Integer operations depend on both the operation and the signs of the numbers. Addition combines changes, subtraction finds a difference or means adding the opposite, and multiplication or division follows sign rules. A good strategy is to identify the starting value, translate each change into a signed integer, and write a number sentence before solving.
The final answer should include a unit and should make sense in the situation.
Key Facts
- A positive integer represents a gain, increase, deposit, above-zero value, or movement right on a number line, such as .
- A negative integer represents a loss, decrease, withdrawal, below-zero value, or movement left on a number line, such as .
- To add integers with the same sign, add the absolute values and keep the common sign, such as .
- To add integers with different signs, subtract the smaller absolute value from the larger absolute value and keep the sign of the number with greater absolute value, such as .
- To subtract an integer, add its opposite, so and .
- The product or quotient of two integers with the same sign is positive, such as and .
- The product or quotient of two integers with different signs is negative, such as and .
- In a word problem, a total change can be written as .
Vocabulary
- Integer
- An integer is a whole number, its opposite, or zero, such as , , or .
- Positive number
- A positive number is greater than and often represents a gain, increase, or amount above a starting point.
- Negative number
- A negative number is less than and often represents a loss, decrease, or amount below a starting point.
- Absolute value
- Absolute value is a number's distance from on a number line, written as .
- Opposite
- Opposite numbers are the same distance from but on different sides, such as and .
- Net change
- Net change is the overall increase or decrease after all signed changes are combined.
Common Mistakes to Avoid
- Treating every word like loss as subtraction is wrong because a loss is often represented by a negative number that may be added, such as .
- Forgetting that subtracting a negative becomes addition is wrong because , so .
- Adding integers with different signs by adding their absolute values is wrong because means compare distances, giving , not .
- Dropping the negative sign in multiplication or division is wrong because different signs give a negative result, such as .
- Writing only a number without a unit or context is incomplete because word problem answers need meaning, such as degrees or dollars owed.
Practice Questions
- 1 A submarine is at meters and rises meters. What is its new depth?
- 2 Maya has \12\. Write an integer expression and find her balance.
- 3 The temperature was in the morning and dropped by night. What was the night temperature?
- 4 A problem says a football team lost yards, then gained yards. Explain why the net change is without just giving a calculation.
Understanding Integer Operations Word Problems Reference
Integers are useful because many situations have a reference point. Zero degrees is a temperature reference. Sea level is a height reference.
A bank balance of zero is a money reference. The sign tells which side of that reference a value lies on. Its absolute value tells the distance from zero.
This distinction matters when comparing values. Negative twelve is less than negative five, even though twelve has the larger distance from zero.
On a number line, negative twelve sits farther left. Drawing a quick number line can make a confusing comparison visible.
Subtraction needs careful reading because it can describe two different ideas. It may mean taking away a change, or it may mean finding the gap between values. For example, the change from a temperature of negative three degrees to four degrees is seven degrees upward.
The two temperatures are on opposite sides of zero, so the distance includes both parts. When a problem says a loss was removed or a debt was canceled, the result moves in the positive direction.
This is why subtracting a negative value increases the result. Thinking in terms of direction is more reliable than trying to memorize a string of signs.
Multiplication and division usually appear when the same signed change happens repeatedly or is shared equally. A submarine that descends three meters each minute for four minutes has a total change of negative twelve meters. The negative sign comes from the direction of each minute of motion.
A negative multiplier can represent reversing a direction, reversing a profit or loss, or comparing changes across opposite situations. The sign rules follow patterns that keep multiplication consistent. Division should be connected to multiplication when checking work.
If a total loss of twelve points is split equally across three rounds, each round is a loss of four points. The quotient must rebuild the original total when multiplied by the divisor.
Word problems often include extra words that do not affect the calculation. Mark the quantities, their units, and whether each quantity is a starting amount, a change, a rate, or a final amount. Words such as fell, owed, below, refunded, rose, and gained give clues, but the situation decides the sign.
A refund increases a bank balance, while a refund of a lost point may describe a different context. Write a short statement in words before calculating, such as final balance equals starting balance plus change. Then estimate the direction of the answer.
A result that says a diver rose when every change was downward signals an error. Units provide another check because meters, dollars, points, and degrees should remain attached to the final answer.