Download VF935309

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