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Goal 3 Data Analysis And Probability

Expectation 3.2 The student will apply the basic concepts of statistics and probability to predict possible outcomes of real-world situations.

Indicator 3.2.1 The student will make informed decisions and predictions based upon the results of simulations and data from research.

Extended Constructed Response (ECR) Item - Released in 2004

A city bus company requires exact change or a token to ride a bus. The company conducted a random survey of 100 passengers to determine how they paid their bus fare. The survey results are shown in the table below.

Complete the following in the Answer Book:

  • Last week 2,000 passengers rode the bus. Based on this survey, how many of these passengers would be expected to use exact change to pay the bus fare? Use mathematics to explain how you determined your answer. Use words, symbols, or both in your explanation.
  • Mr. Blake, a bus driver for the city bus company, reported that 44 passengers who rode on his bus one morning paid using a token. Based on this survey, estimate how many passengers rode on his bus that morning. Use mathematics to explain how you determined your answer. Use words, symbols, or both in your explanation.
  • Should a different city use these survey results to predict the number of bus passengers that would be expected to use exact change to ride the bus in their city? Use mathematics to justify your answer.

The following 4 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 number of exact-change passengers (1260) is correct. The explanation supports the solution (multiplying 100 by 20 and you get 2000. Then you multiply both 63 + 37 by 20 because that's how many times 100 goes into 2000). The number of passengers on Blake's bus (114) is incorrect. The explanation reveals a flaw in reasoning. (There's a difference of 7 between 37 + 44 {people using tokens} so I figured that the people paying exact change would go up by 7 too.) Although the student indicates that the survey's results should be used, the justification reveals a flaw in reasoning. This response demonstrates a minimal understanding and analysis of the problem. Compare to Anchor Paper #2.
Sample Student Response #2

image of student response

Score for Sample Student Response #2: Rubric Score 4

Annotation: This response indicates application of a reasonable strategy that leads to a correct solution within the context of the problem. The number of exact-change passengers (1,260) is correct. The explanation is clearly presented, fully developed, and supports the solution (63/100 passengers used exact change. I converted this to 63% or .63. Of the 2000 passengers, 63%, or 1,260 people, could be expected to use exact change). The number of passengers on Blake's bus (around 119) is correct. The explanation is fully developed (37% of the passengers who rode the bus used tokens. This meant that those 44 people were 37% of the people who had ridden the bus. I set up the equation .37x=44 and solved). The student indicates that the survey's results should not be used, and the fully developed justification supports the solution. (The people from that city probably use different methods of transportation, etc. and can be expected to use different methods to pay fares.) This response demonstrates a complete understanding and analysis of the problem. Compare to Anchor Paper #8.
Sample Student Response #3

image of student response

Score for Sample Student Response #3: Rubric Score 2

Annotation: This response indicates application of an incomplete strategy. Both the number of exact-change passengers (1,260) and the number of passengers on Blake's bus (119) are correct. However, explanations are not provided. The student indicates that the survey's results should not be used. The justification supports the solution, but is incomplete (people may buy more tokens there, or they may not have bus tokens). This response demonstrates a conceptual understanding and analysis of the problem.


Sample Student Response #4

image of student response

Score for Sample Student Response #4: Rubric Score 3

Annotation: This response indicates application of a reasonable strategy that leads to some correct solutions. The number of exact-change passengers (1,260) is correct. The explanation supports the solution (finding that 63% used exact change and 37% used tokens. Then I found 63% of 2000 {2000 x .63} and got 1,260 passengers). The number of passengers on Blake's bus (119) is correct, and the explanation supports the solution (doing the equation 44=.37x, letting x= the number of passengers total who rode that morning). The student indicates that the survey's results should be used. The justification reveals a flaw in reasoning (no matter what they factor in {population, not as many buses, etc.}, the percentages will remain the same and could predict how many use exact change). This response demonstrates a clear understanding and analysis of the problem. Compare to Anchor Paper #7.

Additional Resources

Anchor Papers used in scoring

Extended Constructed Response (ECR) Rubric
Print: Scoring Rubric (pdf)
Score 4

The response indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. The representations are 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 3

The response indicates application of a reasonable strategy that may or may not lead to a correct solution. The representations are essentially correct. The explanation and/or justification is generally well developed, feasible, and supports the solution. The response demonstrates a clear understanding and analysis of the problem.

Score 2

The response indicates an incomplete application of a reasonable strategy that 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 application of a reasonable 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 3.2.1:
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