Test Information Guide

Field 63: 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 0018
Prepare an organized, developed analysis on a topic integrating knowledge from at least two of the following: number sense and operations; relations, functions, and algebra; geometry and measurement; probability, statistics, calculus, and discrete mathematics.

Note: For this assignment on the actual test, you may type a response into the on-screen response box and/or handwrite and scan your response using a Response Sheet provided for you. Do not use more than two Response Sheets for this assignment.

Use the information below to complete the assignment that follows.

The fuel efficiency of a certain car can be represented by a quadratic function where fuel efficiency is a quadratic function of the speed at which the car is driven according to the data shown in the table below.

Speed
(miles per hour)		 Fuel Efficiency
(miles per gallon)
45 38 1 third
55 40
65 38 1 third

Use your knowledge of quadratic relationships to develop a response of approximately 150 to 300 words, or 1 to 2 pages, in which you analyze the relationship between fuel efficiency and speed for the car. 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.

1. From the table given, you can see the symmetry of the data and I know it is a quadratic from the information given. This tells me that the (55, 40) is the vertex of the parabola. Using the vertex form of the quadratic equation, I put in the coordinates of the vertex.

y = a times open parens x minus h close parens squared + k where (h, k) is the vertex

y = a times open parens x minus 55 close parens squared + 40 From the chart pick one of the other two given ordered pairs

38 1over3 = a times open parens 65 minus 55 close parens squared + 40 Solving for a, I get a = negative 1over60

The equation is y = negative 1over60times open parens x minus 55 close parens squared + 40 where x is the mph and y is the mpg

2. I substituted x = 0 into the equation and found the y-intercept to be negative 10 5over12 and then put y = 0 into the equation to find the x-intercepts of about 6 open parens 6 point 0 1 close parens and about 104 open parens 103 point 9 9 close parens. Using these and the 3 points given in the table, I made a graph of the parabola.

The maximum tells us that you get the highest fuel efficiency open parens 40 m p g close parens going 55 mph.

The y-intercept tells you that when you are idling open parens going 0 m p h close parens you are getting negative fuel efficiency.

The x-intercepts tell us where we start to have a positive fuel efficiency open parens around 6 m p h close parens and when we no longer get any fuel efficiency open parens 104 m p g close parens.

hand-drawn graph with eight points plotted

Image of hand-drawn graph is shown. Graph of x and y axis. The x axis is labeled M P H and has increments of 10 marked from 10 to 100. The y axis is labeled M P G and has increments of 10 marked from negative 10 to 50. An arc is drawn with arrows at each end that includes the following points: (0, negative 10), (6, 0), (25, 25), (45, 37), (55, 40), (65, 37), (85, 25), (104, 0)

3. Given 16 gallons of gas and going 61 mph: This means we put x = 61 into our equation and solve for y.

y = negative 1over60times open parens x minus 55 close parens squared + 40 put in x = 61

y = negative 1over60times open parens 61 minus 55 close parens squared + 40 or y = negative 1over60times open parens 6 close parens squared + 40 and y = negative 36over60 + 40

y = 39.4 mpg so 16 times 39.4 = 630.4 miles on 16 gallons of gas

4. I want at least a fuel efficiency of 25 mpg so that means I need to put 25 in for y and solve for x.

25 = negative 1over60times open parens x minus 55 close parens squared + 40       25 = negative 1over60times open parens x squared minus 110 x plus 3025 close parens + 40

25 = negative 1over60times x squared + 110over60x − 3025over60 + 2400over60 and multiply both sides by negative 60

negative 1500 = x squared − 110x + 625 Take 1500 from both sides of the equation and solve.

0 = x squared − 110x + 2125 this is open parens x minus 25 close parens times open parens x minus 85 close parens so x = 25 mph or 85 mph

This means you need to go at least 25 mph and no faster than 85 mph to have at least 25 mpg for your efficiency.

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.