The Answer Key That Was Wrong
An Algebra II teacher grades a quiz on absolute-value inequalities. Item 4 asks students to solve |2x - 3| < 7. The answer key, written two years ago, says x < 5. During review, a student shows work: -7 < 2x - 3 < 7, then -4 < 2x < 10, then -2 < x < 5. Two other students got x < 5 by dropping the absolute-value bars and solving only 2x - 3 < 7. A parent has emailed asking why her daughter's correct interval was marked wrong, and grades post to the portal tomorrow at 8 a.m. The teacher has 24 students and one 47-minute planning period today. The department shares a common quiz bank, and the same key has been used by two other teachers.
1. Which solution is mathematically correct, and what exactly is wrong with the key? Show how you would verify rather than assert.
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The correct solution is -2 < x < 5. The key's x < 5 is the right-hand restriction only; it is a superset of the true solution set, so it both marks correct answers wrong and accepts extraneous ones.
Verification by test points is the cleanest check and is also what to show the parent and colleagues:
- x = 0: |2(0) - 3| = 3 < 7, true. Both answers accept it.
- x = 4: |8 - 3| = 5 < 7, true. Both answers accept it.
- x = -4: |-8 - 3| = 11, not < 7, false. The key accepts x = -4; the student's answer rejects it.
So the student's interval and the key disagree, and the key is the wrong one. The underlying content point is that |a| < b is equivalent to -b < a < b precisely because absolute value is a distance from zero; a single linear inequality cannot express a bounded interval. The teacher's judgment here is to own the error quickly and specifically, not to soften it into 'a different method.'
2. The two students who dropped the bars landed on the key's answer. What does that coincidence tell you about their understanding, and how would you probe it without simply handing them the rule?
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The coincidence is diagnostic rather than reassuring. They produced the accepted answer with invalid reasoning, and the key masked it. If the key had been right, their work would have been flagged; because the key was wrong, their misconception was rewarded.
Probe with a counterexample before any re-teaching: ask both students to test x = -10 in |2x - 3| < 7. They will get 23 < 7, which is false, while their own answer says x = -10 is a solution. That creates the cognitive conflict that a stated rule does not.
Then connect to the distance interpretation on a number line: |2x - 3| is the distance between 2x and 3, and 'less than 7' means 2x is within 7 units of 3, i.e., between -4 and 10. That re-derives the compound inequality instead of memorizing it. Finally have them re-solve a same-structure item (|3x + 1| < 8) and one inequality case (|x - 2| > 5) so the bounded-interval idea is distinguished from the unbounded case. The judgment is to treat this as a reasoning repair for a subset of students, not a whole-class reteach.
3. Plan the next 24 hours: portal grades, the parent reply, the two colleagues using the same key, and tomorrow's class. What are the decision points and risks?
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Priority order is dictated by hard deadlines and by who is harmed by delay.
- Before 8 a.m. posting: re-score every student's item 4 against the correct interval, including partial credit decisions, and record who was marked down and who was wrongly credited. Posting known-wrong grades is the one irreversible error available today.
- Parent reply today: state plainly that the key was incorrect, that the daughter's work was right, that the grade is corrected, and what will change. Do not attribute the error to the student or to 'a different method.' Specificity buys credibility.
- Colleagues: notify the two other teachers immediately and in writing, with the corrected solution and the item number, because their students are affected and further quiz-bank errors may exist. Suggest a short audit of the absolute-value items rather than a vague warning.
- Tomorrow's class: open with the x = -10 counterexample as a bellringer, since students who got it right by accident need the same repair.
Risk to manage: the wrongly credited students. Correcting their scores is required for accuracy but will feel arbitrary unless the reasoning is taught first. Sequence the re-teach before the grade correction is discussed, and keep the tone on the mathematics.