Test Information Guide
Field 09: Mathematics
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 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 twodimensional and threedimensional geometry and functions, prepare a response in which you analyze this situation. In your response:
 draw a labeled diagram that clearly presents the given information and any variables needed;
 determine the initial volume of water in the tank;
 derive an equation that expresses the total volume of water in the tank as a function of time;
 sketch a graph of the equation on a set of coordinate axes and describe the meaning of the slope and the yintercept in the context of the situation; and
 calculate how long it will take to drain the tank.
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.
Water tank with water in it. The Water tank is in the shape of a trapezoid prism. The top opening the widest part of the trapezoid is nine inches wider on each side then the smaller base which is eighteen inches. The height of the trapezoid is twenty four inches. The tank is ten feet in length. The water in the tank is filled up half way so twelve inches from the top and twelve inches from the bottom. The water makes a shorter trapezoid with the height being twelve inches and the short base still being eighteen inches. The longest side of the trapezoid made by the water is x inches wider on each side then the smaller base which is eighteen inches.
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 crosssectional 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 = (b_{1} + b_{2})h with b_{1} and b_{2} 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, b_{1} = 2x + 18, b_{2} = 18, and h = 12, so x must be found.
Because of similar triangles , 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 crosssectional area of the water in the tank: A = (27 + 18)12 = 270 square inches.
The volume of the water is the product of the crosssectional 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.
Graph showing v of t on the y axis and t on the x axis. The line starts at open parenthesis zero comma thirty two thousand four hundred close parenthesis and decreases pretty rapidly down to open parenthesis one hundred and eight comma zero end parenthesis
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 yintercept 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 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. 