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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 2006

A cereal company awards a prize to anyone who collects all 5 different game pieces. Each game piece has an equal chance of being placed in a box of cereal. Each box of cereal contains only 1 game piece.

Complete the following in the Answer Book:

  • Design a simulation of at least 30 trials that can be used to estimate how many boxes of cereal a person needs to buy to collect all 5 different game pieces. Describe how to collect and interpret the data so that you have confidence in the estimation.

The following 12 Sample Student Responses represent a range of score points.

Sample Student Response #1

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Score for Sample Student Response #1: Rubric Score 1

Annotation: This response indicates little attempt to apply a reasonable strategy. The student designs a simulation using a spinner with five equal parts, each representing a different game piece. However, the student only describes conducting one trial instead of thirty (spin it 30 times or until your spinner lands on every space at least once). No interpretation of the data is provided. The response demonstrates a minimal understanding and analysis of the problem. (Compare to this Level 1 anchor paper.)


Sample Student Response #2

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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. A simulation is designed using a spinner with five equal sections numbered 1-5. The student understands that each trial should last (until every number has been spun) and that thirty trials should be performed. The student correctly describes how to interpret the data so you have confidence in the estimation (Add up the amount of times it took to get all the numbers on the spinner on all 30 trials. Then divide that total by 30). The response demonstrates a complete understanding and analysis of the problem. (Compare to this Level 3 anchor paper.)


Sample Student Response #3

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Score for Sample Student Response #3: Rubric Score 2

Annotation: This response indicates an incomplete application of a reasonable strategy. The student designs a simulation using a five section spinner, with each section representing a different game piece. The student understands that each trial lasts (until you get every game peice) and that thirty trials should be conducted. However, the interpretation of the data is not complete (count how many times it took you to get all of them). The response demonstrates a conceptual understanding and analysis of the problem. (Compare to this Level 2 anchor paper.)


Sample Student Response #4

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Score for Sample Student Response #4: Rubric Score 2

Annotation: This response indicates an incomplete application of a reasonable strategy. The student designs a simulation using a random number generator and the numbers 1-5. The student understands that each trial lasts (until all 5 numbers appear) and that thirty trials should be conducted. The interpretation of the data is not complete (however many times it takes for all five numbers to appear during a trial would be how many boxes it may take to buy to get all of the prizes). The response demonstrates a conceptual understanding and analysis of the problem. (Compare to this Level 2 anchor paper.)


Sample Student Response #5

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Score for Sample Student Response #5: Rubric Score 1

Annotation: This response indicates little attempt to apply a reasonable strategy. The student designs a simulation using a random number generator with the numbers 1-5 each representing a game piece. The student gives an example of thirty integers generated but this does not show understanding of a trial or that thirty trials should be conducted. The response demonstrates a minimal understanding and analysis of the problem. (Compare to this Level 1 anchor paper.)


Sample Student Response #6

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Score for Sample Student Response #6: Rubric Score 2

Annotation: This response indicates an incomplete application of a reasonable strategy. The student designs a simulation using five colored or numbered pieces of paper in a bag, each representing a different game piece. The student understands that one trial lasts until each game piece is selected (draw out of the bag until I have one of every color) and that thirty trials need to be conducted. The student begins the interpretation of the data but it is not complete (count how many times it took you to get all 5). The response demonstrates a conceptual understanding and analysis of the problem. (Compare to this Level 2 anchor paper.)


Sample Student Response #7

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Score for Sample Student Response #7: Rubric Score 2

Annotation: This response indicates an incomplete application of a reasonable strategy. The student designs a simulation using a random number table, with two numbers representing each game piece. The student understands that each trial should last (untill you have all 5 prizes) and that 30 trials should be conducted. However, the interpretation of the data is incomplete (record how many numbers were used when all 5 prizes were found). The response demonstrates a conceptual understanding and analysis of the problem. (Compare to this Level 2 anchor paper.)


Sample Student Response #8

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Score for Sample Student Response #8: Rubric Score 1

Annotation: This response indicates little attempt to apply a reasonable strategy. The student gives a partial simulation design by assigning each game piece a number and then randomly selecting the numbers. However, there is no description of how to randomly select the numbers. The student understands that each trial should last until (you have recieved all five numbers) and that 30 trials should be performed. There is no indication of how to interpret the data so that you have confidence in the estimation. The response demonstrates a minimal understanding and analysis of the problem. (Compare to this Level 1 anchor paper.)


Sample Student Response #9

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Score for Sample Student Response #9: Rubric Score 3

Annotation: This response indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. A simulation is designed using a random number generator and the numbers 1-5, with each number representing a different game piece. The student understands that each trial should last until you (get numbers 1 through 5) and that thirty trials should be conducted. The student correctly describes how to interpret the data by finding the mean of the thirty trials. The response demonstrates a complete understanding and analysis of the problem. (Compare to this Level 3 anchor paper.)


Sample Student Response #10

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Score for Sample Student Response #10: Rubric Score 2

Annotation: This response indicates an incomplete application of a reasonable strategy. The student designs a simulation using (50 cards equaly marked with numbers 1-5 on them to simulate the 5 different game pieces) in a bowl. The student understands how long each trial should last (until you get all five game pieces) and that 30 trials should be conducted, but the student does not include replacing the card after each time a card is picked out of the bowl. The student interprets the data by using (the measures of central tendency to find the average number of boxes to get all five game pieces). The response demonstrates a conceptual understanding and analysis of the problem. (Compare to this Level 2 anchor paper.)


Sample Student Response #11

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Score for Sample Student Response #11: Rubric Score 1

Annotation: This response indicates little attempt to apply a reasonable strategy. The student designs a simulation using a spinner with 5 equal sections (which each represent a different game peice). It is unclear whether the student understands that each trial lasts until each section is landed on at least once, and that thirty trials should be conducted. There is no indication of how to interpret the data so that you have confidence in the estimation. The response demonstrates a minimal understanding and analysis of the problem.


Sample Student Response #12

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Score for Sample Student Response #12: Rubric Score 2

Annotation: This response indicates an incomplete application of a reasonable strategy. The student designs a simulation using random integers 1 through 5, with each integer representing a different game piece, but does not specifically state how the numbers are randomly selected. The student understands that each trial lasts (until you finally recieve all five numbers) and that thirty trials should be conducted. The interpretation of the data is not complete (record how many times it took you to get all of them). The 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 |