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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 7 Anchor Papers represent a range of score points and are used in conjunction with the rubrics to assess student responses.

Anchor Paper #1

image of student response

Score for Anchor Paper #1: Rubric Score 1

Annotation: This response indicates little attempt to apply a reasonable strategy. The student has correctly selected Betsy's suggestion as giving a sample that is more representative of the apples in the orchard. The justification given ("you can test more trees out") is relevant, but references only the larger sample size of Betsy's method. There is no mention of variety or bias. This response demonstrates a minimal understanding and analysis of the problem.


Anchor Paper #2

image of student response

Score for Anchor Paper #2: Rubric Score 1

Annotation: This response indicates little attempt to apply a reasonable strategy. The student has correctly selected Betsy's suggestion as giving a sample that is more representative of the apples in the orchard. However, in the justification the student states, "it takes one apple from each tree to test." Whether the student is referring to the sample or the entire orchard is unclear. There is some relevant justification for Betsy's method when the student states that testing one apple from each tree would tell you about the rest of the apples on that tree. This response demonstrates a minimal understanding and analysis of the problem.


Anchor Paper #3

image of student response

Score for Anchor Paper #3: Rubric Score 2

Annotation: This response indicates an incomplete application of a reasonable strategy. The student has correctly selected Betsy's suggestion as giving a sample that is more representative of the apples in the orchard. Although the student correctly recognizes that Betsy's method offers a "variety of apples" and addresses sample size "more apples," no mention is made of the problem of bias in Alfred's suggestion or of the fact that the whole orchard should be represented by the sample. Therefore, the justification given supports the solution but is not complete. This response demonstrates a conceptual understanding and analysis of the problem.


Anchor Paper #4

image of student response

Score for Anchor Paper #4: Rubric Score 2

Annotation: This response indicates an incomplete application of a reasonable strategy. The student has correctly selected 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 "uses many different trees," indicating variety. In addition, there is the inference of bias in Alfred's method ("...instead of just one tree, which could even be a bad tree"). To receive a higher score, the idea of bias would need to be more clearly presented. This response demonstrates a conceptual understanding and analysis of the problem.


Anchor Paper #5

image of student response

Score for Anchor Paper #5: Rubric Score 2

Annotation: This response indicates an incomplete application of a reasonable strategy. The student has correctly selected 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 "will have more diversified results," indicating variety. The bias in Alfred's method also is discussed ("Alfred picked one tree the problem with that is maybe the tree had a bad year but all the others did good"). The justification supports the solution but is not complete. The student needs to reference the fact that the whole orchard should be represented by the sample. This response demonstrates a conceptual understanding and analysis of the problem.


Anchor Paper #6

image of student response

Score for Anchor Paper #6: 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 indicates that Betsy's method has variety and represents the whole orchard ("if you have a wider variety of trees then you know more about the whole orchard"). The bias in Alfred's method also is discussed. ("If you take 500 off one tree then you only know about that one tree.") The justification is clearly presented and supports the solution. This response demonstrates a complete understanding and analysis of the problem.


Anchor Paper #7

image of student response

Score for Anchor Paper #7: 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 states that "Betsy's suggestion is more representative of the apples in the orchard" and that "every tree's apples in that entire orchard..." This indicates understanding that the sample should represent the whole orchard. The idea of bias in Alfred's suggestion also is shown ("every trees apples don't taste the same"). The student describes Betsy's method as using "an assortment of trees," indicating variety. The justification is clearly presented and supports the solution. This response demonstrates a complete understanding and analysis of the problem.


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 |