Central to the development of the NAEP assessment in mathematics is the careful selection of test questions or items to measure the content objectives and cognitive areas. The NAEP mathematics assessment consists of multiple-choice, short constructed-response, and extended constructed-response items. Examples of NAEP mathematics items for grades 4, 8, and 12 are provided below. Please refer to the NAEP Web site at www.nces.ed.gov/nationsreportcard for additional items, scoring rubrics, performance data, and sample student responses.

Multiple-Choice Items

               Grade 4

503 – 207 =    A.   206  *B.   296    C.   304    D.   396 [Percent correct: 53%]

N stands for the number of stamps John had. He gave 12 stamps to his sister. Which expression tells how many stamps John has now?    A.   N + 12  *B.   N – 12    C.   12 – N    D.   12 x N [Percent correct: 67%]

In a bag of marbles, 1/2 are red, 1/4 are blue, 1/6 are green, and 1/12 are yellow. If a marble is taken from the bag without looking, it is most likely to be  *A.   red    B.   blue    C.   green    D.   yellow [Percent correct: 25%]

                 Grade 8

A.   10    B.    20    C.    30  *D.   40    E.    50 [Percent correct: 48%]

Which of the following ordered pairs (x, y) is a solution to the equation 2x – 3y = 6?    A.   (6, 3)  *B.   (3, 0)    C.   (3, 2)    D.   (2, 3)    E.    (0, 3) [Percent correct: 41%]

How many hours are equal to 150 minutes?    [Percent correct: 58%] (Note: Fractions appeared with horizontal fraction bars on the exam.)

                 Grade 12

In the figures above, the radius and height of each right circular cylinder are given. If w, x, and y represent the volume of the cylinders, which of the following statements is true? [Percent correct: 30%]     D

                Grade 12

               Grade 12

Short Constructed-Response Items

To provide more reliable and valid opportunities for extrapolating about students’ approaches to problems, recent NAEP assessments have included items that are often referred to as constructed response or open ended. These short-answer items require students to give either a numerical result or the correct name or classification for a group of mathematical objects, draw an example of a given concept, or write a brief explanation for a given result.

Grade 4

Ms. Hernandez formed teams of 8 students each from the 34 students in her class. She formed as many teams as possible, and the students left over were substitutes.

How many students were substitutes?

    Answer: ___________________

[Percent correct: 39%]

 

Scoring Guide

Score and Description    Correct    2    Incorrect    Incorrect response

Grade 4

How many fourths make a whole?

    Answer: ___________________

[Percent correct: 50%]

 

Scoring Guide

Score and Description    Correct    Correct Response    4, or four fourths, or 4 fourths, etc.    Incorrect    Incorrect response

Grade 8

From any vertex of a 4-sided polygon, 1 diagonal can be drawn.
From any vertex of a 5-sided polygon, 2 diagonals can be drawn.
From any vertex of a 6-sided polygon, 3 diagonals can be drawn.
From any vertex of a 7-sided polygon, 4 diagonals can be drawn.
How many diagonals can be drawn from any vertex of a 20-sided polygon?

    Answer: ___________________

[Percent correct: 54%]

 

Scoring Guide

Solution:    17

In this question a student needed to demonstrate an understanding of diagonals of polygons. A diagonal of a polygon is a segment that joins two nonadjacent vertices (a vertex is a common endpoint of two sides of the polygon). To answer the question it was expected that a student would observe that the number of diagonals from any vertex is 3 less than the number of sides. That is, from any vertex of a convex polygon a diagonal can be drawn to any of the other vertices of that polygon except the two adjacent vertices. For a 20- sided polygon, the answer is 20 – 3 = 17.

Grade 8

Hair Color Survey Results

Color of Hair	Percentage
Blond	17
Brown	50
Black	33
Total	100

The table above shows the results of a survey of hair color. On the circle below, make a circle graph to illustrate the data in the table. Label each part of the circle graph with the correct hair color.

[Percent correct: 72%]

Grade 8

Scoring Guide

Score and Description Correct The brown region should be about 1/2 of the circle. The blond region should be about 1/2 the black region. All three regions must be labeled with the correct colors or with the correct percents.   Incorrect   Incorrect answer

Grade 12

In the figure above, ABDG is a parallelogram and CDEF is a rectangle.
If EF = 9 and CG = 10, what is AB to the nearest hundredth?

   Answer: ___________________
   (Students had access to a calculator.)

[Percent correct: 21%]

Scoring Guide

Score and Description Correct   D  Incorrect #1  Incorrect answer other than  Incorrect #2

 

Extended Constructed-Response Items

Extended constructed-response items require students to consider a situation that demands more than a numerical or short verbal response. These items require the student to carefully consider a problem within or across the content strands, understand what is required to “solve” the problem, choose a plan of attack, carry out the attack, and interpret the solution in terms of the original problem. The response mode requires that students provide evidence of their work on some aspects of the problem-solving process and communicate their decisionmaking steps in the context of the problem.

Grade 4

Think carefully about the following question. Write a complete answer. You may use drawings, words, and numbers to explain your answer. Be sure to show all of your work.

There are 20 students in Mr. Pang’s class. On Tuesday, most of the students in the class said they had pockets in the clothes they were wearing.

  D

Which of the graphs most likely shows the number of pockets that each child had?

Explain why you chose that graph.

Explain why you did not choose the other graphs.

Scoring Guide
Solution:

Graph B because it had 20 students and most of the students had pockets.
It could not be graph A because most of the students should have pockets.

It could not be graph C, since there are more than 20 students shown.
OR
It is not likely that there would be the same number of students for each number of pockets.
OR
Most clothes don’t have 10 pockets.

 

Score and Description

Extended The student chooses graph B and gives a good explanation why it should be B and explains why it can’t be A or C. The explanation must deal with both the number of students in the class and the fact that most of them have pockets. These explanations may occur in either response. Satisfactory The student chooses graph B and gives a good explanation (which includes the fact that graph B has 20 students and most of the students have pockets) but does not mention the other graphs. OR The student gives a good explanation why it cannot be A or C but does not give a good explanation of why it is B. (See examples in solution above for explanations.) Partial The student chooses graph B but does not give an adequate, relevant explanation. OR The student chooses B but gives no explanation why; student explains why it’s not C or why it’s not A. Minimal The student chooses graph B with no explanation or a weak, nonrelevant (e.g., because it made sense) explanation. OR The student chooses A or C with an explanation that shows some understanding. Incorrect/Off Task The work is completely incorrect, irrelevant, or off task. OR The student answers A or C with no explanation.

[Extended 3%, Satisfactory 7%, Partial 15%, Minimal 23%]

Grade 8

This question requires you to show your work and explain your reasoning. You may use drawings, words, and numbers in your explanation. Your answer should be clear enough so that another person could read it and understand your thinking. It is important that you show all your work.

Radio station KMAT in Math City is 200 miles from radio station KGEO in Geometry City. Highway 7, a straight road, connects the two cities.

KMAT broadcasts can be received up to 150 miles in all directions from the station and KGEO broadcasts can be received up to 125 miles in all directions. Radio waves travel from each radio station through the air, as represented above.

On the next page, draw a diagram that shows the following:

• Highway 7.

• The location of the two radio stations.

• The part of Highway 7 where both radio stations can be received.

Scoring Guide Solution:

There is a 75-mile part of Highway 7 that is within both broadcast areas. It starts 75 miles outside Math City and ends 150 miles outside Math City.

Score and Description

Extended Correct answer. (75 miles must be stated.) Satisfactory Map with cities or stations and 200 miles labeled (or a clear and correct application of scale) and identifies common broadcast area on Highway 7 but omits length of common area. Partial Map with cities or stations and 200 miles labeled (or some attempt at using a scale): the highway should be shown as straight, and identifies incorrect common broadcast area (e.g. not on Highway 7) or insufficiently identifies an area. (Insufficiently means that there is not enough information labeled to determine the length of the common broadcast area.) Bounds of common area may or may not be labeled. Minimal Map with cities or stations and 200 miles labeled (or some attempt to use a scale). Highway should be shown as straight. There is no indication of how student determined common broadcast area. (It may, for example, be represented as a single point or not at all.) OR Map that uses some but not all of given information with no indication of how common broadcast area was determined. Incorrect/Off Task The work is completely incorrect, irrelevant, or off task.

Note:

A student’s map must include enough information concerning the lengths given in the question to justify that 75 miles is the common broadcast area. If pertinent information concerning lengths is missing, the maximum score that can be awarded is Partial.

[Extended 1%, Satisfactory 4%, Partial 13%, Minimal 22%]

Grade 12

This question requires you to show your work and explain your reasoning. You may use drawings, words, and numbers in your explanation. You answer should be clear enough so that another person could read it and understand your thinking. It is important that you show all your work.

The table below shows the daily attendance at two movie theaters for 5 days and the mean (average) and the median attendance.

 

	Theater A	Theater B
Day 1	100	72
Day 2	87	97
Day 3	90	70
Day 4	10	71
Day 5	91	100
Mean (average)	75.6	82
Median	90	72

(a) Which statistic, the mean or the median, would you use to describe the typical daily
     attendance for the 5 days at Theater A? Justify your answer.

(b) Which statistic, the mean or the median, would you use to describe the typical daily
     attendance for the 5 days at Theater B? Justify your answer.

Scoring Guide
Solution:

Selects and provides appropriate explanation for why the mean is a better measure for the typical attendance for Theater B and the median is the better measure for Theater A.

An explanation for Theater A should include the idea that the attendance on day 4 is much different than the attendance numbers for any other days for Theater A.

An appropriate explanation for Theater B should include the following ideas:

• There are two clusters of data.

• The median is representative of only one of the clusters while the mean is representative of both.

OR

• a justification that conveys the idea that 82 is a better indicator of where the "center" of the 5 data points is located

Scoring Guide

In this question, a student has to look at the data and determine which measure, the median or the mean, would best describe the typical daily attendance at each theater. A student has to have an understanding of the meaning of mean and median in order to provide a correct answer and explanation. For full credit, a student has to answer the median for part a and include an explanation that would include that day 4's attendance is significantly different than the rest of the days and the mean for part b with an explanation that shows an understanding that the mean is a better indicator because all of the attendance numbers for Theater B are clustered. Varying levels of partial credit (satisfactory, partial, and minimal) could be earned depending on how well the student reasons and communicates the correct answer.

 

Score and Description

Extended Indicates the better measure for each theater and gives a complete explanation for each measure. Satisfactory Indicates the better measure for each theater and gives a complete explanation for one measure. Partial Indicates mean for Theater B and median for Theater A with either no explanation or an incomplete explanation. OR The student selects the better measure for one theater and gives an appropriate explanation. Minimal Indicates the mean for Theater B with no explanation or an incomplete explanation. OR The student indicates the median for Theater A with no explanation or an incomplete explanation. Incorrect Incorrect response

[Extended 1%, Satisfactory 3%, Partial 10%, Minimal 28%]

 

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