NAEP Mathematics
Grade 12
Sample Assessment Block
West Virginia Board of Education
2011-2012
L. Wade Linger Jr., President
Gayle C. Manchin, Vice President
Robert W. Dunlevy, Secretary
Michael I. Green, Member
Priscilla M. Haden, Member
Lloyd G. Jackson II, Member
Lowell E. Johnson, Member
Jenny N. Phillips, Member
William M. White, Member
Brian E. Noland, Ex Officio
Chancellor
West Virginia Higher Education Policy Commission
James L. Skidmore, Ex Officio
Chancellor
West Virginia Council for Community and Technical College Education
Jorea M. Marple, Ex Officio
State Superintendent of Schools
West Virginia Department of Education
NAEP Mathematics - Grade 12
Table of Contents
Calculator Information ................................................................................................ 2
Sample Assessment Booklet
Reference Sheet................................................................................................... 3
Assessment Questions......................................................................................... 5
NAEP Mathematics Framework Overview Information......................................... 18
Individual Item Analysis.......................................................................................... 20
1 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
For NAEP calculator-active blocks of items, the instructions listed below will appear on the
first page of the section of assessment items. Students are allowed to bring whatever calculator,
graphing or otherwise, they are accustomed to using in the classroom with some restrictions for
test security purposes. Restrictions are similar to WESTEST 2 calculator rules. For students
who do not bring a calculator to use on the assessment, NAEP will provide a scientific calculator
(TI-30 Challenger).
YOU WILL NEED A CALCULATOR FOR THIS SECTION.
REMEMBER: You will have to decide whether to use the calculator. For some questions using
the calculator is helpful, maybe even necessary, but for other questions the calculator may not be
helpful. You may either use your own calculator or you will be provided with one to use in this
section. If you decide to use the calculator that is provide, the information below explains how to
use it.
ON/AC
To turn on the calculator, press
.
To clear the calculator, press ON/AC or press CE/C twice.
EXAMPLES:
Example
1.
2.
To Solve
4x7.3/2
(80-14)x6
3.
4.
√29
π
Enter/Press These Keys
4 x 7.3 ÷ 2 =
80 − 14 = x 6 =
or
( 80 − 14 ) x 6 =
29 √
π
The Display Will Be
14.6
396
5.3851648
3.1415927
Clear/Press
ON/AC
ON/AC
ON/AC
ON/AC
If necessary, you can refer to the QUICK REFERENCE CARD on the lid of the calculator
provided for more information
2 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
This section has 13 questions. Mark your answer in your booklet. You will have to fill in an
oval or write your answer as directed. In those questions where you must write an answer, it
is important that your answer be clear and complete and that you show all of your work since
partial credit may be awarded.
You will receive a reference sheet with this test booklet. The reference sheet contains
mathematical formulas that may be useful for answering some of the questions in this section.
You may use a calculator for any of the questions in this section. If you are asked to round your
answer, do not round any numbers except your final answer.
1. The table above shows the high and low temperatures on October 1st for five cities. Which
city had the greatest temperature range?
A. City A
B. City B
C. City C
D. City D
E. City E
5 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
2. John is going to cover an attic floor with insulation. The floor measures 25 feet by 35 feet. If one roll of insulation will cover 64 square feet, how many rolls of insulation does John need?
A. 1
B. 2
C. 8
D. 14
E. 110
6 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
3. The population P of a certain town is given by the equation P = 50,000(1 + r)t, where r is
the annual rate of population increase and t is the number of years since 1990.
(a) What was the population in 1990?
Answer: ____________________
(b) In 2001 the population was 100,000. What was the annual rate of population increase?
Answer: ____________________
7 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
4. Quadrilateral ABCD is inscribed in circle O, and C is a right angle, as shown above.
Segment AB is not parallel to segment DC. Which of the following statements must be true?
A.
A= B
B. B = D
C. B is a right angle.
D. AC is a diameter of circle O.
E. BD is a diameter of circle O.
8 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
5. The manager of a company has to order new engines for its delivery trucks after the trucks have been driven 150,000 miles. One of the delivery trucks currently has 119,866 miles on it. This truck has the same delivery route each week and is driven an average of 40,000 miles
each year. At this rate, the manager should expect this truck to reach 150,000 miles in
approximately how many months?
A. Less than 4 months
B. Between 4 and 6 months
C. Between 6 and 8 months
D. Between 8 and 10 months
E. More than 10 months
9 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
6. Angie has a bag containing n apples. She gives 4 to her brother and keeps 5 for herself. She
then divides the remaining apples equally among 3 friends. Which of the following
expressions represents the number of apples each friend receives?
A. n - 4 - 5
3
B. n - 4 - 5
3
+
C. 4 5 - n
3
D. n - 4 - 5
3
E. n - 5 - 4
3
10 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
7. Prove that AC = DC and give a reason for each statement in your proof.
11 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
8. The first four terms in a sequence are shown below.
40, 8, 24, 16, . . .
Each term after the first two terms is found by taking one-half the sum of the two preceding
terms. Which term is the first odd number in this sequence?
A. The 5th term
B. The 6th term
C. The 7th term
D. The 8th term
E. The 9th term
12 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
9. The scatterplot above shows data for groups R and S. Which of the following statements is
true about the correlation between the x and y values of group R and the correlation
between the x and y values of group S?
A. The x and y values appear to be negatively correlated in both R and S.
B. The x and y values appear to be positively correlated in both R and S.
C. The x and y values appear to be negatively correlated in R, but positively correlated in S.
D. The x and y values appear to be positively correlated in R, but negatively correlated in S.
E. The x and y values appear to be more highly correlated in R than in S.
13 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
10. The following chart indicates the maximum number of connecting line segments y that can
be drawn connecting x points, where no three points lie on the same line.
(a)The relationship between x and y is represented by the equation y = kx(x−1) for any positive number of points x. Use the information in the table to determine the value of the real number k.
Answer: k = ____________________
(b)Use the equation from part (a) to determine the maximum number of line segments
that can be drawn connecting 100 points, no three of which lie on the same line.
14 | Page
Answer: ____________________
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
11. On level ground from a distance of 200 feet, the angle of elevation to the top of a building is 21°, as shown in the figure above. What is the height h of the building, to the nearest foot?
A. 72
B. 77
C. 187
D. 201
E. 521
15 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
12. Bob is going on a trip. He will be taking a taxi, a flight, and then a train. Bob chose the
following three companies based on their claims.
• Tom’s Taxi Service claims that it is on time 95 percent of the time.
• Friendly Flyer Airlines claims that it is on time 93 percent of the time.
• Rapid Railways claims that it is on time 98 percent of the time.
Based on the three companies’ claims, what is the approximate probability that all three parts
of Bob’s trip will be on time, assuming that all three probabilities are independent?
Answer: ____________________
16 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
13. Perform the following two transformations on the graph below. After each transformation, draw the resulting image of segment PQ.
(1)Rotate the segment 90° counterclockwise (
with a 1.
) about point P. Label the resulting segment
(2) Reflect the segment you drew in part (1) across the x -axis. Label the resulting segment
with a 2.
17 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
NAEP Mathematics Framework Overview Information
The National Assessment of Educational Progress (NAEP) assesses students’ understanding of
mathematical content. The framework for the mathematics assessment is anchored in five broad
areas of mathematics.
Board Area of Mathematics
Number Properties and Operations
Measurement
Geometry
Data Analysis, Statistics, and Probability
Algebra
Includes but is not limited to
Cumputation
Understanding of number concepts
Use of instruments
Application of processes
Concepts of area and volume
Spatial reasoning
Applying geometric properties
Graphical display
Stastistics
Representations
Relationships
These divisions are not intended to separate mathematic into discrete elements. Rather, they
are intended to provide a helpful classification scheme that describes the full spectrum of
mathematical content assessed by NAEP. Classification of items into one primary content area
is not always clear-cut, but it helps ensure that important mathematical concepts and skills are
assessed in a balanced way.
­­­
Item Distribution by Content Area
Content Area
Number Properties and Operations
Measurement
Geometry
Data Analysis, Statistics, and Probability
Algebra
18 | Page
Grade 4
40%
20%
15%
10%
15%
Grade 8
20%
15%
20%
15%
30%
Grade 12
10%
30%
25%
35%
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
Mathematical Complexity
Items are the NAEP mathematics assessment are categories by mathematical complexity.
Mathematical complexity is an indication of the demands on students’ thinking. The demand
on thinking that an item expects—what it asks the student to recall, understand, reason about,
and do—assume that students are familiar with the mathematics of the task. Mathematical
complexity deals with what the students are asked to do in a task. It does not take into account
how they might undertake it.
NAEP has three levels of complexity—high, moderate, low—forming an ordered description of
the demands an item may make on a student. Items at the low level of complexity, for example,
may ask a student to recall a property. At the moderate level, an item may ask the student
to make a connection between two properties; at the high level, an item may ask a student to
analyze the assumptions made in a mathematical model.
At each grade level, the percent of testing time at each complexity level is the same. One
half the testing time is expected to be spent on moderate complexity level. One quarter of the
testing time is expected to be spent on low complexity items and the remaining quarter on high
complexity items.
Item Type
NAEP assessment in mathematics has three formats or item types: multiple-choice, short
constructed response, and extended constructed response.
• Multiple-choice items require students to read, reflect, or compute and then to select the
alternative that best expresses the answer. Multiple-choice items for grade 4 have four choices,
and at grades 8 and 12, there are five choices.
• Short constructed-responses require students to give either a numerical result or the correct
name or classification for a group of mathematical objects, draw an example of a given concept,
or possibly write a brief explanation for a given result.
• Extended Constructed-response items require students to consider a situation that requires
more than a numerical response or a short verbal communication. The student may be asked, for
example, to describe a situation, analyze a graph or table of values or an algebraic equation, or
compute specific numerical values.
The NAEP assessment is divided evenly between multiple-choice and both types of constructedresponse items. 50% of the testing time is expected to be spent of multiple-choice items and
the other 50% on the constructed-response items. Note: No one student takes the entire
mathematics assessment. Some blocks of assessment items may not contain all types of items,
particularly the extended constructed response.
19 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
1. Data Analysis and Probability
A*
B
C
D
E
Omitted
West Virginia
56%
14%
2%
4%
23%
1%
National Public
62%
14%
2%
4%
18%
Rounds to Zero
The table above shows the high and low temperatures on October 1st for five cities. Which city
had the greatest temperature range?
A. City A
B. City B
C. City C
D. City D
E. City E
2. Measurement
A
B
C
D*
E
Omitted
West Virginia
14%
19%
8%
56%
2%
1%
National Public
9%
11%
7%
70%
2%
1%
John is going to cover an attic floor with insulation. The floor measures 25 feet by 35 feet. If one
roll of insulation will cover 64 square feet, how many rolls of insulation does John need?
A. 1
B. 2
C. 8
D. 14
E. 110
20 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
3. Algebra
Incorrect
Partial 2
Partial 1
Correct
Omitted
Off Task
West Virginia
37%
Rounds to Zero
47%
3%
9%
4%
National Public
35%
Rounds to Zero
46%
9%
8%
2%
The population P of a certain town is given by the equation =50,000 (1+ r)t, where r is the annual
rate of population increase and t is the number of years since 1990.
(a)What was the population in 1990 ?
(b)In 2001 the population was 100,000. What was the annual rate of population increase?
Scoring Guide
Correct
Both parts correct
Sample Correct Responses:
(a) Answer: 50,000
Solution (not required in response):
In 1990, t = 0.
P = 50,000 (1 + r)0 = 50,000(1) = 50,000
(b) Answer: 6.5% (or 0.065) (Accept responses from 6% to 7% inclusive)
Solution (not required in response):
In 2001, t=11
100,000 = 50,000(1 + r)11
2=(1 + r)11
11
2 = 1+r
11
2 -1 = r
r ≈ 0.065 ≈ 6.5%
Partial 1
Part (a) correct only
wwPartial 2
Part (b) correct only
Incorrect
Incorrect response
*Note: Separate “incorrect” and “partial” categories do not indicate one incorrect or partial answer is scored
different from other incorrect or partial answers. “Popular” partial and incorrect answers are separated to better
inform instructional decisions by educators.
21 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
Correct - Student Response
Exemplar 1
The population P of a certain town is given by the equation: P = 50,000 (1+ r)t, where r is the
annual rate of population increase and t is the number of years since 1990.
(a)What was the population in 1990?
Answer: ____________________
(b)In 2001 the population was 100,000. What was the annual rate of population increase?
Answer: ____________________
Scorer Comments:
This response is correct, with a correct answer of P = 50,000 for part (a) and a correct exact
value for r in part (b).
Exemplar 2
The population P of a certain town is given by the equation: P = 50,000 (1+ r)t, where r is the
annual rate of population increase and t is the number of years since 1990.
(a)What was the population in 1990?
Answer: ____________________
(b)In 2001 the population was 100,000. What was the annual rate of population increase?
Answer: ____________________
Scorer Comments:
This response is correct, with a correct answer of 50,000 for part (a) and a correct approximation
of 6.50% for part (b).
22 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
Partial 1 - Student Response
Exemplar 1
The population P of a certain town is given by the equation: P = 50,000 (1+ r)t, where r is the
annual rate of population increase and t is the number of years since 1990.
(a) What was the population in 1990?
Answer: ____________________
(b) In 2001 the population was 100,000. What was the annual rate of population increase?
Answer: ____________________
Scorer Comments:
This response is correct, with a correct answer of P = 50,000 for part (a) and a correct exact
value for r in part (b).
Exemplar 2
The population P of a certain town is given by the equation: P = 50,000 (1+ r)t, where r is the
annual rate of population increase and t is the number of years since 1990.
(a) What was the population in 1990?
Answer: ____________________
(b) In 2001 the population was 100,000. What was the annual rate of population increase?
Answer: ____________________
Scorer Comments:
This response is correct, with a correct answer of 50,000 for part (a) and a correct approximation
of 6.50% for part (b). Although the response shows a correct equation for part (b), the value of r
was not found.
23 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
Partial 2 - Student Response
Exemplar 1
The population P of a certain town is given by the equation: P = 50,000 (1+ r)t, where r is the
annual rate of population increase and t is the number of years since 1990.
(a) What was the population in 1990?
Answer: ____________________
(b) In 2001 the population was 100,000. What was the annual rate of population increase?
Answer: ____________________
Scorer Comments:
This response is partially correct, with an incorrect answer of 0 for part (a) and an incorrect
answer of 0.065 for part (b).
Exemplar 2
The population P of a certain town is given by the equation: P = 50,000 (1+ r)t, where r is the
annual rate of population increase and t is the number of years since 1990.
(a) What was the population in 1990?
Answer: ____________________
(b)
In 2001 the population was 100,000. What was the annual rate of population increase?
Answer: ____________________
Scorer Comments:
This response is partially correct, with an incorrect answer of 65,000 for part (a) and an
acceptable approximation of 6% for part (b).
24 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
4. Geometry
A
B
C
D
E*
Omitted
West Virginia
17%
13%
12%
7%
51%
1%
National Pubic
15%
11%
9%
4%
59%
2%
Quadrilateral ABCD is inscribed in circle O, and C is a right angle, as shown above. Segment
AB is not parallel to segment DC. Which of the following statements must be true?
A. A = B
B. B = D
C. B is a right angle.
D. AC is a diameter of circle O.
E. BD is a diameter of circle O.
5.
Number Properties and Operations
A
B
C
D*
E
Omitted
West Virginia
12%
15%
17%
46%
9%
1%
National Pubic
14%
12%
13%
51%
9%
1%
The manager of a company has to order new engines for its delivery trucks after the trucks have been
driven 150,000 miles. One of the delivery trucks currently has 119,866 miles on it. This truck has the
same delivery route each week and is driven an average of 40,000 miles each year. At this rate, the
manager should expect this truck to reach 150,000 miles in approximately how many months?
A. Less than 4 months
B. Between 4 and 6 months
C. Between 6 and 8 months
D. Between 8 and 10 months
E. More than 10 months
25 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
6. Algebra
West Virginia
National Pubic
A
13%
10%
B*
52%
57%
C
18%
14%
D
11%
12%
E
6%
7%
Omitted
Rounds to Zero
1%
Angie has a bag containing n apples. She gives 4 to her brother and keeps 5 for herself.
She then divides the remaining apples equally among 3 friends. Which of the following
expressions represents the number of apples each friend receives?
A. n - 4 - 5
3
B. n - 4 - 5
3
C. 4 + 5 - n
3
D. n 4 - 5
3
E. n 5 - 4
3
7. Geometry
Incorrect
Minimal
Partial
Satisfactory
Extended
Omitted
Off Task
West Virginia
70%
1%
Rounds to Zero
Rounds to Zero
Rounds to Zero
24%
4%
National Pubic
70%
2%
2%
1%
Rounds to Zero
21%
3%
Prove that AC = DC and give a reason for each statement in your proof.
26 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
Scoring Guide
Extended
Acceptable and complete proof with correct reason for each step
Sample Correct Responses:
Proof:
Statment
Reason
Given
C is the midpint of BE
Given
B and E are right angles
BE = EC
B= E
ACB = DCE
ACB = DCE
AC = DC
Definition of midpoint
Right angles are congruent
Vertical angles are congruent
Angle-Side-Angle (or Leg-Angle)
Corresponding part of congruent triangles
are congruent
Notes:
• Repeating the given information is not required in the response.
• Repeating the given information is not credited as correct statements.
• To be a correct statement-reason pair, the reason must be based on the given information
and/or based on previous statements in the proof.
Satisfactory
Acceptable proof with one missing statement, reason, or statement-reason pair, or
imprecise notation or language
Partial
Response does not prove that AC = DC , but has 3 or more correct statement-reason pairs
Minimal
Response does not prove that AC = DC , but has 2 correct statement-reason pairs
Incorrect
Incorrect response
27 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
Extended - Student Response
Exemplar 1
Prove that AC = DC and give a reason for each statement in your proof.
Scorer Comments:
This response is "Extended," providing a complete proof with a correct reason for each step.
Prove that AC
= DC vand give a reason for each statement in your proof
28 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
Exemplar 2
Scorer Comments:
This response is "Extended," providing a complete proof with a correct reason for each step.
29 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
Satisfactory - Student Response
Exemplar 1
Prove thatAC = DC and give a reason for each statement in your proof.
Scorer Comments:
This response is "Satisfactory," providing a proof in which most steps are correct, but the last
step fails to state that the congruence of the triangles makes their corresponding sides equal.
30 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
Exemplar 2
Prove that AC = DC and give a reason for each statement in your proof.
Scorer Comments:
This response is "Satisfactory," providing a complete proof with one incorrect reason. The
correct reason for statement 5 is the Side-Angle-Side Theorem (SAS), not SAA.
31 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
Partial - Student Response
Exemplar 1
Prove that AC
= DC and give a reason for each statement in your proof.
Scorer Comments:
This response is "Partial" since it does not provide sufficient arguments to prove that the triangles
are congruent, but it does contain three correct statement-reason pairs. Step 2 is a correct
statement-reason pair that leads to a proof that the two segments are congruent, but the response
does not explicitly state the congruences that are needed to draw this conclusion.
32 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
Exemplar 2
Prove that AC = DC and give a reason for each statement in your proof.
Scorer Comments:
This response is "Partial" since it does not provide sufficient arguments to prove that the triangles
are congruent, but it does contain three correct statement-reason pairs (steps 1, 3, 4, and 5). The
statement in step 2 is correct but the reason is incorrect, and therefore is not counted as a correct
statement-reason pair.
33 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
Minimal - Student Response
Exemplar 1
Prove that AC
= DC and give a reason for each statement in your proof.
Scorer Comments:
This response is "Minimal," as two correct statement-reason pairs were given, but the statements
on their own are not enough to prove that the two segments are congruent.
34 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
Exemplar 2
Prove that AC = DC and give a reason for each statement in your proof.
Scorer Comments:
This response is "Minimal." Steps 2 and 3 provide correct statement-reason pairs. Although steps
1 and 4 provide correct statements, a valid reason is not provided for these statements.
35 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
8. Algebra
West Virginia
10%
16%
54%
12%
6%
1%
A
B
C*
D
E
Omitted
National Pubic
9%
15%
60%
10%
5%
2%
The first four terms in a sequence are shown below.
40, 8, 24, 16, . . .
Each term after the first two terms is found by taking one-half the sum of the two preceding terms.
Which term is the first odd number in this sequence?
A. The 5th term
B. The 6th term
C. The 7th term
D. The 8th term
E. The 9th term
9. Data Analysis and Probability
A
B
C*
D
E
Omitted
West Virginia
6%
21%
53%
13%
7%
1%
National Pubic
3%
16%
61%
12%
5%
3%
The scatterplot above shows data for groups R and S . Which of the following statements is true about
the correlation between the x and y values of group R and the correlation between the x and y values
of group S?
A. The x and y values appear to be negatively correlated in both R and S .
B. The x and y values appear to be positively correlated in both R and S .
C. The x and y values appear to be negatively correlated in R , but positively correlated in S.
D. The x and y values appear to be positively correlated in R , but negatively correlated in S.
E. The x and y values appear to be more highly correlated in R than in
36 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
10. Algebra
Incorrect
Partial 3
Partial 2
Partial 1
Correct
Omitted
Off Task
West Virginia
57%
1%
Rounds to Zero
5%
12%
20%
6%
National Pubic
44%
2%
Rounds to Zero
9%
22%
19%
4%
The following chart indicates the maximum number of connecting line segments y that can be
drawn connecting x points, where no three points lie on the same line.
(a)The relationship between x and y is represented by the equation y= kx(x−1) for any positive
number of points x . Use the information in the table to determine the value of the real number k.
(b)Use the equation from part (a) to determine the maximum number of line segments that can be
drawn connecting 100 points, no three of which lie on the same line.
37 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
Scoring Guide
Correct
Both parts correct
Sample Correct Responses:
1
(a) Answer: k = −
2
Solution (not required in response):
y = kx(x−1)
Using the pair (2, 1),
1= k(2)(2−1)
1= 2k
k = −12
(b) Answer: 4,950
Solution (not required in response):
y = −12 ×100 ×(100−1) =4,950
Partial 1
Part (a) correct only
Partial 2
Part (b) correct only
Partial 3
Part (b) correct based on incorrect value of k in part (a)
Incorrect
Incorrect response
*Note: Separate “incorrect” and “partial” categories do not indicate one incorrect or partial
answer is scored different from other incorrect or partial answers. “Popular” partial and incorrect
answers are separated to better inform instructional decisions by educators.
38 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
Correct - Student Response
Exemplar 1
The following chart indicates the maximum number of connecting line segments y that can be
drawn connecting x points, where no three points lie on the same line.
(a) The relationship between x and y is represented by the equation y= kx(x−1) for any positive
number of points x. Use the information in the table to determine the value of the real number k.
Answer: k = ____________________
(b)Use the equation from part (a) to determine the maximum number of line segments that can be
drawn connecting 100 points, no three of which lie on the same line.
Answer: ____________________
Scorer Comments:
This response is correct, with correct answers of 1/2 for part (a) and 4,950 for part (b).
39 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
Exemplar 2
The following chart indicates the maximum number of connecting line segments y that can be
drawn connecting x points, where no three points lie on the same line.
(a) The relationship between x and y is represented by the equation y= kx(x−1) for any positive
number of points x. Use the information in the table to determine the value of the real number k.
Answer: k = ____________________
(b)Use the equation from part (a) to determine the maximum number of line segments that can be
drawn connecting 100 points, no three of which lie on the same line.
Answer: ____________________
Scorer Comments:
This response is correct, with correct answers of .5 for part (a) and 4,950 for part (b).
40 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
Partial 1 - Student Response
Exemplar 1
The following chart indicates the maximum number of connecting line segments y that can be
drawn connecting x points, where no three points lie on the same line.
(a) The relationship between x and y is represented by the equation y= kx(x−1) for any positive
number of points x. Use the information in the table to determine the value of the real number k.
Answer: k = ____________________
(b)Use the equation from part (a) to determine the maximum number of line segments that can be
drawn connecting 100 points, no three of which lie on the same line.
Answer: ____________________
Scorer Comments:
This response is partially correct, with a correct answer of 1/2 for part (a) and an incorrect answer
of 99,000 for part (b).
41 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
Exemplar 2
The following chart indicates the maximum number of connecting line segments y that can be
drawn connecting x points, where no three points lie on the same line.
(a) The relationship between x and y is represented by the equation y= kx(x−1) for any positive
number of points x. Use the information in the table to determine the value of the real number k.
Answer: k = ____________________
(b) Use the equation from part (a) to determine the maximum number of line segments that can
be drawn connecting 100 points, no three of which lie on the same line.
Answer: ____________________
Scorer Comments:
This response is partially correct, with a correct answer of .5 for part (a) and an incorrect answer
of 1200 for part (b).
42 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
Partial 2 - Student Response
Exemplar 1
The following chart indicates the maximum number of connecting line segments y that can be
drawn connecting x points, where no three points lie on the same line.
(a) The relationship between x and y is represented by the equation y= kx(x−1) for any positive
number of points x. Use the information in the table to determine the value of the real number k.
Answer: k = ____________________
(b) Use the equation from part (a) to determine the maximum number of line segments that can
be drawn connecting 100 points, no three of which lie on the same line.
Answer: ____________________
Scorer Comments:
This response is partially correct, with an incorrect answer of y ÷ x = x - 1 for part (a) and a
correct answer of 4950 for part (b).
43 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
Exemplar 2
The following chart indicates the maximum number of connecting line segments y that can be
drawn connecting x points, where no three points lie on the same line.
(a) The relationship between x and y is represented by the equation y= kx(x−1) for any positive
number of points x. Use the information in the table to determine the value of the real number k.
Answer: k = ____________________
(b) Use the equation from part (a) to determine the maximum number of line segments that can
be drawn connecting 100 points, no three of which lie on the same line.
Answer: ____________________
Scorer Comments:
This response is partially correct, with no answer for part (a) and a correct answer of 4950 for
part (b).
44 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
Partial 3 - Student Response
Exemplar 1
The following chart indicates the maximum number of connecting line segments y that can be
drawn connecting x points, where no three points lie on the same line.
(a) The relationship between x and y is represented by the equation y= kx(x−1) for any positive
number of points x. Use the information in the table to determine the value of the real number k.
Answer: k = ____________________
(b)Use the equation from part (a) to determine the maximum number of line segments that can be
drawn connecting 100 points, no three of which lie on the same line.
Answer: ____________________
Scorer Comments:
This response is partially correct, with an incorrect answer of 1 for part (a) and an answer of
9900 for part (b), which is the value of y = kx(x - 1) when k = 1 and x =100.
45 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
Exemplar 2
The following chart indicates the maximum number of connecting line segments y that can be
drawn connecting x points, where no three points lie on the same line.
(a) The relationship between x and y is represented by the equation y= kx(x−1) for any positive
number of points x. Use the information in the table to determine the value of the real number k.
Answer: k = ____________________
(b) Use the equation from part (a) to determine the maximum number of line segments that can
be drawn connecting 100 points, no three of which lie on the same line.
Answer: ____________________
Scorer Comments:
This response is partially correct, with an incorrect answer of 13 for part (a) and an answer of
128700 for part (b), which is the value of y =kx(x - 1) when k = 13 and x =100.
46 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
11. Measurement
A
B*
C
D
E
Omitted
West Virginia
7%
27%
28%
26%
13%
Rounds to Zero
National Pubic
9%
29%
25%
23%
10%
4%
On level ground from a distance of 200 feet, the angle of elevation to the top of a building is 21°,
as shown in the figure above. What is the height h of the building, to the nearest foot?
A. 72
B. 77
C. 187
D. 201
E. 521
47 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
12. Data Analysis and Probability
West Virginia
National Pubic
Incorrect
84%
76%
Correct
3%
12%
Omitted
5%
8%
Off Task
8%
5%
Bob is going on a trip. He will be taking a taxi, a flight, and then a train. Bob chose the following
three companies based on their claims.
• Tom’s Taxi Service claims that it is on time 95 percent of the time.
• Friendly Flyer Airlines claims that it is on time 93 percent of the time.
• Rapid Railways claims that it is on time 98 percent of the time.
Based on the three companies’ claims, what is the approximate probability that all three parts of
Bob’s trip will be on time, assuming that all three probabilities are independent?
Scoring Guide
Correct
Answer of 87%
Sample Correct Responses:
Answer: 87% or 0.87
(Accept answers from 0.86 to 0.87, inclusive.)
Solution (not required in response):
0.95×0.93×0.98 = 0.8658 ≈ 0.87=87%
Incorrect
Incorrect response
Correct - Student Response
Exemplar 1
Bob is going on a trip. He will be taking a taxi, a flight, and then a train. Bob chose the following
three companies based on their claims.
• Tom’s Taxi Service claims that it is on time 95 percent of the time.
• Friendly Flyer Airlines claims that it is on time 93 percent of the time.
• Rapid Railways claims that it is on time 98 percent of the time.
Based on the three companies’ claims, what is the approximate probability that all three parts of
Bob’s trip will be on time, assuming that all three probabilities are independent?
Scorer Comments:
This response is correct, with a correct answer of 87%.
48 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
Exemplar 2
Bob is going on a trip. He will be taking a taxi, a flight, and then a train. Bob chose the following
three companies based on their claims.
• Tom’s Taxi Service claims that it is on time 95 percent of the time.
• Friendly Flyer Airlines claims that it is on time 93 percent of the time.
• Rapid Railways claims that it is on time 98 percent of the time.
Based on the three companies’ claims, what is the approximate probability that all three parts of
Bob’s trip will be on time, assuming that all three probabilities are independent?
Scorer Comments:
This response is correct, with a correct answer of 86.6%.
49 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
13. Geometry
Incorrect
Partial 2
Partial 1
Correct
Omitted
Off Task
West Virginia
74%
10%
6%
8%
Rounds to Zero
2%
National Pubic
58%
20%
6%
14%
Rounds to Zero
1%
Perform the following two transformations on the graph below. After each transformation, draw
the resulting image of segment PQ .
(1) Rotate the segment 90° counterclockwise ( ) about point P . Label the resulting segment with a 1.
(2)Reflect the segment you drew in part (1) across the x -axis. Label the resulting segment with a 2.
50 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
Scoring Guide
Correct
Both transformations performed correctly
Sample Correct Responses:
Notes:
• After performing the first transformation correctly, segment PQ will have coordinates P’(1,2) and Q’(-1,3).
The coordinates are not required in the response.
• After performing the second transformation correctly, segment PQ will have coordinates P”(1,-2) and Q”(-1, -3). The coordinates are not required in the response.
• If the response includes dots in the figures, then the dots must contain the points listed above.
Partial 1
First transformation performed correctly only
Partial 2
Second transformation performed correctly based on incorrect first transformation
Incorrect
Incorrect response
*Note: Separate “incorrect” and “partial” categories do not indicate one incorrect or partial
answer is scored different from other incorrect or partial answers. “Popular” partial and incorrect
answers are separated to better inform instructional decisions by educators.
51 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
Correct - Student Response
Exemplar 1
Perform the following two transformations on the graph below. After each transformation, draw
the resulting image of segment PQ .
(1) Rotate the segment 90° counterclockwise ( ) about point P . Label the resulting segment with a 1.
(2) Reflect the segment you drew in part (1) across the x -axis. Label the resulting segment with a 2.
Scorer Comments:
This response is correct, as both of the required transformations were performed correctly.
52 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
Exemplar 2
Perform the following two transformations on the graph below. After each transformation, draw
the resulting image of segment PQ .
(1) Rotate the segment 90° counterclockwise ( ) about point P . Label the resulting segment with a 1.
(2) Reflect the segment you drew in part (1) across the x -axis. Label the resulting segment with a 2.
Scorer Comments:
This response is correct, as both of the required transformations were performed correctly.
53 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
Partial 1 - Student Response
Exemplar 1
Perform the following two transformations on the graph below. After each transformation, draw
the resulting image of segment PQ .
(1) Rotate the segment 90° counterclockwise ( ) about point P . Label the resulting segment with a 1.
(2) Reflect the segment you drew in part (1) across the x -axis. Label the resulting segment with a 2.
Scorer Comments:
This response is partially correct, as the first transformation was performed correctly but the
second transformation was performed incorrectly.
54 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
Exemplar 2
Perform the following two transformations on the graph below. After each transformation, draw
the resulting image of segment PQ .
(1) Rotate the segment 90° counterclockwise ( ) about point P . Label the resulting segment with a 1.
(2) Reflect the segment you drew in part (1) across the x -axis. Label the resulting segment with a 2.
Scorer Comments:
This response is partially correct, as the first transformation was performed correctly but the
second transformation was performed incorrectly.
55 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
Partial 2 - Student Response
Exemplar 1
Perform the following two transformations on the graph below. After each transformation, draw
the resulting image of segment PQ .
(1) Rotate the segment 90° counterclockwise ( ) about point P . Label the resulting segment with a 1.
(2) Reflect the segment you drew in part (1) across the x -axis. Label the resulting segment with a 2.
Scorer Comments:
This response is partially correct, since the first transformation was performed incorrectly, but
based on this incorrect answer the second transformation was performed correctly.
56 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
NAEP Mathematics - Grade 12
Exemplar 2
Perform the following two transformations on the graph below. After each transformation, draw
the resulting image of segment PQ .
(1) Rotate the segment 90° counterclockwise ( ) about point P . Label the resulting segment with a 1.
(2) Reflect the segment you drew in part (1) across the x -axis. Label the resulting segment with a 2.
Scorer Comments:
This response is partially correct, since the first transformation was performed incorrectly, but
based on this incorrect answer the second transformation was performed correctly.
57 | Page
SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2009 Mathematics Assessment.
Jorea M. Marple, Ed.D.
State Superintendent of Schools