# Study Guide

##
Field 111: Advanced Mathematics

Sample Constructed-Response Assignment

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The following materials contain:

- Test directions for the constructed-response assignment
- A sample constructed-response assignment
- An example of a strong and weak response to the assignment, and a rationale for each
- The performance characteristics and scoring scale

### Test Directions for the Constructed-Response Assignment

This section of the test consists of one constructed-response assignment. You will be asked to prepare a written response of approximately 300 to 600 words on the assigned topic.

Read the assignment carefully before you begin your response. Think about how you will organize your response. You may use the erasable sheet(s) to make notes, write an outline, or otherwise prepare your response. start bold and italics

end bold and italicsHowever, your final response to the assignment must be either:

- typed into the on-screen response box,
- written on a response sheet and scanned using the scanner provided at your workstation, or
- 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.
start bold

Instructions for scanning your response sheet(s) are available by clicking the "Scanning Help" button at the top of the screen.end boldAs a whole, your response must demonstrate an understanding of the knowledge and skills of the field. In your response to the assignment, you are expected to demonstrate the depth of your understanding of the content area through your ability to apply your knowledge and skills rather than by merely reciting factual information.

Your responses to the assignments will be evaluated based on the following criteria.

start bold

PURPOSE:end bold the extent to which the response achieves the purpose of the assignment

start boldSUBJECT KNOWLEDGE:end bold appropriateness and accuracy in the application of subject knowledge

start boldSUPPORT:end bold quality and relevance of supporting evidence

start boldRATIONALE:end bold soundness of argument and degree of understanding of the subject areaThe constructed-response assignment is intended to assess subject matter knowledge and skills, not writing ability. However, your response must be communicated clearly enough to permit valid judgment of the scoring criteria. Your response should be written for an audience of educators in this field. The final version of your response should conform to the conventions of edited American English. Your response must 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 topic. Remember to review your work and make any changes you think will improve your response.

Any time spent responding to the 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 start uppercase NOT end uppercase be scored.

### Sample Constructed-Response Assignment

####
roman numeral 6

Pedagogical Content Knowledge

start bold **Use the data provided to
complete the task that follows.** end bold

Using the data provided, prepare a response of approximately 300 to 600 words in which you:

- identify a significant mathematical strength related to the given standard that is demonstrated by the student, citing specific evidence from the exhibits to support your assessment;
- identify a significant area of need related to the given standard that is demonstrated by the student, citing specific evidence from the exhibits to support your assessment; and
- describe an instructional intervention that builds on the student's strengths and that would help the student improve in the identified area of need.

start bold **Class Context** end bold

Students in a Geometry class are working with similar figures. In previous class sessions, they used scale factors to find side lengths, perimeters, and areas of similar two-dimensional figures. Currently, they are extending these ideas to find side lengths, surface areas, and volumes of similar three-dimensional figures. The teacher asks students to solve problems involving scale factors and volume. One student's work is shown in an exhibit.

start bold **Excerpt from Lesson Plan
** end bold

start bold

Course:end bold Geometrystart bold

Oklahoma Academic Standards for Mathematics:end bold Geometry: Three-Dimensional Shapes (G.3D.1.2)Use ratios derived from similar three-dimensional figures to make conjectures, generalize, and to solve for unknown values such as angles, side lengths, perimeter or circumference of a face, area of a face, and volume.

start bold

Lesson Objective:end bold Students will use scale factors to determine side lengths and volumes of similar 3-dimensional figures.start bold

Previous Knowledge Needed:end bold setting up and solving proportions; the definition of similar figures; the formulas for volumes of 3-dimensional figuresstart italics

Source: Oklahoma Academic Standards 2016–2017, Oklahoma State Department of Educationend italics

start bold **Student Work** end bold

A shipping company uses rectangular boxes of various sizes. The small box has length 10 inches, width 8 inches, and height 4 inches. A larger box is similar to the small box with a scale factor of 3 to 2.

- What is the volume of the larger box? l over 10 equals 3 over 2 w over 8 equals 3 over 2 h over 4 equals 3 over 2 2 L equals 30 2 W equals 24 2 H equals 12 L equals 15 W equals 12 H equals 6 volume of the box equals 15 times 12 times 6 equals 1080 cubic inches
- The company would like a box of similar shape to have six times the volume of the small box. What would the dimensions of that box be? L over 10 equals 6 over 1 W over 8 equals 6 over 1 =
^{6}/_{1}H over 4 equals 6 over 1 L equals 60 W equals 48 H equals 24 The box would be 60 inches long, 48 inches wide, and 24 inches high.

### Sample Strong Response to the Constructed-Response Assignment

start bold **Please note: The sample
response provided below is for review purposes only and should not be used in a
response on an operational exam. Use of the exact words and phrases presented in
this sample response will result in a score of "U" (Unscorable) due to lack of original
work.** end bold

The student's work displays three strengths: setting up and solving proportions,
applying scale factors correctly to lengths of sides of rectangular prisms, and
finding the volumes of these prisms. For example, in the proportion
L over ten equals three over 2
the student correctly uses the scale factor of 3 to 2 to relate
the length of the larger box, *L*, to the length of 10 in the smaller box
and cross multiplies to get
2 l equals 30
and
l equals 15.
The student repeats
this process to find the width and height of the box and then correctly multiplies
length times width times height
to get 1080 cubic inches.

The student's work shows that the student does not understand that the scale factor relating the volumes of two similar shapes is not equal to the scale factor relating its linear measurements. The student uses the volume scale factor of 6 to 1 to set up L over ten equals six over one and conclude that the length of the larger box is 60. The student does not calculate the new volume 60 times 48 times 24 to compare it to the volume of the small box which is 10 times 8 times 4 Doing this would have shown the student that the new volume is not six times the volume of the small box, but 216 times as large.

There are several ways to help this student. One activity could use manipulatives such as unifix (or other) cubes. Ask the student to build a series of similar rectangular prisms starting with a small one left paren like 2 times 3 times 4 right paren and using linear scale factors of 2, 3, 4, etc. Then ask the student to determine, by comparing the prisms, how many of the small prism are contained in each of the larger prisms. The student should find 8 of the original in the one with scale factor 2, 27 of the original in the one with scale factor 3, and so on. This is particularly easy to see if different colors are used to create the extensions, but takes a lot of cubes!

An activity which uses the student's strengths would be to make a table of values for the volumes of similar rectangular prisms given the dimensions of the original.

For example:

Continue with various scale factors. Look for the relationship of the ratio of the volumes to the scale factor in each case to determine that the ratio of the volumes is the cube of the scale factor.

A third option uses an algebraic approach. Since the student uses linear scale factors
correctly, let the scale factor be unknown, say *k*. Then
L over ten equals k
and
L equals 10k.
Similarly
W equals 8k
and
H equals 4k.
New volume
equals 10k times 8k times 4k equals 320k cubed.
The new volume is 6 times the original or 1920, so
320k cubed equals 1920 ,
k cubed equals 6
and
*k* equals cubed root of six.
The dimensions are
10 cubed root of six,
8 cubed root of six,
and
4 cubed root of six.

#### Rationale for the Sample Strong Response

Please note that the response is evaluated based upon the four performance characteristics of Purpose, Subject Matter Knowledge, Support, and Rationale. Please also note how the score point descriptions are based upon how the examinee attends to the performance characteristics. You should be very familiar with the CEOE performance characteristics and score scale and refer to them when reviewing this rationale.

The response fulfills the purpose of the assignment (refer to the instructions for
the assignment) by adequately responding to all elements of the assignment. The
response addresses the first bullet by identifying the student's strengths (i.e.,
setting up and solving proportions, applying scale factors correctly, and finding
the volumes of prisms). Moreover, the response cites specific evidence related to
the student's strengths from the exhibits (i.e., solving appropriately for *L*).
The response reflects a general understanding of the student's weakness by providing
some relevant examples. The response correctly identifies the student's weakness
in understanding the ratio of dimensions of similar figures when the ratio of volumes
is known. In addition, the response adequately addresses the third bullet of the
assignment by describing an appropriate plan of intervention for the student. In
fact, this response elaborates on the plan by providing multiple ways to help the
student. For instance, the response explains that the teacher could have the student
use "manipulatives," "a table of values," or an "algebraic approach" to help the
student improve on his or her identified weakness. Overall, this response reflects
an adequate understanding of the subject matter.

### Sample Weak Response to the Constructed-Response Assignment

This is a Geometry class where we have a lesson objective that states, "Students will use scale factors to determine side lengths and volumes of similar 3-dimensional figures." In order to meet this objective students needed to be here for the last class where they learned how to set up and solve proportions; what similar figures are; and the formulas for 3-dimensional figures." The last class was an important one, so I hope all students were there for it.

To see what this student's strengths and weaknesses are, we will look at the student work sample that was given. By looking at the student work, we see that the student has a strength in setting up ratios to solve for parts of similar figures and to calculate the volume of a box. This is evident in their correct solution to number 1 on the student work sample (exhibit 2).

The student work sample also shows us that the student has some needs. The student did not recognize that the ratio of side lengths is different than the ratio of volumes. This is evident in their incorrect answer to number 2.

As the teacher, I should work to help the student turn their weakness into a strength. In order to do that, an intervention I could use would be to prompt the student to multiply the dimensions and see if that agrees with six times the volume of the small box.

#### Rationale for the Sample Weak Response

Please note that the response is evaluated based upon the four performance characteristics of Purpose, Subject Matter Knowledge, Support, and Rationale. Please also note how the score point descriptions are based upon how the examinee attends to the performance characteristics. You should be very familiar with the CEOE performance characteristics and score scale and refer to them when reviewing this rationale.

The purpose of this assignment is not achieved. The response begins with an elaborate restatement of the prompt. While the response identifies the student's strength and weakness, the evidence cited is minimal. The plan for intervention reflects little appropriate application of subject matter knowledge, as it was minimally developed and lacked mathematical depth and detail. The response provides minimal supporting details and lacks any rationale. Overall, the response provides few relevant examples and reflects little or no reasoning about or understanding of the topic.

### Performance Characteristics

The following characteristics guide the scoring of responses to the constructed-response assignment.

### Scoring Scale

Scores will be assigned to each response to the constructed-response assignment according to the following scoring scale.