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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 2003

The Knitwear Company wants to know what sock colors teenagers will buy. The company is considering the three options below for collecting data.

Option 1: Spend five days in the sock department of a randomly selected store and ask every person who enters what color socks they buy.

Option 2: Spend one day in the sock departments of five randomly selected stores in randomly selected cities and ask every teenager who enters what color socks they buy.

Option 3: Spend one day in eight randomly selected shopping centers and ask ten randomly selected teenagers at each shopping center what color socks they buy.

Complete the following in the Answer Book:

  • Which option gives the company the most representative sample of sock colors that teenagers will buy?
  • Use mathematics to justify why the other two options do not give the company the most representative sample.

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 correctly chooses Option 2 as the most representative sample of sock colors. The justification hints at the concept of variety ("asking different people in different cities to get different opinions"), but lacks the necessary specificity that would apply to this problem. 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 correctly chooses Option 2 as the most representative sample of sock colors. The statement "this is because with option 2 your spending the day asking every teenager not just random selections such as in option 3" is an unclear justification for why Option 3 is not the most representative. The justification for Option 1 is logically sound. ("Also, option one won't be that good either because in this one you asking everyone, meaning even non-teenagers.") 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 1

Annotation: This response indicates little attempt to apply a reasonable strategy. The student correctly chooses Option 2 as the most representative sample of sock colors. The justification for Option 3 is a restatement of the question. ("Option 3 says randomly ten teenagers what color sock they like.") The justification for Option 1 ("Option 1 says every person and they want teenagers") identifies that the wrong population is being sampled. This response demonstrates a minimal 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 correctly chooses Option 2 as the most representative sample of sock colors. No justification is needed for why Option 2 gives the most representative sample, but the student correctly states that Option 2 "is representing many cities + not just one." This shows an understanding that variety is necessary in a representative sample. The justification for Option 1 correctly states that the wrong population is being sampled ("they are asking every person that enters the store; so that isn't representative of teenagers"). The justification for Option 3 is plausible, but not fully developed, since it does not recognize that the sample size for Option 3 is smaller than the sample size for Option 2. 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 correctly chooses Option 2 as the most representative sample of sock colors. The justification for Option 3 is that the sample size is too small ("only giving them eighty teenagers"). The justification for Option 1 is that the wrong population is sampled ("in option one they were going to ask every person who walks into the store. Even if they were seventy or five and they're just looking for teenagers"). Although both of these justifications support the solution, neither addresses the concept of variety. 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 chooses Option 2 as the most representative sample of sock colors. The response shows a clear understanding that variety is necessary in a representative sample ("sample 2 takes place in 5 randomly selected stores. That would incorporate a wide variety of opinion"). The justification for Option 3 ("Option 3 would only be asking 80 teens total") is clearly presented and indicates that the sample size is too small. Fully developed justification is given for Option 1 ("Option 1 asks everybody who enters the 1 store whether they're teenagers or not"), which indicates that the wrong population is being sampled. 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 chooses Option 2 as the most representative sample of sock colors. A fully developed justification, which indicates that the wrong population is being sampled, is given for Option 1. ("Option 1 says ask every person who enters; every person is not a teenager.") The justification for Option 3 is logically sound ("they're only getting eighty opinions") and indicates that the sample size is too small. The student also shows an understanding that variety is necessary to a representative sample by referencing the fact that Option 2 takes place in multiple cities. ("They would be getting the opinion of more teenagers in different cities.") 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 |