Test Information Guide
Field 47: Mathematics (Middle School)
Sample OpenResponse Item
The following materials contain:
 Sample test directions for the openresponse item
 A sample openresponse item
 An example of a strong response to the openresponse item
 The scoring rubric
Sample Test Directions for OpenResponse Items
This section of the test consists of two openresponse item assignments. You will be asked to prepare a written response of approximately 150–300 words, or 1–2 pages, for each assignment.
Read the assignments carefully before you begin your responses. Think about how you will organize your responses. You may use the erasable sheet(s) to make notes, write an outline, or otherwise prepare your responses. However, your final response to each assignment must be either:
 typed into the onscreen response box,
 written on a response sheet and scanned using the scanner provided at your workstation, or
 provided using both the onscreen response box (for typed text) and a response sheet (for calculations or drawings) that you will scan using the scanner provided at your workstation.
Instructions for scanning your response sheet(s) are available by clicking the "Scanning Help" button at the top of the screen.
As a whole, your response to each assignment must demonstrate an understanding of the knowledge of the field. In your response to each assignment, you are expected to demonstrate the depth of your understanding of the subject area by applying your knowledge rather than by merely reciting factual information.
Your responses to the assignments will be evaluated based on the following criteria.
 PURPOSE: the extent to which the response achieves the purpose of the assignment
 SUBJECT KNOWLEDGE: appropriateness and accuracy in the application of subject knowledge
 SUPPORT: quality and relevance of supporting evidence
 RATIONALE: soundness of argument and degree of understanding of the subject area
The openresponse item assignments are intended to assess subject knowledge. Your responses must be communicated clearly enough to permit valid judgment of the evaluation criteria by scorers. Your responses should be written for an audience of educators in this field. The final version of each response should conform to the conventions of edited American English. Your responses should be your original work, written in your own words, and not copied or paraphrased from some other work.
Be sure to write about the assigned topics. Remember to review your work and make any changes you think will improve your responses.
Any time spent responding to an assignment, including scanning the response sheet(s), is part of your testing time. Monitor your time carefully. When your testing time expires, a popup message will appear onscreen indicating the conclusion of your test session. Only response sheets that are scanned before you end your test or before time has expired will be scored. Any response sheet that is not scanned before testing ends will NOT be scored.
Sample OpenResponse Item
Objective 0021
Prepare an organized, developed analysis on a topic related to one or more of the
following: number sense and operations; patterns, relations, and algebra; geometry
and measurement; data analysis, statistics, and probability; and trigonometry, calculus,
and discrete mathematics.
Use the information below to complete the exercise that follows.
A company is considering two bonusplan options for its employees for the next 20 years. The two options are explained in the following chart.
Option 1: Receive $2 the first year. Every year thereafter receive twice the bonus amount of the previous year.
Option 2: Receive $200 the first year. Every year thereafter receive $200 more than the bonus amount of the previous year.
Use your knowledge of exponential and linear functions to develop a response in which you analyze the bonus received each year during a 20year period under each option. In your response:
 create a data table representing the bonus received each year over a 12year period for each option;
 graph the data from both tables on the same coordinate grid and connect the data with the line or curve that best fits the data;
 compare the bonus plans over the 12year period, including a discussion of the significance of the point of intersection of the two graphs;
 explain what type of function, exponential or linear, models each option;
 find equations that describe each option; and
 identify an expression that represents the difference between the bonuses received under the two options in the twentieth year.
Be sure to show your work and explain the reasoning you use in analyzing and solving this problem.
Sample Strong Response to the OpenResponse Item
The sample response below reflects a strong knowledge and understanding of the subject matter.
The data tables representing the bonuses received each year over a 12year period for the two options are shown below.
Option 1
Year 1 2 3 4 5 6 7 8 9 10 11 12 Bonus
(dollars)2 4 8 16 32 64 128 256 512 1024 2048 4096 Option 2
Year 1 2 3 4 5 6 7 8 9 10 11 12 Bonus (dollars) 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 The data from the tables can be graphed on a coordinate grid as shown below.
Graph representing the data from the tables. The x axis is labeled Number of Years. It displays a series of fourteen tick marks equally spaced, with the labels, in numerals, of four, eight, and twelve at those tick marks. The y axis is labeled Bonus open parenthesis Dollars close parenthesis. It displays a series of eight tick marks equally placed, with the labels, in numerals, one thousand, two thousand, three thousand, and four thousand at the second, fourth, sixth, and eighth tick marks, respectively. The graph has two lines. The line labeled option one is a curved line that starts close to the x axis and increases very slowly at first then starts increasing sharply around nine years to an almost vertical increase. The line labeled option two is a straight increasing line. The two lines intersect at point p between years eleven and twelve at about two thousand four hundred dollars.
The two graphs intersect at point P, where they both pay the same bonus. For all years before this point (approximately years 1 through 11), option 2 yields a higher bonus. For all years after this point (years 12 and beyond), option 1 yields a higher bonus.
The bonus plan offered in option 1 increases by a constant factor of 2 and therefore is modeled by an exponential equation. The exponential equation that models this bonus plan is y = 2^{x}. The bonus plan offered in option 2 increases by a constant rate of $200 each year and therefore is modeled by a linear equation. The linear equation that models this bonus plan is y = 200x.
In the 20^{th} year, the bonus offered by option 1 exceeds that offered by option 2 by 2^{20} – 4000 dollars.
Scoring Rubric
Performance Characteristics
The following characteristics guide the scoring of responses to the openresponse item(s).
Purpose  The extent to which the response achieves the purpose of the assignment. 

Subject Matter Knowledge  Accuracy and appropriateness in the application of subject matter knowledge. 
Support  Quality and relevance of supporting details. 
Rationale  Soundness of argument and degree of understanding of the subject matter. 
Scoring Scale
The scoring scale below, which is related to the performance characteristics for the tests, is used by scorers in assigning scores to responses to the openresponse item(s).
Score Point  Score Point Description 

4 
The "4" response reflects a thorough knowledge and understanding of the subject matter.

3  The "3" response reflects an adequate knowledge and understanding of the subject matter.

2  The "2" response reflects a limited knowledge and understanding of the subject matter.

1  The "1" response reflects a weak knowledge and understanding of the subject matter.

U  The response is unrelated to the assigned topic, illegible, primarily in a language other than English, not of sufficient length to score, or merely a repetition of the assignment. 
B  There is no response to the assignment. 