The Maryland State
High School Assessment Initiative:

 Sample Questions for Algebra

 

Below are sample test items from the prototype assessment for Algebra.

Sample selected response item:

 

Look at the table below.

x

y

-1

7

0

3

1

-1

2

-5

Which equation describes the relationship between x and y?

  1. y = x + 3

  2. y = x - 4

  3. y = 4x - 3

  4. y = -4x + 3
     
    Answer: d
   
    Mathematics Goal, Expectation, Learning Indicator:
    1.1.2 The student will represent patterns and/or functional relationships in a table, as a graph, and/or by mathematical expression.
     
    Instructional Implications:
    Students will need extensive experience with these types of selected response items, with activities presented to help them understand the design of such questions. Instructors should be especially sensitive to helping students understand the "embedded" elements of such items. For example, this question requires that students have been coached well in the application of data to equation solution, including the use of tables and other graphic representations. They also should have had extensive work with the construction and solution of equation problems. Implicit in a single item like this is a whole range of background experiences that all students should have, including the following: investigating, interpreting, and communicating solutions to mathematics and real-world problems using patterns, functions, and algebra.

Sample brief constructed response item: Time limit - 5 minutes

 

Hanz and Martha went to a sale at a music store where all CDs were one price and all cassettes were another price. Hanz bought 2 CDs and 2 cassettes for $40, while Mario bought 1 CD and 4 cassettes for $44.

The equations below represent these purchases, where x is the cost of a CD and y is the cost of a cassette.

2x + 2y = 40

x + 4y = 44

What are the costs of a single CD and a single cassette? Solve the system of equations by either constructing a graph in the grid or by using an algebraic process. Explain how you determined the costs. Use words, symbols, or both in your explanation.

                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
                             
   
    Solution:
    The cost of a single CD is $12 and the cost of a single cassette is $8.
For a brief constructed response, the student will be graded in a rubric
scale of 0-3. The criteria includes:
  • Analysis - Did the student show understanding of the problem?
  • Representation - Did the student appropriately display and identify information?
  • Application - Did the student use appropriate concepts and strategies to solve the problem?
  • Explanation - Did the student use words and/or symbols to explain the process used to solve the problem?
  • Justification - Did the student use definitions, postulates, theorems of mathematics to verify the solution to the problem?
     
    Scoring Rubric:
   
  • A score of 3 indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. The representations are correct. The explanation and/or justification are logically sound, clearly presented, fully developed, support the solution, and do not contain significant mathematical errors. The response demonstrates a complete understanding and analysis of the problem.
  • A score of 2 indicates application of a reasonable strategy that may be incomplete or underdeveloped. The representations are essentially correct. The explanation and/or justification support the solution, but may not be well developed, are plausible, and or may be incomplete. The response demonstrates a conceptual understanding and analysis of the problem.
  • A score of 1 indicates little or no attempt to apply a reasonable strategy that may be incomplete or underdeveloped. It may or may not have the correct answer. The representations are fundamentally correct. The explanation and/or justification may
    be incomplete or missing. The response demonstrates a minimal understanding and analysis of the problem.
  • A score of 0 indicates the work is completely incorrect or irrelevant. There may be no response or the response may state "I don't know."
     
    Mathematics Goal, Expectation, Learning Indicator:
    1.2.3 The student will solve and describe if and where two straight lines intersect using numbers, symbols, and/or graphs.
1.2.5 The student will apply formulas and/or use matrices (arrays of numbers) to solve real-world problems.
     
   

Instructional Implications:
   

These brief constructed responses in mathematics require that students have extensive instructional opportunities to work with mathematical and real-world problems using measurement, algebraic, and and geometric models. In this example, students should have had a rich set of experiences modeling and interpreting real-world situations, using the language of mathematics and appropriate technology, including applying formulas and using matrices and graphic representations to solve real-world problems. Teachers must become familiar with the scoring rubric used and practice using the rubric when grading teacher-designed classroom-based tests.

Sample brief constructed response item: Time limit - 5 minutes

 

Ted wants to buy a bike. The bike costs $400. He has two options for payment.

Option One:

Ted can borrow the $400 from his father and repay him $40 a month for a year.

Option Two:

The bike shop will finance the bike for one year at a 15% annual interest rate. The formula for the total amount paid (a) is a = p + prt, where p is the amount borrowed, r is the rate of interest, and t is the time in years.

Which option would cost Ted the least amount of money? Explain how you determined your answer. Use words, symbols, or both in your explanation.
     
    Solution:
    Financing the bike through the bike shop ($460) will cost less than borrowing the money from his father ($480).

For a brief constructed response, the student will be graded in a rubric scale of 0-3. The criteria includes:

  • Analysis - Did the student show understanding of the problem?
  • Representation - Did the student appropriately display and identify information?
  • Application - Did the student use appropriate concepts and strategies to solve the problem?
  • Explanation - Did the student use words and/or symbols to explain the process used to solve the problem?
  • Justification - Did the student use definitions, postulates, theorems of mathematics to verify the solution to the problem?
     
    Scoring Rubric:
   
  • A score of 3 indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. The representations are correct. The explanation and/or justification are logically sound, clearly presented, fully developed, support the solution, and do not contain significant mathematical errors. The response demonstrates a complete understanding and analysis of the problem.
  • A score of 2 indicates application of a reasonable strategy that may be incomplete or underdeveloped. The representations are essentially correct. The explanation and/or justification support the solution, but may not be well developed, are plausible, and or may be incomplete. The response demonstrates a conceptual understanding and analysis of the problem.
  • A score of 1 indicates little or no attempt to apply a reasonable strategy that may be incomplete or underdeveloped. It may or may not have the correct answer. The representations are fundamentally correct. The explanation and/or justification may
    be incomplete or missing. The response demonstrates a minimal understanding and analysis of the problem.
  • A score of 0 indicates the work is completely incorrect or irrelevant. There may be no response or the response may state "I don't know."
     
    Mathematics Goal, Expectation, Learning Indicator:
    1.2.3 The student will solve and describe if and where two straight lines intersect using numbers, symbols, and/or graphs.
1.2.5 The student will apply formulas and/or use matrices (arrays of numbers) to solve real-world problems.
     
    Instructional Implications:
    Once again, this example emphasizes that the Maryland state assessments in mathematics will require students to engage in ongoing real-world applications of mathematical algorithms and problem-solving processes. In all constructed responses, students must demonstrate accurate understanding and use of mathematical operations, provide clear and complete evidence to justify responses, and display the ability to synthesize complex information. Teachers must become familiar with the scoring rubric used and practice using the rubric when grading teacher-designed classroom-based tests.

Sample student produced response  item: 

 

Sarah has $135 in her savings account. Tracy has $150 in her savings account. Each week Sarah saves $8 and Tracy saves $5. After how many weeks will they each have the same amount of money?
     
    Answer: 5 weeks
   
    Mathematics Goal, Expectation, Learning Indicator:
    1.1.2 The student will represent patterns and/or functional relationships in a table, as a graph, and/or by mathematical expression.

1.2.3 The student will solve and describe if and where two straight lines intersect using numbers, symbols, and/or graphs.
     
    Instructional Implications:
    Students must learn parameters of and correctly use special (bubble) grid answer forms in practice situations and on classroom based tests.

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This site was developed by the Department of Staff Development, in collaboration with the Division of Instruction. Questions, comments, and other inquiries may be addressed to Allene Chriest (achriest@pgcps.org) or Jeff Maher  (jmaher@pgcps.org).