Anchor Papers ~ Algebra/Data Analysis ~ Item 11


	Item 11 Anchor Papers

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.

Score Level 1 Anchor Paper
This response indicates little application of a reasonable strategy. The number of exact-change passengers (1260) is correct, and the explanation supports the solution. (I found the ratio {63/100} of the 63 people who paid exact to the 100 random passengers. Then I multiplied the ratio by 2000 to get the ratio 1260/2000.) The student provides no response to the second or third parts of the question. This response demonstrates a minimal understanding and analysis of the problem.
Score Level 1 Anchor Paper
This response indicates little application of a reasonable strategy. The number of exact-change passengers (1260) is correct. The explanation supports the solution (63% of 2000/63 people using exact change/2000-people rode the bus). The number of passengers on Blake's bus (2,140) is incorrect. The explanation reveals an inappropriate strategy (44% of 2000 + 63% of 2000). 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.
Score Level 2 Anchor Paper
This response indicates application of an incomplete strategy. The number of exact-change passengers (1260) is correct, and the explanation supports the solution. (I multiplied 2000 by .63.) The number of passengers on Blake's bus (114) is incorrect. The explanation reveals a flaw in reasoning (subtracting 37 from 44, then I added that answer {7} to 63 and got 70. I added 44 to 70 and got 114 passengers). The student indicates that the survey's results should not be used, and the justification supports the solution (people in different cities might do different things. In another city most may use a token instead of exact change). This response demonstrates a conceptual understanding and analysis of the problem.
Score Level 2 Anchor Paper
This response indicates application of an incomplete strategy. The number of exact-change passengers (1260) is correct. The explanation supports the solution (dividing 2000 by 100, the number of passengers surveyed and multiplying 20 by 63). The number of passengers on Blake's bus (about 72) is incorrect, and the explanation reveals a flaw in reasoning (If 63% of 44 is 28, then adding 28 and 44, about 72 passengers rode the bus). The student indicates that the survey should not be used. The justification supports the solution (the other cities could differ in the amount of people who ride the bus, the price of the ride, etc.). This response demonstrates a conceptual understanding and analysis of the problem.
Score Level 3 Anchor Paper
This response indicates application of a reasonable strategy that leads to some correct solutions. The number of exact-change passengers (1,260) is correct, and the explanation supports the solution. (I made a proportion with 63 over 100 and x over 2,000. I multiplied 63 · 2000 and then divided that by 100.) The number of passengers on Blake's bus (119) is correct. The explanation supports the solution (doing a proportion of 37 over 100 and 44 over x. I multiplied 100 · 44 and divided it by 37). No response is given to the third part of the question. This response demonstrates a clear understanding and analysis of the problem.
Score Level 3 Anchor Paper
This response indicates application of a reasonable strategy that leads to some correct solutions. The number of exact-change passengers (1260) is correct. The generally well developed explanation supports the solution. (If you divide 63 by 100, you get .63. You then multiply .63 by 2000.) The number of passengers on Blake's bus (119) is correct, and the explanation supports the solution. (If you divide 37 {# of people who use tokens} by 100, you get .37. Divide 44 by .37 to get 118.9.) Although the student indicates that the survey's results should be used, the justification reveals a flaw in reasoning. (Subtract 44 from 119 to see how many people use exact change.) While this strategy will provide the number of passengers on Blake's bus that paid using exact change, the student does not justify why the survey results can be applied to a different population. This response demonstrates a clear understanding and analysis of the problem.
Score Level 4 Anchor Paper
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 as well as how many passengers (~119) rode on Mr. Blake's bus. The symbolic explanations are clearly presented, fully developed, and support the solution; the student sets up appropriate proportions and solves for x. The student indicates that the survey's results should be used. The fully developed justification supports the solution. (Each city is different, and its people are different.) This response demonstrates a complete understanding and analysis of the problem.
Score Level 4 Anchor Paper
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 and fully developed. The student points out that 63 out of 100 passengers who used exact change (63/100) would equal x amount of people out of 2,000, sets up a proportion, and solves for x. The number of passengers on Blake's bus (~119) is correct. The explanation is fully developed and supports the solution (37 out of 100 people used tokens, and it would be equal to 44 people out of x people who rode the bus). The student then sets up a proportion and solves for x. The student indicates that the survey's results should not be used. The fully developed justification supports the solution (they might have a different population or another form of transportation. The environment is different, so the data will be different). This response demonstrates a complete understanding and analysis of the problem.

Anchor Papers ~ Algebra/Data Analysis ~ Item 11

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.

Score Level 1 Anchor Paper

This response indicates little application of a reasonable strategy. The number of exact-change passengers (1260) is correct, and the explanation supports the solution. (I found the ratio {63/100} of the 63 people who paid exact to the 100 random passengers. Then I multiplied the ratio by 2000 to get the ratio 1260/2000.) The student provides no response to the second or third parts of the question. This response demonstrates a minimal understanding and analysis of the problem.

Score Level 1 Anchor Paper

This response indicates little application of a reasonable strategy. The number of exact-change passengers (1260) is correct. The explanation supports the solution (63% of 2000/63 people using exact change/2000-people rode the bus). The number of passengers on Blake's bus (2,140) is incorrect. The explanation reveals an inappropriate strategy (44% of 2000 + 63% of 2000). 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.

Score Level 2 Anchor Paper

This response indicates application of an incomplete strategy. The number of exact-change passengers (1260) is correct, and the explanation supports the solution. (I multiplied 2000 by .63.) The number of passengers on Blake's bus (114) is incorrect. The explanation reveals a flaw in reasoning (subtracting 37 from 44, then I added that answer {7} to 63 and got 70. I added 44 to 70 and got 114 passengers). The student indicates that the survey's results should not be used, and the justification supports the solution (people in different cities might do different things. In another city most may use a token instead of exact change). This response demonstrates a conceptual understanding and analysis of the problem.

Score Level 2 Anchor Paper

This response indicates application of an incomplete strategy. The number of exact-change passengers (1260) is correct. The explanation supports the solution (dividing 2000 by 100, the number of passengers surveyed and multiplying 20 by 63). The number of passengers on Blake's bus (about 72) is incorrect, and the explanation reveals a flaw in reasoning (If 63% of 44 is 28, then adding 28 and 44, about 72 passengers rode the bus). The student indicates that the survey should not be used. The justification supports the solution (the other cities could differ in the amount of people who ride the bus, the price of the ride, etc.). This response demonstrates a conceptual understanding and analysis of the problem.

Score Level 3 Anchor Paper

This response indicates application of a reasonable strategy that leads to some correct solutions. The number of exact-change passengers (1,260) is correct, and the explanation supports the solution. (I made a proportion with 63 over 100 and x over 2,000. I multiplied 63 · 2000 and then divided that by 100.) The number of passengers on Blake's bus (119) is correct. The explanation supports the solution (doing a proportion of 37 over 100 and 44 over x. I multiplied 100 · 44 and divided it by 37). No response is given to the third part of the question. This response demonstrates a clear understanding and analysis of the problem.

Score Level 3 Anchor Paper

This response indicates application of a reasonable strategy that leads to some correct solutions. The number of exact-change passengers (1260) is correct. The generally well developed explanation supports the solution. (If you divide 63 by 100, you get .63. You then multiply .63 by 2000.) The number of passengers on Blake's bus (119) is correct, and the explanation supports the solution. (If you divide 37 {# of people who use tokens} by 100, you get .37. Divide 44 by .37 to get 118.9.) Although the student indicates that the survey's results should be used, the justification reveals a flaw in reasoning. (Subtract 44 from 119 to see how many people use exact change.) While this strategy will provide the number of passengers on Blake's bus that paid using exact change, the student does not justify why the survey results can be applied to a different population. This response demonstrates a clear understanding and analysis of the problem.

Score Level 4 Anchor Paper

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 as well as how many passengers (~119) rode on Mr. Blake's bus. The symbolic explanations are clearly presented, fully developed, and support the solution; the student sets up appropriate proportions and solves for x. The student indicates that the survey's results should be used. The fully developed justification supports the solution. (Each city is different, and its people are different.) This response demonstrates a complete understanding and analysis of the problem.

Score Level 4 Anchor Paper

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 and fully developed. The student points out that 63 out of 100 passengers who used exact change (63/100) would equal x amount of people out of 2,000, sets up a proportion, and solves for x. The number of passengers on Blake's bus (~119) is correct. The explanation is fully developed and supports the solution (37 out of 100 people used tokens, and it would be equal to 44 people out of x people who rode the bus). The student then sets up a proportion and solves for x. The student indicates that the survey's results should not be used. The fully developed justification supports the solution (they might have a different population or another form of transportation. The environment is different, so the data will be different). This response demonstrates a complete understanding and analysis of the problem.