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Stage I — Foundations

LESSON 06 / 100

Adding & Subtracting Integers

The arithmetic of negative numbers — the operation most students get wrong and most algebra depends on.

Why This Lesson is Critical

Every algebraic manipulation — solving equations, simplifying expressions, working with polynomials — requires you to add and subtract integers accurately and automatically. A shaky understanding here creates errors that compound invisibly through every future lesson.

We’ll build this from the ground up using the number line as our guide, then establish clear rules you can apply mechanically when intuition isn’t enough.

The Number Line Model

Think of arithmetic on the number line as movement. You start at some position, then move:

Movement Rules Adding a positive number → move RIGHT (increase)
Adding a negative number → move LEFT (decrease)
Subtracting a positive number → move LEFT (decrease)
Subtracting a negative number → move RIGHT (increase)

Case 1 — Adding Two Numbers with the Same Sign

When both numbers have the same sign, add their absolute values and keep that sign.

Both Positive 4 + 3 = 7 Add: 4 + 3 = 7. Keep positive.

Both Negative (−4) + (−3) = −7 Add: 4 + 3 = 7. Keep negative.

Case 2 — Adding Numbers with Different Signs

When the numbers have different signs, subtract the smaller absolute value from the larger, and keep the sign of whichever had the larger absolute value.

(−8) + 3 = ?
|−8| = 8 | |3| = 3 | Larger: 8 (negative) | 8 − 3 = 5 | Keep negative: −5 The larger absolute value was negative, so the answer is negative.

Larger Absolute Value: Negative (−8) + 3 = −5 8 − 3 = 5, keep − because |−8| > |3|

Larger Absolute Value: Positive 8 + (−3) = 5 8 − 3 = 5, keep + because |8| > |−3|

Subtraction — The Master Rule

Here is the single most important rule in integer arithmetic. Once you understand this, subtraction of integers becomes trivial:

a − b = a + (−b) Subtracting a number is identical to adding its opposite. Always convert subtraction to addition.

The Two-Step Method When you see a subtraction: KEEP · CHANGE · CHANGE

Keep the first number as-is.
Change the subtraction sign to addition.
Change the sign of the second number (positive → negative, or negative → positive).

Then apply the addition rules above.

✦ Example 1 — Keep · Change · Change

①

5 − 9 Keep 5 · Change − to + · Change 9 to −9 → 5 + (−9) → different signs: 9−5=4, negative wins → −4

②

−3 − 7 Keep −3 · Change − to + · Change 7 to −7 → (−3) + (−7) → same signs: 3+7=10, both negative → −10

③

−6 − (−2) Keep −6 · Change − to + · Change −2 to +2 → (−6) + 2 → different signs: 6−2=4, negative wins → −4

④

4 − (−10) Keep 4 · Change − to + · Change −10 to +10 → 4 + 10 → same signs: 14, both positive → 14

Subtracting a Negative = Adding a Positive The pattern a − (−b) = a + b appears constantly in algebra. Two negatives in a row always produce a positive. This is not a coincidence — it is a fundamental property of arithmetic.

Worked Examples — Mixed Operations

✦ Example 2 — Multi-Step Calculations

①

−2 + 9 − 4 Left to right: (−2 + 9) = 7, then 7 − 4 = 3

②

−5 − (−3) + (−1) Convert: −5 + 3 + (−1) → (−5 + 3) = −2 → (−2) + (−1) = −3

③

10 − 14 + (−3) − (−6) Convert: 10 − 14 + (−3) + 6 → collect: (10 + 6) + (−14 + −3) → 16 + (−17) = −1

✦ Example 3 — Real-World Integer Problems

①

Temperature drops from −3°C to −11°C. By how much did it drop? Change = −11 − (−3) = −11 + 3 = −8 → it dropped 8 degrees

②

Account balance: −R250. Deposit R400. New balance? −250 + 400 = 150 → new balance is R150 (in credit)

③

Altitude: starts at 1200m, descends 1500m. Final altitude? 1200 + (−1500) = −300 → 300m below sea level

Quick Reference — All Four Rules

Same Sign — Add pos + pos = pos
neg + neg = neg Add absolute values, keep the sign

Different Signs — Subtract pos + neg = ?
neg + pos = ? Subtract abs values, keep sign of larger

Subtracting Positive a − b = a + (−b) Keep · Change · Change, then add

Subtracting Negative a − (−b) = a + b Two negatives → positive. Always.

Analogy — Elevator Buttons

Think of integers as an elevator. Positive numbers take you up; negative numbers take you down. Adding is pressing a button; subtracting a negative is like the elevator reversing direction.

Starting at floor 3 and subtracting −4 floors means the elevator goes up 4 floors instead of down: 3 − (−4) = 7. You end at floor 7.

— ✦ —

✦ Practice Exercises — Lesson 06 ★★☆ Intermediate

6.1 Calculate: (−6) + (−4)

−10 (same signs: add 6+4=10, keep negative)

6.2 Calculate: (−7) + 12

5 (different signs: 12−7=5, positive wins because |12| > |−7|)

6.3 Calculate: −5 − 8

−13 [KCC: −5 + (−8) → same signs: 5+8=13, keep negative]

6.4 Calculate: −3 − (−9)

6 [KCC: −3 + 9 → different signs: 9−3=6, positive wins]

6.5 Calculate: 4 − (−4)

8 [KCC: 4 + 4 = 8]

6.6 Evaluate: −10 + 3 − (−2) + (−5)

Convert all: −10 + 3 + 2 + (−5)
Positives: 3 + 2 = 5
Negatives: −10 + (−5) = −15
Combined: 5 + (−15) = −10

6.7 A diver is at −18m. She ascends 7m, then descends 12m. What is her final depth?

−18 + 7 + (−12) = −18 + 7 − 12 = −23mShe ends up 23m below the surface.

6.8 ★ Challenge: Without calculating, explain why a − (−a) = 2a for any integer a. Then verify with a = 5 and a = −3.

a − (−a) = a + a = 2a (subtracting the negative flips it to addition)

a = 5: 5 − (−5) = 5 + 5 = 10 = 2(5) ✓
a = −3: −3 − (−(−3)) = −3 − 3 = −6 = 2(−3) ✓This identity reappears in algebraic simplification — notice it when you see it.

Lesson Summary

What You Learned in Lesson 6