FOCUS ON THE CONTENT AREAS

NCES 2007–460

National Assessment of Educational Progress

U.S. Department of Education

Mathematics 2005

Performance in Puerto Rico
What is The Nation’s Report CardTM?

The Nation’s Report CardTM informs the public about the academic achievement of elementary and secondary 

students in the United States. Report cards communicate the fi ndings of the National Assessment of 

Educational Progress (NAEP), a continuing and nationally representative measure of achievement in various 

subjects over time. The Nation’s Report CardTM compares performance among states, urban districts, public 

and private schools, and student demographic groups.

individual students is protected, and the identities 

of participating schools are not released. 

NAEP is a congressionally mandated project of 

the National Center for Education Statistics within 

the Institute of Education Sciences of the U.S. 

Department of Education. The Commissioner of 

Education Statistics is responsible for carrying out the 

NAEP project. The National Assessment Governing 

Board oversees and sets policy for NAEP.

For over three decades, NAEP assessments 

have been conducted periodically in reading, 

mathematics, science, writing, history, geography, 

and other subjects. By making objective 

information available on student performance 

at the national, state, and local levels, NAEP 

is an integral part of our nation’s evaluation of 

the condition and progress of education. Only 

information related to academic achievement and 

relevant variables is collected. The privacy of 

Contents

 1 Executive Summary

 MARCH 2007

 2 The Mathematics Assessment

 2 Student Performance

 4 Number Properties and Operations

 10 Measurement

 16 Geometry

 22 Data Analysis and Probability

 28 Algebra

 34 Technical Notes and Data Appendix
Performance is summarized across the entire 

NAEP mathematics assessment and for fi ve content 

areas on a 0–500 scale. Average scores for students 

in Puerto Rico at grades 4 and 8 for 2005 are 

presented in fi gures 1 and 2. Averages for public 

school students in the nation (excluding Puerto 

Rico) are shown for comparison. Because the 

scales were developed separately for each content 

area within each grade, direct comparisons should 

not be made between the average scores for the 

content areas. Question-level results presented in 

the report provide specifi c examples of student 

performance within each of the fi ve areas. For 

each sample question, the percentage of correct 

responses is reported for Puerto Rico and the 

nation.

At grade 4, students in Puerto Rico scored lower, 

on average, than students in the nation overall and 

within each content area. There was no signifi cant 

difference between the performance of male and 

female students in Puerto Rico overall. Female 

students in Puerto Rico scored higher than male 

students in geometry, but there was no difference 

between the performance of male and female 

students in the other content areas.

Students in Puerto Rico also scored lower at 

grade 8 than students in the nation overall and 

within each content area. There was no signifi cant 

difference between the performance of male and 

female students in Puerto Rico overall. Female 

students in Puerto Rico scored higher than male 

students in the data analysis and probability 

content area, but there was no difference between 

the performance of male and female students in 

the other content areas.

Executive Summary

In 2005, public school students in Puerto Rico at grades 4 and 8 participated in a Spanish-language version of the 

National Assessment of Educational Progress in mathematics for the second time. Approximately 3,000 students 

in 100 public schools were assessed at each grade. Puerto Rico also participated in NAEP in 2003, but because 

changes were implemented for the 2005 administration, results for the two years cannot be compared.

191

243

188

241

197

236

181

236

175

235

183

237

0 100 125 150 175 200 225 250 275 300 500

Puerto Rico Nation

Average scale score

OVERALL

NUMBER PROPERTIES

AND OPERATIONS

MEASUREMENT

GEOMETRY

DATA ANALYSIS AND

PROBABILITY

ALGEBRA

Figure 1

Average fourth-grade NAEP mathematics scores in 2005, by 

content area

MATHEMATICS 2005 PERFORMANCE IN PUERTO RICO 1

222

281

224

280

225

275

198

274

218

276

218

278 OVERALL

NUMBER PROPERTIES

AND OPERATIONS

MEASUREMENT

GEOMETRY

DATA ANALYSIS AND

PROBABILITY

ALGEBRA

0 100 125 150 175 200 225 250 275 300 500

Puerto Rico Nation

Average scale score

Figure 2

Average eighth-grade NAEP mathematics scores in 2005, by 

content area

SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics, 

National Assessment of Educational Progress (NAEP), 2005 Mathematics Assessment.
2 THE NATION’S REPORT CARD

The main content areas of the NAEP mathematics 

framework for grades 4 and 8 are 

 number properties and operations, 

 measurement,

 geometry,

 data analysis and probability, and

 algebra. 

The NAEP mathematics framework specifi es the 

percentage of questions to be assessed in each content area. The distribution of questions across content 

areas for grades 4 and 8 is shown in table 1.

Although some topics in these fi ve content areas 

have changed across assessment years, the general 

focus of the mathematics assessments has remained 

consistent. In Puerto Rico, topics in the Estándares 

de Excelencia Programa de Matemáticas (Departamento de Educación 2000) are organized around the 

same fi ve content areas as the Mathematics Framework for the 2005 National Assessment of Educational Progress (National Assessment Governing 

Board 2004).

The entire NAEP mathematics assessment consists 

of 10 sections of mathematics questions at each 

grade. Each section includes 14 to 21 questions covering all fi ve content areas. Because the assessment 

covers a breadth of content and includes more questions than any one student could reasonably answer, 

each student takes just a portion of the assessment, 

answering two sections of subject area questions. All 

of the data from the questions that students answer 

are combined to produce an average score for students in Puerto Rico.

Students are asked multiple-choice questions and 

constructed-response questions that require them 

to produce their own answers. Some questions at 

both grade levels incorporate the use of calculators, 

geometric shapes, rulers (at grade 4), or ruler/protractors (at grade 8). Geometric shapes are available 

for selected questions at both grade levels to help 

assess students’ understanding of topics such as area 

and perimeter.

Student Performance

The performance of public school students in 

Puerto Rico on a Spanish-language version of 

the NAEP mathematics assessment at grades 4 

and 8 is documented in three reports. This report 

focuses on their performance in 2005 within each 

of the fi ve mathematics content areas covered by 

the assessment. For comparison purposes, results 

are also shown for public school students in the 

nation (excluding Puerto Rico). A brief Highlights

report presents average scores and achievementlevel percentages for 2003 and 2005. A detailed 

Technical Report that describes the 2003 and 

2005 assessments—including the translation of 

test questions into Spanish, the process of putting 

performance results onto the NAEP scale, and 

plans for future assessments in Puerto Rico—is 

forthcoming. All three reports may be accessed at 

http://nationsreportcard.gov/puertorico_2005/. 

The Mathematics Assessment 

The content of all NAEP assessments is determined by subject-area frameworks that are developed by the 

National Assessment Governing Board in a comprehensive process involving a broad spectrum of interested 

parties, including teachers, curriculum specialists, subject-matter specialists, school administrators, parents, and 

members of the general public. Frameworks in NAEP are developed to survey student understanding over a broad 

range of content. The frameworks are available at http://www.nagb.org/pubs/pubs.html/. 

CONTENT AREA GRADE 4 GRADE 8

Number properties and operations 40 20

Measurement 20 15

Geometry 15 20

Data analysis and probability 10 15

Algebra 15 30

Table 1

Target percentage distribution of NAEP mathematics 

questions in 2005, by grade and content area

SOURCE: U.S. Department of Education, National Assessment Governing Board, Mathematics 

Framework for the 2005 National Assessment of Educational Progress, 2004.
0 125 150 175 200 225 250 275 300 500 100

PUERTO RICO

All students

Male

Female

NATION

All students

Male

Female

 237

236

238

183

184

183

Average scale score

Figure 3

Average fourth-grade NAEP mathematics scores overall 

in 2005, by gender

0 125 150 175 200 225 250 275 300 500 100

PUERTO RICO

All students

Male

Female

NATION

All students

Male

Female

 278

277

278

218

217

220

Average scale score

Figure 4

Average eighth-grade NAEP mathematics scores overall 

in 2005, by gender

SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for 

Education Statistics, National Assessment of Educational Progress (NAEP), 2005 Mathematics 

Assessment.

MATHEMATICS 2005 PERFORMANCE IN PUERTO RICO 3

NAEP results are based on samples of student 

responses, and there is a margin of error associated 

with every result. Any differences that are 

mentioned in the text as “higher” or “lower” are 

statistically signifi cant at the .05 level. Statistical 

signifi cance is not marked in the fi gures and tables 

in the body of this report, but in all cases, the 

average scores and the percentages of correct or 

most complete responses for Puerto Rico were 

lower than those for the nation.

NAEP mathematics results are reported for grades 

4 and 8 on a 0–500 scale. Scores for all students 

and for male and female students at each grade 

are presented in fi gures 3 and 4. At both grades, 

students in Puerto Rico scored lower, on average, 

than public school students in the nation overall. 

At grade 4, the scores for male and female students 

in Puerto Rico were not signifi cantly different, 

while male students scored higher than female 

students in the nation. The same results for Puerto 

Rico and the nation were seen at grade 8.

The sections that follow provide more details 

about each of the fi ve mathematics content areas. 

In addition to average scores, sample questions 

are shown as concrete examples of what students 

in Puerto Rico know and are able to do within 

selected subtopics. The percentages of students 

answering in each response category are presented 

in a table for each question. The row for the correct 

or most complete response is highlighted. 

For a multiple-choice question, the response 

categories are the answer choices for the question. 

For a constructed-response question, the response 

categories are defi ned in the scoring guide for the 

question. 

Results for selected questions from the 2005 NAEP 

mathematics assessment are listed in the appendix. 

In this appendix, average scores for the nation that 

are statistically different from those for Puerto Rico 

are marked with an asterisk. All of the questions 

referenced in this report are publicly available and 

will not be used in future assessments. 
Photo by Norma Curet, Communications Offi ce, Puerto Rico Department of Education

4 THE NATION’S REPORT CARD

Number Properties and Operations

Numbers are our main tools for describing the world quantitatively, so the ability to use number 

properties and operations is an important expectation of the 2005 NAEP mathematics framework. 

This content area focuses on students’ understanding of ways to represent, calculate, and estimate 

numbers. In addition to basic computation skills, NAEP assesses students’ ability to order and 

compare numbers and to solve problems in real-world settings using arithmetic operations.

At grade 4, students are expected to have a solid grasp of whole numbers and a beginning 

understanding of fractions. At grade 8, students are expected to be able to work with rational 

numbers (both fractions and decimals), ratios and proportional reasoning, scientifi c notation, 

and naturally occurring irrational numbers such as square roots and pi (). 

Subtopics in the number properties and operations content area are

Number sense

Estimation

Number operations

Ratios and proportional reasoning

Properties of number and operations
MATHEMATICS 2005 PERFORMANCE IN PUERTO RICO 5

Number Properties and Operations

Overall Results

At grade 4, the average score for students 

in Puerto Rico in the number properties and 

operations content area was 175. For Puerto 

Rico students at grade 8, the average scale score 

in this content area was 218. These scores were 

lower than the average scores for public school 

students in the nation.

In Puerto Rico, at both grades, the average score 

for male students was not signifi cantly different 

from that of female students for the number 

properties and operations content area. For 

students nationally at both grades, the average 

score of male students was higher than that of 

female students in this content area.

The next few pages contain sample questions from 

the number properties and operations content 

area in the 2005 NAEP mathematics assessment. 

These sample questions do not represent the entire 

range of content assessed in this content area and 

are for illustrative purposes only. For each sample 

question, the percentages of students answering 

in each response category are reported for both 

Puerto Rico and the nation. The row for correct or 

most complete response is highlighted.

Figure 6

Average eighth-grade NAEP scores for number 

properties and operations in 2005, by gender

SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for 

Education Statistics, National Assessment of Educational Progress (NAEP), 2005 Mathematics 

Assessment.

0 125 150 175 200 225 250 275 300 500 100

PUERTO RICO

All students

Male

Female

NATION

All students

Male

Female 274

278

276

218

217

218

Average scale score

0 125 150 175 200 225 250 275 300 500 100

PUERTO RICO

All students

Male

Female

NATION

All students

Male

Female

 235

234

237

175

178

173

Average scale score

Figure 5

Average fourth-grade NAEP scores for number 

properties and operations in 2005, by gender
6 THE NATION’S REPORT CARD

Sample question 1 addresses the number sense subtopic, which includes questions about place value, 

ordering and comparing numbers, and using models and representations of numbers. Sample question 2 

addresses the number operations subtopic, which includes questions about computation, the effects of 

operations on numbers, relationships between operations, and application problems involving numbers 

and operations.

Sample Multiple-Choice Question

Sample question 1 asks the student to identify the standard representation of a number given 

in an expanded form. The framework objective measured in this question is 

“Compose or decompose whole quantities by place value.”

In Puerto Rico, 50 percent of the grade 4 

students answered this question correctly. 

The incorrect answer choices for this 

question refl ected place value errors in the 

hundreds, tens, or units place.

Grade 4

Percentage of fourth-grade students in each response 

category

NOTE: Detail may not sum to totals because of rounding.

SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for 

Education Statistics, National Assessment of Educational Progress (NAEP), 2005 Mathematics 

Assessment.

PUERTO RICO NATION

Choice A 12 3

Choice B 50 86

Choice C 15 6

Choice D 22 4

Omit 1 1

Which of these is equal to

8,000 + 800 + 8 ?

A 8,088

 8,808

C 8,880

D 8,888

Sample question 1 
MATHEMATICS 2005 PERFORMANCE IN PUERTO RICO 7

Number Properties and Operations

Sample Short Constructed-Response Question

Sample question 2 is an example of a computation question in a realworld setting. This is a multi-step problem that requires the student to 

fi nd the number of tickets already sold (264) and then subtract this from 

the total number of tickets the club needs to sell (625). The framework 

objective measured in this question is “Solve application problems 

involving numbers and operations.”

This short constructed-response question was scored as either 

“Correct” or “Incorrect.” The correct response for this question 

is 361, and it was given by 6 percent of the grade 4 students in 

Puerto Rico.

A number of incorrect answers that were commonly given 

by students to this question were tracked during scoring. 

The correct and incorrect answers that were captured are 

described below:

Correct 361

Incorrect #1 Incorrect answers other than those 

specifi ed below

Incorrect #2 264, the total number of tickets already 

sold

Incorrect #3 889, the sum of the three numbers in 

the question

Incorrect #4 104 or 441 or 545, all of which result 

from subtracting two of the numbers 

in the question

A club needs to sell 625 

tickets. If it has already 

sold 184 tickets to 

adults and 80 tickets to 

children, how many more 

does it need to sell?

 Answer: ____________

Sample question 2 

NOTE: Detail may not sum to totals because of rounding.

SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for 

Education Statistics, National Assessment of Educational Progress (NAEP), 2005 Mathematics 

Assessment.

PUERTO RICO NATION

Correct 6 43

Incorrect #1 65 38

Incorrect #2 4 4

Incorrect #3 4 3

Incorrect #4 8 11

Omit 12 2

Percentage of fourth-grade students in each response 

category
8 THE NATION’S REPORT CARD

Grade 8

There were 90 employees in a 

company last year. This year the 

number of employees increased by 

10 percent. How many employees 

are in the company this year?

A 9

B 81

C 91

 99

E 100

Sample question 3 

Percentage of eighth-grade students in each response 

category

NOTE: Detail may not sum to totals because of rounding.

SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for 

Education Statistics, National Assessment of Educational Progress (NAEP), 2005 Mathematics 

Assessment.

PUERTO RICO NATION

Choice A 4 10

Choice B 6 5

Choice C 5 6

Choice D 11 35

Choice E 71 43

Omit 3 1

Sample question 3 addresses the ratios and proportional 

reasoning subtopic, which includes questions about ratios, 

proportions, and percents. Sample question 4 addresses the 

number sense subtopic, which includes questions about 

place value, ordering and comparing numbers, and 

translating between different representations of numbers. 

The emphasis of this subtopic is on rational numbers and 

includes scientifi c notation and absolute value.

Sample Multiple-Choice Question

Sample question 3 asks students to solve a 

real-world problem using percent change. 

When determining a percent change, it is 

important to identify the “base” for 

computing the percent, which in this 

case is 90 employees. The framework 

objective measured in this question is 

“Solve problems involving percentages.”

In Puerto Rico, 11 percent of the grade 8 students 

answered this question correctly. Some misconceptions and errors 

represented by the incorrect answer choices to this question are given below:

• Finding 10 percent of 90 (choice A)

• Decreasing 90 by 10 percent (choice B)

• Increasing 90 by 1 percent (90.9) and rounding (choice C)

• Increasing 90 by 10 (choice E)
# The estimate rounds to zero.

NOTE: Detail may not sum to totals because of rounding.

SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for 

Education Statistics, National Assessment of Educational Progress (NAEP), 2005 Mathematics 

Assessment.

PUERTO RICO NATION

Correct 49 88

Incorrect #1 25 4

Incorrect #2 12 4

Incorrect #3 5 2

Incorrect #4 1 #

Omit 8 1

Percentage of eighth-grade students in each response 

category

Sample question 4 

MATHEMATICS 2005 PERFORMANCE IN PUERTO RICO 9

Number Properties and Operations

On the number line above, 

what number would be located 

at point P ?

Answer: __________________

Sample Short Constructed-Response Question

Sample question 4 asks students to interpret a scale for rational numbers 

on a number line. The framework objective measured in this question is 

“Model or describe rational numbers or numerical relationships using 

number lines and diagrams.” 

This short constructed-response question was scored as either “Correct” or 

“Incorrect.” The correct response for this question, 6.0, was given by 

49 percent of the grade 8 students in Puerto Rico. 

The specifi c incorrect responses that were tracked when this question 

was scored represent common student errors in interpreting the 

information displayed on the number line as described below:

Correct 6.0 or 6

Incorrect #1 Incorrect answers other than those 

specifi ed below

Incorrect #2 5.8 or 6.1, which is based on the 

misconception that each tick mark 

represents 0.1 of a unit either to the 

right of 5.6 or to the left of 6.2

Incorrect #3 5.10 (but not 5.1), which represents an 

understanding that each tick mark represents 

0.2 of a unit, but does not correctly convert 

5.6 + 0.4 to 6.0

Incorrect #4 58, 60, 61, or 62, which represent place 

value errors combined with a possible 

misinterpretation of the scale on the 

number line
Photo by Norma Curet, Communications Offi ce, Puerto Rico Department of Education

10 THE NATION’S REPORT CARD

Measurement

Measuring is the process by which numbers are assigned to describe the world quantitatively. The 

2005 NAEP mathematics framework includes measurement attributes such as capacity, weight or 

mass, time, and temperature, as well as the geometric attributes of length, area, and volume. 

At grade 4, customary units such as inch, quart, pound, and hour, and common metric units 

such as centimeter, liter, and gram are emphasized, as well as the geometric attribute of 

length. At grade 8, the emphasis is on the use of square units for measuring area and surface 

area, cubic units for measuring volume, degrees for measuring angles, and rates. More 

emphasis is placed on area and angle measures than on linear measurements.

Subtopics in the measurement content area are

Measuring physical attributes

Systems of measurement
MATHEMATICS 2005 PERFORMANCE IN PUERTO RICO 11

Measurement

Overall Results

At grade 4, the average score in the measurement 

content area for students in Puerto Rico was 181. 

At grade 8, the average score in measurement 

was 198 for students in Puerto Rico. These scores 

were lower, on average, than the scores for public 

school students in the nation.

While the national average score in the 

measurement content area was higher for male 

students than for female students at both grades, 

the average scores in measurement for male 

and female students in Puerto Rico were not 

signifi cantly different. 

The next few pages contain sample questions from 

the measurement content area in the 2005 NAEP 

mathematics assessment. 

0 125 150 175 200 225 250 275 300 500 100

PUERTO RICO

All students

Male

Female

NATION

All students

Male

Female

 236

234

238

181

181

182

Average scale score

Figure 7

Average fourth-grade NAEP scores for measurement in 

2005, by gender

Figure 8

Average eighth-grade NAEP scores for measurement in 

2005, by gender

SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for 

Education Statistics, National Assessment of Educational Progress (NAEP), 2005 Mathematics 

Assessment.

0 125 150 175 200 225 250 275 300 500 100

NATION

All students

Male

Female 272

277

274

Average scale score

PUERTO RICO

All students

Male

Female

 198

 200

 196
12 THE NATION’S REPORT CARD

Replace photo

Sample questions 5 and 6 address the measuring physical attributes subtopic. This subtopic includes 

questions about identifying attributes that can be measured; comparing objects or estimating the size 

of an object with respect to a given attribute such as length, time, or temperature; using appropriate 

measurement instruments; and solving problems involving the perimeter of plane fi gures or the area of 

squares and rectangles.

Sample Multiple-Choice Question

Sample question 5 asks students to identify an attribute that can be 

measured with a meter stick. The framework objective 

measured in this question is “Identify the 

attribute that is appropriate to measure in a 

given situation.”

In Puerto Rico, 36 percent of the 

grade 4 students answered this 

question correctly. 

The incorrect answer choices for this 

question are attributes of a swimming 

pool that could be measured using other 

tools (e.g., a thermometer or scale) or by 

counting. 

Grade 4

Sample question 5 

NOTE: Detail may not sum to totals because of rounding.

SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for 

Education Statistics, National Assessment of Educational Progress (NAEP), 2005 Mathematics 

Assessment.

Percentage of fourth-grade students in each response 

category

PUERTO RICO NATION

Choice A 36 77

Choice B 19 511

Choice C 19 6

Choice D 25 5

Omit 1 1

Which of these could be measured using 

a meter stick? 

 The length of a swimming pool 

B The temperature of the water in a 

swimming pool 

C The weight of the water in a 

swimming pool 

D The number of people in a 

swimming pool 
MATHEMATICS 2005 PERFORMANCE IN PUERTO RICO 13

Measurement

What is the area of the 

shaded figure? 

A 9 square centimeters

 11 square centimeters

C 13 square centimeters

D 14 square centimeters

Sample question 6 

Sample Multiple-Choice Question

Sample question 6 asks students to determine the area of a fi gure 

drawn on a grid. The correct solution requires students to consider 

both the number of squares that are fully shaded and the number 

of squares that are partially (one-half) shaded. The framework 

objective measured in this question is “Estimate the size of an 

object with respect to a given measurement attribute (e.g., length, 

perimeter, or area using a grid).” 

In Puerto Rico, 17 percent of the grade 4 students answered this 

question correctly. Some misconceptions and errors represented by 

the incorrect answer choices in this question are given below:

• Counting only the squares that are fully shaded (choice A)

• Counting the total number of squares that are fully or 

partially shaded (choice C)

• Incorrectly counting the total number of squares that 

are fully or partially shaded (choice D)

NOTE: Detail may not sum to totals because of rounding.

SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for 

Education Statistics, National Assessment of Educational Progress (NAEP), 2005 Mathematics 

Assessment.

Percentage of fourth-grade students in each response 

category

PUERTO RICO NATION

Choice A 25 20

Choice B 17 47

Choice C 46 25

Choice D 9 7

Omit 3 1

1

cm

1 cm
14 THE NATION’S REPORT CARD

Sample questions 7 and 8 address the measuring physical attributes 

subtopic. This subtopic includes questions about comparing objects 

or estimating the size of an object with respect to a 

measurement attribute such as length, angle, 

weight, or mass; using appropriate measurement 

instruments; solving problems involving the 

perimeter or area of plane fi gures; and solving 

problems involving the volume or 

surface area of solids.

Sample Multiple-Choice Question

Sample question 7 asks students to fi nd the 

length of a rectangular region given its perimeter

and width. The framework objective measured in 

this question is “Solve mathematical or real-world 

problems involving perimeter or area of plane fi gures such 

as triangles, rectangles, circles, or composite fi gures.” In 

Puerto Rico, 21 percent of the grade 8 students answered this 

question correctly. Some misconceptions and errors represented by 

the incorrect answer choices in this question are given below:

• Dividing 390 by 75, which is the length of a rectangle that has an area of 390 square feet and 

a width of 75 feet (choice A)

• Dividing 390 by 4, which is the length of a side of a square that has a perimeter of 390 feet (choice B)

• Using a correct strategy with a subtraction error (choice D)

• Subtracting twice the width from the perimeter, which gives an answer that is twice the length 

of the rectangle (choice E)

Grade 8

A rectangular playground has 

a perimeter of 390 feet. The 

width of the playground is 

75 feet. What is its length?

A 5.2 feet

B 97.5 feet

 120 feet

D 130 feet

E 240 feet

Sample question 7

Percentage of eighth-grade students in each response 

category

NOTE: Detail may not sum to totals because of rounding.

SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for 

Education Statistics, National Assessment of Educational Progress (NAEP), 2005 Mathematics 

Assessment.

PUERTO RICO NATION

Choice A 9 19

Choice B 23 11

Choice C 21 39

Choice D 18 9

Choice E 27 19

Omit 2 2
NOTE: Detail may not sum to totals because of rounding.

SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for 

Education Statistics, National Assessment of Educational Progress (NAEP), 2005 Mathematics 

Assessment.

PUERTO RICO NATION

Correct 16 43

Partial 4 13

Incorrect 71 41

Omit 8 2

Percentage of eighth-grade students in each response 

category

Sample question 8 

MATHEMATICS 2005 PERFORMANCE IN PUERTO RICO 15

Measurement

Sample Short Constructed-Response Question

Sample question 8 is a short constructed-response question that 

asks students to draw a geometric fi gure meeting specifi ed criteria. 

Students are provided with a ruler/protractor for this question. 

The framework objective measured in this question is “Select or use 

appropriate measurement instrument to determine or create a given length, 

area, volume, angle, weight, or mass.”

Student responses for this question were rated using the following 

three-level scoring guide:

Correct Response included a picture and frame drawn with 

the correct dimensions to a tolerance of ± 1

8

 inch on 

all measurements. (It was not necessary for the 

student to draw a picture of the sun and a fl ower, 

or anything else, in the inner rectangle in 

order to earn credit.)

Partial Response showed either a rectangular picture 

with correct dimensions and incorrect frame 

dimensions, or a rectangular picture with 

incorrect dimensions but one-inch frame 

drawn correctly.

Incorrect All incorrect responses. 

In Puerto Rico, 16 percent of grade 8 student responses 

were rated “Correct.”

The fi gure above shows a 

picture and its frame.

In the space below, draw a 

rectangular picture 2 inches 

by 3 inches and draw a 

1-inch-wide frame around it.
16 THE NATION’S REPORT CARD

Photo by Norma Curet, Communications Offi ce, Puerto Rico Department of Education

Geometry

School geometry roughly mirrors the historical development of geometry, which began as a 

practical collection of rules for calculating lengths, areas, and volumes of common shapes. This 

expanded over time to include the study of the possible structures of space and ideas of symmetry 

and transformation. 

Students at grade 4 are expected to be familiar with a library of simple fi gures and their 

attributes, both in a plane and in space. At grade 8, students are expected to be familiar with 

the properties of plane fi gures, especially parallel and perpendicular lines, angle relations in 

polygons, cross sections of solids, and the Pythagorean theorem. 

Subtopics in the geometry content area are

Dimension and shape

Transformation of shapes and preservation of properties

Relationships between geometric fi gures

Position and direction

Mathematical reasoning
MATHEMATICS 2005 PERFORMANCE IN PUERTO RICO 17

Geometry

Overall Results

At grade 4, the average score in geometry for 

students in Puerto Rico was 197. At grade 8, the 

average score in geometry for Puerto Rico students 

was 225. These scores were lower, on average, than 

the scores for public school students in the nation. 

In Puerto Rico at grade 4, the average score in 

geometry was higher for female students than 

for male students, while the results for students 

in the nation showed no signifi cant difference 

in the performance of male and female students 

in geometry. At grade 8, the average score in 

geometry for male students was not signifi cantly 

different from that of female students in either 

Puerto Rico or the nation. 

The next few pages contain sample questions 

from the geometry content area in the 2005 NAEP 

mathematics assessment. 

0 125 150 175 200 225 250 275 300 500 100

PUERTO RICO

All students

Male

Female

NATION

All students

Male

Female

 236

236

235

197

195

199

Average scale score

Figure 9

Average fourth-grade NAEP scores for geometry in 2005, 

by gender

Figure 10

Average eighth-grade NAEP scores for geometry in 2005, 

by gender

SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for 

Education Statistics, National Assessment of Educational Progress (NAEP), 2005 Mathematics 

Assessment.

0 125 150 175 200 225 250 275 300 500 100

NATION

All students

Male

Female 275

275

275

Average scale score

PUERTO RICO

All students

Male

Female

 225

 224

 227
18 THE NATION’S REPORT CARD

Grade 4

Sample question 9

NOTE: Detail may not sum to totals because of rounding.

SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for 

Education Statistics, National Assessment of Educational Progress (NAEP), 2005 Mathematics 

Assessment.

Percentage of fourth-grade students in each response 

category

PUERTO RICO NATION

Choice A 32 22

Choice B 16 53

Choice C 11 9

Choice D 41 15

Omit 1 1

A piece of metal in the shape 

of a rectangle was folded as 

shown above. In the fi gure 

on the right, the “?” symbol 

represents what length? 

A 3 inches 

 6 inches 

C 8 inches 

D 11 inches 

14 in

11 in

6 in

3 in

?

Sample question 9 addresses the transformation of shapes and preservation of properties subtopic, which 

includes questions about symmetry, refl ections, translations, and rotations, and how shapes change or remain 

the same under transformations. Sample question 10 addresses the relationships between geometric fi gures 

subtopic, which includes questions about patterns of geometric fi gures, properties of simple and compound 

fi gures, and two-dimensional faces of three-dimensional shapes.

Sample Multiple-Choice Question

Sample question 9 involves visualizing the result of folding a two-dimensional fi gure into a threedimensional form. The framework objective measured in this question is “Recognize which 

attributes (such as shape and area) change or don’t change when plane fi gures are 

cut up and rearranged.”

Sixteen percent of the grade 4 students in Puerto Rico 

answered this question correctly. The incorrect 

answer choices for this question represent 

misconceptions about the preservation of 

length and the relationship between the 

dimensions in the two fi gures. These 

incorrect choices refl ect the following 

misconceptions and errors:

• Selecting other dimensions shown in one 

of the fi gures (choice A and choice D)

• Finding the difference in the lengths of the 

sides in the original rectangle (choice C)
MATHEMATICS 2005 PERFORMANCE IN PUERTO RICO 19

Geometry

Using 5 of your ti

arrange the tiles. Trace the tiles to show each 

les, show 3 other different ways to 

fi gure. 

Show the lines separating the individual squares.

Sample question 10 

Sample Extended Constructed-Response Question

Sample question 10 is an extended constructed-response question that 

asks students to arrange fi ve squares according to specifi ed criteria. 

Students are provided with 10 paper squares to aid them in answering 

this question. The framework objective measured in this question is 

“Analyze or describe patterns of geometric fi gures by increasing 

number of sides, or changing size or orientation (e.g., polygons with 

more and more sides).”

Student responses for this question were rated using the following

fi ve-level scoring guide: 

Extended Responses were those in which students drew three 

correct arrangements with lines shown separating 

the squares. 

Satisfactory Responses were those in which students drew two 

correct arrangements with lines shown separating 

the squares.

Partial Responses were those in which students drew 

one correct arrangement with lines shown 

separating the squares or drew two of the 

fi gures given with a translation (must be 

translations or fl ips of two different fi gures).

Minimal Responses were those in which students 

made an attempt with fi ve squares but 

showed no examples of a different way 

to arrange the squares (e.g., repeated one 

or more of the examples) or drew an 

arrangement with fi ve squares that failed 

to meet criteria, such as .

Incorrect All incorrect responses. 

In Puerto Rico, 2 percent of grade 4 student 

responses were rated “Extended.”

NOTE: Detail may not sum to totals because of rounding.

SOURCE: U.S. Department of Education, Institute of Education Sciences, 

National Center for Education Statistics, National Assessment of Educational 

Progress (NAEP), 2005 Mathematics Assessment.

Percentage of fourth-grade students in each response 

category

PUERTO RICO NATION

Extended 2 11

Satisfactory 3 13

Partial 13 20

Minimal 23 19

Incorrect 24 19

Omit 36 17

It is possible 

5 tiles so that at least one 

to arrange 

side of each tile completely 

shares one side of another tile. 

Here are 3 different ways to do 

this.

Two fi gures are not considered different 

if one fi gure can be turned or fl ipped to 

match the other.

The fi gures below are not examples of 

proper arrangements or new arrangements.

A

B

C

Tiles do not share 

whole sides.

NO

This is the same 

as C turned.

NO
20 THE NATION’S REPORT CARD

Sample question 11 addresses the position and direction subtopic, which includes questions about relative 

positions of points and lines including midpoints, parallel and perpendicular lines, and points of intersection; 

cross sections of solids; and the representation of geometric fi gures in a rectangular coordinate plane. Sample 

question 12 addresses the dimension and shape subtopic, which includes questions about identifying, describing, 

and drawing two- and three-dimensional geometric objects.

Sample Multiple-Choice Question

Sample question 11 asks students to fi nd the midpoint of a line segment in the coordinate plane. The 

x-coordinate of the midpoint is the average of the x-coordinates of the endpoints, and the y-coordinate 

of the midpoint is the average of the y-coordinates of the endpoints. One way to fi nd the average of 

two numbers is to fi nd one-half of the sum of the two numbers. The framework objective measured in 

this question is “Describe relative positions of points and lines using the geometric ideas of midpoint, 

points on common line through a common point, parallelism, or perpendicularity.”

In Puerto Rico, 17 percent of the grade 8 students answered this question correctly. Some 

misconceptions represented by the incorrect answer choices in this question are given below:

• Finding one-half of the x-coordinate of the second point (4) and subtracting 

the x-coordinate of the fi rst point (2), which is 2; similarly, fi nding 

one-half of the y-coordinate of the second point (4 1

2) and subtracting 

the x-coordinate of the fi rst point (1), which is 3 1

2 (choice A)

• Finding one-half of the positive difference of the x-coordinates and 

one-half of the positive difference of the y-coordinates (choice B)

• Finding the average of 1 and 8 for the x-coordinate of the midpoint, 

and fi nding the average of 2 and 9 for the y-coordinate of the 

midpoint (choice D)

• Finding the sum of the x-coordinates and the y-coordinates 

(choice E)

Grade 8

The endpoints of 

a line segment are 

the points with 

coordinates (2, 1) 

and (8, 9). What 

are the coordinates 

of the midpoint of 

this line segment?

A (2, 31

2)

B (3, 4)

 (5, 5)

D (41

2, 51

2)

E (10, 10)

Sample question 11

Percentage of eighth-grade students in each response 

category

NOTE: Detail may not sum to totals because of rounding.

SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for 

Education Statistics, National Assessment of Educational Progress (NAEP), 2005 Mathematics 

Assessment.

PUERTO RICO NATION

Choice A 25 7

Choice B 21 28

Choice C 17 38

Choice D 21 16

Choice E 15 8

Omit 2 3
Sample question 12 

MATHEMATICS 2005 PERFORMANCE IN PUERTO RICO 21

Geometry

Sample Multiple-Choice Question

Sample question 12 asks students to identify which of the given fi gures could be folded into a triangular 

prism. The framework objective measured in this question is “Demonstrate an understanding about the 

two- and three-dimensional shapes in our world through identifying, drawing, modeling, building, or 

taking apart.”

In Puerto Rico, 79 percent of the grade 8 students answered this question correctly. Some 

misconceptions represented by the incorrect answer choices in this question are given below:

• Not taking into account the scale of the fi gure when visualizing how it 

folds; one of the rectangular faces is not wide enough to 

completely meet all of its adjoining edges when folded 

(choice A)

• Not taking into account the total number of faces in the 

prism; in these instances, there are only four, instead 

of fi ve, faces (choice B and choice C)

• Not taking into account the manner in which the faces 

need to be connected to fold up into the prism that is 

shown (choice D) Which of the following can be 

folded to form the prism above?

A

B

C

D

Percentage of eighth-grade students in each response 

category

# The estimate rounds to zero.

NOTE: Detail may not sum to totals because of rounding.

SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for 

Education Statistics, National Assessment of Educational Progress (NAEP), 2005 Mathematics 

Assessment.

PUERTO RICO NATION

Choice A 5 4

Choice B 5 4

Choice C 6 2

Choice D 3 2

Choice E 79 87

Omit 2 #
22 THE NATION’S REPORT CARD

Photo by Norma Curet, Communications Offi ce, Puerto Rico Department of Education

Data Analysis and Probability

Data analysis is the process of collecting, organizing, summarizing, and interpreting data. This is the 

heart of the discipline called statistics. In the context of data analysis, probability can be thought of 

as the study of potential patterns in outcomes that have not yet been observed. 

At grade 4, the data analysis and probability content area focuses on students’ understanding 

of how data are collected and organized, how to read and interpret various representations 

of data, and basic concepts of probability. At grade 8, the emphasis is on students’ ability to 

use a variety of techniques for organizing and summarizing data (including tables, charts, 

and graphs), to analyze statistical claims, and to use more formal terminology related to 

probability and data analysis. 

Subtopics in the data analysis and probability content area are

 Data representation

 Characteristics of data sets

 Experiments and samples

 Probability
MATHEMATICS 2005 PERFORMANCE IN PUERTO RICO 23

Data Analysis and Probability

0 125 150 175 200 225 250 275 300 500 100

PUERTO RICO

All students

Male

Female

NATION

All students

Male

Female

 241

240

241

188

188

188

Average scale score

Figure 11

Average fourth-grade NAEP scores for data analysis and 

probability in 2005, by gender

Figure 12

Average eighth-grade NAEP scores for data analysis and 

probability in 2005, by gender

SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for 

Education Statistics, National Assessment of Educational Progress (NAEP), 2005 Mathematics 

Assessment.

0 125 150 175 200 225 250 275 300 500 100

NATION

All students

Male

Female 280

280

280

Average scale score

PUERTO RICO

All students

Male

Female

 224

 220

 228

Overall Results

At grade 4, the average score in data analysis and 

probability for students in Puerto Rico was 188. For 

Puerto Rico students at grade 8, the average score 

in data analysis and probability was 224. These 

scores were lower, on average, than the scores for 

public school students in the nation. 

While the results for public school students in the 

nation at grade 4 showed a higher average score 

for male students than for female students in data 

analysis and probability, the results for Puerto Rico 

students showed no signifi cant difference between 

the average scores for male and female students 

in this content area. The pattern was different at 

grade 8, where in Puerto Rico, the average score in 

data analysis and probability was higher for female 

students than for male students, but there was no 

signifi cant difference between the scores of male 

and female students in the nation.

The next few pages contain sample questions from 

the data analysis and probability content area in the 

2005 NAEP mathematics assessment. 
24 THE NATION’S REPORT CARD

Sample question 13 addresses the data representation subtopic, which includes questions about data 

presented in pictographs, bar graphs, circle graphs, line graphs, line plots, tables, and tallies. Sample 

question 14 addresses the probability subtopic, which includes questions about informal probabilistic 

thinking and counting or representing the outcomes of given events.

Sample Multiple-Choice Question

Sample question 13 asks students to select an appropriate title for a bar graph. The framework 

objective measured in this question is “Read or interpret a single set of data.”

In Puerto Rico, 32 percent of the grade 4 students answered this 

question correctly. The following incorrect answer 

choices to this question are based on 

misinterpretations of what could be 

represented by each axis of the graph: 

• The units on the scale are not appropriate 

(choice B).

• The number of categories does not match 

the number of bars on the graph (choice C 

and choice D).

Grade 4

Sample question 13 

NOTE: Detail may not sum to totals because of rounding.

SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for 

Education Statistics, National Assessment of Educational Progress (NAEP), 2005 Mathematics 

Assessment.

Percentage of fourth-grade students in each response 

category

PUERTO RICO NATION

Choice A 32 53

Choice B 17 520

Choice C 18 10

Choice D 32 16

Omit 2 1

Jin made the graph above. 

Which of these could be the 

title for the graph? 

 Number of students who 

walked to school on Monday 

through Friday

B Number of dogs in fi ve 

states

C Number of bottles collected 

by three students 

D The number of students in

ten clubs

5

6

7

8

9

10

Number
MATHEMATICS 2005 PERFORMANCE IN PUERTO RICO 25

Data Analysis and Probability

In a gumball machine 

there are 100 red, 75 blue, 

50 green, and 125 yellow 

gumballs. These 350 gumballs 

are mixed up. Sam puts money 

in and one gumball comes out. 

Which color is most likely to 

come out? 

A Red

B Blue

C Green

 Yellow

Sample question 14 

Sample Multiple-Choice Question

Sample question 14 asks students to identify the most likely 

outcome of a random event. The framework objective measured 

in this question is “Use informal probabilistic thinking to describe 

chance events (i.e., likely and unlikely, certain and impossible).”

In Puerto Rico, 57 percent of the grade 4 students answered this 

question correctly. The incorrect answer choices in this question are 

the other colors of the gumballs. 

# The estimate rounds to zero.

NOTE: Detail may not sum to totals because of rounding.

SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for 

Education Statistics, National Assessment of Educational Progress (NAEP), 2005 Mathematics 

Assessment.

Percentage of fourth-grade students in each response 

category

PUERTO RICO NATION

Choice A 17 4

Choice B 11 2

Choice C 15 3

Choice D 57 91

Omit 1 #
26 THE NATION’S REPORT CARD

Sample question 15 addresses the characteristics of data sets subtopic, which includes questions about statistical 

measures that describe data sets, such as the mean, median, mode, range, interquartile range, and standard 

deviation; the effect of outliers; and scatterplots. Sample question 16 addresses the probability subtopic, which 

includes questions about theoretical and experimental probability; simple, compound, independent, and dependent 

events; and sample spaces.

Sample Multiple-Choice Question

Sample question 15 asks students to determine the median 

of a data set. The median of four numbers is found by 

listing the numbers from least to greatest and then 

fi nding the average of the two middle numbers. 

The framework objective measured in this question 

is “Calculate, use, or interpret mean, median, 

mode, or range.”

In Puerto Rico, 15 percent of the grade 8 students 

answered this question correctly. The incorrect 

choices for this question refl ect the following 

misconceptions and errors:

• Listing the numbers from least to greatest and 

then selecting one of the middle numbers 

(choice A and choice D)

• Using a correct strategy with an arithmetic error 

(choice C)

• Finding the average of the two middle numbers in the original list 

(choice E)

Grade 8

The prices of gasoline in a 

certain region are $1.41, 

$1.36, $1

per gallon. What 

.57, and $1.45

is the median 

price per gallon for gasoline in 

this region? 

A $1.41

 $1.43

C $1.44

D $1.45

E $1.47

Sample question 15

Percentage of eighth-grade students in each response 

category

NOTE: Detail may not sum to totals because of rounding.

SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for 

Education Statistics, National Assessment of Educational Progress (NAEP), 2005 Mathematics 

Assessment.

PUERTO RICO NATION

Choice A 23 10

Choice B 15 51

Choice C 19 16

Choice D 25 15

Choice E 16 8

Omit 2 1
NOTE: Detail may not sum to totals because of rounding.

SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for 

Education Statistics, National Assessment of Educational Progress (NAEP), 2005 Mathematics 

Assessment.

PUERTO RICO NATION

Correct 1 17

Partial 10 30

Incorrect 68 48

Omit 21 4

Percentage of eighth-grade students in each response 

category

Sample question 16 

MATHEMATICS 2005 PERFORMANCE IN PUERTO RICO 27

Data Analysis and Probability

Sample Short Constructed-Response Question

Sample question 16 is a short constructed-response question that 

requires students to reason about and explain the probability of a 

particular outcome for a problem in a real-world setting that involves 

dependent events. This question measures a student’s ability to 

analyze a situation that involves the probability of an event.

Student responses for this question were rated using the following 

three-level scoring guide: 

Correct Responses presented a correct answer and explanation. 

For example:

 Bill is incorrect because two of the candies that Bill 

ate were blue, which leaves only 8 blue candies. 

A total of 28 candies remain, which gives a 

probability of 8

28 (or

4

14 or 

2

7 ) or 0.29.

Partial Responses showed some understanding 

of sampling without replacement 

(i.e., that the number of blue 

candies has decreased because 

he already removed two), 

but the explanation was not 

complete enough to earn full credit.

Incorrect All incorrect responses.

One percent of grade 8 student responses 

in Puerto Rico were rated as “correct.”

A package of candies contained only 

10 red candies, 10 blue candies, and 

10 green candies. Bill shook up the 

package, opened it, and started 

taking out one candy at a time and 

eating it. The fi rst 2 candies he 

took out and ate were blue. Bill 

thinks the probability of getting 

a blue candy on his third try is

10

30 or

1

3.

Is Bill correct or incorrect?

Explain your answer. 
28 THE NATION’S REPORT CARD

Algebra

The concepts of algebra provide students with a powerful tool for understanding mathematics. 

A main focus in the study of algebra is representation, for example, using variables, functions, 

and coordinate geometry. Symbolic algebra easily and effi ciently provides methods to represent 

problems and solve equations. 

At grade 4, students are assessed in their understanding of algebraic representation, patterns, 

and rules; graphing points on a line or a grid; and the use of symbols to represent unknown 

quantities. Much of the emphasis of the algebra content area at grade 4 is on recognizing, 

describing, and extending patterns and rules. At grade 8, the emphasis is on students’

understanding of patterns and functions; algebraic representations; algebraic expressions, 

equations, and inequalities; and linearity, including slope. 

Subtopics in the algebra content area are

 Patterns, relations, and functions

 Algebraic representations

 Variables, expressions, and operations

 Equations and inequalities

Photo by Norma Curet, Communications Offi ce, Puerto Rico Department of Education
MATHEMATICS 2005 PERFORMANCE IN PUERTO RICO 29

Algebra

Overall Results

At grade 4, the average score in the algebra 

content area was 191 for students in Puerto Rico. 

At grade 8, the average score in algebra for Puerto 

Rico students was 222. These scores were lower, 

on average, than the scores for public school 

students in the nation. 

The results by gender at grade 4 showed a higher 

average score in algebra for male students than for 

female students in the nation. However, in Puerto 

Rico, there was no signifi cant difference between 

the average scores in algebra for male and female 

students. At grade 8, there were no signifi cant 

differences between male and female students’

performance in algebra in Puerto Rico or in the 

nation. 

The next few pages contain sample questions 

from the algebra content area in the 2005 NAEP 

mathematics assessment. 

0 125 150 175 200 225 250 275 300 500 100

PUERTO RICO

All students

Male

Female

NATION

All students

Male

Female

 243

242

244

191

190

192

Average scale score

Figure 13

Average fourth-grade NAEP scores for algebra in 2005, 

by gender

Figure 14

Average eighth-grade NAEP scores for algebra in 2005, 

by gender

SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for 

Education Statistics, National Assessment of Educational Progress (NAEP), 2005 Mathematics 

Assessment.

0 125 150 175 200 225 250 275 300 500 100

NATION

All students

Male

Female 281

280

281

Average scale score

PUERTO RICO

All students

Male

Female

 222

 220

 224
30 THE NATION’S REPORT CARD

Grade 4

Sample question 17

NOTE: Detail may not sum to totals because of rounding.

SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for 

Education Statistics, National Assessment of Educational Progress (NAEP), 2005 Mathematics 

Assessment.

Percentage of fourth-grade students in each response 

category

PUERTO RICO NATION

Choice A 39 26

Choice B 12 4

Choice C 30 61

Choice D 17 8

Omit 3 1

N stands for the number of 

hours of sleep Ken gets each 

night. Which of the following 

represents the number of hours 

of sleep Ken gets in 1 week? 

A N + 7 

B N – 7 

 N × 7 

D N ÷ 7 

Sample question 17 addresses the variables, expressions, and operations 

subtopic, which includes questions about representing unknown quantities 

with symbols and expressing simple mathematical relationships in number 

sentences. Sample question 18 addresses the patterns, relations, and 

functions subtopic, which includes questions about recognizing, describing, 

and extending patterns and rules.

Sample Multiple-Choice Question

Sample question 17 asks students to identify the 

mathematical expression that represents a situation 

described in words. The framework objective 

measured in this question is “Use letters and 

symbols to represent an unknown quantity in a 

simple mathematical expression.”

In Puerto Rico, 30 percent of the grade 4 students 

answered this question correctly. 

The incorrect answer choices for this question 

represent a misunderstanding of the mathematical 

relationship (multiplication) between the two 

quantities (hours per night and nights per week) 

given in the question. 
MATHEMATICS 2005 PERFORMANCE IN PUERTO RICO 31

Algebra

Which rule describes the pattern 

shown in the table? 

A + 5 =

B + =

+ + 1 =

D + + 2 =

Sample question 18 

Sample Multiple-Choice Question

Sample question 18 requires students to identify a rule, given 

symbolically, that describes a pattern given in the table. 

The framework objective measured in this question is 

“Given a pattern or sequence, construct or explain a rule 

that can generate the terms of the pattern or sequence.”

In Puerto Rico, 15 percent of the grade 4 students answered 

this question correctly. 

The incorrect answer choices represent the following 

misconceptions and errors:

• Finding a rule that describes only the relationship 

in the fi rst row in the table (choice A)

• Recognizing that the number in the fi rst column is 

doubled, but adding an incorrect constant value 

(choice B and choice D)

NOTE: Detail may not sum to totals because of rounding.

SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for 

Education Statistics, National Assessment of Educational Progress (NAEP), 2005 Mathematics 

Assessment.

Percentage of fourth-grade students in each response 

category

PUERTO RICO NATION

Choice A 53 43

Choice B 15 20

Choice C 15 24

Choice D 12 11

Omit 4 2

4

5

6

7

9

11

13

15
32 THE NATION’S REPORT CARD

Sample question 19 addresses the algebraic representations 

subtopic, which includes questions about analyzing, interpreting, 

and translating among different representations (symbolic, 

graphical, tabular, verbal, and pictorial) of a linear relationship; 

representing points in a rectangular coordinate system; and 

recognizing common nonlinear relationships in meaningful 

contexts. Sample question 20 addresses the equations and 

inequalities subtopic, which focuses on questions about linear 

equations and inequalities.

Sample Multiple-Choice Question

Sample question 19 presents a graph in the xy-coordinate 

plane and requires students to estimate the x-intercept 

of the graph. The framework objective measured in 

this question is “Graph or interpret points that are 

represented by ordered pairs of numbers on a 

rectangular coordinate system.”

In Puerto Rico, 22 percent of the grade 8 students 

answered this question correctly. 

Incorrect answer choices refl ect the following 

misconceptions and errors:

• Finding the y-intercept (choice A)

• Misreading the scale or incorrectly estimating the 

x-coordinate (choice B, choice D, and choice E)

x

y

O 1

1

–1

Grade 8

On the curve above, what is the 

best estimate of the value of x

when y = 0?

A –2.0

B 1.1

 1.4

D 1.7

E 1.9

Sample question 19

Percentage of eighth-grade students in each response 

category

NOTE: Detail may not sum to totals because of rounding.

SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for 

Education Statistics, National Assessment of Educational Progress (NAEP), 2005 Mathematics 

Assessment.

PUERTO RICO NATION

Choice A 27 18

Choice B 28 17

Choice C 22 49

Choice D 8 8

Choice E 11 6

Omit 3 1
Sample question 20 

MATHEMATICS 2005 PERFORMANCE IN PUERTO RICO 33

Sample Multiple-Choice Question

Sample question 20 asks students about the 

relationship between two variables when the value 

of one of the variables is changed. The framework 

objective measured in this question is “Interpret ‘=’

as an equivalence between two expressions and use 

this interpretation to solve problems.”

In Puerto Rico, 26 percent of the grade 8 students 

answered this question correctly. The incorrect 

answer choices represent misconceptions 

about the relationship between the 

expressions 4x and 4(x + 2), 

which is equal to 4x + 8. 

Algebra

In the equation y = 4x, if the value 

of x is increased by 2, what is the 

effect on the value of y ?

 It is 8 more than the 

 original amount.

B It is 6 more than the original 

 amount.

C It is 2 more than the original 

 amount.

D It is 16 times the original 

 amount.

E It is 8 times the original 

 amount.

Percentage of eighth-grade students in each response 

category

NOTE: Detail may not sum to totals because of rounding.

SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for 

Education Statistics, National Assessment of Educational Progress (NAEP), 2005 Mathematics 

Assessment.

PUERTO RICO NATION

Choice A 26 33

Choice B 23 9

Choice C 29 42

Choice D 9 4

Choice E 11 10

Omit 2 2
34 THE NATION’S REPORT CARD

Tables A-1 and A-2 list the NAEP 2005 

mathematics questions for grades 4 and 8 that 

were released to the public after the assessment. 

They are organized by content area and increasing 

order of diffi culty for students in Puerto Rico. 

Student results are presented in terms of the 

average score for each question. An average score 

for a question is expressed as a fraction of the 

maximum possible score and ranges from 0.00 to 

1.00. 

The average score provides a way of comparing 

the diffi culty of multiple-choice and constructedresponse questions. For a multiple-choice question 

or a constructed-response question that is scored 

correct or incorrect, the average score is the 

percentage of correct responses expressed as a 

decimal. For a constructed-response question in 

which students could earn partial credit if they 

did not have a completely correct response, the 

average score is computed by adding the percent 

of students receiving full credit to a fraction of the 

percent of the students receiving partial credit. 

An example of computing an average score for 

a constructed-response question can be provided 

for the grade 8 measurement question shown on 

page 15 that asks students to draw one rectangular 

region enclosed by another. Responses to this 

question were scored “Correct,” “Partial,” or 

“Incorrect.” For Puerto Rico, 16 percent of the 

students gave a fully correct answer, and an 

additional 4 percent of the students gave a partial 

answer. The average score for this question was 

computed as: 16 + 1⁄2 (4) = 18 or 0.18 when 

expressed as a decimal. The partial results were 

weighted by 1⁄2 because there were two levels of 

credit (correct and partial) for the question. Partial 

responses to a question with four levels of credit 

would receive weights of 1⁄4 (minimal), 1⁄2 (partial), 

and 3⁄4 (satisfactory). The fractions applied to 

partial responses are derived from the reciprocal 

of the number of credit levels for the question.

The average score for both multiple-choice 

and constructed-response questions takes into 

account those students who answered the 

question incorrectly, as well as those who reached 

the question but did not attempt to answer it. 

However, students who did not reach the question 

are not included in the calculation of an average 

score. (A student is considered as not reaching 

a question when neither that question nor any 

question following it in the section has been 

answered.) 

For purposes of comparison, average scores for 

each question are also presented for public school 

students in the nation (excluding Puerto Rico). 

Signifi cant differences between the average scores 

for a question for the nation and Puerto Rico are 

noted. 

Technical Notes and Data Appendix
Table A-1. Average score on selected fourth-grade NAEP mathematics questions in 2005, by content area 

Number properties and operations

Puerto 

Rico Nation

Fill in missing numbers on a number line 0.51 0.77*

Identify a number given in expanded notation 0.50 0.86*

Solve a multi-step division problem 0.48 0.57*

Given a solution, determine the numbers in the problem 0.41 0.68*

Subtract fractions with common denominators 0.36 0.53*

Subtract two-digit number from three-digit number1 0.36 0.75*

Determine number of pieces from cutting wholes into fi fths 0.34 0.53*

Subtract two-digit numbers to solve a story problem 0.22 0.79*

Identify number sentence matching a situation 0.22 0.33*

Determine the value of a point on a number line1 0.17 0.55*

Identify numbers rounded to whole numbers 1 0.14 0.45*

Determine the greatest even number less than 20 0.07 0.72*

Solve a multi-step story problem1

 0.06 0.43*

Measurement

Approximate fraction of an hour given minutes 0.37 0.49*

Determine which attribute could be measured with a meter stick 0.36 0.77*

Determine the area of shaded region on grid 1 0.17 0.47*

Geometry

Identify which shapes are cylinders 1 0.69 0.87*

Determine how many angles are less than 90 degrees 1 0.18 0.44*

Complete a set of written directions from a map 0.17 0.67*

Arrange tiles in different ways to satisfy given condition 0.16 0.36*

Determine the width of a rectangle after it is folded 0.16 0.53*

Mark a piece of paper to satisfy a given condition1 0.10 0.31*

Data analysis and probability

Puerto 

Rico Nation

Determine the most likely outcome in a story problem 0.57 0.91*

Identify the appropriate title for a graph 0.32 0.53*

Complete a bar graph from a description of data1 0.13 0.46*

Algebra

Identify the missing fi gure in a pattern 1 0.32 0.72*

Represent a situation with an algebraic expression 0.30 0.61*

Determine next number in given pattern 0.23 0.68*

Identify equation to describe pattern given in table 0.15 0.24*

Determine missing numbers in number sentence 0.10 0.44*

Extend a pattern on a grid1 0.08 0.38*

* Signifi cantly different (p < .05) from Puerto Rico. 1

 Question was also administered to students at grade 8.

NOTE: Regular type denotes a constructed-response question. Italic type denotes a multiple-choice question. The average score for a question is expressed as a fraction of the maximum possible score and 

ranges from 0.00 to 1.00.

SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics, National Assessment of Educational Progress (NAEP), 2005 Mathematics Assessment.

MATHEMATICS 2005 PERFORMANCE IN PUERTO RICO 35
Number properties and operations

Puerto 

Rico Nation

Subtract two-digit number from three-digit number1 0.73 0.88*

Determine the value of a point on a number line1 0.49 0.88*

Solve a multi-step story problem1

 0.43 0.76*

Convert a written number to decimal form 0.38 0.67*

Identify numbers rounded to whole numbers 1 0.32 0.85*

Identify piece of information not needed 1 0.26 0.62*

Solve a story problem with multiple operations 0.24 0.58*

Recognize expanded form of number 0.24 0.40*

Determine distance between points from a written 

 description 0.20 0.22

Shade a fraction of a fi gure1 0.16 0.72*

Find the percent of a tip on a restaurant bill 0.15 0.29*

Identify a point on a number line 0.13 0.41*

Solve a story problem involving percent increase 0.11 0.35*

Write mathematical word problem given a scenario 0.02 0.11*

Measurement

Determine the area of shaded region on grid 1 0.34 0.77*

List angle measures from smallest to largest 0.33 0.72*

Draw arrow to represent direction on a fi gure 0.25 0.44*

Find length of a rectangle given perimeter and width 0.21 0.39*

Draw one rectangular region enclosed by another 0.18 0.50*

Calculate temperature increase on a thermometer 0.16 0.69*

Construct a fi gure on a grid 0.06 0.18*

Find area then construct another fi gure with same area 0.02 0.26*

Determine how many boxes of tile are needed 0.02 0.15*

Geometry

Identify which shapes are cylinders 1 0.90 0.93*

Identify which fi gure can be folded to form a prism 0.79 0.87*

Identify plane fi gure that results from opening a tube1

 0.51 0.86*

Draw the refl ection of a fi gure 0.50 0.77*

Shade a grid to form symmetric pattern 0.35 0.61*

Find the measure of an acute angle 0.35 0.47*

Determine how many angles are less than 90 degrees 1 0.31 0.68*

Determine how many colors are needed to paint a cube 1 0.28 0.79*

Mark a piece of paper to satisfy a given condition1 0.23 0.61*

Determine which shape cannot be formed by 2 overlapping

 tiles 0.21 0.53*

Calculate midpoint of a line segment 0.17 0.38*

Use properties of quadrilaterals to solve a problem 0.14 0.24*

Construct a fi gure using tiles 0.08 0.34*

Data analysis and probability

Puerto 

Rico Nation

Read and interpret a pie chart 0.63 0.87*

Complete a graph given a set of data1 0.59 0.90*

Complete a bar graph from a description of data1 0.59 0.79*

Determine median price for a gallon of gasoline 0.15 0.51*

Given probability, determine faces of a cube labeled R 0.14 0.39*

Solve problem involving dependent events 0.06 0.32*

Algebra

Determine the location of a city on a grid 0.66 0.85*

Reason to order quantities 1 0.60 0.85*

Identify the missing fi gure in a pattern 1 0.56 0.89*

Determine the 6th term in a pattern 0.34 0.59*

Extend a pattern on a grid1 0.28 0.63*

Determine effect of increasing the value of one variable 0.26 0.33*

Use graph of two linear equations to solve a problem 0.24 0.44*

Determine an equation given a table of x and y values 0.23 0.53*

Estimate the x-coordinate from the graph of a curve 0.22 0.49*

Determine coordinates to complete a rectangle 0.17 0.59*

Identify the graph of whole numbers less than 5 0.17 0.36*

Extend a pattern involving perfect squares 0.15 0.39*

Identify an equivalent algebraic expression 0.15 0.42*

Reason about pattern on grid using concept of slope 0.02 0.24*

Table A-2. Average score on selected eighth-grade NAEP mathematics questions in 2005, by content area 

* Signifi cantly different (p < .05) from Puerto Rico. 1

 Question was also administered to students at grade 4.

NOTE: Regular type denotes a constructed-response question. Italic type denotes a multiple-choice question. The average score for a question is expressed as a fraction of the maximum possible score and 

ranges from 0.00 to 1.00.

SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics, National Assessment of Educational Progress (NAEP), 2005 Mathematics Assessment.

36 THE NATION’S REPORT CARD
THIS PAGE INTENTIONALLY LEFT BLANK.
Margaret Spellings

Secretary

U.S. Department 

of Education

Grover J. Whitehurst

Director

Institute of 

Education Sciences

Mark Schneider

Commissioner

National Center for 

Education Statistics

Peggy Carr

Associate Commissioner

National Center for 

Education Statistics

www.ed.gov

“ O U R M I S S I O N I S T O E N S U R E E Q U A L A C C E S S T O E D U C A T I O N A N D T O 

PROMOTE EDUCATIONAL EXCELLENCE THROUGHOUT THE NATION.”

THE NATIONAL ASSESSMENT GOVERNING BOARD

In 1988, Congress created the National Assessment Governing Board to set policy for the National 

Assessment of Educational Progress, commonly known as The Nation's Report Card TM. The Board 

is an independent, bipartisan group whose members include governors, state legislators, local and 

state school officials, educators, business representatives, and members of the general public.

Darvin M. Winick, Chair 

President

Winick & Associates

Austin, Texas 

Amanda P. Avallone, Vice Chair 

Assistant Principal and 

Eighth-Grade Teacher

Summit Middle School 

Boulder, Colorado 

Francie Alexander 

Chief Academic Officer, 

Scholastic, Inc. 

Senior Vice President, 

Scholastic Education 

New York, New York 

David J. Alukonis 

Chairman

Hudson School Board

Hudson, New Hampshire 

Barbara Byrd-Bennett 

Executive Superintendent-inResidence

Cleveland State University 

Cleveland, Ohio 

Gregory Cizek 

Professor of Educational 

Measurement 

University of North Carolina

Chapel Hill, North Carolina 

Shirley V. Dickson 

Educational Consultant 

Aliso Viejo, California 

Honorable David P. Driscoll 

Commissioner of Education 

Massachusetts Department 

of Education

Malden, Massachusetts 

John Q. Easton 

Executive Director

Consortium on Chicago 

School Research

University of Chicago

Chicago, Illinois

Alan J. Friedman 

Consultant

Museum Development and Science 

Communications

New York, New York 

David W. Gordon 

County Superintendent of Schools 

Sacramento County Office of 

Education 

Sacramento, California 

Robin C. Hall 

Principal 

Beecher Hills Elementary School

Atlanta, Georgia 

Kathi M. King 

Twelfth-Grade Teacher

Messalonskee High School

Oakland, Maine

Honorable Keith King 

Member

Colorado House of Representatives 

Denver, Colorado

Kim Kozbial-Hess 

Fourth-Grade Teacher

Hawkins Elementary School

Toledo, Ohio

James S. Lanich 

President 

California Business for 

Education Excellence

Sacramento, California 

Honorable Cynthia Nava 

Senator 

New Mexico State Senate 

Las Cruces, New Mexico 

Andrew C. Porter 

Director, Learning Sciences 

Institute 

Peabody College,

 Vanderbilt University 

Nashville, Tennessee 

Luis A. Ramos 

Community Relations Manager

PPL Susquehanna

Berwick, Pennsylvania 

Mary Frances Taymans, SND

Executive Director

Secondary Schools Department

National Catholic Educational 

Association 

Washington, D.C.

Oscar A. Troncoso

Principal

Socorro High School

El Paso, Texas

Honorable Thomas J. Vilsack

Former Governor of Iowa

Des Moines, Iowa

Honorable Michael E. Ward 

Former North Carolina 

Superintendent of 

Public Instruction

Hattiesburg, Mississippi

Eileen L. Weiser 

Former Member, 

State Board of Education 

Michigan Department of 

Education

Lansing, Michigan

Grover J. Whitehurst 

(Ex officio)

Director

Institute of Education Sciences

U.S. Department of Education

Washington, D.C.

Charles E. Smith

Executive Director 

National Assessment 

Governing Board

Washington, D.C.

T H E N A T I O N ’ S 

REPORT CARD

MATHEMATICS 2005

A s s e s s m e n t o f 

S t u d e n t P e r f o r m a n c e 

in Puerto Rico

March 2007

MORE INFORMATION

The report release site is 

http://nationsreportcard.gov.

The NCES web electronic 

catalog is http://nces.ed.gov/

pubsearch.

For ordering information, 

write to

U.S. Department of Education

ED Pubs

P.O. Box 1398

Jessup, MD 20794-1398

or call toll free 1-877-4ED-Pubs

or order online at http://www.

edpubs.org

S U G G E S T E D 

CITATION

Dion, G.S., Haberstroh, J.G.,

Dresher, A.R. (2007).

The Nationís Report Card: 

Mathematics 2005

Performance in Puerto Ricoó

Focus on the Content Areas

(NCES 2007ñ460). U.S. 

Department of Education, 

National Center for Education 

Statistics. Washington, D.C.: 

U.S. Government Printing Office.

CONTENT CONTACT

Emmanuel Sikali

202-502-7419

emmanuel.sikali@ed.gov

U.S. DEPARTMENT OF EDUCATION

The National Assessment of Educational Progress (NAEP) is a congressionally mandated project sponsored by the U.S. Department 

of Education. The National Center for Education Statistics, a department within the Institute of Education Sciences, administers 

NAEP. The Commissioner of Education Statistics is responsible by law for carrying out the NAEP project.