My 6th grader was working on tonight's r
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My 6th grader was working on tonight's ratio and proportion homework.
The problem states: "A recipe uses 3 cups of milk for every 5 cups of flour. If you use 9 cups of milk to make a larger batch, how many cups of flour do you need?"
He reasoned: "9 is 6 more than 3 (3 + 6 = 9), so I just add 6 to the flour: 5 + 6 = 11 cups."
He is totally convinced his logic is sound because "both sides increased by 6".
I know he is treating a multiplicative scaling relationship as additive differences, but if I just tell him the answer is 15 or explain cross-multiplication, he won't truly understand why addition breaks the recipe taste/proportion.
Can you provide:
1. The exact name of this misconception.
2. 2 diagnostic/Socratic questions I can ask him at the table so he realizes why adding doesn't work, without me giving him the final number.
3. One parallel problem with different numbers (no answer included) for him to try afterward?
Answer
I'll peek at the relevant misconception card so I name this in the way you can repeat at the table, then answer your three asks.
Here's your three asks, in the order you asked.
**1. Misconception name (in words you can say out loud):**
"Additive reasoning on a proportional relationship." He's comparing the two quantities by their *difference* (+6), when the recipe is built on a *multiplier*. In math-ed shorthand this is just "additive vs. multiplicative reasoning" — the well-known middle-school hurdle of switching from comparing amounts by how much they differ to comparing them by how many times over.
**2. Two diagnostic questions, each with how you'll know it landed:**
**Q1.** "If you used 6 cups of milk — double the 3 — how much flour would you use?"
- *How you'll know it landed:* If he still says "add 3 to the flour" (→ 8), ask him to picture both batters side by side. Does the 6-milk batch feel like the same recipe? The original is 3 milk : 5 flour; the 6-milk batch should still feel like that same batter, same thickness — and 6 milk : 8 flour is noticeably thicker. If he can defend "8" after that, the misconception is deeper; if he hesitates, he's starting to see it. The self-check is a *taste/thickness* question, not arithmetic, so it never carries a number.
**Q2.** "What if you used 1 cup of milk — a really small batch. How much flour?"
- *How you'll know it landed:* If he follows his own rule he lands at 1 milk → 6 flour. Have him imagine that batter — one cup of milk with six cups of flour is basically dough, not batter. If he laughs or says "no way," he knows the additive pattern is breaking down. From there he can rebuild the rule with multiplication on his own, and you still haven't said a number.
**3. Parallel problem (different numbers, no answer):**
"A paint mix uses 2 cups of red paint for every 7 cups of white paint. If you use 8 cups of red paint for a larger batch, how many cups of white paint do you need?"
Hand it to him cold. If he falls back into "8 = 2 + 6, so 7 + 6," that's the same misconception repeating — and you'll know to come back to the feel/thickness check on this one too, not jump to giving him the number.