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Student: The majority of the 50 students in our class answered at least 80% of the questions correctly on last year’s Algebra I final exam. If these final exam scores do accurately measure a student’s level of understanding, Marc must have learned less about algebra last year than most other students in our class, because he answered only 75% of the questions correctly on last year’s Algebra I final exam.
The student argues that since more than half of the students in the class scored an 80% or better on the Algebra exam, and since Marc only scored a 75%, then Marc must have “learned less” about Algebra than most of the other students in the class. Unfortunately for the student, this requires a dangerous assumption: because Marc ended the year slightly behind most other students in terms of percentage score (75% to 80%), he must have made less progress during the year than most other students. In other words, the phrase “learned less” implies that someone makes less progress over time, and that may not necessarily be the case here.
Let’s consider an example:
Say you were to ask five people to train for a one-mile race for two months. At the end of those two months, you time them as they run the mile and you record their results. Runners 1, 2, 3, and 4 each finish in exactly 6 minutes. Runner 5, however, takes 10 minutes to complete the race. Would it be fair to conclude that Runner 5 was the least improved runner over the course of those two months? Not necessarily. What if Runners 1-4 could already run a mile in 7 minutes prior to any training, whereas Runner 5 needed 30 minutes to run a mile two months ago? Now it seems clear that, while Runner 5 can still be described as the slowest runner in the group, saying that he or she is the least improved would be inaccurate. So the key when trying to gauge progress is to have a starting point to reference so you can truly measure how far someone has come.And the same is true of Marc in the stimulus. Certainly, he was outscored on the exam by most of the students, but does that mean he learned less over the course of the year? We cannot conclude that unless we know where he started relative to everyone else. So to weaken this student’s claim that Marc learned less, we need an answer choice that suggests he made more progress (started further back) than the majority of his classmates.
Quote:
A) Seven students answered less than 75% of the questions correctly on the final exam in Algebra I last year.
This answer choice places Marc fairly low in the group of 50 students (only 7 of 50 scored worse than him), but this still does not impact the idea of how much he learned. Hence, this answer does not weaken the argument.
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(B) Marc is one of four students in the class who did not take an introductory-level algebra course offered by the school two years ago.
This is the correct answer. If Marc and three other students did not take the introductory-level Algebra course, and the other 46 students all did, then it seems likely that Marc would have started the Algebra I class knowing less about the subject than his classmates. If that is
the case then his final score of 75% could certainly represent much more learned (greater progress) over the course of the year than his classmates who scored 80% or better. Again, numbers can often make these ideas easier to grasp. Say that Marc, having missed the introductory course, began the year only knowing 10% about Algebra I and finished with a 75% (65% improvement). Most of his classmates however, having taken the introductory course, started the year at 50%. Even if they all finished at 90%, that’s still only a 40% improvement, which pales in comparison to Marc’s 65% increase. Clearly, even though Marc may not have finished in the top-half of his class, he still could have learned more than those who outscored him.
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(C) Marc is one of three students who answered exactly 75% of the questions correctly on the final exam in Algebra I last year.
This answer choice, like (A), only addresses where Marc finished relative to some of his classmates. Since we need an answer choice related to Marc’s progress over the course of the year, this answer cannot be correct.
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(D) The teacher estimated that last year’s ninth-grade Algebra I final exam was roughly twice as difficult as this year’s Algebra I final exam.
The overall difficulty of the exam relative to other exams is completely irrelevant to Marc performance or his progress relative to his classmates.
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(E) Only three students spent less time than Marc spent answering the questions on last year’s Algebra I final exam.
The amount of time that Marc (or anyone else) spent answering questions is also irrelevant to how much he ultimately learned during the course relative to his classmates, so this answer is incorrect.