My 8th-grade son is doing math homework.
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My 8th-grade son is doing math homework. For the problem -2x + 7 <= 15, he subtracted 7 to get -2x <= 8, then divided both sides by -2 and wrote x <= -4.
I need to sign off tonight, but I want him to catch the mistake himself instead of me handing him the answer key.
Can you give me:
1. What the misconception name is.
2. 1-2 diagnostic questions I can ask him out loud, followed by how I will know each question landed.
3. One changed-number parallel problem (withholding the final answer) for him to redo tonight.
4. One clarifying question if anything would change how we approach this?
Answer
Here's what you asked for, in the order you asked it.
**The misconception name** (so you can say it out loud in one sentence):
"Forgot to flip the inequality sign when dividing both sides by a negative number." He went `-2x ≤ 8` → `x ≤ -4`, but the ≤ should turn into ≥ because -2 is negative.
**Two diagnostic questions — each followed by how you'll know it landed:**
1. *"You divided both sides by -2. What does the rule say happens to the ≤ when you divide by a negative?"*
- **How you'll know it landed:** He says the sign flips, or draws the arrow turning the other way. If he just says "nothing happens," the rule isn't in his head yet — that's a different conversation to have before any redo.
2. *"Try x = -5 in the original -2x + 7 ≤ 15. Does it work?"*
- **How you'll know it landed:** He computes -2(-5)+7 = 17 and writes 17 ≤ 15, sees it's false. He realises x = -5 shouldn't be in the answer, so the answer can't end at -4 with everything smaller included. The boundary is right; the direction is wrong.
**Changed-number parallel (answer withheld):**
Solve `-3x + 4 ≤ 13`.
(Same shape: subtract 4, divide by -3, and that division step is the one with the sign flip.)
**One clarifying question that would change tonight's approach:**
"Did he write the sign-flip on paper, or just do the divide in his head?" If he skipped writing it, the habit to fix tonight isn't the rule — it's "write the sign rule next to every divide-by-negative step." That's worth a sentence of yours before he starts the redo.
One last thing for the sign-off: after he gets the parallel right, have him plug his answer back into the original. That's the cheapest way to make the rule stick without you re-teaching it.