My 6th grader brought home math homework
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My 6th grader brought home math homework tonight and on fraction addition, they wrote:
1/3 + 1/4 = 2/7
and
2/5 + 1/3 = 3/8
They insisted: "You just add the top numbers and add the bottom numbers, just like regular addition!"
I am sitting with them at the kitchen table. I do NOT want to just lecture them or give the final answer.
Please give me:
1. The exact name of this conceptual misconception.
2. 2 speakable diagnostic questions I can ask them out loud right now, each followed by how I will know their answer landed conceptually.
3. One parallel changed-numbers practice item (withhold the answer).
4. One brief follow-up question that would change how we approach this.
Answer
The misconception is called **treating the numerator and denominator as two independent whole numbers** (the "add the tops, add the bottoms" shortcut). In words you can actually say at the table: *"the bottom number tells you how many pieces the whole was cut into — different bottoms mean different-sized pieces, so the bottom numbers can't just add."*
**1. Diagnostic question (size of pieces)**
"Picture two identical candy bars. We cut the first into 3 equal pieces and you take 1. We cut the second into 4 equal pieces and you take 1. Did you get the same amount of candy from each bar, or different amounts?"
*How you'll know it landed:* They say something like "different amounts — I got more from the first because the pieces were bigger." That means the bottom number is doing real work in their head. If they answer "same amount, because 1 + 1 = 2," the denominator-as-piece-size idea isn't there yet.
**2. Diagnostic question (sum vs. addend)**
"You already have 1/2 of a candy bar and a friend hands you 1/3 of a same-size bar. Do you end up with more than 1/2 of a bar, or less than 1/2?"
*How you'll know it landed:* They say "more than 1/2." That gets the size direction. If they say "less than 1/2," or they write 2/5 of a bar and call it done, the shortcut is still running.
**3. Changed-numbers practice item (no answer)**
Have them try: `1/4 + 1/6 = ?`
*How you'll know it landed:* If they write 2/10 or 1/5, the same mistake is still there. If they find a way to rewrite both fractions over the same denominator first (cut both bars into 12 pieces, or any other common denominator), the idea has clicked.
**4. Follow-up that would change the approach**
"Did the teacher introduce common denominators (LCD) in class this week, or is this homework ahead of the lesson?" If LCD hasn't been taught, tonight is better spent drawing two same-size bars cut into fourths and sixths so they can *see* why the pieces don't just add — push the LCD procedure on a night when it's been introduced, not now.
One thing I can't do from here: I don't know whether the teacher has introduced LCD yet, or whether your child has fraction strips or circles at the table. If you tell me, I can sharpen the next step.