Adding and Subtracting Negative Numbers

For Elementary Students

The Number Line

All of these rules are easier with a number line!

... −5  −4  −3  −2  −1   0   1   2   3   4   5 ...
    ←─────────────────|─────────────────→
    Negative side     |  Positive side

Left = Smaller numbers (negative) Right = Bigger numbers (positive)

Think about it like this: It's like a thermometer! Going left is colder (negative), going right is warmer (positive)!

Adding a Positive Number

Adding a positive means move RIGHT on the number line!

Example 1: −3 + 5 = ?

Start at −3, move 5 spaces RIGHT
−3 → −2 → −1 → 0 → 1 → 2
Answer: 2

Example 2: −7 + 4 = ?

Start at −7, move 4 spaces RIGHT
−7 → −6 → −5 → −4 → −3
Answer: −3

Adding a Negative Number

Adding a negative means move LEFT on the number line!

It's the same as subtracting!

Example 1: 4 + (−3) = ?

Start at 4, move 3 spaces LEFT
4 → 3 → 2 → 1
Answer: 1

Same as: 4 − 3 = 1

Example 2: −2 + (−5) = ?

Start at −2, move 5 spaces LEFT
−2 → −3 → −4 → −5 → −6 → −7
Answer: −7

Same as: −2 − 5 = −7

Memory trick: Plus a negative = Subtract!

Subtracting a Positive Number

Subtracting a positive means move LEFT on the number line!

Example 1: 3 − 5 = ?

Start at 3, move 5 spaces LEFT
3 → 2 → 1 → 0 → −1 → −2
Answer: −2

Example 2: −1 − 4 = ?

Start at −1, move 4 spaces LEFT
−1 → −2 → −3 → −4 → −5
Answer: −5

Subtracting a Negative Number

Subtracting a negative means move RIGHT on the number line!

Two negatives make a positive!

Example 1: 5 − (−3) = ?

Start at 5, move 3 spaces RIGHT
5 → 6 → 7 → 8
Answer: 8

Same as: 5 + 3 = 8

Example 2: −2 − (−6) = ?

Start at −2, move 6 spaces RIGHT
−2 → −1 → 0 → 1 → 2 → 3 → 4
Answer: 4

Same as: −2 + 6 = 4

Memory trick: Minus a negative = Add!

Quick Rules Summary

Super simple:

• Two signs the SAME → Add (and the answer is positive)
• Two signs DIFFERENT → Subtract

Adding Two Negative Numbers

When you add two negatives, you get a MORE negative number!

Example: −4 + (−6) = ?

Think: Both are negative, so add their sizes and keep the negative sign!

4 + 6 = 10
Answer: −10

Another way: −4 − 6 = −10

For Junior High Students

Understanding Integer Operations

When working with negative numbers, visualize the number line or use algebraic rules.

Number line model:

• Adding moves you in a direction
• Positive values move RIGHT
• Negative values move LEFT

Algebraic model:

• Signs determine operations
• Consecutive signs combine according to rules

Adding a Positive Number

Adding a positive number means moving right (toward larger values).

Rule: a + (+b) = a + b

Examples:

Example 1: −3 + 5

Start at −3
Move 5 units right: −3 → −2 → −1 → 0 → 1 → 2
Result: 2

Example 2: −7 + 4

Start at −7
Move 4 units right: −7 → −6 → −5 → −4 → −3
Result: −3

Algebraic interpretation: When magnitudes differ, subtract the smaller from the larger and take the sign of the larger magnitude.

Adding a Negative Number

Adding a negative number is equivalent to subtracting the corresponding positive.

Rule: a + (−b) = a − b

Why? Adding debt is the same as losing money (subtracting).

Examples:

Example 1: 4 + (−3)

= 4 − 3
= 1

Example 2: −2 + (−5)

= −2 − 5
= −7

Number line: Adding −5 means moving 5 units LEFT from −2.

Subtracting a Positive Number

Subtracting a positive number means moving left (toward smaller values).

Rule: a − (+b) = a − b

Examples:

Example 1: 3 − 5

Start at 3
Move 5 units left: 3 → 2 → 1 → 0 → −1 → −2
Result: −2

Example 2: −1 − 4

Start at −1
Move 4 units left: −1 → −2 → −3 → −4 → −5
Result: −5

Subtracting a Negative Number

Subtracting a negative is equivalent to adding the corresponding positive.

Rule: a − (−b) = a + b

Why? Removing debt is the same as gaining money (adding).

Examples:

Example 1: 5 − (−3)

= 5 + 3
= 8

Interpretation: If you owe $−3 (meaning someone owes YOU $3), and that debt is removed, you gain $3.

Example 2: −2 − (−6)

= −2 + 6
= 4

Number line: Subtracting −6 means moving 6 units RIGHT from −2.

Sign Rules Summary

Pattern recognition:

• Same signs: Add the values (++ or −−)
• Different signs: Subtract the values (+− or −+)

Adding Two Negative Numbers

When both addends are negative, add their absolute values and apply a negative sign.

Rule: −a + (−b) = −(a + b)

Examples:

Example 1: −4 + (−6)

= −(4 + 6)
= −10

Example 2: −15 + (−3)

= −(15 + 3)
= −18

Intuition: Combining two debts results in a larger debt.

Subtracting Two Negative Numbers

Rule: −a − (−b) = −a + b

Examples:

Example 1: −8 − (−5)

= −8 + 5
= −3

Example 2: −3 − (−10)

= −3 + 10
= 7

Comparison: Which is larger, −3 or −10?

• −3 > −10 (−3 is closer to zero, so it's larger)
• Result has sign of larger magnitude: 7 (positive)

Multi-Step Expressions

Simplify from left to right, applying sign rules sequentially.

Example: −8 + 5 − (−3) + (−2)

Step 1: −8 + 5 = −3

Step 2: −3 − (−3) = −3 + 3 = 0

Step 3: 0 + (−2) = −2

Answer: −2

Verification:

−8 + 5 − (−3) + (−2)
= −8 + 5 + 3 − 2
= (−8 − 2) + (5 + 3)
= −10 + 8
= −2 ✓

Alternative: Rewrite as Addition

Convert all operations to addition by changing signs.

Original: 5 − (−3) + (−7) − 4

Rewrite:

= 5 + 3 + (−7) + (−4)

Group positives and negatives:

Positives: 5 + 3 = 8
Negatives: −7 + (−4) = −11

Combine:

8 + (−11) = 8 − 11 = −3

Answer: −3

Real-Life Applications

Temperature: Temperature drops 5°F then rises 8°F

−5 + 8 = 3°F net change

Banking: Account has $−20 (overdrawn). You deposit $50.

−20 + 50 = $30 balance

Elevation: Start at 30 m below sea level (−30), climb 75 m

−30 + 75 = 45 m above sea level

Football: Lose 5 yards, then lose 3 more

−5 + (−3) = −8 yards total loss

Debt cancellation: Owe $50 (−50), debt forgiven (subtract the debt)

−50 − (−50) = 0

Common Mistakes

Mistake 1: Treating double negatives incorrectly

❌ 5 − (−3) = 2 ✓ 5 − (−3) = 5 + 3 = 8

Mistake 2: Forgetting to change signs when adding negatives

❌ −4 + (−6) = −2 ✓ −4 + (−6) = −10

Mistake 3: Confusing number line direction

❌ Thinking adding negative moves right ✓ Adding negative moves LEFT

Mistake 4: Sign confusion in multi-step problems

❌ −3 − (−5) + (−2) = −3 − 5 + 2 = −6 ✓ −3 − (−5) + (−2) = −3 + 5 − 2 = 0

Tips for Success

Tip 1: Draw a number line for visual reference

Tip 2: Rewrite all operations as addition first

Tip 3: Remember: subtracting a negative = adding a positive

Tip 4: Group positives and negatives separately, then combine

Tip 5: Check your answer: does the direction make sense?

Tip 6: Practice with temperature, elevation, or money contexts

Absolute Value Perspective

Think in terms of distance (absolute value) and direction (sign).

Example: −7 + 4

• Distance from zero: |−7| = 7, |4| = 4
• Directions: negative and positive
• Since |−7| > |4|, result stays negative
• Difference: 7 − 4 = 3
• Answer: −3

Example: −2 + 8

• Distance: |−2| = 2, |8| = 8
• Since |8| > |−2|, result is positive
• Difference: 8 − 2 = 6
• Answer: 6

Properties of Integer Addition

Commutative: a + b = b + a

−3 + 5 = 5 + (−3) = 2

Associative: (a + b) + c = a + (b + c)

(−2 + 3) + (−4) = −2 + (3 + (−4))
1 + (−4) = −2 + (−1)
−3 = −3 ✓

Identity: a + 0 = a

−5 + 0 = −5

Inverse: a + (−a) = 0

7 + (−7) = 0

Practice