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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 2002

Apples from an orchard need to be sampled to see if they are sweet enough for juice. The orchard has 25,000 apple trees. Each tree has at least 1,000 apples.

Alfred suggests, “Randomly choose one tree and test 500 apples from that tree.”

Betsy suggests, “Randomly choose 1,000 trees and test one apple from each tree.”

Complete the following in the Answer Book:

  • Whose suggestion gives a sample that is more representative of the apples in the orchard? Use mathematics to justify your answer.

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 application of a reasonable strategy. The student correctly selects Betsy's suggestion as giving a more representative sample of the apples in the orchard. The justification given is relevant ("you will have to take 500 more apples than Alfred's idea") but only references the larger sample size of apples in Betsy's method. There is no mention of variety or bias. This 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 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 selects Betsy's suggestion as giving a sample that is more representative of the apples in the orchard. In the justification the student states that Betsy uses "a wide range of trees," indicating variety. The bias in Alfred's method also is discussed ("If you test one tree, you may see the same results in all 500 apples, good or bad"). The student indicates that Betsy's suggestion better represents the "whole" orchard ("...you'll get a better idea of the orchards quality of apples"). The justification is fully developed and supports the solution. This response demonstrates a complete understanding and analysis of the problem.


Sample Student Response #3

image of student response

Score for Sample Student Response #3: Rubric Score 2

Annotation: This response indicates an incomplete application of a reasonable strategy. The student correctly selects Betsy's suggestion as giving a sample that is more representative of the apples in the orchard. In the justification, the student states that Betsy's method gives "a variety" which is needed for a representative sample. However, while this justification supports the solution, it is incomplete. In addition, the student references the bias in Alfred's suggestion ("one tree may not produce good apples and you could get that one"). However, no mention is made of the fact that the sample should represent the whole orchard. This response demonstrates a conceptual 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 |