Item 19 Anchor Papers    
Elizabeth joins a CD buyer's club. She receives 10 free CDs when she joins this club. She must buy 3 CDs each month. Complete the following in the Answer Book:
  • Write an equation that represents the number of CDs (y) Elizabeth will receive from the CD buyer's club after x months.
     
  • What is the y-intercept of your equation? What does the y-intercept mean in the context of this problem?
     
  • Elizabeth wants to receive no more than 55 CDs from this club. What is the maximum number of months Elizabeth will remain in the CD buyer's club? Use mathematics to explain how you determined your answer. Use words, symbols, or both in your explanation.

Score Level 1 Anchor Paper

 

This response indicates little application of a reasonable strategy. The representation is incorrect (y=3x). The value of the y-intercept is not provided, and the contextual meaning is incomplete (the number of CDs). The maximum number of months (18) is incorrect. The symbolic explanation reveals a reasonable strategy, replacing y with 55 and solving for x. The incorrect equation generates an incorrect solution. This response demonstrates a minimal understanding and analysis of the problem.

image of student response

Score Level 1 Anchor Paper

 

This response indicates little application of a reasonable strategy. The equation, the value of the y-intercept, and the intercept's contextual meaning are not provided. The maximum number of months is correct, and the incomplete explanation supports the solution (she starts out with 10 free which = 45; 45÷3=15). This response demonstrates a minimal understanding and analysis of the problem.

image of student response

Score Level 2 Anchor Paper

 

This response indicates an incomplete application of a reasonable strategy. The representation is correct (y=10+3x). While the value of the y-intercept (10) is correct, the contextual meaning is not given. The maximum number of months (15) is correct, but an explanation is not provided. This response demonstrates a conceptual understanding and analysis of the problem.

image of student response

Score Level 2 Anchor Paper

 

This response indicates an incomplete application of a reasonable strategy. The representation, in the form of an expression rather than an equation, is correct (10+3x). The value of the y-intercept is not provided, and the contextual meaning is incomplete (number of CDs). A correct maximum number of months (15) is given with the calculation (10+3·15=55) providing justification, rather than explanation. This response demonstrates a conceptual understanding and analysis of the problem.

image of student response

Score Level 3 Anchor Paper

 

This response indicates application of a reasonable strategy that leads to some correct solutions. The representation is correct (y=3x+10). The student provides the correct y-intercept, although it is given as "y=10." Rather than supplying a contextual meaning, the student supplies the definition of a y-intercept (where the line hits the y-axis). The maximum number of months (15) is correct; however, the calculations (3x15=45+10=55) provide justification, rather than explanation. This response demonstrates a clear understanding and analysis of the problem.

image of student response

Score Level 3 Anchor Paper

 

This response indicates application of a reasonable strategy that leads to some correct solutions. The representation is correct (y=3x+10). Both the value of the y-intercept (ten) and the contextual meaning (she starts with 10 CDs) are correct. Although the maximum number of months (15) is correct, no explanation is given. This response demonstrates a clear understanding and analysis of the problem.

image of student response

Score Level 4 Anchor Paper

 

This response indicates application of a reasonable strategy that leads to a correct solution within the context of the problem. The representation is correct (y=3x+10). The student provides the correct y-intercept (10), contextual meaning (it is the initial amount of CDs Elizabeth gets from the buyers club), and maximum number of months (15). The explanation is clearly presented and fully developed; the student constructs a table showing monthly x and y values for Elizabeth's membership, stopping at the 15th month when the number of CDs stood at 55. This response demonstrates a complete understanding and analysis of the problem.

image of student response

Score Level 4 Anchor Paper

 

This response indicates application of a reasonable strategy that leads to a correct solution within the context of the problem. The representation is correct (y=3x+10). The student provides both the correct y-intercept (10) and the contextual meaning (she starts out with 10 free CDs). The maximum number of months (15) is correct. The explanation is clearly presented and fully developed; the student replaces the y in the inequality with 55 and solves for x. This response demonstrates a complete understanding and analysis of the problem.

image of student response
 

Anchor Papers ~ Algebra/Data Analysis ~ Item 19

Elizabeth joins a CD buyer's club. She receives 10 free CDs when she joins this club. She must buy 3 CDs each month. Complete the following in the Answer Book:
  • Write an equation that represents the number of CDs (y) Elizabeth will receive from the CD buyer's club after x months.
     
  • What is the y-intercept of your equation? What does the y-intercept mean in the context of this problem?
     
  • Elizabeth wants to receive no more than 55 CDs from this club. What is the maximum number of months Elizabeth will remain in the CD buyer's club? Use mathematics to explain how you determined your answer. Use words, symbols, or both in your explanation.

 

Score Level 1 Anchor Paper

 

This response indicates little application of a reasonable strategy. The representation is incorrect (y=3x). The value of the y-intercept is not provided, and the contextual meaning is incomplete (the number of CDs). The maximum number of months (18) is incorrect. The symbolic explanation reveals a reasonable strategy, replacing y with 55 and solving for x. The incorrect equation generates an incorrect solution. This response demonstrates a minimal understanding and analysis of the problem.

image of student response

 

Score Level 1 Anchor Paper

 

This response indicates little application of a reasonable strategy. The equation, the value of the y-intercept, and the intercept's contextual meaning are not provided. The maximum number of months is correct, and the incomplete explanation supports the solution (she starts out with 10 free which = 45; 45÷3=15). This response demonstrates a minimal understanding and analysis of the problem.

image of student response

 

Score Level 2 Anchor Paper

 

This response indicates an incomplete application of a reasonable strategy. The representation is correct (y=10+3x). While the value of the y-intercept (10) is correct, the contextual meaning is not given. The maximum number of months (15) is correct, but an explanation is not provided. This response demonstrates a conceptual understanding and analysis of the problem.

image of student response

 

Score Level 2 Anchor Paper

 

This response indicates an incomplete application of a reasonable strategy. The representation, in the form of an expression rather than an equation, is correct (10+3x). The value of the y-intercept is not provided, and the contextual meaning is incomplete (number of CDs). A correct maximum number of months (15) is given with the calculation (10+3·15=55) providing justification, rather than explanation. This response demonstrates a conceptual understanding and analysis of the problem.

image of student response

 

Score Level 3 Anchor Paper

 

This response indicates application of a reasonable strategy that leads to some correct solutions. The representation is correct (y=3x+10). The student provides the correct y-intercept, although it is given as "y=10." Rather than supplying a contextual meaning, the student supplies the definition of a y-intercept (where the line hits the y-axis). The maximum number of months (15) is correct; however, the calculations (3x15=45+10=55) provide justification, rather than explanation. This response demonstrates a clear understanding and analysis of the problem.

image of student response

 

Score Level 3 Anchor Paper

 

This response indicates application of a reasonable strategy that leads to some correct solutions. The representation is correct (y=3x+10). Both the value of the y-intercept (ten) and the contextual meaning (she starts with 10 CDs) are correct. Although the maximum number of months (15) is correct, no explanation is given. This response demonstrates a clear understanding and analysis of the problem.

image of student response

 

Score Level 4 Anchor Paper

 

This response indicates application of a reasonable strategy that leads to a correct solution within the context of the problem. The representation is correct (y=3x+10). The student provides the correct y-intercept (10), contextual meaning (it is the initial amount of CDs Elizabeth gets from the buyers club), and maximum number of months (15). The explanation is clearly presented and fully developed; the student constructs a table showing monthly x and y values for Elizabeth's membership, stopping at the 15th month when the number of CDs stood at 55. This response demonstrates a complete understanding and analysis of the problem.

image of student response

 

Score Level 4 Anchor Paper

 

This response indicates application of a reasonable strategy that leads to a correct solution within the context of the problem. The representation is correct (y=3x+10). The student provides both the correct y-intercept (10) and the contextual meaning (she starts out with 10 free CDs). The maximum number of months (15) is correct. The explanation is clearly presented and fully developed; the student replaces the y in the inequality with 55 and solves for x. This response demonstrates a complete understanding and analysis of the problem.

image of student response