Ordering Negative Rationals: Diagnosing a Sign-Magnitude Error
A seventh-grade team is moderating a 4-point constructed-response item: 'Order -2/3, -0.6, and -7/10 from least to greatest. Explain your reasoning.' The scoring guide awards 1 point for correct order, 2 for correct order with valid comparison, and 1 additional point for a number-line or equivalent representation. Student A writes '-0.6, -2/3, -7/10' and explains, '0.6 is the smallest decimal, then 0.67, then 0.7, so that is least to greatest.' Student B writes '-7/10, -2/3, -0.6' but her explanation says, 'The absolute values are 0.7, 0.67, and 0.6, so I put them in that order.' The team disagrees about whether Student B's explanation is sufficient and whether Student A should receive any credit. One teacher argues that Student A's error is just a careless sign mistake; another says it reveals a fundamental misconception about negative numbers. The team has 20 minutes to decide scoring and a follow-up task for the whole class.
1. Diagnose Student A's reasoning. What misconception is most likely, and how does it differ from a careless sign error? Recommend a scoring decision consistent with the guide.
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Student A is not making a random sign slip. The explanation shows a consistent rule: compare the positive decimal parts (0.6 < 0.67 < 0.7) and list them in that order, then attach negative signs without reversing the order. On a number line, -0.7 is farthest left, then -2/3 ≈ -0.667, then -0.6. A careless sign error would typically produce an isolated inversion or an otherwise correct comparison; A's explanation justifies the entire reversed order. With the guide as written, A earns 0 of 4: the order is wrong, the comparison is invalid, and no representation is given. The teacher should not award credit for 'effort' under this rubric, but should use the response as evidence of a sign-magnitude misconception, not carelessness.
2. Student B's order is correct, but her explanation says the absolute values are 0.7, 0.67, and 0.6 'so I put them in that order.' Is this a valid justification? Explain what she may be assuming and what a complete response requires.
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B's order is correct, but the explanation is incomplete. Listing absolute values 0.7, 0.67, 0.6 in descending order can be a valid strategy only if B makes the reversal explicit: because 0.7 > 0.67 > 0.6, their opposites satisfy -0.7 < -0.667 < -0.6. As written, 'so I put them in that order' could mean she ordered the absolute values and then kept that order, which would be wrong; the correct final order suggests she may have reversed mentally but did not communicate it. The teacher should award the order point but not full explanation credit unless the reversal is stated or shown on a number line. A good follow-up is to ask B to place -0.7 and -0.6 on a number line and explain which is farther from zero and which is less.
3. Design a 10-minute follow-up that addresses both students without simply reteaching the rule. Include one representation and one question that would surface the misconception.
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Use a vertical number line labeled as elevation or account balance, with 0 at sea level or $0. Have students plot -0.6, -2/3, and -0.7, then ask: 'Which number is farthest from zero? Which is least? Why are those different questions?' Pair A and B to compare their explanations. Then give a boundary case: 'Name a number between -0.7 and -0.6 and explain how you know.' This surfaces the reversal rule without stating it first. Close with a quick exit ticket ordering -3/4, -0.8, and -0.79 and explaining in one sentence. The representation and the boundary question target the misconception rather than just reteaching 'negative numbers are backwards.'