Mathematics Benchmarks & Indicators
with Ohio Graduation Test Questions
Grades 9 – 10
Includes questions from the released
2008, 2007, 2006, 2005, 2004, and 2003
Ohio Graduation Tests
Far East Regional Partnership
for Conceptually Based Mathematics
Youngstown State University
Compiled by A. Crabtree, 2006
Revised by A. Crabtree and L. Holovatick, 2007
Revised by A. Crabtree, J. Lucas, and T. Cameron, 2008
FERPCBM
Geometry and Spatial Sense
Standard
Geometry and Spatial Sense
OGT Problems
A. Formally define geometric figures.
10.1. Formally define and explain key aspects of geometric figures, including:
a. interior and exterior angle of polygons;
b. segments related to triangles (median, altitude, midsegment);
c. points of concurrency related to triangles (centroid, incenter, orthocenter, and
circumcenter);
d. circles (radius, diameter, chord, circumference, major arc, minor arc, sector,
segment, inscribed angle).
10.2. Recognize and explain the necessity for certain terms to remain undefined, such
as point, line and place.
10.6. Identify the reflection and rotation symmetries of two- and three-dimensional
figures.
10.10. Solve problems involving chords, radii, and arcs within the same circle.
OGT 2007 – Problem #10
The diagram below shows two of the faces of a prism.
Ellen is constructing this triangular prism out of cardboard. Ellen plans to draw a
shape (a net of the prism) on a piece of cardboard that she can cut out and fold
to make the prism.
In your Answer Document, sketch a net of the prism that Ellen could use. Place
the dimensions on the net.
Prepared by Aimee Crabtree for FERPCBM, Youngstown, Ohio
Benchmarks with OGT Problems
Summer 2006 (Rev. 2008)
Page 3 of 36
Geometry and Spatial Sense
OGT Problems
OGT 2007 – Problem # 33
The figure shows a three-dimensional object when
viewed from above.
A. pentagonal
pyramid
B. rectangular prism
C. square pyramid
D. triangular prism
Which of these objects could the sketch represent?
OGT 2005 – Problem # 3
Joseph plans to tile his kitchen floor. All the tiles
will be identical regular polygons. Joseph wants
three tiles to meet at each vertex with no space
left over.
A. pentagon
B. hexagon
C. triangle
D. square
What shape of tile should he use?
OGT 2004 – Problem # 13
Daniel cut the corner off a cube as shown in
the diagram below.
A. cone
B. cube
C. triangular prism
D. triangular pyramid
Points A, B and C are the midpoints of the
edges of the cube. What type of three dimensional
figure has been cut off?
Prepared by Aimee Crabtree for FERPCBM, Youngstown, Ohio
Benchmarks with OGT Problems
Summer 2006 (Rev. 2008)
Page 4 of 36
Geometry and Spatial Sense
OGT Problems
OGT 2003 – Problem #5
Four points are connected with line segments, as shown on the coordinate plane
below.
In your Answer Document, find the slope of each side.
Determine if the shape is a parallelogram. Show your work or provide an
explanation to support your answer.
Prepared by Aimee Crabtree for FERPCBM, Youngstown, Ohio
Benchmarks with OGT Problems
Summer 2006 (Rev. 2008)
Page 5 of 36
Geometry and Spatial Sense
OGT Problems
OGT 2003 – Problem #21
The Venn diagram below represents relationships among various kinds of
triangles.
Which of the following triangles is included in the shaded region?
Prepared by Aimee Crabtree for FERPCBM, Youngstown, Ohio
Benchmarks with OGT Problems
Summer 2006 (Rev. 2008)
Page 6 of 36
Geometry and Spatial Sense
OGT Problems
B. Describe and apply the properties of similar and congruent figures; and justify
conjectures involving similarity and congruence.
OGT 2008 – Problem # 37
An airline executive drew a sketch of a logo on a
napkin.
4 .0
y

3.8 10.0
3 .8 6 . 8
B.

y
10
6 .8
y

C.
4.0 10.0
y
10.0

D.
6 .8 3 . 8
A.
She gave the logo to a graphic designer so he could
make a mathematically similar version with a computer
drawing program. The center line of the new logo needs
to be 10 centimeters long. Which of these proportions
could the graphic designer use to find the value of y?
Prepared by Aimee Crabtree for FERPCBM, Youngstown, Ohio
Benchmarks with OGT Problems
Summer 2006 (Rev. 2008)
Page 7 of 36
Geometry and Spatial Sense
OGT Problems
OGT 2007 – Problem # 29
Points E, F, G, and H are midpoints of the sides of
quadrilateral ABCD.
A. 96
If AC = 12 and BD = 8, what is the perimeter of
quadrilateral EFGH?
C. 20
B. 40
D. 4
OGT 2006 – Problem # 37
A. 37°
B. 45°
C. 53°
D. 74°
What is the measure of angle D?
Prepared by Aimee Crabtree for FERPCBM, Youngstown, Ohio
Benchmarks with OGT Problems
Summer 2006 (Rev. 2008)
Page 8 of 36
Geometry and Spatial Sense
OGT Problems
OGT 2005 – Problem # 23
The shadow cast by a one-foot ruler is 8 inches
long. At the same time, the shadow cast by a
pine tree is 24 feet long.
A. 3 feet
B. 16 feet
C. 36 feet
D. 192 feet
What is the height, in feet, of the pine tree?
OGT 2004 – Problem # 36
A student shines a light through a cutout of a triangle held parallel to a wall
several feet straight in front of him, producing a similar image on the wall.
What must the two triangles have in common?
A. equal areas
B. equal heights
C. corresponding sides that are congruent
D. corresponding angles that are congruent
Prepared by Aimee Crabtree for FERPCBM, Youngstown, Ohio
Benchmarks with OGT Problems
Summer 2006 (Rev. 2008)
Page 9 of 36
Geometry and Spatial Sense
OGT Problems
OGT 2003 – Problem #32
The angle of the roof on Kaya’s dollhouse is
56°. She built a scale model of the dollhouse
with a scale ratio of 1 : 4. What is the measure
of the angle of the roof of the model?
A. 14°
B. 34°
C. 56°
D. 224°
C. Recognize and apply angle relationships in situations involving intersecting lines,
perpendicular lines and parallel lines.
10.10. Solve problems involving chords, radii, and arcs within the same circle.
OGT 2008 – Problem # 42
When a marble hits a wall, it bounces off the wall at the
same angle it hits the wall.
A. 44°
B. 68°
C. 136°
D. 15 8°
If a marble hits a wall at a 22 degree angle, what is the
measure of the angle between the two paths of the
marble?
Prepared by Aimee Crabtree for FERPCBM, Youngstown, Ohio
Benchmarks with OGT Problems
Summer 2006 (Rev. 2008)
Page 10 of 36
Geometry and Spatial Sense
OGT Problems
OGT 2007 – Problem # 25
In which figure is the measure of
1 not equal to 60°?
OGT 2006 – Problem # 25
A worker painted stripes for spaces in a parking lot. The
worker first painted a center stripe that marked the front
of the parking spaces. Then he painted parallel stripes
marking the sides.
A.
2 and
3
B.
2 and
5
C.
3 and
5
D.
4 and
5
Which angles will be congruent to angle 1 if all the side
stripes are parallel?
Prepared by Aimee Crabtree for FERPCBM, Youngstown, Ohio
Benchmarks with OGT Problems
Summer 2006 (Rev. 2008)
Page 11 of 36
Geometry and Spatial Sense
OGT Problems
OGT 2005 – Problem # 1
Ms. Chen drew a diagram for a new patio in her
backyard.
OGT 2003 – Problem #1
When framing in a wall, carpenters make sure
that all vertical studs are perpendicular to the
floor and ceiling. They sometimes add a
diagonal brace for added support during
construction (as shown in the drawing).
When the vertical studs are perpendicular to
the floor, which pair of angles will always be
congruent?
Prepared by Aimee Crabtree for FERPCBM, Youngstown, Ohio
Benchmarks with OGT Problems
Summer 2006 (Rev. 2008)
Page 12 of 36
Geometry and Spatial Sense
OGT Problems
D. Use coordinate geometry to represent and examine the properties of geometric
figures.
OGT 2008 – Problem # 13
The coordinates of triangle KLM are K: (–2, –5), L: (0, 1) and M: (6, –2).
What type of triangle is KLM?
A. obtuse isosceles
B. acute scalene
C. right isosceles
D. right scalene
Prepared by Aimee Crabtree for FERPCBM, Youngstown, Ohio
Benchmarks with OGT Problems
Summer 2006 (Rev. 2008)
Page 13 of 36
Geometry and Spatial Sense
OGT Problems
OGT 2006 – Problem # 29
Three vertices of a quadrilateral are (–1, –1), (1, 2) and
(5, 2).
A. (3, 0)
B. (3, –2)
C. (–5, 0)
D. (7, –1)
When used as the last vertex, which point would make
the quadrilateral a trapezoid?
OGT 2005 – Problem # 35
The vertices of a quadrilateral are
(2, 2), (4, 6), (8, 2), and (10, 6).
Which describes this quadrilateral?
A. parallelogram
B. rectangle
C. rhombus
D. square
OGT 2005 – Problem # 41
An electronics engineer is programming the
computer that controls the circuit board cutting
tools. He enters the coordinates of the vertices of
a rectangular circuit board. The first three
coordinates are (2, 0), (2, 6) and (6, 6).
A. (0, 2)
B. (0, 6)
C. (6, 0)
D. (6, 2)
What are the coordinates of the fourth vertex?
Prepared by Aimee Crabtree for FERPCBM, Youngstown, Ohio
Benchmarks with OGT Problems
Summer 2006 (Rev. 2008)
Page 14 of 36
Geometry and Spatial Sense
OGT Problems
OGT 2004 – Problem # 30
Points P, Q and R are shown below.
These points are three vertices of a parallelogram. What are the coordinates of
the fourth vertex of parallelogram PQRS?
A. (4, 6)
B. (5, 2)
C. (8, –1)
D. (9, 1)
Prepared by Aimee Crabtree for FERPCBM, Youngstown, Ohio
Benchmarks with OGT Problems
Summer 2006 (Rev. 2008)
Page 15 of 36
Geometry and Spatial Sense
OGT Problems
OGT 2003 – Problem #5
Four points are connected with line segments, as shown on the coordinate plane
below.
In your Answer Document, find the slope of each side.
Determine if the shape is a parallelogram. Show your work or provide an
explanation to support your answer.
OGT 2003 – Problem #38
A circle is sketched on a coordinate plane with
the center at (6, 8). The circle passes through
the point (0, 5). Which of these expressions
could be used to find r , the radius of the
circle?
Prepared by Aimee Crabtree for FERPCBM, Youngstown, Ohio
Benchmarks with OGT Problems
Summer 2006 (Rev. 2008)
Page 16 of 36
Geometry and Spatial Sense
OGT Problems
E. Draw and construct representations of two- and three-dimensional geometric
objects using a variety of tools, such as straightedge, compass and technology.
10.4. Construct right triangles, equilateral triangles, parallelograms, trapezoids,
rectangles, rhombuses, squares and kites, using compass and straightedge or
dynamic geometry software.
10.5. Construct congruent or similar figures using tools, such as compass, straightedge,
and protractor or dynamic geometry software.
10.7. Perform reflections and rotations using compass and straightedge constructions
and dynamic geometry software.
OGT 2008 – Problem # 4
The figure shows two views of the same object.
Which net will make the figure shown?
Prepared by Aimee Crabtree for FERPCBM, Youngstown, Ohio
Benchmarks with OGT Problems
Summer 2006 (Rev. 2008)
Page 17 of 36
Geometry and Spatial Sense
OGT Problems
OGT 2007 – Problem # 6
The net creates a pyramid.
Which pyramid does this net create?
OGT 2007 – Problem #10
The diagram below shows two of the faces of a prism.
Ellen is constructing this triangular prism out of cardboard. Ellen plans to draw a
shape (a net of the prism) on a piece of cardboard that she can cut out and fold
to make the prism.
In your Answer Document, sketch a net of the prism that Ellen could use. Place
the dimensions on the net.
Prepared by Aimee Crabtree for FERPCBM, Youngstown, Ohio
Benchmarks with OGT Problems
Summer 2006 (Rev. 2008)
Page 18 of 36
Geometry and Spatial Sense
OGT Problems
OGT 2006 – Problem # 11
The figure shows the net for a three-dimensional object.
When folded, which object will this net produce?
OGT 2003 – Problem #13
Penny needs to make a model of an Egyptian pyramid for her history class. She
plans to cut a shape out of thick cardboard and then fold it to make a pyramid
with a square base. Which of these nets or shapes could Penny use to make her
model?
Prepared by Aimee Crabtree for FERPCBM, Youngstown, Ohio
Benchmarks with OGT Problems
Summer 2006 (Rev. 2008)
Page 19 of 36
Geometry and Spatial Sense
OGT Problems
F. Represent and model transformations in a coordinate plane and describe the
results.
10.8. Derive coordinate rules for translations, reflections and rotations of geometric
figures in the coordinate plane.
10.9. Show and describe the results of combinations of translations, reflections and
rotations (compositions); e.g., perform compositions and specify the result of a
composition as the outcome of a single motion, when applicable.
OGT 2008 – Problem # 34
The vertices of Triangle I are (1, 3), (2, 1) and (5, 0). Triangle I is reflected
across the x-axis, resulting in Triangle II. Triangle II is then rotated 180o about
the origin, resulting in Triangle III.
In your Answer Document, draw and label Triangles I, II and III on the same
coordinate plane.
Describe a single transformation that would map Triangle I directly onto Triangle
III.
OGT 2007 – Problem # 37
The quadrilateral below is to be translated 6 units to the
right and 3 units down.
A. (3, –4)
B. (3, –1)
C. (3, 2)
D. (1, –1)
Which ordered pair is not the coordinates for a vertex of
the translated image?
Prepared by Aimee Crabtree for FERPCBM, Youngstown, Ohio
Benchmarks with OGT Problems
Summer 2006 (Rev. 2008)
Page 20 of 36
Geometry and Spatial Sense
OGT Problems
OGT 2007 – Problem # 42
Jake is using a computer program to design parts for a
new car. He has outlined one-half of a part on a
coordinate grid. Rather than drawing the second half,
he plans to have the computer reflect this shape over
the y-axis.
A.
(0, 0); (0, 2);
(−1, 3); (−2, 3);
(−2, 0)
B. (0, 0); (0, 3);
(−1, 3); (−2, 2);
(−2, 0)
C. (0, 0); (0, −3);
(1, −3); (2, −2);
(2, 0)
D. (0, 0); (0, −3);
(−1, −3); (−2, −2);
(−2, 0)
Which of the following represents the 5 vertices of the
reflected shape?
OGT 2005 – Problem # 6
By the end of summer vacation, Callie had completed one-fourth of a quilt, as
shown in the diagram below.
To finish the quilt, Callie plans to reflect the design of the completed portion over
lines p and q until all 4 portions are complete.
In your Answer Document, copy the diagram above. Add the reflections of the
completed portion of the quilt on the same sketch to show how the entire quilt
will look when it is finished.
Prepared by Aimee Crabtree for FERPCBM, Youngstown, Ohio
Benchmarks with OGT Problems
Summer 2006 (Rev. 2008)
Page 21 of 36
Geometry and Spatial Sense
OGT Problems
OGT 2004 – Problem # 10
Triangle DEF has vertices with coordinates D(–2, 1), E(1, 5) and F(2, 3).
In your Answer Document, draw and label triangle DEF on the grid provided.
Draw the triangle D′E′F′ by translating each vertex of triangle DEF three units to
the right and two units down. Appropriately label
triangle D′E′F′.
Draw the triangle D′′E′′F′′ by translating each vertex of triangle D′E′F′ two units to
the left and seven units up. Appropriately label
triangle D′′E′′F′′.
Describe the movements necessary to perform a single translation of each
vertex from triangle DEF to triangle D′′E′′F′′.
OGT 2003 – Problem #28
The quadrilateral STUW has vertices at the coordinates (1, 1), (2, 5), (5, 5), and
(8, 1), as shown.
What are the coordinates of the vertices of quadrilateral STUW when it is
reflected over the x-axis?
A. (1, 1), (2, 5), (5, 5), (8, 1)
B. (–1, 1), (–2, 5), (–5, 5), (–8, 1)
C. (–1, –1), (–2, –5), (–5, –5), (–8, –1)
D. (1, –1), (2, –5), (5, –5), (8, –1)
Prepared by Aimee Crabtree for FERPCBM, Youngstown, Ohio
Benchmarks with OGT Problems
Summer 2006 (Rev. 2008)
Page 22 of 36
Geometry and Spatial Sense
OGT Problems
G. Prove or disprove conjectures and solve problems involving two- and threedimensional objects represented within a coordinate system.
9.3. Analyze two-dimensional figures in a coordiante plane; e.g., use slope and distance
formulas to show that a quadrilateral is a parallelogram.
OGT 2006 – Problem # 40
Triangle ABC is shown on the graph.
In your Answer Document, show that the segment connecting the midpoints of
and
is parallel to
and one-half its length. Show your work or provide
an explanation for your answer.
H. Establish the validity of conjectures about geometric objects, their properties and
relationships by counter-example, inductive and deductive reasoning, and
critiquing arguments made by others.
10.3. Make, test and establish the validity of conjectures about geometric properties and
relationships using counterexample, inductive and deductive reasoning, and
paragraph or two-column proof, including:
a. prove the Pythagoaren Theorem;
b. prove theorems involving triangle similarity and congruence;
c. prove theorems involving properties of lines, angles, triangles and
quadrilaterals;
d. test a conjecture using basic constructions made with a compass and
straightedge or technology.
10.10. Solve problems involving chords, radii and arcs within the same chords.
Prepared by Aimee Crabtree for FERPCBM, Youngstown, Ohio
Benchmarks with OGT Problems
Summer 2006 (Rev. 2008)
Page 23 of 36
Geometry and Spatial Sense
OGT Problems
OGT 2004 – Problem # 33
Which figure is not a parallelogram?
OGT 2003 – Problem #7
Josie claimed that all polyhedra have more vertices than faces. Which of these
polyhedra shows that Josie’s statement is not always true?
Prepared by Aimee Crabtree for FERPCBM, Youngstown, Ohio
Benchmarks with OGT Problems
Summer 2006 (Rev. 2008)
Page 24 of 36
Geometry and Spatial Sense
I.
OGT Problems
Use right triangle trigonometric relationships to determine lengths and angle
measures.
9.1. Define the basic trigonometric ratios in right triangles: sine, cosine, and tangent.
9.2. Apply proportions and right triangle trigonometric ratios to solve problems involving
missing lengths and angle sizes in similar figures.
OGT 2005 – Problem # 15
Susan is flying a kite behind her house. She drops
her string holder, and the kite gets caught in the
top of a tree.
A. 63 feet
B. 65 feet
C. 74 feet
D. 87 feet
If the string makes a 44° angle with the ground,
and the holder is 90 feet from the base of the
tree, how tall is the tree, rounded to the nearest
whole foot?
OGT 2003 – Problem #40
A yacht is anchored 90 feet offshore from the
base of a lighthouse. The angle of elevation
from the boat to the top of the lighthouse is
26 degrees. The distance between the yacht
and the top of the lighthouse is about 100 feet.
A. 25 feet
B. 45 feet
C. 110 feet
D. 135 feet
Which of these is nearest to the height of the
lighthouse?
Prepared by Aimee Crabtree for FERPCBM, Youngstown, Ohio
Benchmarks with OGT Problems
Summer 2006 (Rev. 2008)
Page 25 of 36
Geometry and Spatial Sense
Standard
♦
Answer Key
Geometry and Spatial Sense – Answer Key
OGT Problems
Geometry and Spatial Sense
A
B
C
D
E
Geometry and Spatial Sense
Test
Year
2007
Question #
10
Answer
S.A.
Test
Year
2003
Question #
13
Answer
B
2007
33
C
2008
34
E.R.
2005
3
B
2007
37
B
2004
13
D
2007
42
B
2003
5
S.A.**
2005
6
S.A.
2003
21
B
2004
10
E.R.
2008
37
C
2003
28
D
2007
29
C
2006
40
E.R.
2006
37
A
2004
33
A
2005
23
C
2003
7
B
2004
36
D
2005
15
D
2003
32
C
2003
40
B
2008
42
C
2007
25
A
2006
25
C
2005
1
A
2003
1
C
2008
13
B
2006
29
D
2005
35
A
2005
41
C
2004
30
A
2003
5
S.A.**
2003
38
A
2008
4
A
2007
6
B
2007
10
S.A.
2006
11
D
F
G
H
I
** Scoring Rubric Not Released
Prepared by Aimee Crabtree for FERPCBM, Youngstown, Ohio
Benchmarks with OGT Problems
Summer 2006 (Rev. 2008)
Page 27 of 36
Geometry and Spatial Sense – Answer Key
OGT Problems
GSS – Benchmark F
2008 OGT – Problem # 34 Scoring Guidelines:
Sample Response
The reflection of Triangle I over the y-axis.
Points
Description
4
The focus of this item is to show and describe the results of combinations of
translations, reflections and rotations. The response contains the correct three
triangles labeled I, II and III on the same coordinate grid and contains the
correct single transformation that maps Triangle I to Triangle III.
3
The response clearly addresses the key aspects of the task; however, it
includes minor flaws.
For example, the response may:
- Contain the original triangle and the two transformations. Triangles are
labeled I, II and III. But the response contains an incorrect or missing
description of the single transformation that maps Triangle I to Triangle III.
- Contain a single error in drawing the triangles. (Any subsequent
transformations are consistent with the error.) The single transformation that
maps the student’s Triangle I to the student’s Triangle III is based on the
triangles.
- Contain three correct triangles and a correct description of the single
transformation that maps Triangle I to Triangle III. However, the triangles
are not labeled, and the triangles are placed on separate coordinate
systems.
Prepared by Aimee Crabtree for FERPCBM, Youngstown, Ohio
Benchmarks with OGT Problems
Summer 2006 (Rev. 2008)
Page 28 of 36
Geometry and Spatial Sense – Answer Key
2
OGT Problems
The response provides evidence of a partially correct answer and/or solution
process. The response may adequately address some of the components of
the task but contain gaps or flaws in other components.
For example, the response may:
- Contain Triangle I drawn correctly and one correct transformation. The
description of a single transformation from Triangle I to III is incorrect or
missing.
- Contain an incorrect Triangle I, but both of the transformations based on the
student’s Triangle I are correct. The description of a single transformation
from Triangle I to III is incorrect or missing.
- Contain three correct triangles. But the description of a single transformation
from Triangle I to III is incorrect or missing, the labeling is missing and the
triangles are placed on separate coordinate systems.
- Contain Triangle I drawn correctly, but both of the transformations are
incorrect; however, the description is correct based on Triangle I and the
student’s Triangle III.
1
The response omits significant aspects of the task. There is evidence of
minimal understanding of the concepts involved in the task and/or solution
process; however, the response includes significant errors in most of the
components of the task.
For example, the response may:
- Show Triangle I correctly graphed, but both transformations are incorrect or
missing.
- Describe the correct movements necessary for a single transformation, but
the graphs are omitted or incorrect.
- Show a minimal understanding by completing one of the required
transformations.
- Contain an appropriate description of the transformation from the student’s
Triangle I to the student’s Triangle III.
0
The response indicates inadequate understanding of the task, and the
response does not meet the criteria required to earn one point.
For example, the response may:
- Contain an incorrect drawing of Triangle I and an incorrect attempt at
performing the two transformations on the student’s Triangle I.
- Recopy information provided in the question with no work toward a solution.
- Be blank or the student writes “I do not know” or includes unrelated
statements or work.
Prepared by Aimee Crabtree for FERPCBM, Youngstown, Ohio
Benchmarks with OGT Problems
Summer 2006 (Rev. 2008)
Page 29 of 36
Geometry and Spatial Sense – Answer Key
OGT Problems
GSS – Benchmarks A and E
2007 OGT – Problem # 10 Scoring Guidelines:
Sample Response
OR
Nets – 1020
Prepared by Aimee Crabtree for FERPCBM, Youngstown, Ohio
Benchmarks with OGT Problems
Summer 2006 (Rev. 2008)
Page 30 of 36
Geometry and Spatial Sense – Answer Key
Points
OGT Problems
Description
2
The focus of the item is to sketch a net of a triangular prism with given
dimensions. The response shows a sketch of a net with dimensions labeled
appropriately.
1
The response provides evidence of a partially correct answer and/or solution
process. The response shows understanding of some key elements of the
task but contains gaps or flaws.
For example, the response may:
 Contains a correct net but one or more of the dimensions is labeled
incorrectly or missing.
 Contains a slightly flawed net (e.g., the student may omit the second
triangle, or place the rectangles in an order that does not correspond to the
order of the sides of the triangle.)
0
The response indicates inadequate understanding of the task and the
response does not meet the criteria required to earn one point.
For example, the response may:
• Include unrelated statements or work.
Prepared by Aimee Crabtree for FERPCBM, Youngstown, Ohio
Benchmarks with OGT Problems
Summer 2006 (Rev. 2008)
Page 31 of 36
Geometry and Spatial Sense – Answer Key
OGT Problems
GSS – Benchmark G
2006 OGT – Problem # 40 Scoring Guidelines:
Sample Response
OR
Student may provide a graphic solution by locating the midpoints and showing that the
“rise over run” is the same for both the midpoint segment and the line and
demonstrating that length of the midpoint segment is half the length of side AC using
the Pythagorean theorem or similar triangles.
OR
There is a geometric theorem which states a segment joining the midpoints of two sides
of a triangle is parallel to and equal to 1/2 the length of the third side of the triangle.
Points
Description
4
The focus of this item is the use of coordinate geometry to demonstrate
relationships within a geometric figure. The response provides a
demonstration that the mid-segment is parallel to and one-half the length of
side AC by finding the midpoints of AB and BC and comparing the slope and
length of the segment formed with the slope and length of AC or by
completely stating the theorem.
3
The response clearly addresses the key aspects of the task; however, it
includes errors in completing one or two components.
For example, the response may:
 Correctly locate the midpoints and successfully demonstrate one half of the
relationship. For the other part, the response demonstrates a correct
procedure but contains computational errors or a minor graphing error
Prepared by Aimee Crabtree for FERPCBM, Youngstown, Ohio
Benchmarks with OGT Problems
Summer 2006 (Rev. 2008)
Page 32 of 36
Geometry and Spatial Sense – Answer Key
OGT Problems
(e.g., miscopies initial graph, but correctly demonstrates relationship).
2
The response provides evidence of a partially correct answer and/or solution
process. The response may adequately address some of the components of
the task, but contain gaps or flaws in other components.
For example, the response may:
 Correctly locate the midpoints and demonstrate an attempt to compare
both slope and length but contain computational errors in each.
 Correctly locate the midpoint and correctly compare either the slope or the
length algebraically or graphically, but the other is attempted with an
incorrect procedure or omitted.
 Incorrectly locate the midpoints and attempt to compare both values by
correctly finding values for both slopes and lengths or making an
appropriate graphical comparison for both relationships.
1
The response omits significant aspects of the task. There is evidence of
minimal understanding of the concepts involved in the task and/or solution
process; however, the response includes significant errors in most of the
components of the task.
For example, the response may:
 Correctly find both midpoints algebraically or graphically but contain no
other correct work.
 Incorrectly determine the midpoints but correctly determine the slope or
length of the midsegment based on those values.
 Find the slope and/or length of AC.
0
The response indicates inadequate understanding of the task.
For example, the response may:
 Include unrelated statements or work.
Prepared by Aimee Crabtree for FERPCBM, Youngstown, Ohio
Benchmarks with OGT Problems
Summer 2006 (Rev. 2008)
Page 33 of 36
Geometry and Spatial Sense – Answer Key
OGT Problems
GSS – Benchmark F
2005 OGT – Problem # 6 Scoring Guidelines:
Sample Response
Note: If students fill in the entire grid, focus on the center. Accurate reflections
indicate the reflections are to scale.
Points
Description
2
The focus of the task is to draw reflections of the original design in each of the
quadrants. The response shows the accurate reflections in each of the
remaining three quadrants.
1
The response shows a partial understanding of the solution process or key
elements of the task. The response may contain gaps or flaws in determining
the solution.
For example, the response may:
Show only two accurate reflections of the design in two of the three remaining
quadrants
OR
Show two accurate reflections based upon an initial incorrect reflection
OR
Show three accurate reflections based on a slightly flawed original design
OR
Show accurate translation or rotation of the original design in each of the
quadrants.
0
The response fails to demonstrate minimal understanding of the task.
For example, the response may:
Only copy the original diagram
OR Contain only translation or rotations in fewer than the three remaining
quadrants.
OR Be blank or the student writes, “I do not know” or unrelated statements.
Prepared by Aimee Crabtree for FERPCBM, Youngstown, Ohio
Benchmarks with OGT Problems
Summer 2006 (Rev. 2008)
Page 34 of 36
Geometry and Spatial Sense – Answer Key
OGT Problems
GSS – Benchmark F
2004 OGT – Problem # 10 Scoring Guidelines:
Sample Response
To translate DEF to D"E"F," each vertex should be moved one unit to right and five
units up.
Points
Description
4
The focus of this item requires the student to correctly graph triangle DEF and
perform the two translations in sequence. Finally, the student will identify the
translation moves that would take triangle DEF to the third triangle in the
sequence. Labeling is identified within the item and should be included within
the response.
The response shows triangle DEF and both transformations clearly and
correctly drawn, and all triangles DEF, D′E′F′, and D′′E′′F′′ are appropriately
labeled. An explanation is provided that correctly describes the transformation
from triangle DEF to triangle D′′E′′F′′ by translating each vertex one unit to the
right and five units up. The work shown is organized and completely accurate.
3
The response clearly addresses the key aspects of the task; however, it
includes errors in completing one or two of the components.
For example, the response may:
Show triangle DEF and both transformations correctly graphed and
appropriately labeled, but the description of the transformation from DEF to
D′′E′′F′′ is missing or incorrect.
OR
Show triangle DEF graphed correctly and labeled. One transformation is done
incorrectly, (example: D'E'F' moves to the left, or down) but everything else is
correct, based on this error.
OR
Contain one minor error in any part of the process. For example, triangle DEF
is graphed incorrectly; however, all other parts of the task are correct based
Prepared by Aimee Crabtree for FERPCBM, Youngstown, Ohio
Benchmarks with OGT Problems
Summer 2006 (Rev. 2008)
Page 35 of 36
Geometry and Spatial Sense – Answer Key
OGT Problems
on the incorrect triangle.
OR
Show triangle DEF and both transformations correctly graphed, but the labels
are missing. The description of the transformation from DEF and D"E"F" is
correct.
2
The response provides evidence of a partial correct answer and/or solution
process. The response may adequately address some of the components of
the task, but it contains major gaps or flaws in other components.
For example, the response may:
Show triangle DEF and one of the transformations graphed correctly, but the
other transformation is either done incorrectly or is missing. The description
given is related only to the one transformation or is missing.
OR
Show triangle DEF, and both transformations have been graphed correctly,
but labels are missing and the description for the single transformation is
unclear or missing.
OR
Show triangle DEF incorrectly graphed. Both transformations are consistent
with the original graph, with or without labels. The description is incorrect
based on the transformation or is missing.
OR
Show triangle DEF incorrectly graphed. One of the two transformations has an
error, but is consistent with the original graph with or without labels. The
description is correct based on the transformation.
1
The response omits significant aspects of the task. There is evidence of
minimal understanding of the concepts involved in the task and/or solution
process; however, the response includes significant errors in most of the
components of the task.
For example, the response may:
Show triangle DEF correctly graphed, but both transformations are incorrect
or missing.
OR Describe the movements necessary for a single transformation, but the
graphs are omitted or incorrect.
OR Show triangle DEF incorrectly graphed, but there is one correct
transformation (two congruent triangles that match one of the translations
mentioned in the questions).
0
The response indicates inadequate or no understanding of the task,
and the task does not meet the requirements for one point.
For example, the response may:
Fail to provide evidence of minimal understanding of the concept.
OR Incorrectly plot only the points for triangle DEF.
OR Recopy information provided in the item with no work.
OR Show no apparent understanding of the task.
OR Be blank or the student writes, “I do not know” or includes unrelated
statements or work.
Prepared by Aimee Crabtree for FERPCBM, Youngstown, Ohio
Benchmarks with OGT Problems
Summer 2006 (Rev. 2008)
Page 36 of 36