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Grade 6 Mathematics • Chapter 6: Integers
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Chapter 6: Integers
Chapter 6: Integers
A Grade 6 mission about positive and negative numbers: where they appear in real life, how they move on a number line, and how to use them in calculations.
← You are at the beginning of the lesson
🧠 Concept 12 / 23
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What Are Integers?
Integers are the whole numbers we can place on a number line. They include positive numbers, negative numbers, and zero.
Definition: An integer is any number that can be written without a fractional or decimal component.
ℤ = {..., −3, −2, −1, 0, +1, +2, +3, ...}
ℤ = {..., −3, −2, −1, 0, +1, +2, +3, ...}
Positive Integers+1, +2, +3...greater than zero
Zero0neither positive nor negative
Negative Integers−1, −2, −3...less than zero
🧩 Concept 13 / 23
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Integer Properties
Opposites (Additive Inverses)
Every integer has an opposite that is the same distance from zero on the opposite side. The sum of a number and its opposite is always zero.
Opposite of +5 is −5 | (+5) + (−5) = 0
Closure Property
When you add, subtract, or multiply any two integers, the result is always another integer. (Division may produce a non-integer.)
2 + 3 = 5 (integer) | 2 ÷ 3 = ⅔ (not an integer)
Even & Odd Integers
Even integers are divisible by 2 (..., −4, −2, 0, 2, 4, ...). Odd integers leave a remainder of 1 when divided by 2 (..., −3, −1, 1, 3, ...).
Even: −6, 0, 14 | Odd: −7, 1, 23
Consecutive Integers
Integers that follow one another in order on the number line. They differ by exactly 1.
−2, −1, 0, 1, 2 | n, n+1, n+2
Integers Are Discrete
Unlike fractions or decimals, there are no integers between two consecutive integers (e.g., no integer between 2 and 3).
Between 2 and 3: none | Between −1 and 0: none
The Set is Infinite
There is no largest or smallest integer. The set of integers extends infinitely in both directions on the number line.
... → −1,000,000 → 0 → +1,000,000 → ...
🌍 Concept 14 / 23
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Integers in Daily Life
Integers are not just school numbers. They describe real situations where values can go above or below a reference point.
| Context | Positive Integer (+) | Negative Integer (−) |
|---|---|---|
| Temperature | +30°C — hot summer day | −5°C — freezing winter night |
| Money / Finance | +$50 — deposit into a bank | −$20 — withdrawal or overdraft |
| Elevation | +200 m — above sea level | −15 m — below sea level |
| Sports | +3 — goal difference above opponent | −2 — penalty points deducted |
| Time | +5 years — future date | −5 years — BC or before present |
| Floors in a Building | +10 — 10th floor above ground | −2 — 2nd basement level |
💡 Key Point: The number 0 (zero) is the only integer that is neither positive nor negative. It serves as the neutral reference point on the number line and is the dividing line between positive and negative integers.
🔍 Examples5 / 23
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Identify Integers & Opposites
EXAMPLE 1Identifying Integers
Which of the following numbers are integers? 15, −8, 3.5, 0, −1, ¾, 100, −0.5, −22
Step 1: Check each number — is it a whole number with no fraction or decimal?
15 ✓ whole, positive → integer
−8 ✓ whole, negative → integer
3.5 ✗ has a decimal part → not an integer
0 ✓ whole, neutral → integer
−1 ✓ whole, negative → integer
¾ ✗ is a fraction → not an integer
100 ✓ whole, positive → integer
−0.5 ✗ has a decimal part → not an integer
−22 ✓ whole, negative → integer
✓ Integers: 15, −8, 0, −1, 100, −22
EXAMPLE 2Finding Opposites
Find the opposite of each integer: (a) +7 (b) −15 (c) 0 (d) +100
(a) Opposite of +7 is −7
(b) Opposite of −15 is +15
(c) Opposite of 0 is 0
(d) Opposite of +100 is −100
➡️ Concept 26 / 23
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The Number Line at Work
Zero is the origin. Positive integers move to the right. Negative integers move to the left. Numbers increase as we move right and decrease as we move left.
📌 Key Point: Numbers increase as we move to the right and decrease as we move to the left. Each integer has exactly one unique position on the number line.
📍 Example7 / 23
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Plotting Integers
EXAMPLE 3Plotting Integers on a Number Line
Plot the following integers on a number line and label them: −4, 0, +3, −1, +5
Step 1: Draw a horizontal line. Mark 0 at the centre as the origin.
Step 2: Mark equal intervals — left for negatives, right for positives.
Step 3: −4 → 4 units left Step 4: 0 → at the origin
Step 5: +3 → 3 units right Step 6: −1 → 1 unit left
Step 7: +5 → 5 units right
⚖️ Concept 38 / 23
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Compare and Order Integers
On a number line, the number on the right is greater. The number on the left is smaller.
Greater Than (>)
The number on the left is larger than the number on the right.
5 > 2 | 0 > −4 | −2 > −5
Less Than (<)
The number on the left is smaller than the number on the right.
2 < 5 | −4 < 0 | −5 < −2
Rule for Negative Numbers
When comparing two negative numbers, the one closer to zero on the number line is greater.
−2 > −5 because −2 is closer to 0 than −5
Ascending vs. Descending
Ascending = smallest to largest. Descending = largest to smallest.
Ascending: −3, −1, 0, +2 | Descending: +2, 0, −1, −3
🧮 Examples9 / 23
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Comparing & Ordering
EXAMPLE 4Comparing Integers
Insert >, <, or =: (a) −9 ___ +3 (b) −4 ___ −7 (c) 0 ___ −5 (d) +8 ___ 8 (e) −12 ___ −2
(a) −9 is left of +3 → −9 < +3
(b) −4 is closer to 0 than −7 → −4 > −7
(c) Zero is greater than any negative → 0 > −5
(d) +8 and 8 are the same value → +8 = 8
(e) −12 is farther left than −2 → −12 < −2
EXAMPLE 5Ordering Integers
Arrange in ascending order: +6, −3, 0, −8, +2, −5, +1
Step 1: Identify negatives: −3, −8, −5
Step 2: Order negatives: −8, −5, −3
Step 3: Insert zero: −8, −5, −3, 0
Step 4: Add positives: −8, −5, −3, 0, +1, +2, +6
✓ Ascending: −8, −5, −3, 0, +1, +2, +6
📏 Concept 410 / 23
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Absolute Value
Absolute value means distance from zero, without direction. Distance is never negative.
Notation: The absolute value of n is written as |n|. For any integer n:
|n| = n if n ≥ 0, and |n| = −n if n < 0
|n| = n if n ≥ 0, and |n| = −n if n < 0
Positive Numbers & Zero
For a positive number or zero, the absolute value is the number itself.
|+7| = 7 | |0| = 0 | |+25| = 25
Negative Numbers
For a negative number, the absolute value is its positive opposite.
|−7| = 7 | |−25| = 25 | |−1| = 1
Key Property
Two different integers can have the same absolute value (opposites). The absolute value of any non-zero integer is always positive.
|−5| = |+5| = 5 | |−12| = |+12| = 12
Common Mistake: Absolute value is NOT the same as "removing the sign." It is about distance. The absolute value of any number (except 0) is always positive because distance cannot be negative.
✅ Example11 / 23
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Finding Absolute Value
EXAMPLE 6Finding Absolute Value
Find: (a) |−9| (b) |+15| (c) |0| (d) |−100| (e) |+35|
(a) |−9| = 9 — distance from −9 to 0 is 9 units
(b) |+15| = 15 — distance from +15 to 0 is 15 units
(c) |0| = 0 — zero is at distance 0 from itself
(d) |−100| = 100
(e) |+35| = 35
➕ Concept 512 / 23
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Adding Integers
Adding integers depends on the signs. Same signs work one way; different signs work another way.
Same Signs (Both + or Both −)
Add the absolute values. Keep the common sign.
(+3) + (+5) = +8
(−3) + (−5) = −8
(−3) + (−5) = −8
Different Signs (One +, One −)
Subtract the smaller absolute value from the larger. The answer takes the sign of the number with the larger absolute value.
(−8) + (+3) = −5
(+8) + (−3) = +5
(+8) + (−3) = +5
Adding Zero
Adding zero to any integer leaves it unchanged (identity property).
(+7) + 0 = +7
(−4) + 0 = −4
(−4) + 0 = −4
Adding Opposites
The sum of any integer and its opposite is always zero.
(+6) + (−6) = 0
(−10) + (+10) = 0
(−10) + (+10) = 0
🎯 Examples13 / 23
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Adding: Same Signs & Different Signs
EXAMPLE 7Adding Integers — Same Signs
Calculate: (a) (−6) + (−4) (b) (+7) + (+9) (c) (−12) + (−8)
(a) Same sign (both −). Add: 6 + 4 = 10. Keep −. → −10
(b) Same sign (both +). Add: 7 + 9 = 16. Keep +. → +16
(c) Same sign (both −). Add: 12 + 8 = 20. Keep −. → −20
EXAMPLE 8Adding Integers — Different Signs
Calculate: (a) (−9) + (+5) (b) (+12) + (−7) (c) (−15) + (+20)
(a) |−9|=9, |+5|=5. Subtract: 9−5=4. Sign of −9. → −4
(b) |+12|=12, |−7|=7. Subtract: 12−7=5. Sign of +12. → +5
(c) |−15|=15, |+20|=20. Subtract: 20−15=5. Sign of +20. → +5
➖ Concept 614 / 23
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Subtracting Integers
Subtraction becomes easier when we rewrite it as adding the opposite.
📌 The Golden Rule of Subtraction:
a − b = a + (−b) and a − (−b) = a + b
Subtracting an integer is the same as adding its opposite.
a − b = a + (−b) and a − (−b) = a + b
Subtracting an integer is the same as adding its opposite.
| Expression | Rewrite As | Then Simplify | Result |
|---|---|---|---|
| (+8) − (+3) | (+8) + (−3) | Diff. signs: 8 − 3 = 5, sign of +8 | +5 |
| (+8) − (−3) | (+8) + (+3) | Same sign: 8 + 3 = 11, both + | +11 |
| (−8) − (+3) | (−8) + (−3) | Same sign: 8 + 3 = 11, both − | −11 |
| (−8) − (−3) | (−8) + (+3) | Diff. signs: 8 − 3 = 5, sign of −8 | −5 |
🔄 Example15 / 23
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Subtracting Integers
EXAMPLE 9Subtracting Integers
Calculate: (a) (−10) − (−4) (b) (+6) − (+9) (c) (−3) − (+7) (d) (+5) − (−8)
(a) (−10) + (+4) → Diff. signs: 10−4=6, sign −10 → −6
(b) (+6) + (−9) → Diff. signs: 9−6=3, sign −9 → −3
(c) (−3) + (−7) → Same sign: 3+7=10, both − → −10
(d) (+5) + (+8) → Same sign: 5+8=13, both + → +13
✖️ Concept 716 / 23
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Multiplying & Dividing Integers
Multiplication and division use the same sign rule.
Positive ×/÷ Positive = Positive
(+3) × (+4) = +12
(+12) ÷ (+3) = +4
(+12) ÷ (+3) = +4
Negative ×/÷ Positive = Negative
(−3) × (+4) = −12
(−12) ÷ (+3) = −4
(−12) ÷ (+3) = −4
Positive ×/÷ Negative = Negative
(+3) × (−4) = −12
(+12) ÷ (−3) = −4
(+12) ÷ (−3) = −4
Negative ×/÷ Negative = Positive
(−3) × (−4) = +12
(−12) ÷ (−3) = +4
(−12) ÷ (−3) = +4
📌 Quick Rule: Same signs → positive result | Different signs → negative result
➗ Examples17 / 23
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Multiplying & Dividing
EXAMPLE 10Multiplying Integers
Calculate: (a) (−5)×(+6) (b) (−8)×(−3) (c) (+7)×(+4) (d) (−9)×(+2)
(a) Diff. signs → negative. 5×6=30 → −30
(b) Same signs → positive. 8×3=24 → +24
(c) Same signs → positive. 7×4=28 → +28
(d) Diff. signs → negative. 9×2=18 → −18
EXAMPLE 11Dividing Integers
Calculate: (a) (−20)÷(+5) (b) (−36)÷(−6) (c) (+42)÷(−7) (d) (−48)÷(−8)
(a) Diff. signs → negative. 20÷5=4 → −4
(b) Same signs → positive. 36÷6=6 → +6
(c) Diff. signs → negative. 42÷7=6 → −6
(d) Same signs → positive. 48÷8=6 → +6
📚 Concept 818 / 23
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Order of Operations
When several operations appear in one problem, use PEMDAS. Be extra careful with negative signs.
P — Parentheses
Simplify expressions inside parentheses first.
E — Exponents
Evaluate powers and exponents next.
M/D — Multiply/Divide
From left to right: multiplication and division have equal priority.
A/S — Add/Subtract
From left to right: addition and subtraction have equal priority.
STEP-THROUGHHow PEMDAS Works with Integers
Simplify: 2 + (−3) × 4 − (−6) ÷ 2
Step 1 (P): No grouping needed here — parentheses only wrap signs.
Step 2 (M/D left→right): (−3) × 4 = −12
Step 3 (M/D cont.): (−6) ÷ 2 = −3
Expression becomes: 2 + (−12) − (−3)
Step 4 (A/S left→right): 2 + (−12) = −10
Step 5: −10 − (−3) = −10 + 3 = −7
✓ Result: −7
⚠️ Important: Pay close attention to signs inside parentheses. A negative sign in front of parentheses means multiply by −1.
🧠 Examples19 / 23
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PEMDAS with Integers
EXAMPLE 12Order of Operations
Simplify: (−3) × (4 − 7) + (−12) ÷ (−2)
Step 1 (P): 4 − 7 = −3
Becomes: (−3) × (−3) + (−12) ÷ (−2)
Step 2 (M/D): (−3)×(−3) = +9 and (−12)÷(−2) = +6
Step 3 (A/S): (+9) + (+6) = +15
✓ Result: +15
EXAMPLE 13More Complex Expression
Simplify: −2 × [5 + (−3)] − 4 × (−2)
Step 1 (P): 5 + (−3) = 2
Becomes: −2 × 2 − 4 × (−2)
Step 2 (M): −2 × 2 = −4 and 4 × (−2) = −8
Becomes: (−4) − (−8)
Step 3 (S): (−4) + (+8) = +4
✓ Result: +4
🌡️ Concept 920 / 23
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Integers in Real-World Problems
Word problems usually tell a story: a temperature changes, money is gained or spent, elevation moves up or down, or a score changes.
📌 Translation Guide
🔺 Rise / Gain / Deposit / Ascend / Scored → add a positive number
🔻 Drop / Loss / Withdrawal / Descend / Conceded → add a negative number
📐 How far apart / Change / Difference → subtract (final − initial)
🔺 Rise / Gain / Deposit / Ascend / Scored → add a positive number
🔻 Drop / Loss / Withdrawal / Descend / Conceded → add a negative number
📐 How far apart / Change / Difference → subtract (final − initial)
WORKED EXAMPLETranslate, Then Calculate
A city's temperature was +12°C in the morning. By midnight it had dropped 18°C. What was the midnight temperature, and by how much did it change?
Step 1 — Translate: "Dropped 18°C" → add −18
Step 2 — Midnight temp: (+12) + (−18) = −6°C
Step 3 — Change: final − initial = (−6) − (+12) = −18°C
✓ Midnight: −6°C | Change: −18°C
🏁 Examples21 / 23
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Real-World Integer Problems
EXAMPLE 14Temperature Change
At sunrise, the temperature was −7°C. By noon, it rose by 15°C. What was the temperature at noon?
Step 1: Start: −7°C. Rise → add: +15.
Step 2: (−7) + (+15). Diff. signs: 15−7=8. Sign of +15.
✓ Temperature at noon: +8°C
EXAMPLE 15Bank Balance
Layla has $50. She writes a cheque for $65. New balance?
(+50) − (+65) = (+50) + (−65). Diff. signs: 65−50=15. Sign of −65.
✓ Balance: −$15 (overdrawn)
EXAMPLE 16Elevation Change
A submarine is at −250 m. It descends another 80 m. New depth?
(−250) + (−80). Same sign: 250+80=330, keep −.
✓ New depth: −330 m
EXAMPLE 17Football Score
A team scores 3 goals but concedes 5. Net goal difference?
(+3) + (−5). Diff. signs: 5−3=2. Sign of −5.
✓ Goal difference: −2
⭐ Review22 / 23
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Mission Review
IntegersWhole numbers, their opposites, and zero.
Number LineRight means greater. Left means smaller.
Absolute ValueDistance from zero, always non-negative.
Add/SubtractUse sign rules and add the opposite.
Multiply/DivideSame signs positive. Different signs negative.
Word ProblemsTranslate the situation into integer operations.
✏️ Practice23 / 23
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Final Interactive Practice
Question 1 of 32
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