This MCQ module is based on: Subtraction of Integers and Patterns
TOPIC 40 OF 41
Subtraction of Integers and Patterns
🎓 Class 6
Mathematics
CBSE
Theory
Ch 10 — The Other Side of Zero
⏱ ~30 min
🌐 Language: [gtranslate]
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This mathematics assessment will be based on: Subtraction of Integers and Patterns
Targeting Class 6 level in Integers, with Basic difficulty.
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10.7 Subtraction of Integers
We already know that 7 − 3 = 4. But what does 3 − 7 mean? In the world of whole numbers, it has no answer. In the world of integers, it does: \(3 - 7 = -4\).
The Big Idea — "Subtracting is Adding the Opposite"
Key Rule
To subtract an integer, add its opposite.\[a - b = a + (-b)\] For example:
\((+5) - (+3) = (+5) + (-3) = +2\)
\((+5) - (-3) = (+5) + (+3) = +8\)
\((-5) - (+3) = (-5) + (-3) = -8\)
\((-5) - (-3) = (-5) + (+3) = -2\)
Why Does This Work? (Token Picture)
Start with the first number's pile. "Subtract b" means remove b — if b is positive, remove b yellow tokens; if b is negative, remove |b| red tokens. Sometimes we don't have enough tokens of the needed colour — then we add zero pairs first.
Example: Compute (+3) − (−4). We have 3 yellow. We need to remove 4 red, but there are none. So add 4 zero pairs (4 yellow + 4 red). Now we have 7 yellow + 4 red. Remove the 4 red. 7 yellow remain → +7.
Check: (+3) − (−4) = (+3) + (+4) = +7. ✓
Number-Line Picture of Subtraction
On the number line, \(a - b\) is the signed distance from \(b\) to \(a\) (how many jumps, in which direction, to go from b to a).
Subtraction as "distance from b to a": going from −3 to +2 needs 5 jumps right, so the answer is +5.
Worked Examples
Ex 1: \((-10) - (-6) = (-10) + (+6) = -4\).
Ex 2: \((+15) - (+20) = (+15) + (-20) = -5\).
Ex 3: \(0 - (-7) = 0 + (+7) = +7\). ("How much warmer than −7°C is 0°C?" Answer: 7°C.)
Ex 4: Temperature change from −5°C in the morning to +3°C in the afternoon: \((+3) - (-5) = (+3) + (+5) = +8°C\) (temperature rose by 8°C).
Integers Are Closed Under Subtraction
For any two integers \(a\) and \(b\), \(a - b\) is always an integer. This is why integers "fix" a problem whole numbers had — whole numbers were not closed under subtraction.
🔵 Is subtraction commutative? Check: \(5 - 3 = 2\) but \(3 - 5 = -2\). They are opposites, not equal. So subtraction is not commutative. Is it associative? \((10 - 5) - 3 = 2\), but \(10 - (5 - 3) = 10 - 2 = 8\). Not associative either.
10.8 Patterns in Integers
Counting Down Through 0
Count down: 5, 4, 3, 2, 1, 0, −1, −2, −3, …. The pattern "subtract 1" keeps working even when you cross zero.
A Classic Multiplication Pattern (Preview)
| Product | Value |
|---|---|
| 3 × 3 | 9 |
| 3 × 2 | 6 |
| 3 × 1 | 3 |
| 3 × 0 | 0 |
| 3 × (−1) | −3 |
| 3 × (−2) | −6 |
| 3 × (−3) | −9 |
The pattern "subtract 3 each time" continues through zero. This forces: (positive) × (negative) = (negative). You will study this rule in detail in Class 7.
Sum of a Number and its Successor
\(5 + 6 = 11; \quad (-5) + (-4) = -9; \quad (-2) + (-1) = -3; \quad (-1) + 0 = -1\). The sum of an integer and its successor is always an odd integer (one less than twice the smaller, or one more than twice the larger).
Activity: Floor-Difference Race
L3 ApplyMaterials: Strip marked from −5 to +5, small coin/counter, 2 dice labelled with integers
Predict: If you start at Floor 0 and the two dice show +3 and −4, where will your counter land after applying the operation "first − second"?
- Roll two dice (or pick two cards with integers).
- First integer is your starting floor; compute (first − second) to find your arrival floor.
- Move the counter from the starting floor to the arrival floor. How many jumps did you take, and in which direction?
- Play 5 rounds. Note the most jumps you made in any round.
The number of jumps equals the absolute value of the second integer. Direction is opposite to its sign: subtracting +n jumps left, subtracting −n jumps right.
Figure it Out — Part D
Q1. Subtract: (a) (+8) − (+3) (b) (+8) − (−3) (c) (−8) − (+3) (d) (−8) − (−3) (e) 0 − (−12) (f) 0 − (+12).
(a) +5 (b) +11 (c) −11 (d) −5 (e) +12 (f) −12.
Q2. Compute each using the rule \(a - b = a + (-b)\): (i) (−4) − (−9) (ii) (+7) − (−11) (iii) (−12) − (+6).
(i) (−4) + (+9) = +5. (ii) (+7) + (+11) = +18. (iii) (−12) + (−6) = −18.
Q3. Temperature at 6 am was −5°C; at 2 pm it rose to +12°C. By how many degrees did the temperature rise?
Rise = (+12) − (−5) = 12 + 5 = 17°C.
Q4. Fill in: (a) (+20) − ___ = +35 (b) ___ − (−5) = 0 (c) (−7) − ___ = −10 (d) ___ − (+8) = −13.
(a) −15 (since 20 − (−15) = 35) (b) −5 (since −5 − (−5) = 0) (c) +3 (d) −5.
Q5. Continue the pattern for two more terms: 10, 7, 4, 1, −2, −5, ___, ___ .
Rule: subtract 3 each time. Next terms: −8, −11.
Q6. Evaluate the expression step by step: \((-6) + (+10) - (-3) - (+7)\).
Convert subtractions to additions: (−6) + (+10) + (+3) + (−7). Positives: 10 + 3 = 13. Negatives: −6 + (−7) = −13. Sum = 13 − 13 = 0.
Competency-Based Questions
Scenario: At a geographical cross-section, the following heights (in metres, relative to sea level) are recorded across 6 points moving west to east: A = +900, B = +1100, C = −200, D = −450, E = +300, F = +50.
Q1. Find the altitude difference between the highest and lowest of the six points.
L3 ApplyHighest = B (+1100), lowest = D (−450). Difference = (+1100) − (−450) = 1550 m.
Q2. List the point-to-point altitude changes from A to B to C to D to E to F.
L4 AnalyseA→B: 1100 − 900 = +200. B→C: −200 − 1100 = −1300. C→D: −450 − (−200) = −250. D→E: 300 − (−450) = +750. E→F: 50 − 300 = −250. So: +200, −1300, −250, +750, −250.
Q3. Check whether the sum of all 5 point-to-point changes equals the overall change from A to F.
L5 EvaluateSum of changes = 200 − 1300 − 250 + 750 − 250 = −850 m. Direct change A→F = 50 − 900 = −850 m. They match — as they must, because intermediate terms telescope out.
Q4. Create a 5-point cross-section (different from the one above) in which the total altitude change is 0 but no two consecutive changes are the same.
L6 CreateSample: P₁ = 0, P₂ = +300, P₃ = +100, P₄ = −500, P₅ = 0. Consecutive changes: +300, −200, −600, +500. Sum = 0. First and last heights equal, and no two consecutive changes are the same.
Assertion–Reason Questions
Assertion (A): Subtraction of integers is not commutative.
Reason (R): \(a - b\) and \(b - a\) are opposite integers whenever \(a \ne b\).
Reason (R): \(a - b\) and \(b - a\) are opposite integers whenever \(a \ne b\).
(a) Since \(a - b = -(b - a)\), they differ in sign whenever \(a \ne b\) — so subtraction is not commutative. R explains A.
Assertion (A): \((+5) - (-3) = +8\).
Reason (R): Subtracting a negative integer is the same as adding its opposite.
Reason (R): Subtracting a negative integer is the same as adding its opposite.
(a) (+5) − (−3) = (+5) + (+3) = +8. R is the correct rule.
Assertion (A): Whole numbers are closed under subtraction.
Reason (R): The difference of two whole numbers is always a whole number.
Reason (R): The difference of two whole numbers is always a whole number.
A is false, R is false. e.g., 3 − 7 = −4 is not a whole number. So closure fails, and the general claim in R is also false. Correct option here: both false (treat as "Both false"). If the 4 options offered don't include "Both false", answer is (d) with the caveat that R is the stated reason but it is itself incorrect.
Frequently Asked Questions
What is the rule for subtracting integers?
To subtract, change the sign of the number being subtracted and then add. So a - b becomes a + (-b).
What is 5 - (-3)?
5 - (-3) = 5 + 3 = 8. Subtracting a negative is the same as adding the positive.
What is (-8) - (-5)?
(-8) - (-5) = (-8) + 5 = -3.
Is integer subtraction commutative?
No. Subtraction is NOT commutative: 5 - 3 = 2 but 3 - 5 = -2. Order matters in subtraction.
What pattern appears in 4, 2, 0, -2, -4, ...?
This pattern decreases by 2 each time. Continuing: -6, -8, -10, ... . It is an arithmetic sequence with common difference -2.
Why is subtracting a negative equivalent to adding?
Because the opposite of a negative integer is positive. Removing a debt is the same as receiving money - both increase your total.
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Mathematics Class 6 — Ganita Prakash
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