Why the Sign Rules Work and How to Spot Errors | Grade 7 Math | LevelKite

You already know how to multiply and divide integers. Integers are the counting numbers, zero and the negatives of the counting numbers: $\ldots, -2, -1, 0, 1, 2, \ldots$ But why is a negative times a negative positive? In this lesson you will:

explain why the sign rules for multiplying are true, and
read finished work, find the mistake and fix it.

Four terms will help. A factor is a number being multiplied. A product is the result of multiplying. A number is nonzero when it is not $0$ . The opposite of a number, also called its additive inverse, is the number that adds to it to make $0$ . So $4$ and $-4$ are opposites, because $4 + (-4) = 0$ .

Start with a product that has a clear meaning. $3 \times (-4)$ means 3 groups of $-4$ , which is $(-4) + (-4) + (-4)$ .

Adding $-4$ moves you $|-4| = 4$ units in the negative direction, toward smaller numbers. (The absolute value $|-4|$ is the distance of $-4$ from $0$ , which is $4$ .) Starting at $0$ , you land on $-4$ , then $-8$ , then $-12$ . So $3 \times (-4) = -12$ . A positive number times a negative number is negative.

What about $(-4) \times 3$ ? You cannot make $-4$ groups of something, so use a property instead. The commutative property says that changing the order of two factors does not change their product. We want that property to keep working for integers, so $(-4) \times 3$ must equal $3 \times (-4)$ , which is $-12$ . A negative number times a positive number is negative too.

One case is left: a negative number times a negative number.

What happens as the first factor counts down by 1

Product	Value	Where the value comes from
$3 \times (-4)$	$-12$	3 groups of $-4$
$2 \times (-4)$	$-8$	2 groups of $-4$
$1 \times (-4)$	$-4$	1 group of $-4$
$0 \times (-4)$	$0$	a zero factor gives a zero product
$(-1) \times (-4)$	$4$	the pattern continues: $0 + 4$
$(-2) \times (-4)$	$8$	the pattern continues: $4 + 4$
$(-3) \times (-4)$	$12$	the pattern continues: $8 + 4$

Read down the Value column. In the first four rows, each value is $4$ more than the one above it: $-12$ , $-8$ , $-4$ , $0$ . Here is why. Each row has one fewer group of $-4$ , and taking away a group of $-4$ means subtracting $-4$ . Subtracting a number adds its opposite, so for example $-8 - (-4) = -8 + 4 = -4$ .

If the same pattern continues when the first factor goes below zero, the last three rows are $4$ , $8$ and $12$ . That would make a negative number times a negative number positive.

A pattern is a good clue, but it is not yet a reason. The reason comes from the distributive property: multiplying a number by a sum gives the same result as multiplying it by each addend (each number being added) and then adding those products. For example, $2 \times (3 + 5) = 2 \times 3 + 2 \times 5$ , and both sides equal $16$ . We require this property to stay true when negative numbers join in. The next example shows what that forces.

Worked example

Why a negative times a negative is positive

Use the distributive property to explain why $(-3) \times (-4) = 12$ .

Write 0 as a sum of opposites. The numbers 4 and -4 are opposites, and opposites sum to 0.
$4 + (-4) = 0$
Multiply -3 by that sum. The sum is 0, and any zero factor gives a zero product, so this product equals 0.
$(-3) \times (4 + (-4)) = (-3) \times 0 = 0$
Rewrite the same product with the distributive property, which we require to hold for negative numbers too: multiply -3 by each addend, then add the two products. You only rewrote the expression, so its value is still 0.
$(-3) \times 4 + (-3) \times (-4) = 0$
Work out the product you already know. By the commutative property, -3 times 4 equals 4 times -3, which means 4 groups of -3. Starting at 0, each added -3 moves 3 units, its absolute value, in the negative direction: -3, -6, -9, -12.
$(-3) \times 4 = 4 \times (-3) = (-3) + (-3) + (-3) + (-3) = -12$
Replace the first product in the equation with its value, -12.
$-12 + (-3) \times (-4) = 0$
The unknown product adds to -12 to make 0. Only one number does that: the opposite of -12, which is 12.
$(-3) \times (-4) = 12$

Answer

The product $(-3) \times (-4)$ must equal $12$ . It has to be the opposite of $(-3) \times 4 = -12$ so that the distributive property stays true.

Key concept

Signs come from the properties of operations

The sign rules are the only rules that let the properties of operations, especially the distributive property, keep working with negative numbers. The steps used for $(-3) \times (-4)$ work for any two negative numbers. Multiplying a nonzero number by a negative number flips its sign, and multiplying it by a positive number keeps its sign. So:

When every number is nonzero: count the negative numbers being multiplied or divided. The product or quotient is negative exactly when that count is odd ( $1$ , $3$ , $5$ , ...). When the count is even ( $0$ , $2$ , $4$ , ...), it is positive.
When zero appears: any zero factor gives a product of $0$ . $0$ divided by a nonzero number is $0$ . Division by $0$ is undefined, which means it has no value.

Dividing by a nonzero number undoes multiplying by it. Take two integers $a$ and $b$ . In $a \div b$ , the number $a$ is the dividend, the number $b$ is the divisor, and the result is the quotient. When $b$ is not $0$ , $a \div b$ asks this question: what number times $b$ gives $a$ ?

So $(-12) \div (-4)$ asks what number times $-4$ gives $-12$ . You know $3 \times (-4) = -12$ , so the quotient is $3$ . Quotient signs come straight from product signs. That is why the counting rule for nonzero numbers covers quotients as well as products.

Zero needs its own care:

$0 \div (-5) = 0$ , because $0 \times (-5) = 0$ .
$(-5) \div 0$ is undefined. Any number times $0$ is $0$ , never $-5$ , so no number answers the question.
$0 \div 0$ is undefined too. Every number times $0$ is $0$ , so there is no single answer.

This gives you a check. When the divisor is not $0$ , multiply the quotient by the divisor. You must get the dividend back.

Now use these reasons to judge finished work. Error analysis means finding the first step that goes wrong, saying which rule it breaks, and fixing it. A steady routine helps:

Redo the work yourself, one operation at a time.
Stop at the first line that does not match yours. Later lines may be worked correctly from a wrong number, so the mistake did not start there.
Name the rule that was broken.
Fix that line and carry the new value through to the end.
Check the final sign. If the expression only multiplies and divides nonzero numbers, count the negative numbers. An odd count means a negative result, and an even count means a positive result.

Worked example

Find the first wrong line

A student evaluated $(-36) \div (-4) \times (-2)$ in two lines. Line 1: $(-36) \div (-4) = -9$ . Line 2: $-9 \times (-2) = 18$ . The student's answer is $18$ . Find the first mistake, explain it and give the correct value.

Redo line 1. The parentheses here only hold negative numbers, so you divide and multiply in order from left to right. Dividing first is the correct order. Both numbers are nonzero and both are negative. That is two negatives, an even count, so the quotient is positive: 36 divided by 4 is 9.
$(-36) \div (-4) = 9$
Confirm with the multiplication check. You may use it because the divisor, -4, is not 0. The quotient 9 times the divisor -4 gives back the dividend, -36. The student's -9 fails the check: -9 times -4 is 36, not -36. So line 1 is the first mistake, and its quotient has the wrong sign.
$9 \times (-4) = -36$
Look at line 2. The student's product of -9 and -2 is worked correctly, but it starts from the wrong number. Carry the corrected quotient, 9, forward instead. Both factors are nonzero and one is negative, an odd count, so this product is negative.
$9 \times (-2) = -18$
Check the final sign by counting. The expression only multiplies and divides nonzero numbers, and three of them are negative: -36, -4 and -2. Three is odd, so the value must be negative. That agrees with -18.

Answer

The first mistake is in line 1: the quotient of two negative numbers was written as negative. The correct value of the expression is $-18$ , not $18$ .

Worked example

A dropped sign in the distributive property

A student used the distributive property to find $(-7) \times 98$ . Line 1: $(-7) \times (100 - 2)$ . Line 2: $(-7) \times 100 - 7 \times 2$ . Line 3: $-700 - 14$ . The student's answer is $-714$ . Find the first mistake, explain it and give the correct product.

Line 1 is correct, because 98 equals 100 - 2. Rewrite the subtraction as an addition: subtracting 2 adds its opposite, -2. Now each addend shows its own sign.
$(-7) \times (100 - 2) = (-7) \times (100 + (-2))$
Distribute: multiply -7 by each addend, then add the products. Compare this with line 2. The student subtracts 7 times 2, which puts -14 in the second place. The correct second product is -7 times -2. One negative sign was dropped, so line 2 is the first mistake.
$(-7) \times (100 + (-2)) = (-7) \times 100 + (-7) \times (-2)$
Find each product. All the factors are nonzero, so count the negative factors. The first product has one, an odd count, so it is negative: -700. The second product has two, an even count, so it is positive: 14.
$(-7) \times 100 + (-7) \times (-2) = -700 + 14$
Add the two products. The absolute value of 14 is 14, so adding 14 moves 14 units from -700 in the positive direction, toward larger numbers.
$-700 + 14 = -686$
Check the sign. Both factors of the original product are nonzero and one is negative, an odd count, so the product must be negative, and -686 is. Check the size too: 98 is less than 100 and 7 is positive, so 7 times 98 is less than 7 times 100, which is 700. The student's 714 is too large, and 686 fits.

Answer

The first mistake is in line 2, where a negative sign was dropped from the second product. The correct product is $(-7) \times 98 = -686$ , not $-714$ .

Judge each claim yourself, then read the reason and the verdict

Student's claim	Reason	Verdict
$(-42) \div (-6) = -7$	Both numbers are nonzero and two are negative, an even count. Check: $-7 \times (-6) = 42$ , not $-42$ .	Wrong. The quotient is $7$ .
$56 \div (-8) = -7$	Both numbers are nonzero and one is negative, an odd count. Check: $-7 \times (-8) = 56$ .	Correct.
$(-5) \times 0 \times (-3) = 15$	One factor is $0$ , and any zero factor gives a zero product. Counting negatives is only for nonzero numbers.	Wrong. The product is $0$ .
$0 \div (-9) = -9$	$0$ divided by a nonzero number is $0$ . Check: $0 \times (-9) = 0$ , the dividend.	Wrong. The quotient is $0$ .
$(-9) \div 0 = 0$	The divisor is $0$ , and division by $0$ is undefined. No number times $0$ gives $-9$ .	Wrong. It is undefined.

Takeaway. The sign rules for multiplying are not just rules to memorize. The properties of operations force them: $(-3) \times 4$ and $(-3) \times (-4)$ must add to $0$ , so the second product has to be positive. For nonzero numbers, a product or quotient is negative exactly when its count of negative numbers is odd. A zero factor gives a product of $0$ , $0$ divided by a nonzero number is $0$ , and division by $0$ is undefined. To check work, redo it line by line, find the first line that breaks a rule, name the rule and fix it.