The following 12 Sample Student Responses represent a range of score points.

Score for Sample Student Response #1: Rubric Score 3

Annotation: This response indicates application of a reasonable strategy, demonstrating a clear understanding and analysis of the problem. The correct equations are given for both shops, although the student has not defined d after changing it from x, which was given in the test item. The student provides the correct solution to the system of equations and correctly answers when it is less expensive to use Bicycle City ("It is less expensive to use bicycle city if your bike takes more than 4 days to build…when there is 5 days or more Bicycle City charges alot less"); however, a reference to when it is more expensive is missing. The graphic representation of the equations is plausible, despite the minor error of mislabeling the graph, and supports the solution, but to provide complete justification, the student would need to include statements such as, "The point of intersection makes both equations true" and "The line for Bicycle City is lower after 4 days and higher before 4 days." Compare to Anchor Paper #6.

Score for Sample Student Response #2: Rubric Score 2

Annotation: This response indicates an incomplete application of a reasonable strategy. The representations are fundamentally correct, although the student has not defined y after changing it from C, which was given in the test item. The student also finds one of the values of the point of intersection, although the charge at 4 days is missing. The comparison of the costs before and after the point of intersection, ("It's more expensive up to 3 days, and less expensive to use BC after 3 days"), however, is inconsistent with the student's stated point of intersection ("After 4 days, the prices would be the same"). Justifications are missing; the graph explains, rather than justifies, how the student attempted to arrive at a solution. This response demonstrates a conceptual understanding and analysis of the problem. Compare to Anchor Paper #4.

Score for Sample Student Response #3: Rubric Score 3

Annotation: This response demonstrates a clear understanding and analysis of the problem. The response indicates application of a reasonable strategy that leads to a correct solution. The representations are essentially correct. The student provides the correct solution to the system of equations. The student also demonstrates an understanding of the system of equations past the point of intersection ("If it takes less than 4 days to build the bike then it is less expensive to use Bike town. If it is more than 4 days Bike City is less expensive"). However, the student has not provided any justifications. Compare to Anchor Paper #6.

Score for Sample Student Response #4: Rubric Score 1

Annotation: This response demonstrates a minimal understanding and analysis of the problem. The equations are present but are incomplete, consisting of algebraic expressions that are not set equal to the charge (given in the test item as C). Rather than equations with C, the student gives "Bike city - 80$x + 160$…Bike town - 120$x." There is no indication that the student has applied a reasonable strategy to solve the system of equations. The student's description of when it is less or more expensive to use Bicycle City is unclear and incomplete ("Bike City is cheaper if you are only going to have the bike take 2 days or more to build"), and the justifications are missing. Compare to Anchor Paper #1.

Score for Sample Student Response #5: Rubric Score 4

Annotation: This response demonstrates a complete understanding and analysis of the problem. 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 student provides the correct solution to the system of equations and justifies this solution by replacing the x in each equation with 4, arriving at C=480 for both. The student also demonstrates an understanding of the system of equations past the point of intersection ("It would be less expensive to use Bicycle City to build a bike that would take more than 4 days to put together. It would be more expensive if it took less then 4 days to build the bike."). The justification is well developed since it includes a well-constructed graph and refers to it in the written portion of the response ("Bicycle City's line is below Bike Town's line after 4 days"). Compare to Anchor Paper #8.

Score for Sample Student Response #6: Rubric Score 2

Annotation: In this response, the student demonstrates a conceptual understanding and analysis of the problem. The response indicates an incomplete application of a reasonable strategy. The representations of the equations are fundamentally correct. The student also finds one of the values of the point of intersection ("4 days"), although the cost at 4 days is missing. While the student has demonstrated a conceptual understanding of the point of intersection, he/she does not seem to understand what happens before and after the point of intersection, replying that it is less expensive to use Bicycle City at 19 days, and more expensive at 5 days. Justifications are also missing. Compare to Anchor Paper #3.

Score for Sample Student Response #7: Rubric Score 1

Annotation: This response indicates little application of a reasonable strategy, demonstrating a minimal understanding and analysis of the problem. The representations are incomplete; the student has provided the correct equation for the charge at Bike Town ("$120•x=C"), but the equation for Bicycle City is incorrect ("$160c•x=$80"). The student does not attempt to find the solution to the system of equations. In determining when it is less or more expensive to use Bicycle City than Bike Town, the student seems to indicate that Bike Town will always be less expensive than Bicycle City ("It was less expensive to use the Bike Town rather than to use Bicycle City…"). The attempted justification incorrectly states that "you don't have to add any extra for making the bike." Compare to Anchor Paper #1.

Score for Sample Student Response #8: Rubric Score 3

Annotation: This response indicates application of a reasonable strategy that leads to a correct solution, demonstrating a clear understanding and analysis of the problem. The representations are essentially correct. The student provides the correct solution to the system of equations. The student also understands the system of equations past the point of intersection ("BC is less expensive after [4] days BT is less before [4] days"). The table showing the costs for 4, 5, and 6 days at each store, though lacking labels, serves as justification for the point of intersection and for the student's statement that Bicycle City is less expensive after 4 days. However, because it does not display any values for fewer than 4 days, the table does not justify the statement that Bike Town costs less before 4 days. Compare to Anchor Paper #6.

Score for Sample Student Response #9: Rubric Score 2

Annotation: This response indicates an incomplete application of a reasonable strategy, demonstrating a conceptual understanding and analysis of the problem. The representations of the equations are fundamentally correct. The student also provides the correct solution to the system of equations in indicating "$480 for 4 days." However, the student does not have a clear understanding of a system of equations past the point of intersection ("It is less to use Bicycle City when y <= 0. It is more expensive when y >= 1"). Justifications are missing; the description of calculator steps explains how the student arrived at the solution rather than justifying the solution. Compare to Anchor Paper #4.

Score for Sample Student Response #10: Rubric Score 1

Annotation: This response indicates little application of a reasonable strategy, demonstrating a minimal understanding and analysis of the problem. The representations are incomplete. The correct equations are given for both shops, although the student has not defined y after changing it from C, which was given in the test item. There is no indication that the student has applied a reasonable strategy to solve the system of equations ("It is about 15 day when they whould be the same. Bicycle town was always high that Bicycle City unless the finished the bike in the first day"). Compare to Anchor Paper #1.

Score for Sample Student Response #11: Rubric Score 2

Annotation: This response indicates an incomplete application of a reasonable strategy, demonstrating a conceptual understanding and analysis of the problem. The equations are fundamentally correct, with the minor omission of labels to identify each equation. The student also finds one of the values of the point of intersection ("x=4"), although the cost at 4 days and the justification are missing. "It's less after 4 days" correctly specifies when it is less expensive to use Bicycle City, but the solution to when it is more expensive is missing. The student attempts a graphic representation and points to it as a justification ("look at graph"), but the graph is incorrect. Compare to Anchor Paper #4.

Score for Sample Student Response #12: Rubric Score 4

Annotation: This response indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. The representations are correct. The student provides the correct solution to the system of equations and then justifies the solution by replacing the x in each equation with 4, arriving at C=$480 for each. The student also demonstrates an understanding of the system of equations past the point of intersection ("It is less expensive to use Bicycle City to build a custom-made bicycle than Bike Town after 5 days…. It was more expensive before the four days to use Bicycle City"). There is a minor error: The student should have said "after 4 days" rather than "after 5 days." Replacing x with 5 (days) and 3 (days) in each equation provides justification for the student's cost comparisons ("after 5 days…at Bicycle Town it costs $600 but at Bicycle City it costs $560…it was $400 at Bicycle City and only $360 at Bicycle Town before the four days"). The response demonstrates a complete understanding and analysis of the problem. Compare to Anchor Paper #9.

Anchor Papers used in scoring