Amanda's long distance telephone plan charges a monthly fee of $14.95 and $.05 per minute for the number of minutes (m) that she uses her telephone. The monthly charge for Amanda's long distance telephone plan is represented by the expression shown below.

14.95 + 0.05m

Step A

How much is the monthly charge for Amanda's long distance telephone plan if she talks for 150 minutes in one month?

Step B

• Use what you know about expressions to explain how you determined the monthly charge for Amanda's long distance telephone plan. Use words, numbers, and/or symbols in your explanation.
• Suppose that the cost per minute changes to $.07 per minute. Explain how this change will affect the charge for Amanda's telephone plan. Use words, numbers, and/or symbols in your explanation.

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

Note: 15 "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 irrelevant to the problem. It merely paraphrases given information from the problem. No new, relevant information is given.

Score for Sample Student Response #2:

Annotation for Step B, Using the Rubric: This response demonstrates a minimal understanding and analysis of the problem. The student uses a partial application of a strategy by adding $.05 to $14.95. However, the student uses a process that is logically flawed because the student does not multiply the $.05 by 150 minutes. In addition, the student correctly states that a change in the charge per minute would add "more cents to the total and change the whole amount," but the explanation of how the monthly charge would change lacks support.

Score for Sample Student Response #3:

Annotation for Step B, Using the Rubric: This response demonstrates a minimal understanding and analysis of the problem. The student uses a partial application of a strategy by correctly multiplying $.05 by 150. However, the explanation for the process used to solve the problem reveals a serious flaw in reasoning as the student does not add this product to $14.95. The student explains how the monthly charge would change, "[it] would raise the monthly fee," but the support is missing.

Score for Sample Student Response #4:

Annotation for Step B, Using the Rubric: This response demonstrates a general understanding and analysis of the problem. The student provides supportive information by showing how $0.05 was multiplied by 150 minutes to find $7.50 for the answer. However, the fact that the student neglects to add $7.50 to the constant monthly fee of $14.95 indicates a major flaw in reasoning and a partially developed explanation. The student effectively communicates the extension of the increasing telephone plan charge, and provides some supportive information for the change my multiplying $0.07 by 150 minutes to arrive at $10.50 for the cost of the minutes. Again, the constant monthly fee of $14.95 is ignored.

Score for Sample Student Response #5:

Annotation for Step B, Using the Rubric: This response demonstrates a general understanding and analysis of the problem. The student applies a reasonable strategy to arrive at the monthly charge of $22.45 and provides an explanation for the mathematical process used to solve this portion of the problem. However, the explanation of the relationship between the change in the charge per minute and the monthly charge for the telephone plan is overly general, "The increase of 2 cents will make the bill more expensive."

Score for Sample Student Response #6:

Annotation for Step B, Using the Rubric: This response demonstrates a comprehensive understanding and analysis of the problem. The student applies a reasonable strategy of in the context of the problem. The minor arithmetic error leading to an answer of $21.45 in Step A does not diminish the student's understanding of the problem. The extension concerning the change in the charge per minute and the total monthly charge are clear and stated explicitly. In addition, the explanations are clear and fully developed with appropriate support.

Score for Sample Student Response #7:

Annotation for Step B, Using the Rubric: This response demonstrates a comprehensive understanding and analysis of the problem. The student applies a reasonable strategy in the context of the problem. The student provides numerical support for explanations that are fully developed, and logical. The student explicitly states the extension with regard to raising the per minute charge and the increase that is paid per month, "for every 150 minutes you would have to multiply 2 more cents." The student provides support for this by multiplying 150 minutes by $.02 and comparing the $7.50 charge to the $10.50 charge.

Score for Sample Student Response #8:

Annotation for Step B, Using the Rubric: This response demonstrates a comprehensive understanding and analysis of the problem. The student applies a reasonable strategy in the context of the problem. The student provides support for explanations that are clear, fully developed, and logical. The student provides support for these explanations by showing all work as the expression is solved both when m equals $.05 and when m equals $.07. Compare to Sample Student Response #7.

Score for Sample Student Response #9:

Annotation for Step B, Using the Rubric: This response demonstrates a minimal understanding and analysis of the problem. The student uses a partial application of a strategy by taking "the 5 cents and times it by the 150 minutes and got $750." However, the explanation for the process used to solve the problem reveals a serious logical flaw in as the student omits the decimal point in the product. In addition, the student does not add this product to $14.95. The explanation for how the change in the per minute charge affects the monthly charge is incomplete, and omits the decimal point. The student implies how the cost will change by multiplying 150 by 7 cents to arrive at $1050, but ignores the constant charge as part of the monthly cost. Compare to Sample Student Response #4.

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 student applies a reasonable strategy in the context of the problem. The explanations provided are clear, fully developed, and logical. The student provides numerical support for the original cost of $22.85 and changed cost of $25.45. The extension from the change in the cost per minute to the total monthly cost is stated explicitly, "the money would go up. The answer would come out to be $25.45." Compare to Sample Student Response #8.

Score for Sample Student Response #11:

Annotation for Step B, Using the Rubric: This response demonstrates a minimal understanding and analysis of the problem. The student uses a partial application of a strategy by multiplying $.05 by 150. However, the explanation reveals a logical flaw as the student fails to add this product to $14.95. The student correction states that increasing the per minute charge of the plan "would change the plan by increasing the amount that Amanda pays per month for calling long distance." However, appropriate support for this extension is not provided. Compare to Sample Student Response #2.

Score for Sample Student Response #12:

Annotation for Step B, Using the Rubric: This response demonstrates a comprehensive understanding and analysis of the problem. The student applies a reasonable strategy in the context of the problem. The student applies a reasonable strategy in the context of the problem. The student provides numerical support for explanations that are fully developed, and logical. The student uses a comparison to state explicitly the extension from raising the per minute charge to the amount that is paid per month. Compare to Sample Student Response #7.

Score for Sample Student Response #13:

Annotation for Step B, Using the Rubric: This response is completely incorrect. The strategy of dividing 150 by 5 is inappropriate for this problem.

Score for Sample Student Response #14:

Annotation for Step B, Using the Rubric: This response demonstrates a general understanding and analysis of the problem. The student applies a reasonable strategy in the context of the problem to arrive at the solution to Step A of this problem. The student provides appropriate support by explaining how this total cost was determined; "I multiplied the number of minutes...by the charge per minute...and added it to the monthly fee to get the total money she owed." However, the explanation of the relationship between the change in the charge per minute and the monthly charge for the telephone plan is overly general and lacks support, "It would obviously make the total charge a little more." Compare to Sample Student Response #5.

Score for Sample Student Response #15:

Annotation for Step B, Using the Rubric: This response demonstrates a general understanding and analysis of the problem. The student uses a partial application of a strategy by multiplying 150 by the cost per minute to arrive at $7.50 for the first charge and $10.50 for the increased charge. However, the response is flawed because the student never adds the constant $14.95 fee to either of these products. Thus, the connection between the initial fee and the increased fee is overly general and lacks the appropriate support. Compare to Sample Student Response #4.

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