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Math Released Item 2016 Geometry Reasonable Dimensions VF935309 Prompt Rubric Task is worth a total of 4 points. VF935309 Rubric Score Description 4 Student response includes the following 4 elements. • Computation component = 1 point o Correct value for x: -6 • Reasoning component = 3 points o Establishment of similar triangles by assuming lines are parallel o Correct work shown for solving for x using the given side lengths and proportionality o Recognition that length cannot be negative and that the dimensions are not reasonable. Sample Student Response: If is parallel to than and , since both pairs of angles are alternate-interior angles formed by the parallel lines and the transversals and , respectively. Because two angles in triangle ABC are congruent to two angles in triangle CDE, the triangles ABC and DEC are similar. Because the lengths of corresponding sides of similar triangles are in proportion , and by substituting the given values, 3 2 1 0 The length of one side cannot be negative, so these dimensions are not reasonable. Note: Alternative methods are possible and any valid method is acceptable for full credit. Student response includes 3 of the 4 elements. Student response includes 2 of the 4 elements. Student response includes 1 of the 4 elements. Student response is incorrect or irrelevant. Anchor Set A1 – A10 With Annotations A1 Score Point 4 Annotation Anchor Paper 1 Score Point 4 This response receives full credit. The student includes each of the four required elements. • The response shows the establishment of similar triangles ( ∠ D ≅ ∠ A, ∠ E ≅ ∠ B, by the Alternate interior angles theorem. The triangles are similar by AA). • The correct ratio is shown for the proportional sides of a similar triangle x+2 x+4 ( ). = x x+3 • The correct solution for the variable x is found (x2 + 4x = x2 + 5x + 6, -x = 6, x to equal -6). • A correct conclusion is given about the dimensions being reasonable (If x were -6, then all of the lengths end up being negative, which we know are unreasonable in side lengths). A2 Score Point 4 Annotation Anchor Paper 2 Score Point 4 This response receives full credit. The student includes each of the four required elements. • The response shows the establishment of similar triangles (Because ���� 𝐴𝐴𝐴𝐴 �𝐷𝐷𝐷𝐷, we know that ∠ ABC ≅ ∠ CED and ∠ BAC ≅ ∠ CDE because if two parallel lines are cut by a transversal, then alternate, interior angles are congruent. We also know that ∠ ACB ≅ ∠ ECD because they are vertical angles and vertical angles are congruent…the triangles are similar because all of their angles are congruent). • The correct ratio is shown for the proportional sides of a similar triangle x + 4 ( x + 2) ]. [ = x+3 x • The correct solution for the variable x is found (-6). • A correct conclusion is given about the dimensions being reasonable (x is a length, and no side can have a length of a negative number). A3 Score Point 3 Annotation Anchor Paper 3 Score Point 3 This response receives partial credit. The student includes three of the four required elements. • The correct ratio is shown for the proportional sides of a similar triangle x x+3 ]. [ = x+2 x+4 • The correct solution for the variable x is found (-6). • A correct conclusion is given about the dimensions being reasonable (they are not reasonable because x would be a negative number, ���� an impossible negative distance). giving 𝐴𝐴𝐴𝐴 The establishment of similar triangles is missing. A4 Score Point 3 Annotation Anchor Paper 4 Score Point 3 This response receives partial credit. The student includes three of the four required elements. • • • The correct ratio is shown for the proportional sides of a similar triangle 𝑥𝑥+4 𝑥𝑥+2 (𝑥𝑥+3 = 𝑥𝑥 ). The correct solution for the variable x is found (x = -6) A correct conclusion is given about the dimensions being reasonable (No, they are not reasonable, because x = -6, so sides are negative lengths and that is not possible). The response does not show the establishment of similar triangles. Complete work is shown for determining the value of x, but is unnecessary to receive credit for the second element. A5 Score Point 2 Annotation Anchor Paper 5 Score Point 2 This response receives partial credit. The student includes two of the four required elements. • The correct ratio is shown for the proportional sides of a similar triangle X +2 X ). ( = X +3 X +4 • The correct solution for the variable x is found [X(X+4) = (X + 3)(X+2), X2 + 4X = X2 + 3X + 2X + 6, X2 + 4X = X2 + 5X + 6, 0 = X + 6, -6 = X]. The response does not establish that the triangles are similar although the ratios shown assume that the triangles are similar. The conclusion is incorrect (tHE DIMENSIONS ARE REASONABLE). The response also checks the work and finds each side of the ratio equals 2 when -6 is substituted for x. A6 Score Point 2 Annotation Anchor Paper 6 Score Point 2 This response receives partial credit. The student includes two of the four required elements. • The correct ratio is shown for the proportional sides of a similar triangle x x+3 ]. [ = x+2 x+4 • The correct solution for the variable x is found (-6). The establishment of similarity of the triangles is missing. The response correctly states that the triangles are similar due to AA similarity, but the congruent angles that would justify this conclusion are not included. The conclusion is incorrect (Yes…the 2 triangles should be proportional). A7 Score Point 1 Annotation Anchor Paper 7 Score Point 1 This response receives partial credit. The student includes one of the four required elements. • The response shows the establishment of similar triangles (angel ABC is congruent to angle CED…alternate interiorangles are congruent…angle ACB is congruent to angle ECD because vertical angles are congruent…Therefore, we can use AA similarity Thereom and say that these two triangles are similar). Note: The rubric states that one pair of the alternate interior angles with the vertical angle may be used for credit. The ratios of the sides of the similar triangles are not shown. No solution for the value of x is given. The conclusion is incorrect (these dimensions do make sense). A8 Score Point 1 Annotation Anchor Paper 8 Score Point 1 This response receives partial credit. The student includes one of the four required elements. • The response shows the establishment of similar triangles (Since AB is parallel to DE, that makes AD their transversal. So, the alternate interior angles of each triangle are congruent. Angle CED is congruent to ABC and CDE is congruent to BAC. Then ACB is congruent to ECD because vertical angles are congruent. So these two triangles are similar through AA.). The ratios of the sides of the similar triangles are not shown. No solution for the value of x is given. The conclusion is incorrect (So the corresponding sides would have to be congruent which they are not). A9 Score Point 0 Annotation Anchor Paper 9 Score Point 0 This response receives no credit. The student includes none of the four required elements. The response does not adequately establish the similarity even though they indicate the AA Postulate. To establish similarity of the triangles, two correct congruent angles must be provided. No ratios are provided, no solution for the value of x is shown, and an incorrect conclusion (Yes) without a reason or explanation is given. A10 Score Point 0 Annotation Anchor Paper 10 Score Point 0 This response receives no credit. The student includes none of the four required elements. The establishment of similarity of the triangles is begun with correct angles equal, but incorrectly has line segments equal. These are the steps for showing congruency and not similarity of three triangles. The ratios of the sides are not shown. The value of x is not given. The conclusion with explanation is missing. Practice Set P101 - P105 No Annotations Included P101 P102 P103 P104 P105 Practice Set Paper Score P101 0 P102 2 P103 4 P104 3 P105 1