My 8th-grade son is working on polynomia
提问
My 8th-grade son is working on polynomial multiplication in Algebra 1. On his homework and quiz, he repeatedly writes:
1) (x + 5)^2 = x^2 + 25
2) (2a - 3)^2 = 4a^2 - 9
When I asked him to explain his thinking, he said: "When you square parentheses, you just square each term inside." He is treating exponents as if they distribute across addition/subtraction like multiplication does, completely dropping the middle linear term (+10x and -12a).
Could you format this into our mistake notebook using the strict 5-line cause card format:
- Line 1: 题面 (Problem statement & expressions)
- Line 2: 错法 (Exact error and student's flawed logic)
- Line 3: 修法 (Visual geometric area model / FOIL expansion fix)
- Line 4: tag (Category & topic tags)
- Line 5: 错因 (The precise root cause in one direct sentence)
Please keep it strictly to the 5-line structure without verbose academic filler, plus 1 parallel test problem to verify retention.
回答
The geometric area model and the FOIL rule are explanation — that's homework-tutor's lane, not the card. The card holds five lines only, and the "right move" line is one same-type problem with new numbers, not the rule itself.
Both errors share the same cause (exponent distributing over +/− the way a factor distributes), so they merge into one card, not two. Recorded (`ok:true`, id `86835d311afd9c0c`):
| Line | Content |
|---|---|
| Prompt | (x + 5)² = x² + 25 ; (2a − 3)² = 4a² − 9 — both drop the middle linear term (+10x, −12a). |
| Wrong move | Squares each term inside the parentheses on its own; treats the exponent as distributing over +/− the way a factor distributes. His verbal rule: *"squaring parentheses just squares each term."* |
| Right move | Same-type item with new numbers: (3y + 4)² = (3y + 4)(3y + 4) = 9y² + 24y + 16 by FOIL. Then have him redo one himself, e.g. (2b + 7)². |
| Tag | polynomial multiplication · perfect square · FOIL · exponent distributes |
| Cause | Treats an exponent as distributing over addition/subtraction the way a factor does. |
When this card comes back due, he self-tests on (3y + 4)² first and the fix only appears after — that's the retention check built into the card. If you also want the area-model visual explained in words, send the same prompt to homework-tutor and we'll work it out there.