OHIO GRADUATION TESTS
Mathematics
Scoring Guidelines and Samples of
Scored Student Responses
Spring 2008
© 2008 by Ohio Department of Education
Table of Contents
Item 5: Item and Scoring Guidelines.................................................................................1
Item 5: Samples of Scored Student Responses ................................................................4
Item 11: Item and Scoring Guidelines ............................................................................15
Item 11: Samples of Scored Student Responses ............................................................18
Item 16: Item and Scoring Guidelines ............................................................................29
Item 16: Samples of Scored Student Responses ............................................................32
Item 22: Item and Scoring Guidelines……………………………………………………………42
Item 22: Samples of Scoring Student Responses……………………………………………...45
Item 34: Item and Scoring Guidelines ............................................................................56
Item 34: Samples of Scored Student Responses ............................................................60
Item 40: Item and Scoring Guidelines ............................................................................71
Item 40: Samples of Scored Student Responses ............................................................74
Mathematics
Item 5
Spring 2008
Item and Scoring Guidelines
1
Item
5.
Carmella has a coupon that allows her to buy one pair of shorts at the regular
price and get a second pair at half the regular price. The shorts cost $25.50
per pair.
When Carmella arrives at the store, she finds that the shorts are on sale for 30%
off the regular price. This sale does not allow coupons to be used on the
discounted items.
In your Answer Document, explain whether two pairs of shorts would cost less
by using the coupon or by buying two pairs of shorts at the sale price.
Provide mathematical calculations and/or reasoning to support your answer.
For question 5, respond completely in your Answer Document. (2 points)
Sample Response for Item 5 (Short Answer):
Exemplar:
Two pairs with coupon = $25.00 +
$25.50
= $38.25
2
Sale price for one pair = $25.50 – 0.30 ($25.50) = $17.85
Two pairs = $17.85 x 2 = $35.70
It is cheaper to buy the shorts at the sale price.
OR
Two pairs with coupon:
Two pairs with discount = 2 (0.70 x $25.50) = $37.50
It is cheaper to buy the shorts at the sale price.
OR
Two pairs with coupon = 1.5 x regular price
Two pairs with discount = 2(0.70 of regular price) = 1.4 x regular price.
It is cheaper to buy the shorts at the sale price.
OR
The price of the shorts is about $25.
With coupon = about $25 +
1
($25) = about $37.50
2
Sale price for 2 pairs = about $50 – 0.3($50) = $35
It is cheaper to buy the shorts at the sale price.
2
Scoring Guidelines for Item 5
Score Point
Description
2 points
The focus of this item is to compare the final price of two pairs of shorts
either using a coupon to get the second pair at half price or using a
discount of 30% on both. The response states that the 30% discounted
price is the better buy and provides mathematical support for the
answer.
1 point
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:
-Demonstrate a correct procedure that contains computational errors.
The decision is consistent with the computations.
-Determine two correct prices, but the comparison is incorrect or
missing.
0 points
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:
-State that the 30% discount is the better buy with no support.
-Include unrelated statements or work.
3
Mathematics
Item 5
Spring 2008
Samples of Scored Student Responses
4
Score Point: 0
This response shows a calculation for only the two pairs of shorts with the coupon.
5
Score Point: 0
This response shows a calculation for only one pair of shorts.
6
Score Point: 0
This response shows inadequate understanding of the solution process.
7
Score Point: 1
This response demonstrates a correct procedure that contains computational errors
(30% off and half off), but some correct procedure is evident. The conclusion is
consistent with the computations.
8
Score Point: 1
This response demonstrates a correct procedure that contains a computational error
(30% off). The conclusion is consistent with the computations.
9
Score Point: 1
This response provides evidence of a partially correct solution process (correctly
figures the 30% off price). The conclusion is consistent with the computations and the
student’s interpretation of the question.
10
Score Point: 1
This response demonstrates a correct procedure that contains a computational error
($35.60 instead of $35.70). This response indicates that the 30% would be the better
price.
11
Score Point: 2
This response states that it would cost less to buy two pair of jeans for 30% off and
correctly provides mathematical support for applying the coupon and the 30%
discount. This response compares the final cost of each, reaching the correct
conclusion.
12
Score Point: 2
This response states that Carmella should buy the shorts at 30% off and correctly
provides mathematical support for applying the coupon and the 30% discount. This
response compares the savings of each, reaching the correct conclusion.
13
Score Point: 2
This response states that it is cheaper without [the coupon] and correctly provides
mathematical support for applying the coupon and the 30% discount. This response
compares the final cost of each, reaching the correct conclusion.
14
Mathematics
Item 11
Spring 2008
Item and Scoring Guidelines
15
Item
11.
A sports club is planning a cookout. The food service charges $1.75 per
person plus a flat fee of $200. The club can only spend $500 for food service.
In your Answer Document, determine the maximum number of people the club
will be able to serve. Use a table, graph, equation or inequality to support your
answer.
For question 11, respond completely in your Answer Document. (2 points)
Sample Response for Item 11 (Short Answer):
c = 1.75n + 200
500 = 1.75n + 200
n = 300 ÷ 1.75
n = 171.4
The maximum number is 171 people OR n < 172.
OR
They should be able to serve about 170 people for $500.
16
Scoring Guidelines for Item 11
Score Point
Description
2 points
The focus of the item is to use a table, graph, equation or inequality to
solve a word problem. The response determines the maximum number
of people (171 or 171.4) the club will be able to serve at a cookout for
less than $500 using one of these techniques.
1 point
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:
-Contain a correct equation or inequality but the maximum number of
people is missing or incorrect.
-Contain an incomplete or partially correct linear graph, but the
maximum number of people is missing or incorrect.
-Contain an incomplete or partially correct table, but the maximum
number of people is missing or incorrect.
-Contains the correct answer (171 or 171.4) with incomplete or missing
work.
0 points
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:
-Be blank or give irrelevant information.
17
Mathematics
Item 11
Spring 2008
Samples of Scored Student Responses
18
Score Point: 0
This response contains an incorrect equation, resulting in an incorrect maximum
number of people.
19
Score Point: 0
This response contains only an incorrect equation (should be 1.75p, not “200p”).
20
Score Point: 0
This response contains only an incorrect maximum number of people, with no other
work shown.
21
Score Point: 1
This response contains a correct equation and work, but the maximum number of
people is incorrect. The total of 171.43 is inappropriately rounded to 170.
22
Score Point: 1
This response provides evidence of a partially correct solution process. It contains an
incomplete, partially correct table (“203.05” should be 203.50), but the maximum
number of people is incorrect.
23
Score Point: 1
This response contains only the correct answer (“171”), with missing work.
24
Score Point: 2
This response uses an equation to determine that the maximum number the club can
serve at a cookout is 171.4, then rounds the answer to 171.
25
Score Point: 2
This response uses an inequality to determine that the maximum number of people
the club can serve is 171.
26
Score Point: 2
This response uses a table to determine that the maximum number of people who
can be served, without going over the $500 limit, is 171. This response also includes a
correct inequality.
27
Score Point: 2
This response uses an inequality to determine that the maximum number of people
the club can serve is 171. The answer, 171, is used in place of the variable in the
correct inequality.
28
Mathematics
Item 16
Spring 2008
Item and Scoring Guidelines
29
Item
16.
Before her trip to Canada, Liz exchanged 300 U.S. dollars for Canadian dollars
at a rate of 1 U.S. dollar to 1.35 Canadian dollars.
When Liz arrived in Canada, the exchange rate was 1 Canadian dollar to 0.76
U.S. dollars.
In your Answer Document:
• Determine the amount of money in Canadian dollars that Liz received for
her 300 U.S. dollars.
• Determine whether Liz would have received more Canadian money for her
300 U.S. dollars if she had waited to exchange her money in Canada.
Show your work or provide an explanation for your answers.
For question 16, respond completely in your Answer Document. (2 points)
Sample Response for Item 16 (Short Answer):
Exemplar:
1 Canadian dollar
0.76 U.S. dollar
=
x Canadian dollars
300 U. S. dollars
1 x = 300 × 1.35
x = 405 Canadian dollars
For the second question you would have to compare the amount she got when she
exchanged in the U.S. and if she would have waited to exchange. We know the first
rate was 405 Canadian dollars. So setting up another proportion:
1 Canadian dollar
0.76 U.S. dollar
=
x Canadian dollars
300 U. S. dollars
300 = 0.76x
x= 394.74 Canadian dollars
She would not have received more if she had waited to exchange in Canada.
OR
1.316 Canadian dollars
1 Canadian dollar
=
0.76 U.S.dollar
1 U.S. dollar
This is a lower rate of exchange than the U.S. rate so she would not have received
more money if she had waited.
30
Scoring Guidelines for Item 16
Score Point
Description
2 points
The focus of the item is to use proportions to compare two different
rates. The response states that $300 U.S. can be exchanged for $405
Canadian in the U.S. and that she would not have received more if she
had waited to exchange money with correct work or explanation of
money when exchanged with the first rate and explains that the second
rate would give her less money.
1 point
The response provides evidence of a partially correct answer and/or
solution process. The response shows understanding of most of the key
elements of the task but contains gaps or flaws. For example, the
response may:
-Indicate the correct amount of money for the first or second exchange
rate but fails to or does not correctly explain how she would get less
money for the second exchange rate.
-State the incorrect amount for the first exchange rate but correctly
explains how the second exchange rate would give her less money.
-Indicate an incorrect amount for the first exchange rate, identify the
correct amount for the second rate, and make a correct conclusion
based on the values given in the response.
0 points
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:
-Recopy information provided in the question with no additional work
toward a solution.
-Be blank or the student writes “I do not know” or includes unrelated
statements or work.
31
Mathematics
Item 16
Spring 2008
Samples of Scored Student Responses
32
Score Point: 0
This response indicates an incorrect amount of money (“307.8”) for the first
exchange rate and fails to explain how Liz would get less money for the second
exchange rate.
33
Score Point: 0
This response indicates an incorrect amount of money (“$222.22”) for the first
exchange rate and fails to explain how Liz would get less money for the second
exchange rate.
34
Score Point: 0
This response provides incorrect work and explanation of money when exchanged
with first rate, with inadequate supporting work.
35
Score Point: 1
This response determines the correct amount of money (“$405”) for the first
exchange rate but fails to provide a mathematically accurate reason why Liz would
get less money for the second exchange rate. (“She got $105.00 more” is incorrect.)
36
Score Point: 1
This response determines the correct amount of money (“$405”) for the first
exchange rate but provides an incorrect amount for the second exchange rate
(“$415.53”).
37
Score Point: 1
This response determines the correct amount of money (“405”) for the first exchange
rate but provides an incorrect amount for the second exchange rate (“307.8”) with
an inadequate explanation.
38
Score Point: 2
This response determines that $300 U.S. can be exchanged for $405 Canadian in the
United States and provides a mathematically valid reason why Liz would not have
received more if she had waited to exchange the money in Canada.
39
Score Point: 2
This response determines that $300 U.S. can be exchanged for $405 Canadian in the
United States and provides a mathematical reason why Liz would not have received
more if she had waited to exchange the money in Canada.
40
Score Point: 2
This response determines that $300 U.S. can be exchanged for $405 Canadian in the
United States and provides a mathematical reason why Liz would not have received
more if she had waited to exchange the money in Canada. (Rounding to $395 is
acceptable.)
41
Mathematics
Item 22
Spring 2008
Item and Scoring Guidelines
42
Item:
22.
The transportation department has selected three possible routes for a
new section of highway and wants to know which route landowners
and residents of the affected areas prefer. The transportation
department plans to survey the public by posting the three possible
routes on the department’s Web site with a request that all visitors to the
Web site vote for their preferred route.
In your Answer Document, explain why the design of the survey does
not provide a representative sample of the landowners and residents of
the affected area. Give an example of how the design of the survey
could be changed to better represent the affected population.
For question 22, respond completely in your Answer Document. (2
points)
Sample Response for Item 22 (Short Answer):
Possible Reasons:
Only visitors to the website will be included.
It includes all visitors to the website, not just those in the affected areas.
Some residents/landowners may not have access to the Internet.
Residents/landowners may not know that the survey is posted on the website.
This is a voluntary sample— not everyone will bother to take it. It may be more likely
to include people with strong feelings about the routes.
(Other examples may be acceptable.)
Possible Changes:
Not everyone has Internet access—Provide more options for submitting opinions: mail
surveys or telephone residents.
Survey is based upon a volunteer sample—Go into neighborhoods and survey
residents in person so that everyone has an equal opportunity to be surveyed.
People may not know about the survey or what the options are—The survey should
be advertised widely: newspapers, radio, flyers, signs, etc.
Visitors to the website may not be residents of the area or may vote multiple times—
Visitors should be required to provide their home addresses, and people outside the
area should be excluded.
(Other examples may be acceptable.)
OR
Most people have access to the Internet—either in their home, at the library or
Internet cafes, so this is a good design for the survey. It needs to be widely advertised
so that people will know to go to the website, and people who do not live in the area
should not be included in the survey.
43
Scoring Guidelines for Item 22
Score Point
Description
2 points
The focus of this item is to evaluate and improve the design for a survey
to determine the preferred route for a new highway. The response
explains why the survey design is not representative of the residents in
the affected area and provides a change to the survey design to make
it more representative.
1 point
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:
-Explain why the survey does not represent the residents in the affected
area but fail to provide an example of how to improve the design of the
survey.
-Provide an example of how to change the survey design to better
represent the affected population but fail to explain why the design of
the survey is not representative of the affected population.
0 points
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:
-State that the design may not produce a representative sample with no
support.
-Recopy information provided in the item with no work.
-Be blank or the student writes, “I do not know” or includes unrelated
statements or work.
-Be blank or give irrelevant information.
44
Mathematics
Item 22
Spring 2008
Samples of Scored Student Responses
45
Score Point: 0
This response does not provide a valid argument concerning representation on the
Web site and does not suggest a change that is relevant (“On the website, they
should plot where people’s homes are and where land is not open to build on”).
46
Score Point: 0
This response does not provide a valid argument concerning the design issue (“If you
have them vote on it they will all vote against their own, so they will all be equally
voted”) or an inadequate solution (“If you only put one route up then have people
vote it would be better”).
47
Score Point: 0
This response provides an irrelevant issue with the design of the survey (“the website Just
shows the public’s posting of the three possible routes”) and an invalid change for the
survey (“It should show actual mapped out routes”).
48
Score Point: 1
This response explains why the survey sample is not representative (“They should
take it to the public because not everyone has a computer and if the landowners live
out in the country they might not even have computers”) but does not suggest a
change.
49
Score Point: 1
This response provides a correct explanation of why the sample is not representative
(“They do not know if everyone that is affected has a way to get online so that they
can vote”), but the solution is inadequate (“They could do a door to door survey”).
50
Score Point: 1
This response provides a valid issue with the design of the survey (“on a online survey
Anybody can vote and some people may not have computers”), but the solution is
inadequate because it does not detail who is going to be surveyed (“Go door to
door doing the survey or send it in the mail”).
51
Score Point: 1
This response provides a valid issue with the design of the survey (“The survey isn’t a
good sample because you will get other people opinions who don’t even live near
the affected areas”), but the solution is inadequate (“They should have a meeting for
people who will be affected by a new highway”).
52
Score Point: 2
This response provides a valid issue with the design of the survey (“all visitors on the
website can vote which means people in the non affected area can vote”) and a
valid change to address the issue (“One thing that the survey can improve is by
setting up an identification question that people have to answer before taking the
survey”).
53
Score Point: 2
This response provides a valid issue with the design (“First of all, not everyone has a
computer to use at home”) and valid changes that addresses the issue (“They could
make up papers that had the lay out of all 3 routes and a space for voting all on one
paper and put those papers in everyone’s mailbox to all affected people”).
54
Score Point: 2
This response provides a valid issue with the design (“The design is not good
because anyone visiting the site may vote on the preferred route, not just the
landowners and residents”) and a valid change that addresses the issue (“If they get
all the addresses of the land owners and residents in the area that is concerned, they
could send out a letter to all of them and request that they vote and to send it back
ASAP”).
55
Mathematics
Item 34
Spring 2008
Item and Scoring Guidelines
56
Item
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 180° 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.
For question 34, respond completely in your Answer Document. (4 points)
Sample Response for Item 34 (Extended Response):
Exemplar:
The reflection of Triangle I over the y-axis.
57
Scoring Guidelines for Item 34
Score Point
Description
4 points
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 points
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.
2 points
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.
58
1 point
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 points
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.
59
Mathematics
Item 34
Spring 2008
Samples of Scored Student Responses
60
Score Point: 0
This response does not show evidence of minimal understanding.
61
Score Point: 0
This response does not show evidence of minimal understanding.
62
Score Point: 1
This response shows Triangle I correctly graphed, but neither transformation is
provided.
63
Score Point: 1
This response shows an attempt to transform Triangle I to Triangle III. The x- and yaxes and the labels are not provided.
64
Score Point: 2
This response shows three correct triangles (the origin is assumed), but the
description of a single transformation from Triangle I to III is not provided and the
triangles are labeled incorrectly.
65
Score Point: 2
This response shows Triangle I drawn correctly and one correct transformation (I to
II). An inadequate description of a single transformation from Triangle I to Triangle III
is provided (“A reflection would get I to III”).
66
Score Point: 3
This response shows the original triangle and the two transformations. Triangles are
labeled I, II, and III, but the description of the single transformation that maps
Triangle I to Triangle III is not provided.
67
Score Point: 3
This response shows a single error in drawing the triangles (Triangles II and III are
reversed). The single transformation that maps Triangle I to Triangle III is based on the
student’s triangles drawn (“If triangle 1 was reflected over the X axis it would be
exactly the same as triangle 3”).
68
Score Point: 4
This response shows the correct three triangles, labeled I, II, III, on the same
coordinate grid and contains the correct single transformation that maps Triangle I
to Triangle III (“Reflect [triangle] I over the y-axis to get [triangle] III”).
69
Score Point: 4
This response shows the correct three triangles that are labeled (A'B'C', A"B"C" and
A'''B'''C''') on the same coordinate grid and contains the correct single
transformation that maps Triangle I to Triangle III (“flip over the y axis first”).
70
Mathematics
Item 40
Spring 2008
Item and Scoring Guidelines
71
Item
40.
Selena was given five different cells to measure. The table below shows
Selena’s results.
In your Answer Document, write all of the measurements in scientific notation.
Order the values from smallest to largest.
For question 40, respond completely in your Answer Document. (2 points)
Sample Response for Item 40 (Short Answer):
Exemplar:
9.0 x 10-4, 1.2 x 10-3, 1.5 x 10-3, 4.0 x 10-3, 8.0 x 10-3
Scoring Guidelines for Item 40
Score Point
Description
2 points
The response contains diameters 2, 3 and 5 written correctly in scientific
notation. The five diameters are correctly ordered from smallest to
largest.
1 point
The response provides a partial solution. Diameters 2, 3 and 5 may be
correctly written in scientific notation. The order may be incorrect or
missing.
OR
The scientific notation for the three diameters is incorrect or missing, but
72
the five original values are correctly ordered.
OR
The response contains errors in scientific notation, but the student’s
values obtained are correctly ordered.
OR
The response contains a combination of some correct scientific notation
and some correct ordering that indicate some understanding of the
concepts involved.
0 points
The response fails to demonstrate minimal understanding of the task.
For example, the response may:
-Be blank or give irrelevant information.
-Fail to demonstrate minimal understanding of the task.
73
Mathematics
Item 40
Spring 2008
Samples of Scored Student Responses
74
Score Point: 0
This response does not provide the diameters written correctly in scientific notation
and the order is incorrect (“.008, .0012, .0009, .00015, .0012”).
75
Score Point: 0
This response provides inadequate understanding of the task. Diameters 2, 3 and 5
are not written in scientific notation and the order is incorrect (1, 4, 2, 5, 3).
76
Score Point: 0
This response provides only diameter 3 written correctly in scientific notation, and the
order is incorrect.
77
Score Point: 1
This response provides a partial solution. Diameters 2, 3 and 5 are written correctly in
scientific notation, but the order is incorrect.
78
Score Point: 1
This response provides a partial solution. Diameters 2, 3 and 5 are written correctly in
scientific notation, but the order is incorrect (“.0015, .0012, .0009, .008, .004”).
79
Score Point: 1
This response provides a partial solution. Diameters 2, 3 and 5 are written correctly in
scientific notation, but the order is incorrect.
80
Score Point: 2
This response contains diameters 2, 3 and 5 written correctly in scientific notation,
and the five diameters are correctly ordered from smallest to largest.
81
Score Point: 2
This response contains diameters 2, 3 and 5 written correctly in scientific notation,
and the five diameters are correctly ordered from smallest to largest.
82
Score Point: 2
This response contains diameters 2, 3 and 5 written correctly in scientific notation,
and the five diameters are correctly ordered from smallest to largest.
83