15 of 24 on −7 − (−3)
A teacher gives a six-item exit ticket to 24 students. On the item −7 − (−3), 15 students chose −10, 6 chose −4, and 3 chose 4 - no one chose the correct answer. Work on the −10 papers shows "7 − 3 = 4, keep the negative." Several students say, "Subtracting two negatives makes a positive, so it's 4," then switch to −10 when the teacher says, "Check your sign." The teacher's plan is to spend tomorrow's 50-minute class reteaching "keep-change-change" and drilling 20 problems. The unit test, three days out, includes a temperature-change context item and an order-of-operations item with integers.
1. Analyze the three responses. Is the reteach plan adequate?
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Three different failure modes:
- −10 (15 students): they subtracted the magnitudes and kept a negative sign - treating −(−3) as −3. Consistent with a garbled version of "subtracting a negative makes it positive," and with a rule applied in the wrong direction.
- +4 (3 students): they converted to −7 + 3 but then computed 7 − 3 and dropped the sign. Sign management on the result, not on the operation.
- −4 (6 students): the operation was handled correctly (−7 + 3) but they were outvoted and, on cue, abandoned it. That is an authority-dependence problem, not a knowledge problem.
The plan is inadequate. The mnemonic is close to the source of the confusion: "subtracting two negatives makes a positive" is a misstatement - it is subtracting a negative that adds. Drilling 20 items under a shaky rule rehearses the same error faster. With 15/24 sharing one error, this is a class-wide model problem that needs a representation, not more items.
2. Design the 50 minutes: representation, contrasting pairs, and a context that matches the upcoming test.
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- Number line (10 min). Start at −7. Subtracting −3 means removing a leftward move of 3, so the point moves 3 units right → −4. Physically walk it.
- Counters / zero pairs (10 min). Show −7 as seven negative counters. To subtract −3, remove three negative counters → four remain → −4. Then a case needing zero pairs: 4 − (−4).
- Contrasting pairs (15 min). Compute side by side and sketch each: −7 − 3 vs −7 − (−3); 5 − (−2) vs −5 − 2; −2 − (−6) vs −2 − 6. Students state the direction of movement, not just the answer.
- Context matching the test (10 min). "At 6 a.m. the temperature was −7°F. By noon it had risen 3°F. Write and evaluate an expression for the change." Then the subtraction framing: the change is −4 − (−7); discuss why both representations agree.
- Order of operations (5 min). −7 − (−3) × 2 = −7 − (−6) = −1. Have students identify which operation happens first and why the subtrahend is −6.
Exit ticket: 3 items - one number-line justification, one context, one order-of-operations.
3. How should the unit-test items and your scoring be structured so that answer-only performance does not hide the same misconception?
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Item structure:
- One item requiring a sketch or written justification on the answer line ("show the move on a number line").
- One diagnostic pair - e.g., −6 − 3 and −6 − (−3) - placed adjacently, so the two responses can be compared directly.
- One zero-pair item (4 − (−4)) that cannot be done by magnitude subtraction.
- One context item, since the test includes temperature change.
- One order-of-operations item with a negative subtrahend.
Scoring: use a 2-point rubric per integer item - 1 point for the answer, 1 for a correct representation or explanation. Answer-only credit would rate the 15 students who wrote −10 as "nearly there" when their model is wrong; requiring the justification makes the error visible on the test itself.
Criterion for moving on: roughly 80% of the class producing correct answers with correct representations before the unit test. If justification scores lag answer scores, the class has learned to produce outputs without a model, and the mnemonic will fail on the transfer items.
On the mnemonic: it is not wrong as a shortcut, but it should be introduced after the number-line reasoning as a check, never as the entry point - otherwise the next unfamiliar case produces the same collapse.