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
Directions................................................................................................................... 2
Sample Assessment Booklet
Reference Sheet................................................................................................... 3
Assessment Questions......................................................................................... 5
NAEP Mathematics Framework Overview Information......................................... 19
Individual Item Analysis.......................................................................................... 21
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
Directions for students:
This section has 14 questions. Mark your answers 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.
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
1. 360 x 0.3
A. 10.8
B. 108
C. 120
D. 980
E. 1,080
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. What is the value of h in the figure above?
A. 4 3
B. 8 2
C. 8 3
D. 12 2
E. 12 3
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
Questions 3-4 refers to the following scatterplot.
A random sample of graduates from a particular college program reported their ages and incomes
in response to a survey. Each point on the scatterplot above represents the age and income of a
different graduate.
3. Of the following equations, which best fits the data above?
A.
B.
C.
D.
E.
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. Based on the data in the scatterplot, predictions can be made about the income of a 35
year old and the income of a 55 year old. For which age is the prediction more likely to be
accurate?
35 year old
55 year old
Justify your answer.
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. Which of the following expressions is NOT equivalent to (a+b)(x+y)?
A. (a+b)x+(a+b)y
B. a(x+y)+ b(x+y)
C. (b+a)(y+x)
D. ax+by
E. ax+bx+ay+by
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. Which of the following containers has the greatest liquid capacity?
(1 gallon = 4 quarts = 8 pints = 128 ounces)
A. A 64-ounce orange juice container
B. A 16-pint water jug
C. A 5-quart punch bowl
D. A 2-quart cola bottle
E. A 1-gallon milk bottle
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. The table above shows all the ordered pairs (x,y) that define a relation between the variables
x and y . Is y a function of x ?
Yes
No
Give a reason for your answer.
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 principal of a high school would like to determine why there has been a large decline
during the year in the number of students who buy food in the school’s cafeteria. To do this,
25 students from the school will be surveyed. Which method would be the most appropriate
for selecting the 25 students to participate in the survey?
A. Randomly select 25 students from the senior class.
B. Randomly select 25 students from those taking physics.
C. Randomly select 25 students from a list of all students at the school.
D. Randomly select 25 students from a list of students who eat in the cafeteria.
E. Give the survey to the first 25 students to arrive at school in the morning.
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 graph above shows distance versus time for a race between runners A and B. The race is
already in progress, and the graph shows only the portion of the race that occurred after 11
A.M.
The table on the next page lists several characteristics of the graph. Interpret these
characteristics in terms of what happened during this portion of the race. Include times and
distances to support your interpretation. (A sample interpretation of the y-intercepts is given
in the table.)
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
14 | 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.Carlene told Kyle that a rectangular room measured 16 feet by 12 feet, to the nearest foot.
This means that the length could measure between 15.5 feet and 16.5 feet and the width
could measure between 11.5 feet and 12.5 feet.
Kyle performed the following calculations.
Of the following intervals, which is the smallest interval that contains all possible values of
the area of the room?
A. Between 191.5 and 192.5 square feet
B. Between 191 and 193 square feet
C. Between 179 and 206 square feet
D. Between 178 and 207 square feet
E. Between 165 and 221 square feet
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
11.Which of the following expressions is equal to
?
A.
B.
C.
D.
E.
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
12.In the figure above, the vertices of ABCD are A(-4, -4), B (-2, 2), C (8,4) and D (6, -2).
17 | Page
Give a mathematical justification that ABCD is a parallelogram.
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
Amplitude: 2
Period:
13.Which of the following trigonometric functions has the properties given above?
A.
B.
C.
D.
E.
18 | 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
14.What is the solution to the system of equations
= -7
{3x-2y
x+y=11 ?
Answer: x = ____________________ y = ____________________
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
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.
Broad Area of Mathematics
Number Properties and Operations
Measurement
Geometry
Data Analysis, Statistics, and Probability
Algebra
Includes but is not limited to
Computation
Understanding of number concepts
Use of instruments
Application of processes
Concepts of area and volume
Spatial reasoning
Applying geometric properties
Graphical display
Statistics
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
20 | 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.
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
Individual Item Analysis
For each question in the sample block, the cognitive target is given. In addition, the percentage
of students selecting each answer choice for multiple choice or scoring at each score point/
category are indicated for public school students in West Virginia and the nation. The correct
multiple choice answer is indicated with an asterisk. In order for each question to be considered
“omitted,” the student did not answer the question but answered a question or questions after it.
1. Number properties and operations
A
B*
C
D
E
Omitted
West Virginia
13%
59%
10%
9%
8%
1%
National Public
12%
64%
11%
5%
8%
1%
360 x 0.3
A. 0.8
B. 108
C. 120
D. 980
E. 1,080
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
2. Geometry
West Virginia
12%
29%
38%
12%
7%
2%
A
B
C*
D
E
Omitted
National Public
12%
26%
43%
10%
6%
3%
What is the value of h in the figure above?
A. 4 3
B. 8 2
C. 8 3
D. 12 2
E. 12 3
3. Data Analysis and Probability
A
B
C*
D
E
Omitted
West Virginia
6%
12%
42%
20%
19%
1%
National Public
4%
11%
44%
17%
23%
2%
Of the following equations, which best fits the data above?
A. y= -1,000x+15,000
B. y=1,000x
C. y=1,000x+15,000
D. y=10,000x
E. y=10,000x+15,000
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
4. Data Analysis and Probability
West Virginia
28%
31%
Rounds to Zero
40%
1%
Rounds to Zero
Incorrect 2*
Incorrect 1*
Partial
Correct
Omitted
Off Task
National Public
20%
34%
Rounds to Zero
45%
1%
Rounds to Zero
Based on the data in the scatterplot, predictions can be made about the income of a 35
year old and the income of a 55 year old. For which age is the prediction more likely to be
accurate?
35 year old
55 year old
Justify your answer.
Scoring Rubric
Correct
Correct oval filled in with acceptable justification
Sample Correct Responses:
Correct oval: 35 year old
Justification:
The prediction for the 35 year old is more likely to be accurate because the age 35 is
contained within the interval of the data set. The age 55 is outside the interval of the data set,
so any prediction for the income of a 55 year old would be an extrapolation.
Partial
Incorrect oval or neither oval filled in with justification that supports correct oval
Incorrect 1
Correct oval filled in with incorrect or missing justification
Incorrect 2
Other incorrect responses
*Note: Separate “incorrect” categories do not indicate one incorrect answer is scored different
from other incorrect answers. “Popular” incorrect answers are separated to better inform
instructional decisions.n to educators.
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
Correct - Student Response
Exemplar 1
Based on the data in the scatterplot, predictions can be made about the income of a 35 year old
and the income of a 55 year old. For which age is the prediction more likely to be accurate?
35 year old
55 year old
Justify your answer.
Scorer Comments:
This response is correct, as it correctly states that there are data points for age 35 but no data
points for age 55.
Exemplar 2
Based on the data in the scatterplot, predictions can be made about the income of a 35 year old
and the income of a 55 year old. For which age is the prediction more likely to be accurate?
35 year old
55 year old
Justify your answer.
Scorer Comments:
This response is correct, as it correctly addressed the existence of data for 35 year olds.
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
Partial - Student Response
Exemplar 1
Based on the data in the scatterplot, predictions can be made about the income of a 35 year old
and the income of a 55 year old. For which age is the prediction more likely to be accurate?
35 year old
55 year old
Justify your answer.
Scorer Comments:
This response is partially correct, as an acceptable justification was given for the correct answer,
but the incorrect oval was filled.
Exemplar 2
Based on the data in the scatterplot, predictions can be made about the income of a 35 year old
and the income of a 55 year old. For which age is the prediction more likely to be accurate?
35 year old
55 year old
Justify your answer.
Scorer Comments:
This response is partially correct, since it presented an acceptable justification for the correct
answer, but the incorrect oval was filled in.
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
Incorrect 1 - Student Response
Exemplar 1
Based on the data in the scatterplot, predictions can be made about the income of a 35 year old
and the income of a 55 year old. For which age is the prediction more likely to be accurate?
35 year old
55 year old
Justify your answer.
Scorer Comments:
This response is incorrect as the correct oval is filled in but there is no justification provided.
Exemplar 2
Based on the data in the scatterplot, predictions can be made about the income of a 35 year old
and the income of a 55 year old. For which age is the prediction more likely to be accurate?
35 year old
55 year old
Justify your answer.
Scorer Comments:
This response is incorrect as the correct oval is filled in, but the justification provided was not
acceptable.
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
Incorrect 2 - Student Response
Exemplar 1
Based on the data in the scatterplot, predictions can be made about the income of a 35 year old
and the income of a 55 year old. For which age is the prediction more likely to be accurate?
35 year old
55 year old
Justify your answer.
Scorer Comments:
This response is incorrect because the incorrect oval is filled in and the justification given is
unacceptable.
Exemplar 2
Based on the data in the scatterplot, predictions can be made about the income of a 35 year old
and the income of a 55 year old. For which age is the prediction more likely to be accurate?
35 year old
55 year old
Justify your answer.
Scorer Comments:
This response is incorrect because the incorrect oval is filled in and the justification given is
unacceptable.
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
5. Algebra
A
B
C
D*
E
Omitted
West Virginia
12%
9%
8%
54%
17%
1%
National Public
11%
8%
7%
61%
13%
1%
Which of the following expressions is NOT equivalent to (a+b)(x+y)?
A. (a+b)x+(a+b)y
B. a(x+y)+ b(x+y)
C. (b+a)(y+x)
D. ax+by
E. ax+bx+ay+by
6. Measurement
A
B*
C
D
E
Omitted
West Virginia
National Public
4%
56%
27%
1%
12%
Rounds to Zero
3%
62%
21%
1%
12%
1%
Which of the following containers has the greatest liquid capacity?
(1 gallon = 4 quarts = 8 pints = 128 ounces)
A. A 64-ounce orange juice container
B. A 16-pint water jug
C. A 5-quart punch bowl
D. A 2-quart cola bottle
E. A 1-gallon milk bottle
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
7. Algebra
West Virginia
46%
43%
1%
Rounds to Zero
9%
1%
Rounds to Zero
Incorrect 2
Incorrect 1
Partial 2
Partial 1
Correct
Omitted
Off Task
National Public
43%
38%
2%
Rounds to Zero
14%
3%
Rounds to Zero
The table above shows all the ordered pairs (x,y) that define a relation between
the variables x and y . Is y a function of x ?
Yes
No
Give a reason for your answer.
Scoring Rubric
Correct
Correct oval filled in with acceptable reason
Sample Correct Responses:
Correct oval: Yes
Reason: For each x-value (domain) there is only one y-value (range) that is associated with it
Partial 1*
Neither oval filled in with acceptable reason
Partial 2*
Response of y = x 2 - 1
Incorrect 1*
Correct oval filled in with incorrect or no reason
Incorrect 2*
Other incorrect responses
*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.n to educators
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
Correct - Student Response
Exemplar 1
The table above shows all the ordered pairs (x,y) that define a relation between the variables x
and y. Is y a function of x?
Yes
No
Give a reason for your answer.
Scorer Comments:
This response is correct, as the correct oval is filled in and the response correctly states that for
each x value there is a unique y value. In addition, a particular illustration of this is furnished,
although it is not required for a response to be scored “Correct.” This response also correctly
notes that for the given relation x is not a function of y, and provides a correct reason for this
assertion.
Exemplar 2
The table above shows all the ordered pairs (x,y) that define a relation between the variables x
and y. Is y a function of x?
Yes
No
Give a reason for your answer.
Scorer Comments:
This response is correct, as the correct oval is filled in and the response gives a correct, but
minimal reason that the relation represents a function.
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 1 - Student Response
The table above shows all the ordered pairs (x,y) that define a relation between the variables x
and y. Is y a function of x?
Give a reason for your answer.
Partial 2 - Student Response
Exemplar 1
The table above shows all the ordered pairs (x, y) that define a relation between the variables x
and y . Is y a function of x ?
Yes
No
Give a reason for your answer.
Scorer Comments:
This response is partially correct, as the oval was correctly filled in, but the explanation is not
accurate. It states that the table is the function y = x2 - 1. While the ordered pairs given in the
table do satisfy the equation y =x2- 1, this is not the only such function, since, for example,
also satisfies the relationship in the table. The response provides no reason to support that this is
a functional relationship.
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
The table above shows all the ordered pairs (x, y) that define a relation between the variables x
and y. Is y a function of x?
Yes
No
Give a reason for your answer.
Scorer Comments:
This response is partially correct. It states that the table is a particular quadratic function. While
the ordered pairs given in the table do satisfy the given quadratic function, this is not the only
such function, since plotting the given ordered pairs and joining them in various ways produce
infinitely many possible functions. The response provides no reason to support that this is a
functional relationship.
Incorrect 1 - Student Response
Exemplar 1
The table above shows all the ordered pairs (x, y) that define a relation between the variables x
and y. Is y a function of x?
Yes
No
Give a reason for your answer.
Scorer Comments:
This response is incorrect as the correct oval is filled in but the explanation is unacceptable.
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
Exemplar 2
The table above shows all the ordered pairs (x, y) that define a relation between the variables x
and y. Is y a function of x?
Yes
No
Give a reason for your answer.
Scorer Comments:
This response is incorrect as the correct oval is filled in but there is no reason given.
Incorrect 2 - Student Response
Exemplar 1
The table above shows all the ordered pairs (x, y) that define a relation between the variables x
and y. Is y a function of x?
Yes
No
Give a reason for your answer.
Scorer Comments:
This response is incorrect as the incorrect oval is filled in and the reason is unacceptable.
Exemplar 2
The table above shows all the ordered pairs (x, y) that define a relation between the variables x
and y. Is y a function of x?
Yes
No
Give a reason for your answer.
Scorer Comments:
This response is incorrect as the incorrect oval was filled in and no reason is given.
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
8. Data Analysis and Probability
A
B
C*
D
E
Omitted
West Virginia
National Public
7%
1%
69%
20%
3%
Rounds to Zero
6%
2%
59%
30%
3%
1%
The principal of a high school would like to determine why there has been a large decline
during the year in the number of students who buy food in the school’s cafeteria. To do this,
25 students from the school will be surveyed. Which method would be the most appropriate
for selecting the 25 students to participate in the survey?
A. Randomly select 25 students from the senior class.
B. Randomly select 25 students from those taking physics.
C. Randomly select 25 students from a list of all students at the school.
D. Randomly select 25 students from a list of students who eat in the cafeteria.
E. Give the survey to the first 25 students to arrive at school in the morning.
9. Algebra
Composite
West Virginia
34%
16%
21%
11%
Rounds to Zero
15%
2%
Incorrect
Minimal
Partial
Satisfactory
Complete
Omitted
Off Task
National Public
35%
13%
24%
13%
1%
13%
2%
Interpret characteristics of graph in context A
Unsatisfactory/Incorrect
Partial
Complete
Omitted
Off task
35 | Page
West Virginia
41%
25%
1%
29%
5%
National Public
40%
27%
3%
26%
5%
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
Interpret characteristics of graph in context B
Unsatisfactory/Incorrect
Partial
Complete
Omitted
Off task
West Virginia
35%
31%
15%
17%
3%
National Public
32%
33%
17%
16%
2%
Interpret characteristics of graph in context C
Unsatisfactory/Incorrect
Partial
Complete
Omitted
Off task
West Virginia
29%
13%
34%
20%
3%
National Public
27%
15%
35%
20%
3%
The graph above shows distance versus time for a race
between runners A and B. The race is already in progress, and
the graph shows only the portion of the race that occurred
after 11 A.M.
The table on the next page lists several characteristics of
the graph. Interpret these characteristics in terms of what
happened during this portion of the race. Include times
and distances to support your interpretation. (A sample
interpretation of the y-intercepts is given in the table.)
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
Scoring Rubric
There are three components in this response.
Note: Sample responses for partial credit.
Composite Score:
Student response received one of five possible composite scores (Extended, Satisfactory, Partial,
Minimal, or Incorrect) based on the student’s combined performance on Parts A, B, and C of the
item. For example, a student response of Correct for Part A, Correct for Part B, and Incorrect for
Part C received a composite score of Satisfactory.
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
Parts A, B and C: Complete - Student Response
Exemplar 1
The graph above shows distance versus time for a race between runners A and B. The race is
already in progress, and the graph shows only the portion of the race that occurred after 11 A.M.
The table on the next page lists several characteristics of the graph. Interpret these characteristics
in terms of what happened during this portion of the race. Include times and distances to support
your interpretation. (A sample interpretation of the y-intercepts is given in the table.)
Scorer Comments:
The response for “Slopes” is correct, since it correctly interprets the slopes as the runners’
speeds and gives the correct values for their speeds. This response for “Point of intersection” is
correct because it correctly interprets the point of intersection of the lines as the time when both
runners are the same distance from the finish line and provides the correct values for the time and
distance. This response for “x-intercepts” is correct, since it correctly interprets the x-intercept
as the time when the runners finish the race and provides the correct finishing time for both
runners.
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
Exemplar 2
The graph above shows distance versus time for a race between runners A and B. The race is
already in progress, and the graph shows only the portion of the race that occurred after 11 A.M.
The table on the next page lists several characteristics of the graph. Interpret these characteristics
in terms of what happened during this portion of the race. Include times and distances to support
your interpretation. (A sample interpretation of the y-intercepts is given in the table.)
Scorer Comments:
The response for “Slopes” is correct, since it correctly interprets the slopes as the runners’ speeds
and correctly quantifies their speeds in terms of distance per time. This response for “Point of
intersection” is correct because it correctly interprets the point of intersection of the lines as the
time when both runners are at the same distance from the finish line and provides the correct
values for the time and distance. This response for “x-intercepts” is correct, since it correctly
interprets the x-intercept as the time when the finish line is reached, that is, when the distance to
the finish line is 0, and provides a correct time for both runners.
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
Parts A and C: Complete, Part B: Partial - Student Response
The graph above shows distance versus time for a race between runners A and B. The race is
already in progress, and the graph shows only the portion of the race that occurred after 11 A.M.
The table on the next page lists several characteristics of the graph. Interpret these characteristics
in terms of what happened during this portion of the race. Include times and distances to support
your interpretation. (A sample interpretation of the y-intercepts is given in the table.)
Scorer Comments:
The response for “Slopes” is correct, since it correctly interprets the slopes as the runners’
speeds and gives the correct values for their speeds. This response for “Point of intersection”
is partially correct because it correctly interprets the point of intersection of the lines but fails to
provide numerical values. This response for “x-intercepts” is correct, since it correctly interprets
the x-intercept as the time when the finish line is reached and provides the correct finishing time
for both runners.
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
Parts A, B and C: Partial - Student Response
Exemplar 1
The graph above shows distance versus time for a race between runners A and B. The race is
already in progress, and the graph shows only the portion of the race that occurred after 11 A.M.
The table on the next page lists several characteristics of the graph. Interpret these characteristics
in terms of what happened during this portion of the race. Include times and distances to support
your interpretation. (A sample interpretation of the y-intercepts is given in the table.)
Scorer Comments:
The response for “Slopes” is partially correct, as it correctly identifies the slopes as the runners’
speeds but it does not provide numerical values. This response for “Point of intersection” is
partially correct, as it correctly interprets the point of intersection of the graphs, but it does not
provide numerical values. This response for “x-intercepts” is partially correct, as it correctly
interprets the meaning of the x-intercepts but it does not give numerical values.
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 graph above shows distance versus time for a race between runners A and B. The race is
already in progress, and the graph shows only the portion of the race that occurred after 11 A.M.
The table on the next page lists several characteristics of the graph. Interpret these characteristics
in terms of what happened during this portion of the race. Include times and distances to support
your interpretation. (A sample interpretation of the y-intercepts is given in the table.)
Scorer Comments:
The response for “Slopes” is partially correct, as it correctly identifies the slopes as the runners’
speeds and, although it compares the speeds of the two runners, it does not provide numerical
values. This response for “Point of intersection” is partially correct, as it correctly interprets
the point of intersection of the graphs. The response also provides the distance of the runners
from the finish line, but it does not provide the time. This response for “x-intercepts” is partially
correct as it correctly interprets the meaning of the x-intercepts but it does not give numerical
values.
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
Parts A and C: Partial, Part B: Incorrect - Student Response
The graph above shows distance versus time for a race between runners A and B. The race is
already in progress, and the graph shows only the portion of the race that occurred after 11 A.M.
The table on the next page lists several characteristics of the graph. Interpret these characteristics
in terms of what happened during this portion of the race. Include times and distances to support
your interpretation. (A sample interpretation of the y-intercepts is given in the table.)
Scorer Comments:
The response for “Slopes” is partially correct, since it correctly interprets the slopes as
the runners’ speeds but it does not provide numerical values. This response for “Point of
intersection” is incorrect because it gives an incorrect interpretation of the point of intersection
of the lines and does not provide numerical values. This response for “x-intercepts” is partially
correct, since it correctly interprets the x-intercept as the time when the finish line is reached, but
it does not provide the finishing time for the runners.
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
10.Measurement
A
B
C
D*
E
Omitted
West Virginia
National Public
10%
13%
23%
22%
31%
1%
10%
14%
21%
26%
27%
2%
Carlene told Kyle that a rectangular room measured 16 feet by 12 feet, to the nearest foot.
This means that the length could measure between 15.5 feet and 16.5 feet and the width
could measure between 11.5 feet and 12.5 feet.
Kyle performed the following calculations.
Of the following intervals, which is the smallest interval that contains all possible values of
the area of the room?
A. Between 191.5 and 192.5 square feet
B. Between 191 and 193 square feet
C. Between 179 and 206 square feet
D. Between 178 and 207 square feet
E. Between 165 and 221 square feet
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
11.Algebra
A
B
C
D*
E
Omitted
West Virginia
36%
9%
29%
18%
7%
1%
Which of the following expressions is equal t to
National Public
31%
9%
26%
26%
6%
2%
?
A.
B.
C.
D.
E.
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
12.Geometry
Unsatisfactory/Incorrect
Partial
Complete
Omitted
Off task
West Virginia
65%
3%
3%
26%
3%
National Public
63%
5%
8%
21%
3%
In the figure above, the vertices of ABCD are A(-4, -4), B (-2, 2), C (8,4) and D (6, -2).
Give a mathematical justification that ABCD is a parallelogram.
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
Scoring Rubric
Correct
Complete justification
Sample Correct Responses:
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
Justification:
Alternate Justifications:
Four other approaches to show that a quadrilateral is a parallelogram follow.
NOTES:
• To be acceptable, a justification must include a complete statement or numerical demonstration
of the criteria.
• A partially correct justification shows a complete and correct process but has an arithmetic
error OR shows only that one set of lines are parallel or congruent, or that one set of angles are
congruent, etc.
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
Partial
Partially correct justification
Incorrect
Incorrect response
Correct - Student Response
Exemplar 1
In the figure above, the vertices of ABCD are A (-4, -4), B (-2, 2), C (8, 4), and D (6, -2).
Give a mathematical justification that ABCD is a parallelogram.
Scorer Comments:
This response is correct, as it indicates that the slopes of both pairs of opposite sides are equal by
giving the value of the slopes.
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
Exemplar 2
In the figure above, the vertices of ABCD are A (-4, -4), B (-2, 2), C (8, 4), and D (6, -2).
Give a mathematical justification that ABCD is a parallelogram.
Scorer Comments:
This response is correct, as it shows that the lengths of both pairs of opposite sides are equal by
computing the lengths of the sides.
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
Partial - Student Response
Exemplar 1
In the figure above, the vertices of ABCD are A (-4, -4), B (-2, 2), C (8, 4), and D (6, -2).
Give a mathematical justification that ABCD is a parallelogram.
Scorer Comments:
This response is partially correct. The response correctly stated that the equality of the lengths of
both pairs of opposite sides would prove the figure a parallelogram. The stated equality, however,
is not supported by numerical calculations. The response also correctly stated that equality of the
slopes of both pairs of opposite sides would prove that the figure is a parallelogram. However,
one pair of slope computations was incorrect (the slope of segments AB and CD is equal to 3, not
-3).
Exemplar 2
In the figure above, the vertices of ABCD are A (-4, -4), B (-2, 2), C (8, 4), and D (6, -2).
Give a mathematical justification that ABCD is a parallelogram.
Scorer Comments:
This response is partially correct, as the response establishes the equality of the slopes for one
pair of opposite sides, but fails to compute the slopes of the second pair of opposite sides of the
figure.
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
13.Algebra
A
B
C
D
E*
Omitted
West Virginia
21%
15%
28%
24%
11%
2%
National Public
20%
13%
21%
29%
14%
4%
Amplitude: 2
Period: 2�
3
Which of the following trigonometric functions has the properties given above?
A.
B.
C.
D.
E.
14.Algebra
Incorrect
Partial
Correct
Omitted
Off task
West Virginia
64%
6%
27%
Rounds to Zero
4%
What is the solution to the system of equations
National Public
54%
6%
37%
Rounds to Zero
3%
= -7
{3x-2y
x+y=11 ?
Answer: x = ____________________ y = ____________________
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
Scoring Rubric
Correct
Both values correct
Sample Correct Responses:
Answer: x=3, y=8 Solution (not required in response):
Partial
One correct value only
Incorrect
Incorrect response
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
Correct - Student Response
Exemplar 1
What is the solution to the system of equations
= -7
{3x-2y
x+y=11 ?
Scorer Comments:
This response is correct, with answers of x = 3 and y = 8.
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
What is the solution to the system of equations
= -7
{3x-2y
x+y=11 ?
Scorer Comments:
This response is correct, with answers of x = 3 and y = 8.
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 - Student Response
Exemplar 1
What is the solution to the system of equations
= -7
{3x-2y
x+y=11 ?
Scorer Comments:
This response is partially correct, with an incorrect answer of x = -29/5 and a correct answer of
y = 8.
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
What is the solution to the system of equations
= -7
{3x-2y
x+y=11 ?
Scorer Comments:
This response is partially correct, with a correct answer of x = 3 and an incorrect answer of
y = 18/5.
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.
NAEP Mathematics - Grade 12
58 | 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
59 | 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
60 | 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