A Hop, Mirrored or Not
Drag a and b, each a signed number. The dashed hop above the line is always the plain, unsigned version; the solid hop on the line mirrors it through zero once for every negative factor — so two negatives mirror it twice, right back to where it started.
2 × −3 = −6
Exactly one factor is negative, so the hop mirrors through zero once.
- a
- 2
- b
- −3
- Mirrors
- 1
- Product
- −6
Try a pair
Ready to try some? Multiplying and dividing integers worksheets
Where it usually goes wrong
"A negative times a negative is a positive" is memorized as a lookup in a sign chart — two minuses, one plus — with nothing showing why two of them should cancel rather than, say, stack into something even more negative. It survives being applied correctly for years without ever stopping being a coincidence of the rule table.
Try this
- Press "3 × 2". Neither factor is negative, so the dashed hop above the line and the solid hop on it land in exactly the same place — nothing to mirror yet.
- Press "3 × −2". Now the solid hop lands on the opposite side from the dashed one, with a curved arrow showing the single mirror through zero that sent it there.
- Press "−3 × −2". The curved arrow is gone, and the solid hop is back on the SAME side as the dashed one — two mirrors, done one after the other, is the same as never mirroring at all.
Where this goes next
This is the same fact behind why (−1)ⁿ alternates between 1 and −1 as n counts up by one each time, and why simplifying a string of nested negative signs is just counting how many mirrors are being applied, not a separate rule to memorize for each length.
More about numbers and proportion
- Why the Sign Flipsmultiplying an inequality's solution set by a negative number reverses the order of every point in it, which turns the ray around — the flipped symbol is only reporting which way it now faces
- Cut Once, Counted Twicea(x + c) = ax + ac because cutting a rectangle's width and adding the two pieces' areas back together is not a special property of multiplication — it is what area already means
- Two Points, One Distancean absolute value equation has two solutions because they are the two points at the same distance from the center, not a positive answer and a negative one