We live in Denver, Colorado. My 7th grad
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We live in Denver, Colorado.
My 7th grade daughter is working on her pre-algebra homework on two-step linear equations. I am sitting next to her and have to sign her assignment tonight.
On two problems, she made the exact same error:
Problem 1: Solve 3x + 12 = 30.
Her work: She divided 3x by 3 to get x, divided 30 by 3 to get 10, and wrote: x + 12 = 10, then x = -2.
Problem 2: Solve 5y + 20 = 45.
Her work: She divided 5y by 5 and 45 by 5, wrote: y + 20 = 9, then y = -11.
When I told her the answers were incorrect, she started crying and said: "I did divide by 3 on both sides! What did I do wrong?!"
1. What is the exact mathematical misconception here, and what is one diagnostic question I can say out loud to her right now that leads her to notice the mistake herself without giving away the answer?
2. What is one changed-number parallel problem (with no final answer revealed) she can try tonight to prove she has mastered the order of operations for solving equations?
Answer
**The misconception.** She applied the divide-by-3 to two of the three terms (3x and 30) but skipped the 12. The rule she's missing is "do the same operation to every term on both sides." She did it across the equals sign but not on each side. After her step the equation no longer equals itself — the left went from 3x + 12 down to x + 12 while the right went from 30 down to 10, and that leftover 12 is why her answers come out negative. Same shape on the second problem: she took the 5 off 5y and divided 45, but left20 behind.
The cleaner first move on this form (ax + b = c) is to undo the addition before the multiplication: subtract 12 from both sides first, then divide. (Or, if she wants to divide first, divide *every* term — including the 12 — by 3.)
**One diagnostic question (out loud).** "You divided both sides by 3. Point to every number on the page that just got divided by 3."
How you'll know it landed: she traces with her finger, points at 3x and 30 (the ones she did), then hesitates over the 12. If she says "I forgot that one" or asks "does the 12 get divided too?" — she's got it. If she points at all three confidently without hesitation, she's still missing it; ask instead: "the 12 lives on the left side. What did you do to the left side?"
**One parallel problem (no answer given).** Try tonight: **2x + 6 = 18.** Same shape as her two. Ask her to write every step where she does the same operation to both sides before she moves to the next line. If she lands on a clean integer, the order-of-operations piece is back. If she lands somewhere else, point back to the diagnostic question above — don't tell her the number.
**For tonight.** Let her settle, then have her work the parallel on scratch paper. Sign the original page with a small note in the margin like "redo 2x + 6 = 18 on back" — that covers your signature line and lets her prove the fix without pressure. To check any of her work later, plug her answer back into the original equation; both sides should give the same number.