Test Information Guide

Field 09: Mathematics
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 0024
Prepare an organized, developed analysis emphasizing problem solving, communicating, reasoning, making connections, and/or using representations on topics related to one or more of the following: number sense and operations; patterns, relations, and algebra; geometry and measurement; data analysis, statistics, and probability; trigonometry, calculus, and discrete mathematics.

Use the information below to complete the exercise that follows.

A tank for watering animals is shaped like a right isosceles trapezoidal prism. The tank is 36 inches across the top and 24 inches deep, has a base of 18 inches, and is 10 feet long. Both the top and the base of the tank are parallel to the ground. The tank initially contains water that is 12 inches deep. Water will be drained from the tank at a rate of 300 cubic inches per minute.

Using your knowledge of two-dimensional and three-dimensional geometry and functions, prepare a response in which you analyze this situation. 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.

A labeled diagram representing the situation is drawn above.

The water in the tank is in the shape of a prism whose volume can be calculated by multiplying the cross-sectional area of the water in the tank by the length of the tank.

The base of the prism is a trapezoid whose area can be calculated by A = one half(b1 + b2)h with b1 and b2 representing the width of the water surface across the tank, and the width of the base of the tank, respectively, and h representing the height of the trapezoid (the depth of the water). In this case, b1 = 2x + 18, b2 = 18, and h = 12, so x must be found.

Because of similar triangles x over twelve equals nine over twenty four, from which x is computed to be 4.5 inches. Thus the width of the water across the tank is 4.5 inches + 18 inches + 4.5 inches = 27 inches. Now find the cross-sectional area of the water in the tank: A = one half(27 + 18)12 = 270 square inches.

The volume of the water is the product of the cross-sectional area of the water in the tank and the length of the tank, or (270 inches)(120 inches) = 32,400 cubic inches. [Note: 10 feet = 120"].

Since the water drains at a constant rate of 300 cubic inches per minute, a linear equation can model the situation: V (t) = 32,400 – 300t with V representing volume of water in the tank and t representing the time that the tank has been draining.

The above graph represents the function V (t). The slope of the function, –300 cubic inches per minute, represents the rate at which the water is draining from the tank with respect to time. The y-intercept on the graph is 32,400, which represents the quantity of water initially present in the tank.

To find the length of time it will take to drain the tank, set V (t) equal to zero and solve for t. Thus 0 = 32,400 – 300t implies –300t = –32,400 and t = 108 minutes. It will take 108 minutes to drain the tank.

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.