Appendix B: Mathematical Complexity Items Scoring Guide

Low Complexity

How many fourths make a whole?

      Answer: ___________________

Source: 1996 NAEP (grade 4)       Percent correct: 50

Scoring Guide
Scoring Rubric
1 - Correct
4, 4 fourths, etc.
0 - Incorrect
Any incorrect response

Which of the following is NOT a property of every rectangle?

      A. The opposite sides are equal in length.

      B. The opposite sides are parallel.

      C. All angles are equal in measure.

    *D. All sides are equal in length.

      E. The diagonals are equal in length.
Source: 1992 NAEP (grade 12)       Percent correct: 71


What is 2/3 of 15 marbles?
    *A. 10 marbles

      B. 9 marbles

      C. 8 marbles

      D. 5 marbles
Source: 1990 NAEP (grade 12)       Percent correct: 74

(Students did not have access to a calculator.)


+6 + –12 =
    *A. –6

      B. +6

      C. –18

      D. +18
Source: 1990 NAEP (grade 8)        Percent correct: 68


N stands for the number of stamps John had. He gave 12 stamps to his Which expression tells how many stamps John has now?
      A. N + 12

    *B. N – 12

      C. 12 – N

      D. 12 x N
Source: 1996 NAEP (grade 4)       Percent correct: 67


How many hours are equal to 150 minutes?
      A. 11/2

      B. 21/4

      C. 21/3

    *D. 21/2

      E. 25/6
Source: 1990 NAEP (grade 12)       Percent correct: 74


Scale showing a weight of 60 pounds

What is the weight shown on the scale?

      A. 6 pounds

      B. 7 pounds

      C. 51 pounds

    *D. 60 pounds
Source: 1992 NAEP (grade 4)        Percent correct: 44

Moderate Complexity

The following shapes were provided to students. (Shapes were larger than shown.)

squareriangle

(Grade 8 version)

    Bob, Carmen, and Tyler were comparing the areas of N and P. Bob said that N and P have the same area. Carmen said that the area of N is larger. Tyler said that the area of P is larger.

    Who was correct? _______________________________

    Use words or pictures (or both) to explain why.

Source: 1996 NAEP (grade 8)        Percent correct: 27

Scoring Rubric
1 - Correct

An adequate explanation with or without Bob.
May say "neither" or "both."

Parts of P overlap N, and part sticks out. The sticking out part is equal to the left out part of N.

Square is on bottom, triangle is placed on top of square with right angle directly on top of the square's bottom left corner. Two triangles are formed, one within square, one outside of square, areas of these triangles are equal.

OR

Two P’s match two N’s therefore they have the same area. (Therefore, one N has the same area as one P.)

One triangle is placed upside down on top of other triangle so hypotenuses are aligned. Line is drawn to divide the rectangle formed; the resulting 2 squares have the area of the 2 triangles.

OR

Areas are equal because height of P is the same as the height of N, and the base of P is twice the base of N.

area=1/2 x (2X)=X squaredarea=X squared

OR

Either of these two figures alone are acceptable.

Square is on bottom, triangle is placed on top of square with right angle directly on top of the square's bottom left corner. Two triangles are formed, one within square, one outside of square, areas of these triangles are equal.       One triangle is placed upside down on top of other triangle so hypotenuses are aligned. Line is drawn to divide the rectangle formed; the resulting 2 squares have the area of the 2 triangles.

1 - Correct
Correct response (Click here to see).
0 - Incorrect
Bob was correct, but explanation not given or inadequate Any response that answers Carmen or Tyler to "Who was correct?" or omits the name and gives no satisfactory explanation.

Cube made up of 8 smaller cubes

In this figure, how many small cubes were put together to form the large cube?

      A. 7

    *B. 8

      C. 12

      D. 24
Source: 1996 NAEP (grade 4)       Percent correct: 33


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: ___________________________
Source: 1996 NAEP (grade 8)       Percent correct: 54

Scoring Guide
Scoring Rubric
1 - Correct
17
Number of diagonals is always 3 less than the number of sides.
0 - Incorrect
Any incorrect response.

    Tracy said, "I can multiply 6 by another number and get an answer that is smaller than 6." Pat said, "No, you can’t. Multiplying 6 by another number always makes the answer 6 or larger."

      Who is correct? Give a reason for your answer.
Source: 1992 NAEP (grades 8 and 12)

Percent correct: 49 (grade 8); 63 (grade 12)

Scoring Guide

Scoring Rubric
1 - Correct

Tracy, with correct answer given.

      OR

No name stated but reason given is correct.

Examples of correct reasons:

  • If you multiply by a number smaller than 1 the result is less than 6.

  • 6 x 0 = 0

  • 6 x 1/2 = 3

  • 6 x ¨C1 = ¨C6
    • 0 - Incorrect

    Tracy with no reason.

           OR

    Any response that states Pat is correct.

           OR

    No name stated and reason given is incorrect.

    High Complexity

    A pattern of dots is shown below. At each step, more dots are added to the pattern. The number of dots added at each step is more than the number added in the previous step. The pattern continues infinitely.

    First step: one row of 2 dots; second step: two rows pf three dots; third step, three rows of four dots.

    Marcy has to determine the number of dots in the 20th step, but she does not want to draw all 20 pictures and then count the dots. Explain or show how she could do this and give the answer that Marcy should get for the number of dots.

    Source: 1992 NAEP (grade 8)       Percent correct: 6

    Scoring Guide

    Explanation should include one of the following ideas with no false statements.

    1. For each successive step, the number of rows and the number of columns is increasing by 1, forming a pattern. For example, the first step forms 1 by 2 rows and columns, the next step 2 by 3, the third step 3 by 4, and so on. Continuing this pattern would mean that the 20th step has 20 by 21 or 420 dots.

    2. Look at the successive differences between the consecutive steps. The differences 4, 6, 8, 10,…form a pattern. There are 19 differences forming the pattern 4, 6, 8, 10,… 38, 40 and the sum is (9 x 44) + 22 or 418. However, 2 must be added for the first step, yielding a response of 420.

    Scoring Rubric
    4 - Extended
    Correct answer. (Must state 420; must tie step 20 back to beginning of pattern in some specific form of generalization.)
    3- Satisfactory
    Correct explanation of pattern but does not include or omits the correct number of dots.
    2- Partial
    A partial (incomplete) correct explanation, i.e., does not tie together.
    1 - Minimal
    Any attempt to generalize OR to draw all 20 pictures in the pattern (with a clear understanding of the pattern).
    0 - Incorrect
    The work is completely incorrect, irrelevant, or off.

    Jaime knows the following facts about points A, B, and C.

    • Points A, B, and C are on the same line, but might not be in that order.

    • Point C is twice as far from point A as it is from point B.
    Jaime concluded that point C is always between points A and B.
      Is Jaime's conclusion correct?
      ovalYes        ovalNo

      In the space provided, use a diagram to explain your answer.

    Source: 1996 NAEP (grade 8)        Percent correct: 23

    Scoring Guide
    The explanation must include either one of the following diagrams:
      First line with C, B, and A underneath; second line with A, B, and C underneath
    Diagram should include or illustrate the idea that B is halfway between A and C.

    Scoring Rubric
    1 - Correct

    Correct response. Both correct diagrams are given

           OR

    If correct diagram is clearly indicated either by circling and/or the incorrect ones are crossed out.

    0- Incorrect

    Any incorrect reason OR if there is no indication of which diagram is correct (by circling or crossing out incorrect one).

    NOTE: Points need to be indicated by some mark, such as a dot or tick mark. A response in which points are indicated by letters only is an automatic incorrect.

    a square, a triangle with long hypotenuse, and a triangle with short hypotenuse

    In Mr. Bell’s classes, the students voted for their favorite shape for a symbol.

    Here are the results.

       

      Class 1

      Class 2

      Class 3

      Shape N

      9

      14

      11

      Shape P

      1

      9

      17

      Shape Q

      22

      7

      2

    Using the information in the chart, Mr. Bell must select one of the shapes to be the symbol. Which one should he select and why?

      The shape Mr. Bell should select: _________________

      Explain:

    Source: 1996 NAEP (grades 4 and 8)       

    Percent correct: 31 (grade 4); 58 (grade 8)

    Scoring Guide
    Scoring Rubric
    1 - Correct

    N, because more students chose it.

           OR

    N, because it was first choice in one class and second choice in the other classes.

    "Majority" is acceptable (taken to mean most). If student says most classes, do not accept.

    0- Incorrect

    Shape Q chosen, with an explanation that refers to a number of votes

           OR

    Shape N chosen, but explanation not given or is inadequate with incorrect computation.

           OR

    Any other incorrect response.



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    Mathematics Framework for the 2005 National Assessment of Educational Progress