This cheat sheet helps students remember the sign of an answer when multiplying or dividing integers. It focuses on the rule that same signs make a positive result and different signs make a negative result. Students need this reference because sign errors are common when working with negative numbers.
The memory aid is especially useful for simplifying expressions, solving equations, and checking arithmetic.
Key Facts
- When multiplying or dividing two integers with the same sign, the result is positive, so and .
- When multiplying or dividing two integers with different signs, the result is negative, so and .
- For division, the same sign rule also applies, so and .
- For division, the different sign rule also applies, so and .
- The absolute values are multiplied or divided first, and then the sign rule is applied, such as .
- A product with an even number of negative factors is positive, such as .
- A product with an odd number of negative factors is negative, such as .
- Zero has no positive or negative sign, so and .
Vocabulary
- Integer
- An integer is a whole number, its opposite, or zero, such as , , or .
- Positive number
- A positive number is greater than and is often written with a plus sign, such as .
- Negative number
- A negative number is less than and is written with a minus sign, such as .
- Product
- A product is the answer to a multiplication problem, such as .
- Quotient
- A quotient is the answer to a division problem, such as .
- Absolute value
- Absolute value is a number's distance from , so .
Common Mistakes to Avoid
- Thinking two negatives make a negative is wrong because for both multiplication and division.
- Using addition rules for multiplication is wrong because , but .
- Forgetting to check both signs is wrong because the sign depends on whether the two signs are the same or different.
- Ignoring the number of negative factors is wrong because an even number of negatives gives a positive product and an odd number gives a negative product.
- Treating zero as positive or negative is wrong because has no sign, and multiplying by always gives .
Practice Questions
- 1 Find the value of .
- 2 Find the value of .
- 3 Find the value of .
- 4 Explain why is positive, but is negative.
Understanding Sign when multiplying or dividing integers Memory Aid
The sign rule comes from patterns and from the meaning of inverse operations. Start with a pattern such as three times three, three times two, three times one, and three times zero. The answers decrease by three each time.
Continuing the pattern gives three times negative one equals negative three. Now reverse the order of the factors and watch a second pattern. Negative one times three is negative three, negative one times two is negative two, and negative one times one is negative one.
To keep the pattern going, negative one times zero is zero, so negative one times negative one must be positive one. This is why two negative factors produce a positive result. It is not an exception to memorize without reason.
Division can be understood as a missing-factor problem. When twelve divided by negative three equals negative four, the statement means negative three times negative four equals twelve. The sign of a division answer must make the related multiplication statement true.
This connection is useful when solving equations. If an equation says negative five times a number equals thirty, divide thirty by negative five.
The number must be negative six because negative five times negative six gives thirty. Thinking backward through multiplication is often safer than trying to recall a rule under pressure.
Integers describe quantities that can move in opposite directions. A positive number can represent money earned, a rise in temperature, or movement forward. A negative number can represent money spent, a temperature drop, or movement backward.
Multiplication becomes useful when the same change happens repeatedly. For example, a loss of four points over six rounds gives a total change of negative twenty four points. In some real situations, two negatives can describe reversing an earlier change.
A correction that removes a debt can increase an account balance. Math signs are not always labels for good or bad. They tell the direction or type of change in the situation.
Pay close attention to parentheses, because they show whether a negative sign belongs to a number. Negative three squared can mean a different calculation from the square of negative three. In multiplication, write each signed factor clearly before finding the size of the answer.
For several factors, count the negative factors first. Pairs of negatives cancel into positive pairs, while one unpaired negative leaves a negative result. Do not confuse subtraction with a negative number.
In eight minus negative two, the operation is subtraction, but the second number is negative. Finally, zero needs special care.
Zero divided by any nonzero integer is zero, but division by zero is undefined. A calculator may display an error because there is no integer that can be multiplied by zero to make a nonzero number.