Integer Rules
In short: Integer rules are the sign patterns that tell you whether the result of adding, subtracting, multiplying or dividing whole numbers and their opposites is positive or negative.
7th GradeNumbers and Operations7.NS.A.1
What is Integer Rules?
The integer rules are a short list of sign patterns that let you compute with positive and negative whole numbers without drawing a number line every time. They cover four operations, and the multiplication and division patterns are the easiest to memorize.
For products and quotients, matching signs produce a positive answer and mismatched signs produce a negative one, as in \((-) \times (-) = (+)\) and \((+) \div (-) = (-)\). Sums behave differently: two negatives pile up into a bigger negative, while a negative and a positive partly cancel and the heavier one wins.
How to spot it
- An expression mixes positive and negative whole numbers, such as -12 times -5 or 18 plus -27.
- A story involves debts and deposits, falling and rising temperatures, or floors above and below ground.
- A subtraction contains a negative sign after a minus, like 9 minus negative 4.
- A multi-operation expression forces you to track signs across several steps.
The method
- Sums: when both signs match, total the sizes and keep that sign; when they clash, take the smaller size away from the bigger and wear the bigger one’s sign.
- Differences: turn \(a – b\) into \(a + (-b)\), then handle it as a sum.
- Products and quotients: ignore signs while you compute, then decide the sign of the answer by comparing the two signs.
- Respect the order of operations and re-check every sign before writing the final answer.
Integer Rules: solved example
Problem
Evaluate \((-12)(-5) + (-48) \div 6 – (-9)\).
Solution
Products and quotients first: \((-12)(-5) = 60\) because the signs match, and \((-48) \div 6 = -8\) because they differ.
Now \(60 + (-8) – (-9) = 60 – 8 + 9 = 61\). The value is \(61\).
Try it
Question 1
Compute \((-81) \div (-9)\) and \((-7) + (-15)\).
Show the answer
Equal signs divide to a positive: \(9\). Equal signs add and keep the sign: \(-22\).
Question 2
A submarine at \(-120\) feet rises 45 feet, then dives 70 feet. Where is it now?
Show the answer
\(-120 + 45 – 70 = -145\), so the submarine is at \(-145\) feet.
Integer Rules vs. Additive Inverse
The integer rules are a toolkit of sign patterns for four operations. The additive inverse is a single idea within that toolkit: the opposite number that makes a sum of zero. A common slip is applying the multiplication pattern to addition, since -3 + (-4) is -7, not 12.
Keep going
Lessons
Worksheets and quizzes
- Adding and Subtracting Rational Numbers – Worksheet
- Adding and Subtracting Rational Numbers – Quiz
- Multiplying and Dividing Rational Numbers – Worksheet
- Multiplying and Dividing Rational Numbers – Quiz
Related words
Whole dictionary: A–Z · 7th Grade · Numbers and Operations