We live in Pennsylvania, US. I'm sitting
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We live in Pennsylvania, US.
I'm sitting with my 7th grader on his math homework tonight.
The problem is: (3/4) ÷ (2/5)
He wrote: (3 ÷ 2) / (4 ÷ 5) = 1.5 / 0.8, and then broke down in tears because he didn't know what to do with decimals inside a fraction.
I am tutoring him tonight. Please DO NOT give the final numerical answer to his problem.
Could you give me:
1. The exact name of this misconception.
2. 1 or 2 diagnostic questions I can say out loud to him, followed by how I will know his answer landed.
3. A parallel problem with changed numbers (without the answer) for him to try independently once he sees the pattern.
Answer
Here are the three things you asked for. You are the tutor tonight — I'm just handing you the script.
**1. The misconception (so you can name it out loud):**
**"Treating division like multiplication."** When multiplying two fractions, the shortcut is: tops together, bottoms together. He tried to apply the same shortcut to division — divide the tops, divide the bottoms — and got stuck when the result wasn't clean. Division needs a different shortcut (keep the first, change ÷ to ×, flip the second), and he doesn't have it yet because he never connected the multiplication shortcut to *what multiplication actually does*.
**2. Diagnostic questions (each with how you'll know it landed):**
**Q1.** *"Walk me through (3/4) × (2/5). What do you do with the 3 and the 2? What do you do with the 4 and the 5?"*
**How I'll know it landed:** He says "multiply the tops together, multiply the bottoms together" without hesitation. If he does, he still knows the × rule. If he pauses, the gap is even earlier than the division one.
**Q2.** *"If you had 3/4 of a chocolate bar and you wanted to know how many 2/5-sized pieces fit inside it, what is that asking you to do — multiplication or division? Why?"*
**How I'll know it landed:** He says division and tries to describe it as "how many groups of 2/5 fit inside 3/4." If he jumps to a shortcut instead of describing the meaning, the misconception is still driving.
Once Q2 lands — he's named that division means "how many groups fit" — then you can introduce the keep-change-flip shortcut as the bridge between that meaning and the calculation. Until then, the shortcut has nothing to attach to.
**3. Parallel problem with changed numbers (your son works it; the answer is his to find):**
$$\frac{2}{3} \div \frac{1}{4}$$
Save it for after he names the pattern from Q2, not before. He needs to articulate the meaning of division first; the numbers don't matter as much as him getting there himself.
**One honest limit:** I can't see his textbook's exact wording for the division rule, so when you do teach keep-change-flip, please use the words and the order his class uses. Different PA districts use "keep-change-flip" vs. "invert and multiply" vs. "multiply by the reciprocal" — match his teacher's vocabulary.
The tears are normal at this exact spot — the multiplication shortcut just stopped being the answer, and he doesn't have a replacement yet. Once he names what division actually means in Q2, the shortcut has somewhere to land.