page and includes scoring information. The "Sample Student Responses" represent a range of score points.

Score for Sample Student Response #1:

Annotation for Step B, Using the Rubric: This response is completely incorrect.

Score for Sample Student Response #2:

Annotation for Step B, Using the Rubric: This response demonstrates a minimal understanding and analysis of the problem. A partial application of a strategy in the context of the problem is indicated. The justification for why the length of side x is correct is logically flawed as the scale is not correctly applied to the given dimensions: "[Divide] 84 by 3 and get 28... 1 foot on the model is 28 feet on the real school." The explanation for how the change in the length of the model will change with value of side x is missing.

Score for Sample Student Response #3:

Annotation for Step B, Using the Rubric: This response demonstrates a minimal understanding and analysis of the problem. An application of a strategy of using proportions is indicated. However, the justification for the value of side x is logically flawed as the proportion is simplified incorrectly. The explanation of how the value of side x will change is minimal: "The scale would then have to bring both measurements up, not just one of them." Supportive information and/or numbers are not provided. Connections to the concept of similarity are missing.

Score for Sample Student Response #4:

Annotation for Step B, Using the Rubric: This response demonstrates a minimal understanding and analysis of the problem. An application of a strategy of using proportions is indicated. The justification for the value of side x is logically flawed: "You divide 3 ft into 84=28. Divide 28/21 = 1.7." The explanation for how the change will affect the value of side x is minimal: "You would multiply the other side by two." Connections to the concept of similarity are missing.

Score for Sample Student Response #5:

Annotation for Step B, Using the Rubric: This response demonstrates a minimal understanding and analysis of the problem. The justification for why the value of side x is correct is missing. The connection to similar polygons having the same shape and proportional sized is implied: "He will have to change the other side or it will make the school look bigger than it really is." Supportive information and/or numbers are not provided.

Score for Sample Student Response #6:

Annotation for Step B, Using the Rubric: This response demonstrates a general understanding and analysis of the problem. A reasonable strategy of using proportional reasoning to find the value for side x is indicated. The justification, presented as an explanation, for the value of side x is partially developed: "84/3 = 28 then 21/28 =.74." The minor calculation error does not detract from the overall understanding of the process used to determine the value of side x. The explanation for how the change will affect the value of side x is clear: 'The length of side x will change to 1.5...84/6=14... 21/14=1.5." Connections to the concept of similarity are missing.

Score for Sample Student Response #7:

Annotation for Step B, Using the Rubric: This response demonstrates a general understanding and analysis of the problem. A reasonable strategy of using proportional reasoning to determine the value of side x is indicated. The justification, presented as an explanation, for why this value is correct is only partially developed: "I divided 3 by 84 to get the scale, then divided 21 by 28. The minor mathematical error of stating "3 is divided by 84" instead of "84 is divided by 3" does not detract from the overall understanding of the process used to solve the problem. This justification is supported by the correct quotient of 28. The explanation for how the change in the length would affect the value of side x is partially developed: "the scale would end up being 14...divide 21 by 14." Connections to the concept of similarity are missing.

Score for Sample Student Response #8:

Annotation for Step B, Using the Rubric: This response demonstrates a general understanding and analysis of the problem. A reasonable strategy of using proportional reasoning to determine the value for side x is indicated. The justification, presented as a numerical explanation, for why this value is correct, is only partially developed: "84/3=28...21/28= . 75." The explanation for how the change will affect the value of side x is clear: "84/6=14...21/14=1.5." Appropriate supportive numbers are provided. However, connections to the concepts of similarity and scale are missing.

Score for Sample Student Response #9:

Annotation for Step B, Using the Rubric: This response demonstrates a comprehensive understanding and analysis of the problem. The application of a reasonable strategy in the context of the problem is indicated. The justification for why the value of side x is correct is clear, fully developed and logical: "Divide 84 by 3 to get 28...divide 21 by 28 to get .75. To check, multiply .75X28 to get 21." The connection to similar polygons is clear and stated explicitly: "The model is the same shape as the school only smaller." The explanation of how the change will affect the values of side x is clear, fully developed and logical: "Divide 84 by 6 instead of 3 to get 14...and 21 divided by 14 is 1.5." Appropriate supportive numbers are provided.

Score for Sample Student Response #10:

Annotation for Step B, Using the Rubric: This response demonstrates a comprehensive understanding and analysis of the problem. The application of a reasonable strategy in the context of the problem is indicated. The justification for how the value for side x was determined is clear, fully developed and logical: "21 feet is one fourth of 84 feet... side x will be one fourth of 3 feet." The connection to the concept of similar polygons is clear as the same scale is used throughout because, "The school and the model are similar objects." The explanation of how the change would affect the value of side x is clear, fully developed and logical: "If Cedric doubled the value of the 3 foot side... the value of side x would double too." Supportive numbers are provided: "It would change from .75 feet to 1.5 feet."

Score for Sample Student Response #11:

Annotation for Step B, Using the Rubric: This response demonstrates a general understanding and analysis of the problem. A reasonable strategy of using proportional reasoning to determine the value of side x is indicated. The justification, presented as an explanation, for why this value is correct is only partially developed: "I divided 84 by 3... it becomes a fraction... ." Connections to the concept of similarity are implied: 'There is 28 feet to one foot of the model." The explanation for how the change in the length would affect the value of side x is partially developed: "It would be 14ft=1ft of the model." Compare to Sample Student Response #7.

Score for Sample Student Response #12:

Annotation for Step B, Using the Rubric: This response demonstrates a minimal understanding and analysis of the problem. The justification for why the value of side x is correct is feasible: "Divide 84ft by 28 to get a quotient of 3ft... do the same for side x." The connection to similar polygons having the same shape and proportionally sized is implied: "When you make a model your sides still have to be equivalent to the normal size." The explanation for how the change will affect the value for side x is missing. Compare to Sample Student Response #5.

Score for Sample Student Response #13:

Annotation for Step B, Using the Rubric: This response is completely incorrect. Compare to Sample Student Response #1.

Score for Sample Student Response #14:

Annotation for Step B, Using the Rubric: This response demonstrates a minimal understanding and analysis of the problem. A partial application of a strategy in the context of the problem is indicated. The justification for why the length of side x is correct is logically flawed but supports the answer in Step A: "I first divided 84ft by 3ft... then I subtracted... 21 from 28." The explanation for how the change in the length of the model will change with value of side x is minimal: "The x's value would increase." Supportive numbers are not provided. Compare to Sample Student Response #2.

Score for Sample Student Response #15:

Annotation for Step B, Using the Rubric: This response demonstrates a comprehensive understanding and analysis of the problem. The application of a reasonable strategy in the context of the problem is indicated. The justification for how the value for side x was determined is clear, fully developed and logical: "21/84= and x/3= so x is 0.75." The connection to the concept of similar polygons is clear: "similar models are in exact proportion to the actual school... This makes the polygons similar." The explanation of how the change would affect the value of side x is clear, fully developed and logical: "x/6 is 14 so x is 1.5." Supportive numbers are provided. Compare to Sample Student Response #10.

Score for Sample Student Response #16:

Annotation for Step B, Using the Rubric: This response demonstrates a minimal understanding and analysis of the problem. An application of a strategy of using proportions is indicated. However, the justification for the value of side x is logically flawed as the proportion is reversed. The explanation of how the value of side x will change is minimal: "The side with the x value will have to increase." Supportive information and/or numbers are not provided. Connections to the concept of similarity are missing. Compare to Sample Student Response #3.

Score for Sample Student Response #17:

Annotation for Step B, Using the Rubric: This response demonstrates a general understanding and analysis of the problem. A reasonable strategy of using proportional reasoning to determine the value for side x is indicated. The justification, presented as a numerical explanation, for why this value is correct, is only partially developed: "84/3=28... 21/28 rounds to .75." The explanation for how the change will affect the value of side x is clear: "84/6=14... 21/14=1.5." Appropriate supportive numbers are provided. However, connections to the concepts of similarity and scale are missing. Compare to Sample Student Response #8.

Score for Sample Student Response #18:

Annotation for Step B, Using the Rubric: This response demonstrates a minimal understanding and analysis of the problem. An application of a strategy of using proportions is indicated. The justification for the value of side x is missing. The explanation for how the change will affect the value of side x is logically flawed but is consistent with the answer given in Step A: "the x-side would be 4 instead of 2... you would have to multiply by 2." Connections to the concept of similarity are missing. Compare to Sample Student Response #4.

Score for Sample Student Response #19:

Annotation for Step B, Using the Rubric: This response demonstrates a comprehensive understanding and analysis of the problem. The application of a reasonable strategy in the context of the problem is indicated. The justification for why the value of side x is correct is clear, fully developed and logical: "[I used] a scale factor... 21 is of 84. So I divided 3 by 4 and got x=.75." The connection to similar polygons is clear and stated explicitly: "Similar polygons share all the same characteristics except for size," The explanation of how the change will affect the values of side x is clear, fully developed and logical: "Divide 6 by 4 and get 1.5 for x." Appropriate supportive numbers are provided. Compare to Sample Student Response #9.

Score for Sample Student Response #20:

Annotation for Step B, Using the Rubric: This response demonstrates a minimal understanding and analysis of the problem. An application of a strategy of using proportions is indicated; however, the justification for the value of side x is logically flawed as the proportion is solved incorrectly. The explanation for how the change will affect the value for side x is minimal: "The value of x will be greater." The connection to similar polygons having the same shape and proportional sized is implied: "You are increasing the size and the scale of the model." Compare to Sample Student Response #5.

Score 3

The response demonstrates a comprehensive understanding and analysis of a problem.

• Application of a reasonable strategy in the context of the problem is indicated.
• Explanation¹ of and/or justification² for the mathematical process(es) used to solve a problem is clear, fully developed, and logical.
• Connections and/or extensions made within mathematics or outside of mathematics are clear and stated explicitly.
• Supportive information and/or numbers are provided as appropriate. ³

Score 2

The response demonstrates a general understanding and analysis of a problem.

• Application of a reasonable strategy in the context of the problem is indicated.
• Explanation¹ of and/or justification² for the mathematical process(es) used to solve a problem is feasible, but may be only partially developed.
• Connections and/or extensions made within mathematics or outside of mathematics are partial or overly general, or may be implied.
• Supportive information and/or numbers are provided as appropriate. ³

Score 1

The response demonstrates a minimal understanding and analysis of a problem.

• Partial application of a strategy in the context of the problem is indicated.
• Explanation¹ of and/or justification² for the mathematical process(es) used to solve a problem is logically flawed or missing.
• Connections and/or extensions made within mathematics or outside of mathematics are flawed or missing.
• Supportive information and/or numbers may or may not be provided as appropriate.³

Score 0

The response is completely incorrect, irrelevant to the problem, or missing.⁴

Note 1:

Explanation refers to students' ability to communicate how they arrived at the solution for an item using the language of mathematics.

Note 2:

Justification refers to students' ability to support the reasoning used to solve a problem, or to demonstrate why the solution is correct using mathematical concepts and principles.

Note 3:

Students need to complete rubric criteria for explanation, justification, connections and/or extensions as cued for in a given problem.

Note 4:

Merely an exact copy or paraphrase of the problem will receive a score of "0".

Rubric Document Date: August 2003