Integers - Sixth Grade
Complete Notes & Formulas
1. Understanding Integers
What are Integers?
Integers = Whole numbers including negatives
NO fractions or decimals!
Types of Integers
| Type | Symbol | Examples |
|---|---|---|
| Positive Integers | + | 1, 2, 3, 4, 5, ... |
| Zero | 0 | 0 (neither positive nor negative) |
| Negative Integers | − | -1, -2, -3, -4, -5, ... |
Set of Integers
ℤ = {..., −3, −2, −1, 0, 1, 2, 3, ...}
Real-World Examples:
• Temperature: 5°C (above freezing), −10°C (below freezing)
• Money: $50 (credit), −$20 (debt)
• Elevation: 100m (above sea level), −50m (below sea level)
2. Integers on Number Lines
Horizontal Number Line
← Negative | 0 | Positive →
Numbers increase from left to right
Vertical Number Line
↑ Positive
0
↓ Negative
Numbers increase from bottom to top
Key Rules
• Numbers to the right are greater (horizontal)
• Numbers to the left are smaller (horizontal)
• Numbers above are greater (vertical)
• Numbers below are smaller (vertical)
3. Opposite Integers
Definition
Opposite integers are the SAME distance from 0
but on OPPOSITE sides of the number line
Formula
Opposite of n = −n
Opposite of −n = n
Examples
| Integer | Opposite | Explanation |
|---|---|---|
| 7 | -7 | Both 7 units from 0 |
| -12 | 12 | Both 12 units from 0 |
| 0 | 0 | Zero is its own opposite |
| -25 | 25 | Both 25 units from 0 |
Key Property: An integer + its opposite = 0
Example: 5 + (−5) = 0 or −8 + 8 = 0
4. Absolute Value
What is Absolute Value?
Absolute value = DISTANCE from 0
Always POSITIVE or ZERO
Notation & Formula
|n| = absolute value of n
If n ≥ 0, then |n| = n
If n < 0, then |n| = −n
Examples
Example 1: |7| = 7 (already positive)
Example 2: |−7| = 7 (distance is 7)
Example 3: |0| = 0 (zero distance)
Example 4: |−25| = 25 (distance is 25)
Example 5: |100| = 100 (already positive)
Key Facts
• Absolute value is never negative
• Opposite numbers have the same absolute value
• |a| = |−a| for any integer a
• Only |0| = 0
5. Comparing Integers
Rules for Comparing
1. All positive integers > 0 > all negative integers
2. For positive integers: larger number is greater
3. For negative integers: number closer to 0 is greater
4. On number line: right side is always greater
Comparison Symbols
| Symbol | Meaning | Example |
|---|---|---|
| > | Greater than | 5 > −3 |
| < | Less than | −8 < 2 |
| = | Equal to | 3 = 3 |
Examples
• 8 > 5 (both positive, 8 is larger)
• 0 > −7 (zero greater than any negative)
• −2 > −10 (−2 is closer to 0)
• −100 < −50 (−100 is farther from 0)
• 15 > −20 (positive always greater than negative)
6. Ordering Integers
Steps to Order
Step 1: Place all numbers on a number line
Step 2: Read from left to right for ascending order (smallest to largest)
Step 3: Read from right to left for descending order (largest to smallest)
Example: Order from Least to Greatest
Problem: Order: 5, −3, 0, −8, 2, −1
Step 1: Identify the most negative: −8
Step 2: Next: −3, then −1
Step 3: Then zero: 0
Step 4: Then positives: 2, then 5
Answer: −8, −3, −1, 0, 2, 5
7. Integer Inequalities with Absolute Values
Steps to Solve
Step 1: Calculate the absolute value of each number
Step 2: Compare the absolute values
Step 3: Apply the comparison symbol
Examples
Example 1: Compare |−7| and |5|
|−7| = 7
|5| = 5
7 > 5
Answer: |−7| > |5|
Example 2: Is |−12| < |−15|?
|−12| = 12
|−15| = 15
12 < 15 ✓ True
Answer: Yes, |−12| < |−15|
8. Word Problems
Example 1: Temperature
Problem: The temperature is −5°C. It rises by 8°C. What is the new temperature?
Start: −5°C
Rise: +8°C
New: −5 + 8 = 3°C
Answer: 3°C
Example 2: Elevation
Problem: A submarine is at −200 meters (below sea level). It rises 50 meters. What is the absolute value of its new depth?
Start: −200 meters
Rise: +50 meters
New position: −200 + 50 = −150 meters
Absolute value: |−150| = 150 meters
Answer: 150 meters
Example 3: Money
Problem: You owe $35 (debt = −35). You owe your friend $20 (−20). Who do you owe more money to? Use absolute values.
|−35| = 35
|−20| = 20
35 > 20
Answer: You owe more to the first person ($35)
Quick Reference: Integer Rules
| Concept | Key Rule |
|---|---|
| Integers | Whole numbers (positive, negative, zero) |
| Opposite | Same distance from 0, opposite sides |
| Absolute Value | Distance from 0 (always positive) |
| Comparing | Right side greater (number line) |
| Ordering | Use number line or rules |
💡 Important Tips to Remember
✓ Integers = whole numbers (no fractions or decimals)
✓ Zero is neither positive nor negative
✓ Opposite integers: same distance from 0, opposite directions
✓ Absolute value is ALWAYS positive (or zero)
✓ On number line: right = greater, left = smaller
✓ Positive > 0 > Negative (always)
✓ For negatives: closer to 0 = greater value
✓ |a| = |−a| for any integer a
✓ Use number line when unsure about ordering
✓ Real-world context: debt, temperature, elevation
🧠 Memory Tricks & Strategies
Integers:
"Integers are whole, from negative to positive's goal!"
Opposite Integers:
"Mirror, mirror, across from zero!"
Absolute Value:
"Distance from zero, always positive, never below!"
Comparing:
"On the number line, right is greater every time!"
Negative Integers:
"The more negative the sign, the smaller the line!"
Temperature:
"Above zero? Positive flow! Below zero? Negative show!"
Master Integers! ➕ ➖ 0️⃣ 🎯
Remember: Absolute value is always positive!