Adding and Subtracting Integers (Negative Numbers)
| Rule / Case | What to Do (Mental Model) | Example |
|---|---|---|
| 1. Same Signs (+) + (+) or (-) + (-) | Add absolute values and KEEP the common sign. | (-4) + (-5) = -9 (+3) + (+6) = +9 |
| 2. Different Signs (+) + (-) or (-) + (+) | TUG-OF-WAR: Subtract the smaller number from the larger number. The answer takes the sign of the larger team! | (-8) + 5 = -3 10 + (-4) = +6 |
| 3. Double Negative a - (-b) | Two negative dashes face-to-face turn into a PLUS (+): Cutting away sandbags shoots UP! | 5 - (-3) = 5 + 3 = 8 (-2) - (-7) = -2 + 7 = 5 |
• + Positive: Add Helium Balloons ➔ Float UP ⬆️
• + Negative: Add Heavy Sandbags ➔ Pull DOWN ⬇️
• - Positive: Remove Balloons ➔ Drop DOWN ⬇️
• - Negative: Cut away Sandbags ➔ Shoot UP ⬆️
-5 - 3 = -8, NOT +15! Only multiply signs when multiplying numbers.• Trap 2 (Double Negative Placement):
-4 - (-6) becomes -4 + 6 = +2, NOT -10.• Trap 3 (Zero Pairs):
(+1) + (-1) = 0. They cancel out cleanly!Adding & Subtracting Integers: The Hot Air Balloon & Number Line Guide
Why Subtracting a Negative Goes UP (The Hot Air Balloon Proof)
Imagine you are piloting a Hot Air Balloon. Sea level is 0.
• Positive numbers (+): Helium gas bags that pull you UP into the sky.
• Negative numbers (-): Heavy iron sandbags that pull you DOWN toward the ocean.
The 3 Rules to Solve Any Integer Problem
Example: (-4) + (-5) = - (4 + 5) = -9Example: (-8) + 5 ➔ 8 - 5 = 3 ➔ Team Negative wins: -3Example: 5 - (-3) = 5 + 3 = 83 Step-by-Step Worked Examples
Master the exact thinking process across the 3 core integer problem types:
6 + 7 = 13 ➔ Keep negative sign ➔ -1315 - 9 = 6- (-8) has two negative signs directly touching.4 - (-8) = 4 + 8 = 12🎯 Active Recall Quiz (Test Your Integer Mastery)
Test your sign rules with 3 quick questions. Instant feedback provided:
Frequently Asked Questions
Think of negative numbers as heavy sandbags on a hot air balloon. If you add sandbags (+ negative), you go DOWN. But if you SUBTRACT (take away) heavy sandbags (- negative), the balloon gets lighter and shoots UP! Therefore, a - (-b) = a + b.
Think of it as a Tug-of-War battle: Subtract the smaller absolute number from the larger number. The answer takes the sign of whichever team had the larger number. For example, in -8 + 5, team negative has 8 and team positive has 5. The difference is 3, and team negative wins: -3.
A zero pair is one positive counter (+1) and one negative counter (-1) paired together. Because (+1) + (-1) = 0, zero pairs cancel each other out without changing the overall value.
The most common mistake is confusing addition rules with multiplication rules. Many students think "two negatives always make a positive" and write -5 - 3 = +8 (WRONG! -5 - 3 = -8). Two negative dashes only merge into a plus when they are directly next to each other in subtraction: 5 - (-3) = 8, or in multiplication: (-5) × (-3) = +15.
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