My 7th-grade son had this ratio word pro
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My 7th-grade son had this ratio word problem:
'If 4 painters take 6 hours to paint a house, how long would it take 8 painters working at the same pace?'
He wrote: '12 hours. Because 4 painters is 6 hours, so double painters means double the time: 6 x 2 = 12.'
When I told him to think about it, he argued: 'That is how proportions work! If you multiply the left by 2, you multiply the right by 2.'
Can you give me the exact name of this mathematical misconception, and give me 2-3 Socratic questions I can ask him at the kitchen table tonight so he realizes his own logic flaw and recalculates without me giving him the final answer?
Answer
**Misconception: confusing direct and inverse proportion (assuming every "more X, more Y" relationship is direct proportionality).**
What he did: he treated "more painters, more time" as a *direct* proportion — multiply the input, multiply the output. What he missed: the house is a fixed amount of work, and "more painters" doesn't add work, it splits it. That's *inverse* proportion — the product (painters × hours) stays constant, not the ratio.
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**Three questions for tonight, each with how you'll know it landed:**
**1. The rule-push test.**
"Your rule is: double the painters, double the time. Try it with 100 painters. How many hours, by your rule?"
*You'll know it landed when he laughs or hesitates — 600 hours to paint one house doesn't survive a second look. Don't say "that's wrong"; let the absurdity do the work.*
**2. The invariant question.**
"Between 4 painters and 8 painters, what's the thing that does NOT change — the size of the house, or how fast each painter paints?"
*You'll know it landed when he points at the house: the work is fixed, only the splitting changes. That's the hinge. If he says "the time stays the same," he's still on the wrong track — ask which number in the original sentence would change if the painters changed.*
**3. The one-painter question.**
"If 4 painters finish the house in 6 hours, how much house does ONE painter finish in one hour? Now imagine one hour with 8 painters working at that same pace — what fraction of the house gets done in that one hour? How many of those hours fill the whole house?"
*You'll know it landed when he names a fraction per hour and divides 1 by it. If he stops at "one painter takes 24 hours" and doesn't go on, prompt: "okay, and eight of him working together?"*
Stop and let him talk between questions. If he revises to 3 hours on his own, you've reached the goal — don't keep pressing to "make sure."
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**Parallel item for him to try (different numbers, no answer here):**
*"If 3 machines stamp 240 parts in 4 hours, how long would 6 machines (same speed) take to stamp 240 parts?"*
Same trap. Watch which rule he reaches for first — that's the diagnostic.