Math Fall Block 2015 Algebra 2 PBA Item #13 System of Inequalities M44085
Prompt
Task is worth a total of 3 points. M44085 Rubric Score Description 3 Student response includes the following 3 elements. Reasoning component = 2 points o o Correct explanation of a method to prove that there are no solutions to the system of inequalities Correct explanation of a change to the system that would result in there being a solution set Computation component = 1 point o Correct ordered pair from the solution set, based on the change described Sample Student Response: In order to prove that the system of inequalities has no solution, you could graph each inequality. 2 The graph of y x + 3 is a parabola that opens upwards with vertex (0, 3). The parabola is a solid line, and its interior is shaded. The graph of y < x + 1 is a dashed line that passes through the 2 point (0, 1), and has slope 1. The graph is shaded below the dashed 2 line. Neither the boundary lines nor the shaded regions of the graphs intersect, so there are no solutions to the system. In order to change the system so that there is a solution set, the second inequality could be changed to y > x + 1. The shading 2 would then be above the dashed line, and the solution set would be the parabola and all of the points in its interior. One point would be (0, 4). Note: The shading or overlap of the solution region must be stated or implied to receive credit for a valid explanation. Graphing or intersection alone does not imply that the shaded areas of the inequalities overlap. Other valid changes and valid points may be accepted. Precision point deductions could include, but are not limited to: imprecise mathematical language, such as labeling the inequalities
as functions or equations; or, incorrect description or drawing of graphed boundary lines such as using a solid line when a dashed line is required. 2 Student response includes 2 of the 3 elements. 1 Student response includes 1 of the 3 elements. 0 Student response is incorrect or irrelevant.
Anchor Set A1 A8
A1 Score Point: 3
Annotations Anchor Paper 1 Score Point 3 This response receives full credit. The student includes each of the following three required elements: There is a correct explanation of a method to prove that there are no solutions to the system of inequalities with the student s drawing of a correct graph of both inequalities with shading. There is a correct explanation of a change to the system that would result in there being a solution set (If you flip the inequality sign in y> x + 1, then you can shade up 2 and have a solution). There is a correct ordered pair from the solution set, based on the change described ((0,6)).
A2 Score Point: 3
Annotations Anchor Paper 2 Score Point 3 This response receives full credit. The student includes each of the following three required elements: There is a correct explanation of a method to prove that there are no solutions to the system of inequalities (when you graph the 2 inequalities, their shaded regions don t combine/cross anywhere). The statement regarding overlapping shaded regions strengthens this response. There is a correct explanation of a change to the system that would result in there being a solution set (If you made the inequality y -x 2 +3 (originally y x 2 +3), they would have a solution because the parabola then faces downward). The use of shading is sufficiently implied from the explanation above. A scoring decision was made that you can use any part of the student s previous work to score the subsequent parts of the item. You can read up for the response, but you cannot read down. There is a correct ordered pair from the solution set, based on the change described ((-3,-4)).
A3 Score Point: 2
Annotations Anchor Paper 3 Score Point 2 This response receives partial credit. The student includes two of the three required elements: There is a correct explanation of a method to prove that there are no solutions to the system of inequalities with the student s drawing that correctly graphs both inequalities, showing no intersection. There is a correct ordered pair from the solution set, based on the change described ((0,5)). The student does not sufficiently explain the change to the system that would result in a solution set (there would of been a solution if the less than sign (<) in the second one was a greater than one (>)). The shading/overlap of the solution region must be stated or implied in the explanation of the change.
A4 Score Point: 2
Annotations Anchor Paper 4 Score Point 2 This response receives partial credit. The student includes all three of the required elements, however a precision point is deducted. There is a correct explanation of a method to prove that there are no solutions to the system of inequalities (when graphed... if the shaded parts do not intersect, the student s claim is correct). There is a correct explanation of a change to the system that would result in there being a solution set (change the +1 to a +5 in the second part of the equation... all points in the shaded region in the 2 nd equation would move up 5 on the graph so they will intersect). There is a correct ordered pair from the solution set, based on the change described ((0,4)). A precision point is deducted for identifying the inequalities as equations. A scoring decision has been made that if a precision point error occurs and the response would have received the top score without the error, a score point will be deducted.
A5 Score Point: 1
Annotations Anchor Paper 5 Score Point 1 This response receives partial credit. The student includes one of the three required elements: There is a correct ordered pair from the solution set, based on the change described ((0,0)). The student provides an incorrect explanation of a method to prove that there are no solutions to the system of inequalities (Put the numbers in the calculator) and does not completely explain a change that would result in a solution set (make y into y and you have a solution). This implies that the first inequality is flipped, therefore the given ordered pair is valid. The shading/overlap of the solution region must be stated or implied in the explanation of the change.
A6 Score Point: 1
Annotations Anchor Paper 6 Score Point 1 This response receives partial credit. The student includes one of the three required elements: There is a correct ordered pair from the solution set, based on the change described ((1,4)). The student provides an incorrect explanation of a method to prove that there are no solutions to the system of inequalities (graph both inequalities to see where they intersect, they do not, therefore there is no solution) and does not correctly explain a change that would result in a solution set (change y< x + 1 to y> x + 1, so that the two inequalities would 2 2 intersect). Intersection alone does not imply that the shaded areas of the inequalities overlap. The shading/overlap of the solution region must be stated or implied in the explanation of the change.
A7 Score Point: 0
Annotations Anchor Paper 7 Score Point 0 This response receives no credit. The student includes none of the three required elements: The student provides an incorrect explanation of a method to prove that there are no solutions to the system of inequalities (graph the inequalities). The student does not correctly explain a change that would result in a solution set (make the x 2 become a 2x). The student provides an incorrect ordered pair based on the change described ((1,3)).
A8 Score Point 0
Annotations Anchor Paper 8 Score Point 0 This response receives no credit. The student includes none of the three required elements. The student does not provide an explanation of a method to prove that there are no solutions to the system of inequalities. The student does not sufficiently explain a change that would result in a solution set (make the top less than). The student does not provide a correct ordered pair from the solution set, based on the change described.
Practice Set P101 - P105
P101
P102
P103
P104
P105
Practice Set Paper Score P101 2 P102 1 P103 0 P104 2 P105 1