Test Information Guide

Field 47: Mathematics (Middle School)
Sample Open-Response Item

The following materials contain:

Sample Test Directions for Open-Response Items

This section of the test consists of two open-response 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:

  1. typed into the on-screen response box,
  2. written on a response sheet and scanned using the scanner provided at your workstation, or
  3. provided using both the on-screen 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.

The open-response 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 pop-up message will appear on-screen 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 Open-Response 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 bonus-plan 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 20-year period under each option. In your response:

Be sure to show your work and explain the reasoning you use in analyzing and solving this problem.

Sample Strong Response to the Open-Response 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 12-year 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.

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 = 2x. 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 20th year, the bonus offered by option 1 exceeds that offered by option 2 by 220 – 4000 dollars.

Scoring Rubric

Performance Characteristics

The following characteristics guide the scoring of responses to the open-response item(s).

Performance Characteristics
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 open-response item(s).

Score Scale with description for each score point.
Score Point Score Point Description
4 The "4" response reflects a thorough knowledge and understanding of the subject matter.
  • The purpose of the assignment is fully achieved.
  • There is substantial, accurate, and appropriate application of subject matter knowledge.
  • The supporting evidence is sound; there are high-quality, relevant examples.
  • The response reflects an ably reasoned, comprehensive understanding of the topic.
3 The "3" response reflects an adequate knowledge and understanding of the subject matter.
  • The purpose of the assignment is largely achieved.
  • There is a generally accurate and appropriate application of subject matter knowledge.
  • The supporting evidence is adequate; there are some acceptable, relevant examples.
  • The response reflects an adequately reasoned understanding of the topic.
2 The "2" response reflects a limited knowledge and understanding of the subject matter.
  • The purpose of the assignment is partially achieved.
  • There is a limited, possibly inaccurate or inappropriate, application of subject matter knowledge.
  • The supporting evidence is limited; there are few relevant examples.
  • The response reflects a limited, poorly reasoned understanding of the topic.
1 The "1" response reflects a weak knowledge and understanding of the subject matter.
  • The purpose of the assignment is not achieved.
  • There is little or no appropriate or accurate application of subject matter knowledge.
  • The supporting evidence, if present, is weak; there are few or no relevant examples.
  • The response reflects little or no reasoning about or understanding of the topic.
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.