School Improvement in Maryland
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Goal 3 Data Analysis And Probability

Expectation 3.1 The student will collect, organize, analyze, and present data.

Indicator 3.1.1 The student will design and/or conduct an investigation that uses statistical methods to analyze data and communicate results.

Assessment Limits:

  • The student will design investigations stating how data will be collected and justify the method.
  • Types of investigations may include: simple random sampling, representative sampling, and probability simulations.
  • Probability simulations may include the use of spinners, number cubes, or random number generators.
  • In simple random sampling each member of the population is equally likely to be chosen and the members of the sample are chosen independently of each other. Sample size will be given for these investigations.

Brief Constructed Response (BCR) Item - Released in 2005

A high school principal is meeting with the PTA to discuss the amount of time students spend doing homework each day. He asked a teacher to gather data for the parents. The teacher is going to survey 25 students from a student body of 1,000 students. Consider the methods of sampling shown below.

Method 1: randomly select 25 students from honors classes

Method 2: select 25 members of the girls' varsity softball team at random

Method 3: use a random number generator to generate 25 random numbers and use these to select 25 students from a numerical list of the students

Method 4: randomly select 25 classrooms and ask the teacher in each classroom to select a student

Complete the following in the Answer Book:

  • Which of the above methods of sampling would give the most representative random sample of the student population? Use mathematics to justify your answer. Include in your justification why you chose that method and why you did not choose each of the other three methods.

The following 3 Sample Student Responses represent a range of score points.

Sample Student Response #1

image of student response

Score for Sample Student Response #1: Rubric Score 1

Annotation: This response indicates little attempt to apply a reasonable strategy. The student incorrectly chooses Method 4 and gives no justification. Because it indicates a lack of variety, the justification for not selecting Method 1 (you would be choosing only the smart people) is plausible. The justification for not selecting Method 2 (girls do more homework than boys, and girls' softball is an all girls sport) indicates bias. The justification for not selecting Method 3 shows a lack of understanding of the method. The response demonstrates a minimal understanding and analysis of the problem.


Sample Student Response #2

image of student response

Score for Sample Student Response #2: Rubric Score 2

Annotation: This response indicates an incomplete application of a reasonable strategy. Although the student incorrectly chooses Method 4, the wrong method, some understanding is shown in the justification, which indicates that variety is something you want in a representative sample (you get results from all different students from all different grades). The justification for not selecting Method 1 (you would only survey honors students) is logical and indicates a lack of variety. The justification for not selecting Method 2 (would only be girls) also indicates a lack of variety. The justification for not selecting Method 3 (it would not be as easy) does not address representation of the sample and is not a valid reason. This response demonstrates a conceptual understanding and analysis of the problem.


Sample Student Response #3

image of student response

Score for Sample Student Response #3: Rubric Score 3

Annotation: This response indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. The student correctly chooses Method 3. The justification (everyone has an equally likely chance of being picked) is logically sound. The justifications for both Methods 1 and 2 are clearly presented and indicate a lack of variety. The justification for not selecting Method 4 (the teacher would probably pick her favorite student) is fully developed and indicates bias. The response demonstrates a complete understanding and analysis of the problem.


Additional Resources

Anchor Papers used in scoring

Brief Constructed Response (BCR) Rubric
Print: Scoring Rubric (pdf)
Score 3

The response indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. The representations are essentially correct. The explanation and/or justification is logically sound, clearly presented, fully developed, supports the solution, and does not contain significant mathematical errors. The response demonstrates a complete understanding and analysis of the problem.

Score 2

The response indicates application of a reasonable strategy that may be incomplete or undeveloped. It may or may not lead to a correct solution. The representations are fundamentally correct. The explanation and/or justification supports the solution and is plausible, although it may not be well developed or complete. The response demonstrates a conceptual understanding and analysis of the problem.

Score 1

The response indicates little or no attempt to apply a reasonable strategy or applies an inappropriate strategy. It may or may not have the correct answer. The representations are incomplete or missing. The explanation and/or justification reveals serious flaws in reasoning. The explanation and/or justification may be incomplete or missing. The response demonstrates a minimal understanding and analysis of the problem.

Score 0

The response is completely incorrect or irrelevant. There may be no response, or the response may state, “I don't know.”

Explanation refers to the student using the language of mathematics to communicate how the student arrived at the solution.

Justification refers to the student using mathematical principles to support the reasoning used to solve the problem or to demonstrate that the solution is correct. This could include the appropriate definitions, postulates and theorems.

Essentially correct representations may contain a few minor errors such as missing labels, reversed axes, or scales that are not uniform.

Fundamentally correct representations may contain several minor errors such as missing labels, reversed axes, or scales that are not uniform.

Last Revised 8/16/00
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Resources for 3.1.1:
Skill Statements | PUBLIC RELEASE ITEMS | Lesson Plans |