Grade 9 – 10 Mathematics Constructed Response Problem Ohio Graduation Test

Mrs. Foyle told Yolanda that her test had 38 problems worth a total of 100 points. Each test problem is worth either 5 points or 2 points. Yolanda wanted to determine how many 2-point and how many 5-point questions are on the test.

In the space below, determine how many questions of each point-value are on the test. Show your work or provide an explanation to support your answer. (2 points) Write your response in the space below.

Grade 9 – 10 Mathematics Constructed Response Problem Ohio Graduation Test

Standard: Patterns, Functions and Algebra Benchmark 8-10: H. Solve systems of linear equations involving two variables graphically and

symbolically.

Scoring Guidelines Description

Points Student Response 2 This problem assesses skills in solving word problems involving two

variables, the number so 2-point problems and the number of 5-point problems, to determine the number of problems for each point value that are on a test that is worth 100 points.

Explanation of Correct Responses: The response receiving maximum points clearly states that there are 8 of

the 5-point questions and 30 of the 2-point questions. This satisfies both requirements that there are 38 found by (30 + 8) questions on the test and that the value of the questions (30 X 2 + 8 X 5) equals 100. This response may be determined by using various techniques, from formal algebra, a system of two equations, to guess and check techniques to support the response. A formal algebra response defines the variables, x = number of 5-point questions and y = number of 2-point questions, and includes two equations, x + y = 38 and 5x + 2y = 100. Solving this system of equations yields the answer x = 8 and y = 30.

Sample Responses: Use a system of equations:

x + y = 38 2x + 5y = 100 ∴ x = 30 2-point questions

y = 8 5-point questions

OR

Use a table to guess-and-check:

Number of 2-points 35 32 30 Number of 5-points 3 6 8 Total number of points 85 94 100

There are 8 5-point questions and 30 2-point questions.

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Grade 9 – 10 Mathematics Constructed Response Problem Ohio Graduation Test

Score Point

Description

2 The focus of the item is to determine the number of 2-point and 5-point questions on the test. The response indicates that there are 30 two-point and 8 five-point questions with supporting work or explanation.

1 The response provides evidence of a partially correct answer and/or solution process. The response shows understanding of some key elements of the task but contains gaps or flaws. For example, the response may:

• Provide the correct answer with missing or incomplete work. • Provide an incorrect answer based on slightly flawed equations. • Provide correct equations or demonstrate a valid process but

contain a calculation error in determining the solution.

0 The response indicates inadequate understanding of the task. For example, the response may:

• Contain one answer (8 or 30) with no correct work. • Include unrelated statements or work.

Grades 9 – 10 Mathematics Constructed Response Problem Ohio Graduation Test

Mrs. Foyle told Yolanda that her test had 38 problems worth a total of 100 points. Each test problem is worth either 5 points or 2 points. Yolanda wanted to determine how many 2-point and how many 5-point questions are on the test.

In the space below, determine how many questions of each point-value are on the test. Show your work or provide an explanation to support your answer. (2 points)

Grades 9 – 10 Mathematics Constructed Response Problem Ohio Graduation Test

Mrs. Foyle told Yolanda that her test had 38 problems worth a total of 100 points. Each test problem is worth either 5 points or 2 points. Yolanda wanted to determine how many 2-point and how many 5-point questions are on the test.

In the space below, determine how many questions of each point-value are on the test. Show your work or provide an explanation to support your answer. (2 points) Write your response in the space below.

Grades 9 – 10 Mathematics Constructed Response Problem Ohio Graduation Test

Mrs. Foyle told Yolanda that her test had 38 problems worth a total of 100 points. Each test problem is worth either 5 points or 2 points. Yolanda wanted to determine how many 2-point and how many 5-point questions are on the test.

In the space below, determine how many questions of each point-value are on the test. Show your work or provide an explanation to support your answer. (2 points) Write your response in the space below.

Grades 9 – 10 Mathematics Constructed Response Problem Ohio Graduation Test

Mrs. Foyle told Yolanda that her test had 38 problems worth a total of 100 points. Each test problem is worth either 5 points or 2 points. Yolanda wanted to determine how many 2-point and how many 5-point questions are on the test.

In the space below, determine how many questions of each point-value are on the test. Show your work or provide an explanation to support your answer. (2 points) Write your response in the space below.

Grades 9 – 10 Mathematics Constructed Response Problem Ohio Graduation Test

Mrs. Foyle told Yolanda that her test had 38 problems worth a total of 100 points. Each test problem is worth either 5 points or 2 points. Yolanda wanted to determine how many 2-point and how many 5-point questions are on the test.

In the space below, determine how many questions of each point-value are on the test. Show your work or provide an explanation to support your answer. (2 points) Write your response in the space below.

Grade 9 – 10 Mathematics Constructed Response Problem Ohio Graduation Test

Cameron had $500 in savings on January 1. Quinn had $800 in savings on January 1. Cameron deposits $20 per week into his savings account. Quinn withdraws $15 per week from his savings account. In the space provided below, write two equations: one for the amount of money in Cameron’s savings x weeks after January 1, and one for the amount of money in Quinn’s savings x weeks after January 1. Determine the number of weeks until Cameron will have more money in his savings account than Quinn. Show your work or provide an explanation for your answer. Write your response in the space below.

Grade 9 – 10 Mathematics Constructed Response Problem Ohio Graduation Test

Standard: Patterns, Functions and Algebra Benchmark 8-10: H. Solve systems of linear equations involving two variables graphically and

symbolically.

Scoring Guidelines Description

Points Student Response 2 The focus of this item is solving systems of equations. Students are asked to write

equations to represent the amounts of money in two different savings accounts after any number of weeks. For example, y = $500 + $20x (Cameron’s account) and y = $800 - $15x (Quinn’s account). The equations are then used to determine when the amount in Cameron’s account is greater than the amount in Quinn’s account. The response that earns the maximum number of Points includes appropriate equations and identifies the number of weeks with accurate and complete supporting work or explanation; e.g., table, graph or computation used to find the number of weeks until the amount in Cameron’s account is greater than that in Quinn’s account (9 weeks).

Score Point

Description

2 points The task includes two components – writing equations that represent the amount of money in each account for a variable number of weeks and determining the number of weeks until the amount of money in Cameron’s account is greater that that in Quinn’s account. The response contains accurate equations for finding the amount in each account for any number of weeks and indicates Cameron will have more money in his savings account that Quinn in week 9 (or other accurate responses such as those provided in sample responses) with clear, accurate work or explanation.

1 point The response provides evidence of a partial answer and/or solution process. The response has an error(s) or does not carry out all parts of the task. For example, the response may: Provide correct equations for each account with an incorrect answer. Supporting work or explanation is incorrect or missing for the number of weeks until Cameron will have more money in his account than Quinn. OR Find the correct number of weeks with supporting work but not provide two correct equations. OR Provide one correct equations one flawed equation, nut then determine an accurate number of weeks based upon the equation. OR Provide one correct equation with no additional work. OR Indicate the 9th week without any supporting work or equations.

Grade 9 – 10 Mathematics Constructed Response Problem Ohio Graduation Test

Score Point

Description

0 points The response indicates inadequate or no understanding of the task and/or incorrect use of the key elements or information; e.g., writing an equation to represent a problem situation and finding a value that when used in both equations meets a specified criteria. The response does not meet the criteria required to earn one point. For example, the response may: State an incorrect number of weeks with no supporting work or explanation OR Give two incorrect equations and no other work or explanation. OR Recopy information provided in the item with no work. OR Show no apparent understanding or relationship to the key components in the task or a possible solution process; e.g., finds the sum of the dollar amount in the item. OR Be blank, or the students writes, “I do not know” or includes unrelated statements or work.

Grade 9 – 10 Mathematics Constructed Response Problem Ohio Graduation Test

Score Point 2 Cameron had $500 in savings on January 1. Quinn had $800 in savings on January 1. Cameron deposits $20 per week into his savings account. Quinn withdraws $15 per week from his savings account. In the space provided below, write two equations: one for the amount of money in Cameron’s savings x weeks after January 1, and one for the amount of money in Quinn’s savings x weeks after January 1. Determine the number of weeks until Cameron will have more money in his savings account than Quinn. Show your work or provide an explanation for your answer.

Score Rationale

The response shows two correct equations for determining the amount in either account after any number of weeks (x). These equations are used to determine when Cameron’s account will surpass Quinn’s, resulting in the answer of nine weeks (x > 8.57). All work is clear and accurate.

Grade 9 – 10 Mathematics Constructed Response Problem Ohio Graduation Test

Score Point 1 Cameron had $500 in savings on January 1. Quinn had $800 in savings on January 1. Cameron deposits $20 per week into his savings account. Quinn withdraws $15 per week from his savings account. In the space provided below, write two equations: one for the amount of money in Cameron’s savings x weeks after January 1, and one for the amount of money in Quinn’s savings x weeks after January 1. Determine the number of weeks until Cameron will have more money in his savings account than Quinn. Show your work or provide an explanation for your answer.

Score Rationale The response provides appropriate equations for both accounts; however, no evidence of an attempt to determine the number of weeks until Cameron’s account will have more money that Quinn’s is shown.

Grade 9 – 10 Mathematics Constructed Response Problem Ohio Graduation Test

Score Point 0 Cameron had $500 in savings on January 1. Quinn had $800 in savings on January 1. Cameron deposits $20 per week into his savings account. Quinn withdraws $15 per week from his savings account. In the space provided below, write two equations: one for the amount of money in Cameron’s savings x weeks after January 1, and one for the amount of money in Quinn’s savings x weeks after January 1. Determine the number of weeks until Cameron will have more money in his savings account than Quinn. Show your work or provide an explanation for your answer.

Score Rationale The response shows incorrect equations and supporting work that results in two different values for x. The response has not demonstrated minimal understanding of the task.