We live in Washington, USA. My 6th grade
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We live in Washington, USA.
My 6th grader was working on dividing fractions homework.
On problem: (2/3) ÷ (4/5), he wrote:
(2/3) × (4/5) = 8/15.
He remembered the operation turns into multiplication, but kept the second fraction as 4/5 instead of inverting it.
I am checking his worksheet before signing off tonight. Please strictly withhold the final numerical answer.
1. What is the name of this misconception?
2. What is 1 diagnostic question I can ask him aloud, and how will I know he landed it?
3. Provide a changed-number parallel problem (without answer) for him to rework.
Answer
**1. The misconception**
This is "reciprocal applied to the wrong object" in fraction division — sometimes called "flipped the operation but not the divisor." He has half the rule ("÷ becomes ×") without the other half ("and the *second* fraction turns upside down").
In words you can say to him: *"You remembered to switch the sign, but you didn't switch the second number. Dividing by a fraction means multiplying by its upside-down — both pieces have to move together."*
**2. One diagnostic question + how you'll know it landed**
Ask him out loud: *"Is 4/5 bigger than 1 or smaller than 1? Now look at your answer 8/15 — is 8/15 bigger than 2/3 or smaller? When you divide something by a number bigger than 1, the result has to be smaller than what you started with. Does yours fit that? Which fraction was supposed to flip?"*
How you'll know it landed: he points at the **second** fraction (4/5) as the one to invert — *and* he can say why (because 4/5 > 1, so dividing by it has to shrink2/3). If he only flips the right number but can't explain the bigger/smaller reasoning, the procedure stuck but the meaning didn't — push for the why before moving on.
**3. Changed-number parallel (no answer)**
`3/4 ÷ 2/5`
Have him rewrite it as a multiplication on his scratch paper, with the second fraction flipped, then finish it. Check the final number yourself afterward. If he writes `3/4 × 2/5`, he's in the same hole and needs the question above one more time; if he writes the second fraction upside-down before multiplying, the rule has clicked.
You gave me everything I need (grade, country, the symptom in his words) — no follow-up question from me tonight.