In this lesson
The sign rule
Multiplication and division both follow the same sign rule. Click each cell below to reveal the sign of the answer, then flip the toggle to see the same rule on division.
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.
Here's why (−)(−) = (+). The right factor drops by 1 each row,
and the product drops by 3 each row. Predict the negative rows.
The only sign that keeps the column smooth is the one the rule
gives.
Multiplication pattern
Each row drops by the same amount. Predict the next three products — the pattern shows you the sign rule.
Multiplication pattern with constant factor −2. The right factor decreases by 1 each row, and you predict the products for the negative right factors so the column stays smooth — the row that lands on (−)(−) shows the rule.- (−2) × 3 =−6
- (−2) × 2 =−4
- (−2) × 1 =−2
- (−2) × 0 =0
- (−2) × (−1) =
- (−2) × (−2) =
- (−2) × (−3) =
A negative as multiplication by −1
Any negative number can be written as −1 times its magnitude. The
minus sign is a multiplier:
So −7 = (−1)(7), and −(−7) = (−1)(−7) = 7. This is the same sign
rule you use for products, written with only one visible factor.
Three or more factors: count the negatives
When you multiply three or more numbers, don't track the sign step by step. Just count the negative factors.
The rule extends to division and to mixed products with positives
in the chain. Each − flips the sign once; the final sign depends
only on whether the flip count is odd or even.
Where it shows up in real life
Tournament golf scores are signed numbers. Each round's score is
written relative to par — −2 if you finished two strokes
under par, +1 if you finished a stroke over. The tournament
total adds the four rounds.
At the Wolf Creek Open near Ponoka, a four-round score of
−2, −2, −2, −2 is a tournament total of
— twice the same factor, written as
multiplication. Three rounds at −3 and one at +1 doesn't share a common factor, so the total
stays a sum. Same answer, two
different operations — multiplication is the shortcut that only works when the rounds repeat.
Worksheet
Try these to put the idea into practice. Two negatives make a positive; one negative flips the sign.
Practice · Not graded
MA.7.NUM.2Practice the idea
01 / 14
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 → −
Show common mistakes
Student says
“Multiplication always makes things bigger; division always makes things smaller.”
What it reveals
Pre-negative intuition. Once a factor can be negative, the product can land on the other side of zero, smaller than the starting number.
Targeted response
Try (−3) × 4 on the pattern builder. The product is −12, smaller than the starting 4. Try (−12) ÷ 4: the quotient is −3, larger than the starting −12. Signs decide direction; the operation symbol does not.
Student says
“(−3)(−4) = −12. I kept the negative sign.”
What it reveals
Applied addition-style sign reasoning to multiplication.
Targeted response
Open the Multiplication Pattern Builder and walk through the −2 × n pattern. (−)(−) = (+) is the only rule that keeps the column smooth.
Student says
“(−2)(3)(−4) = −24. I lost track of the sign count.”
What it reveals
Knows pairs cancel but counts wrong when applying sequentially.
Targeted response
Count out loud. Even number of negatives → positive; odd → negative. Two negatives in this expression, so positive.
Student says
“−24 ÷ (−4) = −6. I kept the negative sign.”
What it reveals
Didn't transfer the multiplication sign rule to division.
Targeted response
Division and multiplication share sign rules because division is the inverse. (−24) ÷ (−4) = 6 because (6)(−4) = −24.
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.