What multiplication and division mean
Multiplication is repeated addition. 3 × 4 means 4 added to itself
3 times: .
Division is the inverse — it asks how many of one number fit into
another. 12 ÷ 4 = 3 because three 4s make 12.
Once negatives are on the line, both operations follow one sign rule:
same signs give a positive, different signs give a negative. It
follows from −a = (−1) · a — multiplying by −1 flips a sign:
So (−3) × 4 = −12, and (−3) × (−4) = +12. Two negatives flip the
sign twice and you're back to positive.
Try it
Click each cell to reveal the sign of the answer.
Tab into the grid and press Enter on a cell to flip it.
Division uses the same rule. (−12) ÷ 4 asks "what number times 4
gives −12?" The answer is −3.
Four ways to write the same multiplication
The notation varies. These are all the same product:
Textbooks use all four. Use whichever is clearest.
Where it shows up in real life
A four-round golf tournament. Each round is scored relative to par — under par is negative, over par is positive, and the tournament total is the sum of the four rounds.
Four steady rounds at −2 each give 4 × (−2) = −8: a multiplication
because the rounds are identical. Vary the rounds and the total is a
plain sum.
Try it
Set each round's score relative to par. Watch the total.
Round 1
-2
vs par
Round 2
-2
vs par
Round 3
-2
vs par
Round 4
-2
vs par
Tournament total
When all four rounds match, the total reads as a multiplication.
Move one round off the others and the readout becomes a sum. Set them all to the same number again and it's a multiplication.
Worksheet
These aren't graded. Get them right, get them wrong — the goal is to feel out where the idea works.
Question 1 of 3
Try it
What is (−3) × 4?
Multiple choice: what is negative three times four? Four answer cards: minus twelve, plus twelve, minus seven, plus one.Sign rule
Same signs → + · Different signs → −
Going further
Multiplying fractions and decimals comes next. Same sign rule, just applied to fractions and decimals instead of whole-number magnitudes.
The sign rule also shows up in slope and rate of change. A line that goes up as you move right has positive slope; one that falls has negative slope. Multiply a slope by a length and you get the rise (or the fall) over that length.