
 
2. 
The graphs of the two equations shown above intersect at the point (x, y) What is the value of x at the point of intersection?  
A) 
 
B) 
5
 
C) 
7
 
D) 
10
 
E) 
20  

6. 
The graph of ƒ(x) = 2x^{3} _{} is shown above. Which of the following is the graph of ƒ(x) ?  
A)  
B)  
C)  
D)  
E)  
7. 
In the solution of the system of equations above, what is the value of x ?  
A) 
 1  
B) 
2  
C) 
3  
D) 
4  
E) 
5  
8. 
What are all values of x such that x^{2} + 7x + 6 ≥ 0 ?  
A)  6 ≤ x ≤ 1  
B)  6 ≤ x ≤ 1  
C)  1 ≤ x ≤ 6  
D)  x ≤ 6 or x ≥ 1  
E)  x ≤ 1 or x ≥ 6  
 
9. 
The graph of ƒ(x) = sin
x _{} is
shown above. Which of the following is the x coordinate of point P ?  
A) 
 
B) 
 
C) 
 
D) 
 
E) 
 
10. 
The first term in a sequence of numbers is . Each term after the first term is 1 more than twice its previous term. What is the 4th term?  
A)  2  
B)  4  
C)  5  
D)  11  
E)  23  
Did you use the calculator on this question? 
11. 
Rosa is twice as old as
Byron.
Fred is one year older than Byron. If Fred's age is represented by F, which of the following represents the ages of Rosa and Byron, respectively?  
A)  ½ (F  1) and F  1  
B)  ½ (F + 1) and F + 1  
C)  2 (F  1) and F  1  
D)  2 (F + 1) and F + 1  
E)  2F and F  1  
Did you use the calculator on this question? 
 
12. 
Which of the following best illustrates a geometric representation of (x + 2) (x + 4) ?  
A)  
B)  
C)  
D)  
E)  
Did you use the calculator on this question? 

14. 
The number of bacteria present in a laboratory sample after t days can be represented by 500(2^{t}). What is the initial number of bacteria present in this sample?  
A)  250  
B)  500  
C)  750  
D)  1,000  
E)  2,000  
Did you use the calculator on this question? 
15. 
Yvonne has studied the cost
of tickets over time for her favorite sports team. She has created a model
to predict the cost of a ticket in the future. Let C represent the
cost of a ticket in dollars and y represent the number of years in
the future. Her model is as follows.
C = 2.50y + 13 Based on this model, how much will the cost of a ticket increase in two years?  
A)  $5  
B)  $8  
C)  $13  
D)  $18  
E)  $26  
Did you use the calculator on this question? 
 
16. 
What number if placed in each box above would make both equations true?  
A)  0  
B)  1  
C)  2  
D)  3  
E)  4  

20. 
 
A)  0.5 × 10^{2}  
B)  2 × 10^{2}  
C)  2 × 10^{0.6}  
D)  0.5 × 10^{2}  
E)  2 × 10^{2}  

25. 
For what value of x is 8^{12} = 16^{x}?  
A)  3  
B)  4  
C)  8  
D)  9  
E)  12  
Did you use the calculator on this question? 

27. 
If k can be replaced by any number, how many different values can the expression k + 6 have?  
A)  None  
B)  One  
C)  Six  
D)  Seven  
E)  Infinitely many  
Did you use the calculator on this question? 



31. 
3^{3} + 4(85) ÷ 6 =  
A)  6.5  
B)  11  
C)  27.5  
D)  29  
E)  34.16  
Did you use the calculator on this question? 

 
33. 
The figure above shows the graph of y = f (x). Which of the following could be the graph of y = f (x)?  
A)  
B)  
C)  
D)  
E)  

37. 
The length of a rectangle is 3 more than its width. If L represents the length, what is an expression for the width?  
A)  3 ÷ L  
B)  L÷ 3  
C)  L× 3  
D)  L + 3  
E)  L  3  
The following question refers to the following
pattern of dotfigures.  
38. 
If this pattern of dotfigures is continued, how many dots will be in the 100th figure?  
A)  100  
B)  101  
C)  199  
D)  200  
E)  201  

The following question refers to the graph shown
below.  
40. 
What is the value of g(1)?  
A)  2  
B)  4  
C)  5  
D)  6  
E)  8  
The following question refers to the graph shown
below.  
41. 
What is the value of f(g(1))?  
A)  2  
B)  4  
C)  5  
D)  6  
E)  8  
43. 
If d = 110 and a = 20 in the formula d = (2t  1), then t =  
A) 
 
B) 
 
C)  5  
D) 
 
E)  6  
Did you use the calculator on this question? 
44. 
How many integers are there between and ?  
A)  Three  
B)  Four  
C)  Five  
D)  Six  
E)  Seven  
Did you use the calculator on this question? 

 
46. 
The graphs of y = f(x) and y = g(x) for 0 x 10 are shown in the figure above. For how many values of x is the product f (x) g( x) = 0 for 0 x 10?  
A)  Two  
B)  Four  
C)  Five  
D)  Six  
E)  Seven  
Did you use the calculator on this question? 
47. 
Suppose that a_{1} , a_{2} , a_{3} , . . . is the sequence of numbers such that a_{1} = 3, a_{2} = + 1, a_{3} = + 1, and, in general, a_{n+1} = + 1 for all n 1. To the nearest hundredth, the value of a_{5} is  
A)  1.63  
B)  2.62  
C)  2.73  
D)  3.24  
E)  5.73  
Did you use the calculator on this question? 


Scoring Guide 
Solution: ƒ(g(x))= (2x+7)^{2} + (2x+7) = 4x^{2} + 28x + 49 + 2x + 7 Any one of these is a correct response = 4x^{2} + 30x + 56 = 2(2x^{2} + 15x + 28) = (2x + 7)(2x + 8) = 2(x + 4)(2x + 7) 
Score & Description  
 
 

Correct  Student Response 
4 
If ƒ(x) = x^{2} + x and g(x) = 2x + 7, what is an expression for ƒ(g(x)) ?  
Scorer Comments: This response received full credit because the student correctly performed the composition of the two given functions. 

Partial  Student Response 
4 
If ƒ(x) = x^{2} + x and g(x) = 2x + 7, what is an expression for ƒ(g(x)) ?  
Scorer Comments: Even though the student has written “f(g(x)),” the student actually found g(f(x)). This response was given partial credit. 

Incorrect  Student Response 
4 
If ƒ(x) = x^{2} + x and g(x) = 2x + 7, what is an expression for ƒ(g(x)) ?  
Scorer Comments: This response received no credit because the answer given is not the correct composition of the two functions. 

Scoring Guide 
Solution: a. 2, 4, 8, 16 b. 6th term = 65 7th term = 129 c. Difference between nth and (n + 1)st term is 2^{n} Value points: a. 1 point b. 2 points  1 point for correct 6th term (65) 1 point for correct 7th term (129) OR 1 point if arithmetic error made when adding 32 to get 6th term or 64 to get 7th term (work must be shown for both terms) c. 1 point for correct expression 2^{n} or 2^{x}

Score & Description  
 
 
 


Correct  Student Response 
5 
Sequence I, shown above, is an increasing sequence. Each term in the sequence is greater than the previous term.  
Scorer Comments: This response received full credit because the student correctly answered all three parts of the problem. 

Partial  Student Response 
5 
Sequence I, shown above, is an increasing sequence. Each term in the sequence is greater than the previous term.  
Scorer Comments: This response received partial credit because the student correctly answered parts a and b of the problem. 

Minimal  Student Response 
5 
Sequence I, shown above, is an increasing sequence. Each term in the sequence is greater than the previous term.  
Scorer Comments: This response received minimal credit because the student correctly answered part a of the problem only. 

Incorrect  Student Response 
5 
Sequence I, shown above, is an increasing sequence. Each term in the sequence is greater than the previous term.  
Scorer Comments: This response received no credit because all three parts were answered incorrectly. 








Scoring Guide 
Solution: Solution 1  Take 80% of the cost at the beginning of each year. Value of car at the end of year 1: $16,000 (20,000 x .8) Value of car at the end of year 2: $12,800 (16,000 x .8) Value of car at the end of year 3: $10,240 (12,800 x .8) Value of car at the end of year 4: $8,192 (10,240 x .8)
Solution 2  Take 20% of the cost at the beginning of each year and subtract that from the cost at the beginning of each year. 20,000 x .2 = 4,000 20,000  4,000 = $16,000 End of 1st year 16,000 x .2 = 3,200 16,000  3,200 = $12,800 End of 2nd year 12,800 x .2 = 2,560 12,800  2,560 = $10,240 End of 3rd year 10,240 x .2 = 2,048 10,240  2,048 = $8,192 End of 4th year
Solution 3  Use depreciation formula.
Solution 4  Use depreciation formula with base e. 20,000e^{k}·1 = 16,000 k = 0.223 20,000e^{0.223t }= 10,000 t = 3.1062...

Score & Description  
 
 
 


Correct  Student Response 
13 
A car costs $20,000. It decreases in value at the rate of 20 percent each year, based on the value at the beginning of that year. At the end of how many years will the value of the car first be less than half the cost?  
Scorer Comments: This response received full credit because the student specified the correct number of years and has shown work to support the answer. 

Partial  Student Response 
13 
A car costs $20,000. It decreases in value at the rate of 20 percent each year, based on the value at the beginning of that year. At the end of how many years will the value of the car first be less than half the cost?  
Scorer Comments: This response received partial credit because the student gave the correct number of years, but provided incomplete work. 

Minimal  Student Response 
13 
A car costs $20,000. It decreases in value at the rate of 20 percent each year, based on the value at the beginning of that year. At the end of how many years will the value of the car first be less than half the cost?  
Scorer Comments: This response received minimal credit because the student has an incorrect number of years, but followed the correct process. 

Incorrect  Student Response 
13 
A car costs $20,000. It decreases in value at the rate of 20 percent each year, based on the value at the beginning of that year. At the end of how many years will the value of the car first be less than half the cost?  
Scorer Comments: This response received no credit because the answer given is incorrect and there is no work to support the answer. 




Scoring Guide 
Solution: The explanation should indicate there are 442 tiles in the 20th
figure. Counting methods are supported by generalizations (verbal or symbolic) that are based on the students' observations about the pattern. Scoring Guide In this question, a student had to use reasoning skills as well as problem solving skills to describe the number of tiles that would be in the 20th figure. A student needed to recognize the pattern that was given and be able to generalize it in some correct way. In addition, the student had to write a description that could be used to describe any figure in the pattern. Different levels of partial credit (satisfactory, partial, and minimal) were earned by a student depending on his or her reasoning skills and the description of the 20th figure. 
Score & Description  
 
 
 
 

Extended  Student Response 
19 
The first 3 figures in a pattern of tiles are shown below. The pattern of tiles contains 50 figures. 
Satisfactory  Student Response 
19 
The first 3 figures in a pattern of tiles are shown below. The pattern of tiles contains 50 figures. 
19 
The first 3 figures in a pattern of tiles are shown below. The pattern of tiles contains 50 figures. 
Partial  Student Response 
19 
The first 3 figures in a pattern of tiles are shown below. The pattern of tiles contains 50 figures. 
19 
The first 3 figures in a pattern of tiles are shown below. The pattern of tiles contains 50 figures. 
Minimal  Student Response 
19 
The first 3 figures in a pattern of tiles are shown below. The pattern of tiles contains 50 figures. 
19 
The first 3 figures in a pattern of tiles are shown below. The pattern of tiles contains 50 figures. 
Incorrect  Student Response 
19 
The first 3 figures in a pattern of tiles are shown below. The pattern of tiles contains 50 figures. 
19 
The first 3 figures in a pattern of tiles are shown below. The pattern of tiles contains 50 figures. 

Scoring Guide 
Score & Description  
 

Correct  Student Response 
24 
If n × n = 729, what does n equal? 
24 
If n × n = 729, what does n equal? 
Incorrect  Student Response 
24 
If n × n = 729, what does n equal? 
24 
If n × n = 729, what does n equal? 
Scoring Guide 
Score & Description  
 

Correct  Student Response 
26 
If f(x) = 4x^{2}  7x + 5.7, what is the value of f(3.5)? 
26 
If f(x) = 4x^{2}  7x + 5.7, what is the value of f(3.5)? 
Incorrect  Student Response 
26 
If f(x) = 4x^{2}  7x + 5.7, what is the value of f(3.5)? 
26 
If f(x) = 4x^{2}  7x + 5.7, what is the value of f(3.5)? 

Scoring Guide 
Score & Description  
 
 
 

*The use of more than one correct category in this
question enabled NAEP to gather data on different ways students responded
correctly to this question. Any one of these responses that fell into one
of these correct categories earned full credit.
*The use of more than one incorrect category in this question enabled NAEP to gather data on common student errors. Any response that fell into one of the incorrect categories earned no credit. 
Correct #1  Student Response 
28 
In the equation above, what is the value of N, rounded to the nearest tenth? 
28 
In the equation above, what is the value of N, rounded to the nearest tenth? 
Correct #2  Student Response 
28 
In the equation above, what is the value of N, rounded to the nearest tenth? 
28 
In the equation above, what is the value of N, rounded to the nearest tenth? 
Incorrect #2  Student Response 
28 
In the equation above, what is the value of N, rounded to the nearest tenth? 
Incorrect #1  Student Response 
28 
In the equation above, what is the value of N, rounded to the nearest tenth? 
28 
In the equation above, what is the value of N, rounded to the nearest tenth? 
Scoring Guide 
Score & Description  
 
 
 

*The use of more than one correct category in this
question enabled NAEP to gather data on different ways students responded
correctly to this question. Any one of these responses that fell into one
of these correct categories earned full credit.
*The use of more than one incorrect category in this question enabled NAEP to gather data on common student errors. Any response that fell into one of the incorrect categories earned no credit. 
Correct #1  Student Response 
29 
The base of the pyramid shown above is a regular hexagon with side of length 12. If point O is the center of the base and the length of OB is 15, what is the degree measure of angle OAB to the nearest whole number? 
Correct #2  Student Response 
29 
The base of the pyramid shown above is a regular hexagon with side of length 12. If point O is the center of the base and the length of OB is 15, what is the degree measure of angle OAB to the nearest whole number? 
Incorrect #2  Student Response 
29 
The base of the pyramid shown above is a regular hexagon with side of length 12. If point O is the center of the base and the length of OB is 15, what is the degree measure of angle OAB to the nearest whole number? 
Incorrect #1  Student Response 
29 
The base of the pyramid shown above is a regular hexagon with side of length 12. If point O is the center of the base and the length of OB is 15, what is the degree measure of angle OAB to the nearest whole number? 
Scoring Guide 
Solution: Yes, it can be 5%: No, it can not be 6%: Examples: (optional) 
Score & Description  
 
 
 
 

Extended  Student Response 
30 
One plan for a state income tax requires those persons with
income of $10,000 or less to pay no tax and those persons with income
greater than $10,000 to pay a tax of 6 percent only on the part of their
income that exceeds $10,000.
A person's effective tax rate is defined as the percent of total income that is paid in tax. Based on this definition, could any person's effective tax rate be 5 percent? Could it be 6 percent? Explain your answer. Include examples if necessary to justify your conclusions. 
Satisfactory  Student Response 
30 
One plan for a state income tax requires those persons with
income of $10,000 or less to pay no tax and those persons with income
greater than $10,000 to pay a tax of 6 percent only on the part of their
income that exceeds $10,000.
A person's effective tax rate is defined as the percent of total income that is paid in tax. Based on this definition, could any person's effective tax rate be 5 percent? Could it be 6 percent? Explain your answer. Include examples if necessary to justify your conclusions. 
Partial  Student Response 
30 
One plan for a state income tax requires those persons with
income of $10,000 or less to pay no tax and those persons with income
greater than $10,000 to pay a tax of 6 percent only on the part of their
income that exceeds $10,000.
A person's effective tax rate is defined as the percent of total income that is paid in tax. Based on this definition, could any person's effective tax rate be 5 percent? Could it be 6 percent? Explain your answer. Include examples if necessary to justify your conclusions. 
Minimal  Student Response 
30 
One plan for a state income tax requires those persons with
income of $10,000 or less to pay no tax and those persons with income
greater than $10,000 to pay a tax of 6 percent only on the part of their
income that exceeds $10,000.
A person's effective tax rate is defined as the percent of total income that is paid in tax. Based on this definition, could any person's effective tax rate be 5 percent? Could it be 6 percent? Explain your answer. Include examples if necessary to justify your conclusions. 
30 
One plan for a state income tax requires those persons with
income of $10,000 or less to pay no tax and those persons with income
greater than $10,000 to pay a tax of 6 percent only on the part of their
income that exceeds $10,000.
A person's effective tax rate is defined as the percent of total income that is paid in tax. Based on this definition, could any person's effective tax rate be 5 percent? Could it be 6 percent? Explain your answer. Include examples if necessary to justify your conclusions. 
Incorrect/Off Task  Student Response 
30 
One plan for a state income tax requires those persons with
income of $10,000 or less to pay no tax and those persons with income
greater than $10,000 to pay a tax of 6 percent only on the part of their
income that exceeds $10,000.
A person's effective tax rate is defined as the percent of total income that is paid in tax. Based on this definition, could any person's effective tax rate be 5 percent? Could it be 6 percent? Explain your answer. Include examples if necessary to justify your conclusions. 
Scoring Guide 
Solution: (10n + 5)^{2} = 100n^{2} + 100n + 25 = 100(n^{2} + n) + 25 Since n is an integer, n^{2} is an integer; n^{2} + n is also an integer and 100(n^{2} + n) is a multiple of 100 that ends in 00. When 25 is added to that integer, the sum ends in 25. Note: If (10n + 5) is squared incorrectly, the other work must be checked. The student can earn a Minimal of Partial depending on the correctness of the work. 
Score & Description  
 
 
 
 

Extended  Student Response 
32 
The examples above suggest the following statement.
Explain why this statement is always true.  
Satisfactory  Student Response 
32 
The examples above suggest the following statement.
Explain why this statement is always true.  
32 
The examples above suggest the following statement.
Explain why this statement is always true.  
Partial  Student Response 
32 
The examples above suggest the following statement.
Explain why this statement is always true.  
Minimal  Student Response 
32 
The examples above suggest the following statement.
Explain why this statement is always true.  
Incorrect/Off Task  Student Response 
32 
The examples above suggest the following statement.
Explain why this statement is always true.  

Scoring Guide 
Solution: a) and b) c) d) Between 3 and 4 seconds. 
Score & Description  
 
 
 
 

Extended  Student Response 
35 
The darkened segments in the figure above show the path of
an object that starts at point A and moves to point C at
a constant rate of 1 unit per second. The object's distance from point
A (or from point C) is the shortest distance
between the object and the point.
In the space below, complete the following steps. a) Sketch the graph of the distance of the object from point A over the 7second period. b) Then sketch the graph of the distance of the object from point C over the same period. 
Satisfactory  Student Response 
35 
The darkened segments in the figure above show the path of
an object that starts at point A and moves to point C at
a constant rate of 1 unit per second. The object's distance from point
A (or from point C) is the shortest distance
between the object and the point.
In the space below, complete the following steps. a) Sketch the graph of the distance of the object from point A over the 7second period. b) Then sketch the graph of the distance of the object from point C over the same period. 
35 
The darkened segments in the figure above show the path of
an object that starts at point A and moves to point C at
a constant rate of 1 unit per second. The object's distance from point
A (or from point C) is the shortest distance
between the object and the point.
In the space below, complete the following steps. a) Sketch the graph of the distance of the object from point A over the 7second period. b) Then sketch the graph of the distance of the object from point C over the same period. 
Partial  Student Response 
35 
The darkened segments in the figure above show the path of
an object that starts at point A and moves to point C at
a constant rate of 1 unit per second. The object's distance from point
A (or from point C) is the shortest distance
between the object and the point.
In the space below, complete the following steps. a) Sketch the graph of the distance of the object from point A over the 7second period. b) Then sketch the graph of the distance of the object from point C over the same period. 
35 
The darkened segments in the figure above show the path of
an object that starts at point A and moves to point C at
a constant rate of 1 unit per second. The object's distance from point
A (or from point C) is the shortest distance
between the object and the point.
In the space below, complete the following steps. a) Sketch the graph of the distance of the object from point A over the 7second period. b) Then sketch the graph of the distance of the object from point C over the same period. 
Minimal  Student Response 
35 
The darkened segments in the figure above show the path of
an object that starts at point A and moves to point C at
a constant rate of 1 unit per second. The object's distance from point
A (or from point C) is the shortest distance
between the object and the point.
In the space below, complete the following steps. a) Sketch the graph of the distance of the object from point A over the 7second period. b) Then sketch the graph of the distance of the object from point C over the same period. 
Incorrect/Off Task  Student Response 
35 
The darkened segments in the figure above show the path of
an object that starts at point A and moves to point C at
a constant rate of 1 unit per second. The object's distance from point
A (or from point C) is the shortest distance
between the object and the point.
In the space below, complete the following steps. a) Sketch the graph of the distance of the object from point A over the 7second period. b) Then sketch the graph of the distance of the object from point C over the same period. 


Scoring Guide 
Score & Description  
 

*The use of more than one incorrect category in this question enabled NAEP to gather data on common student errors. Any response that fell into one of the incorrect categories earned no credit. 
Correct  Student Response 
39 
Explain how you found your answer to question 16. 
Incorrect #1 & 2  Student Response 
39 
Explain how you found your answer to question 16. 
39 
Explain how you found your answer to question 16. 
Scoring Guide 
Score & Description  
 

Correct  Student Response 
42 
What are the roots of 2x^{2} + 5x + 1 = 0? 
42 
What are the roots of 2x^{2} + 5x + 1 = 0? 
Incorrect  Student Response 
42 
What are the roots of 2x^{2} + 5x + 1 = 0? 
42 
What are the roots of 2x^{2} + 5x + 1 = 0? 

Scoring Guide 
Score & Description  
 
 
 

*The use of more than one incorrect category in this question enabled NAEP to gather data on common student errors. Any response that fell into one of the incorrect categories earned no credit. 
Correct  Student Response 
45 
In right triangle ABC shown above, AB = 14. What is BC to the nearest hundredth? 
Incorrect #1  Student Response 
45 
In right triangle ABC shown above, AB = 14. What is BC to the nearest hundredth? 
Incorrect #2  Student Response 
45 
In right triangle ABC shown above, AB = 14. What is BC to the nearest hundredth? 
Incorrect #3  Student Response 
45 
In right triangle ABC shown above, AB = 14. What is BC to the nearest hundredth? 

