Integer rules chart
The integer rules say what sign an answer gets when you add, subtract, multiply or divide positive and negative numbers. Subtracting is adding the opposite, and when you multiply or divide, two signs that match give a positive answer.
Free printable PDF, 6th grade to 7th grade.
One page, black and white, prints the same every time.
What is on the sheet
Adding integers
| Signs | Rule | Example |
|---|---|---|
| Same signs | Add the absolute values and keep the sign. | −4 + (−6) = −10 |
| Different signs | Subtract the smaller absolute value from the larger. Keep the sign of the number with the larger absolute value. | −9 + 4 = −5 |
| Opposites | A number plus its opposite is 0. | 7 + (−7) = 0 |
Subtracting integers, add the opposite
| Problem | Add the opposite | Answer |
|---|---|---|
| 8 − 3 | 8 + (−3) | 5 |
| 3 − 8 | 3 + (−8) | −5 |
| 8 − (−3) | 8 + 3 | 11 |
| −8 − 3 | −8 + (−3) | −11 |
| −8 − (−3) | −8 + 3 | −5 |
Multiplying and dividing integers
| Signs | Answer | Multiply | Divide |
|---|---|---|---|
| Positive and positive | Positive | 3 × 4 = 12 | 12 ÷ 4 = 3 |
| Negative and negative | Positive | (−3) × (−4) = 12 | (−12) ÷ (−4) = 3 |
| Positive and negative | Negative | 3 × (−4) = −12 | 12 ÷ (−4) = −3 |
| Negative and positive | Negative | (−3) × 4 = −12 | (−12) ÷ 4 = −3 |
Keep, change, change
Subtracting an integer is the same as adding its opposite, so every subtraction can be turned into an addition. Many classes call it keep, change, change: keep the first number, change the minus to a plus, and change the sign of the second number. Then use the adding rules. 8 − (−3) becomes 8 + 3, which is 11.
A number line shows why. Adding a positive number moves right and adding a negative number moves left. Subtracting does the reverse, so taking away −3 moves 3 to the right, the same as adding 3.
Where the rules come in
6th grade meets negative numbers on the number line and learns that a number and its opposite are the same distance from 0 (CCSS 6.NS.C.6). 7th grade adds and subtracts them (7.NS.A.1) and then multiplies and divides them (7.NS.A.2). Many classes start adding integers in 6th grade, which is where the site's practice sheets sit.
Common questions
- Why is a negative times a negative positive?
- Multiplying by −1 flips a number to the other side of 0. Flip a negative number and it lands on the positive side, so (−3) × (−4) is the same as 3 × 4, which is 12.
- What about three or more negative numbers?
- Count the negative signs. An even number of them gives a positive answer and an odd number gives a negative one: (−1) × (−1) × (−1) is −1.
- What is the absolute value of an integer?
- Its distance from 0, which is never negative. The absolute value of −9 is 9, and the absolute value of 9 is also 9.