# Study Guide

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Field 225: Middle Level/Intermediate 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 300to600 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

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**Competency 0013**

Analyze a lesson plan for a given content standard in the Oklahoma Academic Standards for Mathematics, including examples of student work and/or assessments, and describe subsequent activities that address student needs on the basis of your analysis.

start bold **Use the information provided in the exhibits to complete the assignment that follows.** end bold

Using the data provided, prepare a response of approximately 300to600 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.

#### Exhibit 1: Class Context

Students are working on solving percent problems using equations involving decimals. In previous lessons, they calculated total costs involving percent markups and percent discounts. They have also calculated total cost by applying sales tax to a subtotal. Students have been encouraged to make use of the distributive property to calculate total cost with tax, or subtotal after a discount, using a single operation by identifying relationships such as "a 6% sales tax is the same as 106% of the subtotal" or "20% off means the new cost is 80% of the original." This is the first activity in which students are required to combine their knowledge of discounts and sales tax in multistep problems. For the given activity, students have access to a four-function calculator. One student's work is shown in an exhibit.

#### Exhibit 2: Excerpt from Lesson Plan

Standard: 7.A.2"Recognize proportional relationships in real-world and mathematical situations; represent these and other relationships with tables, verbal descriptions, symbols, and graphs; solve problems involving proportional relationships and interpret results in the original context."

Sub Standard: 7.A.2.2"Solve multi-step problems involving proportional relationships involving distance-time, percent increase or decrease, discounts, tips, unit pricing, similar figures, and other real-world and mathematical situations."

Essential Question:How can we use our understanding of the distributive property to efficiently solve multistep problems involving discounts and sales tax?

Previous Knowledge Needed:Applying the distributive property, converting percents to decimals, finding percents using equations with decimals, and finding total cost after percent discount or percent markup

#### Exhibit 3: Student Work Sample

Practice Solving Multistep Problems Involving

Percent Discounts and Sales TaxCalculate the total cost for the items described below. Remember to show all work and round all answers to the nearest hundredth.

1. A T-shirt that is regularly $27.99 is on sale for 15% off. There is a 4% sales tax applied to the subtotal.

27 point 99 times 0 point 1 5 equals 4 dollars and 20 cents

27 point 99 times 0 point 0 4 equals 1 dollar and 12 cents

27 point 99 minus 4 point 2 0 plus 1 point 1 2 equals 24 dollars and 91 cents total2. A video game that is originally $59.99 is on sale for 30% off. There is a 5% sales tax applied to the subtotal.

59 point 99 times 0 point 3 0 equals 18 dollars

59 point 99 times 0 point 0 5 equals 3 dollars

59 point 99 minus 18 plus 3 equals 44 dollars and 99 cents total3. A store offers a deal where customers can buy five pairs of socks and receive a sixth pair for free. If each pair of socks is originally $4.99, what is the cost of 6 pairs of socks after a 5% sales tax is applied to the subtotal?

5 times 4 point 9 9 equals 24 dollars and 95 cents

24 point 95 times 0 point 0 5 equals 1 dollars and 25 cents

24 point 95 plus 1 point 2 5 equals 26 dollars and 20 cents total

### 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

In this classroom, students are learning to solve percent problems using equations involving decimals (exhibit 1). The lesson plan (exhibit 2) shows that students will be focusing on substandard 7.A.2.2 and will be trying to answer: "How can we use our understanding of the distributive property to efficiently solve multistep problems involving discounts and sales tax?" A student work sample (exhibit 3) is given in order to determine the effectiveness of the lesson plan. The student work sample shows both strengths and weaknesses.

One strength is how the student demonstrates the ability to find the cost of multiple units of the same item as shown in problem 3. This is part of the substandard (i.e., unit pricing). The student multiplied the unit cost by 5 for the buy 5, get the sixth free offer. The student also correctly calculates the sales tax as a percentage of the subtotal. This is shown as a part of all three sample problems shown. In addition, the student successfully converted a percentage to a decimal and correctly multiplied the decimal and the price to get the tax amount. In problems 1 and 2, the student correctly calculated the amount of the discount (i.e., discounts substandard), again converting the percentage to a decimal and multiplying it by the original price.

One area of need is seen in the student work sample in the attempt to apply the tax rate to the corrected price. In problems 1 and 2, the student correctly multiplied the numbers but applied the tax rate to the original price, not the subtotal after the discount. The tax should be applied to the discounted price, not the original price. This is related to the multi-step problems aspect of the stated standard.

One method to improve this area of need is to use real life examples of discounts. The student could be asked to bring in a receipt from a store that shows a discount. The student would need to go to the store with a parent or guardian. An alternative could be the teacher provides xerox copies of a receipt. Have the students circle the original price and draw a square around the discounted price. Ask the student to calculate the sales tax on both the regular and discounted price then identify which amount matches what is on the receipt. Referring to PEMDAS, ask the student to determine how you could use parentheses to mathematically determine the correct way to determine the total price of an item on sale. If further practice is needed, students could work independently or with a partner to pair-share a worksheet and practice additional multi-step problems.

#### 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 generally achieves the purpose of the assignment by adequately addressing all bullets in the prompt. This response also demonstrates generally accurate subject matter knowledge as is evidenced by the use of supporting evidence throughout the response (e.g., the response cites directly from the student work sample, specifically identifying strengths and weaknesses from problems 1, 2, and 3). The response generally identifies and explains the strengths of the student work sample by correctly identifying the ability to calculate the sales tax and discount as well as the concept of unit pricing. Additionally, the candidate correctly identifies a significant error and explains not only why it is an error but also the correct way to do the calculation. Lastly, the response provides an appropriate suggestion for instructional intervention. The idea of using receipts is student focused and allows for students with different learning styles or modalities to improve upon their understanding. A strong response would have elaborated in more detail on how the intervention would have worked in the classroom or even outside the classroom. Overall, the response indicates a generally accurate and appropriate understanding of the subject matter.

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

This student demonstrates the ability to calculate discounts and applies that knowledge accurately by subtracting the calculated amount from the original prices. This is indicated in problems 1 and 2. The student converted a percentage to a decimal and multiplied that decimal by the price to determine the discount amount. Later the student subtracted the amount from the original price to find the sale price. This matches the standard about applying discounts. Likewise, the student does correctly determine the tax by converting the percentage to a decimal and then correctly multiplying by the price to determine the sales tax. The student then correctly subtracts the discount and then adds the tax to find the end price.

This student exhibits a misunderstanding of unit pricing. Problem 3 clearly states that the person is purchasing 6 pairs of socks, yet the work sample multiplies the price by 5. The customer will still get the free pair, but the student needs to read more carefully.

I think there are several ways to improve this lesson. The teacher should make sure the students are ready for multi-step problems. Even so the teacher should lead the students through a sample problem and leave the work on the board for students to model when they do their own calculations. The teacher should not be so tricky the first time students try a new skill. He or she should have had the customer buy 5 pairs of socks or make it that if the customer buys 6 pairs they get one free. The teacher could even add "if the customer wants blank number of socks, how many need to be purchased and how many would be free." Using an overhead projector, the teacher could show the students how to circle the important parts of the question and set up.

#### 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 writer attends to each of the performance characteristics. You should be very familiar with the CEOE performance characteristics and score scale and refer to them when reviewing this rationale.

This response partially achieves the purpose of the assignment. While this candidate does address all the bullet points of the prompt, there are several errors in the response. There is evidence of limited and inaccurate subject matter knowledge. While the candidate does point out strengths (e.g., the conversion of percent to decimal and order of steps), they are not part of the objectives for this lesson. The candidate does connect one part of the work to the lesson standard (i.e., application of percent discounts) when they say the student subtracted the discount amount from the original price. However, there are several errors in the analysis. The candidate claims the understanding of unit pricing is wrong, but problem 3 is done correctly. The candidate is unclear as to whether the sales tax rate is applied to the original price or the discounted price. Overall, this candidate shows a limited understanding of the topic. There are some errors and inaccuracies (e.g., determination of correct sales tax). The suggestions for improvement are all teacher based, not student focused. Overall, this response demonstrates a limited understanding of the subject matter.

### 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.