My 7th grade daughter asked me tonight:
Ask
My 7th grade daughter asked me tonight: "Why is a negative number times a negative number a positive number? Why is (-3) * (-4) = +12 instead of -12?"
I didn't want to just say "two negatives make a positive because that's the rule," but every explanation I tried to come up with felt confusing or circular.
Can you explain negative multiplication in one clear, intuitive step with a concrete demonstration that a 7th grader can actually see, followed by a quick check I can try with her tonight?
Answer
**Tonight's goal (10 min):** Your daughter can explain in her own words why two negatives multiply to a positive, and lands on `(-2) × (-6) = +12`.
**The one step she can see — the pattern she already knows.**
She knows `5 × 3 = 15`. Watch what happens as the second number steps down by 1:
```
5 × 3 = 15
5 × 2 = 10
5 × 1 = 5
5 × 0 = 0
5 × -1 = -5
5 × -2 = -10
5 × -3 = -15
```
Each row subtracts 5, so the answer keeps falling. She can see for herself that `5 × (-3) = -15`.
Now flip **both** signs:
```
-5 × 3 = -15
-5 × 2 = -10
-5 × 1 = -5
-5 × 0 = 0
-5 × -1 = +5 ← here's the flip
-5 × -2 = +10
-5 × -3 = +15
```
The pattern doesn't break — each row still adds 5. The sign flips exactly where you'd expect: once both factors are negative, the running total crosses zero and goes positive.
**Why it works in one breath:** a negative sign means *the opposite of*. Two opposites cancel — like walking forward 3 steps, then doing the "opposite" 4 times (backward), and the opposite of backward is forward again. So `(-3) × (-4) = -(3 × -4) = -(-12) = +12`.
**Tonight's check — hand her this and let her fill it in:**
```
-2 × 6 = __
-2 × 5 = __
-2 × 4 = __
-2 × 3 = __
-2 × 2 = __
-2 × 1 = __
-2 × 0 = __
-2 × -1 = __
-2 × -2 = __
-2 × -3 = __
-2 × -4 = __
-2 × -5 = __
-2 × -6 = __
```
She should land on **+12**. If she hesitates at the sign flip across0, point back at the second table — same shape, same flip.
**If she still wants more:** the `(-1) × (-1) = +1` case is the cleanest to talk through, because "opposite of opposite" is just one breath. If she nails tonight's row, she's got the rule — no further drills needed.