Does the Japanese multiplication method actually work or is it just a visual trick

The Japanese multiplication method works because it's literally computing the product using the distributive property and place value. For $12 \times 34$: $$12 \times 34 = (10 + 2)(30 + 4) = 10 \times 30 + 10 \times 4 + 2 \times 30 + 2 \times 4 = 300 + 40 + 60 + 8 = 408$$ The "lines and intersections" method encodes this geometrically: - 3 lines (tens of 34) crossed with 1 line (tens of 12) gives $3 \times 1 = 3$ intersections in the hundreds region - The diagonal bands group intersections by place value **Does it work for zeros?** Yes, but zeros require drawing a "line" that looks like a dotted or dashed line (representing the zero digit), which then has zero intersections. It gets messy. **Is it really Japanese?** The method is known as "dot multiplication" or "Chinese multiplication" in some cultures. Its historically used in various forms across Asia but isnt "the standard method" in Japanese schools today — they use the same column multiplication as everyone else. **Bottom line:** Its a neat visual representation of the distributive property, not a separate mathematical discovery.