2007-2008 Grade 8 Mathematics Item and Scoring Sampler

The Pennsylvania System
of School Assessment
Mathematics
Item and Scoring Sampler
2007–2008
Grade 8
Pennsylvania Department of Education Bureau of Assessment and Accountability 2007–2008

TABLE OF CONTENTS
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
General Description of Mathematics Scoring Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Mathematics Reporting Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Description of Sample Items . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Formula Sheet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Multiple-Choice Items. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
First Open-Ended Item . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Item-Specific Scoring Guideline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Open-Ended Item Responses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Second Open-Ended Item. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Item-Specific Scoring Guideline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Open-Ended Item Responses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Grade 8 Mathematics Item Sampler 2007–2008
i

MATHEMATICS
INTRODUCTION
General Introduction
The Department of Education provides districts and schools with tools to assist in delivering focused instructional
programs aligned to the state assessment system. These tools include assessment anchor documents, assessment
handbooks, and content-based item and scoring samplers. This 2007–2008 Mathematics Item and Scoring Sampler is a
useful tool for Pennsylvania educators in the preparation of local instructional programs and the statewide PSSA.
What Is Included
This item and scoring sampler contains mathematics multiple-choice items and open-ended items that have been
written to align to the 2007 Assessment Anchor Content Standards (Assessment Anchors). Some of the items are
actual items used to assess student performance on the PSSA. They provide an idea of the types of items that will
appear on the operational Spring 2008 PSSA. Each item has been through a rigorous review process to ensure
alignment with the Assessment Anchors.
Purpose and Uses
The items in this sampler may be used as examples for creating assessment items at the classroom level, and they may
also be copied and used as part of a local instructional program.* Classroom teachers may find it beneficial to have
students respond to the open-ended items in this sampler. Educators can then use the sampler as a guide to score the
responses either independently or together with colleagues within a school or district.
Item Format and Scoring Guidelines
The multiple-choice items have four answer choices. Each correct response to a multiple-choice item is worth
one point.
Each open-ended item is designed to take about ten minutes to complete. During an actual testing event, students are
given additional time as necessary to complete the test items. The open-ended items in mathematics are scored with
item-specific scoring guides on a 0–4 point scale. An item-specific scoring guide with examples of student responses
for each score point is presented with each item.
Also included is the General Description of Mathematics Scoring Guidelines used to develop the item-specific
guidelines. The General Scoring Guidelines should be used to develop any item-specific scoring guidelines created for
use within local instructional programs.*
Item Alignment
All PSSA items are aligned to statements and specifications included in the Assessment Anchor Content Standards
documents. The mathematics content included in the PSSA mathematics multiple-choice items will align with the
Assessment Anchors as defined by the Eligible Content statements. The process skills, directives, and action statements
will also specifically align with the Assessment Anchors as defined by Eligible Content statements.
The mathematics content included in the PSSA mathematics open-ended items will align with content included
in Eligible Content statements. The process skills, directives, and action statements included in the performance
demands of the PSSA mathematics open-ended items will align with specifications included in the Assessment Anchor
statements, the Descriptor statements, and/or the Eligible Content statements. In other words, the verbs or action
statements used in the open-ended questions or stems can come from the Eligible Content, Descriptor, or Anchor.
* The permission to copy and/or use these materials does not extend to commercial purposes.
Grade 8 Mathematics Item Sampler 2007–2008
1

MATHEMATICS
GENERAL DESCRIPTION OF MATHEMATICS SCORING GUIDELINES
4 – The response demonstrates a thorough understanding of the mathematical concepts and procedures
required by the task.
The response provides correct answer(s) with clear and complete mathematical procedures shown and a correct
explanation, as required by the task. Response may contain a minor “blemish” or omission in work or explanation
that does not detract from demonstrating a thorough understanding.
3 – The response demonstrates a general understanding of the mathematical concepts and procedures
required by the task.
The response and explanation (as required by the task) are mostly complete and correct. The response may have
minor errors or omissions that do not detract from demonstrating a general understanding.
2 – The response demonstrates a partial understanding of the mathematical concepts and procedures
required by the task.
The response is somewhat correct with partial understanding of the required mathematical concepts and/or
procedures demonstrated and/or explained. The response may contain some work that is incomplete or unclear.
1 – The response demonstrates a minimal understanding of the mathematical concepts and procedures
required by the task.
0 – The response has no correct answer and insufficient evidence to demonstrate any understanding of the
mathematical concepts and procedures as required by the task for that grade level.
Response may show only information copied from the question.
Special Categories within zero reported separately:
BLK (blank) ...........Blank, entirely erased, or written refusal to respond
OT ..........................Off task
IL ...........................Illegible
LOE .......................Response in a language other than English
Grade 8 Mathematics Item Sampler 2007–2008 2

MATHEMATICS
MATHEMATICS REPORTING CATEGORIES
Mathematics scores are reported in five categories:
A – Numbers and Operations
B – Measurement
C – Geometry
D – Algebraic Concepts
E – Data Analysis and Probability
Examples of multiple-choice and open-ended items assessing these categories are included in this booklet.
DESCRIPTION OF SAMPLE ITEMS
The mathematics multiple-choice items begin on page 5. Each item is preceded by the Assessment Anchor and Eligible
Content coding. The majority of answer options A–D are followed by a brief analysis or rationale. The correct answer
is indicated by an asterisk (*).
Two open-ended items follow the multiple-choice items. Each of these is displayed with an item-specific scoring guide
and examples of responses with scores and annotations.
A calculator is permitted for use in solving items numbered 4–27 in this sampler. Items numbered 1–3 are to be solved
without the use of a calculator. Scratch paper may be used in solving all items.
Grade 8 Mathematics Item Sampler 2007–2008 3

MATHEMATICS
FORMULA SHEET
Formulas that you may need to work questions are found below.
You may use calculator Õ or the number 3.14.
2008
Grade 8
w
P 2l + 2w
A lw
a
c
Pythagorean Theorem :
a 2 b 2 c 2
l
b
h
A bh
Distance :
d rt
b
Simple Interest :
l prt
a
b
h
A 1 h(a b)
2
Degrees in a 3 to 8 sided figure :
Sum of angle measures 180(n 2),
where n number of sides
Temperature Conversions :
c
b
h
d
P b c d
A 1 2 bh
C F
F
Customary Conversions :
1 mile = 5,280 feet (ft)
1 yard (yd) = 3 feet
1 foot = 12 inches (in.)
C
r
s
s
C 2r
A r 2
SA 6s 2
V s s s
1 ton (T) = 2,000 pounds (lb)
1 pound = 16 ounces (oz.)
1 gallon (gal) = 4 quarts (qt)
1 quart = 2 pints (pt)
1pint=2cups(c)
1 cup = 8 fluid ounces
1 day = 24 hours (hr)
1 hour = 60 minutes (min)
1 minute = 60 seconds (sec)
s
l
w
h
SA 2lw 2lh 2wh
V lwh
Metric Conversions :
1 kilometer = 1000 meters
1 hectometer = 100 meters
1 dekameter = 10 meters
1 meter
1 decimeter = 0.1 meter
1 centimeter = 0.01 meter
1 millimeter = 0.001 meter
Grade 8 Mathematics Item Sampler 2007–2008 4

MATHEMATICS
MULTIPLE-CHOICE ITEMS
During an assessment, students would not be permitted to use a calculator on items 1–3.
A.2.1.1
1. Simplify: 8 • 3 2 + 7 • (6 ] 4)
A.3.3.1
3. Multiply:
41-7433
A 62 3 2 = 6
B 86 *
C 110 (8 × 9) + (7 • 6) – 4
D 590 (8 × 3) 2
A 12.48
41.6 × 3
incorrect decimal point
placement
B 123.18 0.6 × 3 = 0.18 with no regroup
C 124.8 *
A.3.2.1
2. Jim earned $127.59 last month. He
spent about 20% of the money. Which
is the closest estimate of the amount of
money Jim spent?
8MSA331
D 1231.8 0.6 × 3 = 1.8 with no regroup
41-7283
A $13 10% of $130
B $20 20% of $100
C $26 *
D $30 20% of $150
Grade 8 Mathematics Item Sampler 2007–2008 5

MATHEMATICS
A.1.1.1
4. Which is 5,291,000 written in scientific
notation?
A.3.1.2
6. Mara has $100. She wants to buy 3 pieces
of luggage priced as shown below.
41-7293
A 5.291 3 10 3
exponent equals number of zeros
B 529.1 3 10 4
incorrect placement of decimal point
C 52.91 3 10 5
incorrect placement of decimal point
D 5.291 3 10 6
*
•
•
•
suitcase $39.95
carry-on bag $24.95
tote bag $16.95
When is an estimate appropriate?
A
B
when Mara is deciding if $100 is
enough to buy all 3 pieces
*
when the salesperson is scanning the
cost of each piece
exact amount
A.3.1.1
5. Lisa needs 56 envelopes. She buys the
envelopes in packages of 12. What is the
minimum number of packages Lisa should
buy to get 56 envelopes?
8MSA311
A 3 56 rounded to 60 and 12 to 20; 60 4 20
B 4 56 4 12; rounded
C 5 *
D 6 56 rounded to 60 and 12 to 10; 60 4 10
C
D
8MSA312
when Mara is told how much money
is due
exact amount
when the salesperson is counting out
Mara’s change
exact amount
Grade 8 Mathematics Item Sampler 2007–2008 6

MATHEMATICS
B.1.1.3
7. Danny babysat for 5 hours yesterday. For
how many seconds did Danny babysit
yesterday?
A 300 minutes, 5 × 60
B 3,600 seconds in 1 hour, 60 × 60
C 7,200 minutes in 5 days, 5 × 24 × 60
B.2.1.3
9. The sum of the measures of the interior
angles of a polygon is 9008. Exactly how
many sides does the polygon have?
8MSB213
A 3 900 4 180 – 2
B 5 900 4 180
C 7 *
D 10 900 4 180 × 2
8MSB113
D 18,000 *
B.1.1.4
8. The thermometer in Molly’s backyard
read 21°F. What was the approximate
temperature in degrees Celsius?
41-0141
A –20.3°C (5/9)(21) – 32
B –6.1°C *
C 11.6°C (5/9)(21)
D 69.8°C (9/5)21 + 32
Grade 8 Mathematics Item Sampler 2007–2008 7

MATHEMATICS
C.1.1.1
10. A cone is shown below.
Which is a net for the cone?
A
B
*
C
D
8MSC111
Grade 8 Mathematics Item Sampler 2007–2008 8

MATHEMATICS
C.1.1.2
11. Angles J and K are vertical angles. The
measure of angle J is 468. What is the
measure of angle K?
C.1.1.3
12. In the diagram below, line l is parallel
to line m and line p is parallel to line q.
A 448 complement
B 468 *
C 1348 supplement
2
3
4 5
1
p
q
8MSC112
D 1368 468 + 908
l
Which angle has the same measure
as /1?
m
41-7321
A /2 supplement
B /3 *
C /4 supplement
D /5 supplement
Grade 8 Mathematics Item Sampler 2007–2008 9

MATHEMATICS
C.3.1.1
Use the graph below to answer question 13.
7
6
5
4
3
2
1
−7−6−5−4−3 −2−1 −1
−2
−3
−4
−5
−6
−7
y
1 2 3 4 5 6 7
x
D.1.1.1
14. Fiona created a pattern using numbers as
shown below.
8MSD111
0, 2, 6, 12
The pattern continues. What is the next
number in the pattern?
A 14 0 + 2 = 2; 12 + 2 = 14
B 18 6 + 6 = 12; 12 + 6 = 18
C 20 *
D 24 6 × 2 = 12; 12 × 2 = 24
13. At what point does the line intersect
the y -axis?
41-0120
A ( _ 3, 0) reverse order
B (0, _ 3) *
C (0, 5) reverse order of x-intercept
D (5, 0) x-intercept
D.1.1.2
15. At 3:00 P.M. the temperature in
Pittsburgh was 938F. The temperature
decreased at a rate of 28F every
30 minutes. What was the temperature
at 7:30 P.M.?
A 758F *
B 778F 93 ] (2 × 8)
41-0711
C 848F 93 ] (2 × 4.5)
D 1118F 93 + (2 × 9)
Grade 8 Mathematics Item Sampler 2007–2008 10

MATHEMATICS
D.2.1.1
16. Dora owns a card store. After a full
week, she made $250.00 by selling
cards (c). Using the equation
1.25c = 250, how many cards did Dora
sell that week?
A 125 250 ] 125
B 200 *
C 251 250 + 1
D.2.2.1
18. Which expression represents 4 times
the sum of x squared and 6?
41-7364
A 4x 2 + 6 4 distributed to first term only
B 4(x 2 + 6) *
C 4(x + 6) 2 quantity squared
D (4x + 6) 2 entire expression squared
41-7447
D 312 250 × 1.25
D.2.1.2
17. In which equation is m = 28 the solution?
A
m
}
5 – 3 = 5 m = 40
B m – 3 }
5 = 5 m = 5 3 }
5
C
m – 3
} = 5 *
5
8MSD212
D (m – 3)5 = 5 m = 4
Grade 8 Mathematics Item Sampler 2007–2008 11

MATHEMATICS
D.4.1.1
Use the table below to answer question 19.
x
3
6
9
12
y
11
17
23
29
19. Which graph shows the relationship?
A
y
12
10
8
6
4
2
B
12
10
8
6
4
2
–12–10 –8 –6–4
–2 –2
–4
–6
–8
–10
–12
2 4 6 8 1012
x
x
–12–10 –8 –6–4
–2 –2
2 4 6 8 1012
–4
–6
–8
–10
–12
slope and y-intercept interchanged
x and y values reversed
y
y
y
C
12
10
8
6
4
2
–12–10 –8 –6–4
–2 –2
2 4 6 8 1012
–4
–6
–8
–10
–12
x
D
12
10
8
6
4
2
–12–10 –8 –6–4
–2 –2
2 4 6 8 1012
–4
–6
–8
–10
–12
x
8MSD411
(3, 11) plotted with slope of 3
*
Grade 8 Mathematics Item Sampler 2007–2008 12

MATHEMATICS
D.4.1.3
20. The table below shows a relationship
between the values of x and y.
41-7268
x
y
–5 –9
–2 –6
1 –3
2 –2
4 0
Which equation describes the
relationship?
A y = ]2x ] 1 x = 1; y = ] 3 (3rd row)
B y = ]x + 4 x = 4; y = 0 (5th row)
C y = x ] 4 *
D y = 2x ] 5 x = 1; y = ] 3 (3rd row)
E.1.1.1
21. A student surveyed his classmates
about their favorite choice for a field
trip. The results of the survey are
shown below.
41-2429
21% art gallery
15% museum
16% theater
48% zoo
Which of these displays would best
represent the data?
A line graph 2 variables
B circle graph *
C frequency table tally list
D box-and-whiskers plot data list
Grade 8 Mathematics Item Sampler 2007–2008 13

MATHEMATICS
E.1.1.2
22. The graph below shows the frequency of
days that were sunny, cloudy, rainy, and
snowy in February of 2005.
February 2005
E.3.2.1
23. Matt has three pairs of shorts,
five shirts, and two pairs of shoes.
How many different outfits consisting
of one shirt, one pair of shorts, and
one pair of shoes can he make?
rainy
snowy
sunny
cloudy
41-0379
A 5 number of shirts
B 10 3 + 5 + 2
C 25 5 × (3 + 2)
D 30 *
Which statement about the data is true?
A
B
C
D
It was sunny the same number of days
as it was rainy.
sunny section larger than rainy
It was snowy the same number of
days as it was rainy.
*
It was sunny more days than it was
cloudy.
cloudy section larger than sunny
It was snowy more days than it was
sunny.
sunny section larger than snowy
8MSE112
Grade 8 Mathematics Item Sampler 2007–2008 14

MATHEMATICS
E.4.1.1
Use the scatterplot below to answer
question 24.
Height (ft)
9
8
7
6
5
4
3
2
1
Weight and Height of 25 People
y
25 50 75 100 125 150 175 200 225
Weight (lbs)
24. Which type of correlation is represented
by the scatterplot?
x
E.4.1.2
25. A random survey shows that 14 out of
21 students plan to vote for Jacob for
class president. Which is the best
prediction of the total number of votes
Jacob will receive if 180 students vote?
41-7277
A 35 14 + 21
B 90 180 ÷ 2
C 120 *
D 173 180 – (21 – 14)
41-0051
A
B
C
D
strong positive
*
strong negative
opposite direction
weak positive
incorrect relation of points
weak negative
opposite direction; incorrect relation
Grade 8 Mathematics Item Sampler 2007–2008 15

MATHEMATICS
FIRST OPEN-ENDED ITEM
D.2
26. Some students held a car wash. They charged $3 to wash each car. The students made
a profit of $181 after they spent $11 on supplies for the car wash.
A. How many cars did they wash? Show all your work. Explain why you did
each step.
41-7332
GO TO THE NEXT PAGE TO FINISH THE QUESTION.
Grade 8 Mathematics Item Sampler 2007–2008 16

MATHEMATICS
26. Continued. Please refer to the previous page for task explanation.
B. What is the least number of cars the students must wash to have a profit of
$330 after spending a total of $15 on supplies? Show all your work. Explain
why you did each step.
41-7332
Grade 8 Mathematics Item Sampler 2007–2008 17

MATHEMATICS
ITEM-SPECIFIC SCORING GUIDELINE
Item #26
This item will be reported under Category D, Algebraic Concepts.
Assessment Anchor:
D.2–Represent and/or analyze mathematical situations using numbers, symbols, words, tables, and/or graphs.
Specific Eligible Content addressed by this item:
D.2.2.2–Write and/or solve an equation for a given problem situtation (one variable only).
Scoring Guide:
Score
4
3
2
1
0
Nonscorable
In response to this item, the student—
demonstrates a thorough understanding of how to write and solve an equation for
a given problem situation (one variable only) by correctly solving problems and
clearly explaining procedures.
demonstrates a general understanding of how to write and solve an equation for
a given problem situation (one variable only) by correctly solving problems and
clearly explaining procedures with only minor errors or omissions.
demonstrates a partial understanding of how to write and solve an equation for a
given problem situation (one variable only) by correctly performing a significant
portion of the required task.
demonstrates minimal understanding of how to write and solve an equation for a
given problem situation (one variable only).
The response has no correct answer and insufficient evidence to demonstrate any
understanding of the mathematical concepts and procedures as required by the
task. Response may show only information copied from the question.
BLK (blank) .............. Blank, entirely erased, or written refusal to respond
OT ............................. Off task
IL ............................... Illegible
LOE ........................... Response in a language other than English
Grade 8 Mathematics Item Sampler 2007–2008 18

MATHEMATICS
Item #26
Top Scoring Response:
Part A Answer
64 (cars)
Support
3c ] 11 = 181
3c ] 11 + 11 = 181 + 11
3c = 192
3c
}
3 = 192 }
3
c = 64
AND
Sample Explanation:
To find the number of cars washed I added the money spent to the profit
to find out how much money they really collected. Then I divided by 3
(charge per car) to find out how many cars they washed.
OR equivalent
(2 score points)
0.5 point for correct answer
1.5 points for complete support
Part B Answer
115 (cars)
Support
3c ] 15 = 330
3c ] 15 + 15 = 330 + 15
3c = 345
3c
}
3 = 345 }
3
c = 115
AND
Sample Explanation:
To find the number of cars washed I added the money spent to the profit to
find out how much money they really collected. Then I divided by 3
(charge per car) to find out how many cars they washed.
OR equivalent
(2 score points)
0.5 point for correct answer
1.5 points for complete support
Grade 8 Mathematics Item Sampler 2007–2008 19

OPEN-ENDED ITEM RESPONSES
D.2 Response Score: 4
MATHEMATICS
26. Some students held a car wash. They charged $3 to wash each car. The students made
a profit of $181 after they spent $11 on supplies for the car wash.
A. How many cars did they wash? Show all your work. Explain why you did
each step.
The student has given the correct answer.
The student has shown complete support.
41-7332
GO TO THE NEXT PAGE TO FINISH THE QUESTION.
Grade 8 Mathematics Item Sampler 2007–2008 20

MATHEMATICS
26. Continued. Please refer to the previous page for task explanation.
B. What is the least number of cars the students must wash to have a profit of
$330 after spending a total of $15 on supplies? Show all your work. Explain
why you did each step.
41-7332
The student has given the correct answer.
The student has shown complete support.
Grade 8 Mathematics Item Sampler 2007–2008 21

MATHEMATICS
D.2 Response Score: 3
26. Some students held a car wash. They charged $3 to wash each car. The students made
a profit of $181 after they spent $11 on supplies for the car wash.
A. How many cars did they wash? Show all your work. Explain why you did
each step.
The student has given the correct answer.
The student has shown all correct work.
The student did not give an explanation.
41-7332
GO TO THE NEXT PAGE TO FINISH THE QUESTION.
Grade 8 Mathematics Item Sampler 2007–2008 22

MATHEMATICS
26. Continued. Please refer to the previous page for task explanation.
B. What is the least number of cars the students must wash to have a profit of
$330 after spending a total of $15 on supplies? Show all your work. Explain
why you did each step.
The student has given the correct answer.
The student has shown all correct work.
The student did not give an explanation.
41-7332
Grade 8 Mathematics Item Sampler 2007–2008 23

MATHEMATICS
D.2 Response Score: 3
26. Some students held a car wash. They charged $3 to wash each car. The students made
a profit of $181 after they spent $11 on supplies for the car wash.
A. How many cars did they wash? Show all your work. Explain why you did
each step.
The student has given the correct answer.
The student has shown all correct work.
The student did not explain why each step was done.
41-7332
GO TO THE NEXT PAGE TO FINISH THE QUESTION.
Grade 8 Mathematics Item Sampler 2007–2008 24

MATHEMATICS
26. Continued. Please refer to the previous page for task explanation.
B. What is the least number of cars the students must wash to have a profit of
$330 after spending a total of $15 on supplies? Show all your work. Explain
why you did each step.
The student has given the correct answer.
The student has shown all correct work.
The student did not explain why each step was done.
41-7332
Grade 8 Mathematics Item Sampler 2007–2008 25

MATHEMATICS
D.2 Response Score: 2
26. Some students held a car wash. They charged $3 to wash each car. The students made
a profit of $181 after they spent $11 on supplies for the car wash.
A. How many cars did they wash? Show all your work. Explain why you did
each step.
The student has given the correct answer.
The student has shown all correct work.
The student did not explain why each step was done.
41-7332
GO TO THE NEXT PAGE TO FINISH THE QUESTION.
Grade 8 Mathematics Item Sampler 2007–2008 26

MATHEMATICS
26. Continued. Please refer to the previous page for task explanation.
B. What is the least number of cars the students must wash to have a profit of
$330 after spending a total of $15 on supplies? Show all your work. Explain
why you did each step.
The student has given an incorrect answer.
The student has shown correct work.
The student did not explain why each step was done.
41-7332
Grade 8 Mathematics Item Sampler 2007–2008 27

MATHEMATICS
D.2 Response Score: 2
26. Some students held a car wash. They charged $3 to wash each car. The students made
a profit of $181 after they spent $11 on supplies for the car wash.
A. How many cars did they wash? Show all your work. Explain why you did
each step.
The student has given the correct answer.
The student has shown some correct support.
41-7332
GO TO THE NEXT PAGE TO FINISH THE QUESTION.
Grade 8 Mathematics Item Sampler 2007–2008 28

MATHEMATICS
26. Continued. Please refer to the previous page for task explanation.
B. What is the least number of cars the students must wash to have a profit of
$330 after spending a total of $15 on supplies? Show all your work. Explain
why you did each step.
The student has given the correct answer.
The student has shown all correct work.
The student did not give a complete explanation.
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Grade 8 Mathematics Item Sampler 2007–2008 29

MATHEMATICS
D.2 Response Score: 1
26. Some students held a car wash. They charged $3 to wash each car. The students made
a profit of $181 after they spent $11 on supplies for the car wash.
A. How many cars did they wash? Show all your work. Explain why you did
each step.
The student has given an incorrect answer.
The student has shown some correct support.
41-7332
GO TO THE NEXT PAGE TO FINISH THE QUESTION.
Grade 8 Mathematics Item Sampler 2007–2008 30

MATHEMATICS
26. Continued. Please refer to the previous page for task explanation.
B. What is the least number of cars the students must wash to have a profit of
$330 after spending a total of $15 on supplies? Show all your work. Explain
why you did each step.
The student has given an incorrect answer.
The student has shown some correct support.
41-7332
Grade 8 Mathematics Item Sampler 2007–2008 31

MATHEMATICS
D.2 Response Score: 1
26. Some students held a car wash. They charged $3 to wash each car. The students made
a profit of $181 after they spent $11 on supplies for the car wash.
A. How many cars did they wash? Show all your work. Explain why you did
each step.
The student has given an incorrect answer.
The student has shown some correct support.
41-7332
GO TO THE NEXT PAGE TO FINISH THE QUESTION.
Grade 8 Mathematics Item Sampler 2007–2008 32

MATHEMATICS
26. Continued. Please refer to the previous page for task explanation.
B. What is the least number of cars the students must wash to have a profit of
$330 after spending a total of $15 on supplies? Show all your work. Explain
why you did each step.
The student has given an incorrect answer.
The student has shown some correct support.
41-7332
Grade 8 Mathematics Item Sampler 2007–2008 33

MATHEMATICS
D.2 Response Score: 0
26. Some students held a car wash. They charged $3 to wash each car. The students made
a profit of $181 after they spent $11 on supplies for the car wash.
A. How many cars did they wash? Show all your work. Explain why you did
each step.
The student has given an incorrect answer.
The student has shown no correct support.
41-7332
GO TO THE NEXT PAGE TO FINISH THE QUESTION.
Grade 8 Mathematics Item Sampler 2007–2008 34

MATHEMATICS
26. Continued. Please refer to the previous page for task explanation.
B. What is the least number of cars the students must wash to have a profit of
$330 after spending a total of $15 on supplies? Show all your work. Explain
why you did each step.
The student has given an incorrect answer.
The student has shown no correct support.
41-7332
Grade 8 Mathematics Item Sampler 2007–2008 35

MATHEMATICS
SECOND OPEN-ENDED ITEM
A.2
27. Train L and train M leave a station at the same time heading in opposite directions.
Train L is going north at an average speed of 75 miles per hour (mph). Train M is going
south at an average speed of 60 mph.
A. How long does it take each train, in hours, to travel 225 miles along the track?
Show all your work. Explain why you did each step.
41-7289
GO TO THE NEXT PAGE TO FINISH THE QUESTION.
Grade 8 Mathematics Item Sampler 2007–2008 36

MATHEMATICS
27. Continued. Please refer to the previous page for task explanation.
B. What is the distance, in miles, between the two trains after each train travels
for 6 hours? Show all your work. Explain why you did each step.
41-7289
Grade 8 Mathematics Item Sampler 2007–2008 37

MATHEMATICS
ITEM-SPECIFIC SCORING GUIDELINE
Item #27
This item will be reported under Category A, Numbers and Operations.
Assessment Anchor:
A.2–Understand meanings of operations, use operations, and understand how they relate to each other.
Specific Eligible Content addressed by this item:
Scoring Guide:
A.2.2.2–Represent or solve rate problems (e.g., unit rates, simple interest, distance, etc.). Students may be
asked to solve for any term (formulas provided on the reference sheet for distance and interest).
Score
4
3
2
1
0
Nonscorable
In response to this item, the student—
demonstrates a thorough understanding of how to represent or solve rate
problems (e.g., unit rates, distance, etc.) by correctly solving problems and
clearly explaining procedures.
demonstrates a general understanding of how to represent or solve rate
problems (e.g., unit rates, distance, etc.) by correctly solving problems and
clearly explaining procedures with only minor errors or omissions.
demonstrates a partial understanding of how to represent or solve rate
problems (e.g., unit rates, distance, etc.) by correctly performing a
significant portion of the required task.
demonstrates minimal understanding of how to represent or solve rate
problems (e.g. unit rates, distance, etc.).
The response has no correct answer and insufficient evidence to demonstrate
any understanding of the mathematical concepts and procedures as required
by the task. Response may show only information copied from the question.
BLK (blank) .............. Blank, entirely erased, or written refusal to respond
OT ............................. Off task
IL ............................... Illegible
LOE ........................... Response in a language other than English
Grade 8 Mathematics Item Sampler 2007–2008 38

MATHEMATICS
Item #27
Top Scoring Response:
Part A Answer
Train L : 3 (hours)
Train M: 3.75 (hours) or
3 3 } (hours) or
4
3 hours and 45 minutes
Train L :
Support
d = r • t
225 = 75 • t
225
}
75 = 75t }
75
3 = t
Train M :
225 = 60 • t
225
}
60 = 60t }
60
3.75 = tmi
AND
(2 score points)
0.5 point for 2 correct times
1.5 points for complete support
Sample Explanation:
I used the distance formula (distance equals rate times time) for each
train. I put in 225 miles for the distance and then put in each train’s
speed. I then solved each equation for t to determine each train’s time
to travel 225 miles.
OR equivalent
Part B Answer
810 (miles)
Train L :
Train M :
Support
d = r • t
d = 75 • 6
d = 450
d = 60 • 6
d = 360
450 + 360 = 810
AND
(2 score points)
0.5 point for correct answer
1.5 points for complete support
Sample Explanation:
I put the trains’ time (6) and each train’s rate into the distance formula. By
multiplying rate times time, I found the distance each train traveled in 6 hours.
Then, since the trains are going in opposite directions, I added the distances together
to determine the total distance between them.
OR equivalent
Grade 8 Mathematics Item Sampler 2007–2008 39

OPEN-ENDED ITEM RESPONSES
A.2 Response Score: 4
MATHEMATICS
27. Train L and train M leave a station at the same time heading in opposite directions.
Train L is going north at an average speed of 75 miles per hour (mph). Train M is going
south at an average speed of 60 mph.
A. How long does it take each train, in hours, to travel 225 miles along the track?
Show all your work. Explain why you did each step.
The student has given the correct answer.
The student has shown complete support.
41-7289
GO TO THE NEXT PAGE TO FINISH THE QUESTION.
Grade 8 Mathematics Item Sampler 2007–2008 40

MATHEMATICS
27. Continued. Please refer to the previous page for task explanation.
B. What is the distance, in miles, between the two trains after each train travels
for 6 hours? Show all your work. Explain why you did each step.
The student has given the correct answer.
The student has shown complete support.
41-7289
Grade 8 Mathematics Item Sampler 2007–2008 41

MATHEMATICS
A.2 Response Score: 3
27. Train L and train M leave a station at the same time heading in opposite directions.
Train L is going north at an average speed of 75 miles per hour (mph). Train M is going
south at an average speed of 60 mph.
A. How long does it take each train, in hours, to travel 225 miles along the track?
Show all your work. Explain why you did each step.
The student has given an incorrect answer due
to a calculation error.
The student has shown complete support.
41-7289
GO TO THE NEXT PAGE TO FINISH THE QUESTION.
Grade 8 Mathematics Item Sampler 2007–2008 42

MATHEMATICS
27. Continued. Please refer to the previous page for task explanation.
B. What is the distance, in miles, between the two trains after each train travels
for 6 hours? Show all your work. Explain why you did each step.
The student has given the correct answer.
The student has shown complete support.
41-7289
Grade 8 Mathematics Item Sampler 2007–2008 43

MATHEMATICS
A.2 Response Score: 3
27. Train L and train M leave a station at the same time heading in opposite directions.
Train L is going north at an average speed of 75 miles per hour (mph). Train M is going
south at an average speed of 60 mph.
A. How long does it take each train, in hours, to travel 225 miles along the track?
Show all your work. Explain why you did each step.
The student has given the correct answer.
The student has shown some correct support.
41-7289
GO TO THE NEXT PAGE TO FINISH THE QUESTION.
Grade 8 Mathematics Item Sampler 2007–2008 44

MATHEMATICS
27. Continued. Please refer to the previous page for task explanation.
B. What is the distance, in miles, between the two trains after each train travels
for 6 hours? Show all your work. Explain why you did each step.
The student has given the correct answer.
The student has shown complete support.
41-7289
Grade 8 Mathematics Item Sampler 2007–2008 45

MATHEMATICS
A.2 Response Score: 2
27. Train L and train M leave a station at the same time heading in opposite directions.
Train L is going north at an average speed of 75 miles per hour (mph). Train M is going
south at an average speed of 60 mph.
A. How long does it take each train, in hours, to travel 225 miles along the track?
Show all your work. Explain why you did each step.
The student has given the correct answer.
The student has shown some correct support.
41-7289
GO TO THE NEXT PAGE TO FINISH THE QUESTION.
Grade 8 Mathematics Item Sampler 2007–2008 46

MATHEMATICS
27. Continued. Please refer to the previous page for task explanation.
B. What is the distance, in miles, between the two trains after each train travels
for 6 hours? Show all your work. Explain why you did each step.
The student has given an incorrect answer.
The student has shown some correct work.
41-7289
Grade 8 Mathematics Item Sampler 2007–2008 47

MATHEMATICS
A.2 Response Score: 2
27. Train L and train M leave a station at the same time heading in opposite directions.
Train L is going north at an average speed of 75 miles per hour (mph). Train M is going
south at an average speed of 60 mph.
A. How long does it take each train, in hours, to travel 225 miles along the track?
Show all your work. Explain why you did each step.
The student has given only 1 correct answer.
The student has shown complete support.
41-7289
GO TO THE NEXT PAGE TO FINISH THE QUESTION.
Grade 8 Mathematics Item Sampler 2007–2008 48

MATHEMATICS
27. Continued. Please refer to the previous page for task explanation.
B. What is the distance, in miles, between the two trains after each train travels
for 6 hours? Show all your work. Explain why you did each step.
The student has given an incorrect answer.
The student has shown some correct work.
41-7289
Grade 8 Mathematics Item Sampler 2007–2008 49

MATHEMATICS
A.2 Response Score: 1
27. Train L and train M leave a station at the same time heading in opposite directions.
Train L is going north at an average speed of 75 miles per hour (mph). Train M is going
south at an average speed of 60 mph.
A. How long does it take each train, in hours, to travel 225 miles along the track?
Show all your work. Explain why you did each step.
The student has given the correct answer.
The student has shown some correct support.
41-7289
GO TO THE NEXT PAGE TO FINISH THE QUESTION.
Grade 8 Mathematics Item Sampler 2007–2008 50

MATHEMATICS
27. Continued. Please refer to the previous page for task explanation.
B. What is the distance, in miles, between the two trains after each train travels
for 6 hours? Show all your work. Explain why you did each step.
The student has given an incorrect answer.
The student has shown incorrect and incomplete support.
41-7289
Grade 8 Mathematics Item Sampler 2007–2008 51

MATHEMATICS
A.2 Response Score: 1
27. Train L and train M leave a station at the same time heading in opposite directions.
Train L is going north at an average speed of 75 miles per hour (mph). Train M is going
south at an average speed of 60 mph.
A. How long does it take each train, in hours, to travel 225 miles along the track?
Show all your work. Explain why you did each step.
The student has given an incorrect answer.
The student has shown incomplete support.
41-7289
GO TO THE NEXT PAGE TO FINISH THE QUESTION.
Grade 8 Mathematics Item Sampler 2007–2008 52

MATHEMATICS
27. Continued. Please refer to the previous page for task explanation.
B. What is the distance, in miles, between the two trains after each train travels
for 6 hours? Show all your work. Explain why you did each step.
The student has given an incorrect answer.
The student has shown some correct support.
41-7289
Grade 8 Mathematics Item Sampler 2007–2008 53

MATHEMATICS
A.2 Response Score: 0
27. Train L and train M leave a station at the same time heading in opposite directions.
Train L is going north at an average speed of 75 miles per hour (mph). Train M is going
south at an average speed of 60 mph.
A. How long does it take each train, in hours, to travel 225 miles along the track?
Show all your work. Explain why you did each step.
The student has given an incorrect answer.
The student has shown incomplete support.
41-7289
GO TO THE NEXT PAGE TO FINISH THE QUESTION.
Grade 8 Mathematics Item Sampler 2007–2008 54

MATHEMATICS
27. Continued. Please refer to the previous page for task explanation.
B. What is the distance, in miles, between the two trains after each train travels
for 6 hours? Show all your work. Explain why you did each step.
The student has given an incorrect answer.
The student has shown incomplete support.
41-7289
Grade 8 Mathematics Item Sampler 2007–2008 55

Mathematics
Grade 8
Item and Scoring Sampler
Copyright © 2007 by the Pennsylvania Department of Education. The materials contained in this publication may be
duplicated by Pennsylvania educators for local classroom use. This permission does not extend to the duplication
of materials for commercial use.