Subtracting integers is easier when you remember that subtraction can be changed into addition of the opposite. The phrase Keep Change Change helps you rewrite the problem without changing its value. You keep the first number, change the subtraction sign to addition, and change the sign of the second number.
This matters because integer subtraction appears often in algebra, number lines, temperatures, money, and coordinate problems.
The reason Keep Change Change works is that subtracting a number means adding its additive inverse. For example, 8 - (-3) becomes 8 + (+3), because the opposite of -3 is +3. Once the expression is rewritten as addition, you can use the normal rules for adding integers.
The mnemonic is a memory aid, but the main idea is that every subtraction problem can be converted into an equivalent addition problem.
Understanding Math: How to subtract integers (Keep Change Change)
A useful way to understand integer subtraction is to think about what number must be added to reach a target. In the statement negative four minus six, start at negative four and ask which move changes it into a value six less. The move is six units left, so the result is negative ten.
When the number being subtracted is negative, the direction reverses. Removing a debt of six has the same effect as gaining six. This is why a pair of minus signs can lead to a positive result.
The rule is not a trick that changes signs for no reason. It records the direction of a change.
A number line gives a concrete check for each answer. Adding a positive number moves right. Adding a negative number moves left.
Subtracting a positive number moves left because the value decreases. Subtracting a negative number moves right because a negative amount is being removed. For example, beginning at two and subtracting negative five ends at seven.
The move goes five spaces right. Drawing a quick number line is especially helpful when an answer feels surprising. It can show that a result should be larger or smaller before any detailed calculation is finished.
Integer subtraction appears in situations where values represent gains, losses, positions, or changes. A bank balance of negative twenty means twenty dollars are owed. Subtracting a charge makes the balance lower.
Subtracting a fee that was cancelled makes the balance higher. In temperature work, a change of negative four degrees means cooling by four degrees. Subtracting that change describes undoing the cooling, which raises the temperature.
In coordinate graphs, subtracting signed values helps find horizontal or vertical change between points. The signs describe direction, not just whether a number looks positive or negative.
The most common mistake is changing only the operation while forgetting that the entire second number must be replaced by its opposite. Parentheses make this clearer. In an expression such as negative nine minus negative two, negative two is one complete number.
Its opposite is positive two. Another mistake is treating two nearby signs as a single symbol before deciding what each sign means. First identify the subtraction operation.
Then identify whether the following integer is positive or negative. After rewriting the expression, pause and estimate the direction. Subtracting a positive value should decrease the result.
Subtracting a negative value should increase it. This quick check catches many sign errors in arithmetic and later algebra.
Key Facts
- Keep Change Change means keep the first number, change subtraction to addition, and change the sign of the second number.
- a - b = a + (-b)
- 8 - (-3) = 8 + 3 = 11
- -5 - 7 = -5 + (-7) = -12
- The opposite of a positive integer is negative, and the opposite of a negative integer is positive.
- After using Keep Change Change, solve the new problem using integer addition rules.
Vocabulary
- Integer
- An integer is a whole number, its opposite, or zero, such as -4, 0, or 9.
- Opposite
- The opposite of a number is the number the same distance from zero on the other side of the number line.
- Additive inverse
- An additive inverse is the number that adds with the original number to make zero.
- Subtraction
- Subtraction is an operation that can be rewritten as adding the opposite of the number being subtracted.
- Keep Change Change
- Keep Change Change is a mnemonic for rewriting integer subtraction as addition of the opposite.
Common Mistakes to Avoid
- Changing only the minus sign to plus is wrong because the sign of the second number must also change to make an equivalent expression.
- Changing the first number is wrong because Keep means the first number stays exactly the same.
- Forgetting parentheses around a negative second number can cause confusion because 8 - (-3) is not the same as 8 - 3.
- Using Keep Change Change on an addition problem is wrong because the rule is for subtraction problems only.
Practice Questions
- 1 Use Keep Change Change to solve 12 - (-5). Show the rewritten addition problem.
- 2 Use Keep Change Change to solve -9 - 4. Show the rewritten addition problem.
- 3 Explain why 6 - (-2) and 6 + 2 have the same value using the idea of opposites.