Item 19 Practice Set B Responses    
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 3 Paper

 

This response indicates application of a reasonable strategy that leads to some correct solutions. The representation is correct (y=10+3x). The student provides the correct y-intercept (ten); however, instead of a contextual meaning, an explanation of how to use the y-intercept (that's the number you have to add when you solve the equation for x) is given. While the maximum number of months is correct, the calculations {10+(3·15)=55 and 3·15=45 and 45+10=55} provide justification, rather than explanation. This response demonstrates a clear understanding and analysis of the problem. (Compare to this Level 3 anchor paper.)

image of student response

Score Level 4 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). Both the y-intercept (10) and the contextual meaning (Elizabeth receives 10 CDs for joining the club) are correct. A correct number of months (15) is provided. The symbolic explanation is clearly presented and fully developed; the student replaces the y in the equation with 55 and solves for x. This response demonstrates a complete understanding and analysis of the problem.

image of student response

Score Level 1 Paper

 

This response indicates little application of a reasonable strategy. The representation, in the form of an expression rather than an equation, is (3x+10). The value of the y-intercept (10,0) is incorrect but relevant and the student provides the definition of a y-intercept, rather than the contextual meaning (y-intercept is the amount when a line crosses through the y-axis and x=0). The maximum number of months and an explanation are not given. This response demonstrates a minimal understanding and analysis of the problem.

image of student response

Score Level 3 Paper

 

This response indicates application of a reasonable strategy that leads to some correct solutions. The representation is correct (y=10+3x). While the student provides the correct y-intercept (10), an explanation (I know this is the y- intercept because y=10+3x is the y=mx+b formula, and the b is your y-intercept), rather than the contextual meaning, is given. The maximum number of months is correct, and the symbolic explanation supports the solution. The student replaces the y in the equation with 55 and solves for x. This response demonstrates a clear understanding and analysis of the problem. (Compare to this Level 4 anchor paper.)

image of student response

Score Level 4 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=10+3x). The student provides the correct y-intercept (10), contextual meaning (she receives 10 free CDs when she joins the club), and maximum number of months (15). The symbolic explanation is clearly presented and fully developed; the student replaces the y in the equation with 55 and solves for x. This response demonstrates a complete understanding and analysis of the problem.

image of student response

Score Level 2 Paper

 

This response indicates an incomplete application of a reasonable strategy. The representation is correct (y=10+3x). The value of the y-intercept (13) is incorrect, and the student gives the definition of the variable y, rather than the contextual meaning of the y-intercept (number of CDs Elizabeth receives after x amount of months). The maximum number of months (15) is correct. The explanation supports the solution, but is incomplete (selecting a random number then replacing x with that number). This response demonstrates a conceptual understanding and analysis of the problem. (Compare to this Level 2 anchor paper.)

image of student response

Score Level 1 Paper

 

This response indicates little application of a reasonable strategy. The representation is incorrect (y=3x). No value for the y-intercept is given, and the contextual meaning is incorrect (number of CDs that Liz has currently purchased after joining the music club). The maximum number of months (19) is incorrect. The explanation reveals a reasonable strategy (dividing 3 into 55 and came out with 18.3), using the student's incorrect equation. This response demonstrates a minimal understanding and analysis of the problem.

image of student response

Score Level 2 Paper

 

This response indicates an incomplete application of a reasonable strategy. The representation is incorrect (y=10x+3). However, the y-intercept (positive three) is consistent with the student's equation. While an explanation of the y-intercept is given (In a y-intercept, x always equals zero. When you substitute 0 for x, you get zero + 3=y), the equation creates an incorrect contextual meaning (the number of CDs Elizabeth must buy each month to get 10 CDs free). The maximum number of months (5.2) is incorrect, but consistent with the student's equation. The symbolic explanation reveals a reasonable strategy; the student replaces the y in the equation with 55 and solves for x. This response demonstrates a conceptual understanding and analysis of the problem.

image of student response
 

Practice Set B 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 3 Paper

 

This response indicates application of a reasonable strategy that leads to some correct solutions. The representation is correct (y=10+3x). The student provides the correct y-intercept (ten); however, instead of a contextual meaning, an explanation of how to use the y-intercept (that's the number you have to add when you solve the equation for x) is given. While the maximum number of months is correct, the calculations {10+(3·15)=55 and 3·15=45 and 45+10=55} provide justification, rather than explanation. This response demonstrates a clear understanding and analysis of the problem. (Compare to this Level 3 anchor paper.)

image of student response

 

Score Level 4 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). Both the y-intercept (10) and the contextual meaning (Elizabeth receives 10 CDs for joining the club) are correct. A correct number of months (15) is provided. The symbolic explanation is clearly presented and fully developed; the student replaces the y in the equation with 55 and solves for x. This response demonstrates a complete understanding and analysis of the problem.

image of student response

 

Score Level 1 Paper

 

This response indicates little application of a reasonable strategy. The representation, in the form of an expression rather than an equation, is (3x+10). The value of the y-intercept (10,0) is incorrect but relevant and the student provides the definition of a y-intercept, rather than the contextual meaning (y-intercept is the amount when a line crosses through the y-axis and x=0). The maximum number of months and an explanation are not given. This response demonstrates a minimal understanding and analysis of the problem.

image of student response

 

Score Level 3 Paper

 

This response indicates application of a reasonable strategy that leads to some correct solutions. The representation is correct (y=10+3x). While the student provides the correct y-intercept (10), an explanation (I know this is the y- intercept because y=10+3x is the y=mx+b formula, and the b is your y-intercept), rather than the contextual meaning, is given. The maximum number of months is correct, and the symbolic explanation supports the solution. The student replaces the y in the equation with 55 and solves for x. This response demonstrates a clear understanding and analysis of the problem. (Compare to this Level 4 anchor paper.)

image of student response

 

Score Level 4 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=10+3x). The student provides the correct y-intercept (10), contextual meaning (she receives 10 free CDs when she joins the club), and maximum number of months (15). The symbolic explanation is clearly presented and fully developed; the student replaces the y in the equation with 55 and solves for x. This response demonstrates a complete understanding and analysis of the problem.

image of student response

 

Score Level 2 Paper

 

This response indicates an incomplete application of a reasonable strategy. The representation is correct (y=10+3x). The value of the y-intercept (13) is incorrect, and the student gives the definition of the variable y, rather than the contextual meaning of the y-intercept (number of CDs Elizabeth receives after x amount of months). The maximum number of months (15) is correct. The explanation supports the solution, but is incomplete (selecting a random number then replacing x with that number). This response demonstrates a conceptual understanding and analysis of the problem. (Compare to this Level 2 anchor paper.)

image of student response

 

Score Level 1 Paper

 

This response indicates little application of a reasonable strategy. The representation is incorrect (y=3x). No value for the y-intercept is given, and the contextual meaning is incorrect (number of CDs that Liz has currently purchased after joining the music club). The maximum number of months (19) is incorrect. The explanation reveals a reasonable strategy (dividing 3 into 55 and came out with 18.3), using the student's incorrect equation. This response demonstrates a minimal understanding and analysis of the problem.

image of student response

 

Score Level 2 Paper

 

This response indicates an incomplete application of a reasonable strategy. The representation is incorrect (y=10x+3). However, the y-intercept (positive three) is consistent with the student's equation. While an explanation of the y-intercept is given (In a y-intercept, x always equals zero. When you substitute 0 for x, you get zero + 3=y), the equation creates an incorrect contextual meaning (the number of CDs Elizabeth must buy each month to get 10 CDs free). The maximum number of months (5.2) is incorrect, but consistent with the student's equation. The symbolic explanation reveals a reasonable strategy; the student replaces the y in the equation with 55 and solves for x. This response demonstrates a conceptual understanding and analysis of the problem.

image of student response