Test Information Guide

Field 65: Middle School 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 to 300 words, or 1 to 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 0014
Prepare an organized, developed analysis on a topic cited from the Massachusetts Mathematics Curriculum Framework grades 5 to 8.

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.

Standard

The Massachusetts Mathematics Curriculum Framework for grade 5 provides the following content standard:

Number and Operations—Fractions (5.NF)

4. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

Problem Situation

In the products shown, the numerals are correct but the decimal placement is not necessarily correct.

First product is 1.01 times 9 which is shown to equal 9.09

Second product is 3.8 times 1.2 which is shown to equal 76, which is then added to 380 to produce 45.6

Third product is 4.71 times 5.5 which is shown to equal 2355, which is then added to 23550 to produce 259.05

Assignment

Use your knowledge of mathematics to develop a response of approximately 150 to 300 words, or 1 to 2 pages, in which you analyze the number system with respect to content standard 5.NF.4. In your response:

Be sure to show your work and explain the reasoning you use in analyzing the standard.

Sample Strong Response to the Open-Response Item

The sample response below reflects a strong knowledge and understanding of the subject matter.

Prerequisites:

Understanding the meaning of multiplication as repeated addition and that A times B means A groups of B added together.

Need to know multiplication facts.

Meaning of a fraction and the meaning of numerator and denominator.

Understanding addition of fractions with the same denominator.

Ability to represent fractions with physical models (ex. Fraction pieces), pictures and interpreting simple word problems that have a fraction as the answer.


Representations:

Numerical:
5 times 1 eighth means to add 5 groups of 1 eighth
1 eighth + 1 eighth + 1 eighth + 1 eighth + 1 eighth = 5 eighths
3 times 5 twelfths = 5 twelfths + 5 twelfths + 5 twelfths = 15 twelfths

Pictorial:

Draw a rectangle cut into 8 equal parts and understanding the numerical representation above, shade in 5 of those equal parts.

The diagram below shows that 1 eighth + 1 eighth + 1 eighth + 1 eighth + 1 eighth = 5 eighths and that 5 times 1 eighth = 5 eighths

An image of a hand drawn rectangle is shown. It is divided into 8 equal rectangles, 4 on the top and 4 on the bottom. 5 of the 8 rectangles are shaded. On the top, the first 3 rectangles are shaded and the last rectangle is not shaded. On the bottom, the first 2 rectangles are shaded and the last two rectangles are not shaded.

The answer is the shaded part of the rectangle which is 5 eighths or 5 parts of the 8 which make the whole.

Physical:

Using fractions pieces have the student use a whole and the pieces which are equal to 1 eighth’s, discuss again the meaning of multiplication as adding 1 eighth five times or 5 groups of 1 eighth and have them place the pieces on the “whole”. The answer will be what part of the whole is covered.

Another way to show multiplication of fractions is to use paper folding. Using a rectangular piece of paper, fold it in half horizontally, unfold and color half of the paper. Now fold the paper vertically into four equal pieces, unfold and shade 3 of the vertical pieces a different color. The answer to 1 half times 3 quarters will be the part that has both colors, 3 eighths.


Critique:

The problem given is not a good example for understanding multiplication of common fractions.

Decimals are a specific type of fractions that all have denominators using powers of ten. The denominators are not shown. This is not helpful in understanding the concepts involved with fraction operations because both the numerator and denominator need to be there. Students need to see how the numerator and denominator interact to give a solution and discover the pattern. Fractions can have any denominator. Conceptual understanding of how fractions work requires them seeing what happens with fractions of different sized denominators.

The example given can be solved without understanding fractions at all but simply using estimation skills.

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.